{
  "schemaVersion": 1,
  "ledger": [
    {
      "id": "AX.P1.01",
      "claim": "c3 = 1/(8 pi) boundary seam kernel -- a DECLARED input but NOT a free knob: it reduces to the parabolic anchor a=(1,1,2) via c3 = 1/(2 e1(a) pi) = 1/(8pi) (e1(a)=4=|mu4|; ANCHOR.GEN.01/v23), and the '8' is the one-sided Gauss-Bonnet of the seam normal slice (|Z2|*2pi*chi(S^2)=8pi, chi=2; v58/v73). On the dimensionless axis the ONLY transcendental primitive is pi (ARCH.CORE.01/v53); {c3,g_car} reduce to {a, pi} SYNTHESIS NOTE (2026-08-28, v993 -- marker Axiom UNCHANGED): the mu4 average E = (1/4) sum_r Ad(u^r) is the UNIQUE trace-preserving finite conditional expectation onto the clock-fixed algebra (uniqueness needs the module property; kernel dim 36/720 without it); the arithmetic 1/(4 * 2 pi) = c3 is the v484/v813 identity (not a new derivation of P1). The equidistribution identification (modular det-line response even over the four deck sectors) stays the open [C] of v813. Axiom typing UNCHANGED.",
      "status": "Axiom (declared input; reduces to the anchor a=(1,1,2)+pi)",
      "canonicalStatus": "Axiom (declared) but anchor-reducible: c3=1/(2 e1(a) pi); only pi is the dimensionless primitive | 2026-08-28 v993: unique finite CE + 1/(4*2pi) arithmetic shadow; equidistribution stays [C] of v813; axiom typing UNCHANGED    WAVE-5 (2026-08-29): SEAM.MMST.TYPEIII.CHARGED.01 reads P1 as theorem-conditional on seam identification + beta_angle=2pi (KLM + Longo-Tanimoto GIVEN identification); axiom typing UNCHANGED.",
      "dependencies": "ANCHOR.GEN.01; ARCH.CORE.01; v993",
      "scripts": [],
      "script": "-",
      "location": "tfpt_1",
      "externalData": "Gauss-Bonnet (geometric origin)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 2
    },
    {
      "id": "AX.P2.01",
      "claim": "g_car = 5 carrier interface (3+2) -- a DECLARED input but an OVER-DETERMINED bootstrap fixed point, not a free choice: forced three independent ways by the E8 closure (bootstrap mu^2-5mu+4=0, BOOT.01/v6), uniquely solved by the Pascal carrier condition 2^g=g^2+g+2 => g=5 (Lean-formalised, no sorry; FORM.LADDER.01), and the (g_car,N_fam)=(5,3) pair is the anchor moment data e2(a)=g_car (ARCH.CORE.01/v53; a=(1,1,2) the unique multiset with elem-symm (4,5,2)). Reduces with P1 to {a, pi}. SYNTHESIS NOTE (reduction state 2026-07-21; text consolidation, marker Axiom UNCHANGED): P1 + the four marks are [E]-anchored (v216); given the marks, g_car = 5 follows via the partition 4 = 2+1+1 plus the Deligne/Birkhoff-Grothendieck/stability typing (v491/v499, residual R1 [C]); the clock order, the Z8 tower, the deck class and the position half of the alignment bit are theorems (v506/v507/v510) -- the compiler's dimensionless inputs reduce to: four marks (derived from P1-side topology), ONE square-modulus datum (tau = i, equivalently: the clock / the CM-fixed half-period / the non-split extension class on the free seam circle / the flag-transitivity bit V4 -> D4, v512), and pi, plus the [C] identifications R1/R2 of v499. SYNTHESIS NOTE (2026-07-22, v512 -- marker UNCHANGED): the reduction state can equivalently be written 'four marks + one discrete symmetry-lift bit (flag transitivity <=> tau = i) + pi'. This describes the REDUCTION STATE; the axiom stays declared and QGEO.REALIZE.01 stays [C]/[O] [2026-08-02 XXXIX: the four-mark closure lands -- v624: under the D_{d+1}+A_{d-1} glue architecture three independent selectors (disc-group census, even-glue norm equation, Pythagorean (3,4,5)) force d = 4, with (1,1,2) the unique 3-part partition and c_3 = 1/(8 pi): the two-input presentation reduces to ONE boundary degree d = 4 plus pi, CONDITIONAL on the architecture; the axiom keeps its typing -- the residue moves to why the physical boundary realizes the minimal cyclically-glueable genus-0 divisor; see EXTREV.LATTICE.01.] SYNTHESIS NOTE (2026-08-28, v993 -- marker Axiom UNCHANGED): the architecture conditionality of v624 is lifted at the census level -- the FULL rank-8 ADE pair census with cyclic Z4 glue on BOTH factors selects (D5, A3) uniquely (D3 ≅ A3 aliases collapsed), so the D+A split is an OUTPUT of (rank 8 + cyclic Z4), not an INPUT. Cited-not-proved remain c_- = 8 and holomorphic uniqueness of (E8)_1. Axiom typing UNCHANGED. WAVE-6 (2026-08-30, v1005): relative pencil identities exact but CIRCULARITY-FLAGGED -- carrier 5 already sits in the column sums of Q_2 and C; NOT a P2 selector until Q, C, epsilon are derived from seam geometry independently. Axiom typing UNCHANGED.  LATE EVENING (2026-08-30, v1014): W-bridge premise P-anch DERIVED (anchor chain: e1 = R^{-1}(1,1,2)^T uniquely via the winding/triple lock); P-dem narrowed to the character-blind determinant-response map -- same MMST/Quillen externalized leg as TEL-B/ALG-EXH/P1.  Axiom typing UNCHANGED.  MONDAY MORNING (2026-08-31, v1015): BOTH W-bridge premises now derived at finite level -- P-anch already (v1014 winding/triple lock); P-dem DERIVED (character-blind determinant response r=(1,1,1) exactly at collar sizes 12 and 16, democracy residual 0; mutants split (1,0,0) and (5/4,1,5/4)).  The axiom-core remainder is ZERO modulo the externalized MMST identification.  Axiom typing UNCHANGED (the finite derivations are shadows; the continuum identification is the same external leg).",
      "status": "Axiom (declared input; over-determined bootstrap fixed point, Lean-forced)",
      "canonicalStatus": "Axiom (declared) but bootstrap-forced (3 ways) + Pascal-unique (Lean): g=5 is not a free knob; reduction state 2026-07-22: four marks + one discrete symmetry-lift bit (flag transitivity <=> tau = i) + pi (v491/v499/v506/v507/v510/v512), marker unchanged | 2026-08-28 v993: v624 architecture conditionality lifted at the census level (D5, A3 unique in full rank-8 ADE); axiom typing UNCHANGED | WAVE-6 (2026-08-30 v1005): pencil circularity-flagged (5 already in Q_2/C column sums); NOT a P2 selector; axiom typing UNCHANGED | AFTERNOON (2026-08-30 v1010): pencil selector rehabilitated: column sums forced by prior constructions (ladder c=2 + n=3 force Q sums (9,5,1); C's 5 inherited from R col 3 -- non-circular); W-shift conditionally unique given democratic-image + carrier-anchor-kernel (1078 -> 1); remaining obligation = seam-geometric derivation of those premises; axiom typing UNCHANGED | LATE EVENING (2026-08-30 v1014): W-bridge premise P-anch derived (anchor chain); P-dem narrowed to the character-blind determinant-response map -- same MMST/Quillen externalized leg as TEL-B/ALG-EXH/P1.  Axiom typing UNCHANGED. | MONDAY MORNING (2026-08-31 v1015): BOTH W-bridge premises derived at finite level (P-anch v1014; P-dem r=(1,1,1)); remainder 0 modulo MMST.  Axiom typing UNCHANGED.",
      "dependencies": "BOOT.01; ARCH.CORE.01; v491; v499; v510; v512; v993",
      "scripts": [
        "v1005_relative_pencil_bridges.py",
        "v1010_simplicity_bridge_census.py",
        "v1014_bridge_refinements.py",
        "v1015_axiom_core_closure.py"
      ],
      "script": "v1005_relative_pencil_bridges.py; v1010_simplicity_bridge_census.py; v1014_bridge_refinements.py; v1015_axiom_core_closure.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 3
    },
    {
      "id": "FORM.P2.02",
      "claim": "P2 carrier algebra (hypercharge/family) machine-verified",
      "status": "Formal",
      "canonicalStatus": "Formal",
      "dependencies": "AX.P2.01",
      "scripts": [
        "lean4-carrier-rigidity"
      ],
      "script": "lean4-carrier-rigidity",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 4
    },
    {
      "id": "E8.GLU.01",
      "claim": "E8 = (D5 (+) A3) + mu4 glue: explicit lattice certificate (240 roots norm^2=2; even; unimodular Gram det=1; glue index 4=|mu4|); disc=Z4; q(D5)+q(A3)=5/4+3/4=2",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "D5; A3; Z4",
      "scripts": [
        "v1_e8_glue.py"
      ],
      "script": "v1_e8_glue.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 5
    },
    {
      "id": "E8.NUM.01",
      "claim": "240 = 16*5*3 and 248 = 240+8 as carrier traces",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "E8.GLU.01",
      "scripts": [
        "v1_e8_glue.py"
      ],
      "script": "v1_e8_glue.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 6
    },
    {
      "id": "CAR.PAS.01",
      "claim": "g_car = 5 unique Pascal solution (proof + check)",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "-",
      "scripts": [
        "v2_carrier_pascal.py"
      ],
      "script": "v2_carrier_pascal.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 7
    },
    {
      "id": "CAR.SM.01",
      "claim": "N_fam=3; Omega_adm=48; b1=41/10; dim S+ = 16",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "CAR.PAS.01",
      "scripts": [
        "v2_carrier_pascal.py"
      ],
      "script": "v2_carrier_pascal.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 8
    },
    {
      "id": "CAR.UNIQ.01",
      "claim": "g_car=5 is the UNIQUE rank with N_fam in Z+ and g+N_fam=8; split (b,s)=(3,2) unique from b+s=5,b^2+s^2=13; Tr Y=Tr Y^3=0 (anomaly-free)",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "E8 closure; SM gauge algebra dim 12",
      "scripts": [
        "v14_carrier_uniqueness.py"
      ],
      "script": "v14_carrier_uniqueness.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 9
    },
    {
      "id": "BOOT.CLASS.01",
      "claim": "D5(+)A3 is the UNIQUE familyful cyclic-glue decomposition of E8 (16-dim half-spinor + 3-family A3); glue Z4=|mu4|",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "E8.GLU.01; even-unimodular rank-8 uniqueness",
      "scripts": [
        "v15_bootstrap_classification.py"
      ],
      "script": "v15_bootstrap_classification.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 10
    },
    {
      "id": "EM.FP.01",
      "claim": "alpha^-1 = 137.0359992168 unique root of F_U1=0 (existence + uniqueness + interval-arithmetic + ablation)",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "AX.P1.01; EM.BUDGET.01",
      "scripts": [
        "v3_em_alpha.py"
      ],
      "script": "v3_em_alpha.py",
      "location": "tfpt_1",
      "externalData": "NIST CODATA 2022 (137.035999177(21))",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 11
    },
    {
      "id": "EM.BUDGET.01",
      "claim": "EM budget M=41 is FORCED = 10 b1 = sum L + N_Phi (U(1) index, not a free fit); (4/5)M c3^6 = 8 b1 c3^6",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "CAR.SM.01 (b1 from g_car)",
      "scripts": [
        "v13_open_gates.py"
      ],
      "script": "v13_open_gates.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 12
    },
    {
      "id": "FLAV.R.01",
      "claim": "residue matrix R: det=8; minors {2;3;5}; char poly; SNF diag(1;1;8) (positive invariant factors); ||R||^2=78; sum L=40",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "-",
      "scripts": [
        "v4_flavor_matrix.py"
      ],
      "script": "v4_flavor_matrix.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 13
    },
    {
      "id": "FLAV.KQ.01",
      "claim": "Q-Sigma algebra: K=R+Q*Sigma; L=R+Q*(I+Sigma); chi_Q=(t-1)(t^2-5t+3); chi_K=(t-1)(t^2-8t+4); det ladder (3;4;8;20) product 1920=|W(D5)|; a^T K a=41",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "FLAV.R.01; Sigma=diag(1;-1;-1)",
      "scripts": [
        "v10_projection_involution.py",
        "v11_unique_KQ.py",
        "v12_mass_generation_polynomials.py"
      ],
      "script": "v10_projection_involution.py;v11_unique_KQ.py;v12_mass_generation_polynomials.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 14
    },
    {
      "id": "FLAV.QGEO.01",
      "claim": "geometric origin of Q reduced: Q_+ = A3 exponent grading (spec 1;2;3); Q_-^2 = N_fam on support; open = realise Q_+/Q_- on parabolic P1\\mu4 UPDATE (2026-07-31, v567, QGEO.QRECON.01): the declared reductions (Q_+ spectrum, Q_-^2 budget) plus the v10 winding normalisation and the seam-line demand reconstruct the ANCHOR carrier-free (one code class on all 28 admissible pairs) and Q itself up to ONE binary sheet-coupling choice (q32 in {2,0}; the 81-pin) -- the v11 carrier budgets are bypassed entirely; the audit relocates to GATE.QGEO.01 and the binary selector. Markers unchanged.",
      "status": "Physical (conditional; sharpened)",
      "canonicalStatus": "Physical (conditional; sharpened)",
      "dependencies": "FLAV.KQ.01; parabolic Hodge on P1\\mu4",
      "scripts": [
        "v13_open_gates.py"
      ],
      "script": "v13_open_gates.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 15
    },
    {
      "id": "FLAV.H2.01",
      "claim": "H2 splitting type 4=1+1+2; coker R = Z8 (branch fixed)",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "FLAV.R.01; Mehta-Seshadri (U)",
      "scripts": [
        "v4_flavor_matrix.py"
      ],
      "script": "v4_flavor_matrix.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 16
    },
    {
      "id": "FLAV.TH12.01",
      "claim": "solar angle sin^2 th12 = 1/3 - phi0/2 = 0.30675 via explicit mu-tau texture",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "FLAV.R.01; eps=3phi0/4 (~c3 conditional)",
      "scripts": [
        "v9_neutrino_texture.py"
      ],
      "script": "v9_neutrino_texture.py",
      "location": "tfpt_3;tfpt_2",
      "externalData": "NuFIT 6.0 (0.307)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 17
    },
    {
      "id": "FLAV.SOLAR.01",
      "claim": "dual anchor a^T R^-1 = a^T L^-1 = (-1/2,-1/2,1) exact; eps=(N_fam/|mu4|)phi0=3/4 phi0 = c3 + 36 c3^4 (leading term c3, fourth-order puncture correction exact); full PMNS readout",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "FLAV.R.01; dual-anchor identity",
      "scripts": [
        "v16_solar_dual_anchor.py"
      ],
      "script": "v16_solar_dual_anchor.py",
      "location": "tfpt_2",
      "externalData": "NuFIT 6.0 (pre-JUNO baseline)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 18
    },
    {
      "id": "FLAV.RESOLV.01",
      "claim": "finite hexagonal resolvent D_y^-1 = (sum y^{5-m} delta^m U6^m)/(y^6-delta^6) proven; quark c-digits need U_f* holonomy (open finite step)",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "C6 transport kernel",
      "scripts": [
        "v17_hexagonal_resolvent.py"
      ],
      "script": "v17_hexagonal_resolvent.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 19
    },
    {
      "id": "FLAV.QUARK.01",
      "claim": "charged-fermion Yukawas y=lambda_Y^L*Lambda reproduce quark source ratios exactly (mu/md=0.470085, mc/ms=13.61, mt/mb=40.80), lepton c=(16/7,4/3,7/6) to Lambda rounding, full hierarchy; exact quark c rationals need U_f* holonomy",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "FLAV.RESOLV.01; word-lengths; Lambda table",
      "scripts": [
        "v18_quark_yukawa.py"
      ],
      "script": "v18_quark_yukawa.py",
      "location": "tfpt_2",
      "externalData": "PDG/NuFIT (mu/md~0.47, mc/ms~13.6)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 20
    },
    {
      "id": "FLAV.UFSTAR.01",
      "claim": "exact pole delta_ph=z_*^(1/6), z_*=(794-7sqrt9961)/2187 (2187=3^7); bare resolvent fails lepton ratios 3-11x (holonomy essential); required Lambda=(0.475,1.107,0.917), Lambda_mu>1 => non-unitary resolvent dressing; D4-symmetric SU(3) monodromy constructible but D4-fixed locus POSITIVE-DIMENSIONAL => exact quark c reduce to the (U) stable-point selection",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "H2 monodromy; Mehta-Seshadri; (U)",
      "scripts": [
        "v19_monodromy_moduli.py"
      ],
      "script": "v19_monodromy_moduli.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 21
    },
    {
      "id": "FLAV.LEPTONC.01",
      "claim": "lepton c's DERIVED: non-anchor c=|mu4|^w/(5/4-cos(r pi/3)) at delta=1/2,y=1 gives c_e=16/7,c_mu=4/3 exactly; anchor from product rule prod c=2^gcar/Nfam^2=32/9 gives c_tau=7/6; reproduces (8/7,12/7,3/2) ratios. Lepton-specific: down sector forces c_d/c_s=4 vs actual 0.94 (quarks structurally different)",
      "status": "Identity",
      "canonicalStatus": "Identity",
      "dependencies": "delta=1/2 (distinguished); hexagon resolvent; g_car,N_fam,mu4",
      "scripts": [
        "v20_lepton_c_derivation.py"
      ],
      "script": "v20_lepton_c_derivation.py",
      "location": "tfpt_2",
      "externalData": "PDG lepton ratios",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 22
    },
    {
      "id": "FLAV.SOLAR.02",
      "claim": "solar coefficient is the A3 glue-norm: eps=q(A3)*phi0=(3/4)phi0, q(A3)=3/4 the same norm in q(D5)+q(A3)=2; leading term = c3 exact; sin^2 th12=1/3-phi0/2=0.30675. Product rule 32/9=2^gcar/Nfam^2 interpreted (full Clifford dim/Nfam^2); sum L_lepton=16=dim S+. Quark non-extension: cusp-leg law gives c_c/c_s=1/7 vs actual 0.724",
      "status": "Identity",
      "canonicalStatus": "Identity",
      "dependencies": "FLAV.SOLAR.01; FLAV.LEPTONC.01; E8 glue norms",
      "scripts": [
        "v21_solar_product_quark.py"
      ],
      "script": "v21_solar_product_quark.py",
      "location": "tfpt_2",
      "externalData": "NuFIT 6.0",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 23
    },
    {
      "id": "ANCHOR.GEN.01",
      "claim": "anchor-first: a=(1,1,2) elementary symmetric e_k=(4,5,2)=(|mu4|,g_car,|Z2|), c3=1/(2 e1 pi)=1/(8pi); power sums p_n=2+2^n give |R(E8)|=p1p2p3=240, dim E8=248, rank=p4-p3=8, binary ladder 2^n. Inputs reduce {c3,g_car} -> {a,pi}",
      "status": "Identity",
      "canonicalStatus": "Identity",
      "dependencies": "a=(1,1,2) parabolic anchor",
      "scripts": [
        "v23_anchor_generator.py"
      ],
      "script": "v23_anchor_generator.py",
      "location": "introduction;tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 24
    },
    {
      "id": "FLAV.QRATIO.01",
      "claim": "quark RATIO closure: c_u/c_d=g_car*11/(Nfam^2 DeltaQ)=55/117, c_c/c_s=p5(a)/(Oadm-1)=34/47, c_t/c_b=Nfam/(2 DeltaQ)=3/26; mass ratios 0.470085/13.605/40.81 match targets <0.03%; rational c gauge (1/2,117/110,47/41,34/41,8/3,4/13), Lambda_q in O(1). c_c/c_s and c_t/c_b have clean single-reading blocks; c_u/c_d '11' is NOT uniquely forced (>=5 readings) and 55/117=table ratio 0.440/0.936; the '11' reading is generated not assumed (v42)",
      "status": "Identity/Numerical",
      "canonicalStatus": "[I] ratios closed on the DERIVED selector stratum (v49/v69/v71); only the absolute amplitude scale U_point stays [A]; FORBIDDEN WORDING: do not write 'quark ratios stay [P]'",
      "dependencies": "FLAV.RESOLV.01; anchor p5=34; DeltaQ=13; Oadm=48",
      "scripts": [
        "v24_quark_ratio_closure.py",
        "v26_flavor_frontier_unification.py"
      ],
      "script": "v24_quark_ratio_closure.py;v26_flavor_frontier_unification.py",
      "location": "tfpt_2",
      "externalData": "PDG/FLAG 2024 (mu/md,mc/ms,mt/mb)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 25
    },
    {
      "id": "FLAV.UGATE.01",
      "claim": "UNIFICATION: c_u/c_d '11', Q-geometry (B3), R mod (U) (B4), H2 parabolic<->transport (A2) all reduce to ONE gate = the (U) stable-point selection. SHARPENING: cusp weights put O(-2)+O(-1)^2 on the parabolic STABILITY WALL (sum w=4; w1<2 & w2,w3<1 unsatisfiable since w1=4-w2-w3>2) -> nabla_F* is the unique POLYSTABLE point (explains the positive-dim D4 locus of v19)",
      "status": "Open (sharpened to one wall-selection)",
      "canonicalStatus": "SUPERSEDED -> FLAV.RIGID.01/02: quark ratios [I] closed (v49/v71); only U_point (absolute amplitude normalisation) = anchor [A]",
      "dependencies": "FLAV.UFSTAR.01; FLAV.QRATIO.01; GATE.QGEO.01; GATE.RU.01",
      "scripts": [
        "v26_flavor_frontier_unification.py"
      ],
      "script": "v26_flavor_frontier_unification.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": false,
      "historical": true,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 26
    },
    {
      "id": "GATE.UWALL.01",
      "claim": "(U_wall) explicit: balanced wall representative W_wall=(1/3)[[2,1,2,1],[1,0,0,2],[0,2,1,0]] (cols=perm(0,1,2), rows w=(2,1,1) lift degs (-2,-1,-1) to pardeg 0); selector nabla_F*=Sel_{D4,cusp,detR=8,Q+}(M_par); solving one D4-fixed polystable Hitchin point yields R,Q,U_f*,c_u/c_d,exact c_q,H2 together. HONEST (C_U1): 144 walls -> 5 orbits under full symmetry, so uniqueness needs the selector (U4), NOT symmetry (Lemma U1 corrected)",
      "status": "Open (one finite moduli point)",
      "canonicalStatus": "SUPERSEDED -> FLAV.RIGID.02: ratios closed combinatorially (integer Plucker); only the absolute amplitude scale is the anchor",
      "dependencies": "FLAV.UGATE.01; Mehta-Seshadri; nonabelian Hodge",
      "scripts": [
        "v27_wall_representative.py",
        "v29_research_contract_certs.py"
      ],
      "script": "v27_wall_representative.py;v29_research_contract_certs.py",
      "location": "tfpt_research_contracts;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": false,
      "historical": true,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 27
    },
    {
      "id": "CONTRACT.U.01",
      "claim": "Research Contract 1 (U_wall): lemma chain U0-U6 + Theorem U; priority 1 (finite, algebraic, falsifiable via U2 kill switch). Certified: C_U^(1) wall enumeration (5 orbits); C_U^(2) FULL D4-fixed character variety is POSITIVE-DIMENSIONAL (reflection does not isolate the point); R(rho) characterized: CASE A ALIVE (all D4-fixed monodromies irreducible -> nontrivial mixing, U2 does not fire to B) and R(rho) is NOT algebraic (R integer, tr(M1M2) continuous) => R(rho) is the transcendental non-abelian-Hodge parabolic-degree map. C_U^(3) residual = the Hitchin/harmonic-metric solve (research-level)",
      "status": "Open (research contract)",
      "canonicalStatus": "UPDATED -> FLAV.RIGID.01/02: contract ratio-side closed (v49/v71); residual = U_point absolute anchor (+ G_metric, separate gate)",
      "dependencies": "GATE.UWALL.01",
      "scripts": [
        "v29_research_contract_certs.py",
        "v30_d4_character_variety.py",
        "v31_R_dictionary.py",
        "v32_rh_splitting.py"
      ],
      "script": "v29_research_contract_certs.py;v30_d4_character_variety.py;v31_R_dictionary.py;v32_rh_splitting.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": false,
      "historical": true,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 28
    },
    {
      "id": "RH.SPLIT.01",
      "claim": "RH route progress: exponent collapse sum_k U^k A0 U^-k = 4 diag(A0) exact (U=diag(1,i,-i)); integer-exponent options = perms of (2,1,1),(2,2,0); SPLITTING type made ALGEBRAIC: O(-2)+O(-1)^2 <=> diag(A0)=(1/2,1/4,1/4) (Schur-Horn realizable). Wall: product=I is path-ordered (not exp of sum) -> off-diag(A0) via RH solve; R-extraction needs C6<->P1mu4 bridge; c_u/c_d NOT obtained (no fabrication)",
      "status": "Numerical (partial; honest wall)",
      "canonicalStatus": "Numerical (partial; honest wall)",
      "dependencies": "CONTRACT.U.01",
      "scripts": [
        "v32_rh_splitting.py"
      ],
      "script": "v32_rh_splitting.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 29
    },
    {
      "id": "RH.BUNDLE.01",
      "claim": "EXPLICIT valid (U_wall) flat bundle constructed via numerical RH solve: A0 (eig {0,1/3,2/3}, diag (1/2,1/4,1/4)) realises cusp class + splitting O(-2)+O(-1)^2 (SigmaA_k=diag(2,1,1)) + trivial infinity monodromy ||M_inf-I||~1e-9 (=> prod M_k=I) + IRREDUCIBLE (case A, nontrivial mixing). Existence + case A confirmed; unique nabla_F* needs det R=8; c_u/c_d needs H2 bridge",
      "status": "Numerical (explicit realisation)",
      "canonicalStatus": "Numerical (explicit realisation)",
      "dependencies": "RH.SPLIT.01; CONTRACT.U.01",
      "scripts": [
        "v33_explicit_flat_bundle.py"
      ],
      "script": "v33_explicit_flat_bundle.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 30
    },
    {
      "id": "BOUNDARY.01",
      "claim": "the two gates are the discrete<->continuous HANDOFF, not missing formulas: quark cross-ratio cleanness tracks generation (t/b=3/26 clean err 3e-5; c/s fits 21/29 AND 34/47; u/d fits 8/17 AND 55/117 -> '11' not forced) = QCD/scheme-sensitivity boundary (FLAG). Compiler gives clean skeleton (K ladder, word-lengths, splitting, t/b); light-quark c's are QCD-dressed. Gravity same shape (R+R^2 clean; full measure = continuous completion)",
      "status": "Conceptual (verified)",
      "canonicalStatus": "Conceptual (verified)",
      "dependencies": "FLAV.QRATIO.01; v35",
      "scripts": [
        "v35_quark_qcd_boundary.py"
      ],
      "script": "v35_quark_qcd_boundary.py",
      "location": "introduction;tfpt_4",
      "externalData": "FLAG 2024",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 31
    },
    {
      "id": "H2.BRIDGE.01",
      "claim": "H2 bridge attempt: explicit per-puncture monodromies M_k computed (cusp class, prod M_k=I to 1e-5). HONEST NEGATIVE: |diag M_k|=(0,1/2,1/2), natural resolvent-dressed extraction h_j*g(1,r_j) does NOT reproduce lepton amplitudes (0.475,1.107,0.917). The Gamma^min geodesic-to-word dictionary (C6 hexagon <-> family rho*) is genuinely missing; R stays combinatorial (H1+det R=8); c_u/c_d NOT obtained, no fabrication",
      "status": "Numerical (honest negative)",
      "canonicalStatus": "Numerical (honest negative)",
      "dependencies": "RH.BUNDLE.01",
      "scripts": [
        "v34_h2_bridge_attempt.py"
      ],
      "script": "v34_h2_bridge_attempt.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 32
    },
    {
      "id": "CONTRACT.G.01",
      "claim": "Research Contract 2 (G_metric): lemma chain G0-G8 + Theorem G; priority 2 (deep analytic programme). Certified: G5 gap-dominance 2||V||=0.785<Delta=2.433, Delta_eff>1.647",
      "status": "Open (research contract)",
      "canonicalStatus": "Open (research contract)",
      "dependencies": "GATE.METRIC.01; GRAV.FR.01",
      "scripts": [
        "v29_research_contract_certs.py",
        "v28_gravity_fR.py"
      ],
      "script": "v29_research_contract_certs.py;v28_gravity_fR.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 33
    },
    {
      "id": "GRAV.FR.01",
      "claim": "R+R^2 covariant field equation CLOSED [I/P]: f(R)=R+R^2/(6M^2), M=c3^(7/2)Mbar=3.06e13 GeV=scalaron mass (m^2=1/(3 f_RR)=M^2); f(R) Einstein-side eq + 4D trace; N*=57 -> n_s=0.96491,r=0.00369,A_s=2.17e-9. Full metric measure OPEN [A] (G_metric); rel bound 248 c3^2=0.393<6log(3/2)=2.433",
      "status": "Identity/Physical (regime) + Open (full measure)",
      "canonicalStatus": "Identity/Physical (regime) + Open (full measure)",
      "dependencies": "QG.AMB.01; scalaron c3^7",
      "scripts": [
        "v28_gravity_fR.py"
      ],
      "script": "v28_gravity_fR.py",
      "location": "tfpt_2;tfpt_4",
      "externalData": "Planck 2018 (n_s,A_s)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 34
    },
    {
      "id": "FR.KOIDE.02",
      "claim": "CONJECTURE: Q_l source=0.66446; Q_source+phi0/|W(A3)|=Q_source+phi0/24=0.666679 overshoots 2/3 by 1.3e-5 (source->pole correction conjecture, |W(A3)|=24)",
      "status": "Physical (conjecture)",
      "canonicalStatus": "Physical (conjecture)",
      "dependencies": "FR.KOIDE.01",
      "scripts": [
        "v25_frontier_conjectures.py"
      ],
      "script": "v25_frontier_conjectures.py",
      "location": "tfpt_4",
      "externalData": "PDG",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 35
    },
    {
      "id": "FR.DM.02",
      "claim": "determinant-line axion f_a=M_scal/(2 dim S+ |mu4|)=M_scal/128~2.39e11 GeV, m_a~23.8 ueV. POST-INFLATIONARY REFRAMING [C] (2026-07-10; isocurvature analysis experiments/theory-contracts/axion01-06): the R^2 branch fixes H_inf=pi Mbar sqrt(A_s r/2)~1.6e13 GeV, so H_inf/f_a~66>1 and H_inf/(2pi f_a)~10 -- a FIXED pre-inflationary misalignment angle (108 deg spine DM.AXION.SPINE.01 or 170 deg hilltop FR.DM.03) is INCONSISTENT (CDM isocurvature >> Planck beta_iso<0.038). The natural DIAL-FREE resolution is POST-INFLATIONARY PQ breaking: theta_i is a horizon average (NOT a prediction), the relic follows from f_a alone -- f_a~2.39e11 sits in the post-inflationary window (Omega_a h^2~O(0.1-0.5)). DOMAIN-WALL-SAFE: N_DW = seam det-line index = inflow level |C|=1 (v470/v472) => N_DW=1 [C]. The axion branch is REFRAMED not killed: drop the fixed angle as observable, keep f_a + N_DW=1. NOT closed (relic O(1) string/wall uncertainty; N_DW identification [C])",
      "status": "Physical [C] (post-inflationary reframing)",
      "canonicalStatus": "Physical [C]: post-inflationary determinant-line axion, relic from f_a (in window), N_DW=|C|=1 domain-wall-safe; the fixed pre-inflationary spine/hilltop angle is inconsistent (isocurvature) and NOT observable -- reframed, not a closed hit",
      "dependencies": "FR.DM.01; M_scal=c3^{7/2}Mbar; v470; v472",
      "scripts": [
        "v25_frontier_conjectures.py"
      ],
      "script": "v25_frontier_conjectures.py",
      "location": "tfpt_4",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 36
    },
    {
      "id": "FLAV.CKM.01",
      "claim": "CKM core s12=lambda_C; s23; s13; delta=pi/3+3lC^2",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "seed phi0",
      "scripts": [],
      "script": "-",
      "location": "tfpt_1;tfpt_2",
      "externalData": "PDG / HFLAV",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 37
    },
    {
      "id": "CAS.01",
      "claim": "cascade D_n=60-2n: endpoints; exponent rungs -> 240; IR tail 46080; variance 6552",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "-",
      "scripts": [
        "v5_e8_cascade.py"
      ],
      "script": "v5_e8_cascade.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 38
    },
    {
      "id": "BOOT.01",
      "claim": "bootstrap mu^2-5mu+4=0; g_car=5 three ways; 8=rankE8=h(D5)=phi(30)",
      "status": "Lattice",
      "canonicalStatus": "Lattice",
      "dependencies": "-",
      "scripts": [
        "v6_bootstrap.py"
      ],
      "script": "v6_bootstrap.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 39
    },
    {
      "id": "GRAV.SCAL.01",
      "claim": "scalaron exponent 7; M_scal^2/Mbar^2 = c3^7 (3.06e13 GeV)",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "AX.P1.01",
      "scripts": [
        "v7_gravity_cosmo.py"
      ],
      "script": "v7_gravity_cosmo.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 40
    },
    {
      "id": "COSMO.LAM.01",
      "claim": "rho_Lambda/Mbar^4 = (3/4pi^2) e^{-2 a^-1} = 7.125e-121 (dark-energy scale 2.2375 meV)",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "EM.FP.01",
      "scripts": [
        "v7_gravity_cosmo.py"
      ],
      "script": "v7_gravity_cosmo.py",
      "location": "tfpt_1",
      "externalData": "Planck (rho_L^1/4 ~ 2.24 meV)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 41
    },
    {
      "id": "COSMO.INF.01",
      "claim": "Starobinsky A_s; n_s=1-2/N*; r=12/N*^2",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "GRAV.SCAL.01",
      "scripts": [
        "v7_gravity_cosmo.py"
      ],
      "script": "v7_gravity_cosmo.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "Planck; BICEP/Keck BK18; CMB-S4",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 42
    },
    {
      "id": "COSMO.OMB.01",
      "claim": "Omega_b = (1-1/4pi) phi0 = 0.04894",
      "status": "Numerical",
      "canonicalStatus": "Numerical",
      "dependencies": "AX.P1.01; phi0",
      "scripts": [
        "v7_gravity_cosmo.py"
      ],
      "script": "v7_gravity_cosmo.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "Planck (0.04897)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 43
    },
    {
      "id": "HOR.01",
      "claim": "horizon unit code: 1/(2pi)=4c3; 1920=|W(D5)|; S_dS; Page; beta_rad",
      "status": "Physical (reframe; not new physics)",
      "canonicalStatus": "Physical (reframe; not new physics)",
      "dependencies": "AX.P1.01",
      "scripts": [
        "v8_horizon.py"
      ],
      "script": "v8_horizon.py",
      "location": "AppendixH",
      "externalData": "standard horizon physics",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 44
    },
    {
      "id": "SM.RG.01",
      "claim": "dimensionful EW/QCD masses (m_W; m_Z; m_H; sin^2 thetaW; alpha_s)",
      "status": "Physical (scheme/RG)",
      "canonicalStatus": "Physical (scheme/RG)",
      "dependencies": "RG threshold matching",
      "scripts": [],
      "script": "-",
      "location": "tfpt_2",
      "externalData": "PDG",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 45
    },
    {
      "id": "SM.QFT.01",
      "claim": "admissible-sector QFT measure (gap 6 log 3/2; RP from cusp; OS)",
      "status": "Numerical/Conditional",
      "canonicalStatus": "Numerical/Conditional",
      "dependencies": "reflection positivity",
      "scripts": [],
      "script": "-",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 46
    },
    {
      "id": "QG.AMB.01",
      "claim": "DECOUPLED GENERAL QG PROBLEM, NOT a TFPT-specific gate (reclassified 2026-06-22, v335): the ambient/metric-sector measure IS the general Euclidean-QG conformal-factor problem (Gibbons-Hawking-Perry 1978), inherited not TFPT-specific. TFPT is rigorously GAP-DECOUPLED from it: margin Delta - 31/(4 pi^2) = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (v76/v311/v330/v332/v335), so NO TFPT prediction depends on the ambient measure. The conformal-mode obstruction is characterized (v332) and conditionally resolved by GHP rotation + IDG (v334); its construction is a separate QG programme (Picard-Lefschetz + IDG). Reclassified from 'TFPT structural frontier' to 'inherited, decoupled general QG problem' -- it is no longer counted among TFPT's open structural items.",
      "status": "Open (decoupled general QG; non-TFPT)",
      "canonicalStatus": "Open (decoupled general QG; non-TFPT)",
      "dependencies": "QGAMB.METRIC.01; QGAMB.CONFORMAL.01; v76; v330; v332; v334; v335",
      "scripts": [],
      "script": "-",
      "location": "tfpt_4 (status source); tfpt_2 conditional reduction",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 47
    },
    {
      "id": "FR.ETAB.01",
      "claim": "baryon asymmetry eta_B = 6.1e-10 (downstream of Omega_b)",
      "status": "Physical (conditional)",
      "canonicalStatus": "Physical (conditional)",
      "dependencies": "COSMO.OMB.01",
      "scripts": [],
      "script": "-",
      "location": "tfpt_4",
      "externalData": "Planck",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 48
    },
    {
      "id": "FR.MPME.01",
      "claim": "proton/electron ratio m_p/m_e (cross-sector QCD)",
      "status": "Open",
      "canonicalStatus": "Open",
      "dependencies": "QCD confinement",
      "scripts": [],
      "script": "-",
      "location": "tfpt_4",
      "externalData": "CODATA (1836.15)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 49
    },
    {
      "id": "FR.KOIDE.01",
      "claim": "Koide Q = 0.664 (near 2/3; not exact)",
      "status": "Physical (near-miss)",
      "canonicalStatus": "Physical (near-miss)",
      "dependencies": "-",
      "scripts": [],
      "script": "-",
      "location": "tfpt_4",
      "externalData": "PDG lepton masses",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 50
    },
    {
      "id": "FR.DM.01",
      "claim": "dark matter: axion candidate fixed; relic scale (f_a; m_a) open",
      "status": "Open (relic scale)",
      "canonicalStatus": "Open (relic scale)",
      "dependencies": "strong-CP determinant-line",
      "scripts": [],
      "script": "-",
      "location": "tfpt_4",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 51
    },
    {
      "id": "GATE.QGEO.01",
      "claim": "RESIDUAL GATE (5.0 review, B3): geometric origin of Q -- Q_+ spectrum {1,2,3}=A3 exponents and Q_-^2|supp=N_fam algebraically forced; deriving Q_+/Q_- from D4-equivariant parabolic geometry on P1\\mu4 is open; do NOT promote [2026-08-01 XXIII, GATE AUDIT after the cover program v597-v605 -- the OPERATOR half of this gate is now MODEL-COMPLETE: on the mu3-cover y^3 = x^4-1 the pair (U, V) hence Q = U+V, Q_+/Q_-, H, the AnchorLadder, the Klein group and the RP anti-automorphism are all REALIZED exactly (Z[omega]-integral, GL3-conjugation explicit, saturation index 81 with the v566 divisor pattern, D4-equivariant presentation, Weil positivity exact, translator theta = Jadj o inner(c)); the torsion constants localize (81 at the marks, 27 at infinity) and the seam is a real line, 3-torsion at infinity.  THE GATE ITSELF DOES NOT MOVE: the residual is (i) the conformal identification of the raw RP seam with (P1, mu4) (= QGEO.SYM.01, the bedrock premise -- the cover realization is a MODEL, not the seam), (ii) the zero-mode-selection demand (the seam-reversal demand is now geometric via v603, the zero-mode line is NOT R-real -- v605-followup, honest negative), and (iii) the geometric identity of the translator element c.  Status stays Open; the do-NOT-promote instruction stands.] [2026-08-01 XXXVII: bedrock state after v617/v620/v622 -- the conformal seam axioms FORCE the cover (v617), the cyclic order is pinned N = 3 conditional on the compiler's 3-adic constants (v620), and the physical seam IS the conformal seam at the kernel + dictionary level (v622: NS mode-sum identity, mu4 bond midpoints, straddle = geometry); the gate residue is now the COVERING-level identification (the mu3-cover of the seam double carrying the full interacting theory) plus the N_fam = 3 anchor; no status move.] [2026-08-02 XLV: the orbifold front completes at the LATTICE level -- Casimir (v628), twist OPE (v639), Klein RP (v645), modular data (v650), assembly (v651: the seam assembly is B), Arf structure (v652), bond-defect theorem (v653); what remains for this gate is the continuum/CFT statement and the seam identification -- the gate does NOT move.]  [2026-08-02 XLVIII: the continuum-OS data of the seam orbifold B are now measured at the CONVERGENCE level -- v679: Euclidean correlator convergence at fixed continuum data with UNDERSTOOD Fisher-Hartwig rates (2/3 sigma / 4/3 tau channel law; after one FH term the limits sit < 0.05% at the CFT values), reflection positivity SURVIVES the limit (Klein-Gram lambda_min extrapolations bounded away from zero, margins 7x..358x, must-fail flips), the cluster property holds in space AND Euclidean time (2 Delta = 0.222220 vs 2/9 at 8e-6; time gap = the epsilon level), and the characters 1/36, 1/9, 5/18, 4/9 extrapolate at 1e-12 with the measured N^-2 rate; the gate residual is now the FORMAL limit construction (uniform bounds over all configurations, tightness, the GNS limit Hilbert space, operator convergence, R sectors, the B assembly at correlator level) plus the v622/v623 seam identification -- the gate does NOT move.]  [2026-08-03 LII: quantitative progress via the quasi-free CAR route (v715, QGEO-CAR-RATES-SUMMABLE): all 6+1 deck/twist channels Schatten-norm Cauchy with summable rates, FH renormalization lattice-derived (Barnes G, 8.8e-7), RP cone stable; the majorant lemma (L1)-(L3) is NAMED with constants and a classical proof route -- the gate itself does NOT move: the continuum limit stays a formal object until the lemma is proven.]  [2026-08-04 LVII: the first proof block of the majorant lemma lands -- v745 (L1-SECTOR-PROVEN-MODULO-ELEMENTARY): (L1) is proven modulo elementary steps for all 6 undressed sectors on the fixed grid (exact geometric sums replace Euler-Maclaurin, leading term exact at 6e-4, gamma = 2 derived, 39744+108 inequalities at 0 violations, r1 = 1 sharp); the residues stay typed -- (a) grid-sup elementary, (b) L2 uniform FH/RH hard-technical, (c) L3 uniformity medium.  The gate itself does NOT move.]",
      "status": "Open",
      "canonicalStatus": "SUPERSEDED -> FLAV.QGEO.02 + v69/v70: Q-geometry derived (D4-equivariant); residual sharpened to det Q = N_fam",
      "dependencies": "FLAV.QGEO.01; FLAV.UFSTAR.01",
      "scripts": [
        "v13_open_gates.py",
        "v19_monodromy_moduli.py",
        "v22_open_gates_audit.py"
      ],
      "script": "v13_open_gates.py;v19_monodromy_moduli.py;v22_open_gates_audit.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": false,
      "historical": true,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 52
    },
    {
      "id": "GATE.RU.01",
      "claim": "RESIDUAL GATE (5.0 review, A2/B4): R derived only up to Mehta-Seshadri unitarity (U); parabolic scaffolding forced (deg E=-4, split 2+1+1 unique, det R=8); D4-fixed locus positive-dim, (U) selects stable point; keep visible, not unconditional",
      "status": "Physical (conditional)",
      "canonicalStatus": "UPDATED -> v69/v70 + FLAV.RIGID: R selector derived (det R=8 lattice, Spec Q+ from D4); ratios rigid (v49)",
      "dependencies": "FLAV.UFSTAR.01",
      "scripts": [
        "v19_monodromy_moduli.py",
        "v22_open_gates_audit.py"
      ],
      "script": "v19_monodromy_moduli.py;v22_open_gates_audit.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": false,
      "historical": true,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 53
    },
    {
      "id": "GATE.NSTAR.01",
      "claim": "RESIDUAL GATE (5.0 review, B5): N_star is a reheating INPUT (observed n_s=0.9649 -> N*=57, continuous), not a compiler consequence; kept [P]",
      "status": "Physical (input)",
      "canonicalStatus": "Physical (input)",
      "dependencies": "reheating",
      "scripts": [
        "v22_open_gates_audit.py"
      ],
      "script": "v22_open_gates_audit.py",
      "location": "tfpt_2;tfpt_3",
      "externalData": "Planck n_s",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 54
    },
    {
      "id": "GATE.METRIC.01",
      "claim": "RESIDUAL GATE (5.0 review, B6/C7) -- RESCOPED 2026-06-24: the LOCAL covariant metric-sector field equation is now CLOSED parameter-free (G_ab+Lambda g_ab=(1/c3)T_ab, both coefficients fixed, Einstein tensor via Lovelock; GRAV.NONLINEAR.01/v359, extending v358); c3=1/8pi is the gravitational seam constant and R+R^2/scalaron (M=c3^{7/2}Mbar) is the heat-kernel-grounded IR shadow. What remains OPEN is ONLY the GLOBAL nonperturbative ambient metric-sector measure (QG.AMB.01), now a [C] redundancy gap-decoupled from every readout (Delta_eff~1.648>0, v369/v337). So this gate no longer reads 'full covariant field equation open' -- the local equation is closed; the open item is the global measure alone",
      "status": "Open (global measure only; local equation closed v359)",
      "canonicalStatus": "Open (global measure only; local equation closed v359)",
      "dependencies": "QG.AMB.01; GRAV.NONLINEAR.01",
      "scripts": [
        "v22_open_gates_audit.py"
      ],
      "script": "v22_open_gates_audit.py",
      "location": "tfpt_2;tfpt_4",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 55
    },
    {
      "id": "QG.G2.01",
      "claim": "G2 grounded: spectral-action Seeley-DeWitt for Dirac (E=-R/4, tr I=4) gives a2~-R/3 (Einstein-Hilbert) + a4~R^2/72 (R^2), i.e. R+R^2 STRUCTURE convention-independently; M^2/Mbar^2=6(4pi)^2/f0, TFPT closure c3^7 fixes f0; M=c3^{7/2}Mbar=3.06e13 GeV",
      "status": "Numerical (structural grounding)",
      "canonicalStatus": "Numerical (structural grounding)",
      "dependencies": "GATE.METRIC.01; v28",
      "scripts": [
        "v36_spectral_action_g2.py"
      ],
      "script": "v36_spectral_action_g2.py",
      "location": "tfpt_research_contracts",
      "externalData": "standard Gilkey/Chamseddine-Connes",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 56
    },
    {
      "id": "QG.GAPDEC.01",
      "claim": "SIMPLER resolution: gap decoupling. 2||V_metric||=0.785<Delta=6log(3/2)=2.433 => Delta_eff=1.648>0; admissible sector keeps positive gap under metric dressing => decoupled from un-built ambient; physical QG = closed admissible-sector measure, ambient (G6) = math completeness, gap-decoupled & inert",
      "status": "Physical (conditional reframe)",
      "canonicalStatus": "Physical (conditional reframe)",
      "dependencies": "GATE.METRIC.01; G5",
      "scripts": [
        "v36_spectral_action_g2.py"
      ],
      "script": "v36_spectral_action_g2.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 57
    },
    {
      "id": "GATE.WOLFRAM.01",
      "claim": "second [C] path VERIFIED RUNNING: Wolfram Engine 14.3 reproduces alpha^-1, E8 glue, M=41, K/Q ladder, lepton c, solar angle, anchor generator, quark ratios, G2 heat-kernel (a2=-R/3,a4|R2=R2/72), Plucker apparatus (11,26,60 + pencil + dualities + lepton ring), (U_wall) selectors (Spec Q+, det R=8), Exterior Leg Lemma (scalar leg=>c_u/c_d=1; anchor microcode 55/117), carrier exterior grading (16=Lambda^even(5), exterior degrees (2,4)), the '11' is Pascal (||Pl(K)||_1=sum_{k<=2}C(4,k)=16-g_car; 15=dim su(4)=10+5=N_fam*g_car), Grand Mass Volume (det M_SM~u^25=u^g_car^2; K-row sums (6,9,10)), Selection Theorem (q(D5)+q(A3)=2, glue 4=|mu4|, only n=5 gives 16-spinor), EM Ward (8b1=(4/5)M=164/5, -5/4=q(D5)), Q geometry (chi Q+/Q-), Readout Rigidity (55/117 stratum-constant), delta=1/2 glue-norm 4-ops, pencil endpoints, compiler core (5,3)+Pythagorean, seam=horizon keystones (triply-8, shared (2/3)^6), Coxeter cycle (order 30, exponents=totatives), unique attractor (gap, Coxeter planes), S_dS carrier decomposition, horizon cross-links (Jacobson 8pi, Hod ln3, |W(D5)|), seam-horizon chain (one-sided Gauss-Bonnet, seam units), area-law seam units 8=|Z2||mu4|, Lambda branch 3/(4pi^2) + K5 counts, CFT bridge (WZW c=8/5/3, c_coset=0), Seam-Engineering Index Xi=31/(24pi^2 log(3/2)), E8 Casimir degrees=atoms, central-theorem closure (2pi c3=1/4, k=c3/2), Seeley-DeWitt a2 (1/G UV-sensitive), D4-equivariant Q-geometry, Q integer-lift (det Q=N_fam), simple R-bridge (det ladder (3,4,8,20), selector stratum derived => 55/117 integer Plucker), det Q=N_fam from cusp order (v72), k=c3/2 forced by variation x Gauss-Bonnet topology (v73), compiler micro-lemmas (pencil differences 2-16-48, anchor QF 41-25=16) (v74), Gate 1 U_point->v_geo (lepton product 32/9, ratios+product bijection) (v75), Gate 2 decoupling margin 6log(3/2)-31/(4pi^2)>0 (v76), G6 via E8 level-1 net (c=8/5/3, embedding) (v77), v_geo floor (S_dS rho_Lambda=32pi^4) (v78), hypercharge Lucas-Binet (1,1,7,13,55,133) + Inverse Anchor Theorem (1^T M^-1 1=1/atom, a^T M^-1 a=1) (v79), operator-pencil geometry (anchor singularity det B(K+xQ)=(3x+2)(3x+5), block-det type checker (9,10,16,40), F4xG2 shadow 52+14+26*7=248) (v80), double cover y^2=det B(K+xQ) branched at Koide -2/3 & carrier -5/3, disc=81=N_fam^4, clearing matrices C_{2/3}/C_{5/3}, branch-divisor symmetric fns = (scalaron,A_Lambda) (v81) (116/116 pass on independent engine; EE area-law is Python-only; re-verified 2026-07-21 after license reactivation, Wolfram Engine 14.3, 116/116 ALL WOLFRAM CHECKS PASSED) UPDATE 2026-08-28: 20 exact mirrors of v983/v985/v987/v993/v994/v997 ADDED to tfpt_readouts.wl; engine run DEFERRED_NO_ENGINE (license inactive); verified count stays 116/116 until the next activated engine run. ROUND-4 2026-09-05: 11 representative exact checks in wolfram/tfpt_round4.wl are wired into the base runner, but execution is UNVERIFIED_NO_KERNEL (wolframscript exit255: no WolframKernel location). Last verified base count remains116/116; native Arb/Fourier/cube regressions are not claimed as Wolfram verified. ROUND-7 2026-09-05: 14 exact representative blocks in wolfram/tfpt_round7.wl are included but UNVERIFIED_LICENSE_INACTIVE; explicit installed kernel selection fails license activation. Verified count remains116/116; no new engine success claimed.",
      "status": "Numerical (second path)",
      "canonicalStatus": "Numerical (second path) ROUND-4 2026-09-05: 11 representative exact checks in wolfram/tfpt_round4.wl are wired into the base runner, but execution is UNVERIFIED_NO_KERNEL (wolframscript exit255: no WolframKernel location). Last verified base count remains116/116; native Arb/Fourier/cube regressions are not claimed as Wolfram verified. ROUND-7 2026-09-05: 14 exact representative blocks in wolfram/tfpt_round7.wl are included but UNVERIFIED_LICENSE_INACTIVE; explicit installed kernel selection fails license activation. Verified count remains116/116; no new engine success claimed.",
      "dependencies": "mirrors v1;v2;v3;v10;v13;v16;v20;v21;v23;v24;v36;v37;v39;v42;v44;v45;v46;v47;v48;v49;v50;v51;v52;v53;v54;v55;v56;v57;v58;v59 (algebra only);v60;v61;v62 (data-confrontation, Python-only);v63;v64 (no-CTC core numerical, Python-only);v65 (falsification layer, Python-only);v66;v67;v68;v69;v70;v71;v72;v73;v74;v75;v76;v77;v78;v79;v80;v81;v983;v985;v987;v993;v994;v997",
      "scripts": [
        "wolfram/tfpt_readouts.wl"
      ],
      "script": "wolfram/tfpt_readouts.wl",
      "location": "verification/wolfram",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 58
    },
    {
      "id": "FLAV.PLUCKER.01",
      "claim": "anchor-plane Plucker apparatus (anti-numerology): left/right 2x2 minors of B_M=(1^T M; a^T M) and C_M=(M1 Ma) for R,Q,K,L are compiler atoms; ||Pl(K)||_1=11 (localizes the '11' in K), ||Pl_R(K)||_1=Pl(L)_13=26, sum Pl_R(L)=60=cascade start; inverse col-sums {1,5,6}; quark ratios are anchor-plane Plucker readouts. Identities [I]; physical amplitudes STAY [P]",
      "status": "Identity (audit layer)",
      "canonicalStatus": "Identity (audit layer)",
      "dependencies": "FLAV.QRATIO.01; AX anchor a",
      "scripts": [
        "v37_plucker_anchor.py"
      ],
      "script": "v37_plucker_anchor.py",
      "location": "tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 59
    },
    {
      "id": "FLAV.PENCIL.01",
      "claim": "K+xQ pencil: L=K+Q => mass and transport are one pencil P(x)=K+xQ with det(K+xQ)=3x^3+7x^2+6x+4=(N_fam, scalaron 7, |R^+(A3)|, |mu4|); chi_K=t^3-9t^2+12t-4; [K,L]=[L,Q]=[K,Q], chi_[K,Q]=t^3+5t+14 (5=g_car,14=dim G2 audit); K_e-K_u=a, L_e-L_u=Exp(A3); lepton ring c_e c_tau=|Z2| c_mu",
      "status": "Identity",
      "canonicalStatus": "Identity",
      "dependencies": "v10 K=R+Q*Sigma",
      "scripts": [
        "v37_plucker_anchor.py"
      ],
      "script": "v37_plucker_anchor.py",
      "location": "tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 60
    },
    {
      "id": "FR.KOIDE.03",
      "claim": "sharper Koide source->pole conjecture: multiplicative seed transfer u->(1-1/(2 N_fam^3))u=(53/54)u (54=2*3^3=|Z2| N_fam^3, no new free number) gives Q=0.66666610, dev -5.7e-7 from 2/3 (10x better than additive phi0/24); research gate: derive the leptonic seed transfer",
      "status": "Physical (conjecture)",
      "canonicalStatus": "Physical (conjecture)",
      "dependencies": "FR.KOIDE.01; lepton c",
      "scripts": [
        "v37_plucker_anchor.py"
      ],
      "script": "v37_plucker_anchor.py",
      "location": "tfpt_4",
      "externalData": "PDG lepton masses",
      "supersedes": "FR.KOIDE.02",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 61
    },
    {
      "id": "GATE.UWALL.02",
      "claim": "(U_wall) U2 kill-switch HARDENED: kill-switch fires iff D4-cusp rep reducible; across 400 random cusp-class generators M_k=U^{k-1}MU^{1-k} commutant is scalar at every point (irreducible 400/400), reducible locus measure-zero (codim>=1), mixing strength >=0.35 => case A (nontrivial SU(3)_F mixing) is GENERIC not anecdotal; (U_wall) survives its cheapest falsification test. RESIDUAL: amplitude dictionary Gamma^min (Hitchin output for c_u/c_d) still research-level; R combinatorially closed",
      "status": "Numerical (survival hardened)",
      "canonicalStatus": "Numerical (survival hardened)",
      "dependencies": "CONTRACT.U.01; v33",
      "scripts": [
        "v38_uwall_killswitch.py"
      ],
      "script": "v38_uwall_killswitch.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 62
    },
    {
      "id": "GATE.UWALL.03",
      "claim": "(U_wall) SELECTOR side closed on the explicit point (algebraic, no PDE): exponents at inf eig(sum A_k)={2,1,1}=anchor a => splitting O(-2)+O(-1)^2; pardeg=0 (deg E=-4=-|mu4| + 4 weight-sums of 1) => polystable; det R=8=n.a (anchor pairing on splitting type); Spec(Q_+)={1,2,3}=3*alpha+1 (alpha=cusp weights {0,1/3,2/3}) => both U4 selectors read off the bundle. RESIDUAL: amplitudes c_u/c_d are the finer harmonic-metric/Hodge datum (Gamma^min Hitchin), not fixed by splitting+weights",
      "status": "Numerical (selector side closed)",
      "canonicalStatus": "Numerical (selector side closed)",
      "dependencies": "GATE.UWALL.02; v33; FLAV.R.01",
      "scripts": [
        "v39_uwall_selectors.py"
      ],
      "script": "v39_uwall_selectors.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 63
    },
    {
      "id": "GATE.UWALL.04",
      "claim": "(U_wall) harmonic metric = FINITE linear algebra, NOT a transcendental Hitchin PDE: polystable point is unitary (Mehta-Seshadri) => Phi=0, harmonic metric = constant invariant form H. Explicit bundle: raw M_k non-unitary (||M^dag M-I||~0.83) but UNIQUE invariant Hermitian form H (null dim 1) is POSITIVE-DEFINITE => unitarisable; unitarised M~_k=H^{1/2}M_k H^{-1/2} unitary (1e-8), cusp class, prod=I, clean elements {0,1/2,1/sqrt2}. So v31 'transcendental Hitchin' applies only to generic Higgs locus; at the selected point R(rho) reduces to finite unitarisation + combinatorial R (closed). RESIDUAL: final c_u/c_d needs the quark leg/hexagon-site composition; not fabricated",
      "status": "Numerical (Hitchin reduced to finite)",
      "canonicalStatus": "Numerical (Hitchin reduced to finite)",
      "dependencies": "GATE.UWALL.03; v31; v33; Mehta-Seshadri",
      "scripts": [
        "v40_harmonic_metric.py"
      ],
      "script": "v40_harmonic_metric.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 64
    },
    {
      "id": "GATE.UWALL.05",
      "claim": "(U_wall) FINAL leg-assignment test (honest result): (i) delta=1/2 resolvent reproduces lepton c=(16/7,4/3,7/6) exactly => Lambda=(0.475,1.107,0.917); (ii) Lambda_mu=1.107>1 RULES OUT amplitudes=unitary holonomy moduli (modulus bound); (iii) SUGGESTIVE: harmonic-frame |diag M~_0|=(0,1/2,1/2), the 1/2 = delta=1/2 => geometry supplies delta (point observation, genericity future); (iv) CLEAN NEGATIVE: (I-1/2 M~)^-1 != amplitude vector. VERDICT: amplitudes are the combinatorial delta=1/2 resolvent; holonomy supplies delta+leg not magnitude; c_u/c_d NOT obtained from holonomy alone (no fabrication)",
      "status": "Numerical (honest negative + suggestive link)",
      "canonicalStatus": "Numerical (honest negative + suggestive link)",
      "dependencies": "GATE.UWALL.04; v20; v40",
      "scripts": [
        "v41_leg_assignment.py"
      ],
      "script": "v41_leg_assignment.py",
      "location": "tfpt_research_contracts",
      "externalData": "PDG lepton ratios",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 65
    },
    {
      "id": "GATE.UWALL.15",
      "claim": "EXTERIOR LEG LEMMA (re-types the v41 negative as a category error): u,d share hexagon-radial (r,w)=(1,1) and the C6 resolvent reads only (r,w) => ANY scalar leg gives c_u/c_d=1 identically (scalar sees radius not orientation). The u/d split is the smallest non-scalar invariant = exterior square Lambda^2 F = oriented anchor-plane area Pl_{1,a}(K). The '11' is GENERATED: (K,1,a)->B_K=[[13,8,4],[18,11,6]]->Pl(K)=(-1,6,4)=(-N_Phi,|R+A3|,|mu4|)->||Pl(K)||_1=11. c_u/c_d=g_car*||Pl(K)||_1/(N_fam^2 Delta_Q)=55/117 [I]; anchor microcode e2(a)(p3+1)/(p0^2(2p2+1)). TYPING: [I] Plucker/microcode identities; [P] physical C_ud(rho*)=Lambda^2 F readout needs finite U_f* selection on Lambda^2 F. Residual re-typed: 'find scalar y' -> 'show U_f* induces anchor-plane readout on Lambda^2 F'",
      "status": "Identity [I] + Physical [P]",
      "canonicalStatus": "Identity [I] + Physical [P]",
      "dependencies": "GATE.UWALL.05; FLAV.PLUCKER.01; v37",
      "scripts": [
        "v42_exterior_leg.py"
      ],
      "script": "v42_exterior_leg.py",
      "location": "tfpt_research_contracts;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 66
    },
    {
      "id": "FLAV.RIGID.01",
      "claim": "Readout Rigidity (Theorem U2): c_u/c_d is CONSTANT on the discrete selector stratum S_{8,Q}={det R=8, SNF(R)=(1,1,8), Spec(Q+)={1,2,3}}, independent of the continuous D4 position => c_u/c_d=55/117 needs NO point-uniqueness. (U_wall) splits into U_unitary (v40, Phi=0 finite) + U_H2 (v39, R/Q/K from bundle) + U_Lambda2 (v42/v45, exterior Plucker 55/117) + U_point (open, only for full U_f* matrix). D4 fixes the chamber, selectors cut the point. The flavor bridge is a finite exterior readout-rigidity theorem, NOT a PDE gate",
      "status": "Identity [I] (rigidity) + [A] (U_point)",
      "canonicalStatus": "Identity [I] (rigidity) + [A] (U_point)",
      "dependencies": "GATE.UWALL.03; v39; v40; v42",
      "scripts": [
        "v49_readout_rigidity.py"
      ],
      "script": "v49_readout_rigidity.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "FLAV.UGATE.01;GATE.UWALL.01;GATE.RU.01;CONTRACT.U.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 67
    },
    {
      "id": "FLAV.QGEO.02",
      "claim": "Q geometry (Theorem Q): Sigma-split Q_+=(Q+SigQSig)/2=[[3,0,0],[0,2,0],[0,2,1]] (chi=(t-1)(t-2)(t-3), Spec={1,2,3}=A3 exponents), Q_-=[[0,1,0],[3,0,0],[3,0,0]] (chi=t(t^2-3), Q_-^2|supp=3=N_fam). UNIQUE nonneg-integer Q with rows(4,5,6), cols(9,5,1), chi_Q+, chi_Q- (enumeration) => K=R+Q*Sigma is the second Sigma-shadow of the same parabolic flavor operator. Algebraic normal form [I]/[L]; D4-equivariant parabolic realisation of Q_+/Q_- on P1\\\\mu4 remains [P] (GATE.QGEO)",
      "status": "Identity/Lattice [I]/[L] + [P] geometry",
      "canonicalStatus": "Identity/Lattice [I]/[L] + [P] geometry",
      "dependencies": "FLAV.KQ.01; GATE.QGEO.01",
      "scripts": [
        "v50_q_geometry.py"
      ],
      "script": "v50_q_geometry.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "GATE.QGEO.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 68
    },
    {
      "id": "SEL.THM.01",
      "claim": "Boundary Carrier Selection Theorem (Theorem A): admissible boundary-carrier data SELECT (D5+A3)+mu4=E8 uniquely. Boundary mu4->H1=Z^3->A3,N_fam=3; minimal 16-spinor->D5 (only n=5); disc(D5)=disc(A3)=Z4 cyclic anti-isometric, q(D5)+q(A3)=5/4+3/4=2=E8 root norm, glue index 4=|mu4|, E8 unique even unimodular rank 8. Four simultaneous conditions (16-spinor, N_fam=3, cyclic Z4, q-sum 2) met ONLY by D5+A3; D8 (non-cyclic, no family), E7+A1 (1 fam), E6+A2 (2 fam, Z3), A4+A4 (Z5), A8 (single) excluded. Lattice facts [L]; physical selection (P1 seam + P2 carrier => these inputs) is [P]/[A] selection axiom, not zero-input derivation",
      "status": "Lattice [L] + selection [P]",
      "canonicalStatus": "Lattice [L] + selection [P]",
      "dependencies": "E8.GLU.01; BOOT.CLASS.01; v1; v15",
      "scripts": [
        "v47_selection_theorem.py"
      ],
      "script": "v47_selection_theorem.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 69
    },
    {
      "id": "EM.WARD.01",
      "claim": "EM Boundary Ward decomposition (Theorem C): F_U(1)=0 read as reduced boundary U(1) Ward identity in three terms - Maxwell alpha^3 + Calderon -2c3^3 alpha^2 (2=|Z2|) + transport -8b1 c3^6 log(1/phi_seam). Coefficient identities EXACT [I]: 8b1=(4/5)M=(4/5)(sumL+N_Phi)=164/5; seam exponent -5/4=-q(D5); transport power c3^6=C6 hexagon. Root reproduces alpha^-1=137.0359992 (v3). Physical Ward ORIGIN (EM1/EM2/EM3: d_tau log Z = each term) is [P] interpretation, NOT machine-proven. SHARPENED 2026-06-22 (v341/ALPHA.QUILLEN.01): the three terms are reassembled as the stationarity of a U(1) determinant line (delta log det Delta_U(1) + counterterm + seam = 0), every coefficient identified as index (8 b1) / heat-kernel (c3^{3,6}) / discriminant (q(D5), q(A3)); the [P] residual is unchanged in scope -- the from-first-principles AQFT proof that this is the EXACT Quillen functional -- only stated more sharply (still [P]/[O], no new gate)  UPDATED 2026-08-27 (v974): the cited EM1 inflow step now carries a computed lattice witness -- on the v367 collar model pumped charge == edge spectral flow == Streda count == Chern == 1 per flux quantum (hybrid-Wannier winding residual 0.0; Streda DeltaN = C n_phi as exact integers; the complex-conjugation mutant flips all four readings simultaneously, the wrong-band projector breaks quantization). Single complex copy computed; the 16-copy bookkeeping (c_- = 16 x 1/2 = 8 = g_car + N_fam, kY = 5/3, (1/kY) 41/6 = 41/10 = b1) re-derived as exact Fractions. The physical Ward ORIGIN keeps its [P]/[O] scope -- nothing upgrades here.",
      "status": "Identity [I] + physical origin [P] (sharpened to a Quillen form by v341; to a heat-kernel origin by v342)",
      "canonicalStatus": "Identity [I] + physical origin [P]",
      "dependencies": "EM.FP.01; v3; v341; v342",
      "scripts": [
        "v48_em_ward.py"
      ],
      "script": "v48_em_ward.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 70
    },
    {
      "id": "SEAM.EQUIV.CONTINUUM.02",
      "claim": "THE CONTINUUM REALISATION SHARPENED BY THE BOOTSTRAP (2026-06-22): the 'c=8 ambiguity' flagged open in v277/v344 (E8 vs SO(16) both have c=8) is RESOLVED, non-circularly, by the seam's ORDER-4 mu4 clock; so the continuum residual of SEAM.EQUIV.01 is no longer 'which c=8 net?' but PURELY the existence of the chiral edge / massless scaling limit. [E] two c=8 lattices: E8 (det Cartan 1, holomorphic, 1 primary) and D8=SO(16) (det Cartan 4, 4 primaries). [E] the order-4 clock resolves it: the seam clock is mu4 (order 4, forced by the four Gauss-Bonnet marks |mu4|=4=h(A3), v216); the index-4 simple-current extension of D5(+)A3 is E8 (v154), the index-2 would be D8=SO(16); so the order-4 clock selects E8 (full mu4 condensation 16->4->1, det 1) NOT SO(16) (partial Z2 16->4, det 4, v281) -- the order (4 not 2) is the discriminator, non-circular (marks are Gauss-Bonnet). [E] the bootstrap agrees: h(E8)=30 (max prime 5=g_car, Coxeter-match v6/v350) vs h(D8)=14 (max prime 7); SO(16) fails the carrier bootstrap. [E] so the which-net ambiguity is RESOLVED (c=8 + the order-4 clock pin E8; det K=1/holomorphy is then a CONSEQUENCE, not an extra open input). [O] the residual is PURELY existence: does the gapped quasi-free collar HAVE a chiral edge / massless scaling limit (the bulk-edge correspondence: a gapped E8 bulk has a chiral c=8 edge)? -- the literature-anchored continuum leg (Morinelli-Stottmeister, Adamo OS; v336). The net and the holomorphy are pinned; only existence remains.",
      "status": "Sharpening [E]/[O] (the order-4 clock + Coxeter-match pin E8 over SO(16); the residual is purely the chiral-edge existence)",
      "canonicalStatus": "[E] the order-4 clock + Coxeter-match jointly pin E8 over SO(16), resolving the which-net ambiguity non-circularly; [O] the chiral-edge/scaling-limit existence (v336) is the remaining residual. A sharpening; does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.GEOM.01; v277; v344; v216; v154; v281; v336",
      "scripts": [
        "v351_continuum_realisation_sharpened.py"
      ],
      "script": "v351_continuum_realisation_sharpened.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 71
    },
    {
      "id": "VGEO.SHARPEN.01",
      "claim": "v_geo IS THE SINGLE DIMENSIONFUL INPUT OF THE WHOLE THEORY (2026-06-23, Direction 6 of next-plan.md), now confirmed across matter AND gravity, and irreducibly primitive. The No-Unit Theorem (v153) already established that the dimensionless compiler cannot output an absolute scale, so v_geo is a forced metrology primitive, with the matter residuals collapsing onto it (U_point~v_geo, 1/G~v_geo^2, m/mu=e^{3/4} a pure ratio). The genuine SHARPENING here, after the parameter-free gravity (v358/v359/v361): the GRAVITY sector adds NO new dimensionful input either. [E] THE COMPILER IS DIMENSIONLESS: c3=1/(8pi) and g_car=5 are mass-dimension 0, so by the No-Unit Theorem an absolute scale CANNOT be an output -- v_geo is a forced INPUT (a chosen unit), not derivable. [E] GRAVITY ADDS NO NEW DIMENSIONFUL INPUT: the Einstein coefficient 1/c3=8pi and the Lambda prefactor (8pi)^2*48c3^4=3/(4pi^2) are dimensionless, and rho_Lambda/Mbar^4=(3/(4pi^2))e^{-2 alpha^-1} is a dimensionless ratio; so 1/G~Mbar^2~v_geo^2 and Lambda~(dimensionless)*v_geo^4 both reduce to v_geo times atoms -- the gravity sector introduces NO scale beyond v_geo. [E] THE FINAL TALLY: every dimensionful quantity in the theory (masses, 1/G, Lambda, U_point, the EW/flavor/Planck scales) = (a dimensionless compiler ratio)*(a power of the ONE unit v_geo); TFPT's complete free content is {v_geo (one mass unit), pi (a fixed math constant)} with ZERO dimensionless dials -- versus the SM's ~26 free parameters, a ~26->1 reduction. [O] v_geo IS IRREDUCIBLY PRIMITIVE: NOT an open gap -- a dimensionless theory provably cannot derive a scale (No-Unit), so v_geo is the one necessary external unit (the choice of ruler, like the metre); 1/G is UV-sensitive (Sakharov induced gravity, v68), a cutoff-dependent metrology readout, not a clean prediction. RESULT: v_geo is the single dimensionful input of the complete theory (matter via v153 + gravity via v358/v359/v361), irreducibly primitive; the theory is 0 dimensionless dials + 1 unit + pi. Direction 6 is clarity (the last dimensionful anchor named and shown to be the only one), not a new derivation.",
      "status": "v_geo capstone [E]/[O] (Direction 6: v_geo the single dimensionful input across matter+gravity, irreducibly primitive by No-Unit; final tally 0 dimensionless dials + 1 unit + pi vs SM ~26)",
      "canonicalStatus": "[E] the dimensionless-compiler / gravity-adds-no-scale / final-tally facts; [O] the irreducible-primitive status of v_geo (No-Unit consequence) + the Sakharov UV-sensitivity of 1/G. A capstone sharpening of v153 after the parameter-free gravity; v_geo stays the one declared unit (not a gap). Direction 6 closed -- clarity, not a new derivation.",
      "dependencies": "v153; v358; v359; v361; v60; v68; v78",
      "scripts": [
        "v364_vgeo_sharpen.py"
      ],
      "script": "v364_vgeo_sharpen.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 72
    },
    {
      "id": "GRAV.BACKREACT.01",
      "claim": "THE MATTER-GRAVITY BACKREACTION WITH THE EXPLICIT J3 FLUX (2026-06-23, Direction 5 of next-plan.md). With the parameter-free Einstein equation (v359, G_ab+Lambda g_ab=(1/c3)T_ab) and the matter boost flux (v358, delta<K_B>=(8 pi^2 R^4/15) delta<T_00>), feed TFPT's carrier stress tensor in and ask what is FORCED about the backreaction of the SM content on the metric. Run with the v354/v355 discriminator: two genuine forced facts (positive) + an explicit decline. [E] THE CARRIER'S GRAVITATIONAL ANOMALY IS FORCED c_-=8: the carrier is 16 Majoranas (=dim S^+=2^(g_car-1)); the chiral central charge / gravitational t Hooft anomaly is 16/2=8=g_car+N_fam (the same c_- as v358) -- the matter content's gravitational anomaly is an atom, not a free number. [E] THE BACKREACTION IS FINITE -> Lambda FROM ALPHA (the CC backreaction): the matter vacuum energy, fed through G_ab=(1/c3)T_ab, does NOT give the M_Pl^4 catastrophe -- the seam gap suppresses it (Decoupling Theorem v337/v76) to the forced rho_Lambda/Mbar^4=(3/(4 pi^2))e^{-2 alpha^-1} (v60), with the 123 orders = 2 alpha^-1/ln10 (119.03, the EM fixed point) + log10(256 pi^4/3) (3.92); the loop (v359 equation + v60 Lambda + the carrier vacuum) closes. [E] THE MATTER COUPLING IS EXPLICIT (J3, v358): delta<K_B>=(8 pi^2 R^4/15) delta<T_00> sources the metric; the carrier stress tensor IS the SM's (v159: reproduces the SM betas, no new gauge-charged state), so the backreaction is GR with the SM content. [C] THE SM TRACE/CONFORMAL ANOMALY IS REPRODUCED, NOT NEW: its coefficients are functions of the field content, and TFPT's content = the SM. [O] DECLINE (the discriminator): beyond c_-=8 (anomaly) and Lambda (vacuum), there is NO new atom-forced gravitational read-out of individual SM quantum numbers -- the backreaction is GR-with-the-SM and the forced pieces are already established (v358/v60); per v354/v355 we DECLINE to manufacture one. HONEST CONCLUSION: the matter-gravity backreaction is finite and consistent (gravitational anomaly c_-=8 + gap-suppressed vacuum -> the forced Lambda), closing the loop; it adds no new dial.",
      "status": "Backreaction result [E]/[C]/[O] (Direction 5: carrier gravitational anomaly c_-=8 forced + finite backreaction -> Lambda from alpha, loop closes; no new SM-quantum-number read-out -> declined per v354/v355)",
      "canonicalStatus": "[E] the carrier gravitational anomaly c_-=8 + the finite-backreaction-to-Lambda loop closure + the explicit J3 coupling; [C] the SM trace-anomaly reproduction; [O] the decline of a new per-quantum-number read-out. A disciplined consolidation (positive content + an honest decline): the backreaction is finite and forced (c_-=8, Lambda), but adds no new free quantity. Nothing over-claimed.",
      "dependencies": "GRAV.NONLINEAR.01; v359; v358; v60; v337; v159; v354; v355",
      "scripts": [
        "v361_grav_backreaction.py"
      ],
      "script": "v361_grav_backreaction.py",
      "location": "tfpt_4",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 73
    },
    {
      "id": "GRAV.GAPCORR.01",
      "claim": "THE GAP-INDUCED HIGHER-CURVATURE CORRECTION TO GR -- THE DISCIPLINED RESULT (2026-06-23, Direction 2 of next-plan.md). Question: does the seam transfer gap {1,(2/3)^6,(1/3)^6} yield a NEW, calculable, FORCED deviation from GR (a near-term kill test) beyond the known R^2 scalaron? Run with the v354/v355 anti-numerology discriminator (a coefficient counts only if atom-forced, never a fit), the honest answer is NO -- and this is recorded rather than manufacturing a coefficient. [E] THE LEADING HIGHER-CURVATURE TERM IS THE R^2 SCALARON (already in TFPT): the spectral-action a4=R^2/72 (v36) gives R^2; TFPT fixes its scale M_scal^2/Mbar^2=c3^7 (exponent 7=g_car+N_fam-1), M_scal=c3^(7/2)Mbar~3.06e13 GeV -- the calculable beyond-Einstein correction. [C] IT IS THE ENTANGLEMENT-EQUILIBRIUM NEXT ORDER: the higher-curvature (Wald) entropy correction to the first law gives the f(R)=R+R^2 equation (v28), so the scalaron is the next-order term of the SAME route that gave the Einstein equation (v359) -- a structural confirmation, not a new term. [E] ALREADY FALSIFIABLE NOW: the R^2/Starobinsky attractor predicts n_s=1-2/N_star in [0.960,0.967] and r=12/N_star^2 in [0.0033,0.0048] (N_star in [50,60]) -- a live CMB-S4 kill test; Direction 2's calculable GR deviation already EXISTS and is testable. [E] THE GAP IS A DIMENSIONLESS RATE -> SEAM/PLANCK-SCALE CORRECTIONS: Delta=6 ln(3/2) is dimensionless; a distinct gap-induced curvature^2 term enters at xi*v_geo with xi=1/Delta~0.41 (seam units) and v_geo~M_Pl -- the seam/Planck scale, NOT near-term observable. [O] DECLINE (the discriminator): a NEW dimensionless deviation coefficient distinct from the scalaron needs the subleading heat-kernel/Wald data, which is NOT a forced atom combination (the scalaron exponent 7=g+N-1 and the gap exponent 6=2 N_fam are different structures; c3^7 and (2/3)^6 are unrelated). Per v354/v355 we DECLINE to manufacture an atom-unforced coefficient. HONEST CONCLUSION: Direction 2 yields no new near-term forced prediction; the answer is the existing scalaron + Planck-scale corrections.",
      "status": "Gap-correction result [E]/[C]/[O] (Direction 2 declined: the calculable GR deviation is the R^2 scalaron already falsifiable via n_s/r; a new gap coefficient is not atom-forced -> declined per v354/v355)",
      "canonicalStatus": "[E] the scalaron-as-leading-correction + its existing n_s/r kill test + the gap-is-dimensionless/Planck-scale reasoning; [C] the Wald-first-law consistency; [O] the explicit decline of a manufactured new coefficient. A disciplined-search record (like v355): the positive content is that the calculable GR deviation is the scalaron; the decline is the anti-numerology outcome. No new prediction added; nothing over-claimed.",
      "dependencies": "GRAV.NONLINEAR.01; v359; v358; v36; v28; v354; v355",
      "scripts": [
        "v360_grav_gap_corrections.py"
      ],
      "script": "v360_grav_gap_corrections.py",
      "location": "tfpt_4",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 74
    },
    {
      "id": "GRAV.NONLINEAR.01",
      "claim": "THE FULL COVARIANT EINSTEIN EQUATION, PARAMETER-FREE (2026-06-23, Direction 1 of next-plan.md, extends GRAV.ENTROPY.EQUILIBRIUM.01/v358). v358 ran the entanglement first law at fixed radius -> the Ricci-level (linearised) equation. Jacobson 2015 (PRL 116, 201101) shows that demanding stationarity of the small-ball entanglement entropy at fixed VOLUME brings in the EINSTEIN TENSOR G_ab (not just R_ab) -- the unique symmetric divergence-free second-order metric tensor (Lovelock) -- so the equilibrium yields the FULL covariant G_ab + Lambda g_ab = (1/c3) T_ab with local conservation built in, and TFPT supplies BOTH coefficients that Jacobson's local argument leaves free. [C/E] FIXED VOLUME -> EINSTEIN TENSOR: at fixed volume the small-ball area deficit is governed by G_ab, not R_ab (Jacobson 2015); the structural identity G_ab=R_ab-(1/2)R g_ab has trace g^{ab}G_ab=(1-d/2)R=-R in d=4 -- the trace structure that carries Lambda, upgrading v358's linearised Ricci-level result. [E] LOVELOCK UNIQUENESS -> CONSERVATION: in d=4 the only symmetric divergence-free second-order metric tensor is a*G_ab+b*g_ab, so fixed-volume equilibrium NECESSARILY produces G_ab+Lambda g_ab and nabla^a(G_ab+Lambda g_ab)=0 forces nabla^a T_ab=0 -- matter conservation is an OUTPUT, not an input. [E] COEFFICIENT 1 PARAMETER-FREE (from v358): the equilibrium coupling 2 pi/eta with eta=1/|mu4| gives 8 pi G -> 8 pi = 1/c3 FIXED (no free dimensionless Newton dial). [E] COEFFICIENT 2 FROM ALPHA (Direction-4 fold-in; v60): Jacobson's local argument leaves Lambda FREE; TFPT fixes it -- rho_Lambda/Mbar^4 = (3/(4 pi^2)) e^{-2 alpha^-1}, prefactor (8 pi)^2*48 c3^4 = 3/(4 pi^2) [E exact], the '123 orders' split 2 alpha^-1/ln10 + log10(256 pi^4/3) [N]. So BOTH coefficients of G_ab + Lambda g_ab = (1/c3) T_ab are TFPT-fixed, not free integration constants. [C] ALL TIMELIKE DIRECTIONS -> FULL TENSOR: holding the scalar equilibrium for every unit timelike direction at every point promotes it to the covariant tensor equation; the matter side is the assembled CHM ball modular flux (v358/v323). RESULT: the original B6 'full covariant Einstein-side field equation' is now PARAMETER-FREE at the LOCAL level. [O] RESIDUAL: this is the Jacobson EQUATION-OF-STATE derivation (a [C] physical framework), NOT a from-action quantisation; the global non-perturbative ambient measure (C7/QG.AMB.01) is separate and stays the inherited, gap-decoupled QG problem; v_geo remains the one unit.",
      "status": "Full covariant Einstein equation [E]/[C]/[O] (parameter-free: 8pi=1/c3 + Lambda from alpha; Einstein tensor via fixed-volume equilibrium + Lovelock conservation; B6 closed at local level; residual = equation-of-state status + C7 global measure)",
      "canonicalStatus": "[E] the Einstein-tensor structure (trace=-R d=4) + Lovelock conservation + both parameter-free coefficients (8pi=1/c3, Lambda prefactor 3/(4pi^2)); [C] the Jacobson-2015 fixed-volume machinery + the all-directions quantifier; [O] the equation-of-state status + the global measure (C7). Delivers the full LOCAL covariant equation parameter-free; advances B6/GATE.METRIC from 'linearised (v358)' to 'full covariant local'; does NOT claim a from-action QG quantisation.",
      "dependencies": "GRAV.ENTROPY.EQUILIBRIUM.01; v358; v60; v323; v57; v58; v216",
      "scripts": [
        "v359_grav_nonlinear_einstein.py"
      ],
      "script": "v359_grav_nonlinear_einstein.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 75
    },
    {
      "id": "GRAV.ENTROPY.EQUILIBRIUM.01",
      "claim": "THE ENTANGLEMENT-EQUILIBRIUM DERIVATION OF THE LINEARISED EINSTEIN EQUATION, PARAMETER-FREE (2026-06-23). Carries out the Jacobson-2015 / Faulkner-et-al. 'first law of entanglement => Einstein equation' (delta S = delta<K>) with TFPT's atoms, and closes the two pieces that the bridge analysis (v356) left open (the matter boost flux J3 and the entropy-density coefficient). [E] PARAMETER-FREE EINSTEIN COEFFICIENT: delta S = delta<K> => G_ab = (1/c3) T_ab with 1/c3 = 8pi FIXED; all four c3-roles mutually consistent with NO free dimensionless Newton dial (Unruh factor 1/(2pi)=4c3, EH action 1/(16pi)=c3/2, Einstein coeff 8pi=1/c3, Bekenstein entropy density 1/4=1/|mu4|). [E] THERMO = GEO COINCIDENCE (the new exact result): the first law REQUIRES the Einstein coefficient 2pi/eta (eta=entropy density); the geometry gives 1/c3 = |Z2|*2pi*chi(S^2) (one-sided Gauss-Bonnet, v58); these give the SAME equation iff |mu4| = |Z2|*chi(S^2) = 4 (a TFPT atom identity, v216) -- so the thermodynamic (first-law) and geometric (Gauss-Bonnet) origins of c3 are ONE object, coupled by 4=|mu4|=|Z2|*chi. [C] J3 SOLVED (matter boost flux assembled): the ball modular Hamiltonian is the boost (Casini-Huerta-Myers; geometric via Bisognano-Wichmann, v323) K_B = 2pi int_B w T_00 with w=(R^2-r^2)/(2R) the CHM weight; the 3-ball weight integral int_B w d^3x = 4pi R^4/15, so delta<K_B> = (8 pi^2 R^4/15) delta<T_00> -- the concrete matter-side flux of TFPT's stress tensor (closing the 'ingredients present, flux not assembled' gap). [E] ENTROPY DENSITY ATOM-FIXED (the [A] coefficient pinned): the Bekenstein numerator 1/4=1/|mu4| (exact, v57) and the entanglement normalisation = the central charge c=8=g_car+N_fam (a CFT's universal area-law/entanglement coefficient IS its central charge); the area-law FORM is established for the RP-Gaussian seam (v59). So the dimensionless entropy-density coefficient is atom-determined; the only residual is the absolute area UNIT (v_geo), not a free dimensionless gap. RESULT: the LINEARISED Einstein equation G_ab=(1/c3)T_ab closes PARAMETER-FREE. [O] RESIDUAL: (i) linearised -> full non-linear covariant Einstein (the original B6/GATE.METRIC, all-balls/all-frames + non-linear) and (ii) the absolute scale v_geo (Planck-area unit); NO free dimensionless parameter remains in the gravity coupling.",
      "status": "Entanglement-equilibrium gravity [E]/[C]/[O] (parameter-free linearised Einstein 8pi=1/c3; thermo=geo coincidence via |mu4|=|Z2|chi; J3 assembled; entropy density atom-fixed; residual = nonlinear + v_geo)",
      "canonicalStatus": "[E] the parameter-free Einstein coefficient + the thermo=geo coincidence (the new exact result) + the entropy-density atom-fixing; [C] J3 (cited CHM/BW ball modular Hamiltonian applied to TFPT's matter) + the area-law form; [O] linear->non-linear (B6 full) + the v_geo unit. Closes the LINEARISED Einstein equation parameter-free; advances B6/GATE.METRIC substantially; does NOT claim the full non-linear closure.",
      "dependencies": "SEAM.EQUIV.CONTINUUM.03; v356; v57; v58; v59; v216; v221; v323; v76",
      "scripts": [
        "v358_grav_entropy_equilibrium.py"
      ],
      "script": "v358_grav_entropy_equilibrium.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 76
    },
    {
      "id": "SEAM.EQUIV.CONTINUUM.03",
      "claim": "DIRECTION A (2026-06-23): the continuum chain of SEAM.EQUIV.01 assembled theorem-by-theorem, with the residual reduced to ONE lattice-realization input tied to the seam's one-sidedness (the c3 origin). After the reverse-audit/bandwidth work showed no hidden DISCRETE lever remains, the only fundamental progress is to assemble the CONTINUUM chain -- which this does. THE CHAIN: S1 the gapped (Delta=6 ln(3/2)>0, v302) quasi-free (CAR, v155/v160) 16-Majorana collar -> S2 invertible (a gapped free-fermion 2+1D phase has no intrinsic topological order, Kitaev, v301) -> S3 the NONTRIVIAL invertible phase (c_-=8!=0), not the trivial c_-=0 one -> S4 a NON-gappable anomalous chiral edge (anomaly inflow / bulk-edge: a nontrivial invertible phase cannot have a gapped edge), so edge EXISTENCE is a CONSEQUENCE of S2+S3, NOT a separate assumption -- this directly reduces the v336/v351 'does the gapped collar HAVE a chiral edge?' residual -> S5 the edge is 16 chiral Majoranas whose chiral-CFT scaling limit is given by MMST (Morinelli-Morsella-Stottmeister-Tanimoto, v336; range rank<=c<=D = 8<=8<=16, IN range) -> S6 the order-4 mu4 clock selects the index-4 net (E8)_1 (v351) and the AQFT stack invertible->trivial center->holomorphic->(E8)_1 is cited (v297, LIT-A/B/C). [C] the MMST range; [C] edge-existence-from-invertibility (anomaly inflow); [C] chirality = one-sidedness = the c3 ORIGIN: c_-!=0 is the seam being one-sided, the SAME one-sidedness that makes c3=1/(8pi) a one-sided Gauss-Bonnet (|Z2| oint K, oint K=4pi on S^2; v58/v73); [E] the chain closes modulo S3 (the only non-theorem link); [O] the ONE remaining input -- the abstract collar realized as a genuine LATTICE chiral free-fermion invertible phase (the v297 Flat-Away). NET: SEAM.EQUIV.01 reduced to one realization input tied to the constant that DEFINES the theory; ADVANCES, does not close. SEPARATELY, Direction B (the U_wall Hitchin selector) was found ALREADY closed by v40: at the polystable point Mehta-Seshadri makes the bundle unitary, so the harmonic metric is the constant invariant Hermitian form (finite linear algebra) with Higgs Phi=0 -- NOT a transcendental PDE -- and det R=8 is read off; only the absolute amplitude U_point (an anchor) remains. No new B module was shipped (a redundant v357 was built, found to duplicate/contradict v40, and deleted).",
      "status": "Continuum chain assembly [C]/[E]/[O] (edge-existence reduced via anomaly inflow; residual = lattice realization tied to seam one-sidedness; B already closed by v40)",
      "canonicalStatus": "[C] the MMST range + edge-existence-from-invertibility + chirality=one-sidedness=c3-origin; [E] chain-closes-modulo-S3; [O] the one lattice-realization input. A genuine assembly reducing SEAM.EQUIV.01 to one realization input tied to the seam one-sidedness; does NOT close it. Direction B found already closed by v40 (harmonic metric = finite linear algebra, not a PDE); no redundant B module shipped.",
      "dependencies": "SEAM.EQUIV.01; v301; v302; v336; v351; v297; v58; v40",
      "scripts": [
        "v356_continuum_mmst_applicability.py"
      ],
      "script": "v356_continuum_mmst_applicability.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 77
    },
    {
      "id": "QGAMB.SADDLE.01",
      "claim": "Direction 3 (2026-06-23): the QG measure (gate C7/QG.AMB.01) as one-loop GAUSSIAN fluctuations around the PARAMETER-FREE saddle (v358/v359). Before v358 fluctuations around the gravitational saddle were not even dimensionlessly well-posed (the Newton coupling was a free dial); with the parameter-free saddle the path integral organises as a Gaussian one-loop determinant whose convergence and dimensionless content are atom-fixed. [E] PARAMETER-FREE QUADRATIC FORM: the fluctuation inverse-propagator stiffness is the saddle coefficient 1/c3=8pi (no free dial). [E] GAPPED FLUCTUATION OPERATOR: 2||V||=2*248*c3^2=31/(4pi^2) (31=2^g_car-1) and Delta=6 ln(3/2), so M>=Delta_eff=Delta-2||V||~1.648>0 (v76/v337) => mode-by-mode Gaussian convergence, no zero/negative modes besides the saddle directions. [E] FINITE ONE-LOOP LOG-DET (model): on the OS spectrum {0,6 ln(3/2),6 ln 3} the regularised Tr' log M=6 ln(9/2) and W=3 ln(9/2) are finite (gapped, no IR divergence). [C] CONFORMAL DIRECTION CONVERGENT: the wrong-sign conformal mode (c_conf(4)=-3/2) is made convergent by the GHP rotation int exp(-|c|phi^2)=sqrt(2pi/3) + IDG (no Stelle ghost), v334. [E] ADMISSIBLE PROJECTIVE LIMIT EXISTS: chi=1/(1-(2/3)^6)=729/665 finite (v330) => the one-loop measure is tight on the admissible sector. [E] NO NEW DIAL: the dimensionless content is a function of {c3,Delta,chi,c=8} only (v364). [O] RESIDUAL: the full non-perturbative projective limit (G6/QG.AMB.01) on the ambient sector remains OPEN. Sharpens C7 from diffuse full QG to a one-loop Gaussian problem around a parameter-free saddle; does NOT close it.",
      "status": "QG measure reduced to a one-loop Gaussian problem around the parameter-free saddle [E]/[C]/[O] (residual = the non-perturbative projective limit)",
      "canonicalStatus": "[E] fixed quadratic form (1/c3=8pi) + gap-controlled convergence (Delta_eff~1.648>0) + finite model log-det (6 ln(9/2)) + tightness (chi=729/665) + 0 dials; [C] the GHP conformal contour + the one-loop Gaussian measure on the admissible sector; [O] the full non-perturbative projective limit. Sharpens C7; does NOT close it.",
      "dependencies": "QG.AMB.01; v358; v359; v76; v337; v330; v334; v364; v59",
      "scripts": [
        "v365_qg_oneloop_saddle.py"
      ],
      "script": "v365_qg_oneloop_saddle.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 78
    },
    {
      "id": "SEAM.MMST.INCLASS.01",
      "claim": "Direction 7 (2026-06-23): the seam collar is IN MMST's free-lattice-fermion scaling-limit class, so the chiral scaling limit is a chiral CFT of central charge 8 pinned to (E8)_1 -- reducing SEAM.EQUIV.01 to the single S3 lattice-realisation input. the cited theorem (Osborne-Stottmeister, arXiv:2107.13834, CMP 398 (2023) 219; the internal 'MMST' code-token/claim-id is LEGACY and denotes THIS Osborne-Stottmeister theorem, built on the MMST/Morinelli-Morsella-Stottmeister-Tanimoto OAR framework, arXiv:2010.11121): free lattice fermions (CAR quasi-free) give a chiral-CFT massless scaling limit with the correct central charge, in the range rank<=c<=D (D = number of Majorana copies). [E] CLASS ARITHMETIC / MMST RANGE: D=dim S^+=2^(g_car-1)=16 Majoranas, c=g_car+N_fam=8, rank E8=8 => rank<=c<=D reads 8<=8<=16, IN range (saturating rank=c=8, inside D=16). [E] GAPPED (derived, not assumed): Delta=6 ln(3/2)>0 (v302). [C] QUASI-FREE CAR CLASS: the collar is in the CAR quasi-free class (v155/v160), MMST's hypothesis class. [E] CHIRALITY: c_-=D/2=8=g_car+N_fam!=0 (a non-gappable anomalous edge by inflow, v356 S4). [C] MMST SCALING LIMIT APPLIES: in-class & gapped & chiral & in-range => the MMST theorem gives a chiral CFT of central charge 8 for the 16-Majorana edge (v336). [E] TARGET PINNED TO (E8)_1: c=8 + the order-4 mu4 clock select E8 (det K=1, one primary) over the same-c rival SO(16)_1 (det K=4) -- v351; holomorphy det K=1 is the consequence. [O] RESIDUAL S3: the one open input is the abstract collar realised as a genuine LATTICE chiral free-fermion invertible phase (the one-sidedness / Flat-Away, v297/v356 S3) = the seam one-sidedness that DEFINES c3; given S3 the MMST scaling-limit theorem applies as a cited result and the limit is (E8)_1. Verifies MMST in-class; ADVANCES SEAM.EQUIV.01, does NOT close it.",
      "status": "SEAM.EQUIV.01 reduced to the single S3 lattice-realisation input; MMST in-class verified for the collar [E]/[C]/[O]",
      "canonicalStatus": "[E] class arithmetic 8<=8<=16 + derived gap + chirality c_-=8 + (E8)_1 pinning; [C] the CAR-class typing + the MMST scaling-limit theorem (cited); [O] the S3 lattice realisation. Verifies MMST in-class; ADVANCES SEAM.EQUIV.01, does not close it.",
      "dependencies": "SEAM.EQUIV.01; v336; v356; v351; v302; v155; v160; v297; v301",
      "scripts": [
        "v366_mmst_seam_collar.py"
      ],
      "script": "v366_mmst_seam_collar.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 79
    },
    {
      "id": "GRAV.SPIN2.UNITARITY.01",
      "claim": "Track 2 (2026-06-23): the Barnes-Rivers spin decomposition of the graviton propagator -- perturbative graviton unitarity SECTOR-BY-SECTOR. Extends v304 (scalar pole algebra) to the tensor spin structure and combines it with v334 (spin-0 conformal mode); does not re-derive v304 and does not close QG.AMB.01. [E] PROJECTOR COMPLETENESS: tr P2=(d-2)(d+1)/2, tr P1=d-1, tr P0s=tr P0w=1 sum to d(d+1)/2 (the symmetric-tensor dimension); in d=4 the split is 5+3+1+1=10. [E] EH STRUCTURE: the Einstein-Hilbert graviton propagator ~ P2/p^2 - 1/(d-2) P0s/p^2, spin-2 coefficient +1, spin-0 scalar -1/(d-2)=-1/2 (d=4). [E] THE STELLE GHOST IS A SPIN-2 STATE: a local R+R^2/Weyl^2 dressing 1+p^2/M^2 gives the spin-2 propagator a negative-residue pole at p^2=-M^2. [E] ENTIRE FORM FACTOR => SPIN-2 GHOST-FREE: the un-truncated KMS form factor a(p^2)=e^{p^2/M^2} (v304/v259, entire, nowhere zero) keeps the spin-2 propagator's only pole at p^2=0. [C] SPIN-0 VIA GHP: the wrong-sign conformal/trace mode (c_conf(4)=-3/2, v332) is cured by the GHP contour rotation + IDG (v334) -- a different mechanism. [C] => the full PERTURBATIVE graviton is ghost-free sector-by-sector (spin-2 entire form factor + spin-0 GHP), conditional on v304's entire-analyticity assumption. [O] perturbative-only: inherits the unproven entire-a(Box) assumption, and a ghost-free perturbative graviton is NOT the non-perturbative ambient measure (QG.AMB.01). [O] DECLINE: tr P2=5 in d=4 is the spin-2 polarisation count (dimension counting), NOT g_car=5 (v354/v355).",
      "status": "perturbative graviton unitarity sector-by-sector (spin-2 ghost-free via the entire KMS form factor, spin-0 via GHP) [E]/[C]/[O]",
      "canonicalStatus": "[E] the Barnes-Rivers projector completeness + EH structure + the spin-2 pole algebra; [C] the entire form factor (inherits v304) + GHP (v334) + the sector-by-sector unitarity conclusion; [O] perturbative-only + the open analyticity assumption. Extends v304 to the tensor sector; does NOT close QG.AMB.01. tr P2=5 != g_car (declined).",
      "dependencies": "QG.AMB.01; v304; v334; v332; v259; v278; v354",
      "scripts": [
        "v370_grav_spin2_unitarity.py"
      ],
      "script": "v370_grav_spin2_unitarity.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 80
    },
    {
      "id": "SEAM.S3.LATTICE.01",
      "claim": "Track 1 (2026-06-23): the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1. Turns the abstract S3 residual of v356/v366 ('the collar realised as a genuine lattice chiral free-fermion invertible phase') into a RUNNABLE model with a numerically computed topological invariant, not a re-assertion of det K / c_-. [E] BULK GAPPED: h(k)=sin kx sx + sin ky sy + (M-cos kx-cos ky) sz at M=1 has band gap 2 min|d(k)|=2>0. [E] NON-TRIVIAL CHERN (numerical Fukui-Hatsugai-Suzuki plaquette): |C|=1 at M=1 (topological) and C=0 at M=3 (trivial neg control). [E] c_-=8: each chiral Majorana edge carries c_-=C/2=1/2; N_Maj=dim S^+=2^(g_car-1)=16 copies => c_-=16/2=8=g_car+N_fam (the edge gravitational anomaly). [E] EDGE K-MATRIX: the 16-Majorana edge is SO(16)_1 (det Cartan D8=4) condensed to (E8)_1 (det Cartan E8=1) by the order-4 mu4 clock (v351/v369). [C] the index-4 mu4 simple-current condensation SO(16)_1 -> (E8)_1 (v154/v351) is the seam glue on top. [O] residual: the continuum scaling-limit existence (the lattice model's massless limit IS the chiral CFT, MMST v336) stays open; S3 is now a concrete lattice model, reduced not closed.",
      "status": "S3 realised as an explicit gapped chiral-Majorana lattice model (numerical Chern |C|=1, c_-=8, det K 4->1) [E]/[C]/[O]",
      "canonicalStatus": "[E] the gapped Chern model + the c_-=8 counting + the det K discriminators (numerical Chern + lattice arithmetic); [C] the mu4 condensation to (E8)_1; [O] the continuum scaling limit. Reduces S3 to a runnable lattice model; does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.01; v356; v366; v351; v369; v301; v302; v154",
      "scripts": [
        "v367_seam_s3_lattice.py"
      ],
      "script": "v367_seam_s3_lattice.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 81
    },
    {
      "id": "SEAM.S3.INFLOW.01",
      "claim": "Track 1 (2026-06-23): the bulk-edge / anomaly-inflow leg (S4 of the v356 chain) made CONCRETE on the v367 lattice model -- a finite STRIP of the gapped p+ip model has a REALISED, non-gappable chiral edge, so edge EXISTENCE is a CONSEQUENCE of the bulk being a nontrivial invertible phase (anomaly inflow), NOT a separate assumption. This directly discharges the v336/v351 'does the gapped collar HAVE a chiral edge?' residual on the lattice. [E] TOPO IN-GAP EDGE STATES: for M=1 (|C|=1) the strip spectrum reaches min_kx|E|~0 deep in the bulk gap (a chiral edge branch traverses the gap). [E] TRIVIAL CLEAN GAP (neg control): for M=3 (C=0) min_kx|E|>0.3 stays well above zero. [E] EDGE-LOCALISED: the near-zero in-gap state's weight is concentrated within 4 sites of a boundary (edge>>bulk). [E] BULK-EDGE CORRESPONDENCE: the chiral edge branch count = bulk Chern |C|=1 (one chiral Majorana per edge) => the 16-copy carrier has a 16-Majorana chiral edge, c_-=8. [C] ANOMALY INFLOW: a gapped invertible phase with c_-!=0 cannot have a gappable edge, so the chiral edge of the collar (c_-=8, v301/v356) EXISTS as a consequence of the bulk. [O] residual: the continuum scaling-limit of that lattice edge (MMST v336) stays open; v368 establishes edge EXISTENCE on the lattice, not the scaling limit.",
      "status": "edge existence forced by the invertible bulk (anomaly inflow) -- realised non-gappable chiral edge on the lattice strip [E]/[C]/[O]",
      "canonicalStatus": "[E] the strip edge spectrum + localisation + bulk-edge correspondence + the trivial neg control; [C] the anomaly-inflow edge-existence argument; [O] the continuum scaling limit. Discharges the S4 'edge exists' leg on the lattice; does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.01; v367; v356; v336; v351; v301",
      "scripts": [
        "v368_seam_s3_inflow.py"
      ],
      "script": "v368_seam_s3_inflow.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 82
    },
    {
      "id": "FR.POLE.SOLVE.01",
      "claim": "Track 3 (2026-06-23, F_transfer): the Koide source->pole transfer as a TYPED solver -- promotes experiments/ftransfer/koide_source_to_pole into the ledger with a kill test. Does NOT re-prove the 53/54 operator factor (v183, [E]) or Q*=2/3 (v82/v101); answers the one open question: is the physical source->pole transfer standard QED/EW running? A clean NEGATIVE. [E] Q_pole=Koide(PDG poles)=2/3 to 3e-6; Q_src(phi0-ladder, v93)=0.6644638 (0.33% below 2/3). [E] dQ/deps>0 at the pole: more QED running widens the hierarchy and pushes Q ABOVE 2/3 (Q(eps_QED)>2/3). [E] the eps mapping Q_pole->Q_src is NEGATIVE (brentq) while QED gives eps=+(3/2)(alpha/pi)>0 for any alpha in [1/137,1/128] -- the WRONG SIGN. [C] verdict: F_pole is NOT perturbative running; the transfer is the non-QED operator/Moebius generator whose signature is 53/54 (v183, [E]) and the (2/3)^6 Moebius flow (v82). Stays [C], sharper -- plain running is the kill (excluded).",
      "status": "Koide source->pole: standard QED running EXCLUDED (wrong sign) -- the transfer is the non-QED 53/54 operator/Moebius generator [E]/[C]",
      "canonicalStatus": "[E] the Koide coincidence + the QED-wrong-way sign + the negative required eps (PDG poles + exact ladder ratios, no fit); [C] the verdict that the true transfer is the non-QED generator. A typed solver with a kill test; does NOT promote Q_pole to [E].",
      "dependencies": "FR.POLE.01; v183; v82; v93; v101",
      "scripts": [
        "v371_ftransfer_pole.py"
      ],
      "script": "v371_ftransfer_pole.py",
      "location": "tfpt_4_frontier",
      "externalData": "PDG 2024 lepton pole masses",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 83
    },
    {
      "id": "FR.BOLTZMANN.SOLVE.01",
      "claim": "Track 3 (2026-06-23, F_transfer): eta_B from the INTEGRATED Buchmueller-Di Bari-Pluemacher Boltzmann network at the TFPT-frozen heavy scale -- promotes experiments/ftransfer/leptogenesis_boltzmann, replacing the v326 toy washout (kappa=0.3 tuned). Does NOT re-derive the viability band (v169) or the decuple anchoring (v212); integrates the ODE at the frozen inputs and checks eta_B. Frozen (NO free M_R dial): M1=M_scal phi0^2/A_Lambda~8.65e9 GeV (v212), m~1=m3/A_Lambda=5 meV, eps1 Davidson-Ibarra with the predicted delta_CP=4pi/3 (|sin|=0.866, v9). [E] the full-ODE efficiency kappa_f~0.09 (strong washout K~4.6) agrees with the analytic BDP fit to ~25% -- integrated transport, not a tuned constant. [C] eta_B(ODE)~6.5e-10 at the frozen M1 lands within a factor ~1.1 of the observed 6.1e-10; the M1 reproducing eta_B_obs is within a factor ~1.1 of the frozen M1. [X] kill: a proper solve putting eta_B_obs outside [obs/3, 3obs] at the frozen (M1, m~1) for all seesaw-consistent M_R drops the leptogenesis branch (compiler core + Omega_b readout FR.ETAB.01 untouched).",
      "status": "eta_B from the integrated BDP Boltzmann ODE at the frozen M1 -- near-hit (factor ~1.1) with no M_R dial [C]/[X]",
      "canonicalStatus": "[E] the integrated efficiency; [C] the frozen-scale eta_B near-hit (washout anchored m~1=m3/A_Lambda, M1 via the scalaron route v212); [X] the branch kill test. A typed solver, not a toy; eta_B stays [C] (a transfer target, not a compiler power).",
      "dependencies": "FR.ETAB.01; v169; v212; v9; v7",
      "scripts": [
        "v372_ftransfer_boltzmann.py"
      ],
      "script": "v372_ftransfer_boltzmann.py",
      "location": "tfpt_4_frontier",
      "externalData": "Planck eta_B=6.1e-10",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 84
    },
    {
      "id": "FR.RELIC.SOLVE.01",
      "claim": "Track 3 (2026-06-23, F_transfer): the axion relic abundance from the FINITE-T misalignment ODE -- promotes experiments/ftransfer/axion_relic and DECIDES the two competing TFPT angle branches, replacing the v326 adiabatic toy. Does NOT re-fix the candidate scales (f_a=M_scal/128, m_a; v185/v211); integrates the nonlinear ODE theta'' + (3+dlnH/dN)theta' + (m_a(T)/H)^2 sin theta = 0 with the lattice topological susceptibility m_a(T)=m_a min(1,(T_QCD/T)^{n/2}) (n=8.16) and realistic g_*(T). [I] candidate scales f_a=M_scal/(2 dim S^+ |mu4|)=M_scal/128~2.39e11 GeV, m_a~23.8 ueV, N_DW=1 (determinant-line atoms). [C] the frozen SPINE angle theta_i=3pi/5=108deg=pi N_fam/g_car gives Omega_a h^2~0.12 in [0.08,0.16] (brackets Omega_DM=0.12) with NO tuning -- the robust dominant-DM branch. [X] the HILLTOP theta_i~170deg over-produces (Omega_a h^2~0.65, ~5.4x) -- disfavoured/fine-tuned. [X] kill: a finite-T solve robustly outside [0.08,0.16] for the spine across lattice chi(T)/g_* variations drops the dominant axion-DM branch.",
      "status": "axion relic finite-T solve DECIDES the branch -- spine 3pi/5 lands on Omega_DM (in band), hilltop 170deg over-produces [C]/[X]",
      "canonicalStatus": "[I] the candidate scales; [C] the spine in-band relic (finite-T misalignment + lattice chi(T)); [X] the hilltop over-production + the spine kill test. A typed solver that decides the branch; the relic stays [C] (lattice chi(T) + cosmology inputs).",
      "dependencies": "v185; v211; v25",
      "scripts": [
        "v373_ftransfer_relic.py"
      ],
      "script": "v373_ftransfer_relic.py",
      "location": "tfpt_4_frontier",
      "externalData": "Planck Omega_DM h2=0.12",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 85
    },
    {
      "id": "FR.QCD.BUDGET.01",
      "claim": "Track 3 (2026-06-23, F_transfer): the m_p/m_e transfer as a TYPED uncertainty-budget + kill test -- the falsifiability companion to v262 (the precise 2-loop point), replacing the v326 toy (which hardcodes 1836.15). Does NOT re-derive Lambda^(3) or the b3=-7 beta (v164/v262, [E]); propagates the two declared externals through the transfer m_p/m_e=C_p Lambda^(3)/m_e. [E] b3=-(11-2 n_f/3)=-7 (n_f=6 carrier). [N] two declared externals (v339): alpha_s(M_Z)=0.1179+-0.0009 (PDG) and C_p=2.83+-0.15 (lattice), entering linearly-in-log so d ln Lambda=(2pi/(b0^(5) alpha_s^2)) d alpha_s~5.3% and d C_p/C_p~5.3%. [C] m_p/m_e central ~1824 (v262 2-loop) with a propagated 1-sigma band +-sqrt(5.3^2+5.3^2)%~+-7.5% -> [~1690,~1960], CONTAINING the observed 1836.15 -- no free dial. [X] kill: a future lattice C_p+-0.03 + tighter alpha_s shrink the band to ~+-2%; a band then EXCLUDING 1836.15 drops the QCD-matching transfer route (b3=-7 carrier + the SU(9)-integer firewall v262 untouched).",
      "status": "m_p/m_e uncertainty budget: the observed 1836.15 is inside the propagated +-7.5% (alpha_s, C_p) band; lattice-tightening is the kill [C]/[N]/[X]",
      "canonicalStatus": "[E] b3=-7; [N] the two externals; [C] the observed ratio inside the propagated band (no dial); [X] the lattice-tightening kill test. An uncertainty-budget/falsifiability solver, distinct from v262's point estimate; m_p/m_e stays [C].",
      "dependencies": "FR.MPME.02; v262; v164; v339",
      "scripts": [
        "v374_ftransfer_qcd.py"
      ],
      "script": "v374_ftransfer_qcd.py",
      "location": "tfpt_4_frontier",
      "externalData": "PDG alpha_s(M_Z), lattice C_p, CODATA m_p/m_e",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 86
    },
    {
      "id": "OBSERVATORY.REGISTRY.01",
      "claim": "Track 4 (2026-06-23): the prediction OBSERVATORY -- a status-typed CI over the frozen prediction registry (verification/freeze_file.csv) that makes the falsifiability surface machine-checkable. Does NOT re-derive the physics; verifies (a) the registry is COMPLETE (every frozen signal has a kill criterion), (b) the headline numerical predictions re-derive from the atoms {phi0}, (c) the live scorecard (each headline prediction's sigma-distance to the current measurement) is internally consistent and correctly typed. [E] FALSIFIABILITY COMPLETE: all registry rows carry a non-empty kill_criterion (no orphan claims). [E] HEADLINE VALUES FROM ATOMS: sin^2 theta12=1/3-phi0/2, sin^2 theta13=phi0 e^{-5/6}, beta_rad=phi0/(4pi). [N] LIVE SCORECARD: theta12 vs JUNO first 59.1-day run (0.3092+-0.0087)=0.28 sigma (compatible); theta13 vs NuFIT 6.0 NO (0.02195+-0.00058)=2.0 sigma (the flagged most-tensioned core prediction); beta_rad vs ACT DR6 (0.215+-0.074)=0.37 sigma (a hint, systematics pending); r=12/N*^2~0.004<BK18 0.036 (compatible). [E] REGISTRY INTEGRITY: the single machine-readable falsifiability surface, theta13 flagged in its own row.",
      "status": "a status-typed CI over freeze_file.csv -- falsifiability complete + the live scorecard (theta13 flagged at 2.0 sigma) [E]/[N]",
      "canonicalStatus": "[E] the completeness + atom re-derivation + integrity; [N] the live data scorecard (the sigma-distances are data, updated as experiments report). Tooling that verifies the prediction surface; introduces no new physics claim.",
      "dependencies": "v3; v9; v268; v328; v86; v60",
      "scripts": [
        "v375_observatory_registry.py"
      ],
      "script": "v375_observatory_registry.py",
      "location": "tfpt_5_redteam",
      "externalData": "JUNO first run; NuFIT 6.0; ACT DR6; BK18",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 87
    },
    {
      "id": "SEAM.S3.CENTRALCHARGE.01",
      "claim": "S3 closure stack (2026-06-23): the central charge c=8 extracted NUMERICALLY from the lattice via the Calabrese-Cardy finite-size entanglement-entropy scaling of the critical free-fermion content -- the standard, rigorous lattice->CFT central-charge diagnostic (the quantitative core of the MMST scaling-limit claim), on OUR model, not a re-assertion. [E] the Peschel correlation-matrix EE of one critical tight-binding chain scales as (c/3)ln[(L/pi)sin(pi l/L)] with fitted c=1.0000 (L=258,386; the L=4m+2 sizes avoid the half-filling zero-mode). [E] the collar is N_Maj=dim S^+=2^(g_car-1)=16 Majoranas = 8 complex modes (v367), c additive over decoupled modes => c=8=g_car+N_fam. [E] chiral c_-=8 (bulk Chern |C|=1 per Majorana v367; non-gappable chiral edge v368), so the scaling limit is a CHIRAL c=8 CFT. [O] the specific net (E8)_1 is v377/v378; the abstract continuum EXISTENCE is the cited MMST theorem (v336).",
      "status": "central charge c=8 from finite-size entanglement scaling of the lattice (chiral, c_-=8) [E]",
      "canonicalStatus": "[E] the numerical c=1 per mode (EE scaling) + additive c=8 + the chirality cross-link; [O] the specific-net identification (v377/v378) + the continuum existence (v336). Numerical evidence that the scaling limit has c=8; does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.01; v367; v368; v336",
      "scripts": [
        "v376_seam_s3_centralcharge.py"
      ],
      "script": "v376_seam_s3_centralcharge.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 88
    },
    {
      "id": "SEAM.S3.E8CHARACTER.01",
      "claim": "S3 closure stack (2026-06-23): the chiral c=8 content of the seam is the SPECIFIC net (E8)_1, not just 'some c=8' -- shown by the exact affine character q-series and the mu4/GSO promotion SO(16)_1 -> (E8)_1 (the genus-0 character discriminator that pins the target of the scaling limit). [E] chi = E4/eta^8 = q^{-1/3}(1+248q+4124q^2+...): eta^8=q^{1/3}prod(1-q^n)^8 gives the prefactor q^{-c/24}=q^{-1/3} => c=8; the level-1 coefficient 248=dim E8 (one primary = the vacuum module), computed exactly from E4*prod(1-q^n)^{-8}. [E] the promotion: 248=240+8=(E8 roots)+(Cartan rank), 240=dim S^+*g_car*N_fam=16*5*3 (carrier trace); equivalently 248=120+128=dim SO(16)+spinor(128) -- the order-4 mu4 clock condenses the SO(16)_1 vacuum+spinor into the single (E8)_1 module. [E] discriminator: (E8)_1 has 1 primary and 248 level-1 currents vs SO(16)_1 4 primaries and 120. [O] the continuum existence (v336); the mu4 projection = the seam one-sidedness (S3).",
      "status": "the chiral c=8 content is (E8)_1 (248 currents, 1 primary) via the exact character + mu4/GSO promotion [E]",
      "canonicalStatus": "[E] the exact character q-series (c=8, 248, 4124) + the integer promotion identities (248=240+8=120+128, 240=16*5*3) + the 1-vs-4 primary discriminator; [O] the continuum existence (v336) + the mu4 projection = S3. Pins the target net (E8)_1; does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.01; v336; v351; v83; v1",
      "scripts": [
        "v377_seam_s3_e8character.py"
      ],
      "script": "v377_seam_s3_e8character.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 89
    },
    {
      "id": "SEAM.S3.MODULAR.01",
      "claim": "S3 closure stack (2026-06-23): the GENUS-1 discriminator -- the torus ground-state degeneracy of the seam net is 1 (holomorphic = (E8)_1), via the KLM index condensation and the modular invariance of the single (E8)_1 character. This is exactly the 'hard part' v344 flagged: the genus-0 arguments (a unique plane vacuum) are necessary but NOT sufficient (a plane vacuum is always 1, v87); det K=1 is the torus (genus-1) statement. [E] KLM mu-additivity: the carrier A=(D5)_1 x (A3)_1 has mu(A)=det Cartan(D5)*det Cartan(A3)=4*4=16; the order-4 mu4 glue (|L|=|mu4|=4) gives mu(E8)=16/4^2=1 (torus GSD=1); the index-2 glue gives the rival mu(SO16)=16/2^2=4. [E] |det Cartan(E8)|=1 (one primary), |det Cartan(D8)|=4 (four primaries). [E] the single (E8)_1 character has modular T-eigenvalue e^{2 pi i(h-c/24)}=e^{-2 pi i/3} (h=0,c=8), consistent with v377's q^{-1/3}; a holomorphic theory has a 1-dim modular S,T rep => exactly one character => torus GSD=1. [O] the abstract continuum EXISTENCE (v336).",
      "status": "torus GSD=1 / modular uniqueness -- the genus-1 holomorphy discriminator closes the v344 hard part [E]",
      "canonicalStatus": "[E] the KLM condensation 16->1, the det-Cartan #primaries (1 vs 4), the modular T-eigenvalue / 1-dim rep, and the genus-0-vs-genus-1 distinction; [O] the continuum existence (v336). Pins the net at genus-1; does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.01; v344; v154; v369; v83; v87",
      "scripts": [
        "v378_seam_s3_modular.py"
      ],
      "script": "v378_seam_s3_modular.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 90
    },
    {
      "id": "SEAM.S3.RP.01",
      "claim": "S3 closure stack / C7 (2026-06-23): reflection positivity (the Osterwalder-Schrader axiom) of the EXPLICIT gapped collar measure, verified numerically -- the remaining constructive-QFT ingredient for the OS reconstruction of the boundary net, after which C7/QG.AMB.01 is discharged by the redundancy theorem v369 (its first premise, SEAM.EQUIV.01, being the S3 stack), not built from scratch. [E] Kaellen-Lehmann/OS positivity: the gapped p+ip band gives G(tau)=mean_k e^{-eps_k tau} with eps_k>=gap/2~1>0 (v367) -- a positive spectral measure. [E] REFLECTION POSITIVITY BY AN EXACT POSITIVE-MIXTURE GRAM IDENTITY (2026-07-28 hardening from T133/CERT.FLOOR): the Hankel/reflection Gram matrix M_ij=G(tau_i+tau_j) is built explicitly as M = V^T diag(w) V with V_ki = e^{-eps_k tau_i} and w_k = 1/K identical and STRICTLY POSITIVE over the K=1600 band modes, so x^T M x = sum_k w_k (v_k.x)^2 is a SUM OF SQUARES and M >= 0 holds EXACTLY as an algebraic identity (symbolic PSD, no tolerance); the mixture identity M - V^T D V is zero at the rounding level (1.1e-15 <= 64 u |M| = 7.6e-15) and the sum-of-squares route is nonnegative on 510/510 test directions. THE EIGENVALUE CHECK IS RETAINED AS A DIAGNOSTIC MEASUREMENT, not the certificate: T133 certified that the ASSEMBLED double-precision matrix has a negative direction at -7.75e-18 (60 digits) -- inside the module's old min_eig > -1e-10 tolerance, and the assembled float quadratic form is indeed negative on 2 of the 510 directions -- while the MATHEMATICAL matrix is fine; the Gram identity is the exact rescue. OS reflection positivity holds for the gapped collar. [E] neg control: injecting a growing (ghost) mode e^{+s tau} with negative weight makes the Hankel matrix indefinite (a negative eigenvalue) -- the test is non-vacuous. [C] RP + the gap (v76/v337) + GNS (v240) + c=8 (v376) + (E8)_1 (v377/v378) give the OS reconstruction inputs; C7/QG.AMB.01 is then DISCHARGED by v369. [O] the full non-perturbative ambient measure as a constructive object is the cited constructive-QFT step.",
      "status": "reflection positivity (OS axiom) of the explicit collar verified by an EXACT positive-mixture Gram identity (symbolic PSD: M = V^T diag(1/K) V, sum of squares; eigenvalue check retained as diagnostic, T133 hardening); C7 then discharged via v369 [E]/[C]",
      "canonicalStatus": "[E] the positive spectral rep + the EXACT positive-mixture Gram identity for RP (symbolic PSD; the min-eigenvalue check retained as a DIAGNOSTIC measurement of the assembled float matrix, T133) + the non-vacuous neg control; [C] the OS reconstruction + the C7 discharge via v369; [O] the full constructive measure. Verifies the RP axiom for the explicit collar; does NOT by itself construct C7. Marker unchanged [E] (the Gram rescue carries; no upgrade).",
      "dependencies": "QG.AMB.01; v367; v369; v240; v76; v376; v377; v378; T133",
      "scripts": [
        "v379_seam_s3_rp.py"
      ],
      "script": "v379_seam_s3_rp.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 91
    },
    {
      "id": "FORM.SEAM.MMST.01",
      "claim": "LEAN 4 FORMALISATION of the S3 / SEAM.EQUIV.01 continuum leg (the MMST scaling-limit theorem applied to the seam collar), 2026-06-23 -- new module TfptCarrier/SeamScalingLimit.lean, lake build OK (target TfptCarrier.SeamScalingLimit, 230ms), #print axioms CLEAN. It formalises the status 'CLOSED MODULO A CITED THEOREM': (1) [F] the MMST applicability HYPOTHESES for the collar are PROVABLE arithmetic facts by kernel decide (NO axioms) -- D=16 Majoranas, rank=c=8, the range rank<=c<=D i.e. 8<=8<=16 (mmst_range), chiral c_-=D/2=8!=0 (chiral), and the holomorphy discriminator det K=1 vs the same-c rival SO(16) det K=4 (holomorphic_discriminator); (2) [C] the seam scaling-limit theorem (Osborne-Stottmeister, arXiv:2107.13834, CMP 398 (2023) 219; legacy internal token 'MMST', built on the MMST/Morinelli-Morsella-Stottmeister-Tanimoto OAR framework arXiv:2010.11121) and the Adamo-Moriwaki-Tanimoto OS/holomorphy selector (2024) are declared as NAMED cited axioms (mmst_scaling_limit, adamo_holomorphic_e8); (3) the conclusion SeamIsE8 = (E8)_1 follows by the well-typed composition seamScalingLimit, and #print axioms pins the residual to EXACTLY those two cited theorems + the one carrier-forced gap (collar_gapped) -- no hidden assumption, no sorry. Mirrors the S3 closure stack v376/v377/v378/v379 (which pin the TARGET c=8 / (E8)_1 character / torus GSD=1 / RP numerically). So S3 is closed modulo the two published theorems; every TFPT-internal hypothesis is machine-checked.",
      "status": "Formal [F] (Lean kernel; closed modulo the cited MMST/Adamo theorems)",
      "canonicalStatus": "Formal [F]: the S3 continuum leg machine-formalised as 'closed modulo cited theorems' -- the collar's MMST hypotheses kernel-proved (decide, no axioms), the MMST/Adamo theorems named axioms, the composition's #print axioms clean. The honest closure ceiling for SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.01; v336; v366; v376; v377; v378; v379; FORM.GLUE.01",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/SeamScalingLimit.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/SeamScalingLimit.lean",
      "location": "tfpt_research_contracts",
      "externalData": "MMST (CMP 2022); Adamo-Moriwaki-Tanimoto (2024)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 92
    },
    {
      "id": "GRAV.KMS.HESSIAN.01",
      "claim": "The KMS Entire Hessian (2026-06-23): the Stelle ghost is EXACTLY the Seeley-DeWitt truncation, and resummation pushes it to infinity -- upgrading v304/v370 from the ASSUMPTION 'the resummed graviton form factor is entire' to a DERIVED truncation-artefact + decoupling statement. [E] SEAM CUTOFF -> ENTIRE FORM FACTOR: f(u)=e^{-u} (the beta=1 seam KMS weight, v259) gives the dressed spin-2 propagator e^{-p^2/M^2}/p^2 = 1/(p^2 a) with kinetic form factor a(u)=e^u (u=p^2/M^2) entire and nowhere zero (Weierstrass), so the ONLY pole is p^2=0 (the healthy graviton), no ghost. [E] EACH FINITE TRUNCATION CARRIES THE GHOST: the Seeley-DeWitt truncations are the Taylor partial sums T_n of e^u; the local R+R^2 order T_1=1+u has the Stelle ghost zero u=-1 (p^2=-M^2), the Weyl^2 order T_2=1+u+u^2/2 the complex pair -1+-i; every finite truncation has a zero, the entire a has none. [E] THE GHOST DECOUPLES: the modulus of the nearest zero of T_n grows MONOTONICALLY with n (1, sqrt2, 1.60, 1.95, 2.18, 2.51, 2.76, 3.07 for n=1..8) -> infinity, so keeping more heat-kernel orders pushes the spurious pole to infinity -- 'entire' = 'do not truncate the heat kernel'. [C] => perturbative graviton unitarity (the KMS Entire Hessian theorem), upgrading the v304/v370 [C] assumption. [O] the identification of a(Box)=e^{Box/M^2} with the EXACT off-shell curved-space graviton Hessian of Tr e^{-D^2/Lambda^2} is the cited heat-kernel resummation (Tomboulis 1997 / Biswas-Mazumdar-Siegel); perturbative-only, not the non-perturbative ambient measure (QG.AMB.01).",
      "status": "the Stelle ghost = the Seeley-DeWitt truncation; resummation decouples it (root modulus -> infinity) => entire spin-2 form factor [E]/[C]",
      "canonicalStatus": "[E] the truncation-artefact mechanism (each finite Taylor truncation of e^u has a zero, e^u has none) + the decoupling (nearest truncation-zero modulus grows monotonically with order); [C] the identification with the full off-shell curved-space Hessian (cited heat-kernel resummation); [O] perturbative-only. Closes the v304/v370 analyticity gap at the truncation-tower level; does NOT change QG.AMB.01.",
      "dependencies": "QG.AMB.01; v259; v304; v370; v334",
      "scripts": [
        "v380_grav_kms_hessian.py"
      ],
      "script": "v380_grav_kms_hessian.py",
      "location": "tfpt_5_redteam",
      "externalData": "Tomboulis 1997; Biswas-Mazumdar-Siegel 2006",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 93
    },
    {
      "id": "QGAMB.REDUNDANCY.01",
      "claim": "Track 2 (2026-06-23, next-plan v2): the AMBIENT REDUNDANCY statement -- the holographic route to Gravity-complete that SIDESTEPS building the general Euclidean QG measure (C7/QG.AMB.01). Instead of constructing the non-perturbative ambient measure (the conformal-factor swamp, v332/v334), assemble the established discriminators that prove the ambient sector carries NO physical d.o.f. relative to the holomorphic boundary algebra, so QG.AMB.01 is a non-fundamental CERTIFICATION object (like a coordinate choice in GR), not missing dynamics. [E] HOLOMORPHIC => DHR=Vec: #DHR primaries = |det Cartan| = 1 for E8 (ONE primary = vacuum, trivial braided category) vs 4 for SO(16)/D8 -- no nontrivial superselection charges => no hidden boundary sector a bulk could carry invisibly. [E] NO TORUS GSD: ground-state degeneracy on T^2 = #anyons = |det K| = 1 for E8 => a unique ground state => no topologically protected invisible bulk d.o.f. (the genus-1 obstruction of v344, redundancy reading). [E] FINITE PETZ RECOVERY: the seam transfer deviation subspace contracts at rate (2/3)^6=64/729<1 (v221) => bulk perturbations are boundary-recoverable at a finite gapped rate => Rec_Sigma bounded. [E] GAP-DECOUPLING MARGIN: Delta_eff=6 ln(3/2)-31/(4pi^2)~1.648>0 (v76/v337) => the un-built ambient sector is energetically separated from the admissible (boundary-reconstructible) sector. [C] RECONSTRUCTION MAP: Bisognano-Wichmann / Tomita-Takesaki gives the modular flow as the geometric boost (v258/v309/v323); with holomorphy this supplies Rec_Sigma: (bulk algebra)->A_Sigma^{mu4}. [C] THE REDUNDANCY STATEMENT: under SEAM.EQUIV.01 + BW, every gauge-invariant bulk observable is boundary-reconstructible (O_phys^bulk/Diff ~= A_Sigma^{mu4}) and ambient measures sharing the boundary state are physically equivalent => QG.AMB.01 carries no extra physical d.o.f. [O] RESIDUAL: the theorem consumes exactly TWO premises -- SEAM.EQUIV.01 (the boundary IS (E8)_1; the S3 lattice input, v366) and the BW intrinsicality of the raw collar (v329 NEG); it does NOT construct QG.AMB.01 but provides the ALTERNATIVE (holographic redundancy) route. Gravity-complete = Boundary-complete + Ambient-redundancy.",
      "status": "Ambient measure proved boundary-REDUNDANT (the holographic route); QG.AMB.01 reframed as a non-fundamental certification object [C]/[O] (conditional on SEAM.EQUIV.01 + BW)",
      "canonicalStatus": "[E] det K=1 => DHR=Vec (no superselection) + no torus GSD + finite Petz recovery (2/3)^6 + gap margin Delta_eff~1.648>0; [C] the BW reconstruction map + the redundancy statement O_phys^bulk/Diff ~= A_Sigma^{mu4}; [O] the two premises (SEAM.EQUIV.01 + BW intrinsicality). The holographic route to Gravity-complete; does NOT construct the ambient measure.",
      "dependencies": "QG.AMB.01; SEAM.EQUIV.01; v366; v329; v221; v76; v337; v258; v309; v323; v344; v83",
      "scripts": [
        "v369_qgamb_redundancy.py"
      ],
      "script": "v369_qgamb_redundancy.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 94
    },
    {
      "id": "E8.UNMAPPED.BANDWIDTH.01",
      "claim": "A DISCIPLINED BANDWIDTH SEARCH OF THE UNMAPPED E8 REGION (2026-06-22, the next step after v354). The natural follow-up -- 'search that region for hits and structures we overlooked' -- is itself the numerology temptation v354 warns about: a rich object like E8 yields a numerical match for almost anything. So this scan uses a STRICT discriminator: a candidate is real ONLY if it is a FORCED identity (a theorem about E8), never a coincidence. It also RECONCILES v66 (the 8 degrees coincide with TFPT's load-bearing integers) with v354 (only 2,8,30 are primary readouts). [E] FORCED SET-LEVEL (the genuinely overlooked structure, collective not per-number): sum(degrees)=128=dim S^+ the SPINOR half of 248=120+128 (theorem: sum of degrees=#pos roots+rank=120+8; v66 saw 2^7, the spinor reading is deeper -- the invariant budget equals the fermion content the carrier reads); sum(exponents)=120=#pos roots; product(degrees)=|W(E8)|=696729600; exponents={1,7,11,13,17,19,23,29}=the phi(30)=8 totatives of 30 (v223), so the unmapped degrees' exponents {11,13,17,19,23} are part of the forced totative set; max(degree)=30=h(E8). [C] SUGGESTIVE, NOT A THEOREM (the discriminator at work): 12=h(E6),18=h(E7),30=h(E8) are all degrees of E8 (the 2T/2O/2I McKay ladder, v219), but 'deg(E8) contains its subalgebras' Coxeter numbers' is not a general theorem, so the pattern is flagged, not claimed. [O] DECLINED (the numerology temptation): the per-degree atom matches 14=dim G2, 20=det L, 24=|W(A3)|=4!, 12=|R(A3)|, 18=p4(a)=2+2^4 (v66, which already flagged 12,18 as fittable) each admit >=2 readings from the frozen atoms with no forced selector, and the extra prime 7 in |W(E8)|=2^14 3^5 5^2 7 '=' the scalaron exponent g+n-1 is one of many readings of 7 -- all declined. [E] RECONCILIATION + HONEST CONCLUSION: the unmapped region yields NO new forced physical readout; its genuine content is collective (set-level, already in v66/v223) plus the Coxeter number, and the per-degree atom coincidences are unforced -- so v66 and v354 are both right once forced set/subchain structure is separated from unforced per-degree matches. The disciplined decline IS the anti-numerology result.",
      "status": "Bandwidth search [E]/[C]/[O] (forced set-level structure sum=128=spinor/product=|W|/exponents=totatives; subchain flagged; per-degree matches declined; no new physical hit; reconciles v66+v354)",
      "canonicalStatus": "[E] the set-level forced identities and the reconciliation; [C] the subchain Coxeter pattern; [O] the explicit declines. A self-critical search whose main finding is that there is nothing forced left to find per-number -- the correct outcome of an honest reverse audit; sharpens v354 (no PRIMARY readout, not 'unmapped'), reconciles v66.",
      "dependencies": "E8.REVERSE.AUDIT.01; v354; v66; v223; v219; v305",
      "scripts": [
        "v355_e8_unmapped_bandwidth.py"
      ],
      "script": "v355_e8_unmapped_bandwidth.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 95
    },
    {
      "id": "E8.REVERSE.AUDIT.01",
      "claim": "THE REVERSE NUMEROLOGY AUDIT + WHAT THE GOLDEN RATIO MEANS (2026-06-22). TFPT's forward discipline (v305) is one-directional: every physical READOUT must map onto an E8 structure (anti-fitting). The reverse question (rightly asked): how much E8 structure is mapped onto NOTHING, and why? -- and is the recurring golden ratio numerology? [E] THE REVERSE AUDIT: E8 has 8 Casimir invariant degrees {2,8,12,14,18,20,24,30} (= exponents+1); exactly THREE feed a PRIMARY physical-constant readout -- degree 2 (the quadratic/metric), degree 8 (the rank -> c3=1/(8pi)), degree 30 (the Coxeter number -> g_car=5 and the order-30 cycle) -- and FIVE carry no primary readout {12,14,18,20,24} (they coincide with structural atoms -- A3 roots, dim G2, det L, |W(A3)|, the anchor power sum -- per v66, but those per-degree coincidences are mostly UNFORCED, the numerology temptation reconciled/declined in v355). So 3/8 are primary readouts, 5/8 hull overhead. [E] ECONOMY, NOT PROMISCUITY (the anti-numerology signature): a SMALL fixed set of E8 data (rank 8, h=30, det Cartan 1, the D5(+)A3 maximal sub, the 16 half-spinor) generates the ENTIRE readout set (gauge group, 3 families, hypercharges, alpha^-1, the flavor matrix, the scales); the higher Casimir invariants are NEVER reached into to fit anything. A numerological use of E8 would be PROMISCUOUS (mining the rich structure for matches); TFPT is ECONOMICAL (few invariants, many readouts) and leaves most of E8 untouched. [E] THE GOLDEN RATIO IS UNMAPPED => NOT NUMEROLOGY: phi=2cos(pi/5) appears only in INTERNAL structure -- the affine-E8 attractor subleading eigenvalue (v312) and the icosian lattice/2I (v348) -- and in NO physical readout (no measured constant is phi). Numerology = a number FITTED to an OBSERVATION; phi is fitted to nothing, it is the algebraic signature of the 5-fold (h=30=2*3*5), the carrier ANNOUNCING that it is icosahedral. Unmapped structure cannot be numerology. [O] WHY THE UNMAPPED REMAINDER EXISTS (honest): E8's role is the minimal even-unimodular HULL (consistency container, det=1), not the world group; the hull's job is done by the low data (rank, h, det, decomposition), so the higher Casimirs {12,14,18,20,24} are by-products of unimodular closure. The 5/8 unmapped is the HULL OVERHEAD -- expected for a hull, but stated plainly: where a future higher-invariant readout could live, or genuine over-capacity. TFPT does not pretend E8 is fully used.",
      "status": "Reverse anti-numerology audit [E]/[O] (3/8 Casimir invariants mapped, 5/8 hull overhead; economy not promiscuity; phi is unmapped structure not numerology)",
      "canonicalStatus": "[E] the reverse audit (3/8 mapped, 5/8 unmapped), the economy signature (small fixed invariant set -> all readouts), and phi-as-unmapped-structure (so not numerology); [O] the honest reading of the unmapped remainder as hull overhead. A self-critical anti-numerology audit -- the reverse complement of v305; closes nothing, flags the hull overhead plainly.",
      "dependencies": "WITNESS.INDEP.01; v305; v100; v312; v348; v54",
      "scripts": [
        "v354_e8_reverse_audit.py"
      ],
      "script": "v354_e8_reverse_audit.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 96
    },
    {
      "id": "E8.OTHERSIDE.AUDIT.01",
      "claim": "THE 'OTHER SIDE OF THE SEAM' AUDITED AGAINST THE UNMAPPED E8 STRUCTURE (2026-06-27, v430): the sheet/deck complement of the reverse audit v354/v355. The reverse audit found FIVE Casimir degrees {12,14,18,20,24} with no primary readout ('hull overhead'); the double cover has an 'other side' (the one-sided Z2 collar / conjugate half-spinor S^-). Does that other side correlate with the unmapped E8 structure -- is the leftover where the second sheet lives? The answer is a disciplined NEGATIVE. [E] DECK = DEGREE-2 INVARIANT (matched): the one-sided double cover contributes |Z2|=2 (the factor 1/2 in c3=1/(2*4pi)=1/(8pi), v58/v216); 2=min(E8 degrees)=the quadratic/metric, one of the THREE matched primary readouts {2,8,30} (v354), never one of the unmapped {12,14,18,20,24}. [E] TWO SHEETS = THE 128-SPINOR (matched, collective): dim S^+=dim S^-=16; the full E8 spinor block is 128=rank*dim S^+=8*16=2^(rank-1)=sum(degrees) (v227/v66), the spinor half of 248=120+128; the 'other side' S^- enters 248 as the conjugate (16-bar,4-bar), 128=2*64, not a spare singlet. [E] FORCED-DISJOINT: the natural sheet/deck integer set {|Z2|,dim S^+,dim(S^+ + S^-),128-spinor}={2,16,32,128} has EMPTY intersection with the unmapped degrees {12,14,18,20,24}; its only degree-alphabet contact is the MATCHED degree 2 and the collective spinor budget 128=sum(deg). [O] DISCIPLINED DECLINE (the v355 discriminator applied to the sheet): each unmapped degree, if 'explained' from sheet/deck atoms, needs an UNFORCED coefficient and admits >=2 readings -- exactly the promiscuous mining v355 declines; no forced sheet->unmapped-degree identity. [E] 'S^- = DM' DOWNGRADED + WIMP NO-GO: v227 replaced the over-strong 'S^- is dark matter' by the magnitude/phase typing of 248=120+128; 128=(16,4)+(16-bar,4-bar) is the SAME matter spinor, so there is no spare E8 singlet for a WIMP (frontier) -- dark matter is the determinant-line axion (a PHASE of existing structure). [O] WHERE THE ONLY LIVE 'OTHER SIDE' QUESTION SITS: whether the 128 phase/glue channel hosts a dark sector stays Frontier (v227), but that is MATCHED structure (the spinor 128=sum(deg)), NOT the five unmapped degrees. CONCLUSION: the 'other side of the seam' is forced-disjoint from E8's unmapped Casimir degrees -- a clean structural NEGATIVE complementing v354/v355.",
      "status": "Sheet/deck reverse audit [E]/[O] (the double-cover 'other side' = degree-2 + the 128-spinor, forced-disjoint from the 5 unmapped degrees; the complement of v354/v355)",
      "canonicalStatus": "[E] the deck=degree-2 (matched) and two-sheets=128-spinor (matched, =sum(deg)) identities, and the forced-disjointness {2,16,32,128} cap {12,14,18,20,24}=empty; [O] the disciplined decline (no forced sheet->unmapped identity, v355 discriminator) and the honest residual (the 128 phase/glue dark-sector question concerns matched structure). The sheet/deck complement of v354/v355; closes nothing, records the structural negative.",
      "dependencies": "E8.REVERSE.AUDIT.01; E8.UNMAPPED.BANDWIDTH.01; E8.CHAN.01; E8.CAS.01; v58; v109",
      "scripts": [
        "v430_other_side_reverse_audit.py"
      ],
      "script": "v430_other_side_reverse_audit.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 97
    },
    {
      "id": "E8.DEGREE.LADDER.01",
      "claim": "THE REVERSE-AUDIT 'UNMAPPED' E8 CASIMIR DEGREES ARE THE FORCED TWO-FAMILY DECOMPOSITION (2026-06-27, v431): the structural complement of v354/v355/v430. The reverse audit found FIVE 'unmapped' degrees {12,14,18,20,24} ('hull overhead'); this module shows they are NOT diffuse overhead but the forced two-family decomposition deg(E8) = 6*spine{2,3,4,5} U ({2} U flavor-det-ladder{8,14,20}). [E] TWO-FAMILY SPLIT FORCED BY 30=2*3*5: the E8 exponents are the phi(30)=8 totatives of h=30=2*N_fam*g_car (coprime to 6 => +-1 mod 6), so the degrees occupy ONLY the residue classes {0,2} mod 6. [I] 6k family {12,18,24,30}/6 = {2,3,4,5} = the v91 spine {e3,e1,e2,p0}(a=(1,1,2)). [I] 6k+2 family {2,8,14,20}: {8,14,20}=(det R,det C,det L)=the winding line det M(s,0)=6s+8 of the flavor diamond (v135), C=R+Q diag(1,0,0); 2=the matched quadratic/metric Casimir (8=det R also = rank E8). [E] '18'=6*N_fam is NOT a holdout (spine family, not on the det-ladder; 6s+8=18 has no integer s). [E] SPECIAL TO E8 among the simply-laced exceptionals (E6/E7/D5 fail the clean {0,2}-mod-6 split; F4/G2 share the split but not the spine+det content). [P] HONEST v355 LINE: the arithmetic decomposition is exact [E]/[I], but the FUNCTORIAL claim '6k+2 Casimir stratum = flavor sheet operators' needs a representation-theoretic map and stays [P]; 12,24 admit two canonical readings each (6*spine vs |R(A3)|=12 / |W(A3)|=24), so this is a STRUCTURE theorem, not a forced functorial flavor map. CONCLUSION: reclassifies v354's overhead to ZERO orphan degrees, the functorial flavor identification honestly kept [P]; closes no gate.",
      "status": "E8 degree-ladder structure theorem [E]/[I] + [P] functorial residual (degrees = 6*spine U det-ladder, forced by 2,3,5; reclassifies v354/v355 overhead)",
      "canonicalStatus": "[E]/[I] the forced two-family decomposition (the {0,2}-mod-6 split from h=30=2*3*5, 6k=6*spine, 6k+2={2} U det-ladder) and the '18'=6*N_fam + E8-specialness facts; [P] the functorial flavor-sheet identification (rep-theoretic map open, 12/24 multiplicity). Reclassifies the v354/v355 overhead to zero orphan degrees; closes nothing.",
      "dependencies": "E8.REVERSE.AUDIT.01; E8.UNMAPPED.BANDWIDTH.01; E8.OTHERSIDE.AUDIT.01; E8.CAS.01; v91; v135",
      "scripts": [
        "v431_e8_degree_ladder.py"
      ],
      "script": "v431_e8_degree_ladder.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 98
    },
    {
      "id": "E8.DEGREE.JOINT.01",
      "claim": "THE TWO STRUCTURAL INTEGERS ARE JOINTLY FORCED BY E8 DEGREE-INVARIANTS, AND E8 IS FIXED BY ITS DEGREES (2026-06-29, v437): a CONSOLIDATION of v6/v66/v355/v431 deriving more from the E8 Casimir degrees WITHOUT crossing the v354/v355 anti-numerology line -- no new per-degree coincidence, only a joint lock. [E] E8 IS FIXED BY ITS DEGREE MULTISET: from deg(E8)={2,8,12,14,18,20,24,30} alone -- rank=#degrees=8 (=2nd degree); h=max(deg)=30; #positive roots=sum(exponents)=120; #roots=240; dim E8=#roots+rank=248; |W(E8)|=prod(deg)=696729600; sum(deg)=128=2^7=dim S^+ (the spinor half of 248=120+128). [E] THE JOINT QUADRATIC (the new lock): the two TFPT structural integers (g_car,N_fam) are the UNIQUE root pair of x^2-(rank E8)x+(h/2)=x^2-8x+15=(x-3)(x-5); sum of roots=rank E8=8 (the rank-fill, v6), product=h/2=15=g_car*N_fam (2 g_car N_fam=h, v431); {3,5} is the unique positive-integer factor pair of 15 summing to 8, so BOTH structural integers are forced TOGETHER by rank and Coxeter number. [E] THREE MAPPED DEGREES ARE CANONICAL: the 3 primary-readout degrees (v354) {2,8,30}={min degree (quadratic Casimir), rank, max degree (Coxeter h)} -- the invariants every simple Lie algebra distinguishes; g_car=5=max prime of h=30, N_fam=3=h/(2 g_car). [E] DISCIPLINE UPHELD: no new per-degree mining beyond the joint quadratic (the solution of two pre-existing constraints v6+v431); the five non-primary degrees keep their v431 family homes; the v354/v355 line is not crossed.",
      "status": "E8 degree joint-lock [E] (the two structural integers g_car=5, N_fam=3 = unique roots of x^2-8x+15 with coeffs rank E8=8 and h/2=15; all E8 combinatorics fixed by its degrees; consolidation of v6/v66/v355/v431, no new mining)",
      "canonicalStatus": "[E] the joint quadratic lock (both structural integers as the unique roots of one degree-invariant quadratic) and the 'E8 fixed by its degrees' reconstruction; a consolidation of v6/v66/v355/v431 that respects the v354/v355 anti-numerology discipline (no new per-degree coincidence). Closes no gate; strengthens the over-determination of g_car and N_fam.",
      "dependencies": "E8.REVERSE.AUDIT.01; E8.DEGREE.LADDER.01; v6; v66; v355; v431",
      "scripts": [
        "v437_e8_degree_joint.py"
      ],
      "script": "v437_e8_degree_joint.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 99
    },
    {
      "id": "OVERDET.FLOOR.01",
      "claim": "THE UNCONDITIONAL IMPROBABILITY FLOOR + THE L1<->L2 BRIDGE (2026-06-27, v432): the defensible over-determination number that survives even if a skeptic REJECTS the declared formula grammar -- the complement of the conditional null model (v100) and the multiply-vs-compress framework (v427/v428). [E] DISJOINT MULTIPLICATIVE PIECES (per v428): compression removed (v427's seven readouts are ONE (2,3,5)/E8 object, not multiplied); what honestly multiplies is the input '8' forced FOUR independent ways {rank E8, h(D5)=2(g_car-1), phi(30), Milnor(2,3,5)} plus the FOREIGN witness alpha^-1~137 (prime, outside the (2,3,5) skeleton). [C] CONSERVATIVE FLOOR: multiplying ONLY across disjoint pieces (input over-determination x the alpha grammar census 1/94500 from v100 x the phi0-seed cluster counted ONCE x one weakly-discriminating downstream witness K+->pi+nunu) the floor stays <=1e-6 across a grid of GENEROUS chance assignments, nominal ~1e-10 -- 'extraordinary' WITHOUT the declared grammar. [E] THE BRIDGE: the unconditional floor (~1e-10) sits ~20 orders ABOVE the v100 conditional 10^-30.7; that gap is exactly the evidential value of proving the FORMS forced, so L1 (the alpha form-derivation ALPHA.QUILLEN.EXACT.01/v382) and L2 (this floor) are the SAME lever -- closing L1 moves the alpha factor from the 1/94500 census toward certainty. [C] HONEST SCOPE: bounds ONLY the 'is it numerology?' question, NOT the physics bridge (seam/anchor/transfer, firewall v187); a conservative FLOOR (an upper bound on chance-plausibility), not a posterior.",
      "status": "Unconditional over-determination floor [C] (disjoint pieces only, compression removed; floor ~1e-10 vs v100 conditional 10^-30.7; bounds numerology only, firewall)",
      "canonicalStatus": "[E] the disjoint-pieces accounting (the '8' forced 4 ways + the foreign alpha) and the floor<->ceiling bridge inequality (floor ~1e-10 >> v100 conditional 1e-30.7); [C] the conservative floor value and the bounded scope (numerology only, not the physics bridge). The unconditional complement to v100; closes nothing.",
      "dependencies": "OVERDET.WITNESS.MAP.01; OVERDET.WITNESS.RECLASS.01; v100; v305; ALPHA.QUILLEN.EXACT.01; v187",
      "scripts": [
        "v432_overdet_floor.py"
      ],
      "script": "v432_overdet_floor.py",
      "location": "tfpt_safeguards",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 100
    },
    {
      "id": "OVERDET.FLOOR.02",
      "claim": "HARDENING THE UNCONDITIONAL FLOOR -- THE ASSUMPTION-MINIMAL COUNTING FLOOR (2026-06-29, v436): OVERDET.FLOOR.01 (v432) multiplies the counting census by SUBJECTIVE chance assignments (input-'8' independence 0.1^3, seed 0.012, kaon 0.15) a skeptic can contest; this module shows the 'is it numerology?' verdict does NOT rest on them. [E] ASSUMPTION-MINIMAL FLOOR: the alpha cubic-class census (v100 layer D) is PURE COUNTING -- of N=94500 complexity-matched F_U(1) variants only ONE lands in the 4e-8 CODATA window (the TFPT instance, alpha^-1=137.0359992), so p_alpha <= 1/94500 = 1.06e-5 ~ 4.40 sigma (two-sided) with NO subjective probability, NO prior, NO multi-observable grammar. [E] MONOTONE CONCESSION LADDER: -log10(p) full scorecard+alpha ~30.7 -> scorecard-only ~25.8 -> alpha-census-only ~4.98 (=4.40 sigma); each rung uses strictly less evidence and 'not chance (>4 sigma)' holds at the MOST adversarial rung, so the verdict never rests on one contestable piece. [E] INDEPENDENCE: strip ALL of v432's subjective factors (1.8e-6) and the floor is still the counting census 4.40 sigma -- they only SHARPEN an already-extraordinary floor. [C] HONEST 5-SIGMA GAP: the conventional 5 sigma line (p=5.7e-7 two-sided) is crossed UNCONDITIONALLY by the census times exactly ONE more independent factor <=0.054 (the input-'8' four-fold over-determination or one foreign witness), stated not hidden. [C] FIREWALL: bounds ONLY 'is it numerology?', NOT the physics bridge (firewall v187); a conservative FLOOR (an upper bound on chance-plausibility), not a posterior.",
      "status": "Assumption-minimal counting floor [E]/[C] (the alpha census 1/94500 = 4.40 sigma with no subjective probability; monotone concession ladder; independent of v432's subjective pieces; honest 5-sigma gap; hardens OVERDET.FLOOR.01)",
      "canonicalStatus": "[E] the pure-counting census floor (4.40 sigma), the monotone concession ladder, and independence from v432's subjective pieces; [C] the honest 5-sigma gap (one further independent factor <=0.054) and the bounded scope (numerology only, not the physics bridge). Hardens OVERDET.FLOOR.01 (v432) -- replaces its contestable subjective assignments with an assumption-minimal counting floor; closes nothing.",
      "dependencies": "OVERDET.FLOOR.01; v100; v305; ALPHA.QUILLEN.EXACT.01; v187",
      "scripts": [
        "v436_overdet_floor2.py"
      ],
      "script": "v436_overdet_floor2.py",
      "location": "tfpt_safeguards",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 101
    },
    {
      "id": "TFPT.SELFLOOP.01",
      "claim": "THE BIRD'S-EYE RETHINK (2026-06-22, refining TFPT.IRREDUCIBLE.01/v352). Stepping all the way back: TFPT is not a LINEAR theory (axioms -> theorems) at all -- it is a CLOSED SELF-CONSISTENT LOOP whose outputs fix its own inputs, with ZERO free adjustable dimensionless parameters; pi (the only transcendental) is a fixed mathematical constant of the boundary geometry, not a dial. [E] IT IS A LOOP, NOT A LINEAR FOUNDATION: outputs fix inputs -- g_car=5 = max prime of h(E8)=30 (the output Coxeter number fixes the input rank), the 8 in c3 = rank E8 = g_car+N_fam (the output lattice fixes the input seam constant); inputs fix outputs ({c3,g_car} -> D5(+)A3+mu4 -> E8 -> SM, alpha, scales). The arrows close into a circle (v6 bootstrap) with a UNIQUE fixed point (v56 attractor); 'which is the axiom?' is the wrong question for a loop. [E] ZERO FREE ADJUSTABLE DIMENSIONLESS DIALS: every load-bearing number is forced/over-determined (g_car 3 ways, the 8 four ways = rank E8 = h(D5) = phi(30) = det R, rank E8, c=8, N_fam=3, the (2,3,5) atoms, the golden ratio, alpha^-1 the unique root); the count of free adjustable dimensionless parameters is 0. [E] pi IS A FIXED CONSTANT, NOT A DIAL: pi enters only through the boundary 2-sphere (oint_{S^2} K = 2 pi chi = 4 pi, chi=2), so c3 = 1/(|Z2| oint K) = 1/(8 pi); pi is the sole transcendental but a fixed mathematical constant of the continuous boundary geometry, not tunable. [A] REFINING v352: v352 said 'the irreducibles are the framework + pi'; sharpened -- pi is not a free input (the sphere's constant) and the 'framework' is not a free meta-axiom but a SELECTION PRINCIPLE ('reality is the unique self-consistent discrete reflection-positive boundary loop'); the honest characterisation is a UNIQUE self-consistent loop with ZERO free dials, not 'axioms + a tested meta-axiom'. [O] THE ONE GENUINE RESIDUAL AND THE FORK: discretely the loop CLOSES (unique, over-determined); the only thing not verified is whether it closes in the CONTINUUM -- the existence of the chiral edge / massless scaling limit (SEAM.EQUIV.01, v336/v351). Under STRUCTURAL REALISM (reality IS the self-consistent structure) the loop is complete and this is not a separate question; under CONSTRUCTIVISM (an independent physical world must instantiate the math) the continuum closure is the genuine open theorem. The fork is philosophical, not numerical -- TFPT has no free numbers either way.",
      "status": "Capstone rethink [E]/[A]/[O] (a unique self-consistent loop with zero free dials; pi a fixed geometric constant; the one residual is the continuum closure; the fork is philosophical not numerical)",
      "canonicalStatus": "[E] the loop closure (outputs fix inputs), zero free dials, and pi-as-geometry are machine facts; [A] the refinement of v352's framing (loop, not framework+pi); [O] the continuum closure + the honest philosophical fork. A capstone synthesis; gives the deepest honest characterisation -- a unique self-consistent loop, no free numbers, one continuum residual.",
      "dependencies": "TFPT.IRREDUCIBLE.01; v6; v56; v54; v336; v351",
      "scripts": [
        "v353_selfloop_capstone.py"
      ],
      "script": "v353_selfloop_capstone.py",
      "location": "introduction",
      "externalData": "-",
      "supersedes": "TFPT.IRREDUCIBLE.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 102
    },
    {
      "id": "TFPT.IRREDUCIBLE.01",
      "claim": "THE FRAMEWORK REDUCTION (2026-06-22) -- the deepest honest answer to 'the inputs are not axioms, they are back-determined by the theory'. BOTH P1 (c3=1/(8pi)) and P2 (g_car=5) are bootstrap-forced fixed points, not free postulates; the only genuine irreducibles are the FRAMEWORK (a discrete reflection-positive boundary with a finite carrier) and the transcendental pi. [E] P2 (g_car=5) IS FORCED three ways (v6): rank-fill (g_car+N_fam=8=rank E8), Coxeter-match (g_car=max prime of h(E8)=30), Pascal (2^4=C(5,0)+C(5,1)+C(5,2)=16); a free axiom has one justification, g_car=5 has three. [E] P1 (c3=1/(8pi)) IS FORCED: c3 = 1/(|Z2| oint_{S^2} K) is the boundary Gauss-Bonnet coefficient (v216/v342), and the 8 is over-determined FOUR ways (v54/v6): 8 = rank E8 = h(D5) = phi(30) = det R; the only non-forced ingredient is pi, the transcendental of the boundary geometry. [E] the E8 hull is forced (unique even unimodular rank-8, holomorphic c=8=g_car+N_fam, v83/v154), not posited. [E] so every load-bearing number (g_car, the 8 in c3, rank E8, c=8, N_fam, the (2,3,5) atoms, the golden ratio) is a self-consistency fixed point; the ONLY irreducibles are the FRAMEWORK and pi -- the precise content of 'the inputs are not axioms'. [O] the framework itself ('physics is a discrete reflection-positive boundary with a finite carrier') is NOT reducible by mathematics -- it is the foundational stance (like 'spacetime is a manifold' in GR, 'states are Hilbert rays' in QM), minimal (one structural choice), tested by its consequences (the frozen predictions), not proved. TFPT's true input is ONE framework + pi; the self-consistency does the rest.",
      "status": "Synthesis/inventory [E]/[O] (both axioms bootstrap-forced; irreducibles = framework + pi; the framework is the tested meta-axiom)",
      "canonicalStatus": "[E] both axioms forced (the over-determination counts: g_car three ways, the 8 four ways) + the irreducible inventory {framework, pi}; [O] the framework is the meta-axiom (tested by consequences, not proved). A synthesis confirming 'the inputs are not free axioms' at the deepest level and locating the single irreducible stance.",
      "dependencies": "v6; v54; v83; v154; v216; v342; ANCHOR.VGEO.02",
      "scripts": [
        "v352_framework_irreducible.py"
      ],
      "script": "v352_framework_irreducible.py",
      "location": "introduction",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 103
    },
    {
      "id": "SEAM.EQUIV.BOOTSTRAP.01",
      "claim": "THE HONEST CORRECTION OF v349's FRAMING (prompted by the right objection 'the inputs are not axioms -- they are back-determined by the theory', 2026-06-22). v349 concluded 'the raw seam does not carry the golden ratio (D5 has h=8 -> sqrt2)' and reduced the keystone to 'is g_car=5 a pentagon or a count?'. That framing was WRONG on two counts; this corrects it while keeping v349's true computations. [E] THE INPUTS ARE BOOTSTRAP-FORCED, NOT FREE AXIOMS (v6): g_car=5 is over-determined three ways -- rank-fill (g_car+N_fam=8=rank E8), Coxeter-match (g_car = max prime of h(E8)=30), Pascal/glue (2^4=C(5,0)+C(5,1)+C(5,2)=16). The objection is correct: the numerical inputs are FIXED POINTS of the self-consistency, not free postulates. [E] g_car=5 IS THE ICOSAHEDRAL 5-FOLD, not the D5 rank: g_car = max prime of h(E8)=30, and h(E8)=30 = 2*3*5 is the icosahedral axis data; the atoms (|Z2|,N_fam,g_car)=(2,3,5) ARE the prime factorisation of the icosahedral Coxeter number 30. So the '5' the bootstrap selects is intrinsically icosahedral (golden), NOT the bare D5 rank (h=8) v349 mis-identified it with. [E] THE GOLDEN RATIO IS EMERGENT FROM THE GLUE, not external (correcting v349): D5 (h=8) and A3 (h=4) are individually non-golden, but the mu4 simple-current extension D5(+)A3 -> E8 (theorem v154, index 4=|mu4|) gives h(E8)=30 with 5|30 -> golden; the seam's OWN mu4 clock, gluing the carrier, PRODUCES the golden 5-fold -- v349 evaluated the un-glued pieces, the wrong object. [E] so v349's dichotomy was FALSE: the Coxeter-match bootstrap FORCES the pentagon (g_car=5 = the 5 in h(E8)=30); there is no 'count' alternative consistent with E8-closure. [O] THE RESIDUAL RELOCATES: the golden/icosahedral is NOT open (bootstrap-forced + the v154 extension theorem); the genuinely open thing is ONLY the PHYSICAL CONTINUUM REALISATION (the raw RP quasi-free collar realises the algebraic E8 fixed point, continuum-limit/OS, v336), NOT the golden ratio. HONEST CAVEAT: the bootstrap is self-consistency WITHIN the discrete E8-closure framework (the Coxeter-match uses h(E8)=30), so it is 'no free numerical dials given the framework', not 'from nothing'; the framework is the meta-input.",
      "status": "Correction [E]/[O] (inputs are bootstrap-forced fixed points; g_car=5 IS the icosahedral 5 of h(E8)=30; golden emergent from the mu4-glue; residual = the physical continuum realisation)",
      "canonicalStatus": "Correction [E]/[O] (inputs bootstrap-forced; g_car=5 icosahedral; golden emergent; residual = continuum realisation)",
      "dependencies": "SEAM.EQUIV.GOLDEN.01; v6; v154; v56; v336; v274",
      "scripts": [
        "v350_bootstrap_inputs_correction.py"
      ],
      "script": "v350_bootstrap_inputs_correction.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "SEAM.EQUIV.GOLDEN.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 104
    },
    {
      "id": "SEAM.EQUIV.GOLDEN.01",
      "claim": "THE HONEST TEST OF 'DOES THE RAW SEAM CARRY THE GOLDEN RATIO?' (the v348 question, 2026-06-22) -- answer, computed and honest: NO. The bare carrier (D5, A3) and the dynamical seam data are NOT golden; the golden ratio genuinely enters WITH the icosahedral/E8 identification. A NEGATIVE result, but clarifying: it prevents a false closure and reduces the whole keystone to one almost childlike question. [E] RAW CARRIER NOT GOLDEN: the golden ratio 2cos(pi/5)=phi is the signature of a 5-FOLD ROTATION, appearing in a Coxeter element only when 5 | h; the carrier D5 has h=8 (eigenvalue 2cos(2pi/8)=sqrt2) and A3 has h=4 (eigenvalue 0) -- NO 5-fold, so the bare carrier gives sqrt2, not phi. [E] THE DYNAMICAL SEAM DATA ARE NOT GOLDEN: cusp weights {0,1/3,2/3} rational, recovery (2/3)^6 rational, seed phi0=1/(6pi)+48c3^4 transcendental -- none in Q(sqrt5). [E] GOLDEN IS THE ICOSAHEDRAL/E8 SIDE: 5 | h holds ONLY for A4=SU(5) (h=5), the icosahedral H3 (h=10), H4/E8 (h=30) -- all the OUTPUT/icosahedral side; the golden ratio enters as the icosahedral input, not as a hidden raw feature. [E] phi ALONE INSUFFICIENT: even with a 5-fold, 2I is NOT C5 x mu4 (20 != 120); 2I is a specific (2,3,5) extension, so a 5-fold plus mu4 must satisfy the icosahedral braid relation, not merely coexist. [O] THE RESOLUTION (the simplest form): the keystone L2 reduces to one question -- is g_car=5 a PENTAGON (a genuine 5-fold rotation, hence golden, hence with mu4 the icosahedral 2I=E8 via the icosian ring) or just a COUNT (the rank of D5, h=8, giving sqrt2)? The 'absurdly simple solution' is real but is a READING of axiom P2: the golden ratio is the pentagon content of g_car=5, NOT derivable from P2-as-a-count (D5 gives sqrt2). There is no hidden phi to find -- golden and icosahedral are the SAME input, entering together. L2 is genuinely foundational: either P2 is read as the pentagon (closed, axiom-mode C of v347) or a rigidity theorem must PRODUCE the 5-fold (mode B); not closed here.",
      "status": "Honest negative/clarification [E]/[O] (the raw seam does NOT carry phi; golden is the icosahedral input; L2 = 'is g_car=5 a pentagon or a count?')",
      "canonicalStatus": "[E] the computed negative (D5/A3 give sqrt2; dynamical data not golden; 5|h only on the output side; 2I != C5 x mu4); [O] the resolution that L2 = 'is g_car=5 a pentagon or a count?', an axiom-reading not a hidden derivation. A NEGATIVE/clarification module; closes nothing and prevents a false closure -- the golden ratio is the icosahedral input, not a hidden feature of the raw seam.",
      "dependencies": "SEAM.EQUIV.RIGID.01; SEAM.EQUIV.CLOSURE.01; v312; v313; v219; v246",
      "scripts": [
        "v349_raw_seam_golden_test.py"
      ],
      "script": "v349_raw_seam_golden_test.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 105
    },
    {
      "id": "SEAM.EQUIV.RIGID.01",
      "claim": "ROUTE B (THE RIGIDITY ROUTE) FOR THE ONE OPEN ARROW L2, CARRIED OUT (2026-06-22) -- it lands on a strikingly SIMPLE statement: the whole keystone reduces to ONE question, does the raw seam carry the golden ratio? [E] RIGIDITY CLOSES THE CRYSTALLOGRAPHIC PART: the order-4 clock on a genus-0 surface has a UNIQUE conformal normal form z -> i z (Nielsen realisation theorem / Kerckhoff + uniformisation, v180); Troyanov's prescribed-curvature theorem gives the UNIQUE flat metric on the four-marked sphere (the pillowcase, v284). So the mu4/pillowcase data are rigidly fixed -- this half needs no new input. [E] THE DOWNSTREAM IS A LATTICE IDENTITY: beyond the McKay graph (v219), the Conway-Sloane fact that the ring of icosians (the 120 unit icosians = 2I, with the golden-ratio quaternion norm) is, as a rank-8 Z-module, the E8 LATTICE itself -- so '2I -> E8' is an identity of lattices, the strongest possible downstream link. [E] THE SINGLE BRIDGE IS THE GOLDEN RATIO: the crystallographic restriction allows only rotation orders {1,2,3,4,6} -- the 5-fold is NON-crystallographic; the mu4 clock (order 4) is crystallographic, the icosahedral 2I (5-fold) is not, and phi = 2 cos(pi/5) is EXACTLY what extends one to the other (the quasicrystal scaling; 2I = <order-4, golden>, the 96 golden icosians carry phi). [O] THE KEYSTONE REDUCES TO ONE QUESTION: IF the raw seam carries phi, THEN mu4 + phi generate 2I, the icosian ring gives E8, rigidity fixes the rest, and L2 is closed -- so the ENTIRE keystone reduces to 'does the raw quasi-free seam carry the golden ratio phi (as the icosian/quasicrystal scaling)?', the simplest possible form of L2. [O] THE HONEST SUBTLETY: TFPT's phi currently appears on the E8-SIDE -- the subleading eigenvalue of the affine-E8 attractor (v312) and the icosahedral atom 2 cos(pi/5) (v313), both already referencing E8/the icosahedron; the raw-seam data checked so far (cusp weights {0,1/3,2/3}, recovery (2/3)^6, seed phi0=1/(6pi)+48c3^4) are rational/transcendental, NOT manifestly golden. So showing the RAW seam carries phi NON-circularly (before assuming E8) is the residual -- it IS L2 in its sharpest form. Route B reduces the keystone to 'is there a phi in the raw seam?'; it does not answer it.",
      "status": "Route B / reduction [E]/[O] (rigidity + the icosian-ring identity reduce L2 to one question: does the raw seam carry the golden ratio?; NOT closed)",
      "canonicalStatus": "[E] the rigidity reductions (Nielsen/Troyanov close the crystallographic mu4/pillowcase part), the icosian-ring lattice identity (E8 = icosians), the crystallographic role of phi (the 5-fold is non-crystallographic, phi bridges); [O] the single reduced arrow ('the raw seam carries the golden ratio'), with the honest flag that TFPT's phi is currently E8-side (potential circularity). Route B; closes nothing, reduces the keystone to one question.",
      "dependencies": "SEAM.EQUIV.CLOSURE.01; v180; v284; v219; v312; v313",
      "scripts": [
        "v348_seam_rigidity_route.py"
      ],
      "script": "v348_seam_rigidity_route.py",
      "location": "origin_theory",
      "externalData": "Nielsen realisation (Kerckhoff); Troyanov; Conway-Sloane (the icosian ring = E8)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 106
    },
    {
      "id": "SEAM.EQUIV.CLOSURE.01",
      "claim": "THE CLOSURE-MODE CLASSIFICATION OF THE ONE OPEN ARROW L2 (the seam-orbifold realisation, v346) -- a direct honest answer to 'is there ANY other way to FULLY solve it?' (2026-06-22). [E] THE PRECISE LOCUS OF L2: the seam DIRECTLY gives (from the mu4 clock + four marks) the pillowcase base S^2(2,2,2,2), a EUCLIDEAN orbifold chi_orb = 2 - 4(1-1/2) = 0 (flat, v216/v325); but the 2I/E8 structure is the Seifert fibration over the icosahedral base S^2(2,3,5), a SPHERICAL orbifold chi_orb = 2 - (1/2+2/3+4/5) = 1/30 > 0. So L2 must bridge a FLAT 4-marked base (the seam mu4) to a SPHERICAL 3-marked base (the carrier (2,3,5) atoms) -- two different geometric types; this is WHY L2 needs the carrier input and is nontrivial. [E] FOUR CLOSURE MODES: L2 is a 'physics-realises-geometry' statement, with exactly four closure modes -- A CONSTRUCT (build the continuum scaling limit, R1/v336, the analytic wall); B RIGIDITY (prove the seam data RP+quasi-free+gap+mu4+four marks admit a UNIQUE realisation so C^2/2I is forced with no extra input; the only theorem-route w/o new physics, pursued Troyanov v284 / flat-away v300 / necessity-of-H v331); C AXIOM (accept L2 as a third postulate P3, the seam-identification, exactly as c3=P1 and g_car=P2; then TFPT is COMPLETE relative to three axioms, the role 'c=const' plays in relativity); D EMPIRICAL (test the consequences delta_PMNS=240/theta13/the frozen predictions; physics validates axioms by consequences not proof). [E] NO-NEW-STATE BLOCKS DERIVATION: TFPT adds no gauge-charged state and posits no further dynamical input (tfpt_3/v246), so L2 cannot be DERIVED from additional physics -- the only theorem-route is rigidity (B); otherwise L2 is axiom (C) + empirical (D). [E] VERDICT: there is NO route making L2 a free theorem with zero input; 'fully solve it' = exactly one of (B) a rigidity theorem (the only theorem-route, genuine open math, the most promising new route) or (C) accept axiom P3 (then complete relative to 3 axioms), either way validated by (D) empirics; other routes exist but only these four, and only B closes L2 as a theorem. [O] NOT CLOSED: classifies the modes and locates the difficulty; does NOT prove L2. Actionable: the rigidity route (B) -- 'the seam data uniquely force the flat pillowcase to lift to the C^2/2I orbifold' -- is the single theorem-path worth pursuing.",
      "status": "Analysis/roadmap [E]/[O] (the flat->spherical base locus; four closure modes; no free theorem; only rigidity B or axiom P3; NOT closed)",
      "canonicalStatus": "[E] the orbifold-Euler-characteristic locus (flat seam base vs spherical icosahedral base) + the closure-mode enumeration + the no-new-state blocking; [O] L2 itself. An analysis/roadmap; closes nothing, fabricates nothing -- it answers 'are there other ways to fully solve it' by enumerating all four modes and the honest verdict (only rigidity B as a theorem, else axiom P3).",
      "dependencies": "SEAM.EQUIV.GEOM.01; v216; v284; v300; v331; v246; v336",
      "scripts": [
        "v347_seam_closure_modes.py"
      ],
      "script": "v347_seam_closure_modes.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 107
    },
    {
      "id": "SEAM.EQUIV.GEOM.01",
      "claim": "THE GEOMETRIC BRIDGE AS AN EXPLICIT SIX-LINK CHAIN, WITH THE ONE OPEN ARROW PINNED (2026-06-22) -- the honest capstone of the SEAM.EQUIV.01 investigation (v335/v336/v344/v345). The chain raw-seam => det K=1: L1 [E/C] raw seam -> mu4 clock + four marks (Gauss-Bonnet on the flat pillowcase S^2(2,2,2,2), v216); L2 [O] mu4 + the carrier (2,3,5) atoms -> the binary icosahedral 2I action C^2/Gamma -- THE ONE OPEN ARROW, the geometric content of SEAM.EQUIV.01; L3 [E] 2I -> affine E8 McKay graph (computed from the group via Dixon characters, v219); L4 [E] 2I -> the E8 du Val singularity C^2/2I (Brieskorn/Slodowy, v232); L5 [E] C^2/2I -> the Poincare link S^3/2I with H_1=0 (Smith normal form + 2I perfect, v345); L6 [E] H_1=0 -> det K=1 (holomorphy, v345/v235). [E] FIVE OF SIX LINKS ARE THEOREMS: of L1..L6 exactly ONE (L2, the orbifold realisation) is open; L1 is [E/C] and L3..L6 are [E] cited theorems, so the keystone reduces to the single arrow L2. [E] THE GROUP IS FORCED BY THE SEAM'S OWN ATOMS: the exceptional finite SU(2) subgroups have axis-order signatures (2,3,3)->2T->E6, (2,3,4)->2O->E7, (2,3,5)->2I->E8; the signature (2,3,5) is UNIQUE to 2I, and the TFPT atoms (|Z2|,N_fam,g_car)=(2,3,5) ARE these icosahedral axes (v219/v313), so IF the seam realises a finite SU(2) orbifold its group is FORCED to be 2I -- the seam cannot 'choose' a different ADE type. [E] EVERYTHING DOWNSTREAM OF L2 IS THEN FORCED: given 2I, L3..L6 give E8 (h=2*3*5=30, rank=g_car+N_fam=8), the Poincare link and det K=1 by theorems; so det K=1 <=> the seam realises C^2/2I (L2), the group already pinned. [O] THE IRREDUCIBLE OPEN CONTENT: L2 ('the raw quasi-free seam normal bundle IS an ADE orbifold point C^2/Gamma') is the geometric content of SEAM.EQUIV.01; this does NOT prove it but shows the keystone is EXACTLY this one geometric-realisation arrow, with the group forced by (2,3,5) and all downstream consequences proven. The bridge is one arrow, not a programme.",
      "status": "Capstone/roadmap [E]/[O] (5 of 6 links theorems; the group forced by the icosahedral atoms; the keystone reduced to the single orbifold-realisation arrow L2 = SEAM.EQUIV.01)",
      "canonicalStatus": "[E] the chain bookkeeping (5 of 6 links are theorems) + the (2,3,5)->2I axis selection among exceptional SU(2) subgroups; [O] the single orbifold-realisation arrow L2 (= SEAM.EQUIV.01). A capstone/roadmap; closes nothing, fabricates nothing -- it reduces the keystone to one geometric-realisation statement with the group pinned by the seam's own (2,3,5) atoms.",
      "dependencies": "SEAM.EQUIV.UNIFY.01; SEAM.DETK.02; v216; v219; v232; v345; v313",
      "scripts": [
        "v346_seam_geometric_bridge.py"
      ],
      "script": "v346_seam_geometric_bridge.py",
      "location": "origin_theory",
      "externalData": "McKay correspondence; Brieskorn/Slodowy du Val resolution",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 108
    },
    {
      "id": "SEAM.DETK.02",
      "claim": "EXECUTING THE COMBINATORIAL CORE OF ROUTE R3 (the hypergraph-homotopy route to det K=1, v344) -- the most promising fresh angle, now carried out as far as it computably goes (2026-06-22). Computes explicitly the topology of the (2,3,5)-rewrite attractor graph's plumbing link: it is the POINCARE homology sphere (H_1=0, the det K=1 face), only E8 among ADE. R3's combinatorial half is thereby [E]-complete; the one remaining step is the geometric realisation bridge (= SEAM.EQUIV.01), NOT closed. Standard plumbing topology (Brieskorn/Hirzebruch): a tree plumbing with the ADE Cartan as intersection form has link H_1 = coker(Cartan) (sign-independent: coker(M)=coker(-M)). [E] SMITH NORMAL FORM => coker = H_1(link): the SNF of the E8 Cartan matrix is the identity (all elementary divisors 1), so coker(E8)=0 -- the E8-graph plumbing boundary is an INTEGRAL HOMOLOGY SPHERE, the genus-1/topological meaning of det E8=1 (the discrete graph fixes the link's H_1). [E] ONLY E8 AMONG ADE: coker(A_n)=Z/(n+1), coker(D_n)=Z/4, coker(E6)=Z/3, coker(E7)=Z/2 -- all nontrivial; ONLY E8 has coker=0, so only the E8 rewrite skeleton has a homology-sphere link (= only E8 gives det K=1). [E] pi_1=2I PERFECT => THE POINCARE SPHERE: the link's fundamental group is the binary icosahedral group 2I=SL(2,5) (order 120); the commutator subgroup [2I,2I]=2I (|[G,G]|=120=|G|, computed directly), so 2I is perfect, its abelianisation (=H_1) is trivial, and the link is THE Poincare homology sphere (the unique homology 3-sphere with finite pi_1). [E] (2,3,5) TIE-IN: 2I is the binary (2,3,5) icosahedral group, |2I|=120=|mu4|*h(E8)=4*30, tying the link to the rewrite (2,3,5) data and the order-30 Coxeter cycle (v55/v219/v312). [O] HONEST RESIDUAL: does NOT close det K=1 -- the remaining step is the geometric realisation bridge ('the raw (2,3,5)-rewrite seam attractor IS this E8 plumbing / du Val singularity'), exactly the SEAM.EQUIV.01 content. R3's combinatorial half is now [E]-complete; the geometric half stays [O].",
      "status": "Execution of R3 [E]/[O] (combinatorial core complete: homology-sphere link via SNF + 2I perfect = Poincare sphere; geometric bridge stays SEAM.EQUIV.01)",
      "canonicalStatus": "[E] the SNF/cokernel computation (= H_1 of the plumbing link), the ADE contrast, and the explicit perfectness of 2I=SL(2,5) (commutator subgroup = whole group); [O] the geometric realisation bridge (= SEAM.EQUIV.01). Executes R3's combinatorial core; closes nothing.",
      "dependencies": "SEAM.DETK.01; SEAM.EQUIV.UNIFY.01; v232; v312; v344; v55; v219",
      "scripts": [
        "v345_hypergraph_homotopy.py"
      ],
      "script": "v345_hypergraph_homotopy.py",
      "location": "origin_theory",
      "externalData": "Brieskorn/Hirzebruch plumbing topology (link H_1 = coker intersection form)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 109
    },
    {
      "id": "SEAM.DETK.01",
      "claim": "THE BIRD'S-EYE SYNTHESIS OF THE ONE OPEN KEYSTONE BIT det K=1 (2026-06-22): a full-spectrum map of SEAM.EQUIV.01's only open residual -- it is ONE statement with six equivalent faces; it is hard for ONE precise reason (the genus-1 obstruction); and there are exactly three genus-1 access routes, of which the hypergraph-homotopy route is the most promising fresh angle. [E] SIX FACES: det K=1 <=> holomorphy (1 primary) <=> homology-sphere link (H_1=0) <=> one 1-dim irrep (Gamma perfect) <=> det Cartan=1 <=> the full mu4 condensation 16->4->1 <=> the tau=i CM rigidity (chi_E8=j^{1/3}); unified by |det Cartan| = |H_1(link)| = |Gamma^ab| = #(1-dim irreps) across ADE (A_n->n+1, D_n->4, E6->3, E7->2, E8->1), ONLY E8 -> 1 (2I the unique perfect ADE group). [E] THE GENUS-1 OBSTRUCTION: every easy argument (unique plane vacuum, gap, free-fermion CAR invertibility) is GENUS-0 -- necessary but NOT sufficient (a plane vacuum is always 1 for any chiral net, v87); det K is the TORUS ground-state degeneracy = #primaries, invisible to genus-0 data; so any closing route MUST access genus-1/modular/topological info -- the precise reason det K=1 is the residual. [E] THREE GENUS-1 ACCESS ROUTES: R1 continuum-modular (scaling limit + chi_E8=j^{1/3}, v336); R2 anyon-condensation-torus (degeneracy = #anyons = |det K|, tower 16->4->1, v281); R3 hypergraph-homotopy (torus degeneracy = discrete H_1 of the substrate complex; a Poincare-sphere link gives 1). [C] THE HYPERGRAPH ROUTE = MOST PROMISING FRESH ANGLE: the (2,3,5) rewrite ATTRACTOR is the affine-E8 Dynkin graph (Kac marks = Perron vector, golden 5-fold, v312) and the E8 du Val resolution graph's link is the Poincare sphere (H_1=0, v232), so det K=1 becomes 'the attractor's link homotopy type is the Poincare sphere' -- a finite COMBINATORIAL statement (no continuum limit), unlike R1/R2. [O] HONEST RESIDUAL: none of R1/R2/R3 closes det K=1; the hypergraph route's remaining gap is the combinatorial-to-geometric bridge ('the rewrite attractor (a graph) IS the raw seam's geometric realisation (the du Val singularity)') -- which is exactly the SEAM.EQUIV.01 content. The synthesis SHARPENS the target to one combinatorial statement and ranks the routes; it does NOT prove it. UPDATE 2026-07-14 (v479/SEAM.KRONHEIMER.01): R3's combinatorial-to-geometric bridge is DISCHARGED to Kronheimer 1989 (ALE hyper-Kaehler quotient + Torelli; all finite input data compiler outputs) -- the [O] residual transforms to 'the raw seam supplies the ALE/orbifold datum', the same single arrow as SEAM.EQUIV.01; status unchanged.",
      "status": "Synthesis/roadmap [E]/[C]/[O] (one bit/six faces; genus-1 obstruction; three access routes; hypergraph the fresh one; no closure)",
      "canonicalStatus": "[E] the six-faces equivalence (|det Cartan|=|H_1| ADE arithmetic, computed) + the genus-1 obstruction + the three-route map; [C] the hypergraph route's alignment (attractor=E8 graph v312 + E8 link=Poincare sphere v232); [O] the unclosed combinatorial-to-geometric bridge (= SEAM.EQUIV.01). A synthesis/roadmap; closes nothing, fabricates nothing.",
      "dependencies": "SEAM.EQUIV.UNIFY.01; v232; v235; v219; v281; v282; v312; v336; v87",
      "scripts": [
        "v344_detk_synthesis.py"
      ],
      "script": "v344_detk_synthesis.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 110
    },
    {
      "id": "FOUR.ROUTES.01",
      "claim": "THE FOUR BLACK-HOLE-BIRTH SOLUTION ROUTES INVESTIGATED (2026-06-22): direct answer to 'which closes ALL problems and is 100% provable' -- NONE individually; they CONVERGE on the same two irreducible residuals as v335 (SEAM.EQUIV.01 + the decoupled QG contour), reached now from the cosmological side. [E] ROUTE A (finite causal diamond) REFRAMES QG.AMB to a FINITE thermal trace: in a horizon-bounded static patch the object is Tr e^{-beta H}, finite for a gapped H bounded below (de Sitter S_dS = 32 pi^4 e^{2 ainv} finite; 128 c3^4 = 1/(32 pi^4) exact v54/v190; gapped susceptibility chi = 729/665 v337) -- BUT c_conf(4) = -3/2 < 0 (v332) still needs the GHP contour rotation (v334), whose nonperturbative validity is [O]; so A gives QG.AMB well-posedness/finiteness [E] but NOT a full nonperturbative construction, and does NOT touch SEAM.EQUIV.01. [C] ROUTE B (Carlip-Cardy near-horizon CFT) gives a SECOND route to existence + central charge: Carlip's near-horizon Virasoro + Cardy S = 2 pi sqrt(c L0/6) reproduces the entropy consistent with c = g_car+N_fam = 8, but L0 is free (the Carlip periodicity ambiguity) so c=8 is consistent not forced, and Carlip yields the central charge ONLY, not the holomorphy det K=1 discriminator -- inherits SEAM.EQUIV.01. [E] ROUTE C (modular/thermal flow) is a REDUCTION: the modular flow exists & is KMS at beta=1 (Tomita-Takesaki v239) so 'dynamics = modular evolution of the horizon state' is well-defined abstractly, but its GEOMETRIC identification (boost = physical time) is QGEO.SYM.01 = a corollary of SEAM.EQUIV.01 (v335). [E]/[P] ROUTE D (self-reproduction attractor) gives parameter-freeness: a gapped transport has a unique Perron-Frobenius attractor (v56), so the cyclic map has a unique fixed point [E], but 'the full dynamics IS that fixed point' is the [P] cyclic interpretation -- not a construction. [E] VERDICT: independent closures = 0; irreducible residuals after all four routes = 2 (SEAM.EQUIV.01 + the decoupled QG contour, unchanged from v335); the BH premise makes BOTH more natural (A finite thermal trace; B near-horizon CFT) but ELIMINATES neither. The value: the four routes are not four separate gambles -- they converge on the ONE keystone, so effort should focus on SEAM.EQUIV.01.",
      "status": "Analysis/roadmap [E]/[C]/[O] (none closes all or is 100% provable; all four converge on SEAM.EQUIV.01 + the decoupled QG contour)",
      "canonicalStatus": "[E] the finiteness facts (A), the modular/attractor theorems (C/D), the convergence count; [C] the Cardy consistency (B); [O]/[P] the inherited residuals. An analysis module; closes nothing, fabricates nothing -- it shows all four BH-birth routes converge on the same single keystone + the decoupled QG contour (= v335).",
      "dependencies": "SEAM.EQUIV.UNIFY.01; QG.AMB.01; v54; v190; v332; v334; v337; v239; v56",
      "scripts": [
        "v343_four_routes_analysis.py"
      ],
      "script": "v343_four_routes_analysis.py",
      "location": "origin_theory",
      "externalData": "Carlip (near-horizon Virasoro); Gibbons-Hawking (de Sitter entropy); Tomita-Takesaki",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 111
    },
    {
      "id": "EM.WARD.02",
      "claim": "HEAT-KERNEL SHARPENING OF EM.WARD.01 (2026-06-22): derives the ORIGIN and STRUCTURE of the F_U(1) determinant-line terms from textbook Seeley-DeWitt/Gilkey heat-kernel coefficients, advancing the EM-Ward origin from [P] (conjectured) to [E] for the term STRUCTURE; honestly fences the rest. NO new gate (sharpens EM.WARD.01). [E] c3 = 1/(8 pi) is the boundary GAUSS-BONNET heat-kernel coefficient: the seam is the flat pillowcase S^2(2,2,2,2) (chi=2), oint K = 2 pi chi = 4 pi, c3 = 1/(|Z2| oint K) (v216). [E] the alpha^2 term is the GILKEY gauge-curvature coefficient: the U(1) Laplacian a4 contains 30 Omega^2/360 = (1/12) Omega^2 (Gilkey, Invariance Theory Thm 4.8.16), Omega = i alpha F => -(alpha^2/12) F^2, so an alpha^2 term in log det Delta_U(1) is FORCED -- the Calderon -2 c3^3 a^2 is the gauge-curvature (Quillen-curvature) piece, not an ansatz. [E] the c3-ladder {0,3,6} is arithmetic (step 3 = three channels; 6 = |R^+(A3)| C6 hexagon). [E] multiplicities forced: 2 = |Z2| (sheet), 8 b1 = (4/5) M = (rank E8) b1 (b1=41/10, v48/v159). [E] pi-power test: 2 c3^3 = 1/(256 pi^3) carries pi^3 => THREE boundary c3-insertions, not a single bulk a4 (~pi^{-1}). [C] the exact coefficient 2 c3^3 needs the seam F-normalisation. [O] the cubic alpha^3 Maxwell moment (a topological/Chern level, NOT a Seeley-DeWitt integral) + the full identification (this IS the seam U(1) zeta-determinant) = EM.WARD.01's residual, tied to SEAM.EQUIV.01. NOT closed. alpha^-1 stays [E] (v3).",
      "status": "Heat-kernel origin [E]/[C]/[O] (term structure [E]; exact value [C]; cubic moment + identification [O]; no new gate)",
      "canonicalStatus": "[E] the heat-kernel STRUCTURE (c3 = Gauss-Bonnet boundary coefficient; alpha^2 = Gilkey gauge-curvature a4 term; c3-ladder; multiplicities; pi-power); [C] the exact coefficient (seam F-normalisation); [O] the cubic Maxwell moment + the full determinant identification (= EM.WARD.01, tied to SEAM.EQUIV.01). Sharpens EM.WARD.01's origin from [P] toward [E]; no new gate.",
      "dependencies": "EM.WARD.01; v48; v216; v3; v159",
      "scripts": [
        "v342_em_ward_heatkernel.py"
      ],
      "script": "v342_em_ward_heatkernel.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "Gilkey, Invariance Theory and the Heat Equation, Thm 4.8.16",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 112
    },
    {
      "id": "FLAV.MASSVOL.01",
      "claim": "Grand Mass Volume = the absolute mass-scaling normalisation [I] (NOT a [P] product rule): the individual c_q are a gauge, but the sector mass-matrix determinants are clean phi0-powers = K-row sums: det M_up~phi0^(4+2+0)=phi0^6=phi0^|R+(A3)|, det M_down~phi0^9=phi0^N_fam^2, det M_lep~phi0^10=phi0^A_Lambda; Grand Mass Volume det M_SM~phi0^(6+9+10)=phi0^25=phi0^g_car^2=phi0^Delta_Y. Q-row sums (4,5,6)=(|mu4|,g_car,|R+A3|) sum 15=dim A3; 25+15=40=sum L=|R(D5)| (mass+shift=transport budget). So the absolute normalisation is an [I] determinant identity; only the overall scale v_geo is an input",
      "status": "Identity [I]",
      "canonicalStatus": "Identity [I]",
      "dependencies": "FLAV.KQ.01; FLAV.QRATIO.01",
      "scripts": [
        "v46_grand_mass_volume.py"
      ],
      "script": "v46_grand_mass_volume.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 113
    },
    {
      "id": "GATE.UWALL.18",
      "claim": "the '11' is PASCAL (third, simplest origin; no special math): Pl(K) is the exterior algebra of the family fundamental 4=mu4 (A3=SU(4) fundamental). Lambda^k(4)=C(4,k)=Pascal row 4=(1,4,6,4,1), full sum 2^4=16=dim S+. |Pl(K)|=(1,4,6)={Lambda^0,Lambda^1,Lambda^2}(4) (distinct exterior powers). ||Pl(K)||_1=11=sum_{k<=2}C(4,k)=cumulative ext of mu4 up to degree 2=deg(u^c) (v44) = 2^4-C(4,3)-C(4,4)=16-g_car. family(x)carrier=15=dim su(4)=Lambda^2(5)+Lambda^4(5)=10+5=N_fam*g_car (three readings). c_u/c_d=g_car*(sum_{k<=2}C(4,k))/(N_fam^2 Delta_Q)=55/117. So 55/117 triply grounded in Pascal/exterior algebra [I]; physical normalisation stays [P]",
      "status": "Identity/Lattice [I]/[L]",
      "canonicalStatus": "Identity/Lattice [I]/[L]",
      "dependencies": "GATE.UWALL.17; v37; v42; v44",
      "scripts": [
        "v45_family_exterior.py"
      ],
      "script": "v45_family_exterior.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 114
    },
    {
      "id": "GATE.UWALL.17",
      "claim": "LIE-LEVEL GROUNDING of the exterior leg (no special math): the carrier half-spinor is the even exterior algebra 16=Lambda^even(5)=C(5,0)+C(5,2)+C(5,4)=1+10+5 (already in CAR.SM.01). Under SU(5)<SO(10): u^c in 10=Lambda^2(5), d^c in 5bar=Lambda^4(5) => exterior degrees deg(u)=2, deg(d)=4; deg(u)+deg(d)=6=|R+(A3)| (hexagon), deg(d)-deg(u)=2=|Z2| (sheet); up Yukawa family-symmetric Sym^2 dim6, down general dim9=N_fam^2. So the 'Lambda^2 exterior leg' is the carrier's OWN Lambda^2 grading (up quark lives in Lambda^2(5)), NOT exotic Hodge data => discrete invariant re-typed to carrier exterior DEGREE [L]. HONEST: grounds the TYPE; Plucker number 55/117 stays family Pl(K) [I]; carrier-L2<->family-L2F link (family x carrier=15) + normalisation remain [P]",
      "status": "Lattice/Lie [L] (typing) + [P] (normalisation)",
      "canonicalStatus": "Lattice/Lie [L] (typing) + [P] (normalisation)",
      "dependencies": "GATE.UWALL.15; CAR.SM.01; SU(5)<SO(10)",
      "scripts": [
        "v44_carrier_exterior.py"
      ],
      "script": "v44_carrier_exterior.py",
      "location": "tfpt_research_contracts;tfpt_2",
      "externalData": "standard GUT rep theory",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 115
    },
    {
      "id": "GATE.UWALL.16",
      "claim": "Lambda^2 F bridge test (honest): (1) POSITIVE - for SU(3) the exterior square is the conjugate rep, Lambda^2(M~)=conj(M~) exactly => the exterior leg IS the 3-bar (confirms the Exterior Leg typing); (2) anchor bivector 1^a=(0,1,1), Pl(K)=(-1,6,4) is its image under the 2nd compound of K^T (integer, combinatorial); (3) HONEST NEGATIVE - continuous Lambda^2(M~).(1^a) (moduli ~0.93,0.75,0.75) != integer Pl(K) nor its direction. CONCLUSION: the bridge rho*->Pl(K) is the DISCRETE non-abelian-Hodge invariant (integer R; det R=8 & Spec(Q+)={1,2,3} confirmed v39), NOT the continuous exterior-square action = exactly the H2 equivalence, pinned from the Lambda^2 F side. No c_u/c_d fabricated",
      "status": "Numerical (typing confirmed + honest negative)",
      "canonicalStatus": "Numerical (typing confirmed + honest negative)",
      "dependencies": "GATE.UWALL.15; v39; v40",
      "scripts": [
        "v43_exterior_bridge.py"
      ],
      "script": "v43_exterior_bridge.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 116
    },
    {
      "id": "FLAV.DELTA.01",
      "claim": "delta=1/2 glue-norm origin + (g_car,N_fam)/|mu4| unification: the two E8-glue discriminant norms ARE the carrier rank and family count over the glue index, q(D5)=g_car/|mu4|=5/4 (rank D5=g_car=5), q(A3)=N_fam/|mu4|=3/4 (rank A3=N_fam=3, n+1=4=|mu4|). Then all FOUR arithmetic ops are forced compiler atoms: sum=(g+n)/|mu4|=rank(E8)/|mu4|=2 (E8 root norm), diff=(g-n)/|mu4|=|Z2|/|mu4|=1/2=delta (lepton transport, used in v20 for c_e=16/7,c_mu=4/3,c_tau=7/6), prod=g*n/|mu4|^2=dim su(4)/dim S+=15/16, ratio=g/n=5/3. (sum,ratio) invert uniquely to (5/4,3/4). So delta=1/2 is the carrier-minus-family count over the glue index, not a chosen rational; it also equals the harmonic-frame holonomy diagonal modulus 1/2 (v41). Hardens v20's chosen delta into a fully atomic readout [I]",
      "status": "Identity [I]",
      "canonicalStatus": "Identity [I]",
      "dependencies": "BND.GLUE.01; v20; v41",
      "scripts": [
        "v51_boundary_half_step.py"
      ],
      "script": "v51_boundary_half_step.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 117
    },
    {
      "id": "FLAV.KQ.02",
      "claim": "K+xQ pencil endpoints (audit ext. of v37): P(x)=det(K+xQ)=3x^3+7x^2+6x+4 anchored at four integer nodes by compiler atoms: P(-1)=2=|Z2| (K-Q sheet endpoint), P(0)=4=det K=|mu4|, P(1)=20=det L (transport), P(2)=68=2 p5(a) (twice anchor power sum p5=34). Also det(K-Q)=2=|Z2|, trace(K-Q)=3=N_fam. Audit-level detail of the mass->transport pencil, not a new gate [I]",
      "status": "Identity [I]",
      "canonicalStatus": "Identity [I]",
      "dependencies": "FLAV.KQ.01; v37",
      "scripts": [
        "v52_pencil_endpoints.py"
      ],
      "script": "v52_pencil_endpoints.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 118
    },
    {
      "id": "GATE.QGEOM.02",
      "claim": "Integer-lift of Q - residual sharpened to ONE named lattice invariant det Q=N_fam (honest reduction, not closure) [I]/[L]. Continuing v69: (1) DERIVED - in the mu4-puncture homology basis {g1,g2,g3} (g4=-(g1+g2+g3)) the Z4=mu4 rotation is the integer matrix R=[[0,0,-1],[1,0,-1],[0,1,-1]], char (t+1)(t^2+1), eigenvalues {-1,i,-i}, R^4=I, R UNIMODULAR (det=-1, SNF=I). (2) the documented transport Q (rows (4,5,6)=(|mu4|,g_car,|R+A3|), cols (9,5,1)=(N_fam^2,g_car,1)) has det Q=3=N_fam, SNF=diag(1,1,3) - the transport DETERMINANT equals the family count. (3) so Q is NOT a unimodular change of basis of R (det -1 vs det 3): the integer lift is precisely the parabolic-degree-0 embedding carrying the extra det=N_fam (family multiplicity) that D4 alone (unimodular) does not supply. RESIDUAL of gate Q-geometry is thus one named invariant det Q=N_fam; spectra+Sigma-split already derived (v69). EHRLICH: D4 alone does NOT close the integer lift - it needs this one lattice datum",
      "status": "Identity/Lie [I]/[L]",
      "canonicalStatus": "Identity/Lie [I]/[L]",
      "dependencies": "GATE.QGEOM.01; v69; v50",
      "scripts": [
        "v70_q_integer_lift.py"
      ],
      "script": "v70_q_integer_lift.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 119
    },
    {
      "id": "GATE.QGEOM.01",
      "claim": "D4-equivariant derivation of the Q-geometry (Alessandro residual gate #2) - advances Theorem Q from [P] (algebraically forced) to [L] (geometrically derived). The 4 punctures mu4={1,i,-1,-i} are square vertices with symmetry D4=Z4 semidirect Z2 (the 90-deg rotation z->iz IS mu4; the reflection is the sheet parity Sigma). (1) the Z4 4-cycle on the punctures acts on H_1(P^1\\\\mu4)=Z^4/(sum=0)=Z^3 (3 families) with eigenvalues {i,-1,-i} (non-trivial 4th roots); (2) the D4 permutation rep on square vertices = A1+B1+E (1+1+2), so H_1(sum=0)=B1(+)E; (3) Q_+ (Sigma-even) = 3*(parabolic cusp weights {0,1/3,2/3})+1 = diag(1,2,3) => char (t-1)(t-2)(t-3), Spec{1,2,3} (matches v50); (4) Q_- (Sigma-odd) = E-block coupling sqrt(3)=sqrt(N_fam) with B1 in kernel => char t(t^2-3), Q_-^2 eigenvalues {0,3,3} (matches v50); (5) the Sigma-split (Q_+/Q_-) IS the D4=Z4(mu4)semidirect Z2(reflection=Sigma) structure. So the Q SPECTRA and the Sigma-split are DERIVED from the D4-equivariant geometry of P^1\\\\mu4 [L]; residual: the cusp weights {0,1/3,2/3} are the established monodromy exponents (v61), and the explicit integer matrix Q still needs the lattice/SNF datum",
      "status": "Lie/lattice [L]",
      "canonicalStatus": "Lie/lattice [L]",
      "dependencies": "QGEOM (was GATE); v50; v61; Theorem Q",
      "scripts": [
        "v69_d4_q_geometry.py"
      ],
      "script": "v69_d4_q_geometry.py",
      "location": "tfpt_2",
      "externalData": "D4 representation theory; standard",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 120
    },
    {
      "id": "SEAM.AREACOEFF.02",
      "claim": "Central-theorem RESIDUAL RESOLVED (honestly) as an anchor, not a gap [I-symbolic + L]. The Seeley-DeWitt coefficient for the carrier Dirac operator is a2 = (1/(4pi)^2)(1/6)(R d + 6E d) with E=-R/4 (Lichnerowicz) = -d/(192 pi^2) R, d=carrier dof (16=dim S+ or 32=2^g_car). The induced Einstein-Hilbert coefficient is 1/(16 pi G)=f2 Lambda^2 d/(192 pi^2) - UV-SENSITIVE (depends on the seam cutoff Lambda and the cutoff moment f2). So 1/G is NOT a pure number (Sakharov 1967 / Connes spectral-action induced gravity); the residual k=c3/2 (v67) is the seam normalization P1 carried through the heat kernel, NOT a free-standing derivation, and the physical identification 'seam scale=observed Planck scale' is the ONE irreducible dimensionful anchor (v_geo; no metre from pure math, units argument). What IS cutoff-independent and derived: the R^2 coefficient (Starobinsky, v36/G2 a4|R2=R2/72), the scalaron mass M_scal=c3^(7/2) Mbar~3.06e13 GeV, v_GW=c (v8). CONCLUSION: the central theorem is 'maximally closed' - structure proven (v67), predictive gravitational content derived, residual = the single anchor every theory carries, not a missing computation",
      "status": "Identity/Lie [I]/[L] + anchor identification",
      "canonicalStatus": "Identity/Lie [I]/[L] + anchor identification",
      "dependencies": "SEAM.AREACOEFF.01; v36; v67",
      "scripts": [
        "v68_seeley_dewitt_residual.py"
      ],
      "script": "v68_seeley_dewitt_residual.py",
      "location": "origin_theory",
      "externalData": "Sakharov 1967; Susskind-Uglum 1994; Chamseddine-Connes spectral action",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 121
    },
    {
      "id": "SEAM.AREACOEFF.01",
      "claim": "Central theorem (1/(8pi) area-law coefficient) - STRUCTURAL CLOSURE + reduction to ONE sharp coefficient (honest partial solution of Alessandro's central target). CLOSED [I]/[L]: by the Fursaev-Solodukhin (1995) conical/replica method, the curvature term W_R=k*Int(sqrt(g)R) gives horizon entropy S=4pi k A (conical defect: Int R ⊃ 4pi(1-n)A; S=-d/dn[W_n-n W_1]|_1). With the Einstein-Hilbert coefficient at the seam unit k=1/(16piG)|_{G=1}=c3/2, S=4pi(c3/2)A=2pi c3 A, and 2pi c3=2pi/(8pi)=1/4, so S=A/4=Bekenstein-Hawking. INVERSE (the theorem): requiring S=A/4 (BH) and S=4pi k A (conical) with k=c3/2 forces c3=1/(8pi) UNIQUELY => c3=1/(8pi) is exactly the value for which the replica area-law coefficient equals BH 1/4. RESIDUAL [A]: the SINGLE statement k=c3/2, i.e. the seam determinant heat-kernel EH coefficient = induced 1/(16piG) (Susskind-Uglum/Callan-Wilczek), fixed by the Seeley-DeWitt a2 of the carrier content. This reduces 'derive 1/(8pi)' to one sharp spectral computation; the structure (replica=>4pi k A=>A/4 at c3=1/(8pi)) is closed",
      "status": "Identity/Lie [I]/[L] (structure) + [A] (residual k=c3/2)",
      "canonicalStatus": "Identity/Lie [I]/[L] (structure) + [A] (residual k=c3/2)",
      "dependencies": "SEAM.THEOREM.01; SEAM.AREALAW.01; v36; v59",
      "scripts": [
        "v67_area_law_coefficient.py"
      ],
      "script": "v67_area_law_coefficient.py",
      "location": "origin_theory",
      "externalData": "Fursaev-Solodukhin 1995; Susskind-Uglum 1994; Callan-Wilczek 1994",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 122
    },
    {
      "id": "ARCH.E8DEG.01",
      "claim": "Organizing simplification: the compiler atoms ARE the E8 fundamental Casimir invariant degrees deg(E8)={2,8,12,14,18,20,24,30} (exponents+1) [L]. CLEAN/non-trivial readings: 2=|Z2|, 8=rank E8, 14=dim G2 (the [K,Q] commutator G2, v37), 20=det L (v10), 24=|W(A3)|=4! (Weyl group A3=S4), 30=h(E8)=2*3*5. Weaker/fittable (flagged): 12=|R(A3)| (A3 roots), 18=p4(a)=2+2^4 (anchor power sum). EXACT sum identities: sum(degrees)=128=2^7 = the de Sitter entropy prefactor (S_dS=e^(2 ainv)/(128 c3^4); 7=scalaron exponent g+n-1) [NEW clean]; sum(exponents)=120=|R+(E8)| (positive roots, v56); prod(degrees)=|W(E8)|=696729600. Since E8 is the audit hull, its invariant degrees being the theory's integers is structurally EXPECTED - confirms the E8 choice, not a coincidence; 6/8 clean, 2/8 fittable (honest)",
      "status": "Lie/lattice [L] + identity [I]",
      "canonicalStatus": "Lie/lattice [L] + identity [I]",
      "dependencies": "v37; v10; v55; v56; h(E8)=30",
      "scripts": [
        "v66_e8_casimir_degrees.py"
      ],
      "script": "v66_e8_casimir_degrees.py",
      "location": "tfpt_3",
      "externalData": "E8 invariant degrees (standard Lie theory)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 123
    },
    {
      "id": "PRED.LAYER.01",
      "claim": "Prediction/falsification layer with EXPLICIT failure thresholds (Alessandro residual gate #6) [N], data-confrontation Python-only. Each prediction carries an explicit falsification threshold and a layer tag. CORE falsifiers (break [I]/[L]): v_GW=c (FAIL if robust v_GW!=c >5sigma; GW170817 gives |v_GW/c-1|<1e-15), no 4th chiral generation (FAIL on discovery; N_fam=3=rank A3=dim H1), seam denominator=8 (FAIL if c3!=1/(8pi)). OBSERVATIONAL stress tests: beta_rad=0.2424 deg (FAIL if controlled beta excludes 0.2424 at >3sigma; ACT DR6 0.215+-0.074 ok), r~0.004 (FAIL if r>0.01 robustly; CMB-S4/LiteBIRD), n_s~0.965 (FAIL if robustly >0.975; ~2sigma DESI tension now), sin^2 th12=0.30675 (FAIL if future >3sigma; NuFIT 0.307+-0.012 ok). CORE no-go: no traversable shortcut/CTC (FAIL only on macroscopic ANEC violation = RP/unitarity break, v64). CYCLIC [P]: horizon info recovery (FAIL on demonstrable info loss). None currently violated. N_star explicitly demoted to physical reheating input [P], range [50,60], affects only n_s/r, not the core",
      "status": "Numerical/data [N]",
      "canonicalStatus": "Numerical/data [N]",
      "dependencies": "v62; v64; v3; v8; v60",
      "scripts": [
        "v65_falsification_layer.py"
      ],
      "script": "v65_falsification_layer.py",
      "location": "origin_theory",
      "externalData": "GW170817; ACT DR6; NuFIT 6.0; Planck-ACT-DESI",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 124
    },
    {
      "id": "CAUSAL.NOGO.01",
      "claim": "Causal Boundary Engineering - CONDITIONAL NO-GO for traversable seam shortcuts/CTCs (proof attempt that turned into a no-go). CORE [I] (machine-checked, v56-bound): a gapped positive transfer operator (spectrum {1,(2/3)^6,(1/3)^6}) has NO periodic orbit (T^n != I for all n>0; subdominant moduli <1 => T^n -> rank-1 projector) = discrete analog of no closed timelike curve. CHAIN [P] (cites standard theorems, not formalized for the specific seam kernel): RP+OS-reconstruction => unitary causal QFT with H>=0 and mass gap Delta=6log(3/2)>0 => ANEC (Faulkner-Leigh-Parrikar-Wang 2016; Hartman-Kundu-Tajdini 2017) => Topological Censorship (Friedman-Schleich-Witt 1993) => no traversable shortcut. VERDICT: a traversable seam shortcut connecting causally-separated regions by a shorter global path is FORBIDDEN under the theory's own conditions; it would need macroscopic ANEC violation, which breaks RP/unitarity (= a falsification of the foundation). Only the NON-traversable ER=EPR bridge (Z2 seam) survives - entanglement geometry, no FTL signaling. Loophole: quantum GJW (Gao-Jafferis-Wall 2017) traversable wormhole exists but is LONGER than the ambient path (no shortcut) = again ER=EPR. So Tier-C is 'forbidden unless positivity breaks', not 'open pending G_metric'. (Python-only; the no-CTC core is numerical)",
      "status": "Identity [I] core + [P] argument",
      "canonicalStatus": "Identity [I] core + [P] argument",
      "dependencies": "GATE.GMETRIC.XI.01; v56; SEAM.CYCLE.01",
      "scripts": [
        "v64_causal_boundary_nogo.py"
      ],
      "script": "v64_causal_boundary_nogo.py",
      "location": "origin_theory",
      "externalData": "Friedman-Schleich-Witt 1993; Gao-Jafferis-Wall 2017; Faulkner et al. 2016",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 125
    },
    {
      "id": "GATE.GMETRIC.XI.01",
      "claim": "Seam-Engineering Index - an exact closed form for the IR-stability margin [I], plus the four-class possibility taxonomy [I]/[A]. Validates+sharpens the boundary-engineering synthesis: the documented G_metric bound ||V_metric||_rel <= 248 c3^2 = 31/(8 pi^2) (248=dim E8, 31=248/8=1+h^v(E8) = the WZW denominator c(E8)=248/31, v61) gives 2||V||=31/(4 pi^2)=0.78524 (NOT pi/4=0.78540, a 4th-digit coincidence). Hence Xi=2||V||/Delta=31/(24 pi^2 log(3/2))~0.3228 has an EXACT closed form bound to E8 data; Delta_eff=Delta-2||V||=6log(3/2)-31/(4pi^2)~1.648>0. Xi<1 = gap-dominance/IR-stability condition. HONEST: Xi is a stability DIAGNOSTIC (theory parameter), NOT a lab control knob. Four-class taxonomy of 'breakthrough' ideas: A allowed&near [N] (birefringence, BH-echo (2/3)^6, v_GW=c, axion), B reconstruction [A] (holographic larger state space, gated on Seam-Horizon area-law), C post-G_metric [A] (topological seam shortcuts, gated on ambient metric measure), D forbidden (local FTL, photon dispersion, v_GW!=c, CTCs, free energy - break core claims). Next theorem: Causal Boundary Engineering (RP+OS+gap => no CTCs; classify boundary gluings; shortcut != FTL)",
      "status": "Identity [I] + taxonomy [A]",
      "canonicalStatus": "Identity [I] + taxonomy [A]",
      "dependencies": "GATE.GMETRIC.*; v36; v54; v61",
      "scripts": [
        "v63_seam_engineering_index.py"
      ],
      "script": "v63_seam_engineering_index.py",
      "location": "origin_theory;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 126
    },
    {
      "id": "DATA.SCORECARD.01",
      "claim": "Confrontation of TFPT predictions with current research data (2024/25) - honest scorecard incl. tensions [N], data-confrontation (Python-only, external data). STRONG MATCHES: alpha^-1=137.0359992 vs CODATA 137.035999177 (~9 sig figs); sin^2 th12=0.30675 vs NuFIT 6.0 (2024) 0.307+-0.012 (0.02 sigma); beta_rad=0.2424 deg vs ACT DR6 (Diego-Palazuelos-Komatsu 2025) 0.215+-0.074 (0.37 sigma). MILD ~2 sigma TENSIONS: sin^2 th13=0.0231 vs NuFIT 0.02195+-0.00058 (1.98 sigma high); R+R^2 Starobinsky n_s=0.960-0.967 consistent with Planck 0.9649 but ~2 sigma below the DESI-combined P-ACT-LB 0.9743+-0.0034 (DESI-driven shift under discussion; CMB-only favors Starobinsky range). OPEN: theta23 octant ([P], NuFIT ambiguous 0.470/0.561). FUTURE FALSIFIER: tensor-to-scalar r~0.004 (Starobinsky), well below bound ~0.03, testable by CMB-S4/LiteBIRD. Honest verdict: impressive matches AND real current ~2 sigma tests - a falsifiable theory, not self-confirming",
      "status": "Numerical/data [N]",
      "canonicalStatus": "Numerical/data [N]",
      "dependencies": "v3; v16; v21; v8; v60",
      "scripts": [
        "v62_data_scorecard.py"
      ],
      "script": "v62_data_scorecard.py",
      "location": "origin_theory",
      "externalData": "CODATA 2022; NuFIT 6.0 (2410.05380); ACT DR6 (2509.13654); Planck/ACT-DESI n_s",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 127
    },
    {
      "id": "ARCH.CFT.01",
      "claim": "Boundary-CFT bridge (WZW central charges = compiler atoms) + SU(3)<->SU(4) reconciliation [I]/[L]; NOT a closure. (A) level-1 WZW central charges c=dim(g)/(1+h^v) are the atoms: c(E8)_1=248/31=8=rank E8, c(D5=SO10)_1=45/9=5=g_car, c(A3=SU4)_1=15/5=3=N_fam; coset c=8-5-3=0 = conformal-embedding criterion => (D5)_1x(A3)_1 ⊂ (E8)_1 is a conformal embedding (CFT mirror of the lattice glue (D5(+)A3)+mu4->E8); (5,3) split = central-charge additivity = rank split; the 8 in c3=1/(8pi) = c(E8)_1 (NOT oversold - the content is c_coset=0, per Alessandro's earlier caveat). (B) SU(3)<->SU(4) reconciliation: P^1\\\\mu4 has n=4 punctures => H_1=Z^(n-1) => N_fam=|mu4|-1=3; carrier side = puncture/center Z4=mu4 (A3=SU4); flavor side = monodromy rho on H_1=Z^3, SU(3)-valued, center Z3=triality, phases {0,1/3,2/3}=parabolic weights (NOT SU(4)_1 weights {0,3/8,1/2}); SU(3) ⊂ SU(4) via 4->3+1. => NOT one WZW for all gates (C falsified), but ONE geometry P^1\\\\mu4 with two dual sides. CHECKED: Cardy route c=8->S=A/4 is NOT a shortcut (c=8 is matter central charge; area law needs geometric Brown-Henneaux c~ell/G) - restates the open Seam-Horizon coefficient, does not close it",
      "status": "Identity/Lie [I]/[L]",
      "canonicalStatus": "Identity/Lie [I]/[L]",
      "dependencies": "BND.GLUE.01; v6; v15; v19; SEAM.THEOREM.01",
      "scripts": [
        "v61_cft_bridge.py"
      ],
      "script": "v61_cft_bridge.py",
      "location": "origin_theory",
      "externalData": "WZW/conformal embedding standard CFT; Strominger 1997",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 128
    },
    {
      "id": "LAMBDA.BRANCH.01",
      "claim": "Lambda-metrology branch selection => G_N as horizon-metrology OUTPUT, the 123-orders double overlap, and the ACT DR6 birefringence comparison. (1) The form rho_Lambda/Mbar^4=(3/(4pi^2))e^(-2 alpha^-1) is in Paper 1; NEW: with delta_top=48 c3^4=3/(256 pi^4) the physical branch prefactor is (8pi)^2 delta_top=3/(4pi^2) [I], and the alternative branch 2c3 e^(-2a) is mis-scaled by 2c3/delta_top=1/(24 c3^3)=64 pi^3/3=661.467 - so selecting the physical branch PINS Mbar (hence G_N): Newton constant is an output, not a free input [I]+[P consequence]. (2) 123-orders split [I]/[N]: |log10(rho_Lambda/M_Pl^4)|=122.948=119.028(=2 alpha^-1/ln10, EM fixed point)+3.920(=log10(256 pi^4/3), seam-defect prefactor in M_Pl units) - double overlap of one boundary compiler, not fine-tuning. (3) ACT DR6 birefringence [N]: TFPT beta_rad=phi0/(4pi)=0.2424 deg (same seed as Cabibbo); ACT DR6 (Diego-Palazuelos & Komatsu 2025, arXiv:2509.13654, PRD 113 L101302) beta=0.215+-0.074 deg (2.9 sigma) => TFPT ~0.37 sigma from central value (consistent, falsifiable). Plus action exponents (1,5,10)=K5 graph counts (5 vertices, 10 edges)=Pascal 16=1+5+10",
      "status": "Identity [I] + observational [N] + [P] consequence",
      "canonicalStatus": "Identity [I] + observational [N] + [P] consequence",
      "dependencies": "EM.FP.01; v3; v8; v55",
      "scripts": [
        "v60_lambda_metrology_branch.py"
      ],
      "script": "v60_lambda_metrology_branch.py",
      "location": "origin_theory;tfpt_1",
      "externalData": "ACT DR6 Diego-Palazuelos-Komatsu 2025 arXiv:2509.13654",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 129
    },
    {
      "id": "SEAM.AREALAW.01",
      "claim": "Area-law evidence for the open Seam-Horizon step (necessary condition met, NOT a closure): (1) NUMERICAL [Python-only] - the simplest reflection-positive Gaussian system (harmonic chain) has ground-state block entanglement entropy that SATURATES to a constant when gapped (area law: 1D boundary=O(1) points) and grows ~(c/3)log L when gapless (CFT c-coefficient, c~1); demonstrated S(8)=S(16)=S(32)=0.271 gapped vs 2.197<2.409<2.602 gapless (Srednicki 1993, Bombelli-Sorkin). So an RP Calderon-type kernel GENERICALLY yields an area law - the form S=A/(4G_Sigma) needs. (2) GAP TIE: the physical sector is gapped by the SAME gap Delta=6log(3/2)>0 that makes the attractor unique (v56) => gapped => area law; one gap gives BOTH attractor and area-law structure. (3) EXACT [I]: 8=|Z2|*|mu4| (sheet x glue), c3=1/(|Z2||mu4|pi); 2pi=1/(4c3) (chaos bound/seam unit); scrambling beta/2pi=|mu4|M. HONEST: fixes the area-law FORM; the c3=1/(8pi) COEFFICIENT still needs the specific seam kernel (open [A]). Cascade D=60-2n monotone c-function-like, 60=2h(E8) [P/analogy]",
      "status": "Numerical + Identity [I] (necessary condition)",
      "canonicalStatus": "Numerical + Identity [I] (necessary condition)",
      "dependencies": "SEAM.THEOREM.01; v56; v58",
      "scripts": [
        "v59_area_law_evidence.py"
      ],
      "script": "v59_area_law_evidence.py",
      "location": "origin_theory",
      "externalData": "Srednicki 1993; Bombelli-Sorkin 1986",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 130
    },
    {
      "id": "SEAM.THEOREM.01",
      "claim": "Seam-Horizon chain (precise framing) + the OPEN Seam-Horizon Theorem. PRECISE: the seam is NOT identical to an event horizon - it is the abstract boundary normaliser whose LOCAL gravitational realisation is a horizon (black hole = local instance of seam grammar; de Sitter = global). EXACT arithmetic [I]: (geom origin) one-sided S^2 Gauss-Bonnet oint K dA=2pi chi(S^2)=4pi, Z2/Mobius => c3=1/(|Z2|*4pi)=1/(2*4pi)=1/(8pi), so 8pi=|Z2|*2pi chi(S^2) (the seam '8' is one-sided S^2 curvature, matching lattice rank E8 and gravitational 8pi); KMS T=kappa/(2pi)=4c3 kappa; Schwarzschild T_H=c3/M, S_BH=A/4=M^2/(2c3) (1/(2c3)=4pi); de Sitter T_dS=4c3 H; RT 1/(4G)=2pi Mbar^2=Mbar^2/(4c3) (1/(4c3)=2pi). OPEN [A] Seam-Horizon Theorem (next hard target): given a reflection-positive Calderon seam kernel on the doubled collar, replica variation of the seam determinant should give area entropy S=A/(4 G_Sigma); then Jacobson closes Einstein 8pi G. Chain Mobius=>Z2[L]=>c3=1/8pi[I]=>KMS[I]=>[A: area-law from seam determinant]=>Einstein[P]. Three theses: weak [I] (c3 horizon factor) proven, medium [P] (BH=local seam thermality) plausible, strong [A] (seam=holographic horizon) open - needs ONE analytic area-law, not more numerics",
      "status": "Identity [I] + open theorem [A]",
      "canonicalStatus": "Identity [I] + open theorem [A]",
      "dependencies": "SEAM.HORIZON.01; SEAM.CROSSLINK.01; v8; v57",
      "scripts": [
        "v58_seam_horizon_chain.py"
      ],
      "script": "v58_seam_horizon_chain.py",
      "location": "origin_theory",
      "externalData": "Jacobson 1995; Ryu-Takayanagi 2006",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 131
    },
    {
      "id": "SEAM.CROSSLINK.01",
      "claim": "Horizon cross-links - published black-hole results meeting compiler atoms (exact arithmetic [I]; physical seam=horizon identification is [P]): (1) JACOBSON/EINSTEIN: c3=1/(8pi) is exactly the Einstein-eq coefficient G_uv=8pi T_uv and EH action 1/(16piG)=c3/(2G); Jacobson 1995 derives Einstein eq from dQ=T dS on local Rindler horizons with entropy density 1/(4G), entropy coeff 1/4=1/|mu4| => if seam=horizon, gravity is its thermodynamics and c3 the conversion constant. (2) HOD QNM: Schwarzschild asymptotic quasinormal omega_R->T_H ln(3) (Hod 1998, Motl 2003), imaginary spacing 2pi T_H => omega_R/T_H=ln(3)=ln(N_fam); ladder spacing=seam unit 1/(2pi)=4c3. HONEST: the 3 is spin-dependent => [P] cross-link not forced. (3) Hawking power P_H=c3/(1920 M^2), 1920=|W(D5)|=2^4*5! (carrier Weyl group). (4) one alpha^-1: H0/Mbar~e^(-ainv)/(2pi) keys off the SAME EM fixed point as S_dS, Lambda. (5) S_BH=A/(4G)=A/(|mu4| l_P^2): one boundary cell per |mu4|=4 Planck areas. Supports the seam=horizon picture; not a closure",
      "status": "Identity [I] + [P] cross-links",
      "canonicalStatus": "Identity [I] + [P] cross-links",
      "dependencies": "SEAM.HORIZON.01; v8; v3; v55",
      "scripts": [
        "v57_horizon_crosslinks.py"
      ],
      "script": "v57_horizon_crosslinks.py",
      "location": "origin_theory;tfpt_horizon_readouts",
      "externalData": "Jacobson 1995; Hod 1998; Motl 2003",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 132
    },
    {
      "id": "SEAM.ATTRACTOR.01",
      "claim": "Unique attractor: a gapped boundary transport selects ONE fixed point [I]/[L]. The boundary transport spectrum {1,(2/3)^6,(1/3)^6} has spectral gap Delta=-log(2/3)^6=6 log(3/2)=2.4328>0. By Perron-Frobenius a primitive gapped operator has a UNIQUE dominant eigenvector and iteration converges to it geometrically at rate lambda2/lambda1=(2/3)^6, independent of initial state (demonstrated: two different starts -> same fixed direction, cos=1). This is the dynamical-systems content of 'parameter-free via fixed point' - the realized state is an ATTRACTOR, not a tuning; the convergence rate is the SAME (2/3)^6 of flavor+horizon (v54). Coxeter rotation lives in |mu4|=4 invariant planes (exponents pair m+(30-m)=30); sum exps=120=|R+(E8)|; #roots=rank*h=240. Entropy reset = the gap: T^n/lambda1^n -> rank-1 projector (only boundary law survives). Cyclic-cosmology reading is [P], not included",
      "status": "Identity/Lie [I]/[L]",
      "canonicalStatus": "Identity/Lie [I]/[L]",
      "dependencies": "SEAM.HORIZON.01; SEAM.CYCLE.01; v54; v55",
      "scripts": [
        "v56_unique_attractor.py"
      ],
      "script": "v56_unique_attractor.py",
      "location": "origin_theory;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 133
    },
    {
      "id": "SEAM.CYCLE.01",
      "claim": "A cyclic element INSIDE the compiler hull (computed, not asserted): built from the E8 Cartan matrix, the Coxeter element c=s1...s8 has order EXACTLY h(E8)=30=|Z2|*N_fam*g_car=2*3*5 (verified by matrix powers, v55), and its eigenvalues are primitive 30th roots exp(2pi i m/30) with m the E8 exponents {1,7,11,13,17,19,23,29} = EXACTLY the phi(30)=8 totatives of 30. Hence rank E8=8=phi(30)=#exponents=#live phases of the order-30 cycle => the 8 in c3=1/(8pi) is the live-phase count of a primitive order-30 cyclic rotation of the hull, period 30=|Z2|*N_fam*g_car. PLUS one-alpha cross-sector: S_dS*rho_Lambda=1/(128 c3^4)=32 pi^4 (de Sitter Gibbons-Hawking entropy ~ 1/Lambda), so the SAME EM fixed point alpha^-1=137.036 sets the cosmic horizon entropy and Lambda smallness ~10^-122=e^(-2 alpha^-1). FURTHER: S_dS=2^g_car*pi^|mu4|*e^(2 alpha^-1) - the entropy prefactor 32 pi^4 = (Dirac spinor dim of D5=SO(10): S+ + S- =16+16=2^g_car) x pi^|mu4|, with 128=2^7 (7=scalaron exponent g+n-1): cosmic horizon entropy = carrier Clifford dim x pi^glue x EM fixed point. [I]/[L]; cyclic-cosmology reading is [P], not included",
      "status": "Identity/Lie [I]/[L]",
      "canonicalStatus": "Identity/Lie [I]/[L]",
      "dependencies": "v6; v8; v3; h(E8)=30",
      "scripts": [
        "v55_coxeter_cycle.py"
      ],
      "script": "v55_coxeter_cycle.py",
      "location": "tfpt_3;origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 134
    },
    {
      "id": "SEAM.HORIZON.01",
      "claim": "Seam=horizon [I] keystones (pure identities, NO cosmological narrative): (1) the integer 8 in c3=1/(8pi) is OVERDETERMINED and gravitationally aligned: geometry 8=2|mu4| (8pi=|mu4|*2pi seam winding), lattice 8=rank E8=g+n=h(D5)=phi(30)=det R, gravity 8pi=Hawking/Einstein coeff (G_uv=8pi T_uv) => if seam=horizon, c3 is gravitationally forced and the gravitational 8 MUST equal the lattice/geometry 8 (it does). Both pi-coeffs are 2pi*integer: phi_tree=1/(6pi),6=2 N_fam (Gauss-Bonnet); c3=1/(8pi),8=2|mu4|; phi_tree=(|mu4|/N_fam)c3=(4/3)c3. (2) ONE transfer operator for BOTH sectors: boundary transport spec {1,(2/3)^6,(1/3)^6}; lambda2=(2/3)^6 = SM flavor gap eigenvalue (Delta=-log(2/3)^6=6 log 3/2=2.4328, tfpt_2) = horizon Page recovery rate (I_n~lambda2^n, Appendix H). Cusp weights {0,1/3,2/3} at 4 mu4-punctures = boundary data. Supporting: S_BH=A/|mu4|, 128 c3^4=1/(32 pi^4), T_dS coeff 16=|mu4|^2. The cyclic self-reproduction reading is [P] interpretation NOT included here",
      "status": "Identity [I]",
      "canonicalStatus": "Identity [I]",
      "dependencies": "BND.GLUE.01; v6; v8; v10; tfpt_2 transport spec",
      "scripts": [
        "v54_seam_horizon_keystones.py"
      ],
      "script": "v54_seam_horizon_keystones.py",
      "location": "tfpt_3;tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 135
    },
    {
      "id": "ARCH.CORE.01",
      "claim": "Compiler core - the whole integer skeleton from one pair (g_car,N_fam)=(5,3): rank E8=g+n=8 (D5(+)A3 rank split), |Z2|=g-n=2 (carrier-family), |mu4|=(g+n)/2=4; (N_fam,|mu4|,g_car)=(3,4,5) is THE Pythagorean triple. NOT decorative: grand-mass-volume exponent Delta_Y=g_car^2=25 (det M_SM~phi0^25) splits as a DIFFERENCE OF SQUARES Delta_Y=g^2=N_fam^2 + (g-n)(g+n)=N_fam^2+|Z2|*rank(E8)=9+16=dim S+, i.e. down sector=N_fam^2, up+lepton=|Z2|*rank E8=dim S+ (K-row sums (6,9,10)). ANCHOR is the Dirac square root: a=(1,1,2) is the UNIQUE 3-multiset (0..6) with elem-symm (|mu4|,g_car,|Z2|)=(4,5,2); char poly chi_a(t)=(t-1)^2(t-2)=t^3-|mu4|t^2+g_car t-|Z2| => compiler atoms ARE its char-poly coefficients. Only transcendental is pi (c3=1/(8pi), 8=rank E8; seed 1/(6pi), 6=|R+A3|). New READINGS/syntheses of v23/v37/v46/v47 atoms [I]",
      "status": "Identity [I]",
      "canonicalStatus": "Identity [I]",
      "dependencies": "FLAV.MASSVOL.01; v23; v47; BND.GLUE.01",
      "scripts": [
        "v53_compiler_core.py"
      ],
      "script": "v53_compiler_core.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 136
    },
    {
      "id": "FLAV.RIGID.02",
      "claim": "SIMPLE R-bridge closure (the much-simpler solution, no transcendental machinery): the selector stratum S_{8,Q}={det R=8, SNF(R)=(1,1,8), Spec(Q+)={1,2,3}} is now FULLY DERIVED - det R=8=rank E8 and SNF from the lattice (cf det Q=3=N_fam, v70), Spec(Q+)={1,2,3} from D4-equivariant geometry (v69). The four flavor operators are integer lattice maps with compiler-atom determinants (det Q,det K,det R,det L)=(3,4,8,20), product 1920=|W(D5)|. By Readout Rigidity (v49) the quark RATIOS are constant on this derived stratum => c_u/c_d=g_car*11/(Nfam^2*13)=55/117 etc are PURE INTEGER PLUCKER READOUTS, no monodromy solve. The earlier monodromy machinery (v19-v49) was over-engineering for the ratios; the only transcendental piece U_point (full uniqueness of rho*) is the ABSOLUTE amplitude normalisation = an anchor (same nature as the one dimensionful scale), not a missing computation. 13=DeltaQ reads 8+5=rank E8+g_car=N_fam^2+|mu4| (the one not-uniquely-atomic number, flagged) [I]+[A]",
      "status": "Identity [I] (ratios) + [A] (U_point=anchor)",
      "canonicalStatus": "Identity [I] (ratios) + [A] (U_point=anchor)",
      "dependencies": "FLAV.RIGID.01; v49; v69; v70",
      "scripts": [
        "v71_simple_r_bridge.py"
      ],
      "script": "v71_simple_r_bridge.py",
      "location": "tfpt_research_contracts;tfpt_2",
      "externalData": "-",
      "supersedes": "FLAV.QRATIO.01;FLAV.UGATE.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 137
    },
    {
      "id": "FLAV.QGEO.03",
      "claim": "det Q = N_fam DERIVED from the cusp class (Alessandro Q-residual closed to the same geometry): the integer-lift datum det Q=|coker Q|=3 is the ORDER of the cusp monodromy class (eigenvalues {1,omega,omega^2}, omega^3=1) = the common DENOMINATOR of the cusp weights {0,1/3,2/3} that fix Spec(Q_+)=diag(1,2,3) (v69). So det Q=N_fam is NOT an independent lattice input - it is the cusp-class order = N_fam = rank A3 = dim H1, and coker Q = Z/N_fam is the deck group of the degree-N_fam triality cover. Q-geometry now derived at spectra + Sigma-split + integer lift; only standing input is N_fam=3 itself",
      "status": "Identity [I] chain + [L] non-abelian Hodge",
      "canonicalStatus": "Identity [I] chain + [L] non-abelian Hodge",
      "dependencies": "FLAV.QGEO.02; v69; v70",
      "scripts": [
        "v72_q_det_from_cusp.py"
      ],
      "script": "v72_q_det_from_cusp.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "GATE.QGEO.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 138
    },
    {
      "id": "SEAM.KHALF.01",
      "claim": "k=c3/2 FORCED - the central seam-area coefficient (Alessandro central target met at the dimensionless level): k=c3/2=(1/2)x(1/(|Z2|*2pi*chi(S2))) splits into TWO cutoff-independent pieces, neither a free normalisation. (a) factor 1/2 = universal action<->EOM variational factor (16piG action vs 8piG field eq, forced by delta(sqrt(g)R)->G_uv). (b) c3=1/(|Z2|*2pi*chi(S2))=1/(2*4pi)=1/(8pi) = one-sided Gauss-Bonnet of the seam S2 normal slice (chi(S2)=2 => intK=4pi, TOPOLOGICAL). With k=c3/2, Fursaev-Solodukhin S=4pi k A=2pi c3 A=A/4 exactly. So the central theorem coefficient is internally forced by topology x variation; ONLY the absolute 4D Newton scale (Lambda^2*f2 prefactor of carrier-Dirac a2, v68) stays UV-sensitive = the one irreducible dimensionful anchor",
      "status": "Identity [I] + Lattice/topology [L] (coefficient) + [A] (absolute scale only)",
      "canonicalStatus": "Identity [I]/[L] (coefficient forced) + [A] (absolute scale)",
      "dependencies": "v67; v68; AX.P1.01; SEAM.HORIZON.01",
      "scripts": [
        "v73_k_c3_half.py"
      ],
      "script": "v73_k_c3_half.py",
      "location": "origin_theory;tfpt_4",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 139
    },
    {
      "id": "ARCH.MICRO.01",
      "claim": "Compiler micro-lemmas (review synthesis, all exact identities compressing scattered factors): (1) SPINE QUOTIENT LADDER - the recurring O(1) factors 2/3,3/4,4/5,5/4,4/3 are adjacent quotients of the spine (|Z2|,N_fam,|mu4|,g_car)=(2,3,4,5); feed lambda2=(2/3)^6 (gap=Page rate=Koide target 2/3), eps=3/4 phi0=q(A3)phi0 (solar), phi0 lead (4/3)c3=1/(6pi), q(D5)+q(A3)=5/4+3/4=2. (2) PENCIL DIFFERENCES - P(x)=det(K+xQ)=3x^3+7x^2+6x+4, endpoints (2,4,20,68), consecutive differences (2,16,48)=(|Z2|,dim S+,Omega_adm)=sheet->1gen->3gens. (3) ANCHOR QUADRATIC FORM - 1^TK1=25=g_car^2 (mass volume), a^TKa=41=10b1 (EM budget), a^TKa-1^TK1=16=dim S+ => EM budget = mass volume + one generation. (4) SOLAR DUAL ANCHOR - L=R+6*1*e1^T rank-one winding, a^T R^-1.1=0 => Sherman-Morrison => a^T L^-1=a^T R^-1=(-1/2,-1/2,1): winding invisible to dual anchor, TBM vector stable (mechanism for eps=q(A3)phi0). NOTE fixed doc typo L=R+6 e1 a^T -> L=R+6*1*e1^T",
      "status": "Identity [I]",
      "canonicalStatus": "Identity [I]",
      "dependencies": "ANCHOR.GEN.01; FLAV.KQ.01; FLAV.PENCIL.01; FLAV.SOLAR.01; BND.GLUE.01",
      "scripts": [
        "v74_compiler_micro_lemmas.py"
      ],
      "script": "v74_compiler_micro_lemmas.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 140
    },
    {
      "id": "FLAV.UPOINT.01",
      "claim": "Gate 1 COMPLETE - U_point reduces to v_geo (the absolute flavor amplitude is not an independent gate): quark ratios closed (Plucker, v49/v71) + lepton c exact (16/7,4/3,7/6, v20) + each sector c-product a clean phi0-power (Grand Mass Volume v46: det M ~ phi0^{6,9,10}; lepton product 16/7*4/3*7/6=32/9=2^gcar/Nfam^2). Since (pairwise ratios)+(product)<=>(individuals) is a bijection, the 9 charged-fermion amplitudes are fixed up to ONE overall scale v_geo. So U_point = v_geo = the SAME one irreducible dimensionful anchor as gravity's 1/G (v68): the two [A] anchors collapse to one. Only v_geo (one scale) + pi remain irreducible",
      "status": "Identity [I] (reduction) + [A] (the one scale v_geo)",
      "canonicalStatus": "Identity [I] + [A] (v_geo, shared with 1/G)",
      "dependencies": "FLAV.RIGID.02; FLAV.LEPTONC.01; FLAV.MASSVOL.01; v68",
      "scripts": [
        "v75_upoint_to_vgeo.py"
      ],
      "script": "v75_upoint_to_vgeo.py",
      "location": "tfpt_research_contracts;origin_theory;tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 141
    },
    {
      "id": "GATE.METRIC.02",
      "claim": "Gate 2 attack - two clean tiers (was: diffuse 'full QG missing'). TIER A (physics, CLOSED under RP+gap): Decoupling Theorem - Delta_eff=Delta-2||V||=6log(3/2)-31/(4pi^2)=1.648>0 with ||V_metric||_rel<=248 c3^2=dim(E8) c3^2=31/(8pi^2) (31=2^gcar-1); strictly positive effective gap protects the admissible IR sector => every testable low-energy readout (masses, mixings, alpha^-1, R+R^2) is independent of G6. TIER B (strict TOE, REDUCED not closed): the seam is a FINITE causal boundary => the ambient BULK metric measure is reconstructed from a finite-codimension seam-Calderon BOUNDARY measure (holographic) => G6 is a BOUNDARY projective limit; conditional theorem RP(seam-Calderon)+tightness => G_metric closes. Residual = boundary projective limit (constructive-QFT grade), reduced not closed",
      "status": "Identity [I] (decoupling, cond. on norm bound) + [P]/[A] (boundary G6)",
      "canonicalStatus": "Identity [I] (Tier A) + [P]/[A] (Tier B boundary measure)",
      "dependencies": "CONTRACT.G.01; QG.GAPDEC.01; GATE.METRIC.01",
      "scripts": [
        "v76_gmetric_reduction.py"
      ],
      "script": "v76_gmetric_reduction.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 142
    },
    {
      "id": "GATE.METRIC.03",
      "claim": "G6 route - the seam BOUNDARY measure IS a rigorously constructed conformal net (imports G6 into existing rigor): level-1 central charges c=k dim/(k+h^v) give c(E8)_1=248/31=8=rank E8 (holomorphic c=8 E8-LATTICE VOA, single primary, Frenkel-Kac-Segal / Kawahigashi-Longo / Carpi), c(D5)_1=45/9=5, c(A3)_1=15/5=3; conformal embedding (D5)_1 x (A3)_1 ⊂ (E8)_1 since 5+3=8 => coset c=0. The gap Delta_eff>0 (v76) => exponential clustering => tightness of the boundary projective limit. So G_metric Tier B reduces to identifying the seam-Calderon boundary measure with the (E8)_1 lattice net; residual = the identification + bulk reconstruction",
      "status": "Identity [I] (central charges, embedding) + [P] (net identification)",
      "canonicalStatus": "Identity [I] + [P] (boundary net identification)",
      "dependencies": "GATE.METRIC.02; FLAV.CFT (v61); v76",
      "scripts": [
        "v77_e8_conformal_net.py"
      ],
      "script": "v77_e8_conformal_net.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 143
    },
    {
      "id": "ANCHOR.VGEO.01",
      "claim": "v_geo = CLOSED METROLOGY STRUCTURE (R_+ unit torsor; one dimensionful calibration necessary by dimensional analysis -- the No-Unit theorem v153/v725; honest reframe, not a derived mass and not an open physics gap): no pure number can be dimensionful => one calibration is forced by LOGIC (every theory carries >=1 dimensionful input). ALL TFPT scales are dimensionless ratios times ONE M_bar: rho_Lambda/M_bar^4=(3/4pi^2)e^{-2 ainv} (v60), M_scal/M_bar=c3^{7/2} (v7), masses = (pi/sqrt2)c_f phi0^{k_f} v_geo. Cosmological pinning S_dS*rho_Lambda=1/(128 c3^4)=32 pi^4 (v55) => one measurement sets the unit. The flavor U_point reduces to the SAME v_geo (v75) => one scale shared by flavor+gravity. Theory at the minimum: one scale v_geo + one primitive pi",
      "status": "Identity [I] (ratios) + [A]/[O] (closed metrology: R_+ unit torsor; one calibration by dimensional analysis)",
      "canonicalStatus": "Identity [I] + [A] (one scale, irreducible by logic)  | 2026-08-29 wave-4 retype: CLOSED METROLOGY STRUCTURE (R_+ unit torsor; one dimensionful calibration necessary by dimensional analysis, No-Unit v153/v725).  Display [O] remains the unit/calibration, not a derived mass; no [E] upgrade of physics.  WAVE-5 (2026-08-29): unit choice closed; dimensionless hierarchies (M_Pl/v_EW, M_R/M_scal, Lambda_QCD/v_EW, T_reh/M_scal) remain predictive obligations.",
      "dependencies": "FLAV.UPOINT.01; v60; v55; v68",
      "scripts": [
        "v78_vgeo_floor.py"
      ],
      "script": "v78_vgeo_floor.py",
      "location": "origin_theory;introduction;tfpt_4",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 144
    },
    {
      "id": "FORM.CAR.01",
      "claim": "Theorem A (carrier rank uniqueness g_car=5) machine-formalised in Lean 4: the Pascal carrier condition 2^(g-1)=C(g,0)+C(g,1)+C(g,2), division-free 2^g=g^2+g+2, has the UNIQUE solution g=5 (theorem carrier_rank_pascal_unique: forall g, 2^g=g^2+g+2 <-> g=5; growth lemma pascal_growth for g>=6; g_car_unique ExistsUnique; carrier_closure_integers N_fam=(2^4-1)/5=3, g+N_fam=8=rank E8). AUDIT: PASS (no sorry/admit, no domain axioms, only propext/Classical.choice/Quot.sound). Complements FORM.P2.02 (P2 carrier algebra)",
      "status": "Formal",
      "canonicalStatus": "Formal [F]",
      "dependencies": "AX.P2.01; CAR.UNIQ.01; v6; v14",
      "scripts": [
        "lean4-carrier-rigidity"
      ],
      "script": "lean4-carrier-rigidity",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 145
    },
    {
      "id": "ARCH.REVIEW.01",
      "claim": "Four validated review identities (passed arithmetic AND the anti-numerology/admissible-invariant discipline) + one disciplined rejection: (1) HYPERCHARGE LUCAS-BINET: carrier charges (q+,q-)=(3,-2) [X^2-X-6=(X-3)(X+2), Lean unique_carrier_pair] generate D_n=(3^n-(-2)^n)/5=1,1,7,13,55,133 = (N_Phi, -, scalaron exp 7, Delta_Q 13, quark numerator 55, dim E7 133) in order; c_u/c_d=D_5/(N_fam^2 D_4)=55/117 (cleaner than Plucker 11). (2) INVERSE ANCHOR THEOREM: 1^T M^-1 1 = 1/atom (Q->1/3=1/N_fam, R=K->1/4=1/|mu4|, L->1/10=1/A_Lambda); anchor self-dual a^T R^-1 a=a^T K^-1 a=a^T L^-1 a=1 (a^T Q^-1 a=7/3, so self-duality is exactly residue/mass/transport, not the pure difference Q). (3) MACWILLIAMS: C^+=even-weight code [5,4,2], dual=repetition [5,1,5], weight enum (1,10,5)=Lambda^even(5) (makes the existing Hamming-scheme duality the named MacWilliams/UV-IR transform). (4) 6Y^2-Y-1=(2Y-1)(3Y+1)=(sY-1)(bY+1), roots 1/2 (SU2) and -1/3 (SU3). FIREWALL (recorded): m_p/m_e=1920-84+3lamC^2 (SU(9)=A_8 not in carrier; m_p is QCD confinement, no mechanism) and eta_B=(2/13)c3^6 (model-dependent leptogenesis output) are arithmetically close but REJECTED -> stay [A]/[P] frontier, NOT promoted to [I]",
      "status": "Identity/Lattice [I]/[L]",
      "canonicalStatus": "Identity/Lattice [I]/[L] (4 accepted; m_p/m_e + eta_B firewalled)",
      "dependencies": "CAR.UNIQ.01; FLAV.KQ.01; FLAV.SOLAR.01; v44; FR.MPME.01; FR.ETAB.01",
      "scripts": [
        "v79_review_identities.py"
      ],
      "script": "v79_review_identities.py",
      "location": "tfpt_1;tfpt_2;tfpt_4",
      "externalData": "PDG (m_p/m_e firewall)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 146
    },
    {
      "id": "FLAV.PENCIL.02",
      "claim": "Operator-pencil geometry of (K,Q,R,L) - the anchor plane as a small area in operator space (validated review block; all in admissible classes). STRONG: [2] ANCHOR SINGULARITY THEOREM det B(K+xQ)=9x^2+21x+10=(N_fam x+|Z2|)(N_fam x+g_car)=(3x+2)(3x+5), oriented anchor area degenerates at x=-|Z2|/N_fam=-2/3 (Koide target/gap base (2/3)^6) and x=-g_car/N_fam=-5/3 (D5/A3 glue asymmetry) => 2/3 is the FIRST SINGULAR POINT of the mass<->transport anchor plane, not just a recurring fraction [I/L]; [4] buildup chain (3,7,6,4) partial sums (3,10,16,20)=(N_fam,A_Lambda,dim S+,det L), scalaron 7=A_Lambda-N_fam [I]; [6] type checker det B_Q=9=N_fam^2, det B_K=10=A_Lambda, det B_R=16=dim S+, det B_L=40=|R(D5)| [I/L]. AUDIT (exact values [I], reading decorative [P]): [3] differential spectrum D,D',D'' at -1,0,1,2 = (2,4,20,68),(1,6,29,70),(-4,14,32,50) (29=top E8 exp, 32=2^gcar, 50=A4xA4 block); [5] F4xG2 shadow det B_{R+Q}=52=dim F4, chi_[K,Q]=t^3+5t+14 (14=dim G2), |Pl_R(K)|_1=26, 248=52+14+26*7 = real E8->F4xG2 branching (52,1)+(1,14)+(26,7) - an operator shadow, not a Lie-theoretic embedding proof",
      "status": "Identity/Lattice [I]/[L] + audit [P]",
      "canonicalStatus": "Identity/Lattice [I]/[L] (anchor singularity, type checker) + [P] (atlas/F4xG2 shadow audit)",
      "dependencies": "FLAV.PENCIL.01; FLAV.KQ.01; v37; v52; v74",
      "scripts": [
        "v80_operator_pencil_geometry.py"
      ],
      "script": "v80_operator_pencil_geometry.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 147
    },
    {
      "id": "FLAV.PENCIL.03",
      "claim": "The mass-transport pencil P(x)=K+xQ as a projective line with two anchor-block singularities, and the DOUBLE COVER branched over them (validated review block; all in admissible classes). CENTRAL [I/L]: det B(K+xQ)=(3x+2)(3x+5) is quadratic => y^2=det B(K+xQ) is a 2:1 cover ramified exactly over its zeros, so the Koide/gap point x=-|Z2|/N_fam=-2/3 and the carrier point x=-g_car/N_fam=-5/3 are THE TWO BRANCH POINTS of one double cover; deck involution y->-y of degree 2=|Z2|=the sheet (line-bundle reflection of the Spin(10)->SO(10) half-spinor cover the carrier lives on). Branch locus = -(spine ends 2,5)/N_fam; discriminant = 81 = N_fam^4 (perfect square => rational/split, sqrt=9=N_fam^2=det B_Q); separation = 1 = one transport period (deck-translation x->x+1 maps carrier->Koide). So 2/3 is not a recurring quotient but ONE ramification point of the mass<->transport double cover; sharper open question: why does the leptonic source-to-pole transfer renormalise onto a branch point and which sheet (y->+-) does parity select. DEEPER: the branch divisor symmetric functions ARE the core atoms - sum of branch points = -scalaron/N_fam = -7/3, product = A_Lambda/N_fam^2 = 10/9, integer labels {|Z2|,g_car}={2,5} have sum=7=scalaron, product=10=A_Lambda, diff=3=N_fam, so the scalaron is the TRACE of the branch divisor and all earlier scalaron readouts collapse to this one geometric origin; the fiber y^2=det B is A_Lambda at K and 4A_Lambda at L (transport K->L multiplies the fiber by |Z2|), and the Z2 sheet endpoint x=-1 lies between the branch points where y^2=-|Z2|<0 (the sheet lives on the imaginary cut joining Koide and carrier); the clean cover is a property of the anchor 2-plane <1,a> (the full-3-space cubic det(K+xQ) has disc -1132 with an irrational real root), so the anchor projection regularises the geometry (anchor-first). CLEARING MATRICES [I/L]: Koide C_{2/3}=3K-2Q reproduces D5+A3 glue (tr=15=dim A3, sum=45=dim D5, det=60=D_start), rank-one (forced) anchor block B(C23)=(g_car,7)^T(N_fam^2,11)=(5,7)^T(9,11), sum=240=|R(E8)|; carrier C_{5/3}=3K-5Q charge-neutral (sum=0), chi=(lam+|Z2|)(lam^2+lam+|R+A3|)=(lam+2)(lam^2+lam+6). Scalaron 7 = mixed anchor discriminant/N_fam = (det B_L - det B_K - det B_Q)/N_fam = (40-10-9)/3 (B_L=B_K+B_Q linearity; 7 is an interference area, not a leftover) [I]. c_u/c_d=55/117 diagnostic readout at sheet endpoint K-Q (Plucker v37 canonical) [I]",
      "status": "Identity/Lattice [I]/[L] + audit [P]",
      "canonicalStatus": "Identity/Lattice [I]/[L] (double cover / branch locus, clearing matrices, mixed-disc scalaron) + [P] (per-entry atlas reading of clearing matrices)",
      "dependencies": "FLAV.PENCIL.01; FLAV.PENCIL.02; v37; v52; v74; v80",
      "scripts": [
        "v81_singular_pencil_matrices.py"
      ],
      "script": "v81_singular_pencil_matrices.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 148
    },
    {
      "id": "FR.KOIDE.04",
      "claim": "Koide RG attractor is FORCED, not postulated [I] + ONE [P] identification (sharpens FR.KOIDE.01/02/03 via the v81 double cover): a branch-divisor-preserving Moebius map of the pencil line that fixes BOTH branch points q=2 (Koide) and q=5 (carrier) is, in the cross-ratio coordinate rho=(q-2)/(5-q), EXACTLY rho->mu*rho, hence UNIQUE once mu is fixed [I]. The multiplier is NOT a new input: mu=(2/3)^6 is the subleading eigenvalue lambda2 of the THEORY'S OWN unique gapped boundary transfer operator (v54/v56, spectrum {1,(2/3)^6,(1/3)^6}, gap 6 log(3/2)>0), and the Koide branch x=-2/3=-|Z2|/N_fam IS its cusp weight 2/3={0,1,|Z2|}/N_fam; carrier q=5 -> Koide q=2 under the deck period q->q-N_fam (shared cusp class 2/3). With mu=(2/3)^6 the unique map is the integer Moebius map F(q)=2(569q-3325)/(665q-3517), F(2)=2, F(5)=5, F'(2)=(2/3)^6<1 (Koide is the ATTRACTOR), F'(5)=(3/2)^6>1 (carrier the REPELLER), so q_n->2 i.e. x->-2/3 at the SAME rate that governs SM flavor and horizon recovery. NET: the three 'structural postulates' for a Koide derivation collapse to ONE physical identification - source->pole transfer = branch-preserving relaxation of the established transfer operator; rate, uniqueness and direction then follow. NOT a closure: the single identification stays [P]",
      "status": "Physical (conjecture sharpened) + Identity [I]",
      "canonicalStatus": "Identity [I] (map uniqueness + multiplier = the established transfer gap) + Physical [P] (the one source->pole identification; no closure)",
      "dependencies": "FR.KOIDE.01; FR.KOIDE.03; FLAV.PENCIL.03; v54; v56; v81",
      "scripts": [
        "v82_koide_attractor_splitting.py"
      ],
      "script": "v82_koide_attractor_splitting.py",
      "location": "tfpt_4",
      "externalData": "PDG lepton masses",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 149
    },
    {
      "id": "FLAV.PENCIL.04",
      "claim": "Anchor-placement splitting trichotomy - the clean rational double cover is NON-GENERIC (anchor-first hardening) [I] + [P] (sector id). The splitting-type exponents (2,1,1) (= the anchor (1,1,2); v32 splitting O(-2)+O(-1)^2) placed across the three family slots give three DIFFERENT covers: (1,1,2) heavy-in-fam3 (lepton) det B(K+xQ)=(3x+2)(3x+5) disc=81=N_fam^4 -> SPLIT (individually rational branch labels {2,5}); (2,1,1) heavy-in-fam1 det B=(x+2)(9x+11) disc=49=scalaron^2 -> SPLIT; (1,2,1) heavy-in-fam2 det B=5x^2+10x+3 disc=40=|R(D5)|=det B(L) -> NON-SPLIT (Galois-conjugate IRRATIONAL branch points -1 +/- sqrt(10)/5, centered EXACTLY on the sheet x=-1 (q=3), separation 2 sqrt(A_Lambda)). The three discriminants are themselves compiler atoms (N_fam^4, scalaron^2, |R(D5)|). Negative control (cf. v81): the R-pencil disc=153 is not a square either => Koide & carrier as SEPARATE RATIONAL singularities is a special feature of the canonical anchor placement, not generic. Restates the basis-invariant core T=3 B(Q)^-1 B(K)=[[5,11/3],[0,2]] (Spec{2,5}, tr=7=scalaron, det=10=A_Lambda). The physical fermion-sector identification of the placements is NOT claimed and stays [P]",
      "status": "Identity [I] + Physical [P] (sector id)",
      "canonicalStatus": "Identity [I] (discriminant trichotomy, non-genericity, negative control) + [P] (physical sector identification of the placements)",
      "dependencies": "FLAV.PENCIL.02; FLAV.PENCIL.03; v32; v80; v81",
      "scripts": [
        "v82_koide_attractor_splitting.py"
      ],
      "script": "v82_koide_attractor_splitting.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 150
    },
    {
      "id": "REDTEAM.A.01",
      "claim": "Red Team / Stress Test Target A - adversarial attack on the (E8)_1 boundary-net identification (GATE.METRIC.03). CONFIRMS the conservative wording: central charge is NECESSARY not sufficient - (D8)_1=SO(16)_1 also has c=120/15=8 but is non-holomorphic (4 primaries 1,v,s,c), and the c=8 Narain family is positive-dimensional; (E8)_1 is the UNIQUE holomorphic c=8 net (even unimodular rank-8 lattice = E8), so HOLOMORPHY (mu-index 1) is the load-bearing extra assumption. The conformal embedding c_coset=c(E8)-c(D5)-c(A3)=8-5-3=0 is compatibility, NOT the physical subnet identification; Delta_eff>0 SUPPORTS tightness (clustering) but does not prove it; the constructive seam-Calderon->(E8)_1 map and bulk-reconstruction uniqueness are missing. VERDICT: keep 'reduced to a rigorous boundary-net identification problem', do NOT write 'metric sector closed'",
      "status": "Red-team [P] (reduced, not closed)",
      "canonicalStatus": "Red-team verdict: reduced, not closed [P]",
      "dependencies": "GATE.METRIC.03; v77; v76; v61",
      "scripts": [
        "redteam/rt_A_e8net.py"
      ],
      "script": "redteam/rt_A_e8net.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 151
    },
    {
      "id": "REDTEAM.B.01",
      "claim": "Red Team Target B - attack on the UPSTREAM physical premise of Theorem A (the Lean arithmetic 2^g=g^2+g+2 <=> g=5 is NOT attacked; it is verified [F]). Finding: the carrier rank is selected by the Pascal RULE, not the arithmetic. Perturbing the constant 2^g=g^2+g+c gives {5} only at c=2; the truncation degree K in 2^(g-1)=sum_{k<=K} C(g,k) sets g=2K+1 (K=0->1, 1->3, 2->5, 3->7), so choosing K=2 is the physical input. Over-determined (D5 half-spinor dim S+=2^(g-1)=16 <=> g=5; family closure N_fam=(2^(g-1)-1)/g, g+N_fam=8 also pick 5) but each is itself a postulate. VERDICT: the arithmetic core stands [F]; the Pascal SELECTION must be typed [A]/[P], not [I]",
      "status": "Red-team [F]+[A] (survives, narrowed)",
      "canonicalStatus": "Red-team verdict: survives; Pascal selection now typed [A]/[P]",
      "dependencies": "FORM.CAR.01; CAR.UNIQ.01; v2; v14",
      "scripts": [
        "redteam/rt_B_pascal.py"
      ],
      "script": "redteam/rt_B_pascal.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 152
    },
    {
      "id": "REDTEAM.C.01",
      "claim": "Red Team Target C - dimensional firewall on k=c3/2. The reduced coefficient k_red=c3/2=1/(16pi) is dimensionless; the only legal chain is k_phys=k_red/G, S=4pi k_phys A_phys=A_phys/(4G). The forbidden form k_phys=c3/2 is dimensionally false (legal only at G=1 / Planck units). Firewall scan of v67/v68/v73: 28 'c3/2' mentions, ZERO naked dimensionful identities, each file establishes the reduced/seam-unit/G=1 context; the firewall's teeth are verified against a synthetic violation. VERDICT: safe as worded (reduced coefficient); the absolute 1/G (UV-sensitive, Sakharov/Connes) stays the one dimensionful anchor",
      "status": "Red-team [I]+[A] (survives)",
      "canonicalStatus": "Red-team verdict: survives as worded (reduced coefficient)",
      "dependencies": "SEAM.AREACOEFF.01; SEAM.AREACOEFF.02; v67; v68; v73",
      "scripts": [
        "redteam/rt_C_kc3.py"
      ],
      "script": "redteam/rt_C_kc3.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 153
    },
    {
      "id": "REDTEAM.D.01",
      "claim": "Red Team Target D - stress test of the U_point->v_geo bijection (FLAV.UPOINT.01). For positive reals (ratios,product) <=> (individuals) is a bijection and the TFPT sectors satisfy it (positive rational c, odd size 3=N_fam). Counterexamples off-assumption: even sector size leaves a SIGN ambiguity ({2,3} vs {-2,-3} share ratio 3/2 and product 6); complex amplitudes leave an n-th-root BRANCH ambiguity ({1,1,1},{w,w,w},{w^2,w^2,w^2}, w=e^{2pi i/3}, share all ratios and product 1); a zero amplitude is degenerate. GENUINE RESIDUAL: the bijection fixes amplitude MAGNITUDES only - CP phases delta_CKM/PMNS are complex and are NOT folded into v_geo. VERDICT: holds for TFPT once four hypotheses (real, positive, fixed branch, no zeros / odd sector size) are made EXPLICIT (they are silent in v75). NARROWED (2026-07-29, T177/v561): the magnitude data carry EXACTLY ZERO phase on the TFPT sectors -- arg(prod c) = 0 exactly for both the lepton and the up-quark triple, so the CP residual is not hidden in the product -- and the branch-group blindness is the statement that the branch ORDER divides N_fam = 3 (a mu4 branch already shifts the product phase by -pi/2). The residual is a FIBER datum (the hexagonal CM unit), not a magnitude datum -- a narrowing of the residual, no upgrade.",
      "status": "Red-team [I]+[A] (survives, narrowed)",
      "canonicalStatus": "Red-team verdict: survives; four hypotheses now explicit; CP-phase residual, narrowed to a FIBER datum: arg(prod c)=0 exactly on both sectors, branch blindness = order divides N_fam=3 (T177/v561)",
      "dependencies": "FLAV.UPOINT.01; v75; v20; v24",
      "scripts": [
        "redteam/rt_D_upoint.py",
        "v561_cp_channel_identities.py"
      ],
      "script": "redteam/rt_D_upoint.py; v561_cp_channel_identities.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 154
    },
    {
      "id": "REDTEAM.E.01",
      "claim": "Red Team Target E - single-scale audit of v_geo (ANCHOR.VGEO.01). The closed compiler + protected IR readouts all reduce to (pure number) x v_geo: M_scal=c3^(7/2) Mbar, f_a=(c3/|mu4|)^(7/2) Mbar, masses=(pi/sqrt2) c_f phi0^k v_geo, and 1/G shares the same anchor (v68/v75). The audit SURFACES non-pure-reduction candidates: the seam cutoff Lambda (only an identification seam=Planck collapses it to v_geo), the electroweak matching scale, the reheating temperature, and the leptogenesis scale - all frontier [P]/[A] inputs that must NOT be silently absorbed into v_geo. VERDICT: v_geo is the unique dimensional floor of the CERTIFIED tiers (true by dimensional analysis); full single-scale-ness is conditional on the flagged identifications staying honestly typed",
      "status": "Red-team [I]+[A] (survives, narrowed)",
      "canonicalStatus": "Red-team verdict: floor of certified tiers; frontier scales must stay typed",
      "dependencies": "ANCHOR.VGEO.01; FLAV.UPOINT.01; v78; v68; v60",
      "scripts": [
        "redteam/rt_E_vgeo.py"
      ],
      "script": "redteam/rt_E_vgeo.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 155
    },
    {
      "id": "GATE.METRIC.04",
      "claim": "Red-team Target A residual 1 CLOSED [L] (holomorphy is necessary AND sufficient to pin (E8)_1): at level 1 the number of primaries = det(Cartan) = |fundamental group|, giving A3=4, D5=4, D8=4, E8=1. So holomorphy (single vacuum sector = mu-index 1 = det 1) excludes the SAME-c rival (D8)_1=SO(16)_1 (c=120/15=8 but 4 primaries 1,v,s,c) - equal central charge is necessary not sufficient - AND is sufficient: a holomorphic c=8 chiral CFT is the lattice theory of an even unimodular rank-8 lattice, of which there is exactly ONE, E8, by the Minkowski-Siegel mass formula (mass=1/|W(E8)|=1/696729600=1/(2^14*3^5*5^2*7), and |Aut(E8)|=|W(E8)| saturates it; E8 Gram even, det 1). The (D5)_1x(A3)_1 conformal embedding c=5+3=8=c(E8) is compatibility (16 primaries vs 1), NOT the subnet identification. CONSEQUENCE: with this, the constructive map need only show the seam-Calderon boundary net is holomorphic & c=8 (then E8 is automatic), so Target A drops from 3 residuals to 2: (i) boundary-net holomorphy+c=8 (Delta_eff>0 supports, not proves), (ii) bulk-reconstruction uniqueness - both still [P]/[A]",
      "status": "Lattice [L] (residual 1 closed) + [P]/[A] (residuals 2-3 open)",
      "canonicalStatus": "Lattice [L]: holomorphy => (E8)_1 unique (residual 1 closed); residuals 2 (constructive map) + 3 (bulk uniqueness) remain [P]/[A]",
      "dependencies": "REDTEAM.A.01; GATE.METRIC.03; v77; v66; v1",
      "scripts": [
        "v83_e8net_holomorphic_uniqueness.py"
      ],
      "script": "v83_e8net_holomorphic_uniqueness.py",
      "location": "tfpt_5_redteam",
      "externalData": "E8 lattice / level-1 RCFT (standard)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 156
    },
    {
      "id": "CAR.PASCAL.01",
      "claim": "Red-team Target B residual REDUCED [I]/[L]: the carrier 'K=2 Pascal truncation' flagged by the red team as a free postulate is NOT free. K=(g-1)/2 is the Pascal-row MIDPOINT: sum_{k<=(g-1)/2} C(g,k) = 2^(g-1) for every odd g (verified g=3,5,7,9), i.e. the half-spinor split of the 2^g Dirac spinor. Equivalently the carrier is the even Clifford half-spinor Lambda^even(C^g_car): C(5,0)+C(5,2)+C(5,4)=1+10+5=16=2^(g_car-1)=dim S+. So Target-B's residual is not 'why K=2' but the single standard input 'carrier = half-spinor of Spin(2 g_car)', already used in the D5+A3 glue (v1/v15); the Lean arithmetic 2^g=g^2+g+2<=>g=5 stays [F]",
      "status": "Identity/Lattice [I]/[L] (B residual reduced to the half-spinor input)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: K=(g-1)/2 is the half-spinor midpoint, not a free truncation; B residual reduced to the standard half-spinor carrier",
      "dependencies": "REDTEAM.B.01; FORM.CAR.01; CAR.UNIQ.01; v2; v44",
      "scripts": [
        "v83_e8net_holomorphic_uniqueness.py"
      ],
      "script": "v83_e8net_holomorphic_uniqueness.py",
      "location": "tfpt_5_redteam;tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 157
    },
    {
      "id": "REG.FREEZE.01",
      "claim": "Blind-prediction registry FROZEN 2026-06-09, machine-enforced (review point: pre-registration before JUNO/CMB-S4/LiteBIRD/DESI). predictions_frozen.json freezes every dimensionless prediction of record at 25 significant digits; v84 re-derives each decimal from the two axioms {c3,g_car} on every run, so formula and frozen value are locked together (changing either alone FAILS the suite; the file is covered by manifest.sha256). Registry layers: core [N] (alpha^-1, lambda_C, s23/s13 CKM, sin^2 th13, lepton + quark mass ratios), conditional [N]/[P] with the hypothesis NAMED (th12 seed via seam-misalignment, beta_rad, Omega_b, delta_CKM leading, M_scal/Mbar, rho_Lambda/Mbar^4 branch), assigned (delta_CP^nu=240deg, th23=0.5), conditional BANDS for r and n_s at the declared external N_star in [50,60]. THETA12 FREEZE ENFORCED: exactly ONE prediction of record = seed 1/3-phi0/2 = 0.306747357245; the seam (0.306808) and nonlinear (0.307020) values are typed as derived variants of the SAME texture and may NOT appear as predictions (look-elsewhere ambiguity machine-excluded)",
      "status": "Numerical [N] (registry lock) + typing discipline",
      "canonicalStatus": "Numerical [N] (registry lock)",
      "dependencies": "EM.FP.01; FLAV.TH12.01; PRED.LAYER.01; freeze_file.csv",
      "scripts": [
        "v84_frozen_registry.py"
      ],
      "script": "v84_frozen_registry.py",
      "location": "verification;tfpt_2;introduction",
      "externalData": "JUNO, CMB-S4/LiteBIRD, DESI (future); NuFIT 6.0 / CODATA 2022 / ACT DR6 (current)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 158
    },
    {
      "id": "FLAV.PENCIL.05",
      "claim": "MASTER-COVER THEOREM (answers next.txt P3; resolves P6 negatively; types P4) [I]/[L]: the anchor-block determinant is GL(2)-COVARIANT on the pencil plane span{K,Q} - det B((aK+bQ)+x(cK+dQ)) = (gamma x+alpha)^2 * p_KQ(moebius_G(x)) exactly (symbolic identity), with disc = 81*det(G)^2 = N_fam^4 x square. CONSEQUENCE: there is exactly ONE anchor-block double cover up to Moebius reparametrisation of the line, the master cover y^2=(3x+2)(3x+5) of v81; the whole disc-81 family observed in next.txt P3 (branch {-8/3,-5/3} for L+xQ, {-5/8,-2/5} for K+xL, {-3/2,-3/5} for Q+xK) is its GL(2) orbit, and the conspicuous rung -8/3 = -rank(E8)/N_fam is the carrier branch point shifted by ONE transport period (rank E8 = g_car+N_fam: -(g_car+N_fam)/N_fam = -g_car/N_fam - 1), i.e. no new structure. NEGATIVE CONTROLS (anchor-first hardened): leaving the anchor plane in the R-direction gives non-square discriminants 153 (R+xQ) and 201 (K+xR) - non-split, no rational cover. P6 HONEST NEGATIVE: the natural 4-sheet candidate det B(K+xQ)*det B(L+xQ) factors through the forced square (3x+5)^2, so mu4=Z4 does NOT act as a genuine 4:1 cover of the pencil line; the transport tower generates a LADDER of double covers (rung n: branch {-2/3-n,-5/3-n}) with second level w^2=(3x+2)(3x+8) (disc 324=18^2) connecting Koide to rank E8 at branch separation 2=|Z2|. P4 TYPED: the branch-divisor trace -7/3=-scalaron/N_fam fixes only the SCALE exponent (M_scal=c3^{7/2}Mbar); n_s=1-2/N_star and r=12/N_star^2 are functions of the external N_star alone, so the cover cannot promote the tilt from [P]. Physical readings (Koide branch <-> physical Koide, sheet <-> half-spinor) stay [P] as in v81/v82",
      "status": "Identity/Lattice [I]/[L] (GL(2) covariance + family classification) + [P] physical readings",
      "canonicalStatus": "Identity/Lattice [I]/[L]: one master cover; family = GL(2) orbit; mu4 not a 4:1 cover; tilt stays [P]",
      "dependencies": "FLAV.PENCIL.04; v80; v81; v82",
      "scripts": [
        "v85_master_cover.py"
      ],
      "script": "v85_master_cover.py",
      "location": "verification;next.txt",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 159
    },
    {
      "id": "COSMO.NSTAR.01",
      "claim": "N_star band -> POINT [P] (the announced 'future derivation from a reheating model' executed; demotion of N_star as a compiler output STANDS): the only compiler input is M_scal=c3^{7/2}Mbar=3.0600e13 GeV [I] (v28/v36/v80); everything downstream is imported standard physics - scalaron width Gamma=N_eff M^3/(48pi Mbar^2) with N_eff=4 (SM Higgs doublet; Starobinsky/Vilenkin/Gorbunov-Panin), reheating at H=Gamma => T_reh=(30 rho/(pi^2 g*))^{1/4}=9.55e9 GeV (g*=106.75), pivot matching with w=0 scalaron oscillations + entropy conservation on the EXACT Starobinsky potential V=(3/4)M^2 Mbar^2(1-e^{-sqrt(2/3)phi})^2. RESULT: N_star(k=0.05/Mpc)=51.44 => n_s=1-2/N_star=0.9611, r=12/N_star^2=0.00454. CROSS-CHECK: k=0.002/Mpc gives N_star=54.7 = the known R^2-reheating literature value (~54). ROBUSTNESS: channel ambiguity N_eff=1..100 moves N_star by <0.8 e-folds (log sensitivity); instantaneous-reheating upper bound N_star<=55.7. FREEZE DISCIPLINE: 51.4 lies inside the frozen registry band [50,60]; predictions_frozen.json NOT modified (v84 lock intact). HONEST TENSIONS RECORDED: (1) n_s=0.9611 is +0.90 sigma below Planck 2018 (0.9649+-0.0042) and increases the DESI-combined tension; (2) A_s COHERENCE is sharper: with M_scal fixed A_s=N^2 c3^7/(24pi^2) is also predicted, A_s(51.44)=1.764e-9 = -11.4 sigma vs Planck 2.105(30)e-9; the A_s-preferred N_star=56.20 exceeds even the instantaneous-reheating bound 55.61 by 0.59 e-folds (instantaneous gives 2.06e-9 = -1.5 sigma, compatible) => within standard physics the measured A_s REQUIRES near-instantaneous reheating and the slow Higgs-channel point is A_s-disfavoured. The [P] point is CONDITIONAL on the decay channel; the frozen band [50,60] remains the prediction surface of record. Kill criteria: robust n_s>=0.967 falsifies the reheating chain (not the [I]/[L] core); A_s already arbitrates between fast and slow reheating",
      "status": "Physical [P] (conditional on M_Star=M_scal + standard reheating)",
      "canonicalStatus": "Physical [P]: N_star=51.4 point inside the frozen band; demotion as compiler output stands",
      "dependencies": "GRAV.SCAL.01 (M_scal); PRED.LAYER.01; REG.FREEZE.01",
      "scripts": [
        "v86_nstar_reheating.py"
      ],
      "script": "v86_nstar_reheating.py",
      "location": "origin_theory;tfpt_2;introduction",
      "externalData": "Planck 2018 (n_s=0.9649+-0.0042), DESI-combined, CMB-S4/LiteBIRD (r, future); SM g*=106.75",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 160
    },
    {
      "id": "GATE.METRIC.05",
      "claim": "Red-team Target A residual (ii) CONDITIONALLY CLOSED [L] -> Target A = ONE residual: bulk-reconstruction uniqueness is NOT independent of residual (i). By the AQFT classification of full 2D CFTs over a chiral net (Longo-Rehren 1995; Kawahigashi-Longo-Mueger 2001; Bischoff-Kawahigashi-Longo-Rehren arXiv:1405.7863) the possible bulks = haploid commutative Q-systems = physical modular invariants of Rep(A); for a HOLOMORPHIC net Rep(A)=Vect so the bulk pairing is UNIQUE. MACHINE-CHECKED CONTRAST: the same-c rival SO(16)_1 (discriminant category Z2xZ2 of D8, q=(0,1/2,1,1), mu-index=4, S/T built explicitly, S^2=I, (ST)^3=I) admits exactly SIX modular invariants (Z>=0 integer, Z_00=1, [Z,S]=[Z,T]=0, entries<=2): diagonal, s<->c swap, BOTH E8-extension invariants |chi_0+chi_{s,c}|^2 (= the lattice glue E8=D8 u (D8+s): h_s=1 bosonic/condensable, index 2, det=4/4=1 even unimodular, cf v1/v83) and two twisted pairings => without holomorphy the c=8 bulk is MULTIPLY ambiguous; with one primary trivially Z=(1) unique. CONSEQUENCE: the whole of Target A reduces to the SINGLE theorem 'seam-Calderon boundary net is holomorphic with c=8' - then (E8)_1 (v83/GATE.METRIC.04) AND the unique bulk (this row) follow; the hypothesis stays [P]/[A]. Honest scope: nothing here proves the seam net IS holomorphic",
      "status": "Lattice [L] (classification + machine contrast) + [P]/[A] (the one remaining hypothesis)",
      "canonicalStatus": "Lattice [L]: bulk uniqueness rides on holomorphy; Target A = ONE residual",
      "dependencies": "GATE.METRIC.04; REDTEAM.A.01; v83; v77",
      "scripts": [
        "v87_bulk_uniqueness_reduction.py"
      ],
      "script": "v87_bulk_uniqueness_reduction.py",
      "location": "tfpt_5_redteam",
      "externalData": "AQFT literature (LR/KLM/BKLR; standard)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 161
    },
    {
      "id": "FLAV.CP.01",
      "claim": "Red-team Target D follow-up: CP-phase residual QUANTIFIED [N] (not closed; freeze respected): frozen leading reading delta_CKM=pi/3+3 lam^2=1.198232 rad=68.654 deg (registry DELTA_CKM_RAD untouched) gives J=3.327e-5 [I]. Clean phase probe gamma_PDG=65.7+-3.0 deg: pull +0.98 sigma => leading reading SURVIVES. AUDIT (recorded, explicitly NOT promoted): the data central value sits 0.07 deg from the ALTERNATIVE coefficient reading pi/3+2 lam^2 (|Z2| instead of N_fam) - the residual is -1.02 lam^2; textbook look-elsewhere trap, the registry keeps coefficient N_fam=3 (REG.FREEZE.01/v84 forbid silent switching). DECISION THRESHOLD: the two readings differ by lam^2=2.884 deg => 3-sigma distinguishable once sigma_gamma<=0.96 deg (LHCb upgrade / Belle II). Kill both ways: |gamma-68.65deg|>3sigma at sigma_gamma<1deg kills the N_fam coefficient; gamma->68.65 retires the |Z2| shadow. CAVEAT: J-inversion residual -3.15 lam^2 is magnitude-contaminated (TFPT source-scale s23=0.0434 vs PDG M_Z 0.0408; J pull +1.76 sigma recorded); gamma is the clean probe. Target-D typing unchanged: phases remain outside v_geo [P]",
      "status": "Numerical [N] (audit + thresholds) + [P] (phase mechanism open)",
      "canonicalStatus": "Numerical [N]: leading reading survives at +0.98 sigma; residual = -1.02 lam^2 recorded; decision at sigma_gamma<1 deg",
      "dependencies": "REDTEAM.D.01; REG.FREEZE.01; FLAV.CKM.01",
      "scripts": [
        "v88_cp_phase_audit.py"
      ],
      "script": "v88_cp_phase_audit.py",
      "location": "tfpt_5_redteam;tfpt_2",
      "externalData": "PDG 2024 / CKMfitter (gamma=65.7+-3.0 deg, J=3.08+-0.14e-5)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 162
    },
    {
      "id": "GATE.METRIC.06",
      "claim": "CARRIER INDEX LEMMA [L] - the Gate-A theorem gets an equivalent index-theoretic form (sharper as a target): by Kawahigashi-Longo-Mueger (mu_A=[B:A]^2 mu_B; lattice nets mu=|discriminant|=det Cartan) the carrier inclusion has Jones index [(E8)_1:(D5)_1x(A3)_1]=sqrt(16/1)=4=|mu_4|=|Z2|^2 - the glue-group order IS the inclusion index. The extension is the mu_4 simple-current extension at level 1: adjoint branching 248=(45,1)+(1,15)+(16,4b)+(16b,4)+(10,6)=45+15+64+64+60; the three non-vacuum glue sectors are CURRENTS (h=1 exactly: 1/2+1/2, 5/8+3/8, 5/8+3/8); 1+3=4 sectors = the same Z4 glue as the lattice construction (v1). mu-additivity: an index-4 simple-current extension of the carrier has mu=16/16=1, so HOLOMORPHY FOLLOWS from the index statement - it need not be assumed. CONSEQUENCE: (H) 'seam-Calderon net holomorphic with c=8' <=> (H') 'seam net contains the carrier net with Jones index 4 as its mu_4 simple-current extension'. Why sharper: both ends of (H') are constructed objects (v77), finite index is controllable by Longo index theory, and the gap Delta_eff>0 (v76) supports finiteness. Negative control: mu(SO(16)_1)=4!=1 (v83/v87). Honest scope: the seam-side index computation is NOT done here; the boundary-net identification stays [P]/[A]",
      "status": "Lattice [L] (index arithmetic + reframing) + [P]/[A] (seam-side identification)",
      "canonicalStatus": "Lattice [L]: Gate A <=> index-4 mu_4 simple-current extension of the carrier net",
      "dependencies": "GATE.METRIC.05; GATE.METRIC.04; E8.GLU.01; v77",
      "scripts": [
        "v89_carrier_index_lemma.py"
      ],
      "script": "v89_carrier_index_lemma.py",
      "location": "tfpt_5_redteam;tfpt_research_contracts",
      "externalData": "KLM 2001 (mu-index relation; standard)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 163
    },
    {
      "id": "SEAM.AREACOEFF.03",
      "claim": "Seam-Horizon programme: the Fursaev-Solodukhin factor 4pi is now DERIVED, not imported [I] (removes the literature import of v67): (1) smoothed-cone Gauss-Bonnet - regularising the 2D cone (opening alpha) by a tangent spherical cap (tangency alpha=cos theta0) gives Int K dA = 2pi(1-cos theta0) = 2pi(1-alpha) EXACTLY, independent of the smoothing radius (topological; GB closure Int k_g=2pi alpha, chi(disk)=1; limits: hemisphere->cylinder alpha=0, flat->plane alpha=1); (2) codim-2 lift: Int sqrt(g)R -> 2*2pi(1-alpha)A = 4pi(1-alpha)A; (3) replica definition S=-d_alpha[logZ(alpha)-alpha logZ(1)]|_1 with logZ=k_red Int sqrt(g)R gives S=4pi k_red A with the smooth part dropping out - the FS relation machine-derived; (4) k_red=c3/2 (v73/SEAM.KHALF.01) => S=2pi c3 A, and S=A/4 forces c3=1/(8pi) UNIQUELY (sympy-solved). RESIDUAL ISOLATED [A]: the single remaining step is 'replica variation of the seam determinant (gapped Calderon kernel) produces the EH form k_red Int sqrt(g) R with cutoff-independent k_red' = the Seam-Horizon Theorem proper (SEAM.THEOREM.01); steps 1-4 contain no Bekenstein-Hawking import",
      "status": "Identity [I] (chain derived) + [A] (the one isolated residual)",
      "canonicalStatus": "Identity [I]: S=4pi k A derived; S=A/4 <=> c3=1/(8pi); residual = seam determinant => EH form",
      "dependencies": "SEAM.AREACOEFF.01; SEAM.AREACOEFF.02; SEAM.KHALF.01; SEAM.THEOREM.01",
      "scripts": [
        "v90_conical_defect_chain.py"
      ],
      "script": "v90_conical_defect_chain.py",
      "location": "origin_theory;tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 164
    },
    {
      "id": "ARCH.SPINE.01",
      "claim": "SPINE TETRAHEDRON - the compiler microkernel as ONE finite object (review proposal 2026-06-10, validated and typed): the spine T={2,3,4,5}={|Z2|,N_fam,|mu4|,g_car} equals {e3(a),p0(a),e1(a),e2(a)} for the anchor a=(1,1,2), so the tetrahedron is the anchor plus its zeroth moment, not a second structure [I]. Edge products {6,8,10,12,15,20} = {|R+(A3)|/C6, rank E8=h(D5)=det R, A_Lambda=|E(K5)|, |R(A3)|=dim SM gauge, dim A3, det L} [I]; face products {24,30,40,60} = {|W(A3)|, h(E8), |R(D5)|=SumL, cascade start=dim(10,6) block (v89)} [I]; volume 120=|R+(E8)|=5!=sum E8 exponents=Tr_S+ X^2 [I]. Graph reading: |R(E8)|=240=|mu4|*|E(K4)|*|E(K5)| with |E(K4)|=p2(a)=6, |E(K5)|=p3(a)=10, plus negative control: breaks at K6 (|E(K6)|=15 != p4(a)=18) - specific not generic [I]. DISCIPLINE: the dual cuts (node x opposite face, edge x opposite edge, all =120) are TAUTOLOGIES of complementary partitions - typed as presentation raster, never as evidence; the grammar is a SUB-grammar (7, 16, 41, 48, 240, 248 are NOT subset products: the tetrahedron complements the admissibility filter, it does not replace it). chi_T=t^4-14t^3+71t^2-154t+120 (coefficients = node/edge/face sums + volume; 14=dim G2; spine symmetric about 7/2: chi(0)=chi(7)=120, chi(1)=chi(6)=24; scalaron 7=2+5=3+4 = complementary-pair sum = the same 7 as the branch-divisor trace v80/v81) typed AUDIT, not load-bearing",
      "status": "Identity [I] (closures + negative control) + Audit (chi_T fingerprint; dual cuts = presentation)",
      "canonicalStatus": "Identity [I]: spine tetrahedron closures; dual cuts tautological; sub-grammar honestly typed",
      "dependencies": "ANCHOR.GEN.01; FLAV.PENCIL.05; GATE.METRIC.06 (60=dim(10,6))",
      "scripts": [
        "v91_spine_tetrahedron.py"
      ],
      "script": "v91_spine_tetrahedron.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 165
    },
    {
      "id": "GATE.METRIC.07",
      "claim": "GLUE UNIQUENESS [I]/[L] - the carrier extension tower completely classified (Gate A loses its last structural freedom): extensions of the carrier net/lattice correspond to isotropic subgroups of the discriminant form (Z4xZ4, q=(5x^2+3y^2)/8; q reproduces the level-1 sector weights of v87/v89). Exhaustive enumeration of all 15 subgroups: isotropic = {1, Z2, Z4, Z4}. (a) Exactly TWO Lagrangian glues <(1,1)> and <(1,3)>, each det=16/16=1 even unimodular rank 8 => E8 (unique, Minkowski-Siegel v83); they are swapped by EITHER single spinor conjugation (D5 or A3) and fixed by the simultaneous one => the mu_4 glue is unique UP TO THE SHEET Z2 - the two chiralities, matching exactly the two E8-extension modular invariants |chi_0+chi_{s,c}|^2 of v87. (b) The UNIQUE halfway extension {(0,0),(2,2)} has H_perp/H=Z2xZ2 with induced q={0,0,0,1/2} = the D8 discriminant form => SO(16)_1 is the ONLY intermediate of the tower (and is excluded by holomorphy, mu=4!=1). TOWER: carrier (mu=16) -> SO(16)_1 (mu=4) -> (E8)_1 (mu=1, sheet pair) - nothing else exists. CONSEQUENCE: with v89 (index 4 <=> holomorphy) and this row, Gate A is the BARE index statement 'the seam-Calderon inclusion has Jones index 4'; which extension it then is carries NO freedom. Seam-side identification stays [P]/[A]",
      "status": "Identity/Lattice [I]/[L] (exhaustive classification) + [P]/[A] (seam side)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: extension tower = carrier -> SO(16)_1 -> E8 (sheet pair); no freedom left in WHICH extension",
      "dependencies": "GATE.METRIC.06; GATE.METRIC.05; E8.GLU.01; v83; v87",
      "scripts": [
        "v92_glue_uniqueness.py"
      ],
      "script": "v92_glue_uniqueness.py",
      "location": "tfpt_5_redteam;tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 166
    },
    {
      "id": "FR.KOIDE.05",
      "claim": "Koide relaxation toy (next.txt P2 NARROWED, not closed): (1) BASIN LEMMA [I] - under the canonical dictionary q=N_fam*Q the physical Koide range Q in [1/3,1] (Cauchy-Schwarz) maps to q in [1,3]: the attractor q=2 lies inside, the repeller q=5 outside => EVERY physically possible charged-lepton configuration is in the attractor basin, no fine-tuning of the start needed. (2) EXACT RATE [I]: along the trajectory from the physical source value the cross-ratio rho=(q-2)/(5-q) contracts by EXACTLY mu=(2/3)^6 per F-step (Moebius linearity, machine-checked to 1e-30). (3) SOURCE [I]: the exact ladder gives Q_src=0.6644638161, i.e. rho_src=-0.0021980 = -phi0/24 to 0.8% - the v25 conjecture re-expressed in attractor coordinates: the source sits ONE SEED QUANTUM phi0/24 before the branch point (24=|W(A3)|). (4) POLE [N]: PDG pole masses give Q_pole=0.6666644634 - the branch point is hit to 3.3e-6 at the pole layer. HONEST NEGATIVES (they narrow P2): (5) flow time source->pole t=log_mu(rho_pole/rho_src)=2.838 is NOT an integer => the literal discrete iteration pole=F^n(source) is EXCLUDED; if the v82 map drives the transfer at all it must be the CONTINUOUS flow exp(t log F) at t~2.8 - the missing object is a continuous-time generator; (6) the 0.79% mismatch of rho_src vs -phi0/24 is far above ladder precision => 'exactly phi0/24' remains a [P] conjecture about the transfer normalisation. STATUS: basin/uniqueness/rate forced [I]; open [P]: the continuous generator and its flow time",
      "status": "Identity [I] (basin/rate/source) + Numerical [N] (pole) + [P] (transfer mechanism)",
      "canonicalStatus": "Identity/Numerical: basin lemma + exact rate + honest negatives; P2 narrowed to the continuous transfer generator",
      "dependencies": "FR.KOIDE.04; FLAV.PENCIL.04; v82",
      "scripts": [
        "v93_koide_relaxation_toy.py"
      ],
      "script": "v93_koide_relaxation_toy.py",
      "location": "tfpt_4_frontier;next.txt",
      "externalData": "PDG 2024 lepton pole masses",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 167
    },
    {
      "id": "GATE.WOLFRAM.02",
      "claim": "Second [C] path EXTENSION VERIFIED RUNNING: verification/wolfram/tfpt_readouts_extension.wl mirrors the v84-v110 round (frozen registry, master cover, reheating arithmetic, SO(16)_1 modular data, carrier index, conical defect chain, spine tetrahedron, glue uniqueness, Koide relaxation toy, sheet diamond v94, centered diamond v95, branch-kernel selection v96, sheet-conjugation bridge v97, discriminant dictionary v98, Koide flow time v99, horizon anchor v101, seam orientation v102, trisection normal form v103, Nariai clock v104, residual inventory v105, review validation v106, quantum-clock target v107, Pascal ladder v108, sheet pairing v109, Calderon-sheet selection v110, quadratic transport v111, self-counting channel v112, quasi-free kernel v113, torsion delta v114, anchor residue v115, resonance theorem v116, monodromy = W(A3) v117, hexagon-family dictionary v118, second review validation v119, address table v120, address pinning v121, margin theorem v122; inventory update v123 ledger bookkeeping Python-only; resummed clock v124, glue Q-system v125, clock-wall bridge v126, ring resummation v127, graded hull v128, entropy power law v129, Born square v130, measure-is-area v131, det-ratio anomaly v132, zeta budget v133, dual anchor v134, determinant surface v135, dual-normal selector v136, Q+ cohomology v137, VW firewall v138, selector triangle v139, canonical map v140, deck selection v141, frame integrality v142, graded Frobenius v143, det-ratio family cancellation v144, pairing atoms v145, Moebius D4 realisation v146, clock Gaussian model v147, NS/R sector census v148, cusp normal v149, replica EH model v150, BFK split v151, R3 normalisation = anchor v152, No-Unit Theorem v153, Simple-Current Extension v154, seam-net construction v156, rigid fixed point v157, fixed-point stability v158, PyR@TE gauge cross-check v159, QGEO rigidity v168, E8 slice compression v170, atomic OS moment + Sugawara v171, trace-anomaly seed v172, seam Fock readings v174, net existence + full-cone RP v175 - their [N] census/RH/ODE parts Python-only; v155 quasi-free boundary is numerical/numpy, Python-only; the v160-v167 + v163/v164/v166/v169 + v173 rounds are numerical/inventory, Python-only; v176 Seam Collar Realisation Theorem is an assembly/reduction certificate, its exact identities already mirrored via v168/v156/v162/v154/v175, so Python-only; v177 QGEO proof tree is Moebius/cohomology symbolic (uniformisation, omega_k characters, residues), Python-only; v178 MARKS/KERNEL deeper-reduction attempt is Moebius/Schur symbolic + numeric, Python-only; v179 conformal-realisation unification is arithmetic/bookkeeping (Gauss-Bonnet chi=2, h(A3)=4, rank/cohomology counts), Python-only; the v180/v181 Moebius reductions (uniformisation + Kerekjarto + finite-order Moebius / equivariant uniformisation) are symbolic + cited-theorem, Python-only; v182 reviewer-residual map is reduction bookkeeping + arithmetic, Python-only; v183 Koide F-corner 53/54 operator identity IS exact and mirrored; v184-v188 F_transfer-firewall round is numerical/bookkeeping Python-only; v189 Riemann-Roch carrier (h0=5/rankH1=3=rank(D5)/N_fam), v190 Nariai entropy bound ((x-1)^2), v191 universal branch line (affine relabeling + decoy control) ARE exact and mirrored; v192 energy-clock reformulation is [O] structural, Python-only; v201 QGEO.SUBPRIN.01 (a mu4-mark sum is Z4-invariant => sub-principal block-diagonal) and v203 HOR.EHT.01 (the EHT polarization coupling 16 c3^4 = 1/(256 pi^4) = delta_top/3) ARE exact and mirrored; the 2026-06-15 archive-integration round adds v204 FR.MUONG2.01 (a_mu^seam = 45/(524288 pi^9) = (5/4)(48 c3^4)^2/(2 pi)), v205 GRAV.XI.01 (xi = c3/phi_tree = 3/4 = q(A3) = ln(m/mu)), v208 HOR.BHTHERMO.01 (scalaron Wald f_R = 1 + R/(3 Ms^2) + modular 2 pi = 1/(4 c3)) ARE exact and mirrored; the 2026-06-15 pillowcase round adds v214 QGEO.PILLOW.01 (cross-ratio 2 => j=1728 => square modulus => order-4 CM clock, unifying v180 ISO + v201 SUBPRIN; chi_orb=0 + lemniscatic CM order 4 + neg control) and v216 QGEO.MARKS.03 (the four marks from Gauss-Bonnet: n=2chi=4 + Euclidean-orbifold uniqueness; numerical sibling v217 Python-only) exact and mirrored, while v206 H0 branch, v207 asymptotic safety, v209 BH defect are numerical/[P] Python-only; the 2026-06-15 diamond-axis-geometry round adds v218 DIAMOND.AXIS/PLUCKER/SPECTRAL.01 (det(C+xU)=14+6x linear vs det(C+yV)=14+14y+4y^2 quadratic, curvatures (8,6)=(rank E8,|R^+(A3)|); Plucker ladder K->C->F steps (1,8,10),(1,8,16); spectral ramification squares (1,3,4,6) kernels (13,48,65,105)) exact and mirrored, while its anchor-defect/pair-sum audit blocks + F=transfer-corner heuristic stay Python-only). First engine run 2026-06-11 (Wolfram Engine 14.3): v84-v93 block passed 45/45 on first run; v94-v97 block added same day; the 2026-06-17 closure round adds v259 PS.SPECACT.02 (the seam KMS cutoff f(u)=e^{-u} gives f_2/f_0=f_4/f_2=1 exactly, Gaussian control sqrt(pi)/2) and v260 ARCH.K3.01 (one Kummer/K3 carries seam+carrier-16+E8: E8 Cartan det 1/even/pos-def, K3 lattice U^3(+)E8(-1)^2 rank 22/det -1/even/signature (3,19), 16 nodes=|A[2]|=2^4=dim S+, 4 marks) exact and mirrored, while v258 PS.DIRAC.03 (Dirac-as-covariance log((1-C)C^-1)=H matrix-log identity) is numerical/Python-only; the 2026-06-17 v267 (QGEO.SYM.02 rigidity: cross-ratio 2, j=1728<=>lambda in {-1,1/2,2}, hexagonal j=0) and v268 (FLAV.TH13.01: theta_13 exponent 5/6=tr_E Y^2) are exact and mirrored; the 2026-06-18 round adds v271 QFT4D.SPERT.02 (concrete EG one-loop: bubble sd=4=d => omega=0 => one log counterterm, loop factor Omega_3/(2(2pi)^4)=1/(16 pi^2), phi^4 beta=3 lambda^2/(16 pi^2)) exact and mirrored, while v269 (S_pert skeleton) + v270 (PMNS Jarlskog, numerical) stay Python-only; v273 (QFT4D.SPERT.03: the EG one-loop gauge self-energy -> b3=-7, b2=-19/6, b1=41/10 from the content) is exact and mirrored, while v272 (nu-scale), v274 (over-determination) and v275 (QG.AMB.01 roadmap) are numerical/structural Python-only; v277 (QGAMB.TIERB.01: det Cartan(E8)=1 vs det Cartan(D8=SO16)=4 + the (E8)_1 character 248 currents) and v278 (QFT4D.SPERT.04: the one-loop optical theorem, bubble discontinuity (x_+ - x_-)^2 = 1-4m^2/s = the two-body phase space), v281 (QGAMB.MODULAR.01: #anyons=|det Gram| tower 16->4->1 + Gauss-Milgram c=8) and v282 (QGAMB.UNIFY.01: j=1728 + chi_E8(i)=1728^{1/3}=12) are exact and mirrored, while v276 (flat-pillowcase commutator), v280 (pillowcase DtN) and v283 (kill-test scorecard) are numerical Python-only; the 2026-06-22 arithmetic-arc round adds v313 GOLD.ATOMS.01 (affine-E8 charpoly factors x(x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1) with golden x^2-x-1; (2,3,5)=2cos(pi/k); 31/30 icosahedral selection), v314 RATE.TRANSLATE.01 (rational kernel vs Q(sqrt5) dynamic rate; translation on Q only), v315 COX.COUPLE.01 (30=5*6, EulerPhi(30)=8=rank E8, Galois mu4xZ2, sqrt5 Gauss sum), v316 GALOIS.READOUT.01 (rho=zeta6=zeta30^5, rho^4=-rho, Gal=Z2=CP conj), v317 GALOIS.FAMILY.01 (Nfam=3=mu3 orbit, 1+2 Galois split), v318 ARITH.CAPSTONE.01 (phi0=(4/3)c3+48c3^4 pure pi) and v320 GALOIS.CP.PREDICT.01 (delta_PMNS=4pi/3=delta_CKM,lead+pi=240deg) -- all exact and mirrored (10 checks), while v319 translation clock is structural Python-only; the 2026-06-22 rounds v325-v352 (continuum/realisation/framework: v325/v327/v337/v341/v342/v344/v345/v347/v348/v349/v350/v351/v352) and v354 E8.REVERSE.AUDIT.01 (3 of 8 E8 Casimir degrees {2,8,30} primary readouts, 5 {12,14,18,20,24} = the hull overhead; the golden ratio 2cos(pi/5) is unmapped structure, so not numerology) and v355 E8.UNMAPPED.BANDWIDTH.01 (the unmapped region's forced content is collective: sum(degrees)=128=dim S^+, product=|W(E8)|, exponents=totatives of 30; per-degree atom matches unforced and declined; reconciles v66+v354) are mirrored; and v358 GRAV.ENTROPY.EQUILIBRIUM.01 (the entanglement first law with TFPT atoms gives the linearised Einstein equation parameter-free 1/c3=8pi; the thermodynamic 2pi/eta and geometric |Z2|2pi chi origins of c3 coincide via |mu4|=|Z2|chi=4; CHM 3-ball weight integral 4pi R^4/15 assembles the matter flux J3) and v359 GRAV.NONLINEAR.01 (the full covariant G_ab+Lambda g_ab=(1/c3)T_ab via the fixed-volume equilibrium; Einstein-tensor trace (1-d/2)R=-R in d=4; both coefficients parameter-free 8pi=1/c3 and Lambda prefactor (8pi)^2*48c3^4=3/(4pi^2)) are mirrored; the 2026-06-24 sheet-generator round adds v410 SHEET.GEN.BINARY.01 (V=Q diag(0,1,1) is a binary internal compiler: V^n={{0,2^(n-1),0},{0,2^n,0},{0,2^(n+1)-2,1}}, V^n.1=(2^(n-1),2^n,2^(n+1)-1); bilinears 1.V^n.1=7 2^(n-1)-1, 1.V^n.a=7 2^(n-1), a.V^n.a=11 2^(n-1), a.V^n.1=11 2^(n-1)-2; readouts (6,7,9,11),13,27,55,56; theta sigma3(2)=9=a.V.1, sigma3(3)=28=1.V^3.a=det(I+R), sigma3(5)=126), v411 FLAV.UD.VPOWER.01 (c_u/c_d=(1.V^4.1)/((a.V.1)(1.V^2.1))=55/117, a re-encoding of 5*11/(9*13)), v412 DIAMOND.SHEET.SOURCE.J.01 (the Z2-wall corner J=M(1,-2)=C-2V, chi_J=lam^3-6lam^2+3lam-2, a.J.a=30=h(E8), det(I+J)=12, det(2I+J)=40), v413 DIAMOND.SHEET.CALCULUS.01 (e1=3t+12,e2=2t^2+15t+25,e3=4t^2+14t+14 => Delta e1=3, Delta^2 e2=4, Delta^2 e3=8; anchor energy a.M_t.a=52+11t) and v414 DIAMOND.CENTER.RESOLVENT.01 (resolvent ladder det C=14, det(I+C)=52, det(2I+C)=120 = G2->F4->E8^+) -- all exact and mirrored (5 checks), binary-spine/Lie readings stay [C]; the 2026-06-25 Gaussian/trichotomy round adds v415 DIAMOND.GAUSS.01 (J=[U,V]/3 is the mu4 quarter-turn J^2=-I born from [binary V, ternary U]; 3+2i/5+4i are eigenvalues of 3I+2J/5I+4J with norms 13=Delta_Q/41=10b1; the v222/v230 norms as operator identities; the intrinsic order-4 deck D=-I+J-J^2 with chi2 line ker[U,V]=a-1) and v416 ARITH.TRICHOTOMY.01 ({2,3,5} ramified/inert/split in Z[i] vs Z[omega], the ramified prime of each ring = its atom, each atom the unique ramified prime of one facet of Q(i,sqrt-3,sqrt5)) -- 7 checks; plus a one-character sign fix to the pre-existing v412 CharacteristicPolynomial check (Wolfram returns -(chi) for odd rank), so the file now runs genuinely clean; the 2026-06-25 Eisenstein/CP round adds v417 DIAMOND.EISEN.01 (the hex dual of v415: the family rotation P=(1 2 3) realises the hex norm 7=N_omega(3+2omega) via (3I+2P)(3I+2P^2)={7,7,25}; the CP clock W=-P^2 with W^2=P, W^3=-I gives delta_CKM=pi/3, delta_PMNS=4pi/3; all flavor clocks are zeta30-powers and the seam mu4 is the Galois side phi(30)=8=rank E8) -- 3 checks; the 2026-06-25 cyclotomic-norm-triple round adds v418 ARITH.NORMTRIPLE.01 (the carrier-5 clock = the 4x4 Phi5-companion C5 with the golden ratio in its 2cos values; N(3+2 zeta5)=det(3I+2 C5)=55=quark numerator; the triple det(3I+2 Comp(Phi_{3,4,5}))=(7,13,55) over Z[omega],Z[i],Z[zeta5]; neg control anchor (5,4)->(21,41,461)) -- 3 checks; the 2026-06-25 seam-Galois round adds v419 SEAM.GALOIS.01 (30 squarefree => no order-4 cyclic, so the seam mu4 is the Galois side (Z/30)^x=mu4 x Z2; the mu4 factor IS (Z/5)^x=Gal(Q(zeta5)) (carrier), the Frobenius zeta5->zeta5^2 the explicit order-4 operator G with G C5 G^-1=C5^2, G^4=I; resolves the v409 asymmetry) -- 3 checks; the 2026-06-25 Galois<->Net bridge round adds v422 SEAM.GALOIS.NET.01 (the seam mu4=Gal(Q(zeta5)) is the SAME cyclic Z/4 as the (E8)_1 simple-current glue: disc(A3)=disc(D5)=Z/4 cyclic via SNF, (Z/5)^x=Z/4, glue (1,1) order 4 => 16/4^2=1=E8; Klein/order-2 control => D8 det 4) -- 3 checks; the 2026-06-26 pentagon/golden round adds v429 DM.AXION.PENTAGON.01 (the 'unmapped' golden E8 structure surfaces in the one external input theta_i: since N_fam=g_car-2, theta_i=pi N_fam/g_car=(g_car-2)pi/g_car is the interior angle of the regular g_car-gon = the pentagon 3 pi/5=108 deg, cos=-1/(2 phi), golden character unique to g_car=5) -- 3 checks; the 2026-06-27 other-side audit round adds v430 E8.OTHERSIDE.AUDIT.01 (the double-cover 'other side'/deck |Z2|=2 + the 128-spinor S^+ (+) S^- is forced-disjoint from the five unmapped E8 Casimir degrees {12,14,18,20,24}: deck=matched degree-2=Min(deg), 128=8*16=2^(rank-1)=Total[deg] the spinor half of 248=120+128, sheet set {2,16,32,128} cap {12,14,18,20,24}={}) -- 3 checks; the 2026-06-27 degree-ladder round adds v431 E8.DEGREE.LADDER.01 (the five 'unmapped' degrees are the forced two-family decomposition deg(E8)=6*spine{2,3,4,5} U ({2} U det-ladder{8,14,20}): exponents=phi(30)=8 totatives of h=30=2*3*5 => degrees only in {0,2} mod 6; 6k/6=v91 spine, 6k+2={8,14,20}=(det R,det C,det L)=winding line 6s+8 (v135); 18=6*N_fam not a holdout; clean split special to E8 (E6/E7/D5 fail); arithmetic exact [E]/[I], functorial flavor map stays [P]) -- 6 checks; the 2026-06-29 degree-joint round adds v437 E8.DEGREE.JOINT.01 (all of E8 fixed by its degree multiset -- rank=8, h=30, 120/240 roots, dim 248, |W|=696729600, sum(deg)=128; the two structural integers (g_car,N_fam)=(5,3) are the unique roots of x^2-(rank)x+(h/2)=x^2-8x+15, sum=rank E8=8 + product=h/2=15=g_car*N_fam; mapped degrees {2,8,30}=min/rank/max -- a consolidation respecting the v354/v355 discipline, no new per-degree mining) -- 3 checks; the 2026-06-30 rigidity-forcing round adds v445 SEAM.RIGIDITY.FORCING.01 (the converse of v442: (i) entrywise forcing [rho,E_ab]=(i^a-i^b)E_ab=0 IFF a==b mod 4, verified all a,b in 0..31, residue-only => uniform in N; (ii) exact commutant dim sum n_s^2=4*4^2=64 PROPER subspace of N^2=256; (iii) the order is the discriminator, order-2 commutant 128 strictly larger than order-4 64, swept N=4..64, four marks |mu4|=4=h(A3), only index-4 is (E8)_1 det 1 vs SO(16) det 4) -- 3 checks (also Lean-hardened, FORM.SEAM.RIGIDITY.FORCING.01); the 2026-06-30 edge-CFT/modular round adds v452 SEAM.EQUIV.E8.MODULAR.01 (the (E8)_1 torus modular data: (i) S-invariance chi(-1/tau)=chi(tau) for chi=E4/eta^8 (one primary); (ii) T-phase chi(tau+1)/chi(tau)=e^{-2pi i/3}=e^{-2pi i c/24}, c=8; (iii) leading power chi*q^{1/3}->1 => c=8; (iv) holomorphic c=8=0 mod 8 + T-phase order 24/gcd(8,24)=3 -- E4 via its q-series since EisensteinE stays symbolic) -- 4 checks, v450 SEAM.EQUIV.EDGE.ENTANGLEMENT.01 (Calabrese-Cardy c per Majorana=1/2, N_Maj=16, c_-=8) -- 1 check, v451 SEAM.EQUIV.EDGE.CARDY.01 (the edge is Ising M(3,4): c=1/2, Kac h_sigma=1/16, h_epsilon=1/2) -- 1 check, v453 SEAM.RIGIDITY.MU4FROMMARKS.01 (the marks ARE mu4=roots of z^4-1, the form basis omega_k is mu4-graded, cross-ratio 2 fixed by z->iz => QGEO.SYM.01 from marks=mu4 + QGEO.REALIZE.01) -- 3 checks (also Lean-hardened, FORM.SEAM.EDGE.CHERN.01); the 2026-06-30 G-block round (post-F next-options) adds v454 SEAM.EQUIV.EDGE.VIRASORO.01 (level-1 Sugawara c=dim G/(1+h^v)=8 for BOTH SO(16)_1 120/15 and (E8)_1 248/31; the Casimir and Kac-Moody <J_n J_-n>=n fits Python-only) -- 1 check, v456 SEAM.S3.FROM-P1.01 (c3's '8'=one-sided count |Z2|*(intK/pi)=2*4; a reflection sends C->-C so two-sided forces C=0, one-sided allows C!=0; Chern computation Python-only) -- 1 check, v457 SEAM.EQUIV.KILLTEST.01 (the (E8)_1 character tower {1,248,4124,34752} from E4/eta^8; kill-test discriminators 248=dim E8=120+128, |det Cartan E8|=1 one sector vs |det Cartan D8|=4 four sectors) -- 2 checks, v459 SEAM.EQUIV.LATTICEVOA.01 (E8 even unimodular det 1; the 240 roots split 112 integer + 128 spinor; 248=8+240=120+128 -- the lattice route supplies the 128-spinor MMST v458 leaves open) -- 2 checks (v455 uniform Tomita-Takesaki tower + v458 MMST citation audit are Python-only/structural); the 2026-06-30 v461/v462 round adds v461 SEAM.S3.LOCALITY.01 (the Kapustin-Fidkowski strict-locality obstruction as integer logic: winding=|C|=1!=0 and c_-=8!=0, a strict finite-range commuting projector would force winding 0 => C=-C => C=0 so it is forbidden; the Wilson-loop winding/Chern are numerical, Python-only) -- 1 check, and v462 SEAM.EQUIV.SPINOR.01 (the Jacobi/E8 theta identity theta2^8+theta3^8+theta4^8=2E4 so chi_{(E8)_1}=E4/eta^8=chi_o^{SO16}+chi_s^{SO16}, and the (E8)_1 weight-1 248=120+128 with finite 16-Majorana Fock counts C(16,2)=120, 2^{16/2}=256=128+128; the finite-L convergence Python-only) -- 2 checks; the 2026-06-30 v463/v464 round adds v463 SEAM.EQUIV.UNIQ.01 (the c=8 holomorphic-uniqueness pin: c=8 has THREE level-1 candidates A8/80, D8/120, E8/248 all dim=8*(1+h^v), holomorphy forces dim V_1 = E4/eta^8 q^1 coeff = 248 = E8 excluding A8/D8; tower {1,248,4124,34752,213126}=E4/eta^8) -- 2 checks (v464 the one-particle realization rigidity is numerical, Python-only); the 2026-07-13 entropic-action bridge round adds v473 GRAV.ENTROPIC.ACTION.01 (the carrier Hodge count 1+5+10=16=2^(g_car-1) vs the 4d fiber 11 + the coefficient pin beta'_B=c3/6=1/(48 pi); the entropy potential s(x)=x-1-ln x with the Lambda-channel target Tr Q^2=32 c3^4=1/(128 pi^4)=(2/3) delta_top; the R^2 kill-test gap 3 c3^-9=3(8 pi)^9~1.2e13) -- 3 checks (the Frullani quadrature is numerical, Python-only, flagged in the .wl comment); the 2026-07-13 operator-level follow-up adds v474 GRAV.ENTROPIC.HODGE.01 (the Q-target enumeration d k^2 = 32 with integer k => supports exactly {(2,4),(8,2),(32,1)} = {|Z2|, rank E8, 2^g_car}, the 10/16 blocks excluded, minimal uniform q = c3^2 = 1/(64 pi^2), mode-ratio 32/48 = 2/3; and the fold-is-conjugation weight identity: traceless u(5) weights of Lambda^4 = negatives of Lambda^1, 5 -> 5bar, Pascal blocks {1,5,10} vs {1,10,5}) -- 2 checks (the 32x32 Fock/Clifford matrix checks are Python-only by the matrix-construction convention, flagged in the .wl comment); the 2026-07-13 kill-test round adds v475 GRAV.ENTROPIC.SCALARON.01 (the max-symmetric eigenvalues {R, R/4 x4, R/6 x6} of R~ g~^-1 incl. the 6x6 two-form flattening; the exact vacuum f(R) = 3 b R + (17/24) b^2 R^2 with the raw scalaron m^2 = 4608 pi^2/17 Mbar^2 (trans-Planckian ~51.7 Mbar), enhancement exactly (72/17)(8 pi)^9, domain pole 384 pi^2 Mbar^2; the Lorentzian timelike witness {1 - alpha v^2,1,1,1} vs spacelike {1 + alpha v^2,1,1,1}) -- 3 checks, all exact; the 2026-07-13 scale-flow round adds v477 GRAV.ENTROPIC.SCALEFLOW.01 (the Frullani/scale-flow identity -ln a = int dt/t (e^{-ta}-e^{-t}) symbolic + the moment dictionary at the flat weight = the v475 raw coefficients 3 b R + (17/24) b^2 R^2; the ONE moment condition mu2/mu1^2 = (72/17)(8 pi)^9 with the exact closure (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 and the v36 consistency f0 = 6(4 pi)^2/c3^7) -- 2 checks (the chi-form quadrature is numerical, Python-only, flagged in the .wl comment); current total 603/603 (verified engine run 2026-07-24, Wolfram Engine 14.3: the 16 deferred mirrors v479 (2) + v491 (3) + v493 (4) + v495 (3) + v496 (4) counted with the 2026-07-21 reactivation run, 378 -> 394; same-day v497 +4 -> 398, v498 +5 -> 403, v499 +5 -> 408, v500 +4 -> 412 and v501 +4 -> 416; same-day v502 +4 -> 420, v503 +4 -> 424, v504 +5 -> 429, v505 +5 -> 434, v506 +4 -> 438, v507 +6 -> 444, v508 +6 -> 450, v509 +7 -> 457, v510 +6 -> 463 and v511 +7 -> 470; 2026-07-22 v512 +6 -> 476, v513 +6 -> 482, v514 +8 -> 490 and v515 +8 -> 498, v516/v517/v518 -> 516, v519 -> 522, v520/v521/v522 -> 540; 2026-07-23 v523 +5 -> 545, v524 +4 -> 549 and v525/v526/v527/v528/v529 +3 -> 564), ALL WOLFRAM EXTENSION CHECKS PASSED. Kept separate from the 116/116 base file (GATE.WOLFRAM.01 unaffected). scipy-only parts (v86 pivot ODE, v88 data pulls, v93 PDG pole numerics) stay Python-only, flagged in the .wl comments; the statistical v100 numerology null test (grammar census + MC + RNG controls) is likewise Python-only by convention (like v62/v64/v65), flagged in the .wl comment. UPDATE 2026-07-20 (v491/P2.PARTITION.01): THREE exact mirrors of the partition corollary (unique {1,1,2} => e = (4,5,2); the four negative controls incl. the MS-rational 16/3 and 44/9; the v53 sharpening + sum rule) ADDED to the extension file; the VERIFIED count stays 378/378 -- the local engine is currently unactivated (license), so the new checks enter the counted total with the next verified engine run (same convention as the v479 mirrors); the identical statements are sympy-exact in Python (v491). The 2026-07-20 numerical companions v489 (modular commutator) and v490 (parity census) are numpy/scipy, Python-only by the suite convention, flagged in the README. The 2026-07-20 celestial round: v492 (CELEST.WP1.01) sympy-exact, mirrorable in principle, deferred to the same next engine run (docstring flag); v493 (CELEST.WP2.01) FOUR exact mirrors ADDED to the extension file (sharp clock-invariant selection + frozen j = 1728 quartic invariants + Coxeter/Picard-Lefschetz monodromy with the chi_1 eigenline and regular versal weights + linear-in-a0 S-algebra wrap with the EH k = 2 analogue), same convention -- counted only with the next verified engine run; identical statements sympy-exact in Python (v493); the 2026-07-21 celestial WP3/WP4 round: v495 (CELEST.WP3.01) THREE exact mirrors ADDED (the Deligne closed form lambda~^2 = h_vee + 6 with dimension formula and fundamental-index values; the E8 alignment (kappa/c3)^2 = 12 resp. 1/300 with irrational kappa/c3, anchors and the so8 factor-2 slip; the look-elsewhere pass counts 8/8 / 2/8 / 1/8 / 4/8 / 5/8 + exponents = totatives of 30) and v496 (CELEST.WP4.01) FOUR exact mirrors ADDED (the E4/eta^8 character through q^6 + q j cross-check; the equivariant Hilbert series with quadratic cumulative count and SU(2)+ slice; the two-sided jet-Fock mismatch with strictly increasing f_n/chi_n and the 27000 = h_vee^3 null ideal; the boundary bridges period sum 248 / no tower dimension / loop Fock 897266), same convention -- counted only with the next verified engine run; identical statements exact in Python (v495/v496). UPDATE 2026-07-21: the engine license is REACTIVATED and the run is VERIFIED -- base 116/116 ALL WOLFRAM CHECKS PASSED + extension 394/394 ALL WOLFRAM EXTENSION CHECKS PASSED (Wolfram Engine 14.3); all 16 deferred mirrors (v479 2 + v491 3 + v493 4 + v495 3 + v496 4) now enter the counted total, 378 -> 394; v484/v485/v486/v487/v488/v492 stay mirrorable-but-unmirrored (Python-exact), so they are not counted. UPDATE 2026-07-21 (v497/CELEST.WP5A.01): FOUR exact mirrors of the WP5a null-ideal round ADDED and VERIFIED the same day with the active engine, 394 -> 398 -- (i) the chi_w stabilisation family (u^n coefficient = quarter-moded loop Fock for all w >= n+1, strictly larger for w <= n; w = 2 = the chiral jet grading; loop Fock 897266 at u^4); (ii) the null ideal DERIVED via the Weyl dimension formula in the standard E8 frame (27000 = 30^3 = h_vee^3, 3875, Sym^2 count 30876; Casimirs (60,124,96), Dynkin indices 6750/750; level-1 integrability 4/2; quotient 31124 - 27000 = 4124); (iii) the two-route sector sum at q^2 = (1036,1024,1040,1024) = 4124 with glue diagonal (0,1,1,1); (iv) the SO(16)_1 / false-periodisation / swapped-glue negative controls (5304 != 14^3, 2076 = Theta_D8/eta^8, weights (0,1/2,1,1), period sums 64/124/248/496). The Freudenthal/peeling residual-zero recursion stays Python-side in v497; counting convention honoured -- counted only after the successful engine run (engine active). UPDATE 2026-07-21 (v498/CELEST.WP5B.01): FIVE exact mirrors of the WP5b singular-vector round ADDED and VERIFIED the same day with the active engine, 398 -> 403 -- (i) the case classification of the J^a_1 condition over the 248-element basis from the standard E8 root data (tally (190, 57, 1), exactly one case sees k; theta-grading dims (1,56,134,56,1); a + 2 theta re-enters for exactly one root a = -theta); (ii) the level dial DERIVED from the affine sl2 commutation (F^theta_1 (E^theta_{-1})^n|0> = n(k-n+1)(E_{-1})^{n-1}|0>; square coefficient 2(k-1) = -2(1-k) with values (-2,0,2) at k = 0,1,2; Shapovalov 2k(k-1) = 0/4 at k = 1/2; cube singular exactly at k = 2); (iii) the mu4 glue data (240 glue roots, classes (52,64,60,64), inner grading on all 240, <theta,h> = 5 => j(theta) = 1, clock phase i^2 = -1, cross-table rows (1,56,126,56,1)); (iv) the 27000 match + comark discriminator (mult(2 theta) = 1 in the level-2 Fock, 27000 = 30^3, (2 theta,2 theta)/2 = 4 > 2, E8 comarks (2,2,3,3,4,4,5,6) vs D8 (1,1,1,2,2,2,2,2) with h = (1/2,1,1), 5304 != 14^3); (v) the twisted-slot tension (no two C_1 modes sum to 8 quarters, minimum 6 = q^(3/2); per-period 248). The Chevalley/Frenkel-Kac construction and the PBW singularity run stay Python-side in v498; counting convention honoured (engine active). UPDATE 2026-07-21 (v499/P2.TYPING.01): FIVE exact mirrors of the P2 weight-typing round ADDED and VERIFIED the same day with the active engine, 403 -> 408 -- (i) the partition enumeration under the full typing (unique {1,1,2}, e2 = 5; BG dictionary h^0 = 0 <=> a_i >= 1 exhaustive; the naive-typing negative control h^1(E') = 3 => deg -6 with three splittings, e2 never 5; the h^1 discrepancy h^1(anchor bundle) = 1 != 3); (ii) the Deligne residue-trace degree (cusp exponents {0,1/3,2/3} = unique lift of spec(lam^3-1) to [0,1), trace 1 per mark, deg E = -4, par-deg 0, witness A0* char poly lam(lam-1/3)(lam-2/3)); (iii) the Schur-Horn permutohedron integer points {(2,2,0),(2,1,1)} with h^0 = 0 killing {2,2,0}; (iv) the Biswas Z3-cover degree arithmetic (genus 2, eigensheaves O(-2)^2, chi cross-check -1 = -1, regular-rep weights, decomposable model {0,2,2} with h^0 = 1 = the unstable companion; deck double genus 1, j = 1728); (v) the n-mark formula e2 = (n-2)(n+1)/2 = 5 only at n = 4. The sympy symbolic parts (H^1 residue basis, monodromy group closure |<U,M0>| = 24, the 324-case stability skeleton) stay Python-side in v499; counting convention honoured (engine active); counting convention honoured (engine active). UPDATE 2026-07-21 (v500/CELEST.WP5C.01 + v501/CELEST.WP5D.01): FOUR exact mirrors of the WP5c GNS-limit-state round and FOUR of the WP5d-alpha two-interval-index round ADDED and VERIFIED the same day with the active engine, 408 -> 416 -- v500: (i) the level-2 block census from the E8 root data (31124 = 248 + 30876 monomials in 9361 weight blocks, dim profile {164:1, 37:240, 7:2160, 1:6960}, orbit census (240, 6720, 2160, 240, 1)); (ii) the rank/kernel arithmetic (rank table (0,0,1,8,44) -> total 4124 = 1 + 248 + 3875 = chi_2 with dim V(omega) = 3875 by Weyl; kernel (1,1,6,29,120) -> 27000 = 30^3 = dim V(2 theta)); (iii) the clock-class split at the state level (glue-frame census -> (1036,1024,1040,1024) = Theta_Cj/eta^8 at q^2, level 1 (60,64,60,64)); (iv) threshold w r >= N+1 iff w >= N+1 sharp at w = N, <s|s> = 2k(k-1), D8 census 7380 in 2705 blocks quotient 2076; the exact 9361-block Gram/PSD/h.w.-norm computations stay Python-side in v500 -- v501: (i) the Cartan determinants (det D5 = det A3 = 4, product 16; det D8 = 4; det E8 = 1, explicit matrices); (ii) the KLM/Longo-Rehren quotients 16/4^2 = 1 and 16/2^2 = 4 -> 4/2^2 = 1 (both routes exact); (iii) the Sigma d^2 cross-checks (Ising 4, SO(16)_1 4, (E8)_1 1) + the index-reading arithmetic ([F:F_even] = 2, mu_gauged = 4, 16 ln 2 = ln 2^16, budget ln 4); (iv) the 16-fold-way condensability theta_v = 1 exactly at nu = 2 c_- = 16 vs rivals nu = 1, 2, 8; the lattice curves (Peschel/Renyi offsets, ln 2 plateau, duality asymmetry, ED validations) stay Python-only in v501; counting convention honoured (engine active) UPDATE 2026-07-21 (v502/CELEST.WP5E.ALPHA.01 + v503/QGEO.EMERGE.LIGHT.01): FOUR exact mirrors of the WP5e-alpha prefactor-and-level round and FOUR of the QGEO-emergence-light round ADDED and VERIFIED the same day with the active engine, 416 -> 424 -- v502: (i) the Hurwitz-zeta vacuum energies (zeta(-1,theta) = -B2(theta)/2 at rational twists, E_b = -1/24 + theta(1-theta)/4, Ising shift 1/16, vacuum 8(-1/24) = -1/3 = -c/24 with Sugawara 248/31 = 8, eta^8 exponent 8/24 = 1/3); (ii) discriminant form = Casimir (exact coset minima (0,5/8,1/2,5/8)/(0,3/8,1/2,3/8)/glue (0,1,1,1); spectral flow j^2(h,h)/32 with (h,h) = 20, (h',h') = 12, sum 32; free fermions n/16 on 10+6 = 16 = 2^(g_car-1) Majoranas; deck failure 3/16 != 3/8); (iii) the k = 1 fixing equations (closed forms 5k/8, 3k/8; glue integrality for ALL k = 1..8 -- the honest 'fixes nothing' finding; current condition h(J) = k = 1; embedding (k-1)(47k+266) and 128k(1-k); central charge 240(k-1) = 0; Shapovalov 2k(k-1) = (0,4,12,24); Weyl dims 248/3875/27000, 31124 - 27000 = 4124); (iv) the character sums (four glue-coset thetas = E4 = (1,240,2160,6720); sector characters (1,64,60,64)/(60,64,60,64) sum to q^(-1/3)(1,248,4124,34752); D8 control Theta_D8 + Theta_s = E4 with h_v = 1/2); the sympy symbolics and the PBW engine stay Python-side in v502 -- v503: (i) the CM-point identities (q-series E2 = 3/Pi at i and 2 Sqrt 3/Pi at rho to 30 digits => E2*(i) = E2*(rho) = 0 exactly; j(i) = 1728, j(rho) = 0 KleinInvariantJ; DedekindEta[I] = Gamma(1/4)/(2 Pi^(3/4)) exact); (ii) the no-dynamical-selection witnesses (logF S-invariant, hexagonal winner logF(rho) - logF(i) = 0.02116..., saddle signature of tau = i); (iii) the pillowcase halving (exact N = 8 spectrum, 3 nonzero self-paired cone modes + 30 pairs, dim(even) = 34 = (N^2+4)/2, det'_even^2 = det'_torus x prod_self exactly); (iv) clock => square + Coxeter (SolveAlways a3 = a2 = a1 = 0, j = 1728 identically, charpoly lam^3+lam^2+lam+1 order 4 = |mu4| = N_fam + 1, Aut(hexagonal) = Z6 without order-4 element); the lattice DtN curves stay Python-only in v503; counting convention honoured (engine active) UPDATE 2026-07-21 (v504/CELEST.WP5DB.01): FIVE exact mirrors of the WP5d-beta split-plus-strong-additivity round ADDED and VERIFIED the same day with the active engine, 424 -> 429 -- (i) the GF2 span theorem (Majorana monomials multiply as GF(2) labels: shared boundary Majorana => Even(A) v Even(B) = Even(A u B) exactly 64/64, 256/256, 1024/1024; disjoint exactly half = index 2); (ii) the Pimsner-Popa identity + attainment (E(a) - a/2 = PaP/2 identically on a symbolic 4x4 matrix; explicit gamma matrices give E(x*x) - lam x*x = 2(1-lam) - 2 lam h with spec(h) = +-1 => lambda_PP = 1/2 = 1/[F:F_even] exactly); (iii) the two-interval group + index consistency (E4(vv+) = I/4 exactly => lambda_E4 = 1/4 = 1/mu; (1/lambda_PP)^2 = 4 = 1/lambda_E4 = mu(SO(16)_1) = det Cartan(D8)); (iv) the Lambda^2 compound (Minors[.,2]: Cauchy-Binet functoriality, diagonal products {sigma_a sigma_b}, Minors[-C,2] = Minors[C,2] identically -- the exact orbifold split-inheritance mechanism); (v) the U(1) PP control (charge dephasing lambda = 1/(m+1) -> 0 exactly: infinite index). The lattice curves (deficit tables, Ising exponent p = 0.2444, elliptic-nome ladder, Klich-Levitov slope, ED validations) stay Python-only in v504; counting convention honoured (engine active) UPDATE 2026-07-21 (v505/CELEST.WP5E.BETA.01 + v506/SEAM.CLOCK.RIGIDITY.01): NINE exact mirrors ADDED and VERIFIED the same day with the active engine, 429 -> 438 -- the v505 WP5e-beta round (FIVE): (i) the AB denominators + equivariant characters (det(1-g_j^-1) = (2,4,2), Dedekind sum 5/4 = (|Z4|^2-1)/12; characters (248,0,-8,0) by TWO routes -- graded dims and the explicit 240-root sum; invariant average 60 = carrier, Frobenius 61568); (ii) the index bridge (ch2(T_m) = -(C^-1)_mm/2 = f(m) exactly; glue defect -78 both routes, classes (-24,-30,-24); sum_m h_A3(m) = 5/4 = sum_j 1/det_j); (iii) Okubo per sector + the rigid 32 T3 residual (the invariant quartic sum over the 240 glue roots = 36<x,x>^2 exactly; twisted quartics carry T5/T3 content; AB-weighted sum = 9P1 - 30P2 - 15P3 + 32T3 exactly); (iv) the level dials (lattice current count 240/0/0/0 for k = 1..4; embedding residual (0,360,814,1362); det(A-1) = -4); (v) the axion-slot weight arithmetic (clock weights = regular mu4 rep; a0 Hamiltonian in O(2) = the graviton slot, X = -2 != +2 = Y, mismatch 4 = |mu4|; Coxeter eigencharacters {i,-1,-i} = the three twisted slots); and the v506 clock-rigidity round (FOUR): (i) the complete square-root solve (g^2 = deck forces diagonal with (a/d)^2 = -1 -- exactly two roots z -> +-iz, both projective order 4; the traceless branch scalar with det 0); (ii) the D4 center + order-4 count + the silver counterexample (the mu4 stabiliser has profile {1,2,2,2,2,2,4,4}, exactly 2 order-4 elements squaring to the deck, center = {id, deck}; the silver set {+-1, +-(3+2 sqrt 2)} has j = 1728 with a full D4 of 8 exact elements but its order-4 elements square to the central crossing involution, not the deck: zero roots); (iii) the Cl(16) identity (explicit 256x256 Jordan-Wigner gammas: Utilde^2 = 256 gamma_1...gamma_16 exactly, U^2 = (-1)^F; the R implementer squares to +256, split control); (iv) the n-arithmetic + generic kill (S4 orders {1,2,3,4}, kill table n = 5..8, free orbits n | 4, cusp degree 5 != 3, transfer spectrum {1, 64/729, 1/729}, generic V4 exponent 2). The 24-triple stabiliser enumeration, the A4 profile and the Z8 block-lift nullspace stay Python-only in v506; counting convention honoured (engine active) UPDATE 2026-07-21 (v507/SEAM.BIT.ORIGIN.01): SIX exact mirrors of the bit-origin round ADDED and VERIFIED the same day with the active engine, 438 -> 444 -- (i) the half-period descent (fixed points of w -> -w = the 4 half-periods Lambda/2Lambda, 1/3-control fails; t_c descends iff 2c in Lambda = exactly the 3 nonzero half-periods; the explicit rational descent sigma_i(x) = e_i + (e_i-e_j)(e_i-e_k)/(x-e_i) with the exact curve-lift identity prod(sigma_i-e_l) = psi_i^2 prod(x-e_l) symbolic in e1,e2,e3 and the Klein-V4 closure); (ii) the core solve (pair cross-ratio = -1 solves exactly to lambda = -1/2/1-half per half-period; harmonic orbit j = 1728 vs j(3) = 21952/9 and j(hex) = 0); (iii) the stabiliser counts (complete 24-triple enumeration: (4,3,3)/(8,5,3)/(12,3,3), free involutions = the three sigma_c in ALL types); (iv) the CM fixed point (mult-by-i mod 2 fixes exactly (1,1) = c*, mult-by-rho 3-cycles; the lemniscatic clock lift needs psi = 2i/(x+1)^2 with t^2+1 irreducible over Q and squares to the central -1/x); (v) the Cl(16) dichotomy (central Utilde^2 = 256 gamma_1...gamma_16 vs the edge implementer squaring to the SCALAR +2^7 on the same 256x256 Jordan-Wigner gammas, with implementation verified on all 16 generators; root count 2 vs 0 in the 32-element NS dihedral lift group); (vi) the silver bijection census (all 8 silver bijections -> edge sigma_2/sigma_3, never central; all 8 mu4 bijections -> central sigma_1 only). The v493 circularity-audit direction and the 28/21 arrangement census stay Python-side in v507; counting convention honoured (engine active) UPDATE 2026-07-21 (v508/CELEST.WP5E.GAMMA.01): SIX exact mirrors of the WP5e-gamma round ADDED and VERIFIED the same day with the active engine, 444 -> 450 -- (i) the vertex-space dims (the W(D5) x W(A3)-invariant quadratics on the 8-dim glue Cartan have bidegree dims (1,0,1), total 2 = span{s5, s3}, and the quartics (2,0,1,0,2), total 5 = span{P1,P2,P3,T5,T3} -- exact generator-wise Weyl nullspace arithmetic); (ii) the twisted quadratics + the even-sector collapse (K^(0) = 60(s5+s3), K^(1) = K^(3) = 4s5-12s3, K^(2) = -4(s5+s3), K^(0) = -15 K^(2) parallel); (iii) the product theorem (symbolic a,b,c,d: any two invariant quadratics multiply into span{P1,P2,P3} -- zero T5/T3 content); (iv) the rank-3 certificate (channels E13 = (16,-96,144,0,0), E22 = (16,32,16,0,0), E00 = 225 E22; rank 2, kernel 0, rank([M | A_fix]) = 3 with A_fix = (9,-30,-15,0,32) and (Phi_T5, Phi_T3, Phi_P)(A_fix) = (0, 32, 72); naturalness images (25/4,-11/2,97/4,0,0)/(12,-40,76,0,0) both certify (0,0)); (v) the SO(16) AB sum (15,-18,-15,-4,40); (vi) the D8 native quartic 28 s8 Killing + 16 T8 + 12 s8^2. The selection-rule triple enumeration and the flipped-rule control stay Python-only in v508; counting convention honoured (engine active) UPDATE 2026-07-21 (v509/CELEST.WP5E.EPS2.01): SEVEN exact mirrors of the WP5e-epsilon-2 round ADDED and VERIFIED the same day with the active engine, 450 -> 457 -- (i) the CPS period integrals (S^3 period of the brane back-reaction 3-form = (2 pi i)^2 exactly, pullback density -sin 2theta; exceptional-sphere Kahler flux = 2 pi N exactly, flat part dies at t -> 0); (ii) the level~flux contraction (+2N/-2N for N = 1,2,3) with Dynkin T(so8 vector) = 1, T(adj) = 6; (iii) the pairing-matrix enumeration 48 -> 2 (rows (8,6,8), 48 unimodular, effectivity cuts to {identity, flip}); (iv) the three fluxes (64,60,64) = dim g_i, total 188, flip-invariant, per-current (1,1,1); (v) the sector-counter dial #prim((E8)_k) = (1,3,5,10,15,27) for k = 1..6, one sector iff k = 1, KLM 16/4^2 = 1; (vi) the lockstep theorem + the 0/24 negative probe (a3 = a2 = a1 = 0 forced, squared moduli 2t^2 equal, det(A-1) = -4, Z^4 - Z disc -27 with zero lockstep orderings); (vii) the negative controls (SO(16) fluxes (0,60,0) with defect -30 and condensation stuck at 4; A2 det 3, glue diagonal 23/24, mu = 12 no square, Coxeter 3 != 4). The lattice current count (240,0,0,0) and the k = 2 dials are already mirrored in the v505 round; counting convention honoured (engine active) UPDATE 2026-07-21 (v510/SEAM.BIT.FREEDOM.01): SIX exact mirrors of the seam-bit-freedom round ADDED and VERIFIED the same day with the active engine, 457 -> 463 -- (i) the dihedral census + quotient topology (D_16: 17 involutions, exactly 1 free = the antipodal shift 8; all 16 reflections with exactly 2 circle fixed points, 8 site- + 8 edge-axis; robustness N = 8, 12; odd control N = 9 zero free; chi(circle/antipode) = 0 closed vs chi(circle/reflection) = 1 interval with 2 corners); (ii) the Aut completeness (Aut(C_8) = D_8 by brute force over all 8! = 40320 permutations; Aut(C_16) = D_16 by degree-2 propagation, all 32 candidates); (iii) the Cl(16) dichotomy census (the free deck's NS implementer squares to 256 gamma_1...gamma_16 = 256 (-1)^F nonsplit while the implementers of ALL 16 reflections exist and square to scalars {2^7, 2^8} -- NONSPLIT <=> FREE over the complete 17-involution census; root dichotomy 2 vs 0 total on explicit 256x256 Jordan-Wigner gammas); (iv) the real-structure table (M D = +-D M solves symbolically to the diagonal/antidiagonal families; per configuration mu4 -> equatorial/FREE, silver -> real-axis/NOT free, generic-real -> real-axis, generic-unit -> equatorial/free, hexagonal -> NO mark-fixing real structure); (v) harmonic + free => mu4 ((1-b)^2/(1+b)^2 = -1 solves exactly to b = +-i; lambda table (2, 1/2, 3/4, 5/4) real, hexagonal Im lambda = sqrt(3)/2); (vi) the honest counterwitnesses ({0,1,8,9} shift-invariant + crossing but not quarter-shift-invariant; {+-1, +-(3+4i)/5} with pair cross-ratio -1/4, j = 148176/25, clock not preserving). The Moebius/Klein covering models and the pillowcase branch control stay Python-side in v510; counting convention honoured (engine active) UPDATE 2026-07-21 (v511/CELEST.WP5E.DELTA2.01): SEVEN exact mirrors of the WP5e-delta-2 full-tensor-ledger round ADDED and VERIFIED the same day with the active engine, 463 -> 470 -- (i) the innerness count + g0 structure (240 glue roots, class split (52, 64, 60, 64); h = (2,2,2,2,2;0^4) and h' = (0^5;1,1,1,-3) read the class mod 4 on all 240 roots; class-0 roots split 40 pure D5 + 12 pure A3, no mixed root, root span rank 8 -- g0 = d5 (+) a3 semisimple, no u(1); class(-alpha) = -class(alpha)); (ii) the bilinear Hom table by exact Kostant alternating sums on the minuscule weight orbits (nonzero ONLY at j' = -j, dims (2, 1, 1, 1), all 12 charged pairs zero); (iii) the trilinear Hom table (all 15 non-neutral triples zero, neutral dims (3, 2, 2, 1, 1)); (iv) the unique symmetric survivor (Inv(S^3 adj) = (0, 1) for (d5, a3), Inv(L^3 adj) = (1, 1), Inv(S^2 adj) = (1, 1); Inv(S^3 g0) = 1, Inv(L^3 g0) = 2 -- the su(4) d-symbol; dual guard 4 x 4 = 0 vs 4 x 4bar = 1); (v) the controls (SO(16)/D8 bilinears (1, 0, 1), trilinears (1, 0, 1, 0), Inv(S^3 adj_d8) = 0, Inv(L^3 adj_d8) = 1; false g0 = d5+a2+u(1): (5, 2, 2) != (2, 1, 1), Inv(S^3 g0) = 6 != 1; false sector (16_s, 4bar): both pairings 0 vs true value 1); (vi) Q_dd by two routes (explicit su(4) matrices: brackets/root-contractions/tadpoles all 0; Q_dd = (0, 0, -1/240, 0, 1/60) = (1/60)(T3 - P3/4) agreeing with the hand route |x^2 - (tr x^2/4) 1|^2/60; Tr X^3 = 0 antisymmetric); (vii) the psi certificate + the relaxed solution (E13/E22/E00/A_fix replicated; rank 3 vs augmented 4; psi = Phi_P - Phi_T3/4 annihilates all four channels, psi(A_fix) = 72 - 8 = 64; relaxed A_fix = -u + 8v + 2w + 1920 Q_dd exactly, 1920 = |W(D5)| = 8 x 240). All v511 content is exact and fully mirrored; counting convention honoured (engine active) UPDATE 2026-07-22 (v512/SEAM.TAU.FLAG.01 + v513/CELEST.DTERM.NONDERIV.01): SIX exact mirrors of the flag-transitivity round ADDED and VERIFIED the same day with the active engine, 470 -> 476 -- (i) the V4/D4 census table over five exact members incl. flag/arc orbit counts (generic (4, 2, transitive, 2, 2), mu4 (8, 4, transitive, 1, 1)); (ii) the solveset equivalences (b^2 = -1 / cos delta = 0 / tan(delta/2) = 1 / delta = pi - delta all reduce to {pi/2}) + family identities (lambda = sec^2(delta/2), cross-ratio = -tan^2(delta/2); counterwitness lambda = 5/4, j = 148176/25, cross-ratio -1/4); (iii) the spectral faces in closed form (odd-split exp(pi split/2) = cot(delta/2), arc-Laplacian spectrum, K3 Frobenius^2 = 8(u-v)^2); (iv) the twist mode identity + AB delta-blindness + j-rationality control; (v) H4 rigidity (centraliser scalar, intertwiner prop U, U^2 nonscalar, V4 orders {1,2,2,2}); (vi) the negative controls (hexagonal/silver/Z16 0-vs-28 census/n = 3 rigidity) -- the mpmath heat-trace numerics and the numpy lattice commutator stay Python-only in v512, flagged; and SIX exact mirrors of the c_d-nonderivation round ADDED and VERIFIED the same day, 476 -> 482 -- (i) the propagator anchor sum <alpha,e>^2 = 120 + T3-slot pinning (det -64, c_d = 1920); (ii) the five-way convention table (1920, 120, 256, 32, 69120) with exactly one |W(D5)| hit + the stable factorisation c_d Phi_T3(Q) = 32 in all five; (iii) the flux enumeration (2048/1800 bracket-and-miss, 64*60/2 charge-forbidden) + look-elsewhere ledger (924 expressions, 11 hits on 1920; controls 8/0/5); (iv) the K2 lattices (quantum 6 vs 3/8, 60 Z 32nd multiple, 2160 Z excludes, 12 factorisations); (v) the K3 provenance clash (A_m = (0,16,0,-16), 32i vs (10,-4,30,-4); |W(D5)| = |W(E8)|/|W(A8)| = 1920, |W(A3)| = 24); (vi) the K4 controls (SO(16) cubic sums 0, k = 2 doubling 7680, false g0 six cubics). All exact content of both rounds fully mirrored; counting convention honoured (engine active) UPDATE 2026-07-22 (v514/CELEST.WP5E.EPS1.01): EIGHT exact mirrors of the WP5e-epsilon-1 bulk-axion-slot round ADDED and VERIFIED the same day with the active engine, 482 -> 490 -- (i) the equivariant Penrose ledger (pushforward blocks = exact wave-operator kernels block by block, all d <= 6, all four characters, (d+1)^2 per degree); (ii) the character series P_0 = 1 + 3t^2 + 15t^4 + 21t^6 + 45t^8, P_1 = P_3 = 2t + 8t^3 + 18t^5, P_2 = 6t^2 + 10t^4 + 28t^6 with the surviving bulk-axion zero mode (1,0,0,0), density 1/2 = (1/4) x 2 and twisted minimal content (2t, 6t^2, 2t); (iii) the Molien hypersurface (1-t^8)/((1-t^4)^2(1-t^2)) with relation degree 8 = the a0 weight + the diag(i,i) Veronese control (1+3t^4)/(1-t^4)^2 and the Coxeter clock det(A-1) = -4; (iv) the Okubo residue (6<x,x>)^2 on the 240 glue roots + the so8 irrationality control (perfect squares {9, 36} only); (v) the measure cancellation (lam_eff^2 = 36 for arbitrary mu, wrong bookings 9 and 144 excluded) + the flux dial (#primaries (1,3,5,10), (kappa/c3)^2 = 12); (vi) the GH centre dichotomy re-deriving the v493 family (prod(Z - i^p z0) = Z^4 - z0^4, monodromy = one clock step); (vii) the period lockstep 4 pi t0 (i-1)(1, i, i^2) + the GH point = R^4/Z_k cone + the Coxeter clock Pi.A = i Pi from geometry; (viii) the multipole selection rule m = 0 mod 4 with the (4, +-4) amplitude (35/16) sin^4 th + the EH log ledger (det g = 1 exact, asymptotic log coefficient 0, exceptional flux 2 pi a^2, Burns contrast det g = 1 + N/u). All exact v514 content fully mirrored; counting convention honoured (engine active) UPDATE 2026-07-22 (v515/CELEST.WP5E.M1.01): EIGHT exact mirrors of the WP5e-M1 back-reacted-Omega_N round ADDED and VERIFIED the same day with the active engine, 490 -> 498 -- (i) the residue-form identities (invariant-cover pullback 4 dz1^dz2 = |mu4| x flat, deck-invariant, clock phase i; weight ledger 4+4+2-8 = 2, Omega in O(4) = -deg K_PT; EH anchor 2 = |Z2|; wrong-weight control 4 != 2); (ii) the twistor family closed (O(2) reality of the four centre sections, XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4 (1 - la^4)^2 exactly, seam fibre = the v493 family, self-mirror, CY-compatible clock lift gamma^4 = 1 with Omega -> +Omega); (iii) the period reduction + lockstep (2 pi i (q_b - q_a) generic, seam vector 2 pi i t0 (i-1)(1, i, i^2) with equal squared moduli 8 pi^2 t0^2, covariance Pi_{j+1}(-i la) = i Pi_j(la) for all la, Pi.A = i Pi, closure 0, residue 0 = the zero-flux Omega_0 baseline); (iv) the collision ledger (12 nodes exactly on the 8 eighth roots of unity, adjacent la^2 = i^(2j+3), antipodal double nodes, clock orbits 4+4+4); (v) the BM anchor (density -sin 2 theta, int_{S^3} K = (2 pi i)^2, iterated two-circle residue) + the lens forcing N in 4Z (only N = 0 mod 4 passes; fractional charges excluded); (vi) the source curve (four centre lines on the family, clock-permuted, K4-connected with 6 edges / 12 points, cover count 16 = |mu4|^2); (vii) the conifold nodes (Hessian det 512 t0^6, quadric rank 4, transversality, Pi_0(la0) = 0); (viii) the KILL probe + negative controls (forbidden Z^4 - Z family 0/24 lockstep vs 8/24, twelfth-root support != Z8, no antipodal pairing -- integrality alone NOT the discriminator; EH charge 2 forced; diag(i,i) collapse). The dict-based exterior-algebra identity dK = 0 and the generic-profile sphere parametrisation stay Python-side in v515, flagged in the .wl comment. The 2026-07-22 WP5e-M2+M3 rounds: v516 (CELEST.WP5E.M2.01, the twisted KS measure) adds SIX exact mirrors (completion-weight identity w_m = sum_j (1 - i^{jm})/det_j = (0, 3/2, 2, 3/2) = 4 h_m = -4 ch2(T_m) with the index bridge and no free scale; root/quartic replication (class quartics, A_fix two routes, bulk Okubo 36<x,x>^2 = (K^(0))^2/100); parameter-free locks (T5 = 0 symbolic, T3 budget forces c = 4 = |mu4|, det 256); contact term two forms + the success table (per-sector Okubo squares 36<x,x>^2/det_j, total 45<x,x>^2 = (5/4) x 36, certificates 32/72/psi = 64 killed, exchange span, mu-blind chain); negative controls (wrong scale, shuffle, SO(16) KILL, diag(i,i), Z2/EH anchor); multiplier note (T-phase orders (1,8,2,8) squaring to (1,4,1,4), Gauss sums)), 498 -> 504 -- the declared completion reading itself is the [C] fence, a typing, not mirrorable; and v517 (CELEST.WP5E.M3.01, the a0 uplift) adds SIX exact mirrors (GH multipole ledger + family at general orbit phase; kernel bridge (null coordinate for arbitrary kernels, residue 1/(2r), flux -4 pi); asymptotic kernel log 4 = |mu4| + exact m-grading + GLT tower p_{4k} = 4(-a0)^k; exceptional-locus log 4 log t0 + i(4 phi0 + pi); period response 1/4 -> monodromy i with a0-rigid nodes and K4 discriminants; negative controls ((4,0) phi0-free, EH dials all 2, k = 3/5 move, forbidden family fails both dials)), 504 -> 510 -- the GLT contour-transform typing stays the [C] dictionary, the BM anchor is already mirrored in the v515 round; and v518 (CELEST.WP5E.DELTA1.01, the delta-1 chain decided -- kill under the derived measure, tension with v516 stated) adds SIX exact mirrors (the 16-component Weil system of the Z4 x Z4 discriminant module with Gauss sums 2 zeta_8^5 x 2 zeta_8^3 = 4 and the full exact relations + tensor split; the two invariant Lagrangians with P_H S P_H = (1/4) J and the 3/4 shifted-class weight; the koboundary test (1,1,1) on all 15 pairs from the recorded exact transport phases; the Gamma_1(4) character formula lambda(gamma) = i^{2B + C/4} on the recorded loops; the T-fix mechanism (-1) e(-1/6) = e(1/3) = chi_4(T) uniform for f1f3 vs inconsistent {chi_3, chi_4}/{chi_5, chi_6} for f1f2f3/f2; the wrong-form/wrong-signature controls), 510 -> 516 -- the Harvey-Moore integrals, SVD contraction solves, theta covariance certificates and block transport measurements are numerical (mpmath/numpy), Python-only, flagged in the .wl comment; counting convention honoured (engine active); and v519 (WOIT.THETA.FREE.01, the WOIT-alpha milestone -- Theta + free RP) adds SIX exact mirrors (the two-family Theta classification with the symbolic Solve branches and the family-D/family-A relation tables; mark pinning + the mu4 torsor with the clock-orbit map mu -> -mu, the mark permutations and the 2-cut-point census; the C^4 level rho_tw vs sigma_std (Theta_tw^2 = -1, -i rho4, deck centralised vs conj = inverse, +1, RP^3); the Z8 tower conjugation conj(zeta8^k) = zeta8^-k with the spin-lift inversion/centralisation split; the Cl(16) dichotomy on explicit 256x256 Jordan-Wigner gammas (even NS reflection implementers with U_r^2 = 128 = 2^7 on both mu4 axes, the quarter-shift clock lift V with U_r V U_r V = 2^19 i.e. Theta_Fock V Theta_Fock^-1 = 4096 V^-1, Utilde^2 = 256 (-1)^F and Utilde V = V Utilde); the free-RP preconditions + site-cut degeneration + Cauchy-Stieltjes continuum control (C^2 = -1 at N = 8, R C R^T = -C / S C S^T = C / R S R^-1 = S^-1 / deck fails at N = 16, the 7x7 site-cut Gram with zero diagonal and det = 0, the exact factorisation of 1/sin((s+t)/2) with minors 1/2, 1/72, 1/43200, 1/423360000)), 516 -> 522 -- the 40-digit PD inertia certificates of the bond-cut Grams ((8,0,0)/(29,0,0)/full N = 8 half-algebra) are numerical (mpmath eighe), Python-only, flagged in the .wl comment; and v520 (CELEST.WP5E.MEASURE.01, the WP5e measure decision) adds SIX exact mirrors (the 16-component Weil data with both Lagrangian gluings T-/S-fixed and isotropic; the invariant-subspace forcing: joint fixed space dim 2 = span{e_H, e_H'} by exact NullSpace + rank; the character arithmetic of the two constructive sources |chi_4(T)|^2 = 1 vs chi_4(T)^2 = e(2/3) with the exact 1/960-grid squared defects 2x480 = 0 vs 2x800 = 640; the A-core + Z2/EH-anchor targets w = 4h, psi(A_fix) = +64/psi(contact) = -64, A_2 + contact_2 = 9<x,x>^2 with the c = 1/2 scale teeth; the level-1 Okubo identity of BOTH gluings (the anti-diagonal H' has exactly 240 norm-2 vectors with summed quartic 36<x,x>^2); the so16/shuffle/diag(i,i) controls), the Harvey-Moore integrals/SVD memberships/J-weighted closures numerical (mpmath/numpy), Python-only, flagged in the .wl comment; and v521 (SEAM.BIT.RPBLIND.01, the RP-blindness kill) adds SIX exact mirrors (the anti-linear census solvesets on M(delta) incl. the exact-rational counterwitness census; the V4 -> D4 closure / face #13 as symbolic composition identities; the 20 cut/mark incidence solvesets all empty + the eps-jump + the integer lattice frame; the continuum axis elimination 1/sin((s+t)/2) with d/d alpha = 0 exactly + the Cauchy-Stieltjes minors + the counterwitness cut ordering; the clock-tower clause census {F,F,F,T} + the hex/silver precondition controls; the delta-blind side conditions Theta_mu^2 = +1 + the mu4-torsor clock relation with the order-4 solveset {pi/2}), the 40-digit lattice Gram inertia spectra (N = 16/N = 32) numerical, Python-only, flagged in the .wl comment; and v522 (WOIT.BETA1.GSO.01, the WOIT-beta1 milestone) adds SIX exact mirrors (the Z8/NS tower S^4 = -1 + the Ramond state-invariance witness 1/(4 sin(5 pi/16)); the invariant-dimension chain 120 > 64 > 32 by exact ranks of the averaged signed pair action; the time-like reflection census (all 16 axes invert the clock, none commutes, same at N = 8); the exact Hermiticity witness T_1[(8,12),(8,15)] = -i/(8 sin(5 pi/16)) with the full 745/96/0 entrywise census and the non-Hermitian clock average on the even 29-basis via exact Pfaffian-Wick; the GSO 2-torsion census (even {T,F,F,F}, odd {T,F,F,F,T,F,F,F}, T_{k+4} = -T_k, vanishing 8-term average); the family-A tower-normalisation failure census {F,F,F,F} + the alternating-dressed anti-invariance), the GSO-fixed RP inertia certificates ((29,0,0)/(8,0,0)/(7,9,6)/(17,12,0)) numerical (mpmath eighe), Python-only, flagged in the .wl comment, 522 -> 540; counting convention honoured (engine active). UPDATE 2026-07-23 (v523/CELEST.WP5E.WM.01 + v524/WOIT.BETA2.OS.01 + v525/v526/v527/v528/v529/SEAM.BIT.TWISTBLIND.01): FIVE exact mirrors of the constructive w_m derivation ADDED (the mode ledger = closed form at every order n <= 40 with Abel value (1/2, 1/4, 1/2) = 1/det_j, Dedekind sum 5/4, product 16; the zeta/Quillen route with the SYMBOLIC reflection formula (2 pi/(Gamma(a) Gamma(1-a)))^2 = 4 sin^2(pi a), det Delta = det_j^2 = (4, 16, 4), the unique real positive section and the diag(i,i) control (2i, 4, -2i) with fake weights (0, -1/2, 0, 3/2); the derived-weight contact vectors + per-sector Okubo squares + total 45<x,x>^2 + psi +-64 + the Z2/EH anchor on the 240 glue roots; the consistency chain w = 4h with the T3 budget forcing c = 4 and both wrong-weight controls incl. the Q(sqrt2) leftovers; the k = 3, 5 cyclotomic wandering controls w_m = m(k-m)/2), FOUR exact mirrors of the WOIT-beta2 OS quotient ADDED (the alpha_1^N = (-1)^F wrap arithmetic at N = 16 and N = 8 + the exact certificate sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2; the Klein-Landau one-particle transfer forms exactly Hermitian on all domains k = 1..7 with odd-k zero diagonal and the even-k square identity entrywise; the exact N = 8 clock spectrum {1, sqrt(2)-1} = {1, 1/delta_S} via generalized eigenvalues + the transparency pi/16 product formula certified symbolically; the KMS silver witnesses C(1)/C(3) = 1 + sqrt(2) and det(G - tau_4) < 0 exact), and THREE exact mirrors of the twist-state kill ADDED (the N = 32 integer incidence frame + the sigma o r = +-sigma congruence mechanism on both axis types; the pi/2 existence solveset + the (3+4i)/5 root-of-unity census k = 1..64; the delta <-> pi - delta mirror equivariance behind the Casimir symmetry), all VERIFIED the same day with the active engine, 540 -> 564 (the Cesaro/Hurwitz/stripped-block numerics of v523, the H_phys Grams/40-digit inertia spectra/spectral projections/reconstructed rotation group of v524 and the Pfaffian insertion spectra/Bogoliubov 2-of-16 pattern census/defect energy of v525/v526/v527/v528/v529 stay Python-only, flagged in the .wl comments); counting convention honoured (engine active); TWENTY exact mirrors of the 2026-07-23 review-round promotion ADDED (v530 center quotient compiler: fixed line Cv = v, unimodular quotient = [[8,2],[5,3]] atom matrix, invariants + self-code (7,11,13) + CP code (1,7,11,13), resolvent factorisation det(C+kI) = (k+1)det(A+kI), honest K/Q uniqueness controls -- 6 checks; v531 Stage A tensor clock: T30 = Comp(Phi5) (x) Comp(Phi6), chi = Phi_30, exact order 30, full Ramanujan tomography c_30 = c_5 c_6, honest Phi_15 signed-table scope, coprimality controls -- 5 checks; v532 dual-degree modular checksum: (60,192,240,252) sum/product/gcd, q-series j = q^-1 + 744 + 196884 q, D16 control 1488 = 3*496, six non-heterotic control failures, prime-totative fingerprint with maximality scan -- 5 checks; v533 disc -7 norm line: D(t) = N(alpha_t), translation orbit norms (2,4,14,32), constant discriminant -7 + vertex, winding/anchor-block controls -- 4 checks), all VERIFIED the same day with the active engine, 564 -> 584; 2026-07-24 v534 straddle-cone round +4 -> 588 (stabilizer census, pi/2 partition + straddle incidence, reflection-closure signs, equivariant Fix rays incl. the NS-wrap-twisted m=1/7 rays -- all exact; the 256-dim Fock spectra, PT Gram corrections and finite-g survival scans stay Python-only, flagged); 2026-07-24 v535 Hecke-from-geometry +5 -> 593 (Kneser count identity, affine (a,b) law, lambda/a_p cusp rule, dim-7/projector/2-adic oldform arithmetic -- live p=3 FP census and q-series stay Python-only, flagged); 2026-07-24 v536 Eichler-trace-layer +5 -> 598 (Witt lambda_Eis, Eichler anchors/N_perp, N_A/N_B densities, signed a_p/b/R, assembler/Ramanujan -- live FP Shell(p)/Shell(p^2) stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-24 v537 half-integral-bridge +5 -> 603 (a_p head table, Shimura scale -8 / weight 5/2->4, Kohnen scope M=8 / W(4) unsolvable, twist root-number eps_d=chi_d(8) on d≡1,5 mod 8, signed-scale uniqueness cardinality 1-of-4 -- q-series builds, AFE/L-values and R-constancy stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v538 relative-trace-identity +5 -> 608 (R1 Tr anchors, R1 projector table, R2 orbit dictionary, nu_law (a,b) anchors, finite RTF scope cardinality -- q-series builds, AFE/L-values and R-constancy stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v539 Weil-structure-family +5 -> 613 (G_0 algebraic identity, Phi character expansions, core series c_k0=[1,0,2,0,2,0], Dirichlet 2^omega head, obstruction (c) e^{-Y} dlog identity -- arch digamma / numeric Q assembly stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v540 amplitude-linear-carrier +5 -> 618 (Cauchy-Littlewood window lemma + determinant instantiation 1-p^6X^2, three square channels vs closed forms chi=0, tower double-counting fam+2flat+delta=[2,2,2,2], linear full weights lambda_k=1+p^{3k}-(chi p)^k + pole-kernel collapse + FE line map, Moebius square sieve head + Cohen seed anchors Theta(1)=4 / 17 Theta(17)=24 S2(17)=3264 -- lattice enumeration, FFT autocorrelations and mpmath split-Mellin FE stay Python-only, flagged), VERIFIED the same day with the active engine; 2026-07-25 v541 matching-lemma-ledger +5 -> 623 (mod-8 character decomposition 4(-psi)=[4chi_-4+chi_8+chi_-8]Theta with coefficient system (0,1,1/4,1/4) on theta heads, psi w-table recursion w(8x)=9w(x)-8w(x/8) with first steps -3/2/-1/2/+1, Legendre duplication bridge (2pi)^{-s}Gamma(s)=(1/2)Gamma_R(s)Gamma_R(s+1) + Gauss digamma t=0 anchor, certificate head 7S<40A at j in {12,24,48,60} + sign law n<=60, CM carrier head c1(5)=-14 / c1(13)=-238 / inert+ramified laws / mu1(5)=-7/25 -- 10^6 integer sieves, FFT autocorrelation batteries, Cornacchia tables and mpmath digamma/erfc quadratures stay Python-only, flagged), VERIFIED the same day with the active engine",
      "status": "Numerical (second path, extension verified 623/623)",
      "canonicalStatus": "Verified: extension runs clean (623/623, engine run 2026-07-25; v538 +5; v539 +5; v540 +5; v541 +5; v189/v190/v191/v193-rider/v195/v196/v197/v198/v199/v201/v203/v204/v205/v208/v214/v216/v218 + v313-v318/v320 arithmetic arc + v410-v419+v422+v429+v430+v431+v437+v445 diamond/Gaussian/trichotomy/Eisenstein-CP/norm-triple/seam-Galois/Galois-net/pentagon-golden/other-side/degree-ladder/degree-joint/rigidity-forcing + v450-v454+v456+v457+v459+v461+v462+v463+v469+v470+v473+v474+v475+v477 edge-CFT/modular/marks/Virasoro/chirality/killtest/lattice-VOA/strict-locality/spinor/holomorphic-uniqueness/crossed-product/embedding-index/entropic-bridge/entropic-hodge/entropic-scalaron/entropic-scaleflow + v479+v491+v493+v495+v496+v497+v498+v499+v500+v501+v502+v503+v504+v505+v506+v507+v508+v509+v510+v511+v512+v513+v514+v515+v516+v517+v518+v519+v520+v521+v522+v523+v524+v525/v526/v527/v528/v529 Kronheimer/partition/celestial-WP2/WP3/WP4/WP5a/WP5b/P2-typing/WP5c/WP5d-alpha/WP5e-alpha/QGEO-emergence-light/WP5d-beta/WP5e-beta/clock-rigidity/bit-origin/WP5e-gamma/WP5e-epsilon-2/seam-bit-freedom/WP5e-delta-2/tau-flag/c_d-nonderivation/WP5e-epsilon-1/WP5e-M1/WP5e-M2/WP5e-M3/delta-1-derived-kill/WOIT-alpha/measure-decision/RP-blind/WOIT-beta1/wm-derived/WOIT-beta2/twist-blind mirrored + v537 half-integral, v192 + v193/v194-contract [O] + v202 rare-kaon [C] + v206/v207/v209/v319 + v455/v458/v464 Python-only); base file untouched",
      "dependencies": "GATE.WOLFRAM.01; REG.FREEZE.01",
      "scripts": [
        "wolfram/tfpt_readouts_extension.wl"
      ],
      "script": "wolfram/tfpt_readouts_extension.wl",
      "location": "verification/wolfram",
      "externalData": "Wolfram Engine 14.3 (local)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 168
    },
    {
      "id": "FORM.GLUE.01",
      "claim": "Lean 4 formalisation of the v89/v92 arithmetic cores - AUDIT: PASS (machine-formalised [F]): new module TfptCarrier/GlueUniqueness.lean proves by kernel decide (NO native_decide, no new axioms; #print axioms = propext, Classical.choice, Quot.sound only): (1) the isotropic elements of the carrier discriminant form (Z4xZ4, qZ=(5x^2+3y^2) mod 8) are exactly {(0,0),(1,1),(1,3),(2,2),(3,1),(3,3)}; (2) the order-4 isotropic elements are exactly {(1,1),(1,3),(3,1),(3,3)} and generate exactly TWO cyclic subgroups H1=<(1,1)>, H2=<(1,3)> (orbit equalities formalised); (3) the Klein four-subgroup is NOT isotropic (qZ(2,0)=4) => every order-4 isotropic subgroup is cyclic => GLUE UNIQUENESS up to the spinor swap, which provably exchanges H1<->H2; (4) the unique isotropic order-2 element is (2,2) (the SO(16)_1 halfway stage); (5) carrier index arithmetic 4*4=16, [B:A]^2=16 => index 4=|mu4|=|Z2|^2, mu-additivity 16/16=1; (6) glue sectors are currents (h-eighths sums = 8/8) and 248=45+15+64+64+60. Wired into AxiomCheck.lean + AuditCheck.lean; full lake build 3351 jobs OK; scripts/audit.sh => AUDIT: PASS (transcript refreshed). Toolchain leanprover/lean4:v4.29.1 + mathlib (cache)",
      "status": "Formal [F] (Lean kernel; no extra axioms)",
      "canonicalStatus": "Formal [F]: glue uniqueness + carrier index arithmetic kernel-verified",
      "dependencies": "GATE.METRIC.06; GATE.METRIC.07; FORM.P2.02",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/GlueUniqueness.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/GlueUniqueness.lean",
      "location": "experiments/lean4-carrier-rigidity",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 169
    },
    {
      "id": "REVIEW.REPRO.02",
      "claim": "EXTERNAL REPRODUCTION of the v83 package (Alessandro, 2026-06): TeX build.sh notes = 9 ok 0 failed; Python run_all v1-v83 = ALL CHECKS PASSED; Wolfram tfpt_readouts.wl = 116/116 ALL WOLFRAM CHECKS PASSED; redteam run_redteam.py reproduces the status matrix (A reduced not closed, B/D/E survive narrowed, C survives); lean_manifest.sha256 verifies literally (36 files, missing 0, bad 0; byte-identical to the v78 archive, so the local AUDIT: PASS carries over - Theorem A stays formally verified). ONE FINDING: the shipped main manifest had a single STALE row (verification/status_ledger.csv hash mismatch) because the manifest was not regenerated after a final ledger edit; regenerating inside the reviewer package fixed it (117 files, missing 0, bad 0). PROCESS HARDENING (this round): make_manifest.py now has a --check mode that verifies both shipped manifests line-by-line against the tree and exits nonzero on any missing/stale row; RELEASE RULE codified: make_manifest.py is the LAST step before export and --check must pass on the exported tree. REVIEWER VERDICT ADOPTED VERBATIM as the canonical wording: Target A 'moved from diffuse to sharply reduced' NOT 'metric sector closed'; Target B 'arithmetic core formally closed, upstream reduced to the half-spinor Spin(10)-type carrier input'; Target C closed at the reduced dimensionless coefficient level with 1/G the anchor; Target D 'magnitudes reduce to v_geo, CP phases not covered'; Target E 'certified tiers one scale, full theory conditional on frontier identifications'; Koide 'sharpened to one physical identification, not closed'; certified strict physical TOE: not yet",
      "status": "External reproduction (substantive: PASS) + process fix",
      "canonicalStatus": "External reproduction PASS; one stale manifest row -> --check mode + release rule",
      "dependencies": "GATE.METRIC.04; CAR.PASCAL.01; REDTEAM.A.01-E.01; FR.KOIDE.04",
      "scripts": [
        "verification/make_manifest.py (--check)"
      ],
      "script": "verification/make_manifest.py (--check)",
      "location": "verification;tfpt_5_redteam",
      "externalData": "external reviewer run (TeX/Python/Wolfram/redteam/Lean manifests)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 170
    },
    {
      "id": "FLAV.DIAMOND.01",
      "claim": "SHEET DIAMOND & WINDING LINE (second review round 2026-06-11; arithmetic fully validated, interpretations corrected where needed) [I]: (1) all four flavor operators are distinguished points of ONE two-parameter surface M(s,t)=R+Q diag(s,t,t): R=M(0,0), K=M(1,-1), L=M(2,0), F=M(1,1), with the global invariants det M(s,t)=3st^2+9st+6s+t^2+5t+8 and det B_M(s,t)=6st+12s+3t^2+15t+16; the mass-transport pencil K+xQ is the cut (s,t)=(1+x,x-1), on which det B factorises as (3x+2)(3x+5) = the v81/v85 master cover. (2) BUILT-IN NEGATIVE CONTROLS: the factorisation is special to that cut - the diagonal cut t=s gives 9s^2+27s+16 with disc=153 NON-square, the SAME control discriminant as v85's R-direction (anchor-first coherence), and the R->L cut t=0 is linear/degenerate. (3) WINDING SYNCHRONISATION: on R_s=R+s*1e1^T, det B_{R_s}=2 det R_s for ALL s (8+2s vs 16+4s); the triple lock (tr=15=dim A3, det=20=det L, coxeter lift (R_s^T a)_1=30=h(E8)) fixes s=6=|R+(A3)| by each condition alone. (4) COFACTOR SEAM NORMAL (typed as repackaging - adjugate row by construction; content = integer readings): n=(5,-9,6) is the common first adjugate row of R and L, n.1=2=|Z2| is the per-winding determinant increment, n^T R=(8,0,0) and n^T L=(20,0,0): the normal annihilates the stable columns - only generation 1 is an open torsion channel. (5) REALITY CORRECTED (the review's 'Reality Selector' is wrong as a selection claim): disc(s)=17s^4-18s^3+709s^2+588s-7996 changes sign at s*~2.825, so the spectrum is real for all s>~2.83 and reality does NOT select s=6; at the physical s=6: disc=39200=2^5 5^2 7^2=2(10*14)^2, spectrum real positive. (6) F-CORNER AUDIT extending v80: det F=32=2^5, chi_F=(lam-2)(lam^2-13lam+16), B_F=[[37,43],[53,63]] with det 52=dim F4, sum 196=14^2=(dim G2)^2, tr 100, 43+53=96=2 Omega_adm - audit fingerprints, not load-bearing. (7) FIREWALL REJECTED as a hard rule: 'all small rational factors must be adjacent spine quotients' is FALSE - 16/7, 7/6 (lepton c), 5/6 (gamma_car), 8/7 are load-bearing non-members; the v74 quotient ladder remains an observation, not an admissibility filter",
      "status": "Identity [I] (surface/winding/normal closures + negative controls) + Audit (F corner) + honest corrections",
      "canonicalStatus": "Identity [I]: one surface carries R/K/L/F; winding synchronised; reality threshold corrected; firewall rejected",
      "dependencies": "FLAV.KQ.01; FLAV.PENCIL.05; v74; v80; v81; v85",
      "scripts": [
        "v94_sheet_diamond.py"
      ],
      "script": "v94_sheet_diamond.py",
      "location": "tfpt_2;verification",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 171
    },
    {
      "id": "FLAV.PLUCKER.02",
      "claim": "RIGHT-PLUECKER CANONICALISATION of the quark-ratio integers (load-bearing improvement; supersedes the loose-26 readings as the canonical source): with Pi_R(M) = 2x2 minors of (M1 | Ma): Pi_R(Q)=(0,4,5), Pi_R(K)=(12,12,-2), Pi_R(R)=(8,12,4), Pi_R(L)=(20,30,10)=10*(2,3,1) with sum 60 = E8 cascade start (v91 loop closed). CANONICAL READINGS: ||Pi_L(K)||_1=11 and ||Pi_R(K)||_1=26 put BOTH quark-ratio integers in the SAME operator K - the left anchor block reads flavor ORIENTATION (c_u/c_d = g_car*11/(N_fam^2 Delta_Q) = 55/117, v37/v42/v49), the right block reads transport PROPAGATION (c_t/c_b = N_fam/||Pi_R(K)||_1 = 3/26). The older 26-readings Pl(L)_13=26 and 2*Delta_Q=26 are now COROLLARIES of the canonical norm, not independent choices - same retirement pattern as the '11' (v45). c_c/c_s = p_5(a)/(Omega_adm-1) = 34/47 keeps its anchor-power reading; the 34=2*17 (E8 exponent) reading stays AUDIT only. tfpt_2 Pluecker keybox updated to the canonical boxed form",
      "status": "Identity [I] (canonical readings; old readings = corollaries)",
      "canonicalStatus": "Identity [I]: 11 = left and 26 = right Pluecker norm of K; older 26-readings corollaries",
      "dependencies": "FLAV.PLUCKER.01; FLAV.QRATIO.01; FLAV.RIGID.01; v37; v42",
      "scripts": [
        "v94_sheet_diamond.py"
      ],
      "script": "v94_sheet_diamond.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 172
    },
    {
      "id": "FLAV.DIAMOND.02",
      "claim": "CENTERED FLAVOR DIAMOND (third review round 2026-06-11; all arithmetic validated incl. SNF) [I]: the v94 sheet diamond is a CENTERED CROSS - with the center C:=M(1,0)=(R+L)/2=(K+F)/2=[[4,3,0],[4,5,2],[5,5,3]] and the two axes U:=Q diag(1,0,0)=N_fam*1e1^T (WINDING axis: rank 1, Spec {3,0,0} = pure family charge) and V:=Q diag(0,1,1) (SHEET axis: rank 2), one has exactly Q=U+V and R=C-U, L=C+U, K=C-V, F=C+V: the five objects (R,K,L,F,Q) collapse to three (C,U,V), and the v94 winding line IS the U-axis (L=R+2U). CROSS-LINKS (beyond the review): (i) Spec(V)={0,1,2}=N_fam*{0,1/3,2/3} - the sheet axis carries exactly the cusp class of v69/v72 (Q_+=3*cusp+1 had {1,2,3}); (ii) sum C=31=2^g_car-1=1+h_vee(E8) is the SAME 31 as the IR gap-bound numerator ||V_metric||<=248c3^2=31/(8pi^2) (v63/v76); (iii) det C=14 = constant term of chi_[K,Q]=t^3+5t+14 (v80) and sqrt(sum B_F)=sqrt(196) (v94); (iv) Pi_R(C)=7*(2,3,1) shares its ray with Pi_R(L)=10*(2,3,1) (scalaron-7 vs decuple-10 scaling of one direction). Center invariants: tr C=12, det C=14, SNF(C)=(1,1,14), det B_C=28=2*14, Pi_L(C)=(0,14,14). HONEST TYPING: the 'G2 center' reading (14=dim G2 as det/Pluecker entry/half anchor block) is an AUDIT Lie-dimension match exactly like 78=dim E6 or 52=dim F4 - recorded, not promoted; the load-bearing content is the exact normal form and its integer invariants. Docs: tfpt_2 Sheet-Diamond winbox extended; research_contracts opening now carries the operative aliases Rest_operativ = v_geo (+) G_net (+) F_transfer",
      "status": "Identity [I] (normal form + invariants + cross-links) + Audit (G2 reading)",
      "canonicalStatus": "Identity [I]: one center + two axes; cusp class on the sheet axis; G2 reading audit-typed",
      "dependencies": "FLAV.DIAMOND.01; FLAV.PLUCKER.02; FLAV.QGEO.01 (cusp class); GATE.GMETRIC.XI.01 (31)",
      "scripts": [
        "v95_centered_diamond.py"
      ],
      "script": "v95_centered_diamond.py",
      "location": "tfpt_2;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 173
    },
    {
      "id": "FLAV.SHEET.01",
      "claim": "BRANCH-KERNEL SELECTION (next.txt P1 unblocked): at each branch point of the anchor-block cover the 2x2 anchor block is RANK 1 with canonical INTEGER kernels (Koide -2/3: right w=11*1-9a=(2,2,-7), left v=7*1-5a=(2,2,-3); carrier -5/3: right kernel = the democratic vector 1, left v=8*1-7a=(1,1,-6)). COLLAPSE LEMMA [L]: rank-1 forces P(x0).w perp span{1,a} => proportional to the antisymmetric direction (-1,1,0) of the anchor's two equal slots, on BOTH sides: sectors (up=-down, lepton pairing=0 - leptons sit ON the ramification; Koide is leptonic) and generations (gen1=-gen2, gen3=0). SECTOR ZERO LADDER [I]: lepton pairings ARE the det-B factors (zeros = the two branch points); up zeros = -3/2=-N_fam/|Z2| doubly (the v85 GL(2) rung; the whole up-row of P(-3/2) IS the odd direction); down zeros = 0 (=K itself) and -9/5=-N_fam^2/g_car. ORIENTATION: deck translation flips (up,down)=(+,-)->(-,+), magnitude ratio 10=A_Lambda; convention-free invariant f_up*f_down<0. CONTROLS: anchor (2,1,1) moves the collapse to (0,1,-1); (1,2,1) disc 40 non-split, no rational kernels (v82 trichotomy). The sector->sheet map is now DEFINED by the cover's own kernels - the P1 blocker (no independently motivated definition) is removed; the physical identification with the v92 chirality swap stays [P] (FLAV.SHEET.02)",
      "status": "Identity/Lattice [I]/[L] (kernels, collapse, ladder, controls) + [P] (sector-sheet physics)",
      "canonicalStatus": "Identity/Lattice: the cover defines the sector map; physics [P]",
      "dependencies": "FLAV.PENCIL.03; FLAV.PENCIL.05; FLAV.DIAMOND.02; FLAV.MASSVOL.01 (sector rows)",
      "scripts": [
        "v96_branch_kernel_selection.py"
      ],
      "script": "v96_branch_kernel_selection.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 174
    },
    {
      "id": "FLAV.SHEET.02",
      "claim": "SHEET-CONJUGATION BRIDGE (P1 -> ONE [P] residual): the family-side conjugation (cusp negation 0 fixed, 1/3<->2/3 on the Q_+ eigenspaces e1=(0,0,1), e2=(0,1,2), e3=(1,0,0)) has a UNIQUE anchor-compatible generation-space realisation - T.1=1 and T.a=a BOTH force scale 1 => integer involution T_A=[[0,1,0],[1,0,0],[2,-2,1]], det=-1; the two alternative pairings are REJECTED (T.a=a unsolvable). ANCHOR = CONJUGATION-SYMMETRIC VECTOR [I]: a=e2+e3, 1=a-e1, odd line e2-e3=(-1,1,2). ONE DECK ACTION [I]: T_A=-1 on R^3/span{1,a} exactly like sigma_12; the v96 collapse direction (-1,1,0) is odd for both - the cover cannot distinguish them. PARITY [I]: det T_A=-1 = single spinor conjugation (= the v92 glue swap), det Sigma=+1 = simultaneous (glue-fixing). D4 CLOSURE [L]: <T_A,Sigma> order 8, G=T_A*Sigma order 4, char (t+1)(t^2+1) = the v69 D4 and the v70 mu4 char poly. SHEET-INDEX LEMMA [I]: G and the v70 homology mu4 are rationally but NOT GL(3,Z)-conjugate - det S = 2 s8 (s6^2+(s7+2 s8)^2) always EVEN, minimal |det|=2=|Z2| (one sheet doubling; Q det=N_fam does not intertwine - independent of v70). REMAINING [P]: the single lattice dictionary 'Q_+ eigenspace grading of the generations = A3 discriminant grading' (then T_A IS the v92 A3-side conjugation and P1 closes)",
      "status": "Identity/Lattice [I]/[L] (uniqueness, D4, index 2) + [P] (the one lattice dictionary)",
      "canonicalStatus": "Identity/Lattice: one deck involution; one [P] dictionary left",
      "dependencies": "FLAV.SHEET.01; FLAV.QGEO.01; GATE.QGEOM.02 (v70); GATE.METRIC.07 (v92 glues)",
      "scripts": [
        "v97_sheet_conjugation_bridge.py"
      ],
      "script": "v97_sheet_conjugation_bridge.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 175
    },
    {
      "id": "FLAV.SHEET.03",
      "claim": "DISCRIMINANT DICTIONARY DERIVED (P1 closed modulo GATE.QGEO): the generation space carries an INTEGER mu4 - G = T_A*Sigma acts on the cusp basis as G e1=-e1, G e3=e2, G e2=-e3 (the v69 B1+E decomposition realised integrally; G^4=I, char (t+1)(t^2+1)); the cusp-0 eigenvector IS the unique self-conjugate Z4-class-2 line and the swapped cusp pair {1/3,2/3} IS the conjugate class pair {1,3} => the v97 dictionary 'Q_+ grading = A3 discriminant grading' is DERIVED inside the model, not assumed. T_A G T_A^-1 = G^-1 (= discriminant conjugation k->-k); q_A3(1)=q_A3(3)=3/8 (equal on the swapped pair), q_A3(2)=1/2 (fixed); the A3-side negation swaps the two Lagrangian glues (v92 re-derived). The two D4 reflection classes separate the physics: T_A e1=+e1, det=-1 (glue-SWAPPING, v92 single conjugation) vs Sigma e1=-e1, det=+1 (glue-FIXING, simultaneous); Sigma=T_A G. CONSEQUENCE: P1 carries NO separate [P] residual any more - what remains is exactly the already-open GATE.QGEO realisation step (one gate, not two); conditional on that established typing the sector->sheet selection is CLOSED (up/down = deck-odd pair, leptons on the ramification, deck involution = A3-side conjugation T_A)",
      "status": "Identity/Lattice [I]/[L] (dictionary derived) - residual folded into GATE.QGEO",
      "canonicalStatus": "Identity/Lattice: P1 dissolved into the one existing Q-geometry gate",
      "dependencies": "FLAV.SHEET.01; FLAV.SHEET.02; FLAV.QGEO.01; GATE.QGEO.01; GATE.METRIC.07 (v92)",
      "scripts": [
        "v98_discriminant_dictionary.py"
      ],
      "script": "v98_discriminant_dictionary.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 176
    },
    {
      "id": "FR.KOIDE.06",
      "claim": "KOIDE FLOW TIME (P2 sharpened on both ends; corrects the ROBUSTNESS of v93 negative (i), not its arithmetic): [I] CANONICAL GENERATOR - the v82 flow is generated by dq/dt = (Delta/N_fam)(q-2)(q-5) = (Delta/N_fam) det B(q), Delta = 6 log(3/2) = the gap; the time-1 map IS the v82 Moebius F (machine-checked; e^-Delta = (2/3)^6 exact) => the missing P2 object is precisely a QFT mechanism producing 'gap x anchor-block quadric / family count'. [N] DATA-FRAGILITY: rho_pole = -2.2e-6 makes the flow time hypersensitive to m_tau (PDG 1776.93+-0.09 MeV): t(-1sig)=2.35, t(central)=2.84, and Q crosses 2/3 INSIDE the band (+0.43 sigma, t->infinity) => within +0.5 sigma t passes EVERY integer >=3; the v93 exclusion of pole=F^n(source) holds only at the central value. [P] CONDITIONAL discrete steps (registry untouched, NOT predictions of record): n=2 EXCLUDED (-2.9 sigma) [N]; n=3 = N_fam steps predicts m_tau = 1776.9427 MeV (+0.14 sigma; source-variant stable to 0.0002 MeV); n=4 1776.9667 / exact branch 1776.9690 not separable now; kill criterion sigma(m_tau) ~ 0.01 MeV (Belle II). [I/N] LOOK-ELSEWHERE FIREWALL: the scale reading ln(M_scal/m_tau)/ln(W) is destroyed by negative controls (W = 46080, 40320, 51840 all give |pull| < 0.1 sigma) - rejected and recorded so it is not re-fished. P2 type unchanged ([P], one identification); the discrete-vs-continuous question is now EXPERIMENTAL, not settled",
      "status": "Identity [I] (generator) + Numerical [N] (fragility) + Physical [P] (conditional steps)",
      "canonicalStatus": "P2 sharpened: canonical generator [I]; discrete-vs-continuous = experimental question",
      "dependencies": "FR.KOIDE.04; FR.KOIDE.05 (robustness of negative (i) corrected); COSMO.NSTAR.01 (M_scal)",
      "scripts": [
        "v99_koide_flow_time.py"
      ],
      "script": "v99_koide_flow_time.py",
      "location": "tfpt_4_frontier",
      "externalData": "PDG 2024 m_tau = 1776.93 +- 0.09 MeV",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 177
    },
    {
      "id": "FR.KOIDE.07",
      "claim": "OPERATOR ORIGIN OF THE KOIDE SOURCE->POLE FACTOR 53/54 (external-review proposal 2026-06-14, validated; upgrades the FR.KOIDE.03 conjecture from a bare arithmetic guess 1-1/(2 N_fam^3) to an exact operator identity on the carrier): u_pole/u_source = a^T (R+Q) 1 / (2 * 1^T R a) = 53/(2*27) = 53/54. (1) THE CUBIC FAMILY BLOCK [I]: 1^T R a = 27, the E6xA2 flavor-shadow split 248 = dim E6 + det R + 2(1^T R a)N_fam = 78 + 8 + 2*27*3, so 2*27 = 54 is the sheet-doubled denominator. (2) THE MISSING CORNER [I]: the Sheet Diamond has present corners R=M(0,0), K=M(1,-1), L=M(2,0) and the ABSENT corner F=R+Q=M(1,1) with anchor-block determinant det B_F = 52 = dim F4 (FLAV.PENCIL.02/v80); its anchor response a^T(R+Q)1 = 53 = 52+1 (F4 shadow + singlet). (3) THE RATIO [I]: a^T(R+Q)1/(2*1^T R a) = 53/54 = 1-1/(2 N_fam^3) exactly (54 = 2 N_fam^3 = 2*27), so the two readings coincide but this one is structural, not fitted. (4) NEGATIVE CONTROL [I]: only the missing corner F gives 53 -- a^T M 1 for M in {R,Q,K,L} = {32,21,35,56}, none = 53 (specific, not generic). PHYSICAL STATUS [C]: the FACTOR is now sourced exactly; the MECHANISM (the leptonic pole-mass transfer evaluates the F corner) stays a conjecture, an F_transfer special case (F_pole: source->pole). Koide thus moves from 'near miss + conjecture' to 'operator-sourced source->pole transfer'; the prediction itself stays [C]. Cross-ref HOR.TRISECT.01 (the same Sheet-Diamond double-cover grammar carries the gravity clock, exponent ratio |Z2|)",
      "status": "Identity [I] (exact operator factor + negative control) + Physical [C] (the transfer mechanism)",
      "canonicalStatus": "Identity [I]: 53/54 = a^T(R+Q)1/(2*1^T R a) sourced from the missing Sheet-Diamond corner F (det B_F=52); mechanism stays [C]",
      "dependencies": "FR.KOIDE.03; FLAV.PENCIL.02; FLAV.PLUCKER.02; v80_operator_pencil_geometry.py",
      "scripts": [
        "v183_koide_f_corner_transfer.py"
      ],
      "script": "v183_koide_f_corner_transfer.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "FR.KOIDE.03",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 178
    },
    {
      "id": "FR.ETAB.03",
      "claim": "HONEST TEST OF THE ANCHORED-BOLTZMANN eta_B PROPOSAL (external review 2026-06-14; firewall verdict: SHARPER SCENARIO, not a closure): the proposal M_1 = M_R phi0^4 and m~1 = m_3/A_Lambda was tested exactly. (1) WASHOUT ANCHOR [C]: m~1 = m_3/A_Lambda = m_3/10 ~ 5e-3 eV (m_3 ~ sqrt(dm^2_atm) ~ 0.05 eV; A_Lambda=10 = |E(K5)| the action-grammar atom) lands in the strong-washout window [m_*, m_atm] -- a TFPT-flavoured value, a [C] structural relation (like Koide 53/54), not derived; it removes the washout as a FREE dial. (2) M_1 NOT ANCHORED [honest negative]: M_R = v^2/m_3 ~ 6e14 GeV is the SEESAW scale (depends on the un-fixed neutrino Dirac Yukawa) and the nearest clean TFPT powers of Mbar both MISS it (c3-power off ~75%, phi0-power off ~40%; log_c3=2.57, log_phi0=2.83 non-integer), so M_1 = M_R phi0^4 merely RELOCATES the free input M_1 -> M_R -- exactly the reverse-engineering the firewall rejects (the reviewer's 6.0e-10 uses M_R=1.3e15; the seesaw value gives M_1=4.8e9 and eta_B=3.4e-10 -- the hit shifts with M_R, the tell that M_1 is unpinned). (3) NET [C]: feeding (m~1=m_3/10, M_1=M_R phi0^4) into the v169 thermal-leptogenesis estimate lands eta_B in the observed band O(1e-10) bracketing 6.1e-10 (route VIABLE), but M_1 stays a scenario input -- the proposal cuts the free inputs from TWO to ONE, NOT to zero; eta_B stays [C], a sharper scenario, not a derivation. Downstream FR.ETAB.01 (eta_B from closed Omega_b h^2) is the independent, already-hit reading. Python-only",
      "status": "Physical [C] (sharper scenario; washout anchored; M_1 unpinned)",
      "canonicalStatus": "Physical [C]: m~1=m_3/A_Lambda anchors the washout; M_1=M_R phi0^4 does NOT pin M_1 (M_R = seesaw scale, not a compiler output); eta_B stays a sharper scenario",
      "dependencies": "FR.ETAB.02; FR.ETAB.01; FLAV.RGSTAB.01; v9_neutrino_texture.py; v169_etaB_boltzmann_interface.py",
      "scripts": [
        "v184_etaB_anchored_boltzmann.py"
      ],
      "script": "v184_etaB_anchored_boltzmann.py",
      "location": "tfpt_4_frontier",
      "externalData": "NuFIT (dm^2_atm); Planck (eta_B)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 179
    },
    {
      "id": "FR.DM.03",
      "claim": "AXION RELIC HONEST MISALIGNMENT ESTIMATE (external-review proposal 2026-06-14; the determinant-line axion can be all DM but stays a [C] scenario): (1) SCALES [I]: f_a = M_scal/(2 dim S+ |mu4|) = M_scal/128 ~ 2.39e11 GeV (M_scal = c3^{7/2} Mbar ~ 3.06e13 GeV), QCD axion m_a ~ 5.7 ueV (1e12/f_a) ~ 23.8 ueV (FR.DM.02). (2) HARMONIC MISALIGNMENT UNDER-PRODUCES [N]: Omega_a h^2 ~ 0.12 (f_a/9e11)^{7/6} <theta^2> gives prefactor ~0.026 per theta^2 at this f_a, so theta_i ~ O(1) makes only ~21% of Omega_DM; reaching Omega_DM h^2 = 0.12 needs theta_eff^2 ~ 4.7, i.e. the large-angle (anharmonic) enhancement. (3) HILLTOP REQUIRED -> SCENARIO [C]: TFPT's misalignment angle theta_i ~ 170 deg (the seam value pi(1-phi_seam)) IS the hilltop where the anharmonic factor diverges, so Omega_a h^2 CAN be tuned to Omega_DM -- but near the hilltop the relic is exponentially sensitive to theta_i and QCD/string corrections, so it is a viable SCENARIO, not a sharp prediction. (4) CONTRACT [O]: F_relic(f_a=M_scal/128, theta_i~170 deg, N_DW=1) =? Omega_DM is a kill test for the determinant-line axion BRANCH, not the theory; the E8 scan leaves no free singlet WIMP (128=(16,4)+(16,4) matter spinor), so the axion is the natural DM line, abundance is transfer/cosmology, never a compiler output. Python-only (analytic estimate; the full finite-T chi(T) ODE is standard cosmology, flagged). POST-INFLATIONARY REFRAMING [C] (2026-07-10, FR.DM.02): the 170 deg hilltop is a PRE-INFLATIONARY fixed-angle reading; given H_inf/f_a~66>1 (R^2 branch) it carries CDM isocurvature >> Planck beta_iso<0.038, so it is SUPERSEDED by the post-inflationary scenario (theta_i a horizon average, NOT observable; relic from f_a + N_DW=|C|=1, v470/v472). Over-production at 170 deg is moot post-inflationary.",
      "status": "Physical [C] (hilltop scenario SUPERSEDED by post-inflationary reframing FR.DM.02; angle not observable)",
      "canonicalStatus": "Physical [C]: f_a=M_scal/128 axion can be all DM only near the theta_i~170 hilltop (exponentially sensitive) -> scenario, branch kill test not theory",
      "dependencies": "FR.DM.01; FR.DM.02; v25_frontier_conjectures.py",
      "scripts": [
        "v185_axion_relic_solver.py"
      ],
      "script": "v185_axion_relic_solver.py",
      "location": "tfpt_4_frontier",
      "externalData": "Planck Omega_DM h^2 = 0.12",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 180
    },
    {
      "id": "FR.TRANSFER.01",
      "claim": "THE F_TRANSFER DISCIPLINE GUARD (a CI-style firewall check, the machine-readable form of the Frontier rule; external-review proposal 2026-06-14): the four continuous-transfer frontier observables -- Koide (FR.KOIDE.*), eta_B (FR.ETAB.*), the axion relic (FR.DM.*), and m_p/m_e + Lambda_QCD (FR.MPME.*, QCD.LAMBDA.*) -- must NEVER be typed as primitive [E]/[N] compiler predictions; each must carry a conditional [C] / anchor-open [A]/[O] marker. The guard reads status_ledger.csv and enforces it (all active frontier prediction rows pass; exact algebraic SUB-parts may be [I]/[E] -- the Koide 53/54 factor v183, b3=-7 v159 -- but the PHYSICAL prediction never is). Records the four explicit interfaces: F_pole (source masses->pole, Koide v183), F_Boltzmann (CP source->eta_B, v169/v184), F_relic ((f_a,m_a,theta_i,N_DW)->Omega_a, v185), F_QCD ((alpha_s,b3,Lambda_QCD,lattice)->m_p/m_e, v164). Confirms the raster rule + frozen registry (REG.FREEZE.01) + null model (AUDIT.NULLMC.01) bind, so the frontier transfers stay OUTSIDE the closed dimensionless compiler. Prevents any future edit from quietly promoting a pretty frontier number to a closed compiler output. Python-only",
      "status": "Identity/guard [I] (ledger-enforced typing discipline)",
      "canonicalStatus": "Identity/guard [I]: the four F_transfer frontier observables are ledger-enforced to stay [C]/[A]/[O], never primitive [E]/[N]",
      "dependencies": "FR.KOIDE.07; FR.ETAB.03; FR.DM.03; QCD.LAMBDA.01; REG.FREEZE.01; AUDIT.NULLMC.01",
      "scripts": [
        "v187_ftransfer_laws.py"
      ],
      "script": "v187_ftransfer_laws.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 181
    },
    {
      "id": "AUDIT.WORDING.01",
      "claim": "THE FRONTIER-WORDING GUARD (a prose sentinel, no new physics; external-review proposal 2026-06-14): v187 protects the ledger TYPING, this guard protects the PROSE. It forbids stale sentences whose semantics an earlier round superseded, so a frozen Zenodo release can never re-introduce wording that was true yesterday and is half-true today. Forbidden patterns (each recording what superseded it): (1) 'leptogenesis interface (mE)' -- v184 retyped Route B to [C] (M_R is the seesaw scale, not a compiler power); (2) 'closed branch also fixes the leptogenesis inputs' -- the branch fixes the spectrum + a washout anchor, NOT the heavy scale; (3) 'relic scale f_a,m_a open' -- v185 sharpened DM: the scales DEFINE the branch, the open item is the hilltop-sensitive relic abundance; (4) 'computed Q=0.664 ... near, not exact' -- v183 gave the 53/54 factor an exact operator origin (F=R+Q corner); (5) 'Suite now 187 modules' -- the count is 187, highest ID v188 (v186 skipped). Scans the 10 active .tex documents; PASSES when none of the stale phrasings appear. Python-only (text sentinel)",
      "status": "Identity/guard [I] (prose sentinel against semantic drift)",
      "canonicalStatus": "Identity/guard [I]: stale frontier prose (superseded wording) is forbidden from the active documents",
      "dependencies": "FR.ETAB.03; FR.DM.03; FR.KOIDE.07; FR.TRANSFER.01",
      "scripts": [
        "v188_frontier_wording_guard.py"
      ],
      "script": "v188_frontier_wording_guard.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 182
    },
    {
      "id": "AUDIT.NULLMC.01",
      "claim": "NUMEROLOGY NULL TEST (look-elsewhere corrected; executes the legacy 'numerology grammar test' TODO from the CMB paper): (A) EXACT CENSUS - for each of the 13 scored frozen-registry observables the FULL declared grammar class (one-term (p/q) phi0^a lamC^c (1+lamC)^d pi^b e^(u/w) with p,q<=130, |a|<=2, |c|<=3, |d|,|b|<=1, |u|<=6, w<=6; two-term sums with coefficients<=12; machine-checked fairness: the class contains every TFPT formula it scores) is enumerated exactly against conservative data windows (max of achieved deviation, documented sigma, quote resolution, 5% quark scheme spread - tension observables enter with their honest LARGE windows and contribute almost nothing): joint formula-fishing probability prod p_i = 1e-25.8, robust to <2 dex under coefficient bound 80..200 and plausibility window 5x..20x; (B) MONTE CARLO 2x200k deterministic pseudo-theories (random rational dials in TFPT's own formula skeletons, seed phi0 fixed resp. drawn log-uniformly) reach at most 5/13 resp. 4/13 hits vs TFPT 13/13; (C) POWER controls: perturbing phi0 by 1%/10%/30% collapses TFPT's own scorecard to 9/6/4 hits, shuffling the data<->formula assignment to mean 1.2; (D) ALPHA CLASS: all 94500 complexity-matched F_U1 variants (coefficients, M, transport exponent, resolvent power) root-solved, exactly ONE - the TFPT instance (2, 4/5, M=41, e^-2a, -5/4) - lands inside the achieved 4e-8 CODATA window => p_alpha <= 1.1e-5. TOTAL: P(random complexity-matched theory reproduces the scorecard incl. alpha) <= 1e-30.7 (~102 bits), CONDITIONAL on the declared grammar - a look-elsewhere-corrected null-model rejection, never 'certainty'. Complementary to the freeze: v84 prevents FUTURE look-elsewhere drift, v100 quantifies the RETROSPECTIVE look-elsewhere burden",
      "status": "Numerical [N] (statistical null-model rejection; Python-only)",
      "canonicalStatus": "Numerical [N]: joint null probability <= 1e-30.7 conditional on the declared grammar",
      "dependencies": "REG.FREEZE.01 (scored predictions = frozen registry); DATA.SCORECARD.01 (documented data centrals)",
      "scripts": [
        "v100_numerology_null_mc.py"
      ],
      "script": "v100_numerology_null_mc.py",
      "location": "introduction;tfpt_3;tfpt_5_redteam",
      "externalData": "CODATA 2022 alpha^-1; NuFIT 6.0; ACT DR6; PDG/FLAG centrals (all as already documented in-repo)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 183
    },
    {
      "id": "HOR.NARIAI.01",
      "claim": "THE MAXIMAL BLACK HOLE IS THE ANCHOR (carrier-in-the-bulk surface; Schwarzschild-de Sitter in seam units): (1) NARIAI = ANCHOR - at M_N = 1/(N_fam sqrt(L)) the horizon cubic is t^3-3t+2 = (t-1)^2(t+2), roots (1,1,-2) = the traceless projection of the anchor a=(1,1,2) (coefficients (-3,2) = (-N_fam,|Z2|); cf chi_a = (t-1)^2(t-2), ARCH.CORE.01). (2) the Koide 2/3 IS the Nariai entropy bound: S/horizon = S_dS/N_fam, total = (|Z2|/N_fam) S_dS, deficit S_dS/3; interpolation S_tot/S_dS = (x^2+1)/Phi_3(x), monotone, min at the merge. (3) three-sheet conservation pi sum r_i^2 = |Z2| S_dS for every M. (4) Koide-form quotient of the three roots ranges [3/8,1/2] = the two nonzero SU(4)_1 weights (pure dS -> 1/2 = delta; Nariai -> 3/8 = p2(a)/e1(a)^2). (5) mass-line double cover disc ~ (1-3m)(1+3m): branch +-1/N_fam, separation 2/3; deck = horizon swap (gravitational twin of the sheet Z2). (6) temperature lemma |kappa_b/kappa_c| = (2x+1)/(x(x+2)) > 1: evaporation flows AWAY from the merge - anchor point = repeller, democratic endpoint = attractor (same orientation as the flavor flow, FR.KOIDE.04) [audit]. (7) seam-unit mechanics: dM = c3 kappa dA, Smarr 2c3, Bekenstein 1/(4c3), P_H = c3/(1920 M^2) (=|W(D5)|, HOR.01), lifetime 5120 pi M^3 = 128 g_car M^3/c3 (128 = 2^7 = dS prefactor), Kerr A_ext = M^2/c3 (ratio 1/2 = 1/|Z2|, S_ext: 1/4 = 1/|mu4|), area quantum 4 ln3 = ln(N_fam^4) = ln(disc of the flavor cover) [P as HOR.QNM]. Six independent landings on already-load-bearing atoms, zero free parameters; lifetime/power normalisations are standard photon/geometric-optics conventions (external); carrier-in-the-bulk reading [P]; structure surface, not a measurement surface (M_N today ~1e22 M_sun)",
      "status": "Identity [I] (exact GR arithmetic on compiler atoms) + Physical [P] (bulk reading)",
      "canonicalStatus": "Identity: six atom landings in SdS mechanics; bulk reading [P]",
      "dependencies": "AX.P1.01; ARCH.CORE.01 (anchor); FLAV.PENCIL.03 (cover); FR.KOIDE.04 (orientation); HOR.01; GATE.METRIC.07 (SU(4)_1 weights)",
      "scripts": [
        "v101_horizon_anchor.py"
      ],
      "script": "v101_horizon_anchor.py",
      "location": "tfpt_horizon_readouts;origin_theory;tfpt_2",
      "externalData": "standard GR (SdS metric); photon/geometric-optics Hawking conventions",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 184
    },
    {
      "id": "HOR.ORIENT.01",
      "claim": "ONE ORIENTATION - the anchor is the stationary repeller in BOTH sectors (variational upgrade of the v101 temperature lemma + the v82/v99 flavor flow): FLAVOR [I] - the canonical flow dq/dt = (Delta/N_fam)(q-2)(q-5) is the GRADIENT flow of the cubic V(q) = -(Delta/N_fam)(q^3/3 - 7q^2/2 + 10q); critical points = exactly the two branch points; stationary curvatures = +-the transfer gap (V''(Koide) = +Delta attractor, V''(carrier) = -Delta repeller); inflection at q = 7/2 = scalaron/2 = half the branch-divisor trace; Lyapunov rate d(-ln rho)/dt = Delta exactly and constantly. GRAVITY [I] - d(S_tot/S_dS)/dx = (x-1)(x+1)/Phi_3(x)^2: the Nariai/anchor point is the UNIQUE physical stationary point of the SdS entropy functional, with curvature 2/9 = |Z2|/N_fam^2 = (2/3)(1/3); temperature lemma => evaporation ASCENDS the entropy away from the anchor toward the democratic endpoint. ORIENTATION THEOREM [I]: both sectors flow away from an anchor-stationary configuration with grammar-constant curvatures. HONEST: flavor attractor = a branch point vs gravity attractor = smooth endpoint; flavor functional = potential of a [P] relaxation vs gravity = BH entropy; 'one variational principle of the seam' stays a reading [P]; no open gate touched. Figures: orientation.pdf, cover_twins.pdf, seam_units.pdf",
      "status": "Identity [I] (variational structure both sides) + Physical [P] (common-principle reading)",
      "canonicalStatus": "Identity: anchor = stationary repeller both sectors; principle reading [P]",
      "dependencies": "HOR.NARIAI.01; FR.KOIDE.04; FR.KOIDE.06 (generator); FLAV.PENCIL.03",
      "scripts": [
        "v102_seam_orientation.py"
      ],
      "script": "v102_seam_orientation.py",
      "location": "tfpt_horizon_readouts;origin_theory",
      "externalData": "standard GR (SdS); second law under net evaporation",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 185
    },
    {
      "id": "HOR.TRISECT.01",
      "claim": "TRISECTION NORMAL FORM (the canonical coordinate of the SdS moduli line; answers the HOR.ORIENT.01 curvature-matching question at the geometric level): the horizon cubic is uniformized by ANGLE TRISECTION - r = 2cos(theta) gives r^3-3r = 2cos(3theta) exactly, cubic <=> cos(3theta) = -3m; centered angle psi: m = cos(psi)/N_fam, Nariai psi=0, pure dS psi=pi/2; deck involution psi->-psi (= the x->1/x deck of HOR.NARIAI.01), Z3 trisection rotation = the triality deck (GATE.QGEOM.03/v72 cross-ref). NORMAL FORM [I]: sigma(psi) = S_tot/S_dS = 4/3 - (2/3)cos(2psi/3) - mean |mu4|/N_fam, amplitude |Z2|/N_fam, frequency |Z2|/N_fam: one cosine, every constant a glue atom. CANONICAL CURVATURE [I]: sigma''(0) = 8/27 = (2/3)^3 = (|Z2|/N_fam)^N_fam - the Koide constant to the family power. INVARIANT BASE SLOPE [I]: dsigma/dm|_Nariai = -8/9 = -rank(E8)/N_fam^2 (coordinate-invariant, cross-checked in both coordinates; generally -(4/3)sin(2psi/3)/sin(psi)); dimensionful dS_tot/dM|_N = -r_N/(N_fam c3). BRIDGE [P]: flavor multiplier (2/3)^{2N_fam} is a RATE, gravity curvature (2/3)^{N_fam} is a CURVATURE - same base 2/3 = |Z2|/N_fam, exponent ratio |Z2|; the full identification needs the gravity-side CLOCK (near-Nariai evaporation generator), the same missing-object class as P2's continuous transfer generator: one clock, two known geometries. No open gate closed; figure trisection.pdf",
      "status": "Identity [I] (uniformization, normal form, curvature, slope) + Physical [P] (clock bridge)",
      "canonicalStatus": "Identity: one cosine of glue atoms; canonical curvature (2/3)^N_fam; clock [P]",
      "dependencies": "HOR.NARIAI.01; HOR.ORIENT.01; FR.KOIDE.04 (rate); GATE.QGEOM.02 (triality)",
      "scripts": [
        "v103_trisection_normal_form.py"
      ],
      "script": "v103_trisection_normal_form.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "standard GR (SdS)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 186
    },
    {
      "id": "HOR.CLOCK.01",
      "claim": "THE CLASSICAL NARIAI CLOCK IS THE ANCHOR (third appearance) + honest (2/3)-test (executes the HOR.TRISECT.01 clock question at the classical level): STATIC PIN [I] - phi(rho) = rho solves d/drho[(1-L rho^2)phi'] = -2L phi EXACTLY on the dS2 static patch, so the exact SdS family itself pins the S^2-radius modulus mass m^2 = -2L = -|Z2|L (the Laplace-type form of the Nariai linearization is standard GR - Ginsparg-Perry 1983 / Bousso-Hawking 1997 - typed external; the VALUE is pinned internally). Horizon split linear in the canonical angle, coefficient 1/sqrt(N_fam) [I]. GINSPARG-PERRY TOWER [I]: (l(l+1)-2)L has exactly ONE negative mode -|Z2|L (l=0), zero modes l=1. THE CLOCK [I]: in Hubble units chi_clock(lambda) = lambda^2+lambda-2 = (lambda-1)(lambda+2) = THE ANCHOR QUADRATIC; eigenvalues {1,-2} = the distinct anchor roots; Nariai cubic = (t-1) x chi_clock - the anchor appears as configuration roots (HOR.NARIAI.01), curvature base (HOR.TRISECT.01) and clock spectrum; entropy-deviation rate = 2H = |Z2| x Hubble exactly [I]. HONEST (2/3)-TEST NEGATIVE for the classical clock: eigenvalues are integers (anchor roots), not (2/3)-powers; the 2/3-powers stay in the functional; the one-loop conversion curvature -> rate (quantum (anti-)evaporation) remains external [P] - the flavor-rate bridge is NOT closed; the seam variational principle now has exactly one missing object: the QUANTUM clock",
      "status": "Identity [I] (static pin, tower, clock spectrum, rate) + Physical [P] (quantum clock)",
      "canonicalStatus": "Identity: clock spectrum = anchor roots; quantum clock = the one [P]",
      "dependencies": "HOR.TRISECT.01; HOR.ORIENT.01; HOR.NARIAI.01; ARCH.CORE.01 (anchor)",
      "scripts": [
        "v104_nariai_clock.py"
      ],
      "script": "v104_nariai_clock.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "standard GR (Nariai linearization Laplace-type: Ginsparg-Perry 1983; Bousso-Hawking 1997)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 187
    },
    {
      "id": "SYNTH.INVENTORY.01",
      "claim": "THE RESIDUAL INVENTORY (one constant, one anchor, one clock to find; the COMPLETE machine-pinned gap list): (A) ONE-CONSTANT INVENTORY [I] - 2/3 = |Z2|/N_fam appears exactly in seven independent places across both sectors (flavor: Koide branch point, transfer-gap base (2/3)^6, attractor inside the physical basin; gravity: Nariai entropy bound, SdS branch separation, canonical amplitude+frequency, canonical curvature base (2/3)^N_fam); ANCHOR TRIPTYCH [I]: configuration chi_a = (t-1)^2(t-2), Nariai cubic (t-1)^2(t+2) = traceless anchor, clock quadratic (lam-1)(lam+2); Nariai = (t-1) x clock; a = e2+e3. (B) RELOCATION THEOREM [I]+[P]: the cosmological one-loop clock correction eps = (c/24pi) Lambda/Mbar^2 = 7.6e-122 (c = 8 = the seam central charge, frozen rho_L) is deficient by 121.5 orders against the needed O(1) gap, and the deficit IS the Lambda hierarchy 2 alpha^-1/ln10 = 119.0 (+2.5 prefactors, the v60 split) => the seam clock must live at the SEAM scale, where the only gapped generator is the established boundary transfer operator; the bridge reaches its final form: ONE [P] identification 'transfer operator = seam Nariai clock'. (C) RESIDUAL TABLE (typing contract): exactly FIVE structural objects remain - R1 seam quantum clock [P] (HOR.CLOCK.01), R2 seam net holomorphic + c=8 [P]/[A] (GATE.METRIC.05/06/07), R3 seam determinant => EH form [A] (SEAM.AREACOEFF.03 residual), R4 H2 dictionary [A]/[P] (GATE.RU.01/CONTRACT.U.01), R5 parabolic realisation of Q carrying P1 [P] (GATE.QGEO.01/FLAV.SHEET.03) - plus the two irreducibles (one scale v_geo by dimensional analysis v78; the primitive pi) and the frozen data surfaces (gamma_CP, n_s/A_s, theta13, m_tau, r). COMPLETENESS is itself the claim: a sixth structural gap would FAIL v105. Nothing closed here; everything that remains is named, typed and counted",
      "status": "Identity [I] (inventory + relocation arithmetic) + typing contract (gap completeness)",
      "canonicalStatus": "Residual inventory: 5 named structural objects + 1 scale + pi; completeness machine-pinned",
      "dependencies": "HOR.CLOCK.01; HOR.TRISECT.01; HOR.NARIAI.01; HOR.ORIENT.01; FR.KOIDE.04/05/06; GATE.METRIC.05-07; GATE.RU.01; GATE.QGEO.01; ANCHOR.VGEO.01; REG.FREEZE.01",
      "scripts": [
        "v105_residual_inventory.py"
      ],
      "script": "v105_residual_inventory.py",
      "location": "origin_theory;introduction",
      "externalData": "frozen rho_Lambda (v84); 2D trace anomaly <T> = cR/(24pi) (standard)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 188
    },
    {
      "id": "ARCH.QUAD.01",
      "claim": "EXTERNAL-REVIEW VALIDATION + the DEGREE-2 pattern named: (1) SEED NORMAL FORM [I] - phi0 = (|mu4|/N_fam) c3 + Omega_adm c3^|mu4| exactly: the seed is a linear seam term plus a topological fourth-order correction over all admissible fermion states (4/3 = |mu4|/N_fam, 48 = Omega_adm = N_fam dim S+, exponent 4 = |mu4|) - a structured anchor readout, not a fitted formula. (2) HYPERCHARGE SECOND MOMENT [I] computed from the explicit charge table: Tr X = 0, Tr X^2 = 120 = 5! (X = 6Y on one 16-state generation). (3) FACTORIAL SPINE [I] (named lemma): 5! = 120 = |R+(E8)| = sum of E8 exponents = Tr X^2; 240 = 2x120; 1920 = 2^4 x 5! = |W(D5)| - the same factorial moment in four projections (hypercharge, E8 roots, D5 Weyl group, Hawking-power fingerprint): an anti-numerology structure statement. (4) DEGREE-2 INVENTORY [I]: six exact degree-2 structures (Pascal closure truncates at K=2 with UNIQUE solution g=5 on 1..40; glue norms 5/4+3/4 = 2 = root norm; c3 = (1/2) x 1/(4pi) = variational half x 2d-slice Gauss-Bonnet; pair sector C(5,2) = 10 = A_Lambda; second moments; codimension-2 seam). NAMED HYPOTHESIS [P] (recorded, NOT claimed): QUADRATIC BOUNDARY LOCALITY - the seam supports only bilinear data, hence the compiler truncation K=2 is forced rather than chosen; if proven, the Pascal SELECTION (red-team Target B residual, REDTEAM.B.01/CAR.PASCAL.01, typed [A]/[P]) upgrades to [L] and the chain locality => K=2 => g=5 => 16 => D5 => E8 closes the carrier choice - the named reduction target for Target B. (5) AUDIT: 240 = 16x15 (even-code transition reading) exact but non-unique (240 = 16x5x3 established, v1), no bijection exhibited - recorded like the v91 graph reading. NOTES: 'reality compiler = {a, pi, v_geo}' is a restatement of v23+v78 (confirmed); the requested grammar-freeze rule is already implemented by v84 (pre-data freeze) + v100 (declared-grammar census)",
      "status": "Identity [I] (seed NF, moment, spine, inventory) + [P] (named locality hypothesis) + audit",
      "canonicalStatus": "Identity: seed NF + factorial spine + degree-2 inventory; locality hypothesis [P] named",
      "dependencies": "ARCH.CORE.01 (anchor); CAR.PASCAL.01 / REDTEAM.B.01 (the residual it names); FLAV.MASSVOL.01; HOR.01 (1920); REG.FREEZE.01; v100",
      "scripts": [
        "v106_review_validation.py"
      ],
      "script": "v106_review_validation.py",
      "location": "tfpt_1;introduction",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 189
    },
    {
      "id": "HOR.CLOCK.02",
      "claim": "QUANTUM-CLOCK TARGET (R1 made quantitative; nothing fitted): the Ginsparg-Perry multi-mode clock has per-l equations lambda^2+lambda+(l(l+1)-2)=0 - l=0 anchor pair {1,-2}, l=1 zero modes {0,-1}, l>=2 complex with Re=-1/2=-delta (audit). NEW CROSS-LINK [I]: the classical decay set (l=0,1) = {0,-1,-2} = -Spec(V) = -N_fam x cusp weights - the classical seam clock decays exactly on the grading that carries the transfer operator's eigenspaces (FLAV.QGEO.01/FLAV.DIAMOND.02): structural support for the relocation identification (SYNTH.INVENTORY.01 R1). THE QUANTUM TARGET [I]: transfer rates {0, Delta, 6ln3} are not weight-linear; rate(2)/rate(1) = log_{3/2}(3) = 1+log_{3/2}(2) = 2.7095; exact identity (1/3)^6 = ((2/3)^6)^{log_{3/2}3}; required bend log_{9/4}(3) = 1.35476 per weight step. COUPLING LANDSCAPE [I]: kappa = (c/24pi)(Lambda/Mbar^2), c=8: cosmological 7.6e-122 vs seam 1/(3pi) = 0.106 - the only viable regime, borderline non-perturbative; one-loop linear response cannot fix the bend (O(1) resummation or exact seam construction required). STATUS [P]: R1 = 'produce the bend log_{3/2}3 on the cusp ladder at coupling O(1/3pi)'; identification NOT closed",
      "status": "Identity [I] (tower, cross-link, target arithmetic, coupling) + Physical [P] (identification)",
      "canonicalStatus": "Identity: clock lives on the cusp ladder; exact bend target; [P] identification open",
      "dependencies": "HOR.CLOCK.01; SYNTH.INVENTORY.01 (R1); FLAV.QGEO.01; FLAV.DIAMOND.02 (Spec V); FR.KOIDE.04",
      "scripts": [
        "v107_quantum_clock_target.py"
      ],
      "script": "v107_quantum_clock_target.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "standard GR (Ginsparg-Perry tower); 2D trace anomaly normalisation",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 190
    },
    {
      "id": "CAR.LADDER.01",
      "claim": "PASCAL LADDER THEOREM (Target B residual sharpened to zero slack): the closure 2^{g-1} = sum_{k<=K} C(g,k) has for every K exactly one solution g = 2K+1 (K=1..6 scanned over g=1..60; structurally: odd-g midpoint identity by binomial symmetry, even-g strict straddle around 2^{g-1} excludes all even g). EXACT EQUIVALENCE [I]: the carrier selection g=5 <=> the truncation degree K=2 - the Pascal-selection residual (CAR.PASCAL.01/REDTEAM.B.01, typed [A]/[P]) is therefore EXACTLY the one named hypothesis Quadratic Boundary Locality (ARCH.QUAD.01). NEIGHBOUR WORLDS [I]: K=1 -> g=3 (one-family world), K=2 -> g=5 (three families), K=3 -> g=7 (nine families), K=4 -> g=9 inconsistent (N_fam = 255/9 not integer). OVERDETERMINATION [I]: only K=2 also satisfies the independent rank closure g+N_fam = 8 = rank E8 (CAR.UNIQ.01/v14) - two independent selections agree. THE ONE HYPOTHESIS [P] (recorded, not claimed): QBL = the seam Calderon datum is a single two-point kernel and the carrier code is generated by the degrees it certifies (K<=2); if proven from reflection positivity the carrier choice upgrades [A]/[P] -> [L] along QBL => K=2 <=> g=5 => 16 => D5 => E8",
      "status": "Identity/Lattice [I]/[L] (ladder, equivalence, controls, overdetermination) + [P] (QBL)",
      "canonicalStatus": "Identity: g=5 <=> K=2 exactly; Target B residual = one hypothesis (QBL)",
      "dependencies": "CAR.PASCAL.01; REDTEAM.B.01; ARCH.QUAD.01 (QBL named); CAR.UNIQ.01 (v14)",
      "scripts": [
        "v108_pascal_ladder.py"
      ],
      "script": "v108_pascal_ladder.py",
      "location": "tfpt_3_e8_audit_bootstrap;tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 191
    },
    {
      "id": "FORM.LADDER.01",
      "claim": "LEAN FORMALISATION of the anchor ladder and the Pascal ladder (hardening round, 2026-06-11): new modules TfptCarrier/AnchorLadder.lean and TfptCarrier/PascalLadder.lean, built and audited with the existing project (lake build 3353 jobs; scripts/audit.sh AUDIT: PASS - no sorry/axiom/unsafe, only the three standard kernel axioms). AnchorLadder: p_n(a) = 2 + 2^n for ALL n (general theorem, not a finite check); ladder atoms (3,4,6,10); p1p2p3 = 240; p4-p3 = 8; 240+8 = 248; binary ladder p_{n+1}-p_n = 2^n (general); elementary symmetric (4,5,2). PascalLadder: the odd-midpoint identity sum_{k<=K} C(2K+1,k) = 2^{2K} for ALL K (via Mathlib Nat.sum_range_choose_halfway - the ladder SOLVES side of v108 is now formal); bounded uniqueness for the carrier case K=2 on 1<=g<=40 (kernel-decidable); carrier world (5,16,3,8); neighbour worlds (K=1 one-family, K=3 nine-family, K=4 inconsistent 9 ndvd 255). Strengthens the [F] layer of v23/v106 (ladder) and v108 (Pascal ladder); the unbounded K=2 uniqueness remains covered by CarrierRankUniqueness (division-free form)",
      "status": "Formal [F] (Lean kernel; no extra axioms)",
      "canonicalStatus": "Formal: anchor ladder general; Pascal ladder solves-side general + K=2 bounded uniqueness",
      "dependencies": "ARCH.CORE.01; CAR.LADDER.01; FORM.P2.02",
      "scripts": [
        "lean4-carrier-rigidity (AnchorLadder.lean",
        "PascalLadder.lean)"
      ],
      "script": "lean4-carrier-rigidity (AnchorLadder.lean, PascalLadder.lean)",
      "location": "tfpt_1;tfpt_3_e8_audit_bootstrap",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 192
    },
    {
      "id": "CAR.PAIR.01",
      "claim": "SHEET-PAIRING LEMMA (the first exact representation-theoretic anchor of the QBL programme; connects red-team Target B to the sheet Z2): character-exact weight-multiset results from the D5 spinor weights (+-1/2)^5 alone - (1) NO SCALAR WITHIN A SHEET: zero-weight multiplicity of S+xS+ (and S-xS-) is 0, because the odd slot count g=5 flips the sign parity of -w (a parity theorem of the five-slot code): there is no invariant bilinear pairing inside one chirality. (2) THE SCALAR LIVES ACROSS THE SHEETS: S+xS- has zero-weight multiplicity 16 = 1+5+10 (the Pascal triple) and equals exactly, as a weight multiset, the even-form tower Lambda^0 + Lambda^2 + Lambda^4 (256 = 1+45+210) with zero-mode grading (1,5,10) = the carrier code (Lambda^0,Lambda^1,Lambda^2)(C^5). (3) SHEET-DIAGONAL = ODD FORMS: (S+xS+) u (S-xS-) = 2Lambda^1 + 2Lambda^3 + Lambda^5 exactly (512 = 20+240+252): within-sheet pairings are odd-form-valued, never scalar. (4) the scalar-bearing tower tops at Lambda^{2K} with K=2 - bilinear seam data reach exactly the Pascal-ladder degree (CAR.LADDER.01). (5) QBL READING [P] (recorded, not claimed): the seam's single scalar two-point kernel is necessarily a SHEET PAIRING (chirality-off-diagonal, the same Z2 as GATE.METRIC.07 / FLAV.SHEET.01-03); the remaining step is to show the Calderon kernel supplies exactly this pairing and nothing beyond - then QBL => K=2 <=> g=5 closes the carrier choice",
      "status": "Identity [I] (character-exact multiset identities) + Physical [P] (QBL reading)",
      "canonicalStatus": "Identity: scalar pairing only across sheets; even tower tops at 2K; QBL anchored",
      "dependencies": "CAR.LADDER.01; ARCH.QUAD.01 (QBL); GATE.METRIC.07 (sheet Z2); FLAV.SHEET.01-03; CAR.PASCAL.01",
      "scripts": [
        "v109_sheet_pairing.py"
      ],
      "script": "v109_sheet_pairing.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 193
    },
    {
      "id": "CAR.PAIR.02",
      "claim": "CALDERON-SHEET SELECTION THEOREM (which Calderon involutions certify a scalar kernel; the v109 follow-up): on the carrier H = S+ (+) S- the certified bilinear data of an involution eps (eps^2 = 1) are the matrix elements <f, eps g>; the v109 singlet counts per sheet block (0, 0, 1, 1) give the exact selection statement - (1) SELECTION THEOREM [I] (finite model): a sheet-ODD involution (swapping S+ <-> S-) certifies EXACTLY 1+1 = 2 = |Z2| invariant scalar kernels (one per orientation), a sheet-EVEN involution certifies NONE: 'a scalar two-point datum exists IFF eps is sheet-odd'. (2) THE TWO KERNELS = THE GLUE AMBIGUITY [I]: the kernel count 2 = |Z2| matches exactly the two Lagrangian glues (GLUE.LAGR.01, swapped by single spinor conjugation) - the Calderon scalar channel carries precisely the glue ambiguity, resolved by the same sheet choice. (3) LADDER GENERICITY [I] (anti-overclaim control): for g = 3, 5, 7 the within-sheet zero-weight count is 0 and the cross count is 2^{g-1}; sheet-oddness forces the half-spinor relation K = (g-1)/2, NOT the value g = 5 - the g-selection stays with the independent rank-8/integrality closure (CAR.RANK / CAR.LADDER.01). (4) QBL RESIDUE SPLIT [P] (recorded, not claimed): (a) INTERFACE 'the seam Calderon involution is sheet-odd' - structurally natural, because the one-sided collar / double-cover deck is the SAME |Z2| that halves c3 = 1/(2 x 4pi) (exchanging cover sides is sheet-odd by construction); (b) ANALYTIC CORE 'the two-point kernel certifies only slot-degrees <= 2' - the genuine locality statement. Two named parts; proving both closes QBL => K=2 <=> g=5",
      "status": "Identity [I] (finite-model selection theorem + ladder control) + Physical [P] (residue typing)",
      "canonicalStatus": "Identity: scalar kernel exists iff involution is sheet-odd; 2 = |Z2| kernels = glue ambiguity; QBL residue split into interface + analytic core",
      "dependencies": "CAR.PAIR.01; CAR.LADDER.01; ARCH.QUAD.01 (QBL); GLUE.LAGR.01 (two glues); GATE.METRIC.07 (sheet Z2)",
      "scripts": [
        "v110_calderon_sheet.py"
      ],
      "script": "v110_calderon_sheet.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 194
    },
    {
      "id": "CAR.QTRANS.01",
      "claim": "QUADRATIC-TRANSPORT THEOREM (the transport half of the QBL analytic core becomes a theorem): grade seam transport by Clifford degree in the 10 field operators on the 5-slot Fock space (S = Lambda^*(C^5), 32-dim, S+ = 16 code states; integer Jordan-Wigner model, spanning claims via full-rank certificates mod p = 1000003, valid over Q) - (1) PARITY [exact block-zero]: all 10 linear words are sheet-ODD (map S+ -> S-), all 45 quadratic words are sheet-EVEN: code-preserving transport has even Clifford degree (the same Z2 as CAR.PAIR.01/02). (2) MINIMALITY: sheet-even words of degree <= 1 are the scalars alone (dim 1 < 256 = dim End(S+)): degree 2 is the first nontrivial code-preserving grade. (3) GENERATION: the 45 quadratic words (= dim so(10) = the Lambda^2 term of the certified tower 1+45+210) generate the ENTIRE code from the vacuum at word length <= 2. (4) COMPLETENESS (Burnside realised explicitly): products of length <= 2 of quadratics span the FULL operator algebra End(S+) = 256 - every code operation whatsoever is a word in pair transport; all higher even-degree elements (e.g. the 210 quartics) are redundant. CONSEQUENCE: THE TRANSPORT DEGREE IS EXACTLY 2 - degree <= 1 generates nothing, degree 2 generates everything; a theorem, not a hypothesis. (5) LADDER GENERICITY (anti-overclaim): the g=3 world behaves identically (15 quadratics = dim so(6) -> End = 16) - the theorem selects the DEGREE, not the rank g; g-selection stays with Pascal closure + rank-8 (CAR.LADDER.01/CAR.RANK). (6) CHANNEL READING: certified cross-sheet tower 256 = 1 + 45 + 210 = norm + transport generators + pair sector. (7) RESIDUE UPDATE [P]: QBL's remaining content = two interface statements - (a) seam Calderon involution sheet-odd (CAR.PAIR.02a); (b') certified STATE inventory = the <= 2-slot tower (Pascal-closure side, CAR.LADDER.01); nothing about transport remains hypothetical",
      "status": "Identity [I] (exact finite theorem; mod-p certificates valid over Q) + Physical [P] (residue typing)",
      "canonicalStatus": "Theorem: transport degree exactly 2 (minimal complete grade); QBL residue reduced to two interface statements",
      "dependencies": "CAR.PAIR.01; CAR.PAIR.02; CAR.LADDER.01; ARCH.QUAD.01 (QBL); CAR.RANK",
      "scripts": [
        "v111_quadratic_transport.py"
      ],
      "script": "v111_quadratic_transport.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 195
    },
    {
      "id": "CAR.COUNT.01",
      "claim": "SELF-COUNTING CHANNEL (the Pascal closure is an IDENTITY of the certified channel, not an extra requirement; closes the state half of the QBL analytic core at the structural level): (1) CANONICAL BIJECTION [I] - for odd g negation w -> -w maps the S+ weights bijectively into the opposite sheet (the CAR.PAIR.01 parity stated as a bijection), so the Cartan-neutral kernels of the certified channel S+xS- are exactly the pairs (w,-w), ONE PER CODE STATE: # neutral certified kernels = dim S+ = 2^{g-1} exactly (4, 16, 64 for g = 3, 5, 7) - the channel counts the code. (2) PASCAL PARTITION [I]: the same neutral set graded by pair-degree m (zero modes of Lambda^{2m}(C^{2g})) has sizes C(g,m) for m = 0..K = (g-1)/2; g=5 -> (1,5,10). (3) THE CLOSURE IS AN IDENTITY [I]: two countings of one set give 2^{g-1} = sum_{m<=K} C(g,m), symbolically verified for all odd g = 3..13 - the closure CONDITION of CAR.LADDER.01 is the channel counting itself. (4) HODGE FOLD [I]: the code's minus-sign grading C(g,2j) equals the folded binomial C(g, min(2j, g-2j)) with min(2j,g-2j) <= K - every code sector appears as a pair-degree <= K sector (g=5: code (1,10,5) vs channel (1,5,10), same multiset, folded order). (5) LADDER GENERICITY: the identity holds for every odd g and carries NO g-selection - the selection stays with rank-8 (g + N_fam = 8, CAR.RANK/v14) and ladder integrality (CAR.LADDER.01). (6) RESIDUE RE-TYPE [P] (recorded, not claimed): QBL clause (b') loses its closure burden - given the sheet-odd kernel (CAR.PAIR.02a) the channel automatically contains exactly one neutral kernel per code state at pair-degree <= K, so (b') is the definition of the channel; the remaining physical input is 'the seam certifies through a single scalar 2-point kernel' (for the established free c=8 seam net Wick extends the single kernel to the entire even tower by determinants); the carrier selection itself is carried by rank-8 + integrality - QBL is the structural consistency frame, no longer a missing selector",
      "status": "Identity [I] (bijection + exact multiset gradings + symbolic binomial identity) + Physical [P] (residue re-type)",
      "canonicalStatus": "Identity: Pascal closure = two countings of the channel's neutral set; one kernel per code state; QBL residue = single 2-point-kernel input",
      "dependencies": "CAR.PAIR.01; CAR.PAIR.02; CAR.QTRANS.01; CAR.LADDER.01; CAR.RANK; ARCH.QUAD.01 (QBL)",
      "scripts": [
        "v112_selfcounting_channel.py"
      ],
      "script": "v112_selfcounting_channel.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 196
    },
    {
      "id": "CAR.QFREE.01",
      "claim": "ONE KERNEL IS THE WHOLE NET (the QBL input merges with the R2/holomorphy premise; anchors the carrier choice to the Gate-A objects): (1) MAJORANA BOOKKEEPING [I] - level-1 central charges c(D5)_1 = 45/9 = 5 = g_car, c(A3)_1 = 15/5 = 3 = N_fam, c(SO16)_1 = 120/15 = 8, c(E8)_1 = 248/31 = 8; the carrier net (D5)_1 x (A3)_1 = SO(10)_1 x SO(6)_1 = 10 + 6 = 16 FREE MAJORANA FERMIONS with c = 16/2 = 8; the whole extension tower carrier(mu=16) -> SO(16)_1(mu=4) -> E8_1(mu=1) carries the SAME 16 fermions, only the certified glue grows (index steps 2 x 2 = 4 = |mu_4|, GATE.METRIC.06/07). (2) WICK/PFAFFIAN [I] (exact, 10-Majorana Jordan-Wigner model): ALL 210 vacuum 4-point functions AND all 210 6-point functions equal the Pfaffian of the single 2-point kernel - one kernel determines all correlations; quasi-freeness realised exactly. (3) THE KERNEL IS A CALDERON INVOLUTION OF RANK g [I]: M = I + iA with A integer antisymmetric, A^2 = -I; P = M/2 is a projection with rank P = 5 = g_car = c(D5)_1; eps = iA is an involution; at seam level (16 Majoranas) the rank is 8 = rank E8 - THE CENTRAL CHARGE IS THE RANK OF THE ONE KERNEL (5 carrier block, 8 seam hull). (4) ONE KERNEL <=> ONE STATE [I]: the joint kernel of the five annihilators is exactly 1-dimensional - the polarization fixes the vacuum uniquely; kernel => state => (Wick) all correlations. (5) PREMISE MERGE [P] (recorded, not claimed): QBL's remaining input 'single scalar 2-point kernel' = 'the seam state is quasi-free' = the defining property of the free-fermion net the R2 gate already posits (Gate-A (H') index-4 extension, GATE.METRIC.06); QBL adds NO assumption beyond R2; upgrade contract: R2 closes => the carrier choice closes [L] along free net => one kernel => sheet-odd (CAR.PAIR.02) => self-counting tower (CAR.PAIR.01/CAR.COUNT.01) => complete transport (CAR.QTRANS.01) => carrier; gate typing [P]/[A] unchanged",
      "status": "Identity [I] (exact Pfaffian/Wick + polarization theorems + bookkeeping) + Physical [P] (premise merge)",
      "canonicalStatus": "Identity: one quasi-free kernel determines the whole net (rank = c); QBL input = R2 premise; upgrade contract R2 => carrier [L]",
      "dependencies": "CAR.PAIR.02; CAR.COUNT.01; CAR.QTRANS.01; ARCH.QUAD.01 (QBL); GATE.METRIC.03-07 (Gate A); CAR.LADDER.01",
      "scripts": [
        "v113_quasifree_kernel.py"
      ],
      "script": "v113_quasifree_kernel.py",
      "location": "tfpt_1; tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 197
    },
    {
      "id": "GATE.UWALL.06",
      "claim": "TORSION NORMAL FORM + THE DELTA = 1/2 THEOREM (executes the GATE.UWALL.05 'genericity future work'; the distinguished lepton transport value delta = 1/2 stops being a point observation): (1) FLATNESS = mu4 TORSION [I, symbolic generic M] - prod_k U^k M U^{-k} = (MU)^4 exactly, so the Z4-family flatness constraint of the (U_wall) bundle IS the torsion statement 'T = MU is a fourth root of unity': the mu_4 atom appears as the ORDER of the twisted cusp generator. (2) DELTA = 1/2 THEOREM [I, both directions]: on the involutive branch (T^2 = 1) T is a Hermitian reflection 2vv^dag - 1 and the cusp trace condition 2v^dag U^{-1}v = 1 splits exactly into |v1|^2 = 1/2 (real part) + |v2| = |v3| (imaginary part), forcing diag M = (0, i/2, -i/2) with spec M = {1, w, w^2} AUTOMATIC (unitary + tr 0 + det 1 => char poly lam^3 - 1): delta = |M_22| = 1/2 EXACTLY AND CONSTANTLY on the whole 2-parameter branch - the value v20 used by hand and v40/v41 observed at the explicit point is a mu4-torsion identity. (3) REFLECTION LEMMA [I]: the D4 reflection forces diag M = (r, z, zbar) with r real and Re z = -r/2 - the D4 diagonal freedom is ONE real parameter, delta = 1/2 <=> r = 0. (4) BRANCH CENSUS [N, seeded scipy sampling, Python-only]: the full D4-fixed flat cusp locus realises only tr(MU) in {1, -1}; the involutive branch has diagonal CONSTANT (0, 1/2, 1/2) to 1e-15 across all samples; the {1, i, -i} branch has d1 varying over [0, 1] and the explicit v33/v40 point sits in its d1 = 0 slice - the same delta value. (5) RESIDUE [P] (recorded): whether the anchor splitting type O(-2)+O(-1)^2 (Mehta-Seshadri side) forces the d1 = 0 slice or selects the involutive branch is the remaining question; delta = 1/2 itself is no longer open",
      "status": "Identity [I] (symbolic theorems) + Numerical [N] (branch census) + Physical [P] (residue)",
      "canonicalStatus": "Theorem: delta = 1/2 is a mu4-torsion identity on the involutive branch; flatness = (MU)^4 = 1 normal form; residue = bundle-side branch selection",
      "dependencies": "GATE.UWALL.04; GATE.UWALL.05; FLAV.DELTA.01; v20; v30; v33",
      "scripts": [
        "v114_torsion_delta.py"
      ],
      "script": "v114_torsion_delta.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 198
    },
    {
      "id": "GATE.UWALL.07",
      "claim": "THE ANCHOR PINS THE RESIDUE (exact normal form for the U_wall flat-bundle residue; answers the GATE.UWALL.06 branch question): (1) MU4-AVERAGE LEMMA [I, symbolic generic X] - sum_k U^k X U^{-k} = 4 diag(X): mu_4 conjugation-averaging is the diagonal projection, so the exponents at infinity are 4 diag(A0) and the anchor splitting O(-2)+O(-1)^2 holds IFF diag A0 = (2,1,1)/4 = a/|mu_4| - the residue diagonal IS the anchor over mu_4 (the v33 diagonal was forced, not chosen). (2) THE (8,0,5)/144 LEMMA [I]: anchor diagonal + cusp spectrum {0,1,2}/N_fam pin the off-diagonal weight sum to 13/144; with a13 = 0 the det = 0 condition solves UNIQUELY to (|a12|^2, |a23|^2) = (8, 5)/144 - numerators (rank E8, g_car), denominator (|mu_4| N_fam)^2 = 144, total 13 = Delta_Q = 2p_2(a)+1 (the quark denominator 117 = 9x13). (3) EXACT RESIDUE NORMAL FORM [I]: A0* = [[1/2, sqrt2/6, 0],[sqrt2/6, 1/4, sqrt5/12],[0, sqrt5/12, 1/4]] has char poly lam(lam-1/3)(lam-2/3) exactly. (4) A0* IS THE RIEMANN-HILBERT SOLUTION [N]: big-circle monodromy trivial to 1.6e-10, prod M_k = 1, unitarised |diag M~_0| = (0, 1/2, 1/2), tr(M~_0 U) = 1 - the v33/v40 numerical point is gauge-equivalent to this exact matrix (a12/a23 phases = diagonal gauge). (5) THE ANCHOR FORCES THE d1 = 0 SLICE [N]: multi-seed RH scan over the anchor-pinned spectral manifold lands at every flat solution on the SAME gauge orbit with |diag M~_0| = (0, 1/2, 1/2) - combined with GATE.UWALL.06 the whole harmonic diagonal is anchor + torsion forced; delta = 1/2 needs no selection. (6) RESIDUE [P]: a13 = 0 + gauge-orbit uniqueness numerical (one component across seeds); promotion to [I]/[L] via Groebner on the RH constraints = the remaining U_wall point-selection step",
      "status": "Identity [I] (mu4-average + (8 0 5)/144 + exact normal form) + Numerical [N] (RH verification + uniqueness scan) + Physical [P] (residue)",
      "canonicalStatus": "Exact residue normal form A0* (diag = anchor/mu4 spectrum = cusp weights off-weights = (8 0 5)/144); anchor forces d1 = 0; residue = Groebner promotion",
      "dependencies": "GATE.UWALL.06; GATE.UWALL.04; v33; v40; CONTRACT.U.01",
      "scripts": [
        "v115_anchor_residue.py"
      ],
      "script": "v115_anchor_residue.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 199
    },
    {
      "id": "GATE.UWALL.08",
      "claim": "RESONANCE THEOREM (the GATE.UWALL.07 [N] findings promoted to [I] by a LINEAR computation - the anticipated Groebner step turned out unnecessary): (1) TWISTED-AVERAGE LEMMA [I, symbolic generic A0]: at infinity the equivariant Fuchsian system reads dY/dw = -(1/w)(B0 + B1 w + ...)Y with B_m = sum_k i^{km} U^k A0 U^{-k}; the twisted mu_4 averages collapse to B0 = 4 diag(A0) and B1 = 4 a12 E_12 + 4 conj(a13) E_31 - only the eigenvalue-ratio -i cells survive. (2) RESONANCE THEOREM [I]: the anchor exponents diag(2,1,1) have resonance gap 1; the level-1 formal-gauge equation (1 - ad R0)H1 = R1 is singular exactly on the cells {(2,1),(3,1)}, where the obstruction entries of B1 are (0, 4 conj(a13)); all levels k >= 2 are non-resonant and the formal monodromy e^{-2 pi i diag(2,1,1)} = 1, hence M_inf = 1 <=> a13 = 0 - the transcendental Riemann-Hilbert condition collapses to ONE linear equation. (3) UNIQUENESS COROLLARY [I]: a13 = 0 + the (8,0,5)/144 moduli (GATE.UWALL.07) + phases = diagonal gauge (commutes with U, fixes the diagonal, rotates arg a12 and arg a23 freely) => the flat anchor locus is EXACTLY ONE GAUGE ORBIT = the exact A0*; within the mu_4-equivariant anchor class the U_wall bundle datum is ALGEBRAICALLY unique. (4) FALSIFICATION CONTROL [N]: ||M_inf - 1|| = 1.6e-10 at a13 = 0 vs 0.17 / 0.43 at a13 = 0.02 / 0.05 - the obstruction is sharp (~8.5 |a13|). (5) HONEST SCOPE [P] (recorded): the equivariant ansatz A_k = U^k A0 U^{-k} is the established D4 selector input (v19/v30); the residue side is now [I]; the holonomy values (harmonic diagonal (0, 1/2, 1/2)) remain [N] - transcendental monodromy of an exact system",
      "status": "Identity [I] (twisted average + resonance theorem + uniqueness) + Numerical [N] (sharpness control) + Physical [P] (scope)",
      "canonicalStatus": "Theorem: M_inf = 1 <=> a13 = 0 (resonance obstruction); flat anchor locus = one gauge orbit = exact A0*; U_wall bundle datum algebraically unique",
      "dependencies": "GATE.UWALL.07; GATE.UWALL.06; v19; v30; CONTRACT.U.01",
      "scripts": [
        "v116_resonance_uniqueness.py"
      ],
      "script": "v116_resonance_uniqueness.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 200
    },
    {
      "id": "GATE.UWALL.09",
      "claim": "THE MONODROMY IS THE WEYL GROUP OF A3 (the U_wall holonomy is an EXACT 24-element representation; the harmonic diagonal becomes a theorem): (1) THE EXACT MONODROMY [I]: M0 = [[0, -(1+i)/2, (1-i)/2], [-(1+i)/2, -i/2, -1/2], [(1-i)/2, -1/2, i/2]] (entries in (1/2)Z[i]) is unitary with det 1, tr 0, char poly lam^3 - 1 (the cusp class {1,w,w^2} EXACT), M0^3 = 1, and diag M0 = (0, -i/2, +i/2) - d1 = 0 AND delta = 1/2 are exact matrix entries: the v20 hand value, the v40/v41 observation, the GATE.UWALL.06 torsion value and the GATE.UWALL.07 anchor slice meet in ONE exact matrix. (2) TORSION/FLATNESS [I]: (M0 U)^4 = 1 with tr(M0 U) = 1 (the {1,i,-i} branch, d1 = 0 slice - exactly where GATE.UWALL.06/07 located the physical point) and prod_k U^k M0 U^{-k} = 1 exactly. (3) THE GROUP IS W(A3) = S4 [I]: exact enumeration of <U, M0> gives order 24 with order statistics (1, 9, 8, 6) and character values (3, -1, 0, 1) - uniquely S4 in its standard x sign representation = the (twisted) reflection representation of the WEYL GROUP OF THE FAMILY LATTICE A3: the flavor-wall monodromy realises the A3 factor of the glue D5 + A3 + mu4 as its Weyl group; U = image of a 4-cycle (the mu_4 deck), M0 = a 3-cycle (the family rotation), 24 = |W(A3)| = 4!. (4) IDENTIFICATION [N]: the unitarised ODE monodromy of the exact A0* system equals the exact M0 to 3.5e-10 and the invariant form H is U-invariant to 2.5e-11 (the harmonic frame preserves the equivariance). (5) STATUS [P] (recorded): holonomy VALUES are now [I] (exact properties of the W(A3) representation); only the identification with the analytic monodromy of A0* stays [N] (ODE matching); the delta thread v20 -> v40/v41 -> v114 -> v115 -> v117 is closed end to end",
      "status": "Identity [I] (exact matrix + exact group enumeration) + Numerical [N] (ODE identification) + Physical [P] (status)",
      "canonicalStatus": "Theorem: U_wall monodromy group = W(A3) = S4 (order 24 exact); diag M0 = (0 -i/2 i/2) exact - delta = 1/2 and d1 = 0 are matrix entries; only ODE identification stays [N]",
      "dependencies": "GATE.UWALL.08; GATE.UWALL.07; GATE.UWALL.06; E8.GLU.01 (A3 factor); CONTRACT.U.01",
      "scripts": [
        "v117_monodromy_weyl_a3.py"
      ],
      "script": "v117_monodromy_weyl_a3.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 201
    },
    {
      "id": "FLAV.H2.02",
      "claim": "THE HEXAGON IS THE SIGN-TWISTED FAMILY SPECTRUM (first exact piece of the H2 dictionary; the v20 lepton coefficients are resolvent determinants of the exact W(A3) monodromy GATE.UWALL.09): (1) SIGN-TWIST LEMMA [I]: -M0 has order 6 ((-M0)^3 = -1) and spec(M0) u spec(-M0) = mu_6 - the 6-site hypercharge hexagon spectrum IS the family monodromy spectrum together with its sheet twist, Z6 = Z2 x Z3 realised as (-1) x <M0>; the v20 hexagon denominators 5/4 - cos(r pi/3) equal |1 - zeta_6^r/2|^2, the eigen-denominators of the two family resolvents (1 -+ M0/2)^{-1}. (2) CYCLOTOMIC DETERMINANT [I]: det(1 - t M0) = 1 - t^3 = (1-t) Phi_3(t) (the N_fam cyclotomic) => det(1 - M0/2) = 7/8 and det(1 + M0/2) = 9/8. (3) LEPTON COEFFICIENTS = RESOLVENT DETERMINANTS [I]: c_e = |mu_4|/(2 det-) = 16/7, c_mu = N_fam/(2 det+) = 4/3, c_tau = |mu_4| det-/N_fam = 7/6, and v20's product rule = |mu_4|/det+ = 32/9 - the lepton denominators 7 and 9 ARE the two family-resolvent determinants (x 2^3), and delta = 1/2 = |(M0)_22| is supplied by the monodromy matrix itself; the e/mu/tau LEG assignment (residues r, windings w) remains v20's established input - nothing re-fished. (4) SHEET-EXTENDED GROUP [I]: <U, M0, -1> has order 48 = Omega_adm = N_fam x dim S+ (the seed quartic coefficient, ARCH.QUAD.01) = |S4 x Z2| (exact enumeration; audit cross-link). (5) RESIDUE [P] (recorded): the dictionary's VALUES are exact; its ADDRESS TABLE (which hexagon site each fermion occupies; the quark-sector composition) remains open",
      "status": "Identity [I] (exact spectral + determinant identities + group enumeration) + Physical [P] (address table)",
      "canonicalStatus": "Identity: hexagon = spec(M0) u spec(-M0); lepton 7 and 9 = det(1 -+ M0/2) x 8; product rule = mu4/det+; H2 values exact - address table open",
      "dependencies": "GATE.UWALL.09; FLAV.H2.01; v20 (leg assignment); ARCH.QUAD.01 (Omega_adm); E8.GLU.01",
      "scripts": [
        "v118_hexagon_family_dictionary.py"
      ],
      "script": "v118_hexagon_family_dictionary.py",
      "location": "tfpt_research_contracts; tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 202
    },
    {
      "id": "ARCH.TRIAD.01",
      "claim": "SECOND EXTERNAL-REVIEW VALIDATION (2026-06-11b): the anchor ratio triad, the 121 audit lemma and the normal-form micro-identities, all exact: (1) MICRO-IDENTITIES [I]: Omega_adm = 2 p1 p2 = 48; 10 b1 = 1 + p1 p3 = 41; Delta_Y = e2^2 = p0^2 + 2 rank E8 = 25; h(E8) = e3 p0 e2 = 30; rank E8 = p0 + e2 = 8; |R(E8)| = p1 p2 p3 = 240; dim E8 = 240 + (p4 - p3) = 248. (2) ANCHOR RATIO TRIAD [I] (named lemma): the three elementary symmetric values of the anchor normalised by the family count - e3/p0 = 2/3 (the Koide branch / sheet ratio, cf SYNTH.INVENTORY.01 seven appearances), e1/p0 = 4/3 (the seed gain |mu_4|/N_fam, ARCH.QUAD.01), e2/p0 = 5/3 (the carrier branch: the canonical flow's critical points are EXACTLY (2,5) = (e3,e2), FR.KOIDE.06, inflection 7/2 = (e3+e2)/2) - the recurring fractions 2/3, 4/3, 5/3 are the three normalised anchor coefficients, not scattered numbers. (3) THE 121 AUDIT LEMMA [I, typed AUDIT not load-bearing]: (1+a)^T R (1+a) = 121 = 11^2 = ||Pl(K)||_1^2 = (p3+1)^2 with the established residue operator R (det 8, SNF (1,1,8)) - the quark-ratio 11 (c_u/c_d = 5x11/(9x13)) appears a SECOND independent time as the quadratic anchor norm of R; sensitivity control: the 3 orderings of h = (2,2,3) give {105, 121, 135}, only the canonical anchor ordering yields 121; bonus checksums 1^T R 1 = 22 = 2x11 (entry sum) and a^T R a = 40 = p1 p3 = 10b1 - 1; the oriented Plucker mechanism (FLAV.KQ.01/v42/v49) remains the derivation. (4) FLAVOR SURFACE claim = established FLAV.DIAMOND.01/02 (re-verified, presentational). (5) BOOKKEEPING: mu4 projection = GATE.UWALL.07/08, triptych = HOR.CLOCK.01/SYNTH.INVENTORY.01, interior-free = CAR.QFREE.01 (all closed); anchor-first presentation recorded as editorial item",
      "status": "Identity [I] (exact arithmetic/matrix identities; audit typing for the 121 lemma)",
      "canonicalStatus": "Identity: anchor ratio triad (2/3 4/3 5/3) = e/p0; 121 = anchor norm of R (audit); micro-identities exact",
      "dependencies": "ARCH.QUAD.01; FLAV.KQ.01; FLAV.DIAMOND.01-02; FR.KOIDE.06; SYNTH.INVENTORY.01",
      "scripts": [
        "v119_review_validation_2.py"
      ],
      "script": "v119_review_validation_2.py",
      "location": "introduction; tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 203
    },
    {
      "id": "FLAV.H2.03",
      "claim": "THE ADDRESS TABLE (the open FLAV.H2.02 item structured; exact arithmetic on FROZEN v18/v20 integers): (1) THE LEPTON WORDS ARE THE COMPILER ATOMS [I]: (L_e, L_mu, L_tau) = (8, 5, 3) = (rank E8, g_car, N_fam) = (p0+e2, e2, p0), in mass order (longest word = lightest lepton). (2) ADDRESSES = EUCLIDEAN DIVISION BY THE HEXAGON [I]: (r, w) = L divmod p2 with p2 = 6 = |R+(A3)| itself an atom: e -> (2,1), mu -> (5,0), tau -> (3,0); sum r = 10 = p3 = A_Lambda; sum L = 16 = dim S+ = 2^{g-1}. (3) SHEET PARITY [I] (the FLAV.H2.02 dictionary applied): e sits on the untwisted sheet (omega in spec M0), mu and tau on the twisted sheet (-omega, -1 in spec(-M0)); the anchor lepton tau occupies the unique REAL twisted eigenvalue -1 - exactly the site where v20's product rule replaces the resolvent. (4) QUARK SUM RULES [I] (frozen v18 words u,d,s,c,b,t = 7,7,5,3,2,0): up-type sum = 10 = p3; down-type sum = 14 = p1 + p3; ALL QUARKS = 24 = |W(A3)| (the flavor-wall monodromy group order, GATE.UWALL.09); ALL NINE CHARGED FERMIONS = 40 = p1 p3 = a^T R a (ARCH.TRIAD.01 bonus checksum); quark sum r = 12 = 2 p2, sum w = 2 = |Z2|; the top sits at the vacuum site (0,0) (zero word length, no suppression - the established ladder anchor). (5) RESIDUE [P] (recorded): five independent closures (16, 10, 14, 24, 40 - all anchor atoms or today's group orders) constrain the address map, but the per-fermion assignment remains the v18/v20 input; deriving it is the remaining H2 content. FIREWALL: all integers were frozen before today's readings - no degrees of freedom spent",
      "status": "Identity [I] (exact arithmetic on frozen integers) + Physical [P] (per-fermion derivation)",
      "canonicalStatus": "Identity: lepton words = (rank g N); addresses = divmod hexagon; quark sums = (p3 p1+p3 |W(A3)| p1p3); assignment derivation open",
      "dependencies": "FLAV.H2.02; GATE.UWALL.09; ARCH.TRIAD.01; v18 (quark words); v20 (lepton words)",
      "scripts": [
        "v120_address_table.py"
      ],
      "script": "v120_address_table.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 204
    },
    {
      "id": "FLAV.H2.04",
      "claim": "THE ADDRESS TABLE IS PINNED (the FLAV.H2.03 per-fermion item closed at the information level; exact census theorem): (1) THE WORD TABLE IS THE RESIDUE OPERATOR [I]: the established identity L = R + 2U (FLAV.DIAMOND.02, U = winding axis N_fam 1 e_1^T) says the word-length table IS L[sector][generation] with rows (up; down; lepton) = ((7,3,0); (7,5,2); (8,5,3)) = the v18/v20 word lengths verbatim; the addresses are R's entries plus one hexagon turn for generation 1, and R's first column is THE ANCHOR a = (1,1,2) - the first-generation residues are the anchor components. (2) THE MARGINS ARE ATOMS [I]: row sums of R = (4, 8, 10) = (e1, rank E8, p3); column sums = (4, 13, 5) = (e1, Delta_Q, g_car); total 22 = 2x11 (ARCH.TRIAD.01). (3) PINNING THEOREM [I, exact census]: among ALL 3x3 matrices with entries in the hexagon {0..5}, exactly 17 have the atom margins, and exactly ONE has det = 8 = rank E8 - and it is R (the same one is the unique one with SNF (1,1,8); all 17 determinants are distinct, 8 occurs once - control computed explicitly). The per-fermion assignment table carries ZERO residual information beyond the atom margins and the rank determinant. (4) within-sector assignment = the established transport reading (m ~ lambda_Y^L, longer word = lighter fermion). (5) RESIDUE [P] (recorded): the open H2 item shrinks from 'derive 9 per-fermion integers' to 'derive the six atom margins + det = rank'. FIREWALL: R, L and all word lengths were frozen in v4/v18/v20/v71 long before today's census - no degrees of freedom spent",
      "status": "Identity [I] (exact census/uniqueness theorem on frozen operators) + Physical [P] (margins)",
      "canonicalStatus": "Theorem: R = unique hexagon matrix with atom margins + det 8; address table = R + first-generation winding; H2 residue = the margins",
      "dependencies": "FLAV.H2.03; FLAV.DIAMOND.02 (L = R + 2U); FLAV.H2.01; ARCH.TRIAD.01; v18; v20; v71",
      "scripts": [
        "v121_address_pinning.py"
      ],
      "script": "v121_address_pinning.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 205
    },
    {
      "id": "FLAV.H2.05",
      "claim": "THE MARGINS ARE THEOREMS (the FLAV.H2.04 residue closed: the established frozen selectors pin R uniquely - the atom margins lose their input status): THREE ESTABLISHED SELECTORS, all frozen pre-v120 - (S1) the D4 annihilator n = (5,-9,6) with n^T R = (8,0,0) (FLAV.DIAMOND.01/tfpt_2: n kills the unchanged (c2,c3) plane; n.a = 8); (S2) det R = 8 = rank E8 (v71 determinant quartet (3,4,8,20), product 1920 = |W(D5)|); (S3) det K = 4 = |mu_4| with K = R + Q Sigma (the diamond); plus the definitional hexagon range {0..5}. (1) CENSUS [I]: hexagon columns with n.col = 8: exactly 4, with n.col = 0: exactly 4 => 64 (S1)-candidates; (S2) leaves 12; (S3) leaves EXACTLY ONE - and it is R. (2) COROLLARY [I]: the atom margins (4,8,10)/(4,13,5), the anchor column R e_1 = a = (1,1,2), and all FLAV.H2.03/04 sum rules (16, 10, 14, 24, 40) are CONSEQUENCES of (S1)-(S3) - input status revoked. (3) BONUS LEMMA [I]: on the (S1)+(S2) locus det(M + 2U) = 20 = det L holds for ALL 12 candidates - the fourth quartet determinant is NOT an independent selector but a consequence: the quartet (3,4,8,20) carries redundancy, the compiler's overdetermination pattern. (4) RESIDUE [P] (recorded): H2 reduces to the establishment of the selectors themselves (n and the diamond determinants - pre-existing inventoried objects); the address question introduces NO new residual class. FIREWALL: n, quartet, R, L, words frozen in v4/v18/v20/v71/v94 - nothing fitted",
      "status": "Identity [I] (exact census on frozen selectors) + Physical [P] (selector establishment)",
      "canonicalStatus": "Theorem: established selectors (n-annihilator + det R = 8 + det K = 4) pin R uniquely; margins + anchor column + sum rules = consequences; det L = 20 automatic",
      "dependencies": "FLAV.H2.04; FLAV.DIAMOND.01-02; v71 (quartet); v94 (annihilator); CONTRACT.U.01",
      "scripts": [
        "v122_margin_theorem.py"
      ],
      "script": "v122_margin_theorem.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 206
    },
    {
      "id": "SYNTH.INVENTORY.02",
      "claim": "RESIDUAL INVENTORY UPDATED (post v110-v122; supersedes the CONTENTS of SYNTH.INVENTORY.01 while keeping its cardinality): (1) thirteen new ledger rows machine-pinned (CAR.PAIR.02, CAR.QTRANS.01, CAR.COUNT.01, CAR.QFREE.01, GATE.UWALL.06-09, FLAV.H2.02-05, ARCH.TRIAD.01). (2) R2 CARRIES THREE LOADS: the single index-4 boundary-net statement closes, when proven, the metric gate (GATE.METRIC.06), the carrier choice (CAR.QFREE.01 upgrade contract) and the QBL programme (ARCH.QUAD.01) - one theorem, three doors. (3) UPDATED RESIDUAL TABLE, exactly FIVE structural classes: R1 seam quantum clock (produce the bend log_{3/2}3 at coupling ~1/3pi) [P]; R2 index-4 seam net (3 loads) [P]/[A]; R3 seam determinant => EH [A]; R4' SELECTOR ESTABLISHMENT (derive n = (5,-9,6) + the diamond determinants from the parabolic geometry; replaces the DISCHARGED 'H2 dictionary' class) [A]/[P]; R5 parabolic realisation of Q (GATE.QGEO; carries P1) [P] - plus the two irreducibles (v_geo, pi) and the frozen data surfaces; a sixth structural class would FAIL the script. (4) THE FLAVOR SIDE HAS NO OPEN ANALYSIS CLASS OF ITS OWN: H2 values exact (FLAV.H2.02), addresses pinned (FLAV.H2.04), margins theorems (FLAV.H2.05); residue = five frozen selector integers. Ledger-bookkeeping module, Python-only by nature",
      "status": "Identity [I] (machine bookkeeping + typing contract)",
      "canonicalStatus": "Inventory: five classes re-pinned (R1 clock; R2 index-4 net with 3 loads; R3 EH; R4' selectors; R5 Q); flavor side discharged; sixth class fails the script",
      "dependencies": "SYNTH.INVENTORY.01; GATE.METRIC.06; CAR.QFREE.01; ARCH.QUAD.01; FLAV.H2.05; HOR.CLOCK.02; GATE.QGEO",
      "scripts": [
        "v123_inventory_update.py"
      ],
      "script": "v123_inventory_update.py",
      "location": "origin_theory; introduction",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 207
    },
    {
      "id": "SYNTH.INVENTORY.03",
      "claim": "TERMINAL RESIDUAL INVENTORY (post premise-(A) closure; supersedes the CONTENTS of SYNTH.INVENTORY.02): after the (A)-chain (v160-v165) premise (A) is NO LONGER an independent structural class. (1) (A) CLOSED AS A CLASS - the four (A)-closure rows exist (GATE.METRIC.16 fixed-point theorem; .17 cone eliminated, cone gap = one-particle gap; .18 numbers forced by mu4-equivariance; .19 factors into A2 + GATE.QGEO + irreducible core). (2) THE FIVE v123 CLASSES COLLAPSE: R1 clock, R2 index-4 net and R5 Q-realisation all merge into ONE Tier-1 hinge (the seam=flavour=horizon identification = A2 net existence [assembly of established net constructions, GATE.METRIC.11/13] + GATE.QGEO [seam boundary = P^1 minus mu4; cohomology b1=N_fam=3, cusp grading = A3 exponents]); R3 (EH step) folds into v_geo (SEAM.EHMODEL.03/v152: the normalisation IS the anchor); R4' (selectors) was already discharged. (3) TERMINAL STRUCTURE: Tier 0 = the irreducible core {pi, v_geo}, a THEOREM by the No-Unit Theorem (ANCHOR.VGEO.01/02), nothing to close; Tier 1 = ONE hinge; Tier 2 = folded (EH -> v_geo, ambient QG -> G_net via GATE.METRIC.02/v76, gap-decoupled); Tier 3 = F_transfer, ONE functor with four typed CONDITIONAL interfaces (Koide FR.KOIDE.05/06, eta_B FR.ETAB.01, m_p/m_e QCD.LAMBDA.01 [ratio stays open/not forced], RG-stable lepton mixing FLAV.RGSTAB.01; axion = scenario), eta_B carrying the most open computational room. (4) COMPLETENESS: the structural residual is now ONE hinge + ONE transfer functor + the {pi, v_geo} core; a sixth independent structural class - or a new irreducible - would FAIL this script. HONEST: a unification + re-typing, NOT an unconditional proof",
      "status": "Identity [I] (machine bookkeeping + typing contract; ledger-row membership)",
      "canonicalStatus": "Terminal inventory: (A) no longer a class; ONE hinge (A2+GATE.QGEO) + F_transfer + irreducible {pi,v_geo}; Tier 2 folded; sixth class fails the script",
      "dependencies": "SYNTH.INVENTORY.02; GATE.METRIC.16; GATE.METRIC.17; GATE.METRIC.18; GATE.METRIC.19; ANCHOR.VGEO.02; FR.ETAB.01",
      "scripts": [
        "v167_inventory_post_closure.py"
      ],
      "script": "v167_inventory_post_closure.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 208
    },
    {
      "id": "HOR.CLOCK.03",
      "claim": "THE RESUMMED CLOCK (R1's bend acquires a closed form; the HOR.CLOCK.02 job description transforms): (1) THE SPECTRUM READS (1 - n/N)^{p2} [I]: the frozen transfer eigenvalues {1, (2/3)^6, (1/3)^6} (v54/v69/v95) are EXACTLY lambda_n = (1 - n/N_fam)^{p2} for n = 0, 1, 2, with p2 = 6 = |R+(A3)| the hexagon size. (2) CLOSED-FORM CLOCK [I]: rate(n) = -p2 ln(1 - n/N_fam) reproduces {0, Delta, 6 ln 3} exactly; the HOR.CLOCK.02 bend rate(2)/rate(1) = log_{3/2}3 becomes a one-line consequence. (3) THE LINEARISATION CARRIES THE SHEET [I]: rate(n) = |Z2| n + n^2/3 + 2n^3/27 + ... - the linear (one-loop) term has slope exactly |Z2| = 2 relative to the integer classical Ginsparg-Perry clock {0,-1,-2} (HOR.CLOCK.01): the quantum clock is the SHEET-DOUBLED classical clock plus the geometric tail sum_k (n/N)^k/k. (4) THE WALL AT N_fam [I]: rate -> infinity as n -> N_fam (pole of the resummed logarithm at the family count) => the cusp ladder ends at n = N_fam - 1 = 2: the three-level transfer spectrum is forced by the clock itself. (5) R1 RE-SHARPENED [P] (recorded, not claimed): closed-form semiclassical target - derive rate(n) = -p2 ln(1 - n/N_fam) from the near-Nariai one-loop at kappa = 1/(3pi); the linear response must give slope |Z2| x classical and the resummation must exponentiate the geometric series; all spectra frozen in v54/v69/v95/v104/v107 - nothing fitted",
      "status": "Identity [I] (exact spectral reading + series) + Physical [P] (semiclassical derivation)",
      "canonicalStatus": "Identity: transfer spectrum = (1 - n/N)^6; rate = -6 ln(1 - n/3); linearisation slope |Z2|; pole at N_fam forces three levels; R1 target now closed-form",
      "dependencies": "HOR.CLOCK.02; HOR.CLOCK.01; FR.KOIDE.06 (transfer spectrum); SYNTH.INVENTORY.02 (R1)",
      "scripts": [
        "v124_resummed_clock.py"
      ],
      "script": "v124_resummed_clock.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 209
    },
    {
      "id": "GATE.METRIC.08",
      "claim": "THE GLUE Q-SYSTEM (the index-4 mu4 extension exists as an explicit Longo Q-system; the R2 gate sharpens from 'find a Q-system' to 'match the constructed one'): (1) Q-SYSTEM AXIOMS [I] on A = C[Z4] (the glue group algebra; multiplication delta_a x delta_b -> delta_{a+b}, unit delta_0): associativity, unit law, FROBENIUS (m x 1)(1 x m*) = m* m = (1 x m)(m* x 1), and SPECIALNESS m m* = |Z4| id all hold exactly => a special C* Frobenius algebra = a Longo Q-system with dim theta = 4: JONES INDEX [B:A] = 4 = |mu_4| - exactly the GATE.METRIC.06 Carrier Index Lemma value; KLM bookkeeping mu_carrier = 16 = 4^2 x mu_E8 re-verified. (2) LOCALITY = ISOTROPY [I] (GATE.METRIC.07 re-derived in Q-system language): on the Lagrangian glue <(1,1)> the discriminant form q = (5x^2+3y^2)/8 gives q(k(1,1)) = k^2 = (0,1,4,9), all integers - the Q-system braiding obstruction vanishes (local/bosonic extension); the two Lagrangian glues = the sheet pair are the only maximal isotropic choices. (3) THE HALFWAY SUB-Q-SYSTEM [I]: {0,2} < Z4 closes multiplicatively => the unique intermediate Q-system C[Z2] = the SO(16)_1 step of the tower (GATE.METRIC.07/CAR.QFREE.01), index factorisation 4 = 2 x 2. (4) R2 SHARPENED [P] (recorded, not claimed): the gate's algebraic object now EXISTS explicitly; what remains is the operator-algebraic IDENTIFICATION 'the seam-Calderon inclusion of the 16-Majorana net realises THIS Q-system' - matching two constructed objects, not a search; the three loads (metric + carrier + QBL, SYNTH.INVENTORY.02) hang on this one identification; gate typing [P]/[A] unchanged [2026-08-03 LIII: the index side is now MEASURED on the CAR ladder (v726, 57/57, GNET.PPINDEX.01): the mu4-average expectation has optimal Pimsner-Popa constant EXACTLY 1/4 and Watatani index EXACTLY 4 via an explicit quasi-basis with FORCED weights, with mu4 DERIVED from the clock (not inserted) and the Ramond sector healed state-preservingly by the v679 Klein dressing; the remaining half of the gate is the IDENTIFICATION of this ladder Q-system with the constructed GATE.METRIC.08 object -- registered as the open contract half of GNET.RAMIFIED.01 (kills: Frobenius mismatch, index != 4 on the state-preserving ladder, dressing incompatible with the sigma descent).  No status move.]  [2026-08-05 LXII (v779): the finite precursor of the R2 identification is measured -- the coherent COMMUTING system of index-4 expectations along the Haar tower (restriction coherence, [E_I, E_J] = 0, joint idempotency, all exact) is the finite half of the Longo Q-system identification, and the exact dim law d_c(l) = 4^(l-1) + 2^(l-1) cos(pi c/2) makes the 4-element C[Z4] group-layer quasi-basis obstruction exactly 2^-l at every finite level (d_0 - d_2 = 2^l > 0), restored only asymptotically -- the crossed-product structure is exactly asymptotic; d_2(1) = 0 IS the l = 1 index-3 anomaly.  No status move.]",
      "status": "Identity [I] (exact finite Frobenius/Q-system axioms + isotropy) + Physical [P] (identification)",
      "canonicalStatus": "Identity: C[Z4] is a special Frobenius algebra (Longo Q-system) of index 4 = |mu4|; locality = isotropy; halfway C[Z2] = SO(16)_1; R2 = identification of constructed objects",
      "dependencies": "GATE.METRIC.06; GATE.METRIC.07; CAR.QFREE.01; SYNTH.INVENTORY.02 (R2)",
      "scripts": [
        "v125_glue_qsystem.py"
      ],
      "script": "v125_glue_qsystem.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 210
    },
    {
      "id": "HOR.CLOCK.04",
      "claim": "THE CLOCK AND THE WALL SHARE A SPECTRUM (the v124 resummed-clock weights ARE the Mehta-Seshadri parabolic weights; R1's first checkpoint passes classically): (1) THE WEIGHTS ARE THE PARABOLIC WEIGHTS [I]: the arguments n/N_fam = {0, 1/3, 2/3} of the resummed logarithm are EXACTLY spec A0* (the exact U_wall residue, GATE.UWALL.07); the spectral bridge lambda_n = (1 - alpha_n)^{p2} over alpha in spec A0* reproduces the frozen transfer spectrum {1, (2/3)^6, (1/3)^6} - the horizon clock and the flavor wall share their spectrum through the complement map alpha -> 1 - alpha: the transfer eigenvalues are the hexagon-power of the COMPLEMENTARY parabolic weight. (2) CHECKPOINT 1 PASSES CLASSICALLY [I]: the resummed clock's linear coefficient p2/N_fam = 2 = |Z2| equals the established classical entropy-deviation rate 2H = |Z2| x Hubble (HOR.CLOCK.01) - the slope-2 linear response demanded by HOR.CLOCK.03 is already the classical entropy rate; the genuinely QUANTUM content of R1 isolates to the geometric tail sum_{k>=2} alpha^k/k. (3) THE DETERMINANT IS THE ENTROPY CURVATURE [I]: det(1 - A0*) = (1)(2/3)(1/3) = 2/9 = |Z2|/N_fam^2 - exactly the established gravity entropy curvature at the anchor (HOR.ORIENT.01), a fourth independent appearance; tr(1 - A0*) = 2 = |Z2|; det A0* = 0 (the parabolic degree-0 marker). (4) R1 RE-TARGETED [P] (recorded, not claimed): derive 'rate = -p2 ln(1 - alpha)' with alpha the parabolic weight - first order is classical (HOR.CLOCK.01's entropy rate), the tail is the quantum part; ONE SPECTRUM, TWO GATES, ONE OBJECT A0*",
      "status": "Identity [I] (exact spectral bridge + coefficient identities) + Physical [P] (re-target)",
      "canonicalStatus": "Identity: clock weights = parabolic weights (spec A0*); lambda = (1-alpha)^6; linear term = classical entropy rate |Z2|H; det(1-A0*) = entropy curvature 2/9; quantum content = geometric tail",
      "dependencies": "HOR.CLOCK.03; HOR.CLOCK.01; HOR.ORIENT.01; GATE.UWALL.07 (A0*); SYNTH.INVENTORY.02 (R1)",
      "scripts": [
        "v126_clock_wall_bridge.py"
      ],
      "script": "v126_clock_wall_bridge.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 211
    },
    {
      "id": "HOR.CLOCK.05",
      "claim": "THE CLOCK IS A LOG-DETERMINANT (the geometric tail of HOR.CLOCK.03/04 is the standard ring/RPA resummation; the 'O(1) resummation' demanded by HOR.CLOCK.02 is now TYPED): (1) LOG-DET IDENTITY [I]: for a rank-1 insertion P (tr P^k = 1), -ln det(1 - alpha P) = -ln(1 - alpha) = sum_k alpha^k/k - so the resummed clock is EXACTLY rate = -p2 Tr ln(1 - alpha P): p2 = 6 identical one-loop ring towers, ONE PER HEXAGON SITE (= positive root of A3), with coupling alpha = the Mehta-Seshadri parabolic weight. (2) RING LADDER [I]: the k = 1 ring is the CLASSICAL term (6 alpha = |Z2| n = the HOR.CLOCK.01/04 entropy rate); the k = 2 ring is the first quantum correction (1/3 at weight 1); the partial sums converge to the exact rates {Delta, 6 ln 3} monotonically from below. (3) COUPLING CROSS-LINK [I, audit]: the established seam coupling kappa = (c/24pi)(Lambda/Mbar^2) at Lambda = Mbar^2 with c = 8 (HOR.CLOCK.02) is 8/(24pi) = 1/(3pi) = alpha_1/pi - the seam coupling is the first parabolic weight over pi. (4) THE WALL = RING DIVERGENCE [I]: radius of convergence |alpha| < 1; at alpha -> 1 (n -> N_fam) the sum diverges - the HOR.CLOCK.03 wall at the family count is the standard RPA instability. (5) R1 FINAL FORM [P] (recorded, not claimed): show that the near-Nariai one-loop effective action of the seam mode, with insertion strength = parabolic weight and multiplicity p2, is the ring sum - 'the seam clock is the RPA determinant of one mode on six sites'; the resummation type is identified, the derivation itself remains open",
      "status": "Identity [I] (log-det + ring ladder + coupling arithmetic) + Physical [P] (derivation)",
      "canonicalStatus": "Identity: clock = -p2 Tr ln(1 - alpha P) (ring/RPA determinant; one tower per hexagon site); k=1 ring = classical entropy rate; wall = RPA instability; kappa = alpha_1/pi",
      "dependencies": "HOR.CLOCK.04; HOR.CLOCK.03; HOR.CLOCK.02; FLAV.H2.02 (hexagon); SYNTH.INVENTORY.02 (R1)",
      "scripts": [
        "v127_ring_resummation.py"
      ],
      "script": "v127_ring_resummation.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 212
    },
    {
      "id": "E8.GRADE.01",
      "claim": "THE GRADED HULL + THE ZERO-MODE MULTIPLICITY (both named computations gain their structural anchors): R2 HALF - (1) EXPLICIT COSET CONSTRUCTION [I]: the 240 E8 roots, built as explicit norm-2 vectors in D5 (+) A3 + glue coordinates, decompose over the glue group Z4 as 240 = 52 + 64 + 60 + 64 (C0 = carrier roots 40+12; C1 = (16, 4bar) from norm 5/4 + 3/4; C2 = (10, 6) from 1 + 1; C3 = (16bar, 4)) - exactly the GATE.METRIC.06 branching, now as vectors. (2) THE GRADING IS EXACT [I]: for ALL 6720 root pairs whose sum is again a root, coset(r1+r2) = coset(r1) + coset(r2) mod 4 - E8 IS A Z4-GRADED LIE ALGEBRA over the carrier subalgebra, and the grading group IS the glue = the GATE.METRIC.08 Q-system: the checksum hull is the Q-system's graded module realised in 248 dimensions (not merely 'compatible' with it). R1 HALF - (3) THE MULTIPLICITY IS THE ZERO-MODE COUNT [I]: the Ginsparg-Perry l = 1 sector (the zero modes, lambda in {0,-1}, HOR.CLOCK.01/02) has 2l+1 = 3 modes; sheet doubling gives 2 x 3 = 6 = p2 = |R+(A3)| - the ring multiplicity of the HOR.CLOCK.05 log-det clock equals the sheet-doubled zero-mode count; [P] reading (recorded, not claimed): zero modes are exactly the modes that force nonperturbative resummation in de Sitter one-loop physics - the natural home of the six RPA towers. (4) RESIDUE [P]: R1 = compute the RPA determinant on the six sheet-doubled l=1 zero modes; R2 = identify the seam-Calderon inclusion with the Z4-graded structure now exhibited inside E8 itself",
      "status": "Identity [I] (explicit construction + exact grading + arithmetic) + Physical [P] (anchors)",
      "canonicalStatus": "Identity: E8 = Z4-graded Lie algebra over the carrier (grading = glue Q-system; 6720 pair checks); ring multiplicity 6 = sheet-doubled GP zero modes",
      "dependencies": "GATE.METRIC.08; GATE.METRIC.06; HOR.CLOCK.05; HOR.CLOCK.01; E8.GLU.01",
      "scripts": [
        "v128_graded_hull.py"
      ],
      "script": "v128_graded_hull.py",
      "location": "tfpt_research_contracts; tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 213
    },
    {
      "id": "HOR.CLOCK.06",
      "claim": "THE CLOCK IS AN ENTROPY POWER LAW (the transfer eigenvalues are the Nariai entropy fractions to the hexagon power; the ring coupling is the fractional entropy deficit): (1) THE THREE ENTROPY LEVELS [I]: the established Nariai bookkeeping (HOR.NARIAI.01) has S/S_dS in {1, 2/3, 1/3} (pure dS; two-horizon total; single horizon - deficit S_dS/3 per step) - EXACTLY the complementary cusp weights 1 - n/N_fam. (2) THE CLOCK IS AN ENTROPY POWER LAW [I]: lambda_n = (S_n/S_dS)^{p2} = the frozen transfer spectrum {1, (2/3)^6, (1/3)^6}; rate(n) = p2 ln(S_dS/S_n) = {0, Delta, 6 ln 3} - hexagon-weighted log-entropy deficits; audit: exponent p2 = 6 = 2h reads as the two-point form of a weight h = 3 = N_fam operator (recorded, not claimed). (3) THE COUPLING IS THE ENTROPY DEFICIT [I] (triple identity): alpha = n/N_fam = the Mehta-Seshadri parabolic weight (GATE.UWALL.07/HOR.CLOCK.04) = 1 - S_n/S_dS = DeltaS_n/S_dS - the RPA insertion strength is the fractional entropy deficit of the SdS configuration: the coupling is not a free parameter in ANY of its three readings. (4) ORIENTATION CONSISTENCY [I]: weight 0 <-> pure dS (max entropy, rate 0: the stationary attractor); weight 2 <-> single horizon (lowest level, fastest decay) - monotone, the established repeller -> attractor orientation (FR.KOIDE.05/HOR.NARIAI.01/HOR.ORIENT.01). (5) R1 THERMODYNAMIC FORM [P] (recorded, not claimed): derive Gamma_n ~ (S_n/S_dS)^{p2} from the near-Nariai one-loop - a Gibbons-Hawking-type entropy power law with exponent = the hexagon size; the exponent (weight h = N_fam) is the remaining quantum content",
      "status": "Identity [I] (entropy-level + power-law + triple-coupling identities) + Physical [P] (derivation)",
      "canonicalStatus": "Identity: lambda = (S/S_dS)^p2; rate = p2 ln(S_dS/S); alpha = entropy-deficit fraction = parabolic weight; orientation consistent; R1 = Gibbons-Hawking power law",
      "dependencies": "HOR.CLOCK.05; HOR.CLOCK.04; HOR.NARIAI.01; HOR.ORIENT.01; GATE.UWALL.07",
      "scripts": [
        "v129_entropy_power_law.py"
      ],
      "script": "v129_entropy_power_law.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 214
    },
    {
      "id": "HOR.CLOCK.07",
      "claim": "THE BORN SQUARE (the amplitude weight is h = N_fam = half the zero-mode count; the clock exponent p2 = 2h is its Born square - the HOR.CLOCK.05 multiplicity and the HOR.CLOCK.06 weight are ONE object): (1) EXPONENT BOOKKEEPING [I]: in the standard collective-coordinate treatment each zero mode trades a Gaussian integral for a moduli factor ~ S^{1/2}; with n_zero = 6 the transition AMPLITUDE scales as A ~ (S/S_dS)^{n_zero/2} = (S/S_dS)^3 and the PROBABILITY is its Born square Gamma = |A|^2 = (S/S_dS)^6 = the HOR.CLOCK.06 entropy power law => h = n_zero/2 = 3 = N_fam and p2 = 2h: the weight audit is DERIVED from mode counting; the factor 2 between weight and exponent is the Born rule, nothing else. (2) THE THREE ARE THE SPHERE'S CONFORMAL DIRECTIONS [I]: the l = 1 scalar eigenspace on S^2 has dimension 2l+1 = 3 = the number of proper conformal Killing vectors of the sphere (gradients of the l = 1 harmonics; the classic zero modes of near-Nariai physics, Ginsparg-Perry/Bousso-Hawking class, external standard); the doubling x2 is the HORIZON PAIR = the SdS mass-line deck (HOR.NARIAI.01): n_zero = |Z2| x 3 = 6 = p2 - every factor an established object. (3) CONSISTENCY [I]: multiplicity (E8.GRADE.01) = exponent (HOR.CLOCK.06) = Born square of the amplitude weight; h = N_fam ties the clock's conformal weight to the family count - the same 3 counts generations and the sphere's conformal directions. (4) RESIDUE [P] (recorded, not claimed): justify the per-zero-mode S^{1/2} collective-coordinate normalisation on the Nariai instanton for exactly these six modes - a textbook-class computation; no exotic object remains in R1 (log = ring sum, power law = entropy fractions, exponent = Born-squared mode count)",
      "status": "Identity [I] (exponent bookkeeping + mode-count identities) + Physical [P] (measure normalisation)",
      "canonicalStatus": "Identity: h = n_zero/2 = N_fam; p2 = 2h (Born rule); 3 = proper CKVs of S^2; x2 = horizon pair; R1 residue = standard zero-mode measure",
      "dependencies": "HOR.CLOCK.06; HOR.CLOCK.05; E8.GRADE.01; HOR.NARIAI.01",
      "scripts": [
        "v130_born_square.py"
      ],
      "script": "v130_born_square.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 215
    },
    {
      "id": "HOR.CLOCK.08",
      "claim": "THE MEASURE IS THE AREA (the per-mode S^{1/2} of the Born square is the zero-mode NORM, computed exactly; the R1 residue narrows to one determinant-ratio cancellation): (1) ZERO-MODE NORM = AREA [I]: on the sphere S^2(r) the L^2 norm^2 of each of the three normalised l = 1 harmonics is EXACTLY r^2 = A/(4pi) (all three modes integrated exactly) - the norm of every zero mode IS the horizon area, hence the entropy, up to the fixed 4pi. (2) JACOBIAN BOOKKEEPING [I]: the collective-coordinate Jacobian per zero mode is the mode norm ~ S^{1/2} (standard change of variables, no model input); with the six sheet-doubled modes (HOR.CLOCK.07) the amplitude ratio is (S_n/S_dS)^3 and the Born square gives Gamma_n/Gamma_0 = (S_n/S_dS)^6 = the HOR.CLOCK.06 entropy power law - every exponent factor now has a derivation-level origin: norm = area -> Jacobian = S^{1/2} -> six modes -> amplitude S^3 -> Born -> S^6. (3) UNIT CONSISTENCY [I]: the 4pi in norm^2 = A/(4pi) is the same Gauss-Bonnet 4pi that builds c3 = 1/(2 x 4pi) (ARCH.QUAD.01/P1 hardening); it CANCELS in all ratios - exactly why the clock sees only entropy FRACTIONS S_n/S_dS. (4) THE LAST STEP [P] (recorded, not claimed): show the non-zero-mode fluctuation determinants cancel in the ratio, det'_n/det'_dS = 1 + O(deficit) - plausible (same local near-Nariai geometry) but the one remaining computation; R1 = norm (exact, done) + Jacobian (standard) + Born (rule) + det-ratio (open)",
      "status": "Identity [I] (exact sphere integrals + bookkeeping) + Physical [P] (det-ratio)",
      "canonicalStatus": "Identity: ||Y_1m||^2 = A/(4pi) exactly (all three modes); Jacobian = S^(1/2) per mode; 4pi cancels in ratios; R1 residue = det-ratio cancellation only",
      "dependencies": "HOR.CLOCK.07; HOR.CLOCK.06; HOR.NARIAI.01; ARCH.QUAD.01 (c3 primitive)",
      "scripts": [
        "v131_measure_is_area.py"
      ],
      "script": "v131_measure_is_area.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 216
    },
    {
      "id": "HOR.CLOCK.09",
      "claim": "THE DET-RATIO ANOMALY IS THE KOIDE CONSTANT (the HOR.CLOCK.08 'last step' acquires its first exact invariant): (1) SPECTRAL SETUP (re-pinned): the Ginsparg-Perry sphere operator on the unit S^2 is L = -Delta - 2 (the -2 = -|Z2| = the HOR.CLOCK.01 modulus mass); eigenvalues mu_l = l(l+1) - 2 = (l+2)(l-1), degeneracy 2l+1; kernel = the 3 zero modes (l = 1); one negative mode (l = 0). (2) THE HEAT COEFFICIENT IS MADE OF ATOMS [I]: a_1 = (1/4pi) Int(R/6 - E) dA = 1/3 + 2 = 1/N_fam + |Z2| = 7/3 - the curvature third and the sheet mass, both compiler atoms. (3) THE ANOMALY IS THE KOIDE CONSTANT [I]: zeta(0)|det' = a_1 - dim ker = 7/3 - 3 = -2/3 = -|Z2|/N_fam, independently confirmed by numerical continuation of the heat trace (constant term of Tr' e^{-tL} - 1/t extrapolates to -2/3 at ~1e-7) - the EIGHTH appearance of 2/3 in the inventory (SYNTH.INVENTORY.01 counted seven) and the first as a SPECTRAL anomaly. (4) CONSEQUENCE [I]: det'(cL) = c^{zeta(0)} det'(L): the determinant ratio between configurations is NOT automatically 1 - the S^2-sector share of the scaling budget is exactly -2/3 per Hubble-unit conversion; the dimensionless tower itself is configuration-independent (pure-number GP spectrum, HOR.CLOCK.01): all scale dependence = this one anomaly number + the zero-mode Jacobians (HOR.CLOCK.08). (5) THE BUDGET STATEMENT [P] (recorded, not claimed): show the FULL zeta(0) budget of the 4d Nariai fluctuation problem (dS2 sector + gauge/ghost sectors + this S^2 sector) vanishes or is absorbed into the established power law; the S^2 share is pinned exactly at -|Z2|/N_fam - nothing claimed about the total",
      "status": "Identity [I] (heat-kernel arithmetic + numerical continuation) + Physical [P] (full budget)",
      "canonicalStatus": "Identity: zeta(0) of the det'-operator = -2/3 = -|Z2|/N_fam (a1 = 1/N_fam + |Z2| = 7/3); det-ratio question = one zeta-budget statement; S^2 share pinned",
      "dependencies": "HOR.CLOCK.08; HOR.CLOCK.01; SYNTH.INVENTORY.01 (2/3 inventory); HOR.NARIAI.01",
      "scripts": [
        "v132_detratio_anomaly.py"
      ],
      "script": "v132_detratio_anomaly.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 217
    },
    {
      "id": "HOR.CLOCK.10",
      "claim": "THE ZETA BUDGET, BOTH ROUTES COMPUTED EXACTLY (the reduced seam route carries pure anchor-ratio anomalies; the naive 4d route does not - the budget computation structurally selects the reduced seam reading): Euclidean Nariai = S^2 x S^2, so the temporal dS2 sector continues to the SAME unit sphere and both candidate routes are exactly computable. (1) THE REDUCED ROUTE [I]: each 2d sector carries the HOR.CLOCK.09 operator with zeta(0)|det' = -2/3; two sectors give budget = -4/3 = -e1/p0 = MINUS THE SEED GAIN (the second ARCH.TRIAD.01 ratio; the first, -2/3, is the per-sector share); the six zero modes split 3 + 3 across the two spheres - the sheet doubling of E8.GRADE.01/HOR.CLOCK.07 IS the two-sphere structure of Euclidean Nariai (= the horizon pair, HOR.NARIAI.01). (2) THE NAIVE 4D ROUTE [I]: with the standard unit-S2 heat expansion 1/t + 1/3 + t/15 (verified numerically) and the symmetric split L4 = (-Delta1 - 1) + (-Delta2 - 1): per-factor trace 1/t + 4/3 + (9/10)t => a2(4d) = (4/3)^2 + 2(9/10) = 161/45; zero modes (1,0)+(0,1) = 6; zeta(0)|det'4d = 161/45 - 6 = -109/45 (numerical continuation ~1e-8) - NO anchor-atom reading. (3) ROUTE DISCRIMINATION [I arithmetic + P reading]: reduced-route anomalies = pure anchor ratios (-2/3, -4/3: two of the three triad values); 4d-route anomaly = -109/45, not an atom => the zeta budget structurally favours the REDUCED seam reading, consistent with the theory's 2d-seam architecture throughout; typed as route-selection evidence, not proof. (4) REMAINING [P] (recorded): the graviton/ghost shares on S^2 x S^2 (Volkov-Wipf-class computation, external standard) complete the budget; both scalar-sector shares pinned exactly - the remaining work is a finite list of known heat coefficients",
      "status": "Identity [I] (exact heat-kernel arithmetic both routes + numerics) + Physical [P] (route reading + remaining shares)",
      "canonicalStatus": "Identity: reduced budget = -4/3 = -seed gain (per sector -2/3); 4d route = -109/45 no atom; zero modes 3+3 = horizon pair; route selection -> reduced seam reading",
      "dependencies": "HOR.CLOCK.09; ARCH.TRIAD.01 (triad); E8.GRADE.01; HOR.NARIAI.01",
      "scripts": [
        "v133_zeta_budget.py"
      ],
      "script": "v133_zeta_budget.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 218
    },
    {
      "id": "FLAV.DUAL.01",
      "claim": "THE DUAL ANCHOR LEMMA (the inverse flavor response of the anchor IS the Nariai root; the third purely algebraic leg of the flavor-horizon bridge; external review 2026-06-12, validated): (1) THE DUAL ANCHOR [I]: d := a^T R^-1 = a^T L^-1 = (-1/2, -1/2, 1) with d.1 = 0, d.a = 1 and d = (3/2)(a - (4/3)1)^T - the normalised TRACELESS anchor - and (1,1,-2) = -2d: the inverse flavor response stores EXACTLY the Nariai double-root structure (HOR.NARIAI.01) up to sign and scale; the horizon is not glued on afterwards - it sits inside the flavor compiler as a dual invariant. (2) WHY INVARIANT [I] (Sherman-Morrison, exact): L = R + 6*1 e1^T gives v^T L^-1 = v^T R^-1 <=> v^T R^-1 1 = 0; R^-1 1 = (1,1,-1)/4 (denominator 5/2); a.(R^-1 1) = 0 - the anchor lies on the 2-plane of invariant covectors: the winding deformation lives in the democratic direction 1, which d annihilates. (3) CONTROL [I]: 1, e1 and n = (5,-9,6) do NOT lie on the invariance plane (1/4, 1/4, -5/2) - the membership is special, not generic. (4) THE DUAL NORMAL PAIR [I]: d (sheet normal: kills 1, dual to a, proportional to the Nariai root) and n (torsion normal: n^T R = (8,0,0), n^T L = (20,0,0), n.a = 8, n.1 = 2 = |Z2| cf. FLAV.DIAMOND.02) form the dual basis of the flavor boundary. (5) CONSEQUENCE [P] (recorded): third algebraic bridge leg (besides the shared spectrum HOR.CLOCK.04 and the entropy power law HOR.CLOCK.06); supports the reduced seam reading (HOR.CLOCK.10); the R4' selector programme (FLAV.H2.05) acquires a geometric target: derive the PAIR (d, n), not n alone",
      "status": "Identity [I] (exact dual-vector + Sherman-Morrison + controls) + Physical [P] (bridge reading)",
      "canonicalStatus": "Identity: a^T R^-1 = a^T L^-1 = Nariai root/(-2); invariance <=> orthogonality to R^-1 1; dual normal pair (d n); R4' target = the pair",
      "dependencies": "HOR.NARIAI.01; FLAV.DIAMOND.02; FLAV.H2.05 (selectors); HOR.CLOCK.04; HOR.CLOCK.06",
      "scripts": [
        "v134_dual_anchor.py"
      ],
      "script": "v134_dual_anchor.py",
      "location": "tfpt_2; tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 219
    },
    {
      "id": "FLAV.SURF.01",
      "claim": "THE DETERMINANT SURFACE (the sheet diamond is a 2d surface with iso-volume walls at the discrete atoms; external review 2026-06-12, validated): (1) SURFACE + WALLS [I]: on the established diamond M(s,t) = R + Q diag(s,t,t) (FLAV.DIAMOND.01/02), det M(s,t) = 3s(t+1)(t+2) + t^2 + 5t + 8; two iso-volume WALLS independent of s: t = -2 => det = 2 = |Z2| (the Z2 wall), t = -1 => det = 4 = |mu_4| (the mu_4 wall; K = M(1,-1) lies on it - det K = 4 is a LAYER, not an isolated hit); the WINDING LINE t = 0 has det = 6s + 8 => (det R, det C, det L) = (8, 14, 20); the fourth canonical point closes the pattern: det F = det M(1,1) = 32 = 2^{g_car}. (2) ANCHOR BLOCK [I]: det B_{M(s,t)} = 6s(t+2) + 3t^2 + 15t + 16; on the winding line det B(s,0) = 2 det M(s,0) EXACTLY; own Z2 wall det B(s,-2) = -2 (s-independent); t = -1 is NOT a B-wall (6s+4, recorded honestly). (3) ROW-BUDGET AXES [I]: rowsum(C) = (7, 11, 13), rowsum(U) = (3,3,3) (democratic winding), rowsum(V) = (1,2,3) = Spec(Q_+) verbatim; audit: the centre budget (7, 11, 13) reads (scalaron, ||Pl(K)||_1, Delta_Q) - the three quark constants (audit, not load-bearing). (4) TWO ANCHOR MICRO-IDENTITIES [I]: e1^2 + e2^2 = 16 + 25 = 41 = 10 b1 (the anchor Pythagoras for the abelian coefficient, companion to Delta_Y = 25 = 9 + 16); p1^2 + p2^2 = 16 + 36 = 52 = the carrier coset root count (40 + 12, E8.GRADE.01). (5) READING [P] (recorded): the flavor algebra lives on a 2d surface whose walls are the discrete atoms and whose axes are democratic winding and the Q_+ progression - compiler semantics replacing the matrix list; no gate status changes",
      "status": "Identity [I] (exact surface/wall/budget identities) + Physical [P] (reading)",
      "canonicalStatus": "Identity: det M = 3s(t+1)(t+2)+t^2+5t+8; walls 2 and 4 (s-independent); winding line (8 14 20); det F = 32; det B = 2 det M on t = 0; V-rowsum = Spec(Q+); 41 and 52 micro-identities",
      "dependencies": "FLAV.DIAMOND.01-02; FLAV.H2.04; ARCH.TRIAD.01; E8.GRADE.01 (52); FLAV.KQ.01 (11 13)",
      "scripts": [
        "v135_det_surface.py"
      ],
      "script": "v135_det_surface.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 220
    },
    {
      "id": "GATE.UWALL.10",
      "claim": "THE DUAL-NORMAL SELECTOR ((d,n) pins R columnwise; the R4' residue is bounded; external review 2026-06-12, validated and sharpened): (1) COLUMN NORMAL FORM [I]: d.c_j = a_j and n.c_j = 8 delta_1j for the columns of R; the common kernel of (d,n) is the primitive vector (6,8,7) (audit: p_2, rank E8, scalaron-7), so within the address box {0..5}^3 each line contains at most one lattice point; brute-force census (216 per column): c_1 = (1,1,2) = a, c_2 = (3,5,5), c_3 = (0,2,3) are each the UNIQUE solution - R4' shrinks from a 6^9 matrix census (FLAV.H2.04) to three 1-of-216 column problems: (d,n) => R. (2) HONESTY [I]: K is NOT pinned directly (d^T K = (1,1/2,1), n^T K = (14,1,-6)); det K = 4 follows from R via the diamond (mu_4 wall, FLAV.SURF.01). (3) SPECTRAL NORMAL [I]: the zero mode of the exact anchor residue A0* (GATE.UWALL.07) is v = (-sqrt2,3,-sqrt5)/3 with signed squares = -(2,-9,5) = -(|Z2|, -N_fam^2, g_car); NEW ATOMS: ||v||^2 = 16/9 (numerator 16 = dim S+) and adj(A0*) = (2/9) P_v with 2/9 = |Z2|/N_fam^2 = the SdS entropy curvature (HOR.NARIAI.01). (4) SIGMA RELATIONS [I]: sigma.1 = -|Z2|, sigma.a = N_fam, n.sigma = 121 = ||Pl(K)||_1^2 (the v119 audit constant). (5) ORBIT CONTROL [I]: exact enumeration |W(A3)| = 24 (GATE.UWALL.09); the only real orbit images of sigma are its signed copies; NO image is proportional to n - the naive cofactor/orbit lift FAILS as the review predicted; the step sigma -> n is a genuine mechanism beyond the monodromy group. (6) R4' RESTATED [P] (recorded): derive the PAIR (d,n); then R is forced columnwise; d is structural (FLAV.DUAL.01), the residue is the derivation of n - bounded by the sigma relations and the negative orbit control",
      "status": "Identity [I] (column census + spectral normal + orbit control) + Physical [P] (restated programme)",
      "canonicalStatus": "Identity: (d n) => R columnwise (each column 1-of-216); A0* zero mode = -(2 -9 5) signed squares; n.sigma = 121; W(A3) orbit excludes the naive lift",
      "dependencies": "FLAV.DUAL.01; GATE.UWALL.07; GATE.UWALL.09; FLAV.H2.04 (superseded census); FLAV.SURF.01; HOR.NARIAI.01",
      "scripts": [
        "v136_dual_normal_selector.py"
      ],
      "script": "v136_dual_normal_selector.py",
      "location": "tfpt_research_contracts; tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 221
    },
    {
      "id": "FLAV.QGEO.04",
      "claim": "THE Q+ GRADING IS COHOMOLOGY (the R5/GATE.QGEO dictionary at the spectrum level; external review 2026-06-12, validated and sharpened): (1) BASIS + CHARACTERS [I]: b_1(P^1 minus mu_4) = 3 = N_fam with holomorphic basis omega_k = z^(k-1)dz/(z^4-1), k = 1,2,3 (regular at infinity; the trivial character is absent - k = 4 would be the dlog form); the mu_4 character of omega_k is i^k, so H^1 = chi_1 + chi_2 + chi_3 with each nontrivial character exactly once; the residue vector of omega_k over the punctures is the character vector zeta^k/4 exactly. (2) THE GRADING IS Spec(Q+) [I]: the character degrees (1,2,3) are exactly Spec(Q+) (Q+ = 3*diag(cusp weights)+1, FLAV.QGEO.01); NEW: the cusp weights {0,1/3,2/3} are spec(A0*) (GATE.UWALL.07) - Mehta-Seshadri weights, cusp classes, cohomology characters and the flavor operator Q+ are ONE spectrum in four readouts. (3) A3 EXPONENTS [I]: (1,2,3) are the exponents of A3 (W(A3) = S4, GATE.UWALL.09) with Coxeter number h(A3) = 4 = |mu_4| - the Q+ generation grading IS the A3 exponent grading realised on the seam curve's cohomology: the R5 dictionary statement at spectrum level. (4) RESIDUE [P] (recorded): GATE.QGEO's remainder is the NATURALITY of the identification - the canonical map from the H^1 character basis to the V-axis generation basis (rowsum(V) = (1,2,3) verbatim, FLAV.SURF.01); spectrum equality is [I], the canonical map is the residue",
      "status": "Identity [I] (cohomology basis + characters + residues + spectrum match) + Physical [P] (naturality residue)",
      "canonicalStatus": "Identity: H^1 = chi_1+chi_2+chi_3 with degrees (1 2 3) = Spec(Q+) = A3 exponents; cusp weights = spec(A0*); residue = naturality of the canonical map",
      "dependencies": "GATE.QGEO.01; FLAV.QGEO.01-03; GATE.UWALL.07; GATE.UWALL.09; FLAV.SURF.01",
      "scripts": [
        "v137_qplus_cohomology.py"
      ],
      "script": "v137_qplus_cohomology.py",
      "location": "tfpt_2; tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 222
    },
    {
      "id": "HOR.CLOCK.11",
      "claim": "THE VOLKOV-WIPF FIREWALL (R1 typed closure: the external full-4d graviton/ghost value is pinned and the edge part is exactly two copies of the reduced seam budget; external values verified against arXiv:2506.02142 Table 4.1 which states agreement with Volkov-Wipf Nucl. Phys. B582 (2000) 313 = hep-th/0003081): (1) EXTERNAL DATUM [I typed external]: alpha_bulk = 22/45, alpha_edge = -8/3, alpha_PI = -98/45 for d = 3 (S^2 x S^2, 4d); arithmetic 22/45 - 8/3 = -98/45 exact; the review's arXiv citation 0204021 corrected to 0003081. (2) THE EDGE MATCH [I]: alpha_edge = -8/3 = 2 x (-4/3) - the external edge factor consists of TWO IDENTICAL COPIES (one per horizon) and the per-copy magnitude 4/3 = e_1/p_0 is EXACTLY the reduced seam budget of HOR.CLOCK.10 (= the seed gain; -2/3 per sector); the TFPT reduced route lands on the EDGE sector of the external bulk/edge split; the bulk 22/45 is the ideal graviton gas which the seam clock does not contain; two copies = the horizon pair = the sheet doubling (HOR.NARIAI.01; HOR.CLOCK.10's 3+3 zero-mode split). (3) FIREWALL ARITHMETIC [I]: -98/45 + 60/45 = -38/45 with 38 = 2*19; neither -98/45 nor -38/45 nor 22/45 is an anchor atom (census) - consistent with HOR.CLOCK.10 where the naive scalar 4d route (-109/45) was already non-atomic (different operator content: scalar split vs full graviton + ghosts + isometries, recorded). (4) TYPING DECISION [P] (recorded): R1 = the reduced zero-mode/RPA seam clock = an EDGE-sector object; the full 4d Volkov-Wipf determinant is the GRAVITY FIREWALL (its -98/45 absorbed in the mu_0/(Lambda G)^3 normalisation), not a TFPT atom; per-copy sign/orientation (the edge factor enters inverted) and the Lorentzian edge-mode reading stay [P]",
      "status": "Identity [I] (exact arithmetic vs external values) + Physical [P] (typing decision) + External (VW hep-th/0003081; arXiv:2506.02142)",
      "canonicalStatus": "Identity: -98/45 = 22/45 - 8/3 external; edge = 2 x reduced budget (-4/3); diff -38/45 no atom; typing: seam clock = edge object full 4d determinant = firewall",
      "dependencies": "HOR.CLOCK.10; HOR.NARIAI.01; HOR.CLOCK.03-09 (the clock chain)",
      "scripts": [
        "v138_vw_firewall.py"
      ],
      "script": "v138_vw_firewall.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "Volkov-Wipf Nucl.Phys. B582 (2000) 313 hep-th/0003081; arXiv:2506.02142 Table 4.1",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 223
    },
    {
      "id": "GATE.UWALL.11",
      "claim": "THE SELECTOR TRIANGLE (anchor + A0* zero mode + three atoms => R; the R4' residue shrinks to three pairings): (1) d IS ANCHOR DATA [I]: d = (3/2)a - 2*1 exactly - the dual anchor (FLAV.DUAL.01) has a closed form that never mentions R; the first selector of the pair (d,n) is DERIVED, not residual. (2) THE FRAME [I]: (1, a, sigma) is a basis of Z^3 with det(1|a|sigma) = 11 = ||Pl(K)||_1 - the quark constant as the frame volume (sigma = the A0* zero-mode signature, GATE.UWALL.10). (3) n FROM THREE ATOMS [I]: n = (5,-9,6) is the UNIQUE solution of n.1 = 2 = |Z2|, n.a = 8 = rank E8, n.sigma = 121 = 11^2 - the cofactor definition (FLAV.DIAMOND.02) is replaced by frame data plus three atoms. (4) THE SQUARE SELECTS [I]: on the line n.1 = 2, n.a = 8 (direction 1 x a) the sigma-pairing is 11(11+t) - divisible by 11 for every t and a positive perfect square exactly at t = 0 (window [-20,20]). (5) THE REVIEW LIFT EXACT (audit) [I]: n = reverse(sigma) + |mu_4| e_3 - the review's predicted mechanism (generation flip + mu_4 lift at the third slot) is an exact identity; n - sigma = N_fam e_1 + e_3; audit decompositions, not promoted. (6) RESIDUE [P] (recorded): R4' = derive the three pairings (|Z2|, rank E8, ||Pl(K)||_1^2) of n against the frame; everything else in the chain anchor -> (d,n) -> R (GATE.UWALL.10) is exact",
      "status": "Identity [I] (closed form for d + frame volume + uniqueness + square selection) + Physical [P] (three-pairing residue)",
      "canonicalStatus": "Identity: d = (3/2)a - 2*1 (no R); det(1 a sigma) = 11; n unique from pairings (2 8 121); square selects t = 0; review lift exact",
      "dependencies": "FLAV.DUAL.01; GATE.UWALL.10; FLAV.DIAMOND.02; FLAV.KQ.01 (the 11)",
      "scripts": [
        "v139_selector_triangle.py"
      ],
      "script": "v139_selector_triangle.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 224
    },
    {
      "id": "FLAV.QGEO.05",
      "claim": "THE CANONICAL MAP EXISTS AND IS RIGID (the GATE.QGEO naturality residue collapses to ONE Z_3 choice): (1) THREE mu_4 ACTIONS ONE MATCH [I]: the deck U = diag(1,i,-i) has characters (0,1,3) (invariant line, misses 2 - cannot match H^1); the sheet-twisted deck -U has (2,3,1) and the cusp exponential i^{Q+} has (1,2,3) - both with character SET {1,2,3} = the H^1 set (FLAV.QGEO.04): only the SIGN-TWISTED actions match, the same sheet twist that builds the v118 hexagon (FLAV.H2.02). (2) RIGIDITY (Schur) [I]: both sides are multiplicity-free mu_4-modules with equal character sets, so the equivariant isomorphism exists and is unique up to one scalar per character (commutant = diagonal algebra); the explicit map is the character-matching permutation - the canonical map is WELL-DEFINED; existence and uniqueness are no longer open. (3) GRADING TRANSPORT [I]: under the i^{Q+}-equivariant map the H^1 Euler degree pulls back to Q+ itself (assignment (1,2,3)); under the -U-map to Q+ o rho with rho = the (123) family rotation (assignment (2,3,1)) - the two candidate actions differ by EXACTLY one Z_3 rotation. (4) RESIDUE [P] (recorded): GATE.QGEO collapses from 'prove naturality' to ONE discrete choice - which sign-twisted action is the geometric boundary deck on generation space (-U monodromy sheet twist vs i^{Q+} cusp exponential)",
      "status": "Identity [I] (character algebra + Schur rigidity + grading transport) + Physical [P] (one Z_3 choice)",
      "canonicalStatus": "Identity: only sign-twisted actions match H^1; equivariant iso Schur-rigid; i^Q+ assignment = Q+ and -U = Q+ o rho; residue = one Z_3 choice",
      "dependencies": "FLAV.QGEO.04; FLAV.H2.02 (sheet twist); GATE.UWALL.09 (U); GATE.QGEO.01",
      "scripts": [
        "v140_canonical_map.py"
      ],
      "script": "v140_canonical_map.py",
      "location": "tfpt_2; tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 225
    },
    {
      "id": "FLAV.QGEO.06",
      "claim": "DECK SELECTION THEOREM (R5's Z_3 choice DERIVED, executing the v140 residue): the established integer deck G=T_A*Sigma (v97 anchor-forced conjugation; v98 integer mu4 on generation space) pairs the Q_+=1 cusp line with the SELF-CONJUGATE character 2 (G e1=-e1 exact eigenline; E-plane carries {1,3}); of the three Z_3 rotations only the sheet-twisted assignment (2,3,1)=chars(-U) is compatible - the cusp exponential i^{Q_+} (1,2,3) and the third rotation are EXCLUDED (same eigenvalue multiset, wrong grading pairing: (-1)-eigenlines on Q_+ degrees 1 vs 2). Explicit Q_+=1-line-preserving conjugacy S G S^-1 = -U constructed. SHEET ROBUSTNESS: under k->-k the B1 pairing (1->2) and plane set {1,3} are invariant - the only remaining freedom is the plane orientation = the established sheet Z_2 (v92 chirality pair), NOT a new residual. CONSEQUENCE: the H^1 Euler degree pulls back to Q_+ o rho (rho=(123)=coker-Q triality, v72): cusp grading and Euler grading differ by exactly one family rotation. STATUS: GATE.QGEO's discrete Z_3 residue CLOSED [L]; what remains of R5 is the realisation premise [P] the gate always carried (integer D4 model = parabolic geometry), with no discrete freedom left",
      "status": "Lattice/Lie [L] (derived selection + explicit conjugacy/obstruction) + [P] (realisation premise unchanged)",
      "canonicalStatus": "Lattice [L]: Z_3 choice derived - geometric deck = sheet-twisted class; residual = realisation premise only",
      "dependencies": "FLAV.QGEO.05; FLAV.QGEO.04; v97; v98; v72",
      "scripts": [
        "v141_deck_selection.py"
      ],
      "script": "v141_deck_selection.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 226
    },
    {
      "id": "GATE.UWALL.12",
      "claim": "FRAME INTEGRALITY (R4' narrowed: three pairings -> two): the pairing triples (m.1, m.a, m.sigma) of INTEGER covectors against the frame (1,a,sigma) form an index-11 sublattice of Z^3, cut out by the single congruence x-3y+z = 0 (mod 11) - the index IS 11 = ||Pl(K)||_1 = det(1|a|sigma) (full 11^3 scan, exact). With the two line pairings at their established atoms (x,y)=(|Z2|,rank E8)=(2,8), integrality FORCES the sigma-pairing z = 0 (mod 11): the v139 line z(t)=11(11+t) is exactly the integer line. PRIMITIVITY forces t EVEN (odd t makes every entry of n(t) even); t=0 is the unique even t with z a perfect square for |t|<88 (next square at t=88; global pick still needs the v139 square selector - recorded honestly). CRAMER REDUCTIONS (restructuring, not establishment): n.a = det R = 8 is the Laplace expansion along R's anchor column (v121/v122); n.1 = 8(R^-1 1)_1 = 2 with R^-1 1 = (1,1,-1)/4 (v134) - both consume R. STATUS: R4' reduces from 'derive three pairings' to 'derive the two line pairings (|Z2|, rank E8)'; typing [A]/[P] unchanged, content strictly smaller",
      "status": "Identity [I] (index-11 congruence + parity + reductions) + [P]/[A] (establishment of the two line pairings)",
      "canonicalStatus": "Identity [I]: third pairing integrality-forced mod 11; R4' = two line pairings",
      "dependencies": "GATE.UWALL.11; GATE.UWALL.10; FLAV.H2.05; v134; v139",
      "scripts": [
        "v142_frame_integrality.py"
      ],
      "script": "v142_frame_integrality.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 227
    },
    {
      "id": "GATE.METRIC.09",
      "claim": "GRADED FROBENIUS (R2 identification at the finite Lie level): the Z4-graded hull (E8.GRADE.01/v128) carries (1) EXACT COSET DUALITY coset(-r) = -coset(r) mod 4 for all 240 roots (-C1=C3 bijectively, C0/C2 self-dual; sizes 52,64,60,64) => the Killing form restricts to a NONDEGENERATE pairing g_k x g_{-k} - a Frobenius grading; (2) the glue character average (1/4)sum_k i^{k deg} is EXACTLY the carrier projector (1 on C0, 0 on C1,C2,C3) - the conditional-expectation shadow of the v125 Q-system with quasi-basis index |Z4|=4; (3) graded bracket additivity re-verified self-contained (6720 pairs); (4) each glue sector is ONE W(D5)xW(A3) orbit with multiplicity-one dimensions 64=16*4=(16,4b), 60=10*6=(10,6) - SIMPLE CURRENTS, sector ring = Z4 = the C[Z4] fusion, halfway subgroup {0,2} = SO(16)_1 (v92). CONSEQUENCE: the C[Z4] Q-system (v125) is realised by the 248-dim hull as its graded Frobenius module - R2's identification HOLDS AT THE FINITE LIE LEVEL. RESIDUE [P]: exactly one conformal-net statement remains ('the seam-Calderon inclusion of the 16-Majorana net carries this Q-system'), with no remaining finite freedom",
      "status": "Identity/Lattice [I]/[L] (duality, projector, orbits, fusion) + [P] (the one conformal-net step)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: Q-system realised at the finite Lie level; residual = the net statement",
      "dependencies": "E8.GRADE.01; GATE.METRIC.08; GATE.METRIC.07; CAR.QFREE.01; v125; v128",
      "scripts": [
        "v143_graded_frobenius.py"
      ],
      "script": "v143_graded_frobenius.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 228
    },
    {
      "id": "HOR.CLOCK.12",
      "claim": "DET-RATIO CANCELLATION EXACT WITHIN THE SdS FAMILY (the v131 step closed in the within-family sense): e2-RIGIDITY - the traceless SdS cubic (t^3-3t+2c) forces r_b r_c = 1 - Delta^2/3 EXACTLY (Delta = horizon split; no higher corrections); with the v132 anomaly zeta(0)|det' = -2/3 per sphere, det'(L_r) = r^{4/3} det'(L_1), so the two-sphere non-zero-mode determinant ratio against Nariai is det'_n/det'_N = (1 - Delta^2/3)^{4/3} in CLOSED FORM: NO first-order term in the split (d/dDelta|_0 = 0 exactly) - v131's '1+O(deficit)' DERIVED from e2-rigidity + the established anomaly, not assumed. DEFICIT FORM (audit): eps = 1-c = (3/8)Delta^2 + O(Delta^4) => ratio = 1 - (32/27) eps + O(eps^2), coefficient 32/27 = |mu4|(|Z2|/N_fam)^3 = 4(2/3)^3 (the v103 canonical curvature x glue order; audit-typed). HONEST SCOPE [P]: the finite-weight comparison (S_n/S_dS in {1,2/3,1/3}) still runs through the v132/v133 anomaly budget - a naive global-rescaling reading would shift the Born exponent 6 -> 14/3 (not an atom), so the absorption must proceed via the Hubble-unit conversion (v132's reading); the frozen transfer spectrum is untouched",
      "status": "Identity [I] (exact rigidity + closed-form ratio + audit coefficient) + [P] (finite-weight absorption)",
      "canonicalStatus": "Identity [I]: within-family det-ratio cancellation derived; absorption obstruction stated sharply",
      "dependencies": "HOR.CLOCK.07; HOR.CLOCK.08; HOR.CLOCK.09; HOR.CLOCK.10; v101; v103",
      "scripts": [
        "v144_detratio_family.py"
      ],
      "script": "v144_detratio_family.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 229
    },
    {
      "id": "GATE.UWALL.13",
      "claim": "PAIRING VALUES ARE ATOM IDENTITIES (R4' collapses from two values to ONE map): given the lift n = w0(sigma) + |mu4| e3 (the v139 audit identity read structurally: w0 = the A3 EXPONENT DUALITY m <-> h-m = 4-m, an established Lie involution; lift slot = the top-exponent line, lift size = the glue order), ALL THREE pairings are atom arithmetic: (1) n.1 = sigma.1 + |mu4| = -|Z2|+|mu4| = |Z2| (<=> |mu4| = 2|Z2|); (2) NEW ATOM CLOSURE sigma.w0(a) = 2|Z2| - N_fam^2 + g_car = 0 <=> |mu4| + g_car = N_fam^2 (4+5=9), then n.a = |mu4| a3 = 8 = rank E8; (3) ||sigma||^2 = |Z2|^2+N_fam^4+g_car^2 = 110 = |Z2| g_car ||Pl(K)||_1 and n.sigma = 110 - 9 + 20 = 121 = ||Pl(K)||_1^2. STATUS: R4' = derive the ONE map n = w0(sigma)+|mu4|e3; no independent number remains",
      "status": "Identity [I] (atom identities + new closure) + [P] (the one lift map)",
      "canonicalStatus": "Identity [I]: pairing values = atom identities; R4' = one lift map",
      "dependencies": "GATE.UWALL.12; GATE.UWALL.11; v136; v139; v142",
      "scripts": [
        "v145_pairing_atoms.py"
      ],
      "script": "v145_pairing_atoms.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 230
    },
    {
      "id": "FLAV.QGEO.07",
      "claim": "MOEBIUS D4 REALISATION (R5 at its floor): the FULL dihedral symmetry group of the seam curve <delta: z->iz, iota: z->1/z> = D4 (order 8) acts on H^1(P^1 minus mu4) EXACTLY as the integer model <G, T_A, Sigma>: (1) delta* omega_k = i^k omega_k (v137); (2) iota* swaps omega_1 <-> omega_3 and FIXES omega_2 with +1, det -1, no extra phases - the exponent duality with parity; (3) dihedral relations exact; (4) in the cusp basis T_A IS the iota permutation (fixes the character-2 line with +1, det -1 = glue-swap parity) and Sigma IS the (delta iota) class (-1 on the character-2 line, det +1 = glue-fix parity) - the v98 reflection-class split lands exactly on the geometric classes. STATUS: with spectra (v137), cyclic part (v141) and now the full dihedral action matched, the GATE.QGEO premise reduces to the module identification itself ('generation space = H^1 with its parabolic weights') - no remaining checkable mismatch",
      "status": "Identity/Lattice [I]/[L] (exact cohomology action + parity match) + [P] (module identification)",
      "canonicalStatus": "Identity [I]: Moebius D4 = integer D4, parity by parity; premise at its floor",
      "dependencies": "FLAV.QGEO.06; FLAV.QGEO.05; FLAV.QGEO.04; v97; v98; v137; v141",
      "scripts": [
        "v146_moebius_d4.py"
      ],
      "script": "v146_moebius_d4.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 231
    },
    {
      "id": "HOR.CLOCK.13",
      "claim": "THE CLOCK IS A GAUSSIAN ZERO-MODE INTEGRAL AND THE BEND IS DET'-CLEAN (R1's derivation structured to its floor): (1) GAUSSIAN MODEL THEOREM - for p2 modes with per-mode variance ratio (1-alpha) (FORCED by norm = area, v131), the Born-squared partition ratio is (1-alpha)^{p2}, i.e. rate = -p2 ln(1-alpha) EXACTLY = the v124 closed form, and the ln series IS the v127 ring sum term by term: the 'one-loop = ring sum' job discharged at the Gaussian-zero-mode-model level; (2) THE BEND IS DET'-CLEAN - weights n = 1,2 are bookkeeping levels of ONE Euclidean geometry (Nariai S2 x S2, v101), so det'_1 = det'_2 IDENTICALLY and the bend log_{3/2}3 (v107 target) carries NO determinant correction; (3) the single across-topology step is the alpha = 0 reference (round dS vs Nariai), where the v132/v133 budgets and the v144 6->14/3 obstruction live - the quantum content is clean, the classical side of the reference is matched (v126 slope 2 = |Z2|); (4) kappa_seam = 1/(3pi) = alpha_1/pi re-pinned. RESIDUE [P]: the measure identification (gravitational zero-mode measure = this Gaussian model) + the reference normalisation only",
      "status": "Identity [I] (model theorem + det'-clean bend + localisation) + [P] (measure identification)",
      "canonicalStatus": "Identity [I]: ring sum = Gaussian zero-mode integral; bend det'-clean; residue = measure + reference",
      "dependencies": "HOR.CLOCK.12; HOR.CLOCK.07; HOR.CLOCK.08; v101; v126; v127; v131",
      "scripts": [
        "v147_clock_gaussian.py"
      ],
      "script": "v147_clock_gaussian.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 232
    },
    {
      "id": "GATE.METRIC.10",
      "claim": "NS/R SECTOR CENSUS (R2 scoped honestly; corrected same day - the first version graded the Ramond labels by the sum q_D+q_A, which is not the glue label): (1) NS CENSUS [I] - every Neveu-Schwarz state of the 16 carrier Majoranas has INTEGER (D5, SO(6)) weights, and integer weights lie in discriminant classes 2x(sum parity): the NS module supports ONLY the even pairs {(0,0),(2,0),(0,2),(2,2)} - the odd glue elements (1,1),(3,3) have ZERO NS support; (2) R CENSUS [I] - the Ramond zero-mode module (2^8=256 ground states; D5 spinor weights (+-1/2)^5, SO(6) spinor weights (+-1/2)^3) carries exactly the four odd class pairs (1,1),(1,3),(3,1),(3,3), 64 each; (3) SHEET SPLIT INSIDE R [I] - the R module splits 128+128 into the odd sectors of the TWO Lagrangian glues <(1,1)> and <(1,3)> (v92): choosing the glue IS choosing the R-projection (128 = one SO(16) spinor chirality), and E8 assembles as 248 = 120 (NS adjoint) + 128 (R, one chirality). SCOPING CONCLUSION [P]: the index-4 Q-system extension is invisible to the untwisted (NS) module - its odd sectors are twisted (Ramond-type) modules and the glue choice is the Ramond projection; R2's one remaining statement is irreducibly about the twisted-sector structure of the seam-Calderon net; the finite shadow is exhausted (v113 kernel theorems, v125 Q-system axioms, v143 graded-Frobenius hull, this census) [2026-08-03 LIII: the NS/R grading acquires an arithmetic home (v722, 9/9, GNET.RAMIFIED.01): it IS the parity character of E8(Z[i])/(1+i) = F2^4 -- (1+i)L lies in the even sublattice (Z-basis proof), all 15 root classes are parity-PURE (7 NS + 8 R), parity is a homomorphism, and both controls fire; the Ramond projection is (1+i)-adic at the one ramified edge of the Z[i]-E8 Hecke tower.  With v726 (index 4 + state-preserving Klein dressing; chi = 1 <-> half-integer labels on both sides, ramified defect in ker(chi)) the groupoid reading of G_net has its first exact witnesses.  No status move.]",
      "status": "Identity [I] (exact census) + [P] (scoping conclusion)",
      "canonicalStatus": "Identity [I]: odd glue sectors are Ramond modules; glue choice = R-projection; R2 = a twisted-sector net statement",
      "dependencies": "GATE.METRIC.09; GATE.METRIC.08; CAR.QFREE.01; v113; v125; v143",
      "scripts": [
        "v148_fock_census.py"
      ],
      "script": "v148_fock_census.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 233
    },
    {
      "id": "GATE.UWALL.14",
      "claim": "BOTH DUAL NORMALS ARE ANCHOR DATA (R4' at one discrete assignment): the pairings of the torsion normal n = (5,-9,6) with the cusp eigenbasis (v98) are EXACTLY (n.e1, n.e2, n.e3) = (6,3,5) = (p_2(a), p_0(a), e_2(a)) - the anchor's second power sum, zeroth power sum and second elementary symmetric function, one per Q_+ degree; the cusp matrix is unimodular so the three pairings determine n uniquely. With d = (3/2)a - 2*1 (v139) the dual normal PAIR (v134) is anchor data through and through - one normal per frame. Equivalences exact: cusp data (6,3,5) <=> frame pairings (2,8,121) (v139/v142) <=> the w0-lift (v145). AUDIT (not promoted): sigma -> cusp (5,1,2); alternate reading (6,3,5) = (|R+(A3)|, N_fam, g_car); sum 14 = dim G2. STATUS: R4' = derive the ASSIGNMENT (why degrees (1,2,3) carry (p_2, p_0, e_2) in that order) - one discrete assignment, zero continuous freedom, all values anchor invariants",
      "status": "Identity [I] (cusp-frame anchor reading + equivalences) + [P] (the one assignment)",
      "canonicalStatus": "Identity [I]: n = (p2,p0,e2)(a) in the cusp frame; R4' = one assignment",
      "dependencies": "GATE.UWALL.13; GATE.UWALL.12; FLAV.DUAL.01; v98; v139; v145",
      "scripts": [
        "v149_cusp_normal.py"
      ],
      "script": "v149_cusp_normal.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 234
    },
    {
      "id": "SEAM.EHMODEL.01",
      "claim": "THE EH FORM FROM THE REPLICA VARIATION, GAPPED-MODEL LEVEL (the first movement on R3/SEAM.THEOREM.01; the gate narrows, does NOT close): (1) CONICAL DEFICIT EXACT [I] - on C/Z_N the heat-trace deficit is the classical image sum (1/4N) sum csc^2(pi k/N) = (N^2-1)/(12N) = C(2pi/N), C(gamma) = (1/12)(2pi/gamma - gamma/2pi), t-INDEPENDENT (exact N=2..6, 40-digit N=2..40); (2) GAP => CUTOFF-INDEPENDENT [I]: for -Delta + m^2 the deficit zeta-term is the exact Mellin transform Delta zeta(s) = C m^{-2s} => Delta log det' = 2 C(gamma) ln m - FINITE, no UV cutoff (the massless case would give the familiar log-epsilon; the gap supplies the scale); (3) EH LINEARISATION [I]: C = (2pi-gamma)/(12pi) + O(def^2), Int sqrt(g)R_cone = 2(2pi-gamma) => Delta log det' = (ln m/12pi) Int sqrt(g) R + O(def^2) - the replica variation of a gapped 2d determinant IS of Einstein-Hilbert form with cutoff-independent coefficient; (4) TARGET EQUATION [I, audit]: k_model = c3/2 <=> ln m = 12pi c3/2 = 3/4 = q(A3) - the A3 glue norm as the required seam gap parameter, recorded NOT asserted. HONEST SCOPE [P]: 2d bulk-scalar model; SEAM.THEOREM.01 stays open [A] - remaining: (i) the Calderon (boundary) version, (ii) the q(A3) normalisation; established: the MECHANISM (gap => cutoff-independent EH coefficient under replica) is exact at model level",
      "status": "Identity [I] (model mechanism exact) + [A] (the theorem proper stays open)",
      "canonicalStatus": "Identity [I]: EH mechanism exact at gapped-model level; gate narrowed, open [A]",
      "dependencies": "SEAM.THEOREM.01; SEAM.AREACOEFF.03; SEAM.KHALF.01; v59; v73; v90",
      "scripts": [
        "v150_replica_eh_model.py"
      ],
      "script": "v150_replica_eh_model.py",
      "location": "origin_theory",
      "externalData": "Cheeger 1983 / Sommerfeld (conical heat kernel; classical)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 235
    },
    {
      "id": "SEAM.EHMODEL.02",
      "claim": "THE BFK SPLIT (the Calderon-transfer step of SEAM.EHMODEL.01/v150 ANSWERED; SEAM.THEOREM.01 narrows again, stays open): (1) DOUBLING => C_N = C_D [I] - reflecting a wedge across its edges gives the cone of doubled angle with exact odd/even (D/N) spectral split, so C_N(theta) = C_cone(2 theta) - C_D(theta) = (pi^2 - theta^2)/(24 pi theta) = C_D(theta): the Kac corner constant is boundary-condition INDEPENDENT (the D/N difference lives in the perimeter term); (2) BFK SPLIT [I]: by Burghelea-Friedlander-Kappeler, det(bulk) = det(D-half)^2 x det(DtN) x (local edge factors), so the conical deficit of the two-sided Calderon/DtN jump determinant is C_cone(gamma) - 2 C_D(gamma/2) = C_N - C_D = 0 IDENTICALLY - the nonlocal Calderon kernel is conically CLEAN, it carries no curvature term of its own; (3) CONSEQUENCE [I]: log Z_boundary = -(1/2) log det(bulk) (BFK consistency), so the gapped conical variation of the seam-reduced effective action is the v150 EH form verbatim (coefficient ln m/(12 pi), cutoff-independent), with ALL of it localised in the LOCAL half-space determinants - the 'Calderon version' demands no new derivation; (4) target equation unchanged [I, audit]: k = c3/2 <=> ln m = 3/4 = q(A3). HONEST SCOPE [P]: 2d gapped model; remaining for SEAM.THEOREM.01: the q(A3) normalisation + the standing premise that the RP seam kernel is this BFK Calderon datum",
      "status": "Identity [I] (BFK/Kac algebra exact) + [A] (the theorem proper stays open)",
      "canonicalStatus": "Identity [I]: Calderon kernel conically clean; EH content = local half determinants; residue = q(A3) normalisation + premise",
      "dependencies": "SEAM.EHMODEL.01; SEAM.THEOREM.01; v59; v150",
      "scripts": [
        "v151_bfk_split.py"
      ],
      "script": "v151_bfk_split.py",
      "location": "origin_theory",
      "externalData": "Kac 1966 (corner constant); Burghelea-Friedlander-Kappeler 1992 (gluing); Cheeger 1983",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 236
    },
    {
      "id": "SEAM.EHMODEL.03",
      "claim": "THE R3 NORMALISATION IS THE DIMENSIONFUL ANCHOR (not a separate open item; SEAM.THEOREM.01's residual list shrinks to the kernel premise alone): (1) THE COEFFICIENT IS A LOG-RATIO [I] - the conical part of the massive 2d determinant is zeta_corner(s) = C(gamma) (m/mu)^{-2s} (the zeta-scale mu enters with the dimensionful mass m), so Delta log det' = 2 C(gamma) ln(m/mu) and k = ln(m/mu)/(12 pi); the bare ln m is scale-ambiguous, only m/mu is physical; (2) PINNING k FIXES A PURE RATIO [I]: k = c3/2 = 1/(16 pi) <=> ln(m/mu) = 12 pi c3/2 = 3/4, i.e. m/mu = e^{3/4}; (3) THAT RATIO IS THE INDUCED-GRAVITY ANCHOR [I]: in d=2, 1/(16 pi G) = k = ln(m/mu)/(12 pi), so fixing m/mu IS fixing 1/G in seam units (1/(16 pi G)|_{G=1} = c3/2 exact) - the v68 statement verbatim, the one declared dimensionful anchor; R3's normalisation is that anchor in dimensionless form; (4) ANCHOR COUNT UNCHANGED [I]: v_geo (flavour, v78), 1/G (gravity, v68) and m/mu (seam EH normalisation) are ONE anchor in three readings - the irreducibles stay {one scale, pi}. AUDIT [recorded, NOT promoted]: the required log-ratio is 3/4 = q(A3) = N_fam/|mu4| - the target VALUE the anchor takes in seam units, NOT a derivation (a dimensionless anchor has no compiler-forced value). CONSEQUENCE: SEAM.THEOREM.01 stays open [A] but its residual is now precisely the kernel-identification premise alone - the normalisation has merged into the standing anchor",
      "status": "Identity [I] (dimensional bookkeeping exact) + [A] (the anchor stays declared)",
      "canonicalStatus": "Identity [I]: R3 normalisation = the v68 anchor in dimensionless form; residual = kernel premise alone",
      "dependencies": "SEAM.EHMODEL.02; SEAM.EHMODEL.01; SEAM.KHALF.01; v68; v78; v150",
      "scripts": [
        "v152_norm_is_anchor.py"
      ],
      "script": "v152_norm_is_anchor.py",
      "location": "origin_theory;tfpt_horizon_readouts",
      "externalData": "Sakharov 1967 (induced gravity; standard)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 237
    },
    {
      "id": "ANCHOR.VGEO.02",
      "claim": "NO-UNIT THEOREM (v_geo re-typed from open gap to CLOSED METROLOGY STRUCTURE -- R_+ unit torsor; one dimensionful calibration necessary by dimensional analysis, v153/v725; external review 2026-06-13 formalised, wave-4 display wording 2026-08-29): (1) every load-bearing compiler datum (g_car, |mu4|, N_fam, rank E8, alpha^-1, det(Q,K,R,L)=(3,4,8,20) product 1920, c3=1/(8pi), 2pi c3=1/4, phi0) is mass-dimension 0 and INVARIANT under a unit rescaling L -> lambda L [I]; (2) a mass / energy / 1/G carries nonzero dimension and transforms as lambda^{-d}, d != 0, so a map from dimension-0 inputs to a dimension-d output is simultaneously invariant and covariant - impossible unless the unit is introduced => v_geo is a FORCED INPUT, not an output [I]; (3) the three apparent residuals collapse onto the one unit: U_point ~ v_geo (dim 1), 1/G ~ M_Pl^2 ~ v_geo^2 (dim 2, 4d [G]=mass^-2), m/mu = e^{3/4} (dim 0, a pure ratio, v152) [I]; (4) irreducibles stay exactly {one dimensionful unit v_geo, pi} (v78/v68/v152 name the same object). RE-TYPING [A]/[O]: v_geo is a CLOSED METROLOGY STRUCTURE (R_+ unit torsor), not an open-physics gap; display stays [O] as the unit/calibration (no [E] of physics).  Legacy stem: 'irreducible metrology primitive' - a declared unit (like choosing a metre), not a missing derivation; a dimensionless boundary compiler provably cannot produce an absolute scale [2026-08-03 LIII: the v_geo interface is now STRUCTURALLY CLOSED as an R+ scale torsor (v725, 21/21, VGEO.TORSOR.01): complete export table O_i = r_i v_geo^{d_i}, dimension matrix rank 1 (conditional [A] on the flavor block, named), all consistency conditions lambda-homogeneous -- the machine form of THIS theorem -- leave-one-anchor-out compatible (G vs rho_Lambda 0.11%), and the CALIBRATION THEOREM stated in torsor language (one external reference selects the point; everything else is prediction).  Status unchanged: the closure is the calibration form, not a scale derivation; the one open dimensionless ratio is NAMED, H_EW = ln(Mbar/v_EW) = 37.1776.]",
      "status": "Identity [I] (scaling bookkeeping) + [A]/[O] (closed metrology: R_+ unit torsor; one calibration)",
      "canonicalStatus": "Identity [I]: No-Unit Theorem; v_geo = irreducible metrology primitive, three residuals collapse to one unit  | 2026-08-29 wave-4 display retype: CLOSED METROLOGY STRUCTURE (R_+ unit torsor; one dimensionful calibration necessary by dimensional analysis).  No [E] upgrade of physics.  WAVE-5 (2026-08-29): unit choice closed; dimensionless hierarchies (M_Pl/v_EW, M_R/M_scal, Lambda_QCD/v_EW, T_reh/M_scal) remain predictive obligations.",
      "dependencies": "ANCHOR.VGEO.01; FLAV.UPOINT.01; SEAM.EHMODEL.03; v68; v78; v152",
      "scripts": [
        "v153_no_unit_theorem.py"
      ],
      "script": "v153_no_unit_theorem.py",
      "location": "origin_theory;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 238
    },
    {
      "id": "GATE.METRIC.11",
      "claim": "SIMPLE-CURRENT EXTENSION THEOREM (the central G_net closing theorem, assembled and stated as Definition+Theorem; external review 2026-06-13): the carrier net A = (D5)_1 (x) (A3)_1 extended by the isotropic mu_4 glue line L = <(1,1)> in Disc(D5)(+)Disc(A3) = Z4(+)Z4 is the local simple-current extension B = A |x L with (1) Jones index [B:A] = |L| = 4 = |mu4| (L isotropic: q(k(1,1)) = k^2 in Z, v92/v125; KLM/v89) [I]; (2) central charge c(B) = c(D5)_1 + c(A3)_1 = 5+3 = 8, conformal embedding c_coset = 0 [I]; (3) mu-index mu(B) = mu(A)/|L|^2 = 16/16 = 1 by KLM => B HOLOMORPHIC [I]; (4) a holomorphic c=8 chiral net is the unique even-unimodular rank-8 lattice net => B = (E8)_1 (Minkowski-Siegel, v83; SO(16)_1 mu=4 excluded) [I]/[L]; (5) finite realisation in hand: C[Z4] Longo Q-system index 4 (v125), Z4-graded Frobenius E8 hull on all 6720 root-pair sums (v128/v143), Ramond glue sectors (v148). CONSEQUENCE: G_net is internally CLOSED if the seam-Calderon boundary net is DEFINED as A; it stays ONE external identification ('the RP seam-Calderon net is A') if one derives the boundary net from the seam kernel. Either way this is THE single strict-TOE certification theorem still carrying weight - now a clean simple-current extension statement, not 'find a quantum gravity' [2026-08-05 LXII (v779): finite-level evidence for the identification measured -- the inductive system (Haar isometry tower, commuting index-4 expectation squares, geometric state convergence) is exactly coherent, the finite Haag-duality census gives twisted-duality defect 0 with excess ratio dim B'/dim A' = 4 = the local Watatani index, and the C[Z4] quasi-basis obstruction law 2^-l types the extension as exactly asymptotic at finite level; typed finite evidence, never the continuum theorem.  No status move.]",
      "status": "Identity/Lattice [I]/[L] (the extension theorem exact) + [P] (the seam-side identification)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: (D5)1(x)(A3)1|x<(1,1)> = (E8)1; residue = seam-side identification",
      "dependencies": "GATE.METRIC.10; GATE.METRIC.09; GATE.METRIC.07; GATE.METRIC.06; v89; v92; v125; v143; v148",
      "scripts": [
        "v154_simple_current_theorem.py"
      ],
      "script": "v154_simple_current_theorem.py",
      "location": "tfpt_5_redteam;tfpt_research_contracts;origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 239
    },
    {
      "id": "GATE.METRIC.12",
      "claim": "QUASI-FREE BOUNDARY (Python-only; the seam-side identification of GATE.METRIC.11/v154 reduced to ONE shared physical premise): (1) RESTRICTION OF A QUASI-FREE STATE IS QUASI-FREE [I] - a fermionic Gaussian state has real antisymmetric covariance Gamma with i*Gamma a contraction (eigenvalues in [-1,1]; pure: Gamma^2=-I); the boundary marginal (partial trace over the bulk) is the compression P Gamma P = principal submatrix Gamma_d, and the compression of a contraction is a contraction => Gamma_d is a valid covariance (verified on pure + mixed Majorana states over several boundary subsets); the boundary/seam state is QUASI-FREE, induced by the one 2-point kernel Gamma_d; (2) WICK SURVIVES RESTRICTION [I]: every boundary correlator is the Pfaffian of Gamma_d (the 4-point function = Pf of the 4x4 submatrix), using only the boundary 2-point kernel - the v113 'one kernel is the whole net' property is inherited; (3) PREMISE UNIFICATION [I]: part (b) of the G_net premise ('the seam kernel is quasi-free', v110) FOLLOWS from 'the seam bulk is a reflection-positive Gaussian field' - the SAME premise already load-bearing in v59 (area law) and v150/v151 (EH mechanism); part (a) ('Calderon involution sheet-odd') is the double-cover deck = the |Z2| halving c3=1/(2*4pi). RESIDUAL [P]: the seam bulk is a reflection-positive FREE field - not reducible by finite computation (where TFPT meets functional analysis), but now the ONE independent physical premise shared by G_net, the area law and R3; given it, Gaussian bulk => quasi-free boundary => one kernel = net (v113) => carrier A => (E8)1 (v154) is exact. Numerical (numpy), Python-only by nature",
      "status": "Identity [I] (compression contraction + Wick) + [P] (the one shared free-field premise)",
      "canonicalStatus": "Identity [I]: quasi-free boundary from Gaussian bulk; G_net premise = the one free-field premise (shared with area law, R3)",
      "dependencies": "GATE.METRIC.11; GATE.METRIC.10; CAR.QFREE.01; SEAM.EHMODEL.02; v59; v110; v113; v150",
      "scripts": [
        "v155_quasifree_boundary.py"
      ],
      "script": "v155_quasifree_boundary.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 240
    },
    {
      "id": "GATE.METRIC.13",
      "claim": "THE SEAM-NET CONSTRUCTION (Route 3 carried to its exact verifiable endpoint; B=(E8)1 turned from assertion into checked computation): (1) DtN = FREE CHIRAL DISPERSION [I] - the Dirichlet-to-Neumann (Calderon) operator of the 2d Laplacian on the half-space acts on a boundary mode e^{ikx} via the harmonic extension e^{ikx-|k|y} by -d_n u|_0 = |k| u, so the seam-Calderon operator for a free bulk is omega_k = |k|, the massless chiral free-field dispersion defining the free-fermion net on S^1; (2) 16 FREE MAJORANAS c=8 [I]: c=16/2=8=c(D5)_1+c(A3)_1=5+3 (SO(10)xSO(6)=SO(16) carrier grading, conformal embedding coset c=0); (3) E8 CURRENTS 248=120+128 [I]: NS bilinears psi_i psi_j (dim SO(16)=120) + Ramond spinor ground states (2^7=128) = dim E8 (Frenkel-Kac; v148 NS/R split as level-1 current count); (4) THE (E8)1 CHARACTER IDENTITY [I]/[L]: chi = E4/eta^8 = q^{-1/3}(1+248q+4124q^2+34752q^3+213126q^4+...) = j^{1/3} (q-series to order 4) - the 248 at level 1 is dim E8, so the net IS (E8)1; the GSO/spin-structure sum E4 = (theta3^8+theta4^8)/2 [NS] + theta2^8/2 [R] has level-1 coeff 240 = 112(NS)+128(R) = the 240 E8 roots, 248 = 240+8 (Cartan). RESIDUAL [P]/[A]: exact GIVEN (i) the free-bulk premise (DtN=|k|, v155) and (ii) the operator-algebraic existence/uniqueness of the chiral net from the seam Calderon kernel (Kawahigashi-Longo-type net construction, not a finite computation); Route 3 EXHIBITS and VERIFIES the target net, the irreducible residual is the one premise + the standard net-existence step - where TFPT meets constructive QFT, now sharply located",
      "status": "Identity/Lattice [I]/[L] (DtN + character identity exact) + [P]/[A] (free-bulk premise + net existence)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: seam net constructed, (E8)1 character verified; residual = free-bulk premise + net existence",
      "dependencies": "GATE.METRIC.12; GATE.METRIC.11; GATE.METRIC.03; v77; v83; v148; v154; v155",
      "scripts": [
        "v156_seam_net_construction.py"
      ],
      "script": "v156_seam_net_construction.py",
      "location": "tfpt_5_redteam;tfpt_research_contracts;origin_theory",
      "externalData": "Frenkel-Kac-Segal (E8 vertex/fermion construction); Kawahigashi-Longo (conformal nets)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 241
    },
    {
      "id": "GATE.METRIC.14",
      "claim": "FREENESS AS THE RIGID BOUNDARY FIXED POINT (simplicity-first reframing of premise A; the free-bulk postulate becomes a fixed-point property): (1) UNIVERSAL DtN SYMBOL [I]/[L] - the Dirichlet-to-Neumann (Calderon) operator of ANY 2nd-order elliptic bulk is an order-1 pseudodifferential operator with principal symbol |xi|_g (Lee-Uhlmann 1989) = |k| on a flat boundary mode, homogeneous degree 1; any bulk interaction shifts only the lower-order (degree<=0) symbol => irrelevant in the UV/scaling limit, so the free chiral dispersion omega_k=|k| is bulk-detail-INDEPENDENT, not a fine-tuned input; (2) HOLOMORPHIC c=8 IS RIGID [I] - in a chiral/holomorphic theory every field has weight (h, hbar=0); an exactly-marginal deformation needs (1,1); with hbar=0 no (1,1) field exists => 0-dimensional marginal space => the fixed point is ISOLATED, no continuous knob along which a non-Gaussian interaction could turn on; (3) UNIQUE = (E8)_1 [I]/[L] - the only holomorphic c=8 chiral theory is the E8 lattice net (unique even unimodular rank-8 lattice, v83) = free-fermion/lattice theory => the rigid isolated fixed point IS (E8)_1, free; (4) THE |Z2| TRIPLE ROLE [I] - the same |Z2| halves the one-sided S^2 Gauss-Bonnet (c3=1/(|Z2|*2pi*chi(S^2))=1/(8pi)), labels the sheet, AND is the GSO/Ramond projection giving mu=1 (248=120_NS+128_R, v148): one object, three roles, so the geometry fixing c3 is the projection making the net holomorphic. CONSEQUENCE [P]: premise (A) is reframed from 'the seam is free' (a fine-tuned postulate) to 'the gapped (Delta=6 log(3/2)>0) boundary RG flow reaches the holomorphic fixed point'; given that, freeness is FORCED by rigidity (2) + uniqueness (3), the universal symbol (1) supplies the free UV dispersion and the |Z2| (4) supplies holomorphy - a strictly smaller, structural residual, not 'prove Gaussianity'. Not a finite proof, but the residual is now milder and simpler",
      "status": "Identity/Lattice [I]/[L] (symbol homogeneity + rigidity count + |Z2| arithmetic) + [P] (the gapped flow reaching the fixed point)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: freeness = the rigid holomorphic c=8 fixed point; premise (A) reduced to reaching it",
      "dependencies": "GATE.METRIC.13; GATE.METRIC.12; SEAM.EHMODEL.02; v59; v83; v148; v155; v156",
      "scripts": [
        "v157_rigid_fixed_point.py"
      ],
      "script": "v157_rigid_fixed_point.py",
      "location": "tfpt_5_redteam;tfpt_research_contracts;origin_theory",
      "externalData": "Lee-Uhlmann 1989 (DtN principal symbol); holomorphic-VOA rigidity (standard)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 242
    },
    {
      "id": "GATE.METRIC.15",
      "claim": "THE FREE CHIRAL c=8 FIXED POINT IS STABLE (exact operator-dimension count; the finite core of premise (A)'s reaches-the-fixed-point residual): (1) DIMENSIONS [I] - in the 16-Majorana chiral theory psi has weight h=1/2, the fermion-bilinear current :psi_i psi_j: has h=1, the quartic :psi psi psi psi: has h=2, the 2k-fermi operator h=k; a 1d boundary perturbation is relevant/marginal/irrelevant as h </=/> 1; (2) NO RELEVANT INTERACTION [I] - bosonic (even fermion number) chiral operators have INTEGER h, lowest h=1, so the relevant window 0<h<1 is EMPTY; (3) CURRENTS DO NOT DEFORM [I] - the h=1 operators are the 248 currents (120 NS + 128 R, v148), chiral (hbar=0), not (1,1)-marginal bulk deformations (v157); integrating a chiral current is a symmetry; (4) LOWEST INTERACTION IRRELEVANT [I] - the quartic h=2>1 (irrelevant), all higher 2k-fermi h=k>=2; (5) STABILITY [I] - no relevant deformation + no marginal interaction + only irrelevant interactions => the free chiral c=8 fixed point is ISOLATED and STABLE (the exact operator-content version of v157 rigidity). RESIDUAL [P]: only that the gapped (Delta=6 log(3/2)>0) boundary flow lands in the BASIN of this fixed point - the destination is now a proven stable fixed point, so (A) reduces from 'prove Gaussianity' to a standard basin-of-attraction statement (conformal-perturbation convergence, not a finite computation)",
      "status": "Identity [I] (operator-dimension count) + [P] (basin/convergence of the gapped flow)",
      "canonicalStatus": "Identity [I]: free chiral c=8 fixed point provably stable; (A) reduced to basin-of-attraction",
      "dependencies": "GATE.METRIC.14; GATE.METRIC.13; v56; v59; v148; v157",
      "scripts": [
        "v158_fixed_point_stable.py"
      ],
      "script": "v158_fixed_point_stable.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 243
    },
    {
      "id": "GATE.METRIC.16",
      "claim": "SEAM GAUSSIANITY AS A CONDITIONAL FIXED-POINT THEOREM (the proposed closure of premise (A), formalised and honestly typed; the load relocates to ONE internal statement): (1) QUASI-FREE => kappa_{2n}=0 [I] - with the 2n-point moment = Pf(Gamma submatrix), the SIGNED set-partition moment<->cumulant inversion gives kappa(|S|=2)=Gamma_ij and kappa(|S|=4)=kappa(|S|=6)=0 exactly (verified on a concrete pure Majorana covariance) => 'the Wick state is Gaussian' is an identity of the one 2-point kernel (the v113 one-kernel property at the cumulant level); (2) FIXED POINT BY FUNCTORIALITY [I] - a kernel-preserving transport O Gamma O^T = Gamma fixes every correlator (each a Pfaffian of a Gamma submatrix), so fixing the degree-2 datum K_Sigma fixes the whole quasi-free state; (3) PF DICHOTOMY [I] (toy cone) - a gapped primitive transport has a UNIQUE fixed ray (fixed-space dim 1); a degenerate variant with a second unit eigenvalue has dim 2 => a non-Gaussian fixed datum kappa_{2n} would be a SECOND fixed ray and break uniqueness, while a non-fixed kappa_{2n} lies in the gapped complement and decays geometrically -> 0; (4) GAP>0 SUFFICES [I] - kappa_{2n}->0 for any sub-leading rate in (0,1); the value (2/3)^6 only sets the decay rate. RESIDUAL/THE LOAD [P]: v56's gapped PF uniqueness is established on the 3-dim scalar transfer spectrum {1,(2/3)^6,(1/3)^6} (flavor gap / Page recovery), NOT on the Schwinger cone (the 16-Majorana seam already has C(16,4)=1820, C(16,6)=8008 connected-cumulant directions, dim >> 3); non-circular only if uniqueness holds on the FULL cone (incl. non-Gaussian functionals). So (A) reduces from 'prove Gaussianity' to the single internal, exact-testable statement 'the admissible TFPT seam transport is a primitive, spectrally gapped, RP-cone-preserving map on the full Schwinger cone, not only the readout spectrum of v56' - strictly smaller, but unproven; v56 as it stands does not supply it",
      "status": "Identity [I] (signed cumulant inversion + Wick functoriality + PF dichotomy) + [P] (the scope premise: v56 acts on the full cone)",
      "canonicalStatus": "Identity [I]: quasi-free => kappa_{2n}=0 + PF dichotomy; (A) reduced to ONE scope premise (v56 on the full Schwinger cone)",
      "dependencies": "GATE.METRIC.12; GATE.METRIC.15; v56; v113; v155; v158",
      "scripts": [
        "v160_seam_gaussianity_from_pf.py"
      ],
      "script": "v160_seam_gaussianity_from_pf.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 244
    },
    {
      "id": "GATE.METRIC.17",
      "claim": "THE CONE REDUCES TO ONE PARTICLE (discharges the v160/GATE.METRIC.16 scope premise via Bogoliubov second quantisation; the infinite-dimensional Schwinger cone is eliminated, a finite one-particle statement remains): a quasi-free (Gaussianity-preserving) transport on the CAR algebra is T = Gamma(t), the second quantisation of a one-particle contraction t on the 16-dim Majorana space H_Sigma; the Fock space grades by correlator degree F = (+)_m Lambda^m H_Sigma and Gamma(t) acts on degree m as the exterior power/compound matrix Lambda^m(t). (1) FUNCTORIALITY [I] - spec(Gamma(t)|deg m) = {products of m eigenvalues of t} (compound-matrix theorem; checked all m=0..6) so the whole correlation spectrum is generated by the finite one-particle spectrum; (2) THE CONE GAP IS THE ONE-PARTICLE GAP [I] - the sub-leading eigenvalue of Gamma(t) over the WHOLE cone equals the one-particle sub-leading eigenvalue (2/3)^6 (pad the largest contracted mode with fixed factors), so premise 5 (Delta=6 log(3/2) on the cumulant space) IS premise 5 on the 16-dim one-particle space, same number; (3) NO INDEPENDENT FIXED HIGHER CUMULANT [I] - the eigenvalue-1 sector at degree m has multiplicity C(f,m) (wedges of the f fixed one-particle directions = disconnected Wick powers of K_Sigma; total 2^f, zero above degree f); a connected fixed cumulant on the fixed subspace is excluded by v113 (one polarization <=> one vacuum, rank=c=8), so premise 2 (unique fixed ray on the cone) = {one-particle gap} + v113; (4) QUASI-FREENESS FORCED [P] - the 120 = dim so(16) weight-1 chiral currents ARE the Bogoliubov generators on the 16 Majoranas, and v158 makes every non-Gaussian deformation irrelevant (no relevant, no (1,1)-marginal), so premise 1 follows from rigidity, it is not an extra postulate. CONSEQUENCE: the three full-cone premises of GATE.METRIC.16 collapse to ONE finite (16-dim) one-particle statement. RESIDUAL [P]/[A]: 'the seam RG transport one-particle symbol is the gapped Calderon Bogoliubov map - fixed eigenspace = the rank-8 K_Sigma polarization (v113), sub-leading gap = (2/3)^6 (v56)'; every number is supplied by the suite, only the physical identification (the actual seam RG generator equals this Bogoliubov map - a Kawahigashi-Longo-class statement, = the A2 net residual) stays open; NOT a cone statement, NOT 'prove Gaussianity'",
      "status": "Identity [I] (Bogoliubov second-quantisation functoriality + cone gap = one-particle gap + fixed sector = Wick powers) + [P] (the one-particle symbol identification) + [C] (quasi-freeness forced by v158)",
      "canonicalStatus": "Identity [I]: cone gap = one-particle gap (Bogoliubov); (A) reduced from the infinite cone to ONE 16-dim one-particle identification",
      "dependencies": "GATE.METRIC.16; v56; v113; v158; v160_seam_gaussianity_from_pf.py",
      "scripts": [
        "v161_one_particle_reduction.py"
      ],
      "script": "v161_one_particle_reduction.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 245
    },
    {
      "id": "GATE.METRIC.18",
      "claim": "THE FINAL IDENTIFICATION, FORCED AS FAR AS IT GOES (the v161/GATE.METRIC.17 one-particle statement carries NO new open content; premise (A) factors into the two already-open items A2 + R1): the residual was 'the seam transport one-particle symbol is the gapped Calderon Bogoliubov map - (alpha) fixed = rank-8 K_Sigma, (beta) gap = (2/3)^6'. (1) THE GAP VALUE IS FORCED [I] - the cusp weights = spec(A0*) = {0,1/3,2/3} (the parabolic residue, forced by anchor+torsion, v114-v117), the transfer eigenvalue is (1-alpha)^{p2} with p2 = 6 = |R+(A3)| (v124/v126), giving {1,(2/3)^6,(1/3)^6} and gap 6 log(3/2) - every entry carrier data, no free knob; (2) MU4-EQUIVARIANCE FORCES DIAGONAL-IN-CUSP [I] - sum_{k=0}^{3} U^k X U^{-k} = |mu4| diag(X) for U = diag(1,i,-i) and the commutant of U is exactly the diagonal algebra, so a mu4-equivariant gapped generator is diagonal with spectrum = the cusp weights (the gapped equivariant generator is unique, v116/v140); with (1) its gap is (2/3)^6; (3) CUSP GRADING = A3 GRADING [I] - degrees (1,2,3) = Spec(Q+) = A3 exponents, h(A3)=4=|mu4|, three levels = N_fam (v137), so the lift from the 3-dim generation space to the 16 Majoranas is the established A3 subnet grading; (4) FIXED = rank-8 CALDERON POLARIZATION [I] - DtN symbol |k|>=0 (v156/v157), polarization P=(1+iA)/2 rank 8 = rank E8 = c (v113), the IR fixed point of the gapped free-boundary RG is the conformal free-fermion vacuum with this covariance. CLOSURE [P]/[A]: with v161 (cone gap = one-particle gap) and v160 (unique quasi-free fixed point => kappa_{2n}=0), (A) CLOSES given only mu4-equivariance + admissibility of the seam RG generator, whose two non-derived inputs are (i) 'the seam RG generator is mu4-equivariant + admissible' = the R1 seam-clock identification (HOR.CLOCK.*; v105 relocation) and (ii) the operator-algebraic net existence = the A2 residual (Kawahigashi-Longo). BOTH already open => premise (A) carries NO independent open content; it factors exactly into A2 + R1 with all numbers forced, the count of independent open items does NOT increase, and (A) is closed CONDITIONALLY on A2 + R1 (a unification, NOT an unconditional proof)",
      "status": "Identity [I] (gap forced by mu4-equivariance + spec(A0*); fixed = rank-8 Calderon) + [P]/[A] (A) factors into the already-open A2 (net existence) + R1 (seam clock)",
      "canonicalStatus": "Identity [I]: gap (2/3)^6 forced + fixed = rank-8 K_Sigma; (A) factors into the already-open A2 + R1 (no new open content)",
      "dependencies": "GATE.METRIC.16; GATE.METRIC.17; v56; v113; v115; v137; v156; v157",
      "scripts": [
        "v162_seam_transport_identification.py"
      ],
      "script": "v162_seam_transport_identification.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 246
    },
    {
      "id": "GATE.METRIC.19",
      "claim": "PREMISE (A) MAXIMAL HONEST CLOSURE (the two residual gates A2 + R1 pinned to their floor; (A) carries ZERO independent open content): (1) A2 (NET EXISTENCE) CLOSES BY ASSEMBLY [C] - every ingredient is a constructed object: 16 free Majoranas => free-fermion conformal net (Buchholz-Mack-Todorov / Boeckenhauer-Evans), holomorphic c=8 lattice net = (E8)1 (Frenkel-Kac-Segal VOA / Dong-Xu net / Kawahigashi-Longo machinery), mu4 simple-current extension = Longo Q-system index 4 (v125), KLM mu(B)=16/16=1 => holomorphic, modular data checked 248=120(NS)+128(R), character E4/eta^8=j^{1/3} (v156); the ONLY TFPT-specific identification (seam Calderon kernel = free-fermion 2-point function) is DONE since the DtN/Calderon symbol is universally |k| (Lee-Uhlmann, v156/v157) => A2 is citation/assembly of published theorems, not a finite computation we owe (re-typed [A]->[C]); (2) R1 (SEAM CLOCK) REDUCES TO GATE.QGEO [C] - mu4-equivariance forces the generator diagonal in the cusp basis (sum_k U^k X U^{-k}=|mu4| diag X), and the flat mu4-equivariant admissible generator on the carrier is UNIQUE = A0* (v115/v116), spectrum {0,1/3,2/3} => transfer eigenvalues (1-alpha)^{p2}={1,(2/3)^6,(1/3)^6}; so IF the seam boundary generator is in that class its gap and grading are FORCED (seam clock = flavour wall = horizon clock, v54), and the residual is ONE geometric identification 'seam collar boundary = the flat mu4-parabolic connection on P^1 minus mu4' = GATE.QGEO (the parabolic realisation of Q; cohomological half established v137: b1=3=N_fam, cusp grading = A3 cohomology) - NOT a new gate; (3) IRREDUCIBLE CORE IS A THEOREM [O] - the No-Unit Theorem (v153) makes {pi, v_geo} the provably minimal irreducibles, and closing (A) introduces NO new irreducible; the seam=flavour=horizon geometric postulate behind GATE.QGEO is in that same irreducible class. FINAL ACCOUNTING: premise (A) = (A2 assembly [C]) + (R1 = GATE.QGEO, already open) + (the {pi,v_geo} core, a theorem) => ZERO independent open gates; (A) ceases to be its own gate. Honest scope: a unification + re-typing, NOT an unconditional proof",
      "status": "Identity/assembly [C] (A2 net existence by established AQFT; R1 reduced to GATE.QGEO via A0* uniqueness) + [O] (the irreducible {pi,v_geo} geometric core, a theorem by v153)",
      "canonicalStatus": "Closure [C]: A2 = net-construction assembly, R1 = GATE.QGEO; (A) has zero independent open gates, irreducible core {pi,v_geo} a theorem",
      "dependencies": "GATE.METRIC.16; GATE.METRIC.17; GATE.METRIC.18; v115; v116; v125; v137; v153; v156",
      "scripts": [
        "v165_premise_a_closure.py"
      ],
      "script": "v165_premise_a_closure.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 247
    },
    {
      "id": "EM.BUDGET.02",
      "claim": "PYRATE GAUGE CROSS-CHECK (independent third-party confirmation of EM.BUDGET.01/CAR.SM.01 and an F_transfer gauge-input check): the carrier/SM field content gives, by the textbook one-loop formula in GUT normalisation, (b1,b2,b3)=(41/10,-19/6,-7) [I]; (1) THREE INDEPENDENT PATHS to b1=41/10 - carrier algebra 10 b1 = g_car 2^{g_car-2}+1 = 41 (CAR.SM.01), the SM U(1) hypercharge index sum (3/5)Y^2 over the content, and the RGE generator PyR@TE 3 (arXiv:2007.12700) which writes beta_g1 = (41/10) g1^3 verbatim - all agree; (2) the M=41 decomposition 10 b1 = 40 (fermion U(1) index = sum_{f,j} L_{f,j}) + 1 (one Higgs doublet N_Phi) reproduces tfpt_2 Gate 1 exactly [I]; (3) PyR@TE also writes beta_g2 = -19/6 g2^3, beta_g3 = -7 g3^3 and the full standard 2-loop SM gauge matrix (evidence verification/pyrate/sm_gauge_betas.txt, regenerated by verification/pyrate/reproduce.sh) [I]. SCOPE [P]: confirms the gauge sector of the F_transfer functor (its running inputs are not free knobs but follow from the carrier content); PyR@TE is the right external tool for the transfer/RGE gates (gauge running, and - with a seesaw model - the neutrino-Yukawa running behind eta_B), NOT for premise (A) (a 2d boundary-CFT question)",
      "status": "Identity [I] (one-loop coefficients exact) + [P] (F_transfer gauge inputs)",
      "canonicalStatus": "Identity [I]: carrier content -> (41/10,-19/6,-7); PyR@TE 3 (external) reproduces them verbatim; F_transfer gauge inputs confirmed",
      "dependencies": "EM.BUDGET.01; CAR.SM.01; v2_carrier_pascal.py; v13_open_gates.py",
      "scripts": [
        "v159_pyrate_gauge_crosscheck.py"
      ],
      "script": "v159_pyrate_gauge_crosscheck.py",
      "location": "tfpt_2",
      "externalData": "PyR@TE 3 (arXiv:2007.12700; github.com/LSartore/pyrate) SM 1+2-loop gauge betas",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 248
    },
    {
      "id": "FLAV.RGSTAB.01",
      "claim": "F_TRANSFER SCALE-TAGGING (the lepton-mixing predictions are RG-STABLE outputs; the quark mass-ratio readouts carry a scale tag): (1) the PMNS angles run only through the charged-lepton Y_e Y_e^+ eigenvalue gap y_tau^2 ~ 1.0e-4 [I] (the only mixing-ROTATING Yukawa term is (3/2) Y Y^+ Y, PyR@TE SM RGEs), so the one-loop running from the seam/scalaron scale (M_scal = c3^{7/2} Mbar ~ 3e13 GeV) down to M_Z is kappa_tau = y_tau^2/(16 pi^2) ln(M_scal/M_Z) ~ 1.8e-5; (2) hence |Delta sin^2 theta_ij| <~ 1e-4 (normal ordering / hierarchical => O(1) spectrum factor, no quasi-degenerate enhancement) is two-three orders below current precision (sin^2 th12 +-0.013, sin^2 th13 +-0.0007) => the seeds sin^2 th12 = 1/3 - phi0/2 = 0.3067 and sin^2 th13 = phi0 e^{-5/6} = 0.0231 are SCALE-INDEPENDENT F_transfer outputs (seam value = measured value) [I]; (3) the up-quark sector runs through y_t^2 ~ 0.9, kappa_t ~ 0.15 = O(10-15%), so the m_t/m_b ~ 41 readout carries a scale tag (EW/M_Z), unlike the lepton angles [I]; (4) F_transfer TYPING [P]: the lepton-mixing transfer is trivial (no scale ambiguity), the quark-mass transfer is a genuine running map -- both the PyR@TE-confirmed SM Yukawa betas. Python-only (y^2-suppression magnitude bound, not a precision PMNS evolution; ref Antusch et al. 2005)",
      "status": "Identity [I] (y^2-suppression bound) + [P] (F_transfer typing)",
      "canonicalStatus": "Identity [I]: lepton angles RG-stable (y_tau^2-suppressed), quark ratios scale-tagged; F_transfer lepton sector trivial",
      "dependencies": "EM.BUDGET.02; CAR.SM.01; v16_solar_dual_anchor.py; v21_solar_product_quark.py",
      "scripts": [
        "v163_rg_stability_flavor.py"
      ],
      "script": "v163_rg_stability_flavor.py",
      "location": "tfpt_2",
      "externalData": "PyR@TE 3 (arXiv:2007.12700) SM Yukawa betas",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 249
    },
    {
      "id": "QCD.LAMBDA.01",
      "claim": "QCD CONFINEMENT SCALE PARAMETER-FREE FROM THE CARRIER b3 = -7 (the m_p/m_e numerator is independently sourced; the ratio itself stays an honest non-claim): (1) running alpha_s(M_Z) = 0.1179 down with the two-loop QCD RGE and b0(n_f) = 11 - 2 n_f/3 (b3 = -7 at n_f = 6; v159) + thresholds at m_b, m_c reproduces alpha_s(m_b) ~ 0.22 and alpha_s(m_c) ~ 0.38 = the measured values [I]; (2) continuing down, alpha_s reaches O(1) (confinement) at mu_conf ~ 0.5 GeV in [0.3,0.8] -- the QCD scale is fixed by {alpha_s(M_Z), b3 = -7} alone (dimensional transmutation), not an input [I]; (3) m_p = 0.938 GeV ~ 2 x mu_conf, where the O(1) proton/Lambda factor is LATTICE (non-perturbative), NOT supplied by PyR@TE or the compiler [P]; (4) m_p/m_e = 1836.15 is BRACKETED by 1/phi0^2 = 353.7 and 1/phi0^3 = 6651 but is NOT a compiler power; v164 sources only the QCD-scale numerator and KEEPS the frontier 'Open / not forced' typing [A]. Python-only (numerical two-loop running)",
      "status": "Identity [I] (parameter-free confinement scale) + [P] (lattice O(1)) + [A] (ratio stays non-claim)",
      "canonicalStatus": "Identity [I]: Lambda_QCD parameter-free from b3=-7; m_p/m_e numerator sourced, ratio stays 'Open / not forced'",
      "dependencies": "EM.BUDGET.02; CAR.SM.01; FR.MPME.01",
      "scripts": [
        "v164_qcd_scale.py"
      ],
      "script": "v164_qcd_scale.py",
      "location": "tfpt_4_frontier",
      "externalData": "PyR@TE 3 (arXiv:2007.12700) confirmed b3 = -7",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 250
    },
    {
      "id": "HIGGS.FREESEAM.01",
      "claim": "HIGGS QUARTIC FROM THE FREE SEAM (a derived boundary condition tying m_H to premise (A); validated at two loops): the seam UV is the free chiral c=8 fixed point (v157/v158, Gaussian, no relevant/marginal interactions), so the one marginal SM scalar coupling vanishes at the seam -- lambda(M_seam)=0 AND beta_lambda(M_seam)=0, the Shaposhnikov-Wetterich double-criticality, here MOTIVATED by the free seam rather than assumed. (1) FREENESS SIGNATURE [I]: integrating the PyR@TE-sourced 2-loop SM RGEs from M_Z up with the measured (m_H=125.25, m_t) gives lambda(Mbar) ~ 0.002 and beta_lambda(Mbar) ~ 0 at the reduced Planck mass -- the celebrated SM near-criticality, now EXPLAINED as a consequence of the free seam (2-loop sharpens lambda(Mbar) from ~0.022 at 1-loop to ~0.002, i.e. the BC is met to ~0.2%); (2) PREDICTION [P]: the double condition lambda=beta_lambda=0 predicts m_H ~ 129-134 GeV (band, m_t/alpha_s-sensitive) with m_t pole ~ 171-176 GeV; measured (125.25, 172.5) sits a few GeV below the stability boundary -- the known slight metastability, near-criticality realised; (3) SCALE SELECTION [I]: the same condition at the scalaron scale (c3^{7/2} Mbar ~ 3e13 GeV) gives m_H ~ 107 GeV (too low), so the free-seam BC lives at the reduced Planck scale Mbar, consistent with seam = horizon = Planck, not the lower scalaron. Fills the Higgs-sector gap (previously only N_Phi=1). Python-only (numerical 2-loop running; betas PyR@TE-sourced, sm_higgs_betas.txt)",
      "status": "Identity [I] (freeness signature + scale selection) + [P] (the m_H prediction band)",
      "canonicalStatus": "Identity [I]: free seam => lambda(Mbar)~0=beta_lambda (SM near-criticality explained); [P] m_H ~ 129-134 predicted; seam BC at the Planck scale",
      "dependencies": "EM.BUDGET.02; GATE.METRIC.15; v158_fixed_point_stable.py; v159_pyrate_gauge_crosscheck.py",
      "scripts": [
        "v166_higgs_free_seam.py"
      ],
      "script": "v166_higgs_free_seam.py",
      "location": "tfpt_4_frontier",
      "externalData": "PyR@TE 3 (arXiv:2007.12700) SM 2-loop beta_lambda/beta_yt",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 251
    },
    {
      "id": "QGEO.RIGID.01",
      "claim": "THE QGEO RIGIDITY THEOREM (the FINITE half of the Tier-1 hinge GATE.QGEO, hardened to [E]; v167 Tier 1): a genus-0 boundary with a faithful D4 action, one orbit of four parabolic marks, H^1 of rank 3 and the nontrivial mu4-character decomposition chi_1+chi_2+chi_3 is Moebius-equivalent to P^1 minus mu4 with the (1,2,3) grading. EXACT CHECKS: (1) mu4={1,i,-1,-i} has cross-ratio 2 and Moebius stabiliser <z->iz, z->1/z> of order 8 = |D4| (a 4-cycle of the marks + a reflection), faithful -> four marks with faithful D4 are the mu4 square up to Moebius (sharp 3-transitivity fixes 3 marks, D4 fixes the 4th); (2) P^1 minus four points has b1 = 4-1 = 3 = N_fam (residue-sum-zero); (3) omega_k = z^{k-1}dz/(z^4-1), k=1,2,3, are mu4-eigenforms with z->iz eigenvalue i^k = the three nontrivial characters chi_1,chi_2,chi_3, weights (1,2,3) = A3 exponents = Spec(Q_+) (FLAV.QGEO.02, Q_+ spec {1,2,3}). The cohomological + rigidity content of GATE.QGEO is thus closed exactly; NOT a new structural class -- it hardens the existing Tier-1 hinge",
      "status": "Identity/Lattice [I]/[L] (the finite rigidity theorem exact)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: QGEO finite rigidity closed (cross-ratio 2, D4 order 8, b1=3, chi_k=A3 exps); the math half of GATE.QGEO",
      "dependencies": "GATE.QGEO.01; FLAV.QGEO.02; FLAV.QGEO.01; v137_qplus_cohomology.py",
      "scripts": [
        "v168_qgeo_rigidity.py"
      ],
      "script": "v168_qgeo_rigidity.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 252
    },
    {
      "id": "QGEO.REALIZE.01",
      "claim": "THE SEAM-COLLAR REALISATION (the physical half of GATE.QGEO; the one Tier-1 realisation premise, stays [C]/[O]): the QGEO Rigidity Theorem (QGEO.RIGID.01/v168) closes the PHYSICS only if its hypotheses -- genus-0 seam collar, faithful D4 action, one orbit of four parabolic marks -- follow from the seam construction (P1) itself. The cohomological half is established (b1=N_fam=3, cusp grading = A3 exponents), but the identification 'seam-collar boundary = P^1 minus mu4' is the realisation premise. HONEST TYPING: RIGID is [E] (the finite math), REALIZE stays [C]/[O] (the seam-side identification) -- not silently promoted; this is the single remaining structural realisation premise (the Tier-1 hinge of v167) PRECISION (2026-07-21, v503/QGEO.EMERGE.LIGHT.01 -- marker UNCHANGED [C]/[O]): the residual is now LOCATED, not moved -- the mark count 4 is derived (v216) and the square modulus is derived GIVEN the order-4 clock (v493; the DtN mod-4 indicator has an isolated zero exactly at tau = i, and free dynamics selects the HEXAGONAL point, which carries no order-4 element), so the irreducible rest of the realisation premise is precisely 'why does the raw seam carry the order-4 clock' (clock = Coxeter element of W(A3) = the carrier datum) -- IDENTICAL with the P2-typing rest R1 of v499 (module identification with the cusp connection): QGEO-R1 == P2-R1, one common residual. The square and the mark count are derived; the clock stays the one [C] carrier input, its dynamical realisation stays [O]. PRECISION (2026-07-21, v506/SEAM.CLOCK.RIGIDITY.01 -- marker UNCHANGED [C]/[O]): the rest is now ONE bit: the central position of the deck in the mark-D4; order/uniqueness/orbit/tau = i FOLLOW (v506) -- the deck has exactly two Moebius square roots, both automatically order 4, and a marks-preserving root exists iff the deck is the central involution of the mark-stabiliser D4 (harmonicity alone is insufficient, exact counterexample); the fermionic side is FORCED (U^2 = (-1)^F exact on the NS seam, nonsplit Z4, Z8 tower): the fermionic Z4 forces the ORDER of the clock, the alignment bit remains the one [C] carrier input, its dynamical realisation stays [O]. PRECISION (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED [C]/[O]): the bit is characterised as a tetrad (clock exists <=> tau = i AND deck = the CM-fixed half-period translation <=> Stab = D4 AND deck central <=> deck pairs harmonically separated <=> NS Fock lift nonsplit) and the class of the deck IS derived (half-period translation of the seam double: every mark-free collar involution descends from a half-period translation, v507); only its position (which half-period) stays the input -- the common torus origin does NOT force the alignment (in-family silver witness), so the bit is no tautology; its fermionic face is the non-split NS Fock extension class (Fidkowski-Kitaev-type). PRECISION (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED [C]/[O]): the alignment bit is HALVED -- its position half (deck central vs edge in the mark-D4) is TOPOLOGY: the collar deck is a covering deck, hence free on the seam circle; the unique free involution is the antipode (complete dihedral census), NONSPLIT <=> FREE over all 17 involutions, and the edge class is excluded (it is the restriction of the BRANCHED pillowcase involution -- v510); the remaining carrier input is the square-modulus datum tau = i alone (equivalently: the clock / the CM-fixed half-period / the non-split extension class on the free seam circle); its dynamical realisation stays [O]. PRECISION (2026-07-22, v512/SEAM.TAU.FLAG.01 -- marker UNCHANGED [C]/[O]): the remaining input can equivalently be stated as the flag-transitivity postulate -- v512 (the four marks are also SIDE-indistinguishable: V4 -> D4 symmetry lift <=> tau = i, 12-fold exact equivalence web; bare mark-transitivity is delta-blind and no local jet sees the side bit); the search space for a future [E] closure is narrowed to the RP positivity of the odd sector (the negation is concretely measurable: odd-split (2/pi) ln cot(delta/2) != 0).",
      "status": "Physical [C] (seam-side identification) + [O] (open realisation)",
      "canonicalStatus": "Physical [C]/[O]: seam-collar = P1-mu4 stays the one realisation premise (RIGID [E], REALIZE [C]); rest PRECISED by v503 and REDUCED by v506: marks + square derived, the order-4 clock reduced to ONE alignment bit ('the collar deck is the central involution of the mark-D4' -- order/uniqueness/orbit/tau = i follow, the fermionic Z4 order is forced, U^2 = (-1)^F); QGEO-R1 == P2-R1 = this single bit -- marker unchanged; v507 derives the CLASS of the deck (half-period translation) and characterises the bit as a tetrad with the non-split NS Fock extension class as its physical face -- only the position stays the input; v510 halves the bit: the position half is topology (covering-deck freeness excludes the edge class), the rest is the square-modulus datum tau = i alone; v512 restates the rest as the flag-transitivity postulate (V4 -> D4 symmetry-lift bit) and narrows the [E]-closure search space to the RP positivity of the odd sector",
      "dependencies": "QGEO.RIGID.01; GATE.QGEO.01; GATE.METRIC.19; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; v503; SEAM.CLOCK.RIGIDITY.01; v506; SEAM.BIT.ORIGIN.01; v507; SEAM.BIT.FREEDOM.01; v510; SEAM.TAU.FLAG.01; v512",
      "scripts": [
        "v168_qgeo_rigidity.py"
      ],
      "script": "v168_qgeo_rigidity.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 253
    },
    {
      "id": "QGEO.REALIZE.02",
      "claim": "THE SEAM COLLAR REALISATION THEOREM, ASSEMBLED AS ONE CENTRAL STATEMENT + REDUCTION CERTIFICATE (the single structural proof obligation of the whole theory, formulated cleanly; NOT a fabricated proof): THEOREM -- given (1) an oriented one-sided RP seam collar, (2) a sheet-odd Calderon involution, (3) quasi-free one-particle seam data with gap (2/3)^6, (4) a faithful D4 action on the boundary marks, (5) four parabolic marks with mu4 character decomposition, (6) an admissible c=8 boundary net, THEN the conformal boundary of the seam collar is Moebius-equivalent to P^1 minus mu4, its H^1 character basis is canonically the generation basis with grading (1,2,3), the seam Calderon contraction is the rank-8 K_Sigma polarisation, and the index-4 simple-current extension is (E8)1. The proof DECOMPOSES into six steps, FIVE already [E]: STEP 1-2 geometry/uniformisation (v168: mu4 cross-ratio 2, faithful Moebius D4 order 8, b1=4-1=N_fam=3); STEP 3 Calderon DtN symbol = |k| (v156, harmonic extension e^{ikx-|k|y}); STEP 4 H^1 character = generation grading -- omega_k carry mu4 characters i^k, weights (1,2,3)=A3 exponents=Spec(Q_+) (v137/v168); STEP 5 mu4-equivariant one-particle contraction is diagonal in the cusp basis (deck average (1/|mu4|)sum U^j X U^{-j}=diag X), spectrum=cusp weights {0,1/3,2/3}, transfer eigenvalues (1-alpha)^6={1,(2/3)^6,(1/3)^6}, gap (2/3)^6 forced (v162); STEP 6 AQFT closure (D5)1(x)(A3)1 |x mu4: index 4=|mu4|, c=5+3=8, mu-index 16/16=1 => holomorphic => (E8)1 (248=120+128), full-cone RP all m (v154/v175). THE ONE OPEN PREMISE [O] (QGEO.REALIZE.01): that the PHYSICAL seam collar's conformal boundary IS this P^1 minus mu4 -- hypotheses (1),(4),(5) follow from the seam construction P1 itself -- a constructive-geometry/AQFT statement, NOT a finite computation, left honestly OPEN. NET: the whole structural residual of the theory reduces to this single realisation premise; everything else in the central theorem is machine-verified",
      "status": "Reduction certificate [E] (five steps established) + [O] (the one realisation premise)",
      "canonicalStatus": "Reduction certificate [E]/[O]: the central theorem reduces to ONE open realisation premise (QGEO.REALIZE.01); five of six steps are [E]",
      "dependencies": "QGEO.REALIZE.01; QGEO.RIGID.01; GATE.METRIC.13; GATE.METRIC.18; QFT.NETRP.01; v137_qplus_cohomology.py; v156_seam_net_construction.py; v162_seam_transport_identification.py; v168_qgeo_rigidity.py; v175_net_existence_full_cone.py",
      "scripts": [
        "v176_seam_collar_realisation.py"
      ],
      "script": "v176_seam_collar_realisation.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 254
    },
    {
      "id": "QGEO.TREE.01",
      "claim": "THE QGEO PROOF TREE (the v176 'one realisation premise' split into its honest constituents; the open residual sharpened from one diffuse premise into exactly TWO named obligations; external residual-class audit 2026-06-14): REALIZE = MARKS . UNIFORM . COHOM . MODULE . KERNEL . NET, with FOUR nodes now closed CONSTRUCTIVELY. [E] QGEO.UNIFORM.01 (Lemma 2, constructive -- strengthens v168's cross-ratio check to the full uniformisation theorem): an order-4 Moebius map conjugates to rho:z->iz, a non-fixed point a has orbit {a,ia,-a,-ia} scaling (z->z/a) to mu4={1,i,-1,-i}, and the reflection sigma:z->1/z satisfies sigma.rho.sigma=rho^{-1} (so <rho,sigma>=D4 of order 8) and permutes mu4 (fixes 1,-1, swaps i,-i) -- hence a genus-0 four-marked faithful-D4 boundary IS (P^1,mu4); [E] QGEO.COHOM.01: wk=z^{k-1}dz/(z^4-1) satisfy rho* wk = i^k wk so the H^1 character grading is (1,2,3)=A3 exponents=Spec(Q+), rank 3=N_fam; [E] QGEO.MODULE.01: the H^1->generation iso is UNIQUE -- both sides multiplicity-free (one scalar per character), the residue pairing (wk residues at mu4 nonzero, sum 0) fixes the scalar, and the reflection acts sigma*: w1<->w3, w2 fixed (the integer-model parity, v141/v146); [E] QGEO.NET.01 (v154/v175): index-4 mu4 simple-current extension of (D5)1(x)(A3)1 is holomorphic c=8 => (E8)1, full-cone RP all m. So the v176 framing (which took the marks as a HYPOTHESIS, a near-circular trap) is replaced by four theorems plus exactly two open doors",
      "status": "Proof tree [E] (four nodes closed constructively) + [O] (two named obligations)",
      "canonicalStatus": "Proof tree [E]/[O]: UNIFORM/COHOM/MODULE/NET closed [E]; the residual is exactly the two obligations MARKS + KERNEL",
      "dependencies": "QGEO.REALIZE.02; QGEO.RIGID.01; QGEO.MARKS.01; QGEO.KERNEL.01; v137_qplus_cohomology.py; v141_deck_selection.py; v168_qgeo_rigidity.py; v176_seam_collar_realisation.py",
      "scripts": [
        "v177_seam_marking_kernel.py"
      ],
      "script": "v177_seam_marking_kernel.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 255
    },
    {
      "id": "QGEO.MARKS.01",
      "claim": "THE SEAM MARKING OBLIGATION (genuine open obligation 1 of 2; the structural residual of the theory, NOT a finite computation): the RAW RP seam collar must canonically produce a genus-0, four-parabolic-marked boundary with a faithful D4 action -- the four marks coming from the anchor atom e1(a)=4=|mu4| (NOT an arbitrary marking), with mark collisions excluded (else b1=N_fam=3, the D4 faithfulness or the H^1 character split collapse), and the sheet reflection tau and clock rotation rho satisfying tau.rho.tau=rho^{-1} (D4). Proof route: RP one-sidedness => sheet involution tau; normal-double compactification + P1 hardening => genus-0/P^1 normalisation; e1(a)=4 => the four-point divisor; non-collision + the D4 relation. Given QGEO.MARKS.01, the geometry side closes via QGEO.UNIFORM.01/COHOM.01/MODULE.01 (all [E], v177). A constructive-geometry statement, left honestly OPEN (needs a human argument or proof assistant, not arithmetic)",
      "status": "Physical/geometric [O] (open constructive obligation)",
      "canonicalStatus": "Open [O]: raw RP seam collar -> genus-0 four-marked D4 boundary (one of the two genuine obligations)",
      "dependencies": "QGEO.TREE.01; QGEO.UNIFORM.01; ARCH.SPINE.01; v177_seam_marking_kernel.py",
      "scripts": [
        "v177_seam_marking_kernel.py"
      ],
      "script": "v177_seam_marking_kernel.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 256
    },
    {
      "id": "QGEO.KERNEL.01",
      "claim": "THE SEAM CALDERON-KERNEL OBLIGATION (genuine open obligation 2 of 2; the structural residual of the theory, NOT a finite computation): the RAW RP seam Calderon operator must equal the mu4-equivariant free c=8 gapped one-particle contraction on that boundary AS AN OPERATOR (C_Sigma = U^{-1} C_mu4 U), not merely as the same spectral list. Rails already laid: the DtN/Calderon symbol is universally |k| (free chiral dispersion, v156); a scalar seam datum exists iff the Calderon involution is sheet-odd (v110); the free c=8 fixed point is isolated with no relevant/marginal drift (v158); mu4-equivariance diagonalises the transfer in the cusp basis with weights {0,1/3,2/3} => spectrum {1,(2/3)^6,(1/3)^6} (v162); the fixed space is the rank-8 Calderon polarisation K_Sigma (v113). Proof route: conformal collar normal form (same principal Calderon symbol as the standard mu4 collar) + sheet-odd selection + no-lower-order-drift (c=8 rigidity) + mu4 diagonalisation + rank-8 identification. Given QGEO.KERNEL.01, the net side closes via QGEO.NET.01 (v154/v175, [E]). The operator identity from RAW seam data is NOT a finite computation; left honestly OPEN",
      "status": "Physical/operator [O] (open constructive obligation)",
      "canonicalStatus": "Open [O]: raw RP seam Calderon kernel = the mu4-equivariant free gapped contraction as an operator (one of the two genuine obligations)",
      "dependencies": "QGEO.TREE.01; QGEO.NET.01; GATE.METRIC.13; GATE.METRIC.18; v110; v113; v156_seam_net_construction.py; v158_fixed_point_stable.py; v162_seam_transport_identification.py",
      "scripts": [
        "v177_seam_marking_kernel.py"
      ],
      "script": "v177_seam_marking_kernel.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 257
    },
    {
      "id": "QGEO.REDUCE.01",
      "claim": "DEEPER REDUCTION OF THE TWO OBLIGATIONS (an honest attempt at QGEO.MARKS.01 + QGEO.KERNEL.01; NEITHER is closed -- they are genuine constructive-geometry/AQFT statements, not finite computations -- but each finite/algebraic CORE is now closed [E] and each obligation is reduced to ONE sharper irreducible premise [O]): (1) MARKS FINITE CORE [E] -- given a genus-0 double with an order-4 clock rho (conjugate to z->iz, fixed points {0,infinity}) and sheet reflection tau, the parabolic marks are a generic rho-orbit {1,i,-1,-i}=mu4 (|mu4|=4=e1(a), NOT the fixed points), DISTINCT iff b1=(#marks)-1=3=N_fam (any collision drops b1 below N_fam, excluded by the family count), and tau:z->1/z gives tau.rho.tau=rho^{-1} and permutes the marks => <rho,tau>=D4 of order 8, faithful; so MARKS reduces from 'produce the whole marked D4 boundary' to the SINGLE topological/clock premise 'the seam double is genus-0 and the seam clock is an order-4 conformal automorphism whose marks form one orbit'. (2) KERNEL FINITE CORE [E] (dissolves the spectrum-vs-operator worry on the finite block) -- on the 3-dim H^1 character space the mu4 characters (weights (1,2,3) -> i,-1,-i) are DISTINCT, so by Schur a mu4-equivariant operator is forced DIAGONAL (commutant of distinct-eigenvalue U = diagonal algebra) and is COMPLETELY DETERMINED by its eigenvalue per character; hence two such operators with the same forced transfer spectrum {1,(2/3)^6,(1/3)^6} (v162) and the unique residue-normalised basis Phi (v177) are EQUAL as operators, not merely isospectral -- the 'C_Sigma=U^{-1}C_mu4 U vs same spectral list' distinction VANISHES on the multiplicity-free H^1 block; so KERNEL reduces from the full operator identity to the SINGLE full-operator-reduction premise 'the full seam Calderon DtN on L^2 reduces to this 3-dim gapped mu4-moving transfer block plus the fixed rank-8 Calderon polarisation K_Sigma' (principal symbol universally |k|, Lee-Uhlmann/v156; c=8 fixed point isolated, no relevant/marginal drift, v158). NET (honest): neither obligation closed; both sharpened to milder premises, finite cores [E]; the structural residual of the whole theory is exactly {the topological clock premise, the full-operator-reduction premise}",
      "status": "Deeper reduction [E] (finite cores) + [O] (the two sharper irreducible premises)",
      "canonicalStatus": "Deeper reduction [E]/[O]: MARKS + KERNEL finite cores closed; each reduced to one sharper premise (topological clock / full-operator reduction); neither obligation closed",
      "dependencies": "QGEO.MARKS.01; QGEO.KERNEL.01; QGEO.TREE.01; v156_seam_net_construction.py; v158_fixed_point_stable.py; v162_seam_transport_identification.py; v177_seam_marking_kernel.py",
      "scripts": [
        "v178_marks_kernel_attempt.py"
      ],
      "script": "v178_marks_kernel_attempt.py",
      "location": "tfpt_research_contracts",
      "externalData": "Lee-Uhlmann 1989 (DtN principal symbol)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 258
    },
    {
      "id": "QGEO.CONF.01",
      "claim": "THE SEAM CONFORMAL-REALISATION PREMISE -- the SINGLE structural residual of the whole theory (the two obligations QGEO.MARKS.01 + QGEO.KERNEL.01 collapse onto it; honest unification, the premise itself stays OPEN). The two residual premises left by v178 are NOT independent: (A) MARKS's residual decomposes into three pieces, two of which are already-established TFPT facts and the third forced -- (1) GENUS-0 IS P1: the one-sided Gauss-Bonnet normalisation c3 = 1/(|Z2|*2pi*chi(S^2)) = 1/(2*2pi*2) = 1/(8pi) uses chi(S^2)=2 (TOPOLOGICAL), and chi=2-2g => g=0, so 'genus-0' IS the same chi=2 that supplies the '8' in c3 (v58/v73/SEAM.KHALF.01) -- not a new assumption but P1 itself; (2) ORDER-4 CLOCK IS THE CARRIER: the seam clock is the Coxeter element of the carrier monodromy W(A3)=S4 (v117), order h(A3)=3+1=4=|mu4| -- forced by P2, not assumed; (3) MARKS=ONE ORBIT is FORCED: the parabolic marks are not the two elliptic fixed points {0,infinity} of the order-4 rotation, which acts freely off them, so a free order-4 orbit has exactly 4=e1(a) points; so MARKS's residual is ONLY 'the carrier clock acts conformally (Moebius) on the genus-0 seam double'. (B) KERNEL's residual then FOLLOWS from that same geometry by established theorems: DtN principal symbol universally |k| (Lee-Uhlmann, v156) => the rank-8 (=c=rank E8) continuous 'fixed' polarisation K_Sigma; the c=8 holomorphic fixed point isolated, no relevant/marginal drift (v158); on a finite-volume cusped surface the spectral decomposition (Eisenstein continuous + finite residual) puts the residual/cusp-form data in H^1(P^1\\\\mu4)=C^3=N_fam = the 3-dim gapped mu4-moving transfer block, Schur-diagonal with the forced weights {1,(2/3)^6,(1/3)^6} (v162/v178). (C) UNIFICATION: both reduce to QGEO.CONF.01 -- 'the raw RP seam normal double is conformally (P^1,mu4) with the carrier clock as the order-4 Moebius automorphism z->iz'. Given it, genus-0 (=P1), order-4 (=h(A3)), marks=orbit (forced), the cohomology grading (v177), the DtN block + rank-8 polarisation (v178) and the (E8)1 net (v154/v175) all follow. [O] IRREDUCIBLE: QGEO.CONF.01 is a constructive-geometry statement (the conformal class of the raw seam double IS the cusped sphere with the carrier clock as a Moebius map), NOT a finite computation; it is the single structural residual of the theory, left honestly OPEN",
      "status": "Geometric realisation premise [O] (the single residual; component reductions [I] established)",
      "canonicalStatus": "Geometric [O]: the SINGLE residual is the seam conformal realisation (P^1,mu4) with the carrier clock as the order-4 Moebius map; MARKS+KERNEL both reduce to it",
      "dependencies": "QGEO.MARKS.01; QGEO.KERNEL.01; QGEO.REDUCE.01; AX.P1.01; SEAM.KHALF.01; v58_seam_horizon_chain.py; v73_k_c3_half.py; v117_monodromy_weyl_a3.py; v156_seam_net_construction.py; v158_fixed_point_stable.py",
      "scripts": [
        "v179_conformal_realisation.py"
      ],
      "script": "v179_conformal_realisation.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "Lee-Uhlmann 1989; Selberg/Eisenstein spectral theory (cusped surfaces)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 259
    },
    {
      "id": "QGEO.ISO.01",
      "claim": "THE SEAM-ISOMETRY PREMISE -- the milder, non-circular form of the single structural residual (QGEO.CONF.01 is REPLACED by it via three classical theorems; honest reduction, still OPEN): 'the carrier clock is an order-4 orientation-preserving ISOMETRY of the RP seam metric'. The chain: (1) UNIFORMISATION (Poincare-Koebe) -- a genus-0 Riemann surface is conformally P^1, and the RP seam metric induces a conformal structure on the genus-0 double (chi(S^2)=2 => g=0, P1/v179), so the double IS conformally P^1; the target P^1 is PRODUCED, not assumed; (2) KEREKJARTO (1919; Constantin-Kolev 2003) -- a finite-order orientation-preserving homeomorphism of S^2 is topologically conjugate to a rotation, so the order-4 clock (h(A3)=4, v179) is conjugate to z->iz even topologically; (3) ISOMETRY=>CONFORMAL=>MOEBIUS -- an orientation-preserving isometry of a Riemannian surface preserves angles hence is holomorphic, and Aut(P^1)=PSL(2,C), so an isometric order-4 clock is a Moebius map; (4) ORDER-4 MOEBIUS = z->iz (verified: projective order 4, fixed points {0,infinity}, multiplier i a primitive 4th root, elliptic, free orbit {1,i,-1,-i}=mu4 matching v179). Hence (genus-0 from P1) + (order-4 isometric clock) => the double is conformally P^1 with the clock z->iz, and with v179 (marks = the free orbit = mu4) the full QGEO.CONF.01 follows. So the conformal-class identification is NO LONGER the residual: it is IMPLIED by QGEO.ISO.01. RESIDUAL [O]: QGEO.ISO.01 is milder than QGEO.CONF.01 -- it does NOT name P^1/mu4 (uniformisation produces them) and is structural (the seam transport is metric-compatible, a holonomy preserving the RP inner product, the natural meaning of an RP seam clock). The surface-metric realisation of that isometry from the RAW seam is still a constructive-geometry statement, left honestly OPEN -- the single, now milder, structural residual of the whole theory",
      "status": "Geometric isometry premise [O] (milder residual; the conformal-class identification is now implied) + [E] (the three cited reductions)",
      "canonicalStatus": "Geometric [O]: QGEO.CONF.01 replaced by the milder seam-ISOMETRY premise (the clock is an order-4 isometry); conformal realisation now implied via uniformisation + Kerekjarto + finite-order Moebius",
      "dependencies": "QGEO.CONF.01; QGEO.MARKS.01; AX.P1.01; v117_monodromy_weyl_a3.py; v179_conformal_realisation.py",
      "scripts": [
        "v180_clock_is_mobius.py"
      ],
      "script": "v180_clock_is_mobius.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "Poincare-Koebe uniformisation; Kerekjarto 1919 / Constantin-Kolev 2003; Aut(P^1)=PSL(2,C)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 260
    },
    {
      "id": "QGEO.SYM.01",
      "claim": "A COROLLARY OF SEAM.EQUIV.01 (no independent open premise; reclassified 2026-06-22, v335): per v323/QGEO.BW.01 it is downstream of SEAM.EQUIV.01 + a rotation-invariant vacuum, and the rotation-invariant vacuum is a conformal-NET AXIOM (a chiral net's Moebius-covariant vacuum is the unique invariant positive-energy vector; rotations U(1) < Moebius), so SEAM.EQUIV.01 => QGEO.SYM.01 with NO extra premise -- machine-pinned in Lean as qgeoSymIsCorollary (FORM.QGEO.BW.01). It is therefore NOT a second open bedrock item; the conformal-symmetry/deck face below is its (now corollary) content: the isometry premise QGEO.ISO.01 (v180) is weakened one more genuine step to a CONFORMAL-symmetry premise. (1) EQUIVARIANT UNIFORMISATION (Nielsen realisation; for cyclic groups Kerekjarto + uniformisation): a finite-order orientation-preserving action on a genus-0 surface preserves a complex structure for which it acts by Moebius maps, so a CONFORMAL automorphism -- NOT a full isometry -- already suffices to put the order-4 clock in the form z->iz (the order-4 rotation class is unique up to inverse). (2) STRICT WEAKENING: every isometry is conformal (scale 1) while a conformal map allows any positive scale, so 'conformal automorphism' is strictly weaker than 'isometry' yet (by (1)) still implies the full realisation; hence QGEO.ISO.01 is replaced by the weaker QGEO.SYM.01 'the carrier clock is a conformal automorphism of the seam's conformal structure'. (3) THE CLOCK IS THE mu4 DECK: it is the Coxeter element of the carrier monodromy W(A3)=S4 (v117), order h(A3)=4=|mu4|; a deck transformation preserves the pulled-back conformal structure by definition, so 'the clock is a conformal automorphism' is the DEFINITIONAL property of the mu4 deck once the seam conformal structure is taken to be the mu4-deck structure. (4) BEDROCK [O]: QGEO.SYM.01 reduces to the single identification 'the seam's conformal structure IS the mu4-deck structure of the carrier clock' (the carrier mu4 glue is the conformal monodromy of the seam) -- a definitional/physical statement tying the carrier (P2) to the seam's conformal geometry, NOT a finite computation and NOT reducible further without relabeling. Per v323/QGEO.BW.01 it is DOWNSTREAM of the keystone SEAM.EQUIV.01 (+ a rotation-invariant vacuum, Bisognano-Wichmann/Hislop-Longo), NOT an independent residual: it is the definitional/symmetry face of the single named open arrow, left honestly OPEN only modulo SEAM.EQUIV.01. The reduction chain REALIZE(v175)->theorem(v176)->MARKS+KERNEL(v177)->finite cores(v178)->CONF.01(v179)->ISO.01(v180)->SYM.01(v181) ends here, and the keystone SEAM.EQUIV.01 is the single named open arrow that closes both",
      "status": "Conformal-symmetry/deck face of SEAM.EQUIV.01 [O] (downstream per v323) + [E] (equivariant uniformisation + the weakening)",
      "canonicalStatus": "Face of keystone [O]: QGEO.SYM.01 = the definitional conformal-deck face of SEAM.EQUIV.01; per v323/QGEO.BW.01 downstream of SEAM.EQUIV.01 + rotation-invariant vacuum, not an independent residual",
      "dependencies": "SEAM.EQUIV.01; QGEO.BW.01; QGEO.ISO.01; QGEO.CONF.01; AX.P2.01; v117_monodromy_weyl_a3.py; v180_clock_is_mobius.py",
      "scripts": [
        "v181_clock_is_conformal_symmetry.py"
      ],
      "script": "v181_clock_is_conformal_symmetry.py",
      "location": "tfpt_research_contracts;origin_theory",
      "externalData": "Nielsen realisation (finite cyclic on genus-0); Kerekjarto / Constantin-Kolev; Poincare-Koebe; Bisognano-Wichmann",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 261
    },
    {
      "id": "ARCH.RRCAR.01",
      "claim": "THE RIEMANN-ROCH CARRIER READING (external-review proposal 2026-06-14): (g_car, N_fam)=(5,3) are the two canonical invariants of ONE object, the four-point seam divisor D=mu4 on P^1. [E]: h^0(P^1,O(mu4))=deg+1=5 (function side) and rank H_1(P^1\\\\mu4)=deg-1=3 (cycle side, already QGEO.COHOM.01/v177); g_car=5=rank(D5) (carrier=so(10) half-spinor, rank D5+rank A3=5+3=8=rank E8, v1), N_fam=3=(dim S+ -1)/g_car. [C]: the identification 'carrier = the meromorphic seam-mode space H^0(P^1,O(mu4))' is a geometric reading (less number, more geometry), matched by rank(D5)=5, but the physical identification is conjectural and does NOT replace the over-determined g_car=5 (Pascal v2, bootstrap v6, Lean FORM.LADDER.01) -- it is a fourth, geometric reading; h^0 and rank H_1 are both fixed by deg=4 (one object, two invariants: a unification, not extra over-determination). Exact; Wolfram-mirrored",
      "status": "Identity [E] (RR arithmetic) + Physical [C] (carrier identification)",
      "canonicalStatus": "Identity [E] for h^0=5=rank(D5)=g_car and rankH1=3=N_fam from one mu4 divisor; [C] for the carrier=mode-space identification (does not replace the over-determined g_car=5)",
      "dependencies": "ARCH.CORE.01; AX.P2.01; QGEO.COHOM.01; E8.GLU.01; v177_seam_marking_kernel.py; v1_e8_glue.py",
      "scripts": [
        "v189_riemann_roch_carrier.py"
      ],
      "script": "v189_riemann_roch_carrier.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 262
    },
    {
      "id": "ARCH.RRCAR.02",
      "claim": "RIEMANN-ROCH CARRIER HARDENED FROM g_car TO D5 ITSELF (external-review point 7, 2026-06-14): v189 read g_car=5=h^0(P^1,O(mu4)) and matched rank(D5); this goes one step further -- the 5-dim carrier mode space C_car:=H^0(P^1,O(mu4)) generates the so(10)=D5 half-spinor by its EVEN CLIFFORD ALGEBRA. (1) [I] dim Lambda^even(C^5) = C(5,0)+C(5,2)+C(5,4) = 1+10+5 = 16 = 2^{g_car-1} = dim S^+ (the repo's dim_Splus). (2) [I] so(10)=D5 (rank 5) has Dirac spinor 2^5=32 and half-spinors 2^4=16 each; S^+ = Lambda^even(C_car), so the 5-dim mode space's Clifford module IS the D5 half-spinor. (3) [I] this closes the chain mu4 -> g_car -> D5 from ONE four-point divisor: cycle side rank H_1=deg-1=3=N_fam (A3); function side h^0=deg+1=5=g_car=dim C_car; Clifford spinor Lambda^even=16=D5 half-spinor=carrier -- D5 is the Clifford spinor of the Riemann-Roch carrier, not a separate input. (4) [C] reading the carrier as Cl^even(C_car) (not merely matching rank(D5)=5) is a geometric hardening, but 'carrier = Clifford spinor of the seam meromorphic mode space' is still a physical [C] step (as in v189); does NOT change the over-determined status of g_car=5. Exact; Wolfram-mirrored",
      "status": "Identity [I] (Clifford dim chain) + Physical [C] (carrier = mode-space spinor)",
      "canonicalStatus": "Identity [I]: Lambda^even(C_car=5-dim)=S^+=16=2^{g_car-1}=the D5 half-spinor, closing mu4->g_car->D5; [C] for the carrier=Clifford-spinor-of-the-mode-space identification (hardens v189, does not change over-determined g_car=5)",
      "dependencies": "ARCH.RRCAR.01; AX.P2.01; E8.GLU.01; v189_riemann_roch_carrier.py; v1_e8_glue.py",
      "scripts": [
        "v197_rr_carrier_clifford_d5.py"
      ],
      "script": "v197_rr_carrier_clifford_d5.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 263
    },
    {
      "id": "HOR.NARIAI.02",
      "claim": "THE ONE-LINE NARIAI ENTROPY BOUND (external-review proposal 2026-06-14): the SdS entropy ratio S_tot/S_dS = (x^2+1)/Phi3(x) (x=r_b/r_c; Phi3=x^2+x+1 the N_fam=3 cyclotomic; HOR.NARIAI.01) satisfies S_tot >= (2/3) S_dS with equality exactly at the Nariai merge x=1, via the perfect square 3(x^2+1)-2*Phi3(x)=(x-1)^2 >= 0 -- a variation-free proof of the entropy floor (the global minimum on x>0 is 2/3=|Z2|/N_fam). Consistent with d/dx=(x-1)(x+1)/Phi3^2 vanishing at x=1. Exact; Wolfram-mirrored",
      "status": "Identity [E] (exact variation-free entropy bound)",
      "canonicalStatus": "Identity [E]: S_tot/S_dS >= 2/3 via (x-1)^2 >= 0, equality at the Nariai merge",
      "dependencies": "HOR.NARIAI.01; v101_horizon_anchor.py; v8_horizon.py",
      "scripts": [
        "v190_nariai_entropy_bound.py"
      ],
      "script": "v190_nariai_entropy_bound.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 264
    },
    {
      "id": "HOR.BRANCHLINE.01",
      "claim": "THE UNIVERSAL BRANCH LINE (external-review proposal 2026-06-14; HONESTLY typed [C], an exact affine RELABELING, NOT a theorem): the map q=7/2+(N_fam^2/2)m carries the SdS branch points m=+-1/N_fam onto the flavor branch points {2,5}={|Z2|,g_car} (m=0 -> 7/2, the branch-divisor sum 7 over 2 / scalaron half-exponent). [E] arithmetic, but the map exists for ANY two pairs of points (two intervals are always affinely equivalent), so the slope N_fam^2/2 and midpoint 7/2 are DETERMINED by {-1/3,1/3}->{2,5}, NOT forced independent predictions; a decoy pair {3,4} admits an equally exact map (negative control). The genuine shared content -- the 2/3=|Z2|/N_fam ramification of the mass<->transport double cover -- is already established (FLAV.PENCIL.*, HOR.NARIAI.01). Recorded as a [C] structural alignment, never promoted to [E]",
      "status": "Physical/structural [C] (exact affine relabeling, not a theorem)",
      "canonicalStatus": "Structural [C]: an affine relabeling exhibiting the known 2/3 ramification; NOT an independent theorem (decoy pairs admit equally exact maps)",
      "dependencies": "HOR.NARIAI.01; HOR.NARIAI.02; FLAV.PENCIL.01; v101_horizon_anchor.py",
      "scripts": [
        "v191_universal_branch_line.py"
      ],
      "script": "v191_universal_branch_line.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 265
    },
    {
      "id": "QGEO.ENERGY.01",
      "claim": "THE ENERGY-CONSERVING-CLOCK REFORMULATION (external-review proposal 2026-06-14; HONESTLY typed: RELOCATES the bedrock, does NOT eliminate it): the proposed premise QGEO.ENERGY.01 (rho^* Lambda_Sigma rho = Lambda_Sigma -- the carrier clock preserves the Calderon/DtN energy form of the RP seam double) feeds the existing downstream chain (conformality -> Moebius -> z|->iz -> mu4 -> QGEO.SYM.01, the v180/v181 reduction). The new upstream link (energy invariance => conformality in 2D) is checkable on the DtN operator Lambda=|k| (the rank-8 Calderon polarisation in the net construction), unlike the abstract 'is the conformal deck'. BUT 'rho preserves Lambda' is plausibly EQUIVALENT to 'rho is the conformal deck' (QGEO.SYM.01), so this is a sharper, more falsifiable RESTATEMENT, NOT a theorem, unless rho is defined independently (from the carrier algebra) and the invariance independently proven (open). The bedrock stays [O]; same pattern as v179->v181",
      "status": "Geometric premise [O] (operator-identity restatement of QGEO.SYM.01; relocates, not eliminates)",
      "canonicalStatus": "Geometric [O]: QGEO.ENERGY.01 (rho preserves the DtN energy form) is a sharper operator-identity restatement of QGEO.SYM.01 -- relocates the bedrock, does not eliminate it",
      "dependencies": "QGEO.SYM.01; QGEO.ISO.01; v181_clock_is_conformal_symmetry.py; v180_clock_is_mobius.py",
      "scripts": [
        "v192_qgeo_energy_clock.py"
      ],
      "script": "v192_qgeo_energy_clock.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 266
    },
    {
      "id": "QGEO.ENERGY.02",
      "claim": "THE ENERGY-COMMUTATOR CONTRACT (external-review proposal 2026-06-14; the NON-CIRCULAR sharpening of QGEO.ENERGY.01): the proposed premise is the operator identity [rho, Lambda_Sigma]=0 on the rank-8 Calderon polarisation -- rho the carrier clock, Lambda_Sigma the Dirichlet-to-Neumann (Calderon) energy form of the RAW RP seam double. Three sub-claims make it non-circular: (1) [E] rho is defined from the CARRIER algebra (the Coxeter element of W(A3)=S4, order 4=|mu4|, v117/HOR.CLOCK), NOT imported from the seam geometry; (2) [O] THE NON-CIRCULARITY HINGE -- Lambda_Sigma must be the DtN of the RAW RP seam (definable from RP data alone), NOT the mu4 normal-form operator; if 'raw seam' requires P^1\\\\mu4 assumed it is circular, if from RP data alone the commutator is a genuine rigidity statement (this is the ONE honestly open premise); (3) [E] [rho,Lambda_Sigma]=0 forces the mu4 character grading (1,2,3) on H^1, already established (QGEO.COHOM.01, eigenforms z^{k-1}dz/(z^4-1), v177). EXACT [I] consistency anchor: the induced EH coefficient k=(ln m)/(12 pi) (v150) reproduces c3/2=1/(16 pi) iff ln m=q(A3)=3/4=N_fam/|mu4| (FAMILY glue norm), and q(D5)=5/4=g_car/|mu4| (carrier) gives the WRONG k (off by g_car/N_fam=5/3) -- so the energy form MUST reproduce the family norm, NOT the carrier norm (gravity is family-geometry-induced). VERDICT: a PROOF TARGET, strictly sharper and non-circular IFF sub-claim 2 holds; if proven, QGEO.SYM.01 turns from definitional bedrock into an ENERGY-RIGIDITY THEOREM. Not a proof -- the bedrock stays [O]. Python-only (the q(A3) rigidity is exact and Wolfram-mirrored)",
      "status": "Geometric premise [O] (energy-commutator proof target; sharper, non-circular IFF sub-claim 2)",
      "canonicalStatus": "Geometric [O]: QGEO.ENERGY.02 = [rho,Lambda_Sigma]=0 (energy commutator); non-circular proof target for QGEO.SYM.01 IFF the raw-seam DtN (sub-claim 2) is RP-definable; exact rider: the EH coefficient selects q(A3)=3/4 (family), not q(D5)=5/4 (carrier)",
      "dependencies": "QGEO.ENERGY.01; QGEO.SYM.01; QGEO.COHOM.01; HOR.CLOCK.01; v150_replica_eh_model.py; v192_qgeo_energy_clock.py",
      "scripts": [
        "v193_qgeo_energy_commutator.py"
      ],
      "script": "v193_qgeo_energy_commutator.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 267
    },
    {
      "id": "QGEO.DTN.01",
      "claim": "SUBCLAIM 2 OF QGEO.ENERGY.02 TACKLED -- 'the raw RP-seam DtN is RP-definable' (external-review next-lever 2026-06-14): a REDUCTION, not a closure. (1) FINITE CORE [E]: rho:z|->iz on H^1(P^1\\\\mu4)=<w1,w2,w3>, wk=z^{k-1}dz/(z^4-1), acts as rho* wk = i^k wk (characters i,-1,-i DISTINCT; (iz)^4-1=z^4-1 invariant) -- verified symbolically; distinct eigenvalues => any operator commuting with rho is DIAGONAL on H^1, and the harmonic structure IS rho-equivariant there (v177/COHOM core) => [rho,Lambda_Sigma]=0 on H^1 is [E]. (2) HINGE MET [I]: the DtN/transfer operator is RP-canonical -- Osterwalder-Schrader reconstruction yields T=e^{-H} from the RP state + reflection alone (no coordinates; v54/SEAM.HORIZON.01, tfpt_2 (4)) -- on a quasi-free seam state (v155: the boundary marginal of a Gaussian bulk is the compression P Gamma P, a contraction). So Lambda_Sigma is RP-definable WITHOUT choosing P^1\\\\mu4: subclaim 2's non-circularity hinge is essentially met at the canonical level. (3) RESIDUAL RELOCATES [O]: [rho,Lambda_Sigma]=0 on the FULL boundary L^2 (beyond the 3-dim H^1) <=> the quasi-free seam state's modular flow is GEOMETRIC (Bisognano-Wichmann) -- a named, standard (hard) AQFT property, strictly sharper than 'the seam IS P^1\\\\mu4 (definitional)'. CIRCULARITY WATCH: BW normally presupposes a conformal/Poincare-covariant net (part of what is derived), so the BW step must be established INTRINSICALLY (modular geometricity from RP+quasi-free alone) or it re-circulates -- that intrinsic geometricity is the irreducible analytic content. VERDICT: subclaim 2's hinge met + finite core [E] => QGEO.SYM.01 REDUCES from a definitional postulate to 'intrinsic BW geometric modular covariance of the quasi-free seam state' -- one notch sharper, STILL [O]; the last structural fog point sharpens, does NOT close to [E]. Python-only (H^1 commutation is exact Moebius-symbolic like v177; reduction is structural)",
      "status": "Geometric premise [O] (reduction: subclaim-2 hinge met, finite core [E], bedrock -> BW modular)",
      "canonicalStatus": "Geometric [O]: subclaim 2 (raw-seam DtN RP-definability) met at the canonical OS level + finite core [E]; QGEO.SYM.01 reduces to intrinsic Bisognano-Wichmann modular covariance of the quasi-free seam state -- sharper, still [O], not closed",
      "dependencies": "QGEO.ENERGY.02; QGEO.SYM.01; QGEO.TREE.01; SEAM.HORIZON.01; v177_seam_marking_kernel.py; v155_quasifree_boundary.py; v54_seam_horizon_keystones.py",
      "scripts": [
        "v194_raw_seam_dtn_rp.py"
      ],
      "script": "v194_raw_seam_dtn_rp.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 268
    },
    {
      "id": "QGEO.MARKS.02",
      "claim": "LEFSCHETZ/CHARACTER DERIVATION OF THE FOUR SEAM MARKS (external-review path 2/priority 2, 2026-06-14): instead of POSITING D4 boundary marks (the old QGEO.MARKS.01 finite core), the marks are FORCED to be a free mu4 orbit by orbit-counting plus the H^1 character. (1) [E] rho:z|->iz has order 4 and fixes EXACTLY {0,infinity}; rho^2:z|->-z also fixes only {0,infinity}; so any point off the fixed locus has a FREE order-4 orbit {z,iz,-z,-iz}. (2) [E] NO TRIVIAL CHARACTER: the carrier datum H^1=chi_1+chi_2+chi_3 (the A3 exponents (1,2,3)=Spec Q_+, v177/COHOM) has no trivial component, so no puncture sits at a rho-fixed point. (3) [E] rank H_1(P^1 minus n)=n-1=3 => n=4 punctures, all off the fixed locus => exactly ONE free order-4 orbit => the marks are a free mu4 orbit, Moebius-equivalent to {1,i,-1,-i} (cross-ratio 2). (4) [E] LEFSCHETZ CONSISTENCY: Tr(rho|H^1)=i+i^2+i^3=-1 and on closed P^1 L(rho)=Tr(H^0)+Tr(H^2)=1+1=2=#fixed points. RESIDUAL [O]: the marks no longer rest on a geometric posit (D4 marks) but on the milder representation premise 'the raw seam H^1 carries the A3-exponent rep chi_1+chi_2+chi_3' -- a carrier-algebra statement (A3 exponents are algebraic, independent of the seam geometry). So QGEO.MARKS reduces from geometry to an H^1-rep premise; less circular, still [O]. Exact; Wolfram-mirrored",
      "status": "Identity [E] (orbit/character core) + premise [O] (H^1-rep)",
      "canonicalStatus": "Identity [E]: orbit-counting (order-4 rho fixes {0,inf}; no trivial H^1 char; rank 3 => 4 punctures) forces the marks to a free mu4 orbit; residual [O] relocates from 'seam is P^1\\\\mu4' to 'H^1 carries the A3-exponent rep' -- less circular",
      "dependencies": "QGEO.MARKS.01; QGEO.TREE.01; QGEO.SYM.01; v177_seam_marking_kernel.py; v178_marks_kernel_attempt.py",
      "scripts": [
        "v195_marks_lefschetz_character.py"
      ],
      "script": "v195_marks_lefschetz_character.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 269
    },
    {
      "id": "QGEO.VARI.01",
      "claim": "THE E_fail VARIATIONAL FUNCTIONAL (external-review path 1, 2026-06-14): a concrete, numerically-testable sharpening of the v194/QGEO.ENERGY.02 target. E_fail(Sigma,rho) = ||[rho,Lambda_Sigma]||^2_HS + ||rho^4-1||^2 + ||Theta rho Theta - rho^{-1}||^2, with E_fail=0 <=> the QGEO.SYM.01 conditions hold (energy commutator + order-4 clock + RP-reflection twist). (1) [E] on the finite H^1 block (rho=diag(i,-1,-i), Lambda diagonal=mu4-equivariant, Theta=complex conjugation) all three terms vanish: [rho,Lambda]=0, rho^4=I, conj(rho)=rho^{-1} => E_fail=0. (2) [N] a mu4-breaking off-diagonal Lambda gives [rho,Lambda]!=0 and a non-order-4 rho gives rho^4!=I, both raising E_fail>0 => the mu4 configuration is a true minimiser (verified numerically). (3) [I] the zero locus is exactly the seam-deck conditions of QGEO.SYM.01. (4) [O] minimising E_fail over the FULL raw-seam DtN (beyond the finite H^1) is exactly the v194 statement that the quasi-free seam state's modular flow is geometric (Bisognano-Wichmann) -- so E_fail gives a discretisable variational handle on the open bedrock, a TARGET not a closure. Python-only (finite-block linear algebra; finite vanishing exact, Wolfram-mirrored)",
      "status": "Numerical/identity [N]/[I] (finite vanishing exact) + target [O] (full-operator min = BW)",
      "canonicalStatus": "Numerical/identity [N]/[I]: E_fail=0 exactly at the mu4 config and >0 for perturbations (mu4 minimiser); full-operator minimisation = the v194 Bisognano-Wichmann residual [O] -- a concrete discretisable handle on the bedrock",
      "dependencies": "QGEO.ENERGY.02; QGEO.DTN.01; QGEO.SYM.01; v193_qgeo_energy_commutator.py; v194_raw_seam_dtn_rp.py",
      "scripts": [
        "v196_seam_energy_functional.py"
      ],
      "script": "v196_seam_energy_functional.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 270
    },
    {
      "id": "QGEO.MODULAR.01",
      "claim": "CRACKING THE BEDROCK TO A STATE-INVARIANCE (2026-06-14): the v194 residual -- the full-L^2 commutator [rho,Lambda_Sigma]=0, framed as Bisognano-Wichmann geometric modular flow with a circularity worry -- is (a) made EXACT at the principal-symbol level and (b) reduced via Tomita-Takesaki to a clean state-invariance, removing the circularity. (1) PRINCIPAL SYMBOL [E]: the DtN principal symbol |k|=sqrt(-d^2/dtheta^2) (Lee-Uhlmann, v156) is diag(|n|) and the clock rho:z|->iz (rotation by pi/2) is diag(i^n) in the Fourier basis e^{in theta} -- both DIAGONAL => [rho,|k|]=0 EXACTLY on ALL of L^2 (verified on 33 modes), not just the 3-dim H^1; the leading-order commutation is FREE on the whole boundary, so the open part was never the principal symbol. (2) TOMITA-TAKESAKI [E]: the modular operator Delta_omega is INTRINSIC to (M,Omega); a state-preserving algebra symmetry (rho Omega=Omega, rho M rho*=M) commutes with Delta, hence with the modular Hamiltonian log Delta ~ Lambda_Sigma. So [rho,Lambda_Sigma]=0 FOLLOWS from omega o rho = omega, needing ONLY the state + reflection (which RP supplies) and NO conformal/Poincare covariance -- this REMOVES the v194 Bisognano-Wichmann circularity worry. (3) FINITE BLOCK [E]: on H^1 the covariance is mu4-equivariant (rho* w_k = i^k w_k, v177), so omega o rho = omega already holds there; with (1) the open part is neither the principal symbol nor the finite block. (4) IRREDUCIBLE RESIDUAL [O]: the FULL raw quasi-free seam state is rho-invariant (omega o rho = omega) <= the Gaussian bulk MEASURE is rho(deck)-invariant = the mu4 deck is a symmetry of the raw seam -- the maximally-operational, non-circular form of QGEO.SYM.01 (a STATE/MEASURE invariance, checkable, not a geometric or modular-flow statement). A foundational symmetry postulate cannot be derived from nothing (like c=const); it is now in its sharpest falsifiable form. VERDICT: bedrock cracked to omega o rho = omega -- principal symbol EXACT, Tomita-Takesaki NON-CIRCULAR, finite block invariant; NOT a full closure but the sharpest form, circularity gone. Python-only (principal-symbol Fourier commutation exact, Wolfram-mirrored; Tomita-Takesaki is a standard AQFT theorem)",
      "status": "Geometric premise [O] (bedrock reduced to a non-circular state-invariance; principal symbol exact)",
      "canonicalStatus": "Geometric [O]: bedrock cracked to omega o rho = omega (raw seam state mu4-invariant) -- principal symbol |k| commutes exactly (Fourier diag), Tomita-Takesaki removes the BW circularity, finite block holds; the irreducible residual is the mu4-symmetry of the raw seam measure (operational QGEO.SYM.01), still [O]",
      "dependencies": "QGEO.DTN.01; QGEO.ENERGY.02; QGEO.SYM.01; QGEO.VARI.01; v194_raw_seam_dtn_rp.py; v177_seam_marking_kernel.py",
      "scripts": [
        "v198_modular_commutator_reduction.py"
      ],
      "script": "v198_modular_commutator_reduction.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 271
    },
    {
      "id": "QGEO.STATE.01",
      "claim": "ATTACKING omega o rho = omega DIRECTLY ON THE RAW SEAM (work package A, external-review priority 1, 2026-06-14): a further localising reduction, not a closure. (1) SETUP NON-CIRCULAR [E]: rho is the carrier clock = the Coxeter element of W(A3)=S4 (order 4, rho^4=1, v117), an automorphism of the RAW CAR algebra (16 Majoranas = the carrier net, v156/v175); both rho and the algebra are defined from the CARRIER side with NO seam geometry, so 'rho preserves the seam state' is a genuine, non-circular question. (2) GROUND-STATE REDUCTION [E]: for a quasi-free RP state the covariance is the positive-frequency projection C_Sigma=1/2(1+sgn H_1) of the one-particle Hamiltonian H_1 (~ the DtN), so omega o rho=omega <=> rho preserves C_Sigma <=> [rho,H_1]=0. (3) CHARACTER-BLOCK-DIAGONAL CRITERION [E]: rho has order 4, so the one-particle space splits into the four mu4-character classes {n=r mod 4}; [rho,H_1]=0 <=> H_1 is block-diagonal in these classes -- verified numerically (a character-block-diagonal H commutes, an off-character entry breaks it). (4) [E] the principal symbol |k|=diag(|n|) is diagonal, hence trivially character-block-diagonal, so [rho,|k|]=0 exactly (v198). (5) SHARPENED RESIDUAL [O]: what remains is purely 'the BOUNDED sub-principal symbol of the raw seam DtN has NO off-character (mod 4) matrix elements' -- a bounded, finite-character-class operator condition, not a diffuse geometry; holds on the finite H^1 block (mu4-eigenforms, v177), open on the continuum. VERDICT: omega o rho=omega is confirmed non-circular and reduced to a bounded character-block-diagonal residual; NOT a closure (the foundational symmetry persists), but the sharpest typed form. Python-only",
      "status": "Identity [E] (non-circular setup + reduction) + premise [O] (bounded sub-principal residual)",
      "canonicalStatus": "Identity/premise [E]/[O]: omega o rho=omega is non-circular (rho a carrier automorphism, raw CAR algebra) and reduces to [rho,H_1]=0 <=> H_1 is mu4-character-block-diagonal; residual = the bounded sub-principal symbol has no off-character matrix elements; not a closure",
      "dependencies": "QGEO.MODULAR.01; QGEO.SYM.01; QGEO.DTN.01; v198_modular_commutator_reduction.py; v177_seam_marking_kernel.py",
      "scripts": [
        "v199_seam_state_invariance.py"
      ],
      "script": "v199_seam_state_invariance.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 272
    },
    {
      "id": "QGEO.SUBPRIN.01",
      "claim": "THE v199 BEDROCK RESIDUAL REDUCED ONE GENUINE STEP FURTHER (2026-06-15): block-diagonality of the sub-principal symbol is NOT an independent postulate. (1) [E] the seam DtN is Lambda=|k|+M_f, M_f = multiplication by a curvature function f(theta) on the seam circle (standard Steklov sub-principal structure), so <n|M_f|n'>=f_{n-n'}. (2) [E] M_f is mu4-character-block-diagonal <=> f has Fourier support only on modes ≡0 (mod 4) <=> f is Z4-invariant (the principal |k| already commutes, v198). (3) [E] a mu4-MARK SUM f(theta)=sum_{j=0}^3 g(theta-2pi j/4) has f_m=4 g_m [m≡0 mod 4] for ANY profile g, automatically Z4-invariant, so (v195 marks=mu4 orbit) FORCES the sub-principal symbol block-diagonal. (4) [E] negative controls: 3 equally-spaced marks (Z3) and 4 generic unequal marks both break the commutator -- specific to mu4. (5) [O] NEW RESIDUAL: omega o rho=omega reduces to 'the DtN sub-principal symbol is MARK-LOCAL (seam flat away from the mu4 marks = the conformal-deck structure, v177)'; given it, (v195 marks=mu4 orbit) => Z4-invariant sub-principal => block-diagonal => [rho,H_1]=0 => omega o rho=omega. A genuine reduction; the residual is now structurally definitional (conformal-deck flatness), the narrowest form yet of QGEO.SYM.01. Python-only (exact mark-sum Fourier identity + block-diagonal criterion; Wolfram-mirrored)",
      "status": "Identity [E] (block-diagonality = image of the mu4 marks) + premise [O] (mark-local DtN residual)",
      "canonicalStatus": "Identity/premise [E]/[O]: the v199 sub-principal block-diagonality is FORCED by the v195 mu4-mark orbit under a mark-local DtN (a mark-sum over mu4 points is Z4-invariant => Fourier support on modes ≡0 mod 4 => block-diagonal); residual reduces to DtN mark-locality (conformal-deck flatness), the narrowest form of QGEO.SYM.01; not a closure",
      "dependencies": "QGEO.STATE.01; QGEO.MARKS.02; QGEO.SYM.01; v199_seam_state_invariance.py; v195_marks_lefschetz_character.py; v198_modular_commutator_reduction.py",
      "scripts": [
        "v201_seam_subprincipal_marks.py"
      ],
      "script": "v201_seam_subprincipal_marks.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 273
    },
    {
      "id": "QGEO.PILLOW.01",
      "claim": "PILLOWCASE REDUCTION OF QGEO.SYM.01 (2026-06-15): the two separately-tracked open QGEO residuals -- QGEO.ISO.01 (v180; the carrier clock is an order-4 ISOMETRY of the seam metric) and QGEO.SUBPRIN.01 (v201; the DtN sub-principal symbol is MARK-LOCAL, the seam flat away from the mu4 marks) -- are ONE statement: 'the seam carries the uniformising flat orbifold (pillowcase) metric'; and the order-4 clock is DERIVED from cross-ratio 2 (v168), not assumed. (1) [E] EUCLIDEAN ORBIFOLD: chi_orb(S^2 with four order-2 cone points)=2-4(1-1/2)=0 => flat (Troyanov cone-metric uniformisation, unique up to scale); Gauss-Bonnet four cone deficits pi sum to 4pi=2pi chi(S^2) => int K_smooth=0 (flat away from the marks = the v201 conformal-deck residual). (2) [E] NEW LINK cross-ratio 2 => j=1728: j(lambda)=256(l^2-l+1)^3/(l^2(l-1)^2); j(2)=1728 and all six harmonic cross-ratios {2,-1,1/2} give 1728, so the square modulus (hence the order-4 clock) is a CONSEQUENCE of the cross-ratio 2 that v168 already proved. (3) [E]/[L] j=1728 <=> tau=i (square torus) <=> CM by Z[i] <=> Aut=Z/4; the only other exceptional modulus j=0 (tau=rho) gives Z/6, generic moduli only Z/2. (4) [E] explicit order-4 CM: on y^2=x^3-x the map (x,y)->(-x,iy) negates the defining polynomial (same zero locus) and has order 4 = the z->iz isometry. (5) [E] NEGATIVE CONTROL: a generic cross-ratio lambda=3 gives j=21952/9 not in {0,1728} => Aut=Z/2, no order-4 clock -- specific to mu4. (6) [I] UNIFICATION: 'seam = uniformising flat pillowcase metric' implies BOTH v201 mark-locality (flat away from marks) AND v180 QGEO.ISO.01 (z->iz isometric, from the square modulus), so the two open residuals collapse to ONE canonical-metric statement. (7) [O] does NOT close QGEO.SYM.01: the premise 'RP seam metric = the uniformising constant-curvature representative of its conformal class' stays open -- milder and more canonical than the bare isometry premise, and it SUBSUMES v201; net existence + full-cone RP remain v175's [E]. Wolfram-mirrored (chi_orb=0, cross-ratio 2 => j=1728, harmonic orbit, lemniscatic CM order 4, neg control); Klein-J modular values mpmath-numerical",
      "status": "Identity/Lattice [E]/[L] (cross-ratio 2 => j=1728 => order-4 CM clock) + premise [O] (canonical constant-curvature metric)",
      "canonicalStatus": "Identity/Lattice [E]/[L] + premise [O]: cross-ratio 2 (v168) => j=1728 => square modulus => order-4 CM clock (DERIVED, not assumed); unifies the two open QGEO residuals (v180 ISO + v201 SUBPRIN) into ONE canonical flat-pillowcase-metric statement; does NOT close QGEO.SYM.01 (the constant-curvature representative choice stays open, milder/more canonical than the bare isometry premise)",
      "dependencies": "QGEO.ISO.01; QGEO.SUBPRIN.01; QGEO.RIGID.01; v168_qgeo_rigidity.py; v180_clock_is_mobius.py; v201_seam_subprincipal_marks.py",
      "scripts": [
        "v214_seam_pillowcase.py"
      ],
      "script": "v214_seam_pillowcase.py",
      "location": "tfpt_research_contracts",
      "externalData": "Troyanov 1991 (cone-metric uniformisation); standard elliptic-curve Aut classification (j=1728 -> Z/4, j=0 -> Z/6)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 274
    },
    {
      "id": "QGEO.KILL.01",
      "claim": "BEDROCK AS A FALSIFIABLE KILL-TEST (2026-06-15): QGEO.SYM.01 (omega o rho = omega) is recast as four independently-falsifiable predictions about the raw seam DtN, NOT a proof (a foundational symmetry cannot be derived from nothing, v181/v153; Bisognano-Wichmann would presuppose the very covariance it should produce). Non-circular reduction (v198/v199/v201): omega o rho=omega <=> [rho,C]=0 for the quasi-free covariance C=1/2(1+sgn H_1); the principal symbol |k|=diag(|n|) commutes with rho=diag(i^n) exactly, so the whole residual is the bounded sub-principal M_f, and [rho,M_f]=0 <=> f has Fourier support only on modes ≡0 (mod 4); measure form = the raw Gaussian bulk quadratic form A is mu4-deck-invariant. FOUR LEVERS (carrier-side [E] green, seam-side [O] deferred): K1 EXACTLY 4 MARKS (the order-4 clock = Coxeter of W(A3)=S4 has free orbit mu4, and rank H^1 = #marks-1 = 3 = N_fam => #marks = 4 = |mu4| = e1(a); KILL if #marks!=4, b1!=3); K2 SQUARE CONFIG (cross-ratio(mu4)=2 => j=1728, Aut=Z/4; only j in {0,1728} give Aut>Z2, and clock order h(A3)=4 selects Z/4 NOT the j=0 Z/6; a generic mark set (cross-ratio 3 -> j=21952/9) has only Z/2; KILL if non-square); K3 MOD-4 SUB-PRINCIPAL SUPPORT (a mu4-mark sum has f_m=4 g_m[m≡0 mod4] => M_f block-diagonal => [rho,M_f]=0; off-character breaks it; KILL if a first-principles seam DtN has off-character weight above noise); K4 TRANSFER GAP (cusp weights {0,1/3,2/3} => transfer eigenvalues (1-w)^6={1,(2/3)^6,(1/3)^6}, gap (2/3)^6=64/729; KILL if gap!=64/729). FREEZE-BOUND: freeze_file.csv row 'seam_deck_symmetry' names the four levers (prediction+test locked, v84/v100 discipline). The DECISIVE kill is DEFERRED (QGEO.REALIZE.01): the raw P1 seam-collar DtN must yield exactly 4 square marks WITHOUT inputting them; no first-principles raw-seam construction exists yet (deferred, like axion_DM 'no test possible until the scale is closed'). Carrier-side K1-K4 are green regression guards; a KILL-TEST, NOT a closure of QGEO.SYM.01. Python-only (falsification layer like v65/v100; exact sub-parts j=1728 + gap (2/3)^6 already Wolfram-mirrored via v214/v54/v56)",
      "status": "Open (falsifiable kill-test; carrier-side K1-K4 [E], seam-side emergence deferred [O])",
      "canonicalStatus": "Open / falsifiable: bedrock omega o rho=omega recast as four levers (exactly 4 square marks, mod-4 sub-principal support, transfer gap (2/3)^6), freeze-bound via seam_deck_symmetry; carrier-side [E] regression guards, decisive emergence test deferred to QGEO.REALIZE.01; not a closure and not a Bisognano-Wichmann proof",
      "dependencies": "QGEO.SYM.01; QGEO.PILLOW.01; QGEO.SUBPRIN.01; v168_qgeo_rigidity.py; v201_seam_subprincipal_marks.py; v214_seam_pillowcase.py",
      "scripts": [
        "v215_seam_deck_killtest.py"
      ],
      "script": "v215_seam_deck_killtest.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 275
    },
    {
      "id": "QGEO.MARKS.03",
      "claim": "THE FOUR MARKS EMERGE FROM GAUSS-BONNET (2026-06-15; executes the K1 lever of QGEO.KILL.01): an independent Euclidean-orbifold derivation of the seam-mark structure. (1) [E] MARK COUNT = 2 chi: Z2 branch points (cone angle pi, deficit pi) on a flat sphere (chi=2) give n*pi = 2pi*chi(S^2) = 4pi => n = 2 chi = 4 = |mu4| = e1(a) = N_fam+1; the SAME chi=2 that supplies the 8 in c3 = 1/(|Z2| 2pi chi) = 1/(8pi). (2) [E] the Euclidean sphere 2-orbifolds (chi_orb = 2-sum(1-1/m_i) = 0) are exactly (2,3,6),(2,4,4),(3,3,3),(2,2,2,2) (n<=4 since sum 1/m_i = n-2 <= n/2). (3) [E] ALL-ORDER-2 (= the |Z2| sheet branch) selects (2,2,2,2) uniquely. (4) [E] N_fam=3 selects it: rank H^1 = #marks-1 = 3 only for (2,2,2,2); the three 3-mark orbifolds give 2 families => the three generations pick the 4-mark square over the 3-mark hexagonal (3,3,3); family count and mark count are ONE fact (N_fam = #marks-1). (5) [E] the free order-4 deck selects it: only (2,2,2,2) admits a free order-4 rotation of the marks ((2,4,4) order-4 FIXES its order-4 cone points, (3,3,3)=Z3, (2,3,6)=Z6) -- three independent criteria converge. (6) [O] residual: (2,2,2,2) carries a 1-parameter modulus (the mark cross-ratio); the SQUARE (cross-ratio 2 => j=1728, v214) is selected by the order-4 carrier clock (the one irreducible, over-determined as |mu4|=h(A3)=N_fam+1). So K1 (mark count + equality) is now DERIVED; only the square modulus stays input -- does NOT close QGEO.REALIZE.01, but removes the mark COUNT from the inputs. Exact (sympy); Wolfram-mirrored",
      "status": "Identity/Lattice [E] (Gauss-Bonnet count + Euclidean-orbifold uniqueness) + premise [O] (square modulus)",
      "canonicalStatus": "Identity/Lattice [E] + premise [O]: the mark COUNT (4 = 2 chi) and EQUALITY (all order-2) are DERIVED from Gauss-Bonnet + Z2 branch + flatness (three independent selectors: all-order-2, N_fam=3, free order-4); only the square modulus (cross-ratio 2, the order-4 clock) stays the carrier input; executes K1 of QGEO.KILL.01, does NOT close QGEO.REALIZE.01",
      "dependencies": "QGEO.MARKS.02; QGEO.PILLOW.01; QGEO.KILL.01; AX.P1.01; v195_marks_lefschetz_character.py; v214_seam_pillowcase.py",
      "scripts": [
        "v216_marks_gauss_bonnet.py"
      ],
      "script": "v216_marks_gauss_bonnet.py",
      "location": "tfpt_research_contracts",
      "externalData": "Gauss-Bonnet; Thurston/Conway classification of closed Euclidean 2-orbifolds",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 276
    },
    {
      "id": "QGEO.EMERGE.01",
      "claim": "NUMERICAL EMERGENCE SCAN (2026-06-15; the numerical sibling of QGEO.MARKS.03, as v210 is to v201): with the NUMBER of marks n AND their POSITIONS free, the 4-equally-spaced (square) configuration is the joint minimiser of {clock-invariance + 3-family cohomology + transfer gap (2/3)^6} on the actual seam DtN/state. (1) [N] n=4 equally-spaced marks give a clock-invariant quasi-free state ||rho_4 C rho_4^dag - C|| ~ 1e-14. (2) [N] family count selects n=4: scanning n in {2..8}, only n-1 = 3 = N_fam (n=3 -> 2 families = the hexagonal alternative, excluded; n=5 -> 4) -- the number is not input. (3) [N] the gap: a mu4-equivariant operator is forced diagonal in the cusp basis (deck-average off-diagonal ~1e-16 for V=diag(i,-1,-i)), cusp weights {0,1/3,2/3} => transfer (1-w)^6 = {1,(2/3)^6,(1/3)^6}, gap (2/3)^6 = 64/729. (4) [N] position scan: among 4-mark configs the square minimises the defect (square ~1e-14 vs off-square ~0.4) -- the operator-level v200 theta=pi/2. (5) [N] negative controls: n=3 (Z3/hexagonal) -> 2 families + breaks the order-4 clock; n=5,6 wrong count. (6) [O] numerically confirms the emergence on the real DtN (number scanned, positions free); selector = the 3-family/gap-(2/3)^6 structure (carrier input); executes K1 numerically but does NOT close QGEO.REALIZE.01 (the order-4/N_fam input stays irreducible, v153/v181); complements v216. Python-only",
      "status": "Numerical [N] (emergence confirmation on the real DtN/state) + premise [O] (selector is carrier input)",
      "canonicalStatus": "Numerical [N] + premise [O]: free-n + free-positions scan on the actual DtN/state confirms 4-square is the unique joint minimiser of {clock-invariance, 3 families, gap (2/3)^6}; executes K1 of QGEO.KILL.01 numerically, does NOT close QGEO.REALIZE.01; Python-only",
      "dependencies": "QGEO.MARKS.03; QGEO.KILL.01; v200_seam_variational_scan.py; v210_mark_local_dtn.py; v216_marks_gauss_bonnet.py",
      "scripts": [
        "v217_marks_emergence_scan.py"
      ],
      "script": "v217_marks_emergence_scan.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 277
    },
    {
      "id": "FR.RAREKAON.01",
      "claim": "RARE KAON SECTOR K -> pi nu nubar FROM THE TFPT CKM POINT (2026-06-15, archive integration): the kernel fixes the full CKM matrix (s12=lambda_C, s23=phi0/(1+lambda_C)=|V_cb|, s13=lambda_C^3/3, delta=pi/3+3 lambda_C^2=1.19823 rad; v18/v88) with NO flavor fit; feeding the exact point into the standard Brod-Gorbahn-Stamou short-distance formulas gives BR(K+ -> pi+ nu nubar)=9.45e-11 and BR(K_L -> pi0 nu nubar)=3.33e-11. (1) [I] the CKM point + exact apex (rhobar,etabar)=(0.1374,0.3509) and lambda_t=V_ts* V_td. (2) [C] the branching ratios -- NOT parameter-free: X_t=1.462, P_c, dP_cu, kappa_pm are external electroweak/QCD input (recomputed from the CURRENT CKM point; an earlier scaffold quoted BR(K_L)=3.47e-11 from an Im(lambda_t)/lambda^5 slip, corrected here to 3.33e-11). (3) [N] BR(K+)=9.45e-11 lands ON the NA62 full 2016-2024 combination (9.6 +1.9/-1.8)e-11 (La Thuile 2026), +0.08 sigma (the earlier 2016-2022 obs was (13.0 +3.3/-3.0)e-11, ~1.2 sigma above); a strong consistency hit for the CKM point, but an F_transfer bridge -- NOT a unique TFPT-vs-SM discriminator (the SM (8.6 +- 0.4)e-11 also sits inside the NA62 error). (4) [I] Grossman-Nir ratio BR(K_L)/BR(K+)=0.352 << 4.3 (model-independent isospin corridor). (5) [X] dated kill test: a stable NA62 BR(K+) outside [7,12]e-11, or a KOTO-II BR(K_L) off the GN-plane point, breaks the TFPT flavor bridge for this sector (compiler core untouched); the ~2 sigma delta_CKM tension (v88) makes the corridor discriminating. A downstream FLAVOR readout, guarded [C] by v187 -- NOT a compiler power. Python-only",
      "status": "Physical bridge [C] (CKM point [I], branching ratios external-SD [C], dated kill test [X])",
      "canonicalStatus": "Physical bridge [C]: the TFPT CKM point (no flavor fit) + standard short-distance input give BR(K+)=9.45e-11 and BR(K_L)=3.33e-11; consistent with NA62 at ~1.2 sigma; dated kill test = stable NA62 outside [7,12]e-11 or KOTO-II off the GN point; not a compiler power",
      "dependencies": "v18_quark_yukawa.py; v88_cp_phase_audit.py",
      "scripts": [
        "v202_rare_kaon.py"
      ],
      "script": "v202_rare_kaon.py",
      "location": "tfpt_2_standard_model",
      "externalData": "NA62 (13.0 +3.3/-3.0)e-11; KOTO; Brod-Gorbahn-Stamou SD input",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 278
    },
    {
      "id": "HOR.EHT.01",
      "claim": "EHT ACHROMATIC POLARIZATION-ROTATION INTERCEPT (2026-06-15, archive integration): beta_BH(r) = Q_e^eff Q_m^eff/(256 pi^4 r^2) = 16 c3^4 (Q_e Q_m / r^2) around a horizon-scale black hole. (1) [E] the coupling 16 c3^4 = 1/(256 pi^4) EXACTLY (c3=1/8pi) -- no free coupling. (2) [E] alpha-kernel cross-link: 16 c3^4 = delta_top/3 with delta_top = 48 c3^4 = 3/(256 pi^4), the SAME top-form coefficient that fixes the alpha-kernel precision-zone correction (one row from the cosmic-birefringence seed beta_rad=phi0/4pi) -- EHT polarimetry probes the same admissibility data as the alpha row, projected onto the local horizon collar. (3) [C] TFPT fixes the structured 1/r^2 profile, the achromaticity (no lambda^2 dependence), and the sign-flip under effective E.B reversal; the amplitude Q_e^eff Q_m^eff (local effective E.B structure) is an MHD/GR weight, NOT a compiler output -- so the channel is [C] (shape/sign), never a parameter-free pixel amplitude. (4) [X] dated kill test: the residual intercept chi_0^res = chi_0^obs - chi_0^GRMHD must pass THREE nulls simultaneously (frequency/achromatic, spatial 1/r^2, sign-flip under E.B reversal); a residual statistically consistent with zero after honest GRMHD subtraction, or failing any null, falsifies THIS channel -- the compiler core (the determinant-line coupling shared with alpha/beta_rad) is untouched. Scope: the old UFE closed RN-type metric (D/r^4, regular cores, shadow shift) is NOT imported -- superseded by the Nariai/seam=horizon readouts (v101-v104, v190); only the polarization signature survives. Wolfram-mirrored (the coefficient identity 16 c3^4 = 1/(256 pi^4) = delta_top/3 is exact)",
      "status": "Identity [E] (coupling 16 c3^4=1/(256 pi^4)=delta_top/3) + physical bridge [C] (1/r^2 shape, achromaticity, sign-flip) + dated kill test [X] (three nulls)",
      "canonicalStatus": "Identity [E]: the EHT polarization coupling 16 c3^4 = 1/(256 pi^4) = delta_top/3 is the SAME compiler number as the alpha-kernel correction; the 1/r^2 shape + achromaticity + sign-flip are TFPT-fixed [C], the amplitude an MHD/GR weight; dated [X] kill test = three nulls after honest GRMHD subtraction; NOT the UFE metric (Nariai/seam=horizon supersedes it)",
      "dependencies": "AX.P1.01; HOR.01; EM.FP.01",
      "scripts": [
        "v203_eht_achromatic.py"
      ],
      "script": "v203_eht_achromatic.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "EHT/ngEHT polarimetry (forward GRMHD subtraction)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 279
    },
    {
      "id": "QGEO.VARI.02",
      "claim": "THE DISCRETISED VARIATIONAL TEST (work package B, external-review, 2026-06-14): v196 showed E_fail=0 at the mu4 config on the fixed H^1 block; here the clock angle is a FREE parameter and E_fail is MINIMISED over it. (1) [N] taking rho(theta)=diag(e^{i m theta}) on the H^1 generators m=(1,2,3) and E_fail(theta)=||rho(theta)^4-1||^2 + faithfulness penalty (penalising coincident characters), a scan over theta in (0,pi) has its UNIQUE minimum at theta=pi/2 (E_fail=0): there rho^4=1 AND the three characters i,-1,-i are distinct = the (1,2,3)=A3-exponent grading. (2) [N] decoys lose: theta->0 is trivial (one character), theta->pi gives only two distinct characters (m=1,3 coincide at -1), generic pi/3 and 2pi/3 give E_fail~6 -- so argmin E_fail = the mu4 normal form, no other clock reproduces the faithful order-4 mu4 grading. (3) [E] consistent with v196/v199: theta=pi/2 is the clock z|->iz whose principal-symbol commutator vanishes (v198) and whose H^1 covariance is mu4-equivariant (v177); E_fail=0 there reproduces the v196 finite-block vanishing. (4) [O] this is a numerical sign on the finite H^1 grading, not the full operator-valued minimisation over the raw-seam DtN on the continuum (the v199 bounded-sub-principal residual); strengthens the case that the variational principle selects mu4 but does not close it. Python-only (numerical variational scan)",
      "status": "Numerical sign [N] (argmin = mu4 clock) + open full minimisation [O]",
      "canonicalStatus": "Numerical [N]: argmin E_fail over the clock angle = theta=pi/2 (the mu4 clock z|->iz), decoys lose; a numerical sign that the variational principle selects mu4; full operator-valued minimisation stays the v199 residual [O]",
      "dependencies": "QGEO.VARI.01; QGEO.STATE.01; v196_seam_energy_functional.py; v199_seam_state_invariance.py",
      "scripts": [
        "v200_seam_variational_scan.py"
      ],
      "script": "v200_seam_variational_scan.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 280
    },
    {
      "id": "REVIEW.MAP.01",
      "claim": "THE FOUR EXTERNAL-REVIEW CONCERNS REDUCE TO THE TWO ALREADY-NAMED RESIDUALS + ONE DATA-ONLY ITEM (a careful 2026-06-14 review raised A mapping 2D->4D, B UV-gravity abstractness, C grammar-tuning/meta-look-elsewhere, D Koide Mobius flow; this maps them honestly): three reduce by the same discipline as v175-v181 onto {QGEO.SYM.01 (the seam realisation), F_transfer (the continuous transfer functor / missing dynamical principle)}, and the fourth is experiment-only. (A) MAPPING [I]/[C]: the readout integer is exact -- the Plucker norm 11 = sum_{k<=2}C(4,k) = 1+4+6 = dim S+ - g_car = 16-5 (the QBL-protected boundary state count, v174); the 2D->4D step is the holographic reduction (bulk measure -> seam-boundary measure, gap-decoupled G5, Delta_eff>0); so A = QGEO.SYM.01 + F_transfer, NOT a new gate. (B) UV GRAVITY [I]: the TFPT answer 'gravity is the geometric readout of the boundary' IS QGEO.SYM.01 (the seam conformal deck); B is the bedrock postulate itself, an identity not a new premise. (C) GRAMMAR TUNING [N]/[C]: the strongest honest defence -- the grammar is FROZEN pre-data (v84/REG.FREEZE.01), minimal (anchor a + pi), and OVER-DETERMINED (the same anchor a=(1,1,2) lands in >=6 mutually-independent not-selected-for structures: Nariai roots (1,1,-2) HOR.NARIAI.01; Coxeter-30=|Z2|N_fam g_car SEAM.CYCLE.01; spine tetrahedron {2,3,4,5} ARCH.SPINE.01; EM budget 41=a^T K a; seed 4/3=|mu4|/N_fam; c3=1/(2 e1(a) pi)); a look-elsewhere-resistant signature, but -- the review is correct -- the residual meta-risk falls only to out-of-sample data (JUNO/CMB-S4), never to argument. (D) KOIDE MOBIUS FLOW [I]/[C]: the Mobius nature is not arbitrary -- Aut(P^1)=PSL(2,C) is the Mobius group and the Koide/cusp flow lives on the deck P^1 minus mu4, so its generator is Mobius BY THE GEOMETRY; the non-integer flow time t~2.84 (FR.KOIDE.06) only locates the physical point; the missing QFT RG generator (the beta-function realising the orbit) IS F_transfer; so D = QGEO.SYM.01 + F_transfer, not a new gate. NET: A/B/D collapse onto {QGEO.SYM.01, F_transfer}; C is the single experiment-only meta-question. A unification of the concern structure, NOT a closure of the underlying premises (which stay [O]/[C], honestly)",
      "status": "Reduction map [I]/[C] (A/B/D reduce to the two named residuals; C is experiment-only)",
      "canonicalStatus": "Reduction [I]/[C]: external-review concerns A/B/D reduce to {QGEO.SYM.01; F_transfer}; C is data-only (JUNO/CMB-S4)",
      "dependencies": "QGEO.SYM.01; FR.KOIDE.06; FR.ETAB.02; AUDIT.NULLMC.01; REG.FREEZE.01; QFT.FOCK.01; HOR.NARIAI.01; SEAM.CYCLE.01",
      "scripts": [
        "v182_reviewer_residual_map.py"
      ],
      "script": "v182_reviewer_residual_map.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 281
    },
    {
      "id": "FR.ETAB.02",
      "claim": "ETA_B LEPTOGENESIS TRANSFER OPERATIONALISED (the Tier-3 F_transfer Boltzmann path made falsifiable; NOT a new structural class, NOT a compiler theorem): type-I thermal leptogenesis eta_B ~ 0.96e-2 eps_1 kappa_f (sphaleron a_sph=28/79) fed by TFPT's neutrino sector -- normal ordering (m2,m3 from dm2_sol,dm2_atm) and the predicted phase delta_CP=4pi/3=240deg (|sin|=0.866). (1) Davidson-Ibarra eps_1=(3/16pi)M1 m3/v^2 ~ 8.5e-7 for M1=1e10 GeV [scenario M1]; (2) efficiency kappa_f(m~1) (Buchmueller-Di Bari-Pluemacher) ~ 0.03-0.19 for m~1 in [m_*, m_atm] [scenario washout]; (3) KILL TEST: over natural M1 in [3e9,3e10] GeV and m~1 in [m_*,m_atm], eta_B spans [8.4e-11, 4.7e-9] which BRACKETS the observed 6.1e-10 -- the canonical point (M1=1e10, m~1=5e-3, delta=240deg) gives 6.0e-10 (NOT tuned). So the leptogenesis route fed by TFPT's neutrino sector REACHES the observed asymmetry; but M1 and the washout are SCENARIO inputs (not TFPT outputs), so eta_B stays [C] -- a falsifiable transfer interface, NOT promoted. If a precise Boltzmann/RGE computation excluded the window, the leptogenesis ROUTE falls, NOT the theory (the downstream Omega_b readout FR.ETAB.01 is independent and already hit). Seesaw RGE running (PyR@TE) = an O(1) correction, flagged. Python-only",
      "status": "Physical [C] (viable falsifiable scenario; M1/washout scenario inputs)",
      "canonicalStatus": "Physical [C]: thermal-leptogenesis route viable (band brackets observed 6.1e-10); M1/washout scenario inputs, eta_B stays [C]",
      "dependencies": "FR.ETAB.01; FLAV.RGSTAB.01; v9_neutrino_texture.py",
      "scripts": [
        "v169_etaB_boltzmann_interface.py"
      ],
      "script": "v169_etaB_boltzmann_interface.py",
      "location": "tfpt_4_frontier",
      "externalData": "NuFIT (dm2_sol, dm2_atm); Planck (eta_B = 6.1e-10)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 282
    },
    {
      "id": "E8.SLICE.01",
      "claim": "E8 SLICE COMPRESSION LEMMA (the seven E8 audit slices are ONE projection of two small alphabets; anti-numerology compression, NOT new physics): (1) the anchor power sums p_n(a)=2+2^n give P=(3,4,6,10) whose K4 edge products are all carrier blocks (12=|R(A3)|, 18=N_fam|R+(A3)|, 30=h(E8), 24=|W(A3)|, 40=|R(D5)|, 60=D_start; triples 120,240,720) [I]; (2) the six Sheet-Diamond operators Q,K,R,C,L,F have determinants (3,4,8,14,20,32) summing to 81=N_fam^4 -- the total determinant charge of the flavour space [I]; (3) all SEVEN E8 248-slices (D8, D5xA3, E6xA2, E7xA1, F4xG2, A4xA4, A8) are written from P, D, Delta=p0+p3=13 and (e1,e2,dim S+) [I]; (4) one affine map row(C+sU+tV)=(7,11,13)+s(3,3,3)+t(1,2,3) generates every flavour row budget (democratic winding + A3 exponents) [I]; (5) dim E6 = 78 = p2*Delta = |R+(A3)|(N_fam+A_Lambda) [I]. The E8 atlas is thus a closed grammar over two typed alphabets, not seven trophy lists; the physical reading is AUDIT",
      "status": "Identity/Lattice [I]/[L] (exact integer/lattice identities)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: E8 atlas = projection of power-sum + determinant alphabets; audit bridge, anti-numerology",
      "dependencies": "ARCH.SPINE.01; FLAV.KQ.01; CAR.SM.01",
      "scripts": [
        "v170_diamond_slice_compression.py"
      ],
      "script": "v170_diamond_slice_compression.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 283
    },
    {
      "id": "QFT.OSMOMENT.01",
      "claim": "ATOMIC OS MOMENT LEMMA + SUGAWARA GAP SAFETY (the admissible transfer sector's OS positivity as a positive moment problem, and its gap as an E8 Sugawara inequality): (1) spec(T)={1,(2/3)^6,(1/3)^6}={1,64/729,1/729} makes every scalar two-point correlator C(n)=A+B r^n+C s^n the moment sequence of the positive atomic measure A d1+B dr+C ds, so every OS Hankel matrix factorises as a Vandermonde Gram H=W diag(A,B,C) W^T => positive semidefinite (positive spectrum => moment measure => reflection positivity; a clean proof, not a numerical Gram test) [I]; (2) H=-log T gives a positive discrete Kallen-Lehmann spectrum {0, 6log(3/2), 6log3} with masses m1=2.43279, m2=6.59167 [I]; (3) the exact cluster constant 1/(1-r)=729/665 and correlation length xi=1/(6log(3/2))=0.41105 [I]; (4) SUGAWARA GAP SAFETY: 31=1+h^v(E8)=1+30 and c((E8)1)=248/31=8, so Delta_eff=6log(3/2)-31/(4pi^2)=1.64755>0 reads as |R+(A3)|log(N_fam/|Z2|) > (1+h^v(E8))/(4pi^2) -- the A3 transfer gap beats the E8 Sugawara metric budget",
      "status": "Identity [I] (moment/spectral/Sugawara exact) + [C] (declared V_metric norm)",
      "canonicalStatus": "Identity [I]: OS positivity = positive moment problem; Sugawara gap safety 31=1+h^v(E8); cluster 729/665, xi=0.411",
      "dependencies": "GATE.METRIC.XI.01; SEAM.CYCLE.01; GATE.METRIC.07",
      "scripts": [
        "v171_os_moment_cluster.py"
      ],
      "script": "v171_os_moment_cluster.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 284
    },
    {
      "id": "SEED.ANOMALY.01",
      "claim": "TRACE-ANOMALY ORIGIN OF THE SEED PREFACTOR 4/3 + the shared integer 7 (b3 = scalaron exponent): (1) the retained seed phi0=(4/3)c3+48 c3^4 has base (4/3)c3=1/(6pi); the prefactor 4/3 is the conformal Weyl anomaly of the seam CFT -- over S^2 (chi=2) the (E8)1 (c=8) trace anomaly is c*chi/6=8/3, halved by the one-sided |Z2| Moebius identification to 4/3 = dim S+/dim g_SM = 16/12; the Cabibbo seed prefactor IS the halved (E8)1 Weyl anomaly [I]; (2) the QCD one-loop coefficient |b3|=11-(2/3)N_f=7 (N_f=2 N_fam=6) equals the scalaron exponent 7=Omega_adm-10 b1=48-41 (M_scal=c3^{7/2} Mbar) -- confinement (Lambda_QCD) and inflation (scalaron) are driven by ONE integer (the carrier occupancy deficit) [I]; (3) AUDIT: the external Volkov-Wipf Nariai graviton/ghost coefficient -98/45 (v138) = -2*7^2/45 = -2 (scalaron exp)^2/dim(D5) (dim so(10)=45). Gives the seed prefactor and the QCD/scalaron integer a first-principles CFT/QFT origin, not a fit",
      "status": "Identity [I] (trace anomaly + b3 + VW factorisation exact)",
      "canonicalStatus": "Identity [I]: seed 4/3 = halved (E8)1 Weyl anomaly on S^2; b3=7=scalaron exponent (confinement<->inflation)",
      "dependencies": "CAR.SM.01; FR.MPME.01; ANCHOR.GEN.01; v138_vw_firewall.py",
      "scripts": [
        "v172_trace_anomaly_seed.py"
      ],
      "script": "v172_trace_anomaly_seed.py",
      "location": "tfpt_2",
      "externalData": "Volkov-Wipf Nucl.Phys.B582(2000)313",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 285
    },
    {
      "id": "FLAV.STRONGCP.02",
      "claim": "STRONG CP AS PFAFFIAN REALITY (theta_eff=0 as a model-level CAR/Pfaffian theorem; formalises the tfpt_2 structural chain): in the self-dual CAR (16-Majorana) seam model, (1) the sheet-odd Calderon involution {D_Sigma, Sigma}=0 forces the spectrum into +-lambda pairs [I] (verified: eigenvalues symmetric about 0 to 1e-15); (2) gamma5-Hermiticity D^dag=gamma5 D gamma5 makes conj(det D)=det D, so the fermion-measure Pfaffian is real, arg in {0, pi} [I] (verified: Im(det D)~1e-9); (3) a theta-term gives Z(theta)=e^{i theta}|det D|; reflection positivity (Z>0, OS axiom) excludes theta=pi (Re Z<0) and every theta not in {0,pi} (Im Z != 0), selecting theta_eff=0 [I]. So theta_eff=0 is a topological null theorem -- the antisymmetric Calderon pairing makes the Pfaffian real and RP picks the positive branch; the CAR/Pfaffian form of the polar+gamma5+sheet+RP derivation",
      "status": "Formalised [F] (machine-checked model demonstration)",
      "canonicalStatus": "Formalised [F]: theta_eff=0 via Pfaffian reality (sheet pairing + gamma5-Hermiticity + RP positive branch)",
      "dependencies": "FLAV.QGEO.02; CAR.QFREE.01; GATE.METRIC.13",
      "scripts": [
        "v173_pfaffian_cp_car.py"
      ],
      "script": "v173_pfaffian_cp_car.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 286
    },
    {
      "id": "QFT.FOCK.01",
      "claim": "SEAM FOCK-SPACE READINGS (three exact audit bridges that sharpen the QFT/topological reading of already-used objects): (1) CAR CONE COMPRESSION [I] -- the even CAR algebra of the 16 seam Majoranas has dim Lambda^even(R^16)=2^15=32768 directions (cumulant blocks C(16,4)=1820, C(16,6)=8008, C(16,8)=12870), but a quasi-free Bogoliubov transport is the second quantisation Gamma(t)|_Lambda^m=Lambda^m(t), so the whole cone is generated by the 16-dim one-particle spectrum, compression 2^15/16=2048=2^11 (the explicit count behind v161; 'the QFT cone is large but not independent'); (2) WITTEN INDEX = PLUCKER NORM 11 [I] -- the 4 mu4-corner fermions span a 2^4=16=dim S+ Fock space; QBL (degree<=2) keeps sum_{k<=2}C(4,k)=1+4+6=11=||Pl(K)||_1, discards C(4,3)+C(4,4)=5=g_car, so 11=dim S+ - g_car and the quark ratio c_u/c_d=g_car*11/(N_fam^2*Delta_Q)=55/117 reads the QBL-protected boundary state-space dimension (a topological Witten index, not a free integer); (3) AFFLECK-LUDWIG g-THEOREM RIGIDITY [I]/[L] -- c_UV=c((D5)1)+c((A3)1)=5+3=8=c_IR((E8)1), so Delta c=0 along the seam boundary RG flow; a flow needs Delta c>0 to dissipate, so Delta c=0 forces an exact isometry (no relevant/marginal perturbation), the boundary-entropy form of the v157/v158 rigid fixed point. Exact identities; readings audit",
      "status": "Identity/Lattice [I]/[L] (cone count + Witten index + central-charge conservation)",
      "canonicalStatus": "Identity/Lattice [I]/[L]: CAR cone 2^15->2^11; Witten index 11=Plucker norm=dim S+ - g_car; g-theorem Delta c=0 rigidity",
      "dependencies": "GATE.UWALL.06; CAR.QFREE.01; GATE.METRIC.14; v42_exterior_leg.py; v161_one_particle_reduction.py",
      "scripts": [
        "v174_seam_fock_readings.py"
      ],
      "script": "v174_seam_fock_readings.py",
      "location": "tfpt_2;tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 287
    },
    {
      "id": "QFT.NETRP.01",
      "claim": "NET EXISTENCE + FULL-CONE RP DISCHARGED TO [F] (the heavy artillery; the 'three open QFT theorems' are two [F] + one irreducible [O], nothing fabricated): (1) FULL-CONE REFLECTION POSITIVITY FOR ALL m [F] -- the CAR second-quantisation functor Gamma(t)=(+)_m Lambda^m(t) of the one-particle seam contraction t acts on the COMPLETE Fock space Lambda^*(R^16) (dim 2^16=65536); by the exterior-power spectrum theorem every eigenvalue of Gamma(t) is a subset product of spec(t), verified on all 2^16 products to be in [0,1] (contraction), with eigenvalue-1 (fixed) multiplicity 2^8=256 = wedges of the rank-8 K_Sigma polarisation and sub-leading eigenvalue = the one-particle gap (2/3)^6 -- so full-cone RP/gap reduce, for ALL m (not just m<=6, v161), to the one-particle contraction (Shale-Stinespring CAR functor + quasi-free positivity); the v160 scope premise is DISCHARGED by a theorem; (2) A2 NET EXISTENCE [F] -- the (E8)1 net exists by cited theorems (16 free Majoranas -> SO(16)_1 chiral net, Buchholz-Mack-Todorov/Boeckenhauer-Evans-Kawahigashi; holomorphic c=8 lattice net = (E8)1, Frenkel-Kac-Segal/Dong-Xu/Kawahigashi-Longo; mu4 index-4 Q-system, v125), with machine-verified data: character E4/eta^8=1+248q+4124q^2+34752q^3+213126q^4+1057504q^5 (level-1=248), embedding 248=120+128, E8 Cartan even unimodular (det 1, the unique rank-8 even unimodular lattice, Minkowski-Siegel/v83); (3) BOTH COLLAPSE TO ONE [I] -- full-cone RP -> one-particle contraction, A2 -> cited net existence, both needing the SAME input: the actual seam Calderon kernel = the gapped one-particle contraction on P^1\\\\mu4 = QGEO.REALIZE.01 (finite half proven, v168). (4) IRREDUCIBLE PREMISE [O] -- QGEO.REALIZE.01 (seam boundary CFT = parabolic transport on P^1\\\\mu4) is a constructive-geometry/AQFT statement, NOT closeable by finite computation, left OPEN (needs a human constructive-QFT/geometry argument). NET: net existence + full-cone RP are [F]; the one irreducible structural premise is the seam realisation",
      "status": "Formalised/Identity [F]/[I] (all-m functoriality + verified net data) + [O] (the seam realisation)",
      "canonicalStatus": "Formalised [F]: full-cone RP (all m, CAR functor) + (E8)1 net existence (assembled+verified); the one [O] is the seam realisation QGEO.REALIZE.01",
      "dependencies": "GATE.METRIC.11; GATE.METRIC.13; GATE.METRIC.16; GATE.METRIC.17; QGEO.RIGID.01; QGEO.REALIZE.01; v83_e8net_holomorphic_uniqueness.py; v125_glue_qsystem.py; v161_one_particle_reduction.py",
      "scripts": [
        "v175_net_existence_full_cone.py"
      ],
      "script": "v175_net_existence_full_cone.py",
      "location": "tfpt_research_contracts;tfpt_5_redteam",
      "externalData": "Buchholz-Mack-Todorov; Boeckenhauer-Evans-Kawahigashi; Frenkel-Kac-Segal; Dong-Xu; Kawahigashi-Longo",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 288
    },
    {
      "id": "FR.MUONG2.01",
      "claim": "MUON g-2 SEAM VERTEX (2026-06-15, archive integration of the old muon-g2 note, re-typed for the current kernel): the carrier fixes a second-order topological defect delta_2 = B*gamma*delta_top^2 = (5/4)delta_top^2 (delta_top=Omega_adm c3^4=48 c3^4=3/256pi^4, v2/v23; B*gamma=(3/2)(5/6)=5/4 the carrier compression quotient, v51); the seam-loop projection a_mu^seam = delta_2/(2 pi) = 45/(524288 pi^9) = 2.879e-9. (1) [E] the VALUE is an exact compiler number (trace reading delta_2 = 4! Tr_{S+}(X^2) c3^8, Tr=120=5!). (2) [C] the IDENTIFICATION of delta_2/(2 pi) as a magnetic vertex correction (the 1/2pi=4c3 seam-loop projection) is a physical bridge, NOT a theorem -- so the prediction is [C]. (3) [N] 0.81 sigma vs the dispersive Delta a_mu=(2.49+-0.48)e-9. (4) [C] HONEST: lattice/CMD-3 HVP shrinks Delta a_mu (~1.5e-9), the FIXED TFPT value then sits ~1.5 sigma high -- testable against either HVP outcome, not tuned to the dispersive one. (5) [X] dated kill test: a converged Delta a_mu outside 2.879e-9 +- 0.5e-9 excludes the seam-vertex mechanism (compiler core untouched). A downstream readout, NOT a compiler power. Wolfram-mirrored (the value is exact). (6) KILL FIRED 2026-08-04 (news round): the lattice-side HVP consensus consolidated -- WP25 (Phys. Rep. 2025) adopts the lattice HVP as the SM baseline and the BMW hybrid evaluation (Nature 2026) confirms it, Experiment - SM = 0.5 sigma, i.e. Delta a_mu converged OUTSIDE 2.879e-9 +- 0.5e-9: the seam-vertex identification is excluded in its present form (the dated kill (5) fired exactly as preregistered). The bridge is dissolved; the exact compiler number (1) [E] and the compiler core are untouched -- the watchdog delivered",
      "status": "Physical bridge KILLED: dated kill test [X] fired 2026-08-04 (exact value stays [E]; the [C] vertex identification is excluded in its present form -- bridge dissolved, compiler core untouched)",
      "canonicalStatus": "Physical bridge KILLED [X] fired 2026-08-04: a_mu^seam = delta_2/(2 pi) = 45/(524288 pi^9) = 2.879e-9 stays an exact value [E], but the WP25+BMW lattice-HVP consensus (Experiment - SM = 0.5 sigma) puts the converged Delta a_mu outside 2.879e-9 +- 0.5e-9 -- the [C] vertex identification is excluded in its present form; bridge dissolved, compiler core untouched; not a compiler power",
      "dependencies": "v2_carrier_pascal.py; v23_anchor_generator.py; v51_boundary_half_step.py",
      "scripts": [
        "v204_muon_seam_g2.py"
      ],
      "script": "v204_muon_seam_g2.py",
      "location": "tfpt_4_frontier",
      "externalData": "Fermilab+BNL world average; Aoyama et al. WP2020 dispersive SM; lattice/CMD-3 HVP",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 289
    },
    {
      "id": "GRAV.XI.01",
      "claim": "GRAVITATIONAL 3/4 OVER-DETERMINATION (2026-06-15, archive integration of the old Eliminating-K note): the torsion-compression ansatz kappa^2 = xi phi0/c3^2 with the Einstein limit kappa^2=8 pi G fixes xi = 8 pi c3^2/phi0 = c3/phi0 (since 8 pi c3^2 = c3), and at tree level phi_tree=1/(6 pi) this is EXACTLY 3/4. (1) [E] tree identity xi = c3/phi_tree = (1/8pi)/(1/6pi) = 3/4. (2) [E] Einstein-limit reduction 8 pi c3^2 = c3 => G drops out, G is an output not an input. (3) [E] OVER-DETERMINATION: 3/4 = q(A3) = N_fam/|mu4| = ln(m/mu) (v152 k=c3/2) -- the torsion-compression xi and the gapped EH replica are two structurally independent appearances of the same gravitational 3/4. (4) [N] with the retained seed xi=c3/phi0=0.748303 (-0.226%). (5) [C] the kappa^2=xi phi0/c3^2 ansatz is the [C] bridge; the modern derivation of 3/4 is v152/v68, xi is no new free parameter, 1/G stays the one declared anchor (v60/v153). Wolfram-mirrored (tree identity + over-determination exact)",
      "status": "Identity [E] (tree 3/4 + over-determination) + physical bridge [C] (the kappa^2 ansatz)",
      "canonicalStatus": "Identity [E]: xi = c3/phi_tree = 3/4 = q(A3) = ln(m/mu) -- an independent route to the gravitational 3/4; the kappa^2=xi phi0/c3^2 ansatz is the [C] bridge, G is a metrology output, 1/G the one anchor",
      "dependencies": "v152_norm_is_anchor.py; v68_seeley_dewitt_residual.py; v60_lambda_metrology_branch.py",
      "scripts": [
        "v205_xi_threequarter.py"
      ],
      "script": "v205_xi_threequarter.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 290
    },
    {
      "id": "GRAV.H0.01",
      "claim": "COSMOLOGICAL-CONSTANT BRANCH H0 = 66.5..67.1 (2026-06-15, archive integration of the tfpt-31 CC note; sharpens the qualitative H0-vs-Lambda row): the seam number delta_Sigma = m_1 e^{-2 pi alpha^-1/3} with the lowest-block m_1=Omega_adm=48. (1) [N] delta_Sigma^lead = 48 e^{-2 pi*137.036/3} ~ 1.085e-123. (2) [E] convention split M_Pl^4/Mbar^4 = (8 pi)^2 = 631.65 (unreduced vs reduced Planck density -- must be displayed). (3) [N] rho_Lambda = M_Pl^4 delta_Sigma ~ 2.4e-47 GeV^4 (observed vacuum order). (4) [N] 123-orders cross-link |log10 delta_Sigma| ~ 122.96, consistent with the v60 split 122.948. (5) [C] under the flatness closure H0^lead=66.51, H0^disc=67.10 km/s/Mpc -- 1.6..0.5 sigma BELOW Planck (67.36+-0.54), ~6 sigma below SH0ES (73.0+-1.0): HONEST, the Hubble tension is NOT relieved by raising the vacuum scale; H0 uses imported Omega_b, omega_DM (external cosmology), so it is a [C] readout, not a from-nothing output. Python-only (the (8pi)^2 ratio is on the Wolfram path via v60)",
      "status": "Numerical [N] (delta_Sigma, 123-orders) + physical bridge [C] (H0 flatness readout)",
      "canonicalStatus": "Numerical/bridge [N]/[C]: delta_Sigma=48 e^{-2pi alpha^-1/3}~1.085e-123, (8pi)^2 convention split, |log10|~122.96 (v60 consistent); H0=66.5..67.1 below Planck, does NOT relieve the Hubble tension; [C] readout (imported Omega_b, omega_DM)",
      "dependencies": "v60_lambda_metrology_branch.py; v55_coxeter_cycle.py",
      "scripts": [
        "v206_h0_lambda_branch.py"
      ],
      "script": "v206_h0_lambda_branch.py",
      "location": "tfpt_4_frontier",
      "externalData": "Planck 2018 (H0=67.36+-0.54); SH0ES (73.0+-1.0)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 291
    },
    {
      "id": "GRAV.ASYMP.01",
      "claim": "ASYMPTOTIC SAFETY -- A SECOND [P] ROUTE TO THE METRIC SECTOR (2026-06-15, archive integration of the old Five-Problems note; NOT load-bearing, does NOT close G_metric): the current theory closes QG via the discrete seam-net/AQFT programme (v175; open premise QGEO.REALIZE.01); this records a COMPLEMENTARY continuum-FRG argument that a non-Gaussian UV fixed point exists from the same axioms. Setup d_t g = (2 + eta_R(g)) g, eta_R = -a1 g (torsion K^2 positive-definite => a1>0). (1) [E] the axion coupling is fixed and small: y^2 = 16 c3^2 = 1/(4 pi^2) = 0.025330 exactly (g_agg=-4 c3 from the cubic, scale-invariant). (2) [N] existence by IVT: 2+eta_R is +2>0 as g->0 and <0 at large g (torsion damping) => a root g_*>0 (non-Gaussian UV fixed point). (3) [N] g_*=2/a1>0 finite, Reuter-type O(1), coordinates set by {c3,phi0}+field content, no new free input. (4) [O] HONEST: a continuum-FRG ansatz the discrete-compiler theory did NOT adopt; a second plausibility route only; the primary machine-checked route stays the seam-net (v175); G_metric open either way. Python-only",
      "status": "Open/plausibility [O] (a [P]-typed second QG route; not load-bearing) + [E] (y^2=16 c3^2)",
      "canonicalStatus": "Open/plausibility [O]: a complementary continuum-FRG argument (non-Gaussian UV fixed point exists from {c3,phi0}); y^2=16 c3^2 exact; NOT load-bearing, the seam-net (v175) stays primary, G_metric open",
      "dependencies": "v175_net_existence_full_cone.py; v28_gravity_fR.py",
      "scripts": [
        "v207_asymptotic_safety.py"
      ],
      "script": "v207_asymptotic_safety.py",
      "location": "tfpt_4_frontier",
      "externalData": "Reuter Asymptotic-Safety programme (external)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 292
    },
    {
      "id": "HOR.BHTHERMO.01",
      "claim": "BLACK-HOLE THERMODYNAMICS FROM THE SEAM: modular Hawking temperature + scalaron-corrected Wald entropy (2026-06-15, archive integration of the dropped tfpt-42 stationary-horizons/compact-objects section, re-typed through the Tomita-Takesaki seam work v198/v199). (1) [E] MODULAR T_H: the modular flow Delta^{it}=e^{-2 pi t K_H} (Bisognano-Wichmann/Tomita-Takesaki) gives KMS at beta=2 pi and T_H = kappa/(2 pi); the 2 pi IS the seam unit 1/(4 c3) (1/2pi=4c3, v58), reproducing T_H=c3/M (v57); the seam<->horizon modular identification is [P] (ties to [rho,Lambda]=0, v198/v199). (2) [E] SCALARON WALD ENTROPY: with the induced f(R)=R+R^2/(6 M_s^2) (M_s=c3^{7/2} Mbar, v28), S_W=(f_R A)/(4 G)=(A/4G)(1+R_h/3M_s^2), f_R=1+R/(3M_s^2) -- leading A/4G is the c3 area law (1/4=1/|mu4|), the correction an EXACT scalaron consequence; BH application [P]. (3) [E] SdS T_H=(1/4pi r_h)(1-Lambda r_h^2), 1/4pi=2c3. (4) [E] area law S_BH=M^2/(2c3)=4 pi M^2=A/4G, same 1/(4c3)=2pi factor as the modular beta. Wolfram-mirrored (the f_R Wald identity + modular 2pi=1/4c3 are exact)",
      "status": "Identity [E] (modular 2pi=1/4c3, scalaron Wald factor, area law) + physical bridge [C] (the BH/modular identification)",
      "canonicalStatus": "Identity/bridge [E]/[C]: T_H=kappa/2pi from modular flow (2pi=1/4c3); scalaron Wald entropy S_W=(A/4G)(1+R_h/3M_s^2) exact from v28; area law S_BH=A/4G; the BH/modular identification is [C]",
      "dependencies": "AX.P1.01; HOR.01; v57_horizon_crosslinks.py; v28_gravity_fR.py; v198_modular_commutator_reduction.py",
      "scripts": [
        "v208_bh_thermodynamics.py"
      ],
      "script": "v208_bh_thermodynamics.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "Bisognano-Wichmann; Wald entropy",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 293
    },
    {
      "id": "HOR.BHCORE.01",
      "claim": "BLACK HOLE AS A SEAM FIXED POINT (topological defect), NOT a singularity (2026-06-15, archive integration of the old Five-Problems BH picture; the SUPERSEDED torsion/RN-vortex metric is DROPPED -- RN-type metric superseded by Nariai/seam=horizon v101-v104/v190, cf. v203). Built from established suite atoms. (1) [E] NO SINGULARITY = SEAM FIXED POINT: the gapped attractor multiplier lambda_2=(2/3)^6=64/729 with gap Delta=6 log(3/2)>0 (v56) -- the core is a fixed point (d phi/dt=0), rho->inf replaced by phi->phi_* [P]. (2) [E] MAXIMAL BH = ANCHOR: Nariai cubic t^3-3t+2=(t-1)^2(t+2), roots (1,1,-2) = the traceless projection of the anchor a=(1,1,2) (a-(p1/3)1=-(1/3)(1,1,-2), p1=4; v101/v190). (3) [E] INFORMATION stays in the boundary field: Page recovery decays at lambda_2=(2/3)^6 per step (v54), no interior loss [P]. (4) [E] holographic remnant S_BH=A/4G, 1/4=1/|mu4|, a Planck-scale floor not a point singularity [P]. (5) [P] HONEST: a qualitative structural synthesis (reuses v54/v56/v57/v101/v190), NOT load-bearing, NOT a metric, the old RN/torsion metric NOT resurrected. Python-only",
      "status": "Open/synthesis [O] (a [P]-typed structural reading; not load-bearing) + [E] (the underlying anchor/attractor atoms)",
      "canonicalStatus": "Open/synthesis [O]: BH = seam fixed point (no singularity, phi->phi_*), maximal BH = anchor (Nariai roots (1,1,-2)=traceless a), info via Page recovery (2/3)^6, Planck-scale floor; [P] reading, NOT load-bearing, RN metric not resurrected",
      "dependencies": "v54_seam_horizon_keystones.py; v56_unique_attractor.py; v57_horizon_crosslinks.py; v101_horizon_anchor.py; v190_nariai_entropy_bound.py",
      "scripts": [
        "v209_bh_regular_core.py"
      ],
      "script": "v209_bh_regular_core.py",
      "location": "tfpt_horizon_readouts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 294
    },
    {
      "id": "QGEO.MARKLOCAL.01",
      "claim": "MARK-LOCAL DtN CERTIFIED ON REALISTIC STEKLOV PROFILES + AT THE STATE LEVEL (2026-06-15): a numerical SHARPENING of QGEO.SUBPRIN.01/v201, carried to the quasi-free STATE (the actual omega o rho = omega, not just [rho,Lambda]=0); NOT a closure of QGEO.SYM.01. v201 argued at the abstract mode level; here a real von-Mises curvature bump g(theta) localised at a puncture, summed over the four mu4 marks f(theta)=sum_{j=0}^3 g(theta-j pi/2), builds the Steklov DtN Lambda=|D_theta|+M_f. (1) [E] every off-(mod 4) Fourier coefficient of the REAL f vanishes to <1e-16 (exact 4-fold cancellation), although g itself has all modes. (2) [N] ||[rho,Lambda]||<1e-15 (principal |k| diagonal, v198; sub-principal M_f Z4-invariant). (3) [N] STATE INVARIANCE (the genuinely new step vs v201): the quasi-free covariance C=1/2(1+sgn H_1) satisfies rho C rho^dag = C to <1e-13 and [rho,C]<1e-13 -- the STATE, not just the operator, is clock-invariant. (4) [N] negative controls (a single off-centre bump, a Z3 3-mark source, 4 generic unequal marks) each break BOTH the operator and the state by O(1) -- specific to the mu4 orbit. (5) [N] convergence: mark-local commutator stays <1e-10 for N in {16,32,64}, the Z3 control stays O(1) -- not a truncation artefact. (6) [O] VERDICT: certifies the implication 'mark-local DtN => omega o rho = omega' for concrete Steklov profiles AND at the state level (the operational content of QGEO.SUBPRIN.01); the PREMISE that the physical seam DtN IS mark-local / conformal-deck flat stays [O] -- the narrowest definitional form of QGEO.SYM.01, NOT its closure (net existence + full-cone RP remain the [E] of v175). Python-only (numerical; the exact mark-sum identity is Wolfram-mirrored via v201)",
      "status": "Numerical certification [N] (implication on realistic profiles + state level) + premise [O] (the physical seam being mark-local)",
      "canonicalStatus": "Numerical/premise [N]/[O]: mark-local DtN => omega o rho = omega certified for realistic von-Mises Steklov profiles AND at the quasi-free state level (rho C rho^dag = C to 1e-13); negative controls (off-centre, Z3, generic) break by O(1); the seam being mark-local stays [O], the narrowest form of QGEO.SYM.01, not a closure",
      "dependencies": "QGEO.SUBPRIN.01; QGEO.STATE.01; QGEO.MODULAR.01; v201_seam_subprincipal_marks.py; v198_modular_commutator_reduction.py",
      "scripts": [
        "v210_mark_local_dtn.py"
      ],
      "script": "v210_mark_local_dtn.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 295
    },
    {
      "id": "FORM.QGEO.01",
      "claim": "LEAN 4 FORMALISATION of the QGEO.SYM.01 CONDITIONAL THEOREM (2026-06-15; the algebraic core of v201/v210, machine-proved) -- AUDIT: PASS, no sorry/admit, no domain axioms (#print axioms = propext, Classical.choice, Quot.sound only; the two block-diagonality theorems even use only propext+Quot.sound). New module TfptCarrier/SeamDeckClosure.lean proves: (1) [F] geom_sum_fourth_root: for ζ:ℂ with ζ^4=1, Σ_{j<4} ζ^j = (if ζ=1 then 4 else 0) -- the 4th-root character orthogonality (finite geometric sum) that makes a μ4-mark sum vanish off mod 4; (2) [F] clock_gen_pow_four + mark_sum_residue_nonzero: the clock generator -i has order 4 and the three non-trivial residues give a vanishing mark sum, so a μ4-mark-sourced curvature is supported only on modes ≡0 (mod 4); (3) [F] markLocal_blockDiagonal: a mark-local Toeplitz symbol (support ⊆ {m : (m:ZMod 4)=0}) connects only equal clock-characters (f_{n-n'}≠0 ⇒ (n:ZMod4)=(n':ZMod4)), i.e. [ρ,M_f]=0; (4) [O] SeamDeckPremise (structure): the PHYSICAL premise that the raw seam DtN sub-principal symbol is mark-local, encoded as a typed target exactly like CalderonProjector -- consumed, NOT proved; (5) [F] SeamDeckPremise.clock_invariant: GIVEN the premise, the clock commutes with the DtN at the character level => the quasi-free covariance C=1/2(1+sgn H1) is clock-invariant => omega o rho = omega. So the IMPLICATION mark-local => omega o rho = omega is now [F] (machine-proved); the PREMISE (the physical seam IS mark-local) stays [O] -- the one fundamental seam-identification postulate, NOT closed. Complements FORM.GLUE.01/FORM.LADDER.01; mirrors v201/v210.",
      "status": "Formal [F] (conditional theorem) + [O] (the seam-deck premise)",
      "canonicalStatus": "Formal [F]/[O]: the QGEO.SYM.01 conditional theorem (mark-local DtN => omega o rho = omega, via 4th-root orthogonality + character block-diagonality) is Lean-proved (AUDIT: PASS, clean axioms); the seam-deck premise stays [O], a typed target like CalderonProjector -- not a closure of QGEO.SYM.01",
      "dependencies": "QGEO.MARKLOCAL.01; QGEO.SUBPRIN.01; QGEO.MARKS.01; FORM.GLUE.01",
      "scripts": [
        "lean4-carrier-rigidity"
      ],
      "script": "lean4-carrier-rigidity",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 296
    },
    {
      "id": "FORM.QGEO.02",
      "claim": "LEAN 4 FORMALISATION of QGEO.UNIFORM.01 (the geometric normal form of v177; the UNIFORM node of the QGEO proof tree) -- AUDIT: PASS, no sorry/admit, no domain axioms (only propext, Classical.choice, Quot.sound). New module TfptCarrier/MobiusUniformisation.lean machine-proves the constructive uniformisation content: (1) [F] rho_pow_four: the clock rho:z->iz has rho^4=id; (2) [F] rho_order_exactly_four: rho^2 != id (rho^2(1)=-1), so the clock has order EXACTLY 4; (3) [F] sigma_invol: the reflection sigma:z->1/z is an involution; (4) [F] sigma_rho_sigma: sigma.rho.sigma = rho^{-1} (the dihedral relation => <rho,sigma>=D4 of order 8); (5) [F] orbit_scales_to_mu4: a non-fixed point a!=0 has orbit {a,ia,-a,-ia} that scales (z->z/a) exactly to mu4={1,i,-1,-i}; (6) [F] sigma_perm_mu4: sigma permutes mu4 (fixes 1,-1; swaps i,-i); (7) [F] mult_order_four_iff: a multiplier map z->zeta z has order exactly 4 (zeta^4=1, zeta^2!=1) IFF zeta=i or zeta=-i (the two primitive 4th roots; the inversion conjugates one to the other => canonical rep z->iz). Bundled as uniformisation_normal_form; signature-locked in AuditContract.lean. SCOPE: this is the constructive normal form GIVEN the four marks + clock (the [E] UNIFORM node); the RAW-seam obligation QGEO.MARKS.01 (the seam canonically producing a genus-0 four-marked D4 boundary) stays [O], NOT closed. Mirrors v177 (QGEO.UNIFORM.01); complements FORM.QGEO.01.",
      "status": "Formal [F] (uniformisation normal form) + [O] (the raw-seam marking obligation QGEO.MARKS.01)",
      "canonicalStatus": "Formal [F]/[O]: QGEO.UNIFORM.01 (order-4 Moebius normal form z->iz, D4 relation sigma.rho.sigma=rho^{-1}, orbit->mu4, multiplier classification zeta=+-i) Lean-proved (AUDIT: PASS, clean axioms, signature-locked); the raw-seam marking obligation QGEO.MARKS.01 stays [O]",
      "dependencies": "QGEO.TREE.01; QGEO.MARKS.01; FORM.QGEO.01; FORM.GLUE.01",
      "scripts": [
        "lean4-carrier-rigidity"
      ],
      "script": "lean4-carrier-rigidity",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 297
    },
    {
      "id": "FORM.QGEO.03",
      "claim": "LEAN 4 FORMALISATION of QGEO.COHOM.01 (the character node of v177) + the MODULE reflection parity -- AUDIT: PASS, no sorry/admit, no domain axioms (only propext, Classical.choice, Quot.sound). New module TfptCarrier/CohomologyGrading.lean machine-proves: (1) [F] the three H^1(P^1 minus mu4) eigenforms omega_k = z^{k-1}/(z^4-1) (k=1,2,3) have pullback character rho* omega_k = i^k omega_k under the clock rho:z->iz -- omega1_pullback (i^1=i), omega2_pullback (i^2=-1), omega3_pullback (i^3=-i), each an exact rational-function identity (the denominator is clock-invariant, (iz)^4-1=z^4-1 since i^4=1); (2) [F] character_grading: the eigenvalues (i,-1,-i) = the characters of weights (1,2,3) = the A3 exponents = Spec(Q_+), rank 3 = N_fam; (3) [F] the MODULE reflection parity: the reflection sigma:z->1/z (pullback g(1/z)*(-1/z^2)) satisfies sigma* omega_1 = omega_3, sigma* omega_2 = omega_2, sigma* omega_3 = omega_1 (omega1_reflection/omega2_reflection/omega3_reflection) -- the integer-model parity w1<->w3, w2 fixed (v141/v146/v177); bundled cohom_grading, signature-locked in AuditContract.lean. SCOPE: this is the geometric COHOM/MODULE-parity content GIVEN the marked curve (P^1, mu4) and the clock (the [E] middle nodes); it does NOT close QGEO.MARKS.01/QGEO.KERNEL.01, and the MODULE *uniqueness* (multiplicity-free + residue normalisation) stays the symbolic v177 reading. Mirrors v177 (QGEO.COHOM.01); complements FORM.QGEO.01/02.",
      "status": "Formal [F] (COHOM character grading + MODULE reflection parity) + [O] (the raw-seam obligations)",
      "canonicalStatus": "Formal [F]/[O]: QGEO.COHOM.01 (pullback character rho* omega_k = i^k omega_k, grading (1,2,3)=A3 exponents) + MODULE reflection parity (sigma swaps omega1<->omega3, fixes omega2) Lean-proved (AUDIT: PASS, clean axioms, signature-locked); the MARKS/KERNEL obligations and MODULE uniqueness stay [O]",
      "dependencies": "QGEO.TREE.01; QGEO.MARKS.01; FORM.QGEO.01; FORM.QGEO.02",
      "scripts": [
        "lean4-carrier-rigidity"
      ],
      "script": "lean4-carrier-rigidity",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 298
    },
    {
      "id": "DM.AXION.SPINE.01",
      "claim": "AXION DM SPINE-ANGLE BRANCH theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg (2026-06-15): an ALTERNATIVE, more ROBUST misalignment angle than the determinant-line hilltop theta_i ~ 170 deg (FR.DM.03/v185, which over-produces Omega_a h^2 ~ 0.66 ~ 5.5x). (1) [I] RATIONAL MOTIVE: theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg EXACTLY (the central spine quotient 3/5 = N_fam/g_car, a rational multiple of pi, NOT tuned to the relic). (2) [N] ON TARGET: the v185 converged finite-T solver (normalised theta=1 -> Omega_a h^2 ~ 0.03) reaches Omega_DM h^2 = 0.12 at theta ~ 106 deg; the spine angle 108 deg sits essentially there (~2 deg away), harmonic estimate 0.03 theta^2 = 0.107 confirms the order, a mild anharmonic factor lifts to ~0.12. (3) [C] ROBUST (the genuine advantage over v185): 108 deg lies 62 deg BELOW the 170 deg hilltop, in the MILD-anharmonic regime, so the relic is NOT exponentially sensitive -- a far more robust landing on Omega_DM; IF theta_i is the spine quotient, the determinant-line axion is ALL of dark matter without hilltop tuning. (4) [O] HONEST -- AN ALTERNATIVE ANSATZ: theta_i = 3 pi/5 (spine motive) and theta_i = pi(1-phi_seam) ~ 170 deg (seam-value motive, v185) are MUTUALLY EXCLUSIVE readings of the same angle; neither is forced -- the full finite-T solver decides. A sharper SCENARIO, not a derivation; abundance is cosmology, never a compiler output. (5) [X] KILL TEST: a full finite-T solve at theta_i = 3 pi/5, f_a = M_scal/128, N_DW = 1 -- Omega_a h^2 outside ~[0.08,0.16] demotes the spine branch (compiler core untouched; the no-WIMP E8 scan is independent). Python-only. POST-INFLATIONARY REFRAMING [C] (2026-07-10, FR.DM.02): given H_inf/f_a~66>1 (R^2 branch) a FIXED pre-inflationary spine angle carries CDM isocurvature >> Planck beta_iso<0.038; in the natural POST-INFLATIONARY scenario theta_i is a HORIZON AVERAGE, NOT observable -- so 108 deg = 3 pi/5 is retained as a STRUCTURAL RATIONAL (3/5=N_fam/g_car), NOT a relic prediction; the relic follows from f_a + N_DW=|C|=1 (v470/v472). The [X] angle kill test is SUPERSEDED by the post-inflationary reading (angle not measurable).",
      "status": "Physical [C] (spine angle = structural rational / NOT observable post-inflationary; relic from f_a + N_DW=1; reframed by FR.DM.02)",
      "canonicalStatus": "Physical [C]: theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg (spine quotient) lands on Omega_DM ~106 deg ROBUSTLY (mild-anharmonic, far below the 170 deg hilltop) -- a more robust DM branch than v185; an alternative ansatz, the full solver decides; kill test outside ~[0.08,0.16] demotes the branch",
      "dependencies": "FR.DM.01; FR.DM.02; FR.DM.03; v185_axion_relic_solver.py",
      "scripts": [
        "v211_axion_spine_angle.py"
      ],
      "script": "v211_axion_spine_angle.py",
      "location": "tfpt_4_frontier",
      "externalData": "Planck Omega_DM h^2 = 0.12",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 299
    },
    {
      "id": "DM.AXION.PENTAGON.01",
      "claim": "THE 'UNMAPPED' ICOSAHEDRAL/GOLDEN E8 STRUCTURE SURFACES IN THE ONE EXTERNAL COSMOLOGICAL INPUT theta_i (2026-06-26, v429): a geometric re-reading answering the parsimony question 'why does so much idle E8 structure exist?' for the one case where the idle structure has a job. [E] THETA_I = g_car-GON INTERIOR ANGLE: because N_fam = g_car-2, the spine quotient theta_i = pi N_fam/g_car = (g_car-2)pi/g_car is EXACTLY the interior angle of the regular g_car-gon; for g_car=5 the PENTAGON, theta_i = 3 pi/5 = 108 deg (the carrier polygon, not a bare rational multiple of pi). [E] COS IS GOLDEN: cos(3 pi/5) = (1-sqrt5)/4 = -1/(2 phi), partner 2 cos(2 pi/5) = 1/phi = phi-1, phi = 2 cos(pi/5) = (1+sqrt5)/2 -- the golden ratio v354 called 'unmapped' (and v313/GOLD.ATOMS.01 named the g_car=5 signature) IS theta_i's angular data. [E] GOLDEN CHARACTER UNIQUE TO g_car=5: among small regular n-gons only n=5 has an irrational interior-angle cosine (n=3,4,6 give 1/2,0,-1/2, all rational); so theta_i is golden BECAUSE the carrier is 5 (P2). [C] BRIDGE (NO STATUS CHANGE): connects E8.REVERSE.AUDIT.01/GOLD.ATOMS.01 (phi = the unmapped g_car=5 signature) to DM.AXION.SPINE.01/v211 (theta_i = 3 pi/5 the one external input); REFINES v354 from 'phi in NO physical readout' to 'phi touches ONLY the [C] spine misalignment angle' (conditional, branch-level, never a frozen prediction); a geometric MOTIVE for the spine over the hilltop theta_i ~ 170 deg (whose cosine is not a clean pentagon value), consistent with -- not replacing -- the finite-T solver that decides the branch (v185/v211). Does NOT derive theta_i, does NOT upgrade DM.AXION.SPINE.01 (stays [C]), closes no gate. Wolfram-mirrored, Python (sympy)",
      "status": "Identity [E] (theta_i = regular g_car-gon interior angle; cos = -1/(2 phi); golden character unique to g_car=5) + [C] bridge re-reading (no status change)",
      "canonicalStatus": "Identity [E]: theta_i = 3 pi/5 is the regular g_car-gon (pentagon) interior angle, cos theta_i = -1/(2 phi), and the golden character is unique to g_car=5 -- so the 'unmapped' golden/icosahedral E8 structure (v354/v313) is the geometry of the one external cosmological input theta_i (v211); [C] a re-reading that refines v354 and gives a geometric motive for the spine branch, but does NOT upgrade DM.AXION.SPINE.01 (stays [C]) or close any gate",
      "dependencies": "DM.AXION.SPINE.01; E8.REVERSE.AUDIT.01; GOLD.ATOMS.01",
      "scripts": [
        "v429_axion_pentagon_phi.py"
      ],
      "script": "v429_axion_pentagon_phi.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 300
    },
    {
      "id": "FR.ETAB.04",
      "claim": "LEPTOGENESIS SCALARON-DECUPLE BRANCH (2026-06-15): BOTH Boltzmann inputs share the decuple A_Lambda = 10 = |E(K5)| -- M_1 = M_scal phi0^2/A_Lambda ~ 8.65e9 GeV and m~1 = m_3/A_Lambda ~ 5 meV. A cleaner [C] route than FR.ETAB.03/v184's M_1 = M_R phi0^4, which only RELOCATED the free input to the un-pinned seesaw scale M_R. (1) [I] SHARED DECUPLE: A_Lambda = 10 = |E(K5)| (action-grammar atom) divides BOTH inputs -- M_1 = M_scal phi0^2/A_Lambda, m~1 = m_3/A_Lambda; leptogenesis = scalaron reheating dressed by two seed insertions (phi0^2), over the pair sector (A_Lambda). (2) [N] HEAVY MASS in the window: M_1 = M_scal phi0^2/A_Lambda = 8.65e9 GeV (M_scal = c3^{7/2} Mbar ~ 3.06e13 GeV, phi0 ~ 0.053172), inside the natural thermal window [3e9, 3e10] GeV (v169) and above the Davidson-Ibarra floor ~1e9 GeV. (3) [N] WASHOUT anchor: m~1 = m_3/A_Lambda ~ 5 meV (m_3 ~ sqrt(dm2_atm) ~ 0.05 eV), K = m~1/m_* ~ 4.6 (strong washout, m_* ~ 1.08 meV) -- the SAME A_Lambda = 10 as M_1. (4) [N] eta_B LANDS: Davidson-Ibarra eps_1 = (3/16 pi) M_1 m_3/v^2 ~ 8.5e-7; with kappa_f ~ 0.15 (strong washout) and sphaleron 28/79, eta_B ~ 0.96e-2 eps_1 kappa_f ~ 1.2e-9 -- O(1e-9), bracketing the observed 6.1e-10 (v169 band), no tuning. (5) [C] HONEST: unlike v184's M_1 = M_R phi0^4 (relocates the free input to the un-pinned M_R), here M_1 uses ONLY {M_scal, phi0, A_Lambda} -- no hidden seesaw scale, both inputs share A_Lambda; but the coupling MECHANISM (why M_1 = M_scal phi0^2/A_Lambda) is posited, not derived, and kappa_f is washout-dependent -- so eta_B stays [C], a sharper scenario, not a derivation. (6) [X] KILL TEST: a precise flavored-Boltzmann/RGE solve at M_1 = M_scal phi0^2/A_Lambda, m~1 = m_3/A_Lambda -- an eta_B outside the observed band falls the ROUTE, not the theory (Route A / Omega_b is independent, FR.ETAB.01). Python-only",
      "status": "Physical [C] (sharper scenario; both Boltzmann inputs share A_Lambda=10, no hidden seesaw scale) + [X] kill test",
      "canonicalStatus": "Physical [C]: M_1 = M_scal phi0^2/A_Lambda ~ 8.65e9 GeV and m~1 = m_3/A_Lambda ~ 5 meV both use the decuple A_Lambda=10 (no hidden seesaw scale) -- cleaner than v184's M_R phi0^4; eta_B brackets 6.1e-10 but stays [C] (coupling mechanism posited, not derived)",
      "dependencies": "FR.ETAB.01; FR.ETAB.02; FR.ETAB.03; v169_etaB_boltzmann_interface.py",
      "scripts": [
        "v212_leptogenesis_decuple.py"
      ],
      "script": "v212_leptogenesis_decuple.py",
      "location": "tfpt_4_frontier",
      "externalData": "NuFIT (dm2_atm); Planck (eta_B = 6.1e-10)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 301
    },
    {
      "id": "CONTRACT.F.01",
      "claim": "THE F_transfer FUNCTOR CONTRACT (2026-06-15): the four frontier transfers are ONE typed functor F_transfer = F_observable o F_threshold o F_RG satisfying four structural axioms, each instance a DISCRETE compiler kernel [E/I] composed with an EXTERNAL continuous solver [C/O]. This is the third research contract (alongside U_wall and G_metric), the structural complement of the v187 typing GUARD: v187 enforces the [C]/[O] markers, CONTRACT.F.01 verifies the functor structure. The four instances are F_pole (Koide, v183), F_Boltzmann (eta_B, v212), F_relic (axion, v211), F_QCD (m_p/m_e, v164/FR.MPME). (1) [I] AXIOM 1 mu4-DECK EQUIVARIANCE: the discrete kernels are carrier/deck invariants {|mu4|=4, N_fam=3, g_car=5, A_Lambda=10}; the Koide relaxation multiplier lambda_2 = (2/3)^6 = 64/729 IS the mu4-deck transfer eigenvalue (the deck map fixes the branch divisor, v82/v54), so F_pole commutes with the deck. (2) [I] AXIOM 2 PLUCKER PRESERVATION: where the observable is dimensionless the discrete readout is an Anchor-Plane Plucker reading -- Koide 53 = a^T(R+Q)1 is the F-corner reading (Pl(F)=(1,22,30)), the quark ratio uses ||Pl(K)||_1 = 11; the functor carries these integer Plucker invariants unchanged. (3) [I] AXIOM 3 POSITIVITY/STOCHASTICITY: where Boltzmann/relic densities enter the kernel is positive -- spec(T) = {1,(2/3)^6,(1/3)^6} is positive (RP/OS, v54), the washout kappa_f in (0,1), Omega_a > 0. (4) [I] AXIOM 4 EXTERNAL MODULES EXPLICIT: the continuous content of each instance is a named external module (pole transfer; flavored Davidson-Nardi-Nir Boltzmann; finite-T misalignment ODE with chi(T); non-perturbative lattice C_N), never a compiler number; b_3 = -7 is a carrier output (|b_3| = 11 - (2/3)(2 N_fam) = Omega_adm - 10 b_1 = 48 - 41 = 7). (5) [C] VERDICT: F_transfer is a typed FUNCTOR with four axioms, NOT a bag of open topics; the discrete kernels are [E]/[I] (exact compiler invariants), the external solvers [C]/[O] (standard physics). An architectural consolidation, NOT a closure -- each measured observable still needs its external solver. [2026-08-05 LXII, THE JET ROUTE CLOSES + THE FIBERED REFINEMENT (v777): the executed external-clock contract FTRANSFER.CLOCKS.01 fires its preregistered K1 kill sympy-exactly (transported Schwarzians {-Delta^2/2, -Delta^2/2, 0}; Delta = 6 log(3/2)) -- NO common continuous clock exists for the four jets; the surviving architecture is measured to be a FIBERED functor: one shared DISCRETE PGL2 base (anchor cross-ratio 4/3 exact -- symbolic AND through the frozen g_*s table at dev 6.5e-4; deck unit |Delta sigma| = 1; the deck generator is the unit parabolic translation tr^2 - 4 det = 0, the v632 lattice-explicit discrete half) + channel-specific projective connections whose obstruction is the CONSTANT seam cocycle with values EXACTLY {0, +-Delta^2/2} (antisymmetry + cocycle identity exact on all 27 triples, d/dsigma identically zero); classification matches the predeclared marker: thermal/proper-time clocks {tau, beta} -> coset -Delta^2/2, RG clock {log mu} -> coset 0, relic degenerate NOT classified; torsor difference = Delta^2/2 exactly.  The contract's four functor axioms stand; the jet/continuous-clock unification demand is REPLACED by the fibered form, with the Schwarzian differences stored as the obstruction cocycle -- not fitted away.  The one named proof-grade next step: derive the coset assignment from KMS/modular clock typing.  The four observables stay [C]/[O]; no marker move.] Python-only UPDATE (2026-07-31, v566): a preregistered fit-free cross-ratio check is added to the contract surface -- if the four external solvers export canonical projective states (declared BEFORE evaluation), Moebius transport along the v533 disc -7 norm line forces CR(y_J, y_K; y_C, y_F) = 4/3 exactly (the universal arithmetic-progression cross-ratio; negative control: the raw determinant path gives 28/25 -- the known norm sequence cannot pass); the fourth state is forced from the first three (y_F reconstruction identity). Test design only; no solver states are exported yet; contract status unchanged. [2026-07-31: the preregistered cross-ratio check is EXECUTED on its discrete half by v571 (CR-PASSES; y_F = 53 Moebius-forced by the upstream corners); the continuous-solver half stays open; see FTR.CRDISC.01.] [2026-07-31: the continuous half of the cross-ratio check executed by v575 (UNIFLOW-DEGENERATE -- native exports cannot form four independent projective states, by the v425 single-flow theorem); both halves of the check are now executed; see FTR.CRCONT.01.]  NOTE (2026-08-06, v792 -- marker UNCHANGED): the fibered refinement's clock-fiber classification is completed at theorem level -- the Schwarzian coset carried by a clock chart is a FUNCTION of its KMS/modular typing datum alone (KMS-COSET-THEOREM: modular/thermal clocks carry -Delta^2/2, autonomous RG charts carry 0), with the non-circularity of the classifier gated on its own source; see FTRANSFER.KMSCOSET.01.",
      "status": "Contract [C]/[O] (typed functor: discrete kernels [E], external solvers [C]; not a closure)",
      "canonicalStatus": "Contract [C]/[O]: F_transfer = F_observable o F_threshold o F_RG, four axioms (mu4-deck equivariant, Plucker-preserving, positive/stochastic, external-module-explicit), four instances v183/v212/v211/v164; discrete kernels [E], external solvers [C]; the third research contract alongside U_wall/G_metric, guarded by v187, NOT a closure; 2026-08-05 (v777): the jet route is closed by the executed K1 kill of FTRANSFER.CLOCKS.01 -- the surviving form is the FIBERED functor (discrete PGL2 base, constant seam obstruction cocycle {0, +-Delta^2/2}, thermal-vs-RG coset split, relic degenerate); next named step: coset assignment from KMS/modular clock typing",
      "dependencies": "FR.KOIDE.04; FR.ETAB.04; DM.AXION.SPINE.01; FR.MPME.01; v187_ftransfer_laws.py",
      "scripts": [
        "v213_ftransfer_functor.py"
      ],
      "script": "v213_ftransfer_functor.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 302
    },
    {
      "id": "DIAMOND.AXIS.01",
      "claim": "DIAMOND AXIS CURVATURE LEMMA (2026-06-15): along the two canonical axes of the centered cross (C center, U winding, V sheet, v95) the Sheet-Diamond determinant is LINEAR on the winding axis and QUADRATIC on the sheet axis: det(C+xU)=14+6x (flat, slope 6=|R^+(A_3)|), det(C+yV)=14+14y+4y^2 (2nd difference 8=rank E_8); the anchor block satisfies det B(C+xU)=28+12x=2 det(C+xU) and det B(C+yV)=28+21y+3y^2 (2nd difference 6=|R^+(A_3)|). So winding flows linearly (A_3-driven) and the sheet split curves, with curvatures (8 det, 6 anchor)=(rank E_8, |R^+(A_3)|) -- a discrete differential geometry of the diamond. Exact identity UPDATE (2026-07-31, v566, DIAMOND.PARA.SELFCODE.01): the direction operators U, V generate the full 2|1 parabolic algebra (dim 7); the anchor a = (1,1,2) is the spectrum of I + V(V-I)/2 with chi_H coefficients (4,5,2) = (flag, Levi, radical) -- the double self-code; the staircase-sum 81 is simultaneously the saturated lattice index [P_Z : Z<U,V>] = 3^4 (cokernel (Z/3)^4). Markers unchanged.",
      "status": "Identity (exact)",
      "canonicalStatus": "[E] det(C+xU)=14+6x linear, det(C+yV)=14+14y+4y^2 quadratic; curvatures (8,6)=(rank E_8, |R^+(A_3)|)",
      "dependencies": "v94_sheet_diamond.py; v95_centered_diamond.py",
      "scripts": [
        "v218_diamond_axis_geometry.py"
      ],
      "script": "v218_diamond_axis_geometry.py",
      "location": "tfpt_2_standard_model",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 303
    },
    {
      "id": "DIAMOND.PLUCKER.01",
      "claim": "PLUCKER TRANSFER LADDER (2026-06-15): the anchor-Plucker coordinates lift K->C->F in two exact steps: Pl(K)=(-1,6,4), Pl(C)=(0,14,14), Pl(F)=(1,22,30); Pl(C)-Pl(K)=(1,8,10)=(N_Phi, rank E_8, A_Lambda) and Pl(F)-Pl(C)=(1,8,16)=(N_Phi, rank E_8, dim S^+). Each step adds (1,8) in the first two coordinates; the third adds first the decuple A_Lambda=10 then the full spinor generation dim S^+=16. The Plucker identity is [E]; the F_transfer source->pole reading (a two-stage sheet lift in the Anchor Plane) stays [C]",
      "status": "Identity (exact)",
      "canonicalStatus": "[E] Pl ladder K->C->F: steps (1,8,10) and (1,8,16) = decuple then full generation; transfer reading [C]",
      "dependencies": "v94_sheet_diamond.py; v213_ftransfer_functor.py",
      "scripts": [
        "v218_diamond_axis_geometry.py"
      ],
      "script": "v218_diamond_axis_geometry.py",
      "location": "tfpt_2_standard_model",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 304
    },
    {
      "id": "DIAMOND.SPECTRAL.01",
      "claim": "SPECTRAL RAMIFICATION LADDER (2026-06-15): for Q,K,C,F the characteristic polynomial splits as (t-r)(t^2-bt+c) with r rational, and the full cubic discriminant factors as Disc(chi)=q(r)^2 Disc(q): squares |q(r)| in {1,3,4,6}=(N_Phi,N_fam,|mu4|,|R^+(A_3)|), kernels Disc(q) in {13,48,65,105}=(Delta_Q, Omega_adm, g_car Delta_Q, N_fam g_car 7), with Delta_Q=13=b^2+s^2=|R(A_3)|+1 the carrier-split norm (v14). The missing corner F carries the kernel 105=3*5*7=family*carrier*scalaron. Exact identity. (Audit blocks in the same module: the anchor-defect spectrum Delta_a(Q,K,R,C,L,F)=(8,16,18,21,24,26) and the pair-sum atlas/determinant staircase are AUDIT fingerprints -- the 28=2 dim G_2, 52=dim F_4 labels stay audit-only, not spine; F as transfer-completion corner is a HEURISTIC, not a hard filter)",
      "status": "Identity (exact)",
      "canonicalStatus": "[E] Disc(chi)=q(r)^2 Disc(q) for Q,K,C,F: squares (1,3,4,6), kernels (13,48,65,105); audit/heuristic blocks honestly typed",
      "dependencies": "v14_carrier_uniqueness.py; v94_sheet_diamond.py",
      "scripts": [
        "v218_diamond_axis_geometry.py"
      ],
      "script": "v218_diamond_axis_geometry.py",
      "location": "tfpt_2_standard_model",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 305
    },
    {
      "id": "MCKAY.E8.01",
      "claim": "McKAY BEDROCK (2026-06-16): WHY the three atoms are {2,3,5}. E8 is the exceptional TOP of the McKay tower of finite SU(2) subgroups (2T->affine E6 h=12, 2O->affine E7 h=18, 2I binary icosahedral order 120 -> affine E8 h=30=2*3*5). Built FROM the group, not asserted: (1) the 120 unit-quaternion icosians close to 2I with element orders {1,2,3,4,5,6,10} (it CONTAINS the 2,3,5 rotation-axis orders), 9 conjugacy classes; (2) the 9 irreducible character DEGREES from the class algebra (Dixon) are {1,2,2,3,3,4,4,5,6}, sum=30=h(E8), sum of squares=120=|R^+(E8)|; (3) the McKay adjacency A_ij=<chi_V chi_j,chi_i> from the natural 2-dim rep IS the affine E8 graph (tree, top eigenvalue 2), with Kac marks = the irrep degrees; (4) negative controls 2T->E6, 2O->E7 -- only the icosahedral (2,3,5) gives E8. A BACKWARD CERTIFICATE of the already-closed E8, NOT an independent P2 proof.",
      "status": "Lattice (McKay correspondence, exact)",
      "canonicalStatus": "[E] 2I (order 120) irrep degrees {1,2,2,3,3,4,4,5,6} = affine E8 Kac marks, sum 30=h(E8), sum^2 120=|R^+(E8)|; the McKay graph built from the group; E8 = the (2,3,5) icosahedral top. Backward certificate [E], 'explains (2,3,5)' [C], raw-seam origin [O]",
      "dependencies": "AX.P2.01; E8.GLU.01; ARCH.CORE.01",
      "scripts": [
        "v219_icosahedral_mckay.py"
      ],
      "script": "v219_icosahedral_mckay.py",
      "location": "origin_theory",
      "externalData": "McKay correspondence (Du Val/Slodowy)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 306
    },
    {
      "id": "CMNORM.DUAL.01",
      "claim": "CM-NORM DUALITY (2026-06-16): the two exceptional CM moduli give two central TFPT integers as ideal NORMS. SQUARE seam (Gaussian Z[i], j=1728): N(g_car+i|mu4|)=5^2+4^2=41=10 b1 -- the EM index IS the Gaussian norm of the carrier-glue vector; N(3+2i)=13=Delta_Q. HEX partner (Eisenstein Z[omega], j=0): N(3+2 omega)=3^2-3*2+2^2=7=scalaron exponent. The (3,2)=(colour,weak) split generates (5,6,7,13)=(g_car,|R^+(A3)|,scalaron,Delta_Q) by sum/product/Eisenstein-norm/Gauss-norm; rings rigid (Eisenstein of (5,4)=21!=41, Gauss of (3,2)=13!=7).",
      "status": "Identity (exact ring arithmetic)",
      "canonicalStatus": "[E] 41=N_Zi(5,4)=10 b1, 13=N_Zi(3,2)=Delta_Q, 7=N_Zomega(3,2)=scalaron; (3,2)->(5,6,7,13); the square<->EM, hex<->scalaron architecture bridge [C]",
      "dependencies": "ANCHOR.GEN.01; EM.B1.01; SEAM.KHALF.01",
      "scripts": [
        "v222_cm_norm_duality.py"
      ],
      "script": "v222_cm_norm_duality.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 307
    },
    {
      "id": "COX.CLOCK.01",
      "claim": "COXETER TOTATIVE CLOCK (2026-06-16): the mu4 clock is the order-4 character of the E8 Coxeter cycle. (Z/30)^x={1,7,11,13,17,19,23,29}=the E8 exponents=live Coxeter phases; phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=2*3*5. (Z/30)^x ~ Z/4 x Z/2, order-4 generator 7 (7^2=19,7^4=1 mod 30): a literal mu4 clock <7>={1,7,13,19} on the exponents; conjugate pairs m+(30-m)=30 give |mu4|=4 invariant 2-planes the clock permutes. Only E8 has phi(h)=rank AND h=2*3*5. Complements MCKAY.E8.01 (McKay gives (2,3,5) from the finite group, this gives mu4 from the Coxeter cycle).",
      "status": "Identity (exact)",
      "canonicalStatus": "[E] (Z/30)^x = E8 exponents, order-4 element 7 = mu4 clock on the exponents, 4 invariant planes; seam-clock identification [C]",
      "dependencies": "SEAM.COX.01; AX.P2.01",
      "scripts": [
        "v223_coxeter_totative_clock.py"
      ],
      "script": "v223_coxeter_totative_clock.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 308
    },
    {
      "id": "E8.CHAN.01",
      "claim": "DEGREE/EXPONENT CHANNEL SPLIT (2026-06-16): 248=120+128 as a magnitude/phase channel typing (replaces the over-strong 'S^- is dark matter'). E8 exponents {1,7,11,13,17,19,23,29} sum=120=|R^+(E8)| (MAGNITUDE channel: positive roots/root transitions); E8 degrees {2,8,12,14,18,20,24,30} sum=128=2^(rank-1)=rank*dim S^+ (PHASE/GLUE channel); 248=120+128=adj(D8)+spinor(D8); sum(deg)-sum(exp)=rank=8. The (ratios,product) magnitude bijection (Target D) lives in the 120 channel, the un-covered CP phases in the 128 phase/glue channel; whether 128 hosts a dark sector is Frontier (NOT asserted). STRENGTHENED on the fiber (2026-07-29, T177/v561): the channel typing has an EXACT hexagonal realisation -- sum_k rho^{d_k} = 4 rho (arg = pi/3 exactly) and sum_k rho^{m_k} = 4 (arg = 0 exactly) on the CM unit rho = e^{i pi/3} -- with a parity typing forced by integer parity: all degrees are EVEN (phase channel SHEET-EVEN under rho -> rho^4), all exponents ODD (magnitude channel SHEET-ODD). CONSEQUENCE: the phase-channel angle is sheet-blind -- it carries pi/3 but cannot distinguish quark from lepton (not the CP.MU6.01 sheet split). Honest caveat: reality of the exponent sum alone is not hexagon-specific; only the degree half (= 4 rho) is. The channel typing stays [C].",
      "status": "Identity (exact)",
      "canonicalStatus": "[E] 248=120+128, exponents sum 120=|R^+(E8)|, degrees sum 128=rank*dim S^+; magnitude channel 120 / phase channel 128 [C]; dark-sector reading [O]; exact hexagonal realisation sum rho^{d_k}=4rho (arg pi/3, sheet-even) vs sum rho^{m_k}=4 (arg 0, sheet-odd) [E] -- the phase-channel angle is sheet-blind (T177/v561)",
      "dependencies": "E8.CAS.01; FLAV.CP.01",
      "scripts": [
        "v227_degree_exponent_channel_split.py",
        "v561_cp_channel_identities.py"
      ],
      "script": "v227_degree_exponent_channel_split.py; v561_cp_channel_identities.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 309
    },
    {
      "id": "QGEO.RR.01",
      "claim": "RIEMANN-ROCH INDEX GATE (2026-06-16): P2 as the mode space of a degree-4 seam divisor (bedrock LANGUAGE for P2, NOT a proof from nothing). deg D=4=|mu4| on P^1 => h0(O(D))=deg+1=5=g_car (Riemann-Roch, genus 0); rank H1(P^1 minus 4 marks)=4-1=3=N_fam; h0+rank H1=8=rank E8, h0-rank H1=2=|Z2|, (h0+rank H1)/2=4=|mu4|; Lambda^even(C^5)=1+10+5=16=dim S^+ (the D5 half-spinor, ARCH.RRCAR.02). Negative controls: deg 3->(4,2), deg 5->(6,4), colliding marks lower H1 and break N_fam. Relocates P2 from '5 slots' to 'the genus-0 mode count of a 4-mark divisor', conditional on the raw seam producing the degree-4 divisor (=QGEO.SYM.01).",
      "status": "Identity (exact index) [E]/[C]/[O]",
      "canonicalStatus": "[E] deg-4 divisor => (h0,rankH1)=(5,3)=(g_car,N_fam), sum 8=rank E8, even Clifford 16=dim S^+; mode-space reading [C], the seam->divisor step [O] (=QGEO.SYM.01)",
      "dependencies": "ARCH.RR.01; ARCH.RRCAR.02; QGEO.SYM.01",
      "scripts": [
        "v228_rr_index_gate.py"
      ],
      "script": "v228_rr_index_gate.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 310
    },
    {
      "id": "QEC.SEAM.01",
      "claim": "SEAM FINITE RECOVERABILITY CODE (2026-06-16): the seam as a finite quantum-error-correction-style code with recovery rate (2/3)^6=64/729. Code dimension = dim S^+ = 16 = 2^(g_car-1); the gapped boundary transport T on the cusp-weight 3-space (deviation directions (1,-1,0) and the Nariai anchor (1,1,-2)) is symmetric, doubly stochastic (CPTP), with spectrum {1,(2/3)^6,(1/3)^6}; on the trace-zero deviation subspace ||T delta|| <= (2/3)^6 ||delta||, hence ||T^n delta|| <= (2/3)^{6n} ||delta|| (a Knill-Laflamme-type per-step recovery bound). A free ratio r != (2/3)^6 gives r^n (rate not generic); breaking mu4-equivariance destroys it. 'Finite recoverability code' FIRST, 'holographic code' SECOND.",
      "status": "Numerical (contraction bound) [E]/[C]",
      "canonicalStatus": "[E] code dim 16, transport spectrum {1,(2/3)^6,(1/3)^6}, recovery bound ||T^n delta||<=(2/3)^{6n}||delta||; holographic/Petz reading [C] gated on the Seam-Horizon theorem",
      "dependencies": "SEAM.ATTRACT.01; SEAM.TRANSFER.01; HOR.PAGE.01",
      "scripts": [
        "v221_seam_qecc.py"
      ],
      "script": "v221_seam_qecc.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 311
    },
    {
      "id": "CP.MOD.01",
      "claim": "CP HEXAGONAL MODULUS (2026-06-16): CP lives in the HEXAGONAL phase fiber, not the square seam deck (a geometric sharpening of red-team Target D, the magnitude bijection misses the CP phases). The seam deck is the SQUARE modulus j(i)=1728, automorphism Z/4 (the mu4 clock); its exceptional partner is the HEX modulus j(rho)=0, rho=e^{i pi/3}, automorphism Z/6, CM by Z[omega], arg(rho)=pi/3. The frozen leading CKM reading delta=pi/3+3 lambda_C^2=68.654 deg (FLAV.CP.01, registry untouched) has leading term pi/3=arg(rho); the |Z2| alternative pi/3+2 lambda_C^2 is the frozen look-elsewhere trap. CRITICAL: the seam deck stays Z/4 (QGEO.KILL.01); CP is the Z/6 phase fiber OVER it. SCOPED (2026-07-29, T177/v561): the [C] is relative to a declared branch -- the invariant part of the reading is the leading term arg(rho) = pi/3; the full delta = pi/3 + 3 lambda_C^2 is a branch/frame-relative quantity (it has NO character under the parity/sector involution: a CP-even correction added to a CP-odd angle). The frozen reading itself is untouched (REG.FREEZE.01).",
      "status": "Identity/Numerical (modular) [E]/[C]",
      "canonicalStatus": "[E] j(i)=1728 (Z/4) vs j(rho)=0 (Z/6, arg=pi/3); delta=pi/3+3 lam^2 leading term = the hex CM phase; deck stays Z/4, CP is the Z/6 fiber [C]. NOT a CP derivation; scope narrowed: the invariant part is the leading term pi/3, the full delta is branch/frame-relative (T177/v561)",
      "dependencies": "FLAV.CP.01; QGEO.PILLOW.01; QGEO.KILL.01",
      "scripts": [
        "v220_cp_hexagonal_modulus.py",
        "v561_cp_channel_identities.py"
      ],
      "script": "v220_cp_hexagonal_modulus.py; v561_cp_channel_identities.py",
      "location": "tfpt_5_redteam",
      "externalData": "PDG gamma (look-elsewhere comparison)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 312
    },
    {
      "id": "DUAL.FRAME.01",
      "claim": "DUAL NORMAL FRAME (2026-06-16): the pair (d,n) is one canonical boundary compass for flavor and horizon, with CP as orientation. d=a^T R^{-1}=a^T L^{-1}=(-1/2,-1/2,1)=-1/2 (1,1,-2) (frame-independent; the Nariai/horizon normal; d.1=0, d.a=1); n=(5,-9,6) with n^T R=(det R,0,0)=(8,0,0), n^T L=(det L,0,0)=(20,0,0) (n annihilates the stable columns 2,3 -- the first-generation/torsion selector). Oriented volume det(1,d,n)=21=N_fam*scalaron=3*7. CP AS ORIENTATION: Im det(1,d,rho n)=21 sin(pi/3) for rho=e^{i pi/3} (the hexagonal CM unit) -- exactly where the magnitude bijection (Target D) fails. NARROWED, sharply (2026-07-29, T177/v561): the CP orientation READING is homogeneous of degree +1 in the frame normalisation, so the NUMBER 21 sin(pi/3) is frame data, not an invariant; dividing by the oriented volume removes the frame IDENTICALLY (Im(rho det)/det = Im rho exactly), so the invariant core is the FIBER datum Im(rho) = sin(pi/3) and the SIGN is a declared orientation. The raw functional is deck-invariant only because orientation reversal cancels fiber conjugation -- the deck cannot detect the CP sign.",
      "status": "Identity (exact) [E]/[C]",
      "canonicalStatus": "[E] d=(-1/2,-1/2,1)=-1/2 Nariai, n=(5,-9,6) reads det on the first-gen column, det(1,d,n)=21=3*7; CP orientation READING Im det(1,d,rho n)=21 sin(pi/3) [C], narrowed: degree +1 in the frame normalisation (21 sin(pi/3) is frame data), invariant core = the fiber datum Im rho = sin(pi/3), sign a declared orientation (T177/v561)",
      "dependencies": "HOR.DUALANCHOR.01; FLAV.SELECTOR.01; CP.MOD.01",
      "scripts": [
        "v225_dual_normal_frame.py",
        "v561_cp_channel_identities.py"
      ],
      "script": "v225_dual_normal_frame.py; v561_cp_channel_identities.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 313
    },
    {
      "id": "FTR.PATH.01",
      "claim": "DIAMOND F_TRANSFER PATH (2026-06-16): F_transfer is ONE path on the Sheet Diamond, not four external interfaces. Family M(s,t)=R+Q diag(s,t,t): the transfer corners K=M(1,-1), C=M(1,0), F=M(1,1) lie on the SHEET axis (s=1). The sheet axis is CURVED det M(1,t)=4t^2+14t+14 (dets (K,C,F)=(4,14,32), 2nd diff 8=rank E8), the winding axis det M(s,0)=6s+8 is FLAT (slope 6=|R^+(A3)|); the Plucker ladder Pl(C)-Pl(K)=(1,8,10)=(N_Phi,rank E8,A_Lambda), Pl(F)-Pl(C)=(1,8,16)=(N_Phi,rank E8,dim S^+) (DIAMOND.PLUCKER.01). F_transfer = the K->C->F sheet geodesic, the Koide source->pole its 1-D Moebius projection; the external solver evaluates continuously ALONG the known discrete path. The diagonal cut t=s mixes the axes (negative control).",
      "status": "Identity (exact) [E]/[C]",
      "canonicalStatus": "[E] M(s,t)=R+Q diag(s,t,t), K/C/F on the sheet axis, curved (2nd diff 8) vs flat winding (slope 6), Plucker steps (1,8,10),(1,8,16); F_transfer = the sheet geodesic, Koide its 1-D projection [C]",
      "dependencies": "DIAMOND.PLUCKER.01; DIAMOND.AXIS.01; CONTRACT.F.01",
      "scripts": [
        "v224_diamond_ftransfer_path.py"
      ],
      "script": "v224_diamond_ftransfer_path.py",
      "location": "tfpt_2_standard_model",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 314
    },
    {
      "id": "LEP.FROB.01",
      "claim": "LEPTON ETALE FROBENIUS ALGEBRA (2026-06-16): the three charged-lepton coefficients form a genuine commutative Frobenius algebra over the mu6 resolvent. Exact coefficients (c_e,c_mu,c_tau)=(16/7,4/3,7/6), ring closure c_e c_tau=8/3=|Z2| c_mu, product c_e c_mu c_tau=32/9=2^g_car/N_fam^2. A=Q[t]/(m), m=(t-c_e)(t-c_mu)(t-c_tau): the trace-form Gram G_ij=Tr(t^i t^j) is nondegenerate, det G=discriminant(m)!=0 (A etale/separable) -- a commutative Frobenius algebra with a nondegenerate trace pairing. The C6 shift has characteristic polynomial t^6-1 (spectrum mu6=mu3 family x mu2 sheet). PMNS extension: Aut(A) permutes {e,mu,tau} (real part), the hexagonal CM point j=0 supplies the phase part.",
      "status": "Identity (exact algebra) [E]/[C]",
      "canonicalStatus": "[E] (16/7,4/3,7/6), ring closure c_e c_tau=8/3=|Z2|c_mu, product 32/9; etale Frobenius algebra (det Gram=disc(m)!=0); mu6=mu3 x mu2 spectrum; PMNS extension [C]",
      "dependencies": "FLAV.LEPTON.01; ARCH.HEXRES.01; CP.MOD.01",
      "scripts": [
        "v229_lepton_frobenius_algebra.py"
      ],
      "script": "v229_lepton_frobenius_algebra.py",
      "location": "tfpt_2_standard_model",
      "externalData": "charged-lepton masses (reference)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 315
    },
    {
      "id": "CENTER.NORM.01",
      "claim": "CENTER BUDGET NORMS (2026-06-16): the Sheet-Diamond center budget (7,11,13) is the three LOCAL NORMS of the theory. C=R+Q diag(1,0,0), row sums (7,11,13): 7=N_Z[omega](3+2 omega) (Eisenstein/hexagonal CM norm = scalaron exponent), 13=N_Z[i](3+2 i) (Gaussian/square CM norm = Delta_Q), 11=C(4,0)+C(4,1)+C(4,2)=1+4+6 (QBL boundary Fock count on the |mu4|=4 corners = dim S^+ - g_car = 16-5). So C = (hex norm, boundary Fock count, square norm): the two CM rings (CMNORM.DUAL.01) and the seam boundary count meet in one operator center; the ring assignment is rigid.",
      "status": "Identity (exact) [E]/[C]",
      "canonicalStatus": "[E] C row sums (7,11,13)=(hex norm, QBL Fock count, square norm)=(N_omega(3,2), 1+4+6, N_i(3,2)); the center collects the three local norms [C]",
      "dependencies": "CMNORM.DUAL.01; DIAMOND.AXIS.01; ARCH.QBL.01",
      "scripts": [
        "v230_center_budget_norms.py"
      ],
      "script": "v230_center_budget_norms.py",
      "location": "tfpt_2_standard_model",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 316
    },
    {
      "id": "CP.MU6.01",
      "claim": "CP PHASES AS MU6 POWERS (2026-06-17): both CP phases are mu6 powers of ONE hexagonal CM unit, split by the sheet -- a structural reduction of red-team Target D, sharper than v220/v225 (which only located the residual). With rho=e^{i pi/3} the primitive 6th root (j=0 Eisenstein CM, CP.MOD.01): (1) rho^6=1, rho^3=-1 (the Z2 sheet half-turn); (2) delta_CKM leading = arg(rho^1)=pi/3 (quark, FLAV.CP.01), delta_PMNS = arg(rho^4)=4 pi/3 (lepton, the tfpt_2 neutrino phase lattice); (3) rho^4=-rho => delta_PMNS = delta_CKM,lead + pi = (CM unit) x (Z2 sheet) -- the two CP phases are ONE hexagonal unit on the two sheets, NOT two independent inputs; (4) the C6/mu6 monodromy U6 has charpoly t^6-1 (spectrum mu6 carries rho); (5) the dual-normal-frame orientation (DUAL.FRAME.01) is sheet-flipped: Im det(1,d,rho^1 n)=+21 sin(pi/3) (quark), Im det(1,d,rho^4 n)=-21 sin(pi/3) (lepton); (6) NEG: mu3 (cube roots) has no -1, so it cannot relate the two phases by a half-turn (mu6=mu3 family x mu2 sheet required). REDUCTION: removes one of Target D's two phase inputs (the lepton phase is the quark phase x the sheet); the quark seam correction 3 lambda_C^2 stays the v88 misalignment, the deck stays Z/4 (QGEO.KILL.01). Does NOT close Target D.",
      "status": "Identity (exact) [E]/[C]",
      "canonicalStatus": "[E] delta_CKM,lead=arg(rho)=pi/3, delta_PMNS=arg(rho^4)=4pi/3, rho^4=-rho so the two CP phases = one hexagonal unit x the Z2 sheet; sheet-flipped Jarlskog orientations; reduces Target D's CP inputs from 2 to 1, physical derivation [C]",
      "dependencies": "CP.MOD.01; DUAL.FRAME.01; FLAV.CP.01",
      "scripts": [
        "v231_cp_mu6_phases.py"
      ],
      "script": "v231_cp_mu6_phases.py",
      "location": "tfpt_5_redteam",
      "externalData": "delta_PMNS=4pi/3 (tfpt_2 phase lattice)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 317
    },
    {
      "id": "TOPO.E8.01",
      "claim": "THE SEAM AS THE E8 KLEINIAN SINGULARITY (2026-06-17): a canonical algebraic-geometric MODEL for the open seam-realisation premise QGEO.REALIZE.01, the du Val side of the McKay correspondence (MCKAY.E8.01/v219). (1) the McKay graph of 2I is affine E8 (9 nodes); dropping the unique trivial-rep node (the single Kac mark=1) leaves the FINITE E8 Dynkin on 8 nodes = the dual intersection graph of the 8 exceptional P^1's of the minimal resolution of C^2/2I (one trivalent node, arms of lengths 1,2,4). (2) the number of exceptional P^1's = 8 = rank E8 = g_car+N_fam = #nontrivial 2I irreps (9-1) -- a fourth reading of the '8' in c3=1/(8 pi). (3) the negated resolution intersection form is the E8 Cartan: even (each P^1 a (-2)-curve), unimodular det=1, positive definite -- the seam P^1's carry the E8 lattice as their intersection form. (4) 2I has exactly ONE degree-1 irrep (unique mark=1), so 2I is perfect (abelianisation trivial), hence H_1(S^3/2I)=0: the link is the Poincare homology 3-sphere, pi_1=2I order 120=|R^+(E8)|. (5) SEAM READING: the raw RP seam (a P^1 with marks) is canonically a component of this exceptional P^1-locus, so QGEO.REALIZE.01 gets an algebraic-geometric model (the seam P^1's ARE the E8 du Val curves) rather than a free postulate. Does NOT derive P2: it presupposes E8 (backward), so the carrier<->seam identification stays the open premise.",
      "status": "Lattice/Topology (exact) [E]/[C]",
      "canonicalStatus": "[E] McKay(2I)=affine E8; finite E8 = 8 exceptional P^1's intersection graph (det 1 Cartan); 2I perfect => link is the Poincare homology sphere pi1=2I order 120; [C] the seam is a du Val exceptional curve (model for QGEO.REALIZE.01, presupposes E8, bedrock [O])",
      "dependencies": "MCKAY.E8.01; QGEO.SYM.01; E8.GLU.01",
      "scripts": [
        "v232_e8_kleinian_seam.py"
      ],
      "script": "v232_e8_kleinian_seam.py",
      "location": "origin_theory",
      "externalData": "du Val/Kleinian correspondence; Poincare homology sphere",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 318
    },
    {
      "id": "CP.TRIALITY.01",
      "claim": "CP = THE UNIVERSAL TRIALITY PHASE, SHEET-SPLIT (2026-06-17): one step past CP.MU6.01/v231 in the canonical mu6 = mu3 (family/triality) x mu2 (sheet) factorisation. (1) mu6=mu3 x mu2 (gcd(3,2)=1): the Z3 triality centre {1,omega,omega^2} = the cusp weights {0,1/3,2/3} x 2pi, the sheet is mu2={1,-1}; the hexagonal CM unit factors as rho = omega^2 * (-1). (2) SAME FAMILY CLASS: rho^4 = rho^1 * rho^3 with rho^3 in mu2, so the quark (rho^1) and lepton (rho^4) CP phases have the SAME image in the family quotient mu6/mu2 ~ mu3 (the same triality-centre element) and OPPOSITE sheet ((-1)^1 vs (-1)^4). (3) so the CP phase IS the universal triality phase omega^2 (the 2/3 cusp-weight Z3 centre), read on the two sheets -- NOT a free choice of mu6 power; the power choice from v231 is removed. (4) omega is the mu3 factor of the order-30 (2,3,5) Milnor/Coxeter monodromy that builds E8 (TOPO.E8.01/v232), so CP lives inside the same cycle as the hull. (5) NEG: mu2-only carries no triality, mu3-only carries no sheet; only the full mu6 gives one triality phase on two sheets. Reduces red-team Target D from 'one hexagonal unit + sheet' to 'the universal triality phase + sheet'; the physical CP derivation stays [C] (the quark seam correction 3 lambda_C^2 is the v88 misalignment).",
      "status": "Identity (exact) [E]/[C]",
      "canonicalStatus": "[E] mu6=mu3 x mu2; quark rho^1 and lepton rho^4 share the family triality class omega^2 (the 2/3 cusp centre), opposite sheet; CP = universal triality phase + sheet, power choice removed; physical derivation [C]",
      "dependencies": "CP.MU6.01; FLAV.CP.01; TOPO.E8.01",
      "scripts": [
        "v233_cp_triality_phase.py"
      ],
      "script": "v233_cp_triality_phase.py",
      "location": "tfpt_5_redteam",
      "externalData": "delta_PMNS=4pi/3 (tfpt_2 phase lattice)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 319
    },
    {
      "id": "GATE.HOLO.01",
      "claim": "SEAM-HOLOMORPHY SELECTION CERTIFICATE (2026-06-17): the entire structural residual of TFPT is ONE condition with THREE provably-equivalent faces, all forcing E8 -- so P2/G_net/Target A/QGEO are ONE gate, not four. THE CONDITION: 'the seam carries no nontrivial abelian sector.' Faces: (i) AQFT -- the boundary chiral net is HOLOMORPHIC (mu-index 1, single vacuum sector; Target A, v83/v154); (ii) geometry -- the seam link is a HOMOLOGY 3-sphere <=> the binary group Gamma is PERFECT <=> Gamma=2I (v232); (iii) rep theory -- exactly ONE 1-dim irrep (v219). All three are the SAME number: #(1-dim irreps)=|Gamma^ab|=|H_1(S^3/Gamma)|=#(mark-1 affine-Dynkin nodes), which equals 1 ONLY for affine E8 (A_n->n+1, D_n->4, E6->3, E7->2, E8->1). So 2I is the unique nontrivial perfect finite SU(2) subgroup and the Poincare sphere is the unique homology-sphere spherical space form. And holomorphic c=8=g_car+N_fam => the unique even unimodular rank-8 lattice = E8 (Gram even, det 1). THE CLOSING THEOREM (stated, NOT proved): RP + gap + chirality => quasi-free bulk (v160 fixed-point) => invertible bulk => holomorphic boundary net (no abelian sector) => E8; the single residual analytic step is 'free bulk => holomorphic boundary', after which P2/QGEO/G_net all follow. This certifies that the three faces coincide and force E8; it does NOT prove the analytic step.",
      "status": "Lattice/AQFT selection (exact) [E]/[O]",
      "canonicalStatus": "[E] #mark-1 = |Gamma^ab| = |H1(link)|; only E8->1 (2I unique perfect, Poincare unique homology sphere); holomorphic c=8 unique E8; the three faces (holomorphy/homology-sphere/one-1dim-irrep) are one E8-selector => Target A/G_net/P2/QGEO are ONE gate; closing theorem 'free bulk => holomorphic boundary' stated [O]",
      "dependencies": "E8.GLU.01; MCKAY.E8.01; TOPO.E8.01; GATE.METRIC.06",
      "scripts": [
        "v234_seam_holomorphy_selection.py"
      ],
      "script": "v234_seam_holomorphy_selection.py",
      "location": "tfpt_research_contracts",
      "externalData": "Minkowski-Siegel (unique even unimodular rank-8); holomorphic c in 8Z",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 320
    },
    {
      "id": "GATE.HOLO.02",
      "claim": "SEAM CHERN-SIMONS REALISATION (2026-06-17): the closing step of GATE.HOLO.01 realised in standard abelian Chern-Simons / quantum-Hall language, where 'holomorphic <=> det K = 1' is a THEOREM -- a genuine reduction of the analytic step into anyon condensation, NOT a closure. A free gapped BOSONIC 2+1d bulk = an even integer K-matrix; #anyons (torus degeneracy) = |det K|, edge chiral central charge c = signature(K), and the edge chiral net is holomorphic (single sector) <=> |det K| = 1. So the v234 condition 'no nontrivial abelian sector' = exactly |det K| = 1. The TFPT extension tower (v92/GATE.METRIC.07) IS the anyon-condensation tower read by determinant: carrier D5(+)A3 (even, det 16, c=8, 16 anyons, mu=16) -> SO(16)_1=D8 (det 4, 4 anyons, mu=4) -> (E8)_1 (det 1, 1 anyon, mu=1, HOLOMORPHIC); all rank 8 with c = signature = 8 = g_car+N_fam. Holomorphic <=> det 1 <=> E8 = the unique even unimodular rank-8 K-matrix = the Kitaev E8 quantum-Hall state (edge (E8)_1). CONDENSATION: a Lagrangian (maximal isotropic) glue of order sqrt(16)=|mu4|=4 takes det 16->16/4^2=1 (carrier->E8); an order-2 isotropic only ->16/2^2=4 (D8). So 'holomorphic' = 'condense the order-|mu4| Lagrangian glue', not a partial. NEG: D8 (det 4) is free+bosonic yet has 4 anyons (not holomorphic, the SO(16)_1 rival); an odd-diagonal K is fermionic (spin-CS) -- 'free+bosonic' does NOT imply holomorphic, det=1 is the extra physical condition. RESIDUAL [O]: the closing step = 'the free RP gapped seam condenses the order-|mu4| Lagrangian glue (det->1), not a partial isotropic (det->4)' -- the same sheet/mu4 selection as GATE.QGEO, now with holomorphy<=>det1 a theorem and the residual a single integer condition.",
      "status": "Lattice/CS realisation (exact) [E]/[O]",
      "canonicalStatus": "[E] free gapped bosonic bulk = K-matrix; #anyons=|det K|, c=signature, holomorphic<=>det 1; the v92 tower D5(+)A3(16)->D8(4)->E8(1) is the condensation tower; holomorphic=E8=Kitaev E8 state; residual = 'condense the |mu4| Lagrangian glue' = the sheet/QGEO selection [O]",
      "dependencies": "GATE.HOLO.01; GATE.METRIC.07; TOPO.E8.01; QGEO.SYM.01",
      "scripts": [
        "v235_seam_chern_simons.py"
      ],
      "script": "v235_seam_chern_simons.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kitaev E8 phase; abelian CS bulk-boundary (#anyons=|det K|)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 321
    },
    {
      "id": "TOPO.BRIESKORN.01",
      "claim": "BRIESKORN CAPSTONE (2026-06-17): the (2,3,5) Brieskorn singularity x^2+y^3+z^5 is the ONE generator of the discrete skeleton -- the capstone of the icosahedral round (v55/v219/v223/v232/v233). Its three exponents ARE the atoms (|Z2|,N_fam,g_car)=(2,3,5). (1) MILNOR NUMBER = the '8': mu=(2-1)(3-1)(5-1)=(|Z2|-1)(N_fam-1)(g_car-1)=1*2*4=8=rank E8=#exceptional P^1 = the '8' in c3=1/(8 pi), a FIFTH independent origin (after geometry=2|mu4|, lattice=rank E8, gravity=8pi, Coxeter=phi(30)). (2) MONODROMY = the order-30 Coxeter cycle: the Milnor monodromy eigenvalues lambda=zeta_2^j1 zeta_3^j2 zeta_5^j3 (1<=ji<=ai-1) are the eight primitive 30th roots e^{2pi i m/30}, m in {1,7,11,13,17,19,23,29} = the E8 exponents; characteristic polynomial = Phi_30 (degree phi(30)=8=mu), so the Milnor monodromy IS the E8 Coxeter element of order 30=h(E8) (v55). (3) BOTH CLOCKS are facets of this one monodromy: the monodromy GROUP <h>=Z/30=Z/2(sheet)xZ/3(family/triality)xZ/5(carrier) carries the mu3 triality (the CP phase omega, v233) as the order-3 power h^10; the GALOIS group Gal(Q(zeta_30)/Q)=(Z/30)^x=Z/2xZ/4 carries the order-4 element (x7) = the mu4 clock acting on the spectrum (v223). (4) Milnor lattice=E8 (the resolution intersection form, v232/v234), link=Poincare homology sphere S^3/2I. (5) SYNTHESIS: one singularity x^2+y^3+z^5 generates the atoms (exponents), the 8 (Milnor number), E8 (Milnor lattice), the order-30 cycle (monodromy), both clocks (mu3 in <h>, mu4 in Galois) and the seam (link) -- not many numbers sharing {2,3,5} but one icosahedral singularity read several ways. Unifies the icosahedral round; does NOT close a gate.",
      "status": "Lattice/Topology (exact) [E]/[I]",
      "canonicalStatus": "[E] mu=(2-1)(3-1)(5-1)=8=rank E8 (5th origin of the '8'); Milnor monodromy eigenvalues = E8 exponents/30 = the order-30 Coxeter cycle (charpoly Phi_30); both clocks = mu3 in <h>=Z/30 (CP) + mu4 in Galois (Z/30)^x; one singularity generates the skeleton [I]",
      "dependencies": "MCKAY.E8.01; COX.CLOCK.01; CP.TRIALITY.01; TOPO.E8.01; SEAM.COX.01",
      "scripts": [
        "v236_brieskorn_capstone.py"
      ],
      "script": "v236_brieskorn_capstone.py",
      "location": "origin_theory",
      "externalData": "Brieskorn/Milnor fibration; Pham monodromy theorem",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 322
    },
    {
      "id": "GATE.HOLO.03",
      "claim": "SEAM SRE CLOSURE (2026-06-17): the closing step of GATE.HOLO.01/02 recast as a PHYSICAL, falsifiable condition -- no topological ground-state degeneracy <=> det K=1 <=> the seam bulk is short-range-entangled (SRE/invertible). Sharpens the residual from definitional ('condense the Lagrangian glue', v235) to a yes/no statement about topological order. (1) the abelian K-matrix ground-state degeneracy on a genus-g surface is |det K|^g: carrier D5(+)A3->16, SO(16)_1=D8->4, (E8)_1->1 on the torus; so 'no topological degeneracy on ANY closed surface' <=> |det K|=1. (2) SRE <=> det 1 <=> E8: among rank-8 even positive-definite K (c=signature=8=g_car+N_fam) only E8 has det 1 -- THE unique nontrivial bosonic SRE 2+1d phase (Kitaev E8), edge (E8)_1 holomorphic; D8 and the carrier are long-range-entangled (topologically ordered). (3) PLANE vs TORUS: a unique ground state on the PLANE (RP+gap+cluster+OS, the established admissible-sector facts) is NECESSARY but not sufficient -- the plane has no non-contractible cycles and cannot see |det K|, the torus does; so the residual is exactly 'the seam unique-vacuum property extends to the torus' = SRE. (4) NEG: a topologically-ordered (LRE) bulk has |det K|>1 (D8: 4-fold torus degeneracy + a non-holomorphic boundary 1,v,s,c = SO(16)_1 rival). (5) SHARPENED RESIDUAL: the one open analytic step is now the physical statement 'the free RP gapped seam bulk is short-range-entangled (no topological ground-state degeneracy)' = det K=1 = the Kitaev E8 phase -- sharper than 'condense the Lagrangian glue' (v235) and than 'the seam is P^1 minus mu4' (QGEO.SYM.01). [E] the degeneracy=|det K|^g dictionary + SRE<=>det1<=>E8 selection; [O] the physical SRE premise. NOT a closure.",
      "status": "Lattice/topological-order (exact) [E]/[O]",
      "canonicalStatus": "[E] genus-g GSD=|det K|^g, no topological degeneracy <=> det K=1 <=> SRE; only E8 (det 1) is the bosonic SRE c=8 phase (Kitaev E8); plane unique-vacuum is necessary not sufficient (torus sees det K); residual = 'the seam is SRE' [O], a physical yes/no statement",
      "dependencies": "GATE.HOLO.02; GATE.HOLO.01; QGEO.SYM.01",
      "scripts": [
        "v237_seam_sre_closure.py"
      ],
      "script": "v237_seam_sre_closure.py",
      "location": "tfpt_research_contracts",
      "externalData": "gapped 2+1d topological order (GSD=|det K|^g); Kitaev E8 SRE phase",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 323
    },
    {
      "id": "DYN.SEMIGROUP.01",
      "claim": "STATIC->DYNAMIC FLIP (2026-06-17): the seam's modular flow (reversible) and its recovery channel read as a Markov/Lindblad generator (irreversible), with the dissipative gap EQUAL to the OS mass gap -- the point where the static compiler becomes a dynamics. Companion to v196/v198 (modular) and v221/v64 (recovery), which state the structure STATICALLY (a commutator, a contraction bound, a mass-gap number); this reads the SAME objects DYNAMICALLY. (1) [E] MODULAR FLOW = REVERSIBLE GENERATOR: sigma_t = e^{i t Lambda_Sigma} is a unitary one-parameter group (Tomita-Takesaki); the mu4 clock is conserved, rho sigma_t rho^{-1} = sigma_t for all t <=> [rho,Lambda_Sigma]=0 (the v198 state-invariance), so the static v198 commutator IS the conservation law of modular time. (2) [E] RECOVERY = DISSIPATIVE GENERATOR: L = log T is real-symmetric with off-diagonals >= 0 and zero row/column sums (a doubly-stochastic / CPTP-classical Markov generator); e^{L*1} = T reproduces the v221 channel and e^{L tau} -> the rank-1 democratic projector (the arrow of time, matching v64's T^n -> projector). (3) [E] KEYSTONE: the dissipative spectral gap = -log((2/3)^6) = 6 log(3/2) = the OS/RP mass gap Delta (v64) -- the STATIC recovery bound (2/3)^6 (v221) and the DYNAMICAL mass gap (v64) are ONE number, one exponentiated. (4) [E] GKSL split: G = i Lambda_Sigma + L has reversible (imaginary) + dissipative (real <= 0, gapped) spectrum. NEG: a mu4-breaking Lambda breaks conservation; a free rate r != (2/3)^6 breaks the gap identity. (5) [O] residual: geometric modular flow on the FULL seam algebra (Connes-Rovelli thermal time = physical time = QGEO.SYM.01, v198) and the continuum embedding beyond the finite block -- a target, not a closure",
      "status": "Numerical (finite-block) [E]/[O]",
      "canonicalStatus": "[E] finite block: modular flow = reversible unitary group with mu4 conserved <=> [rho,Lambda]=0; L = log T a doubly-stochastic generator with e^{L}=T; dissipative gap = -log((2/3)^6) = 6 log(3/2) = OS mass gap (v64); GKSL split. [O] geometric modular flow on the full seam algebra (= QGEO.SYM.01) and the continuum embedding -- a target, not a closure",
      "dependencies": "QGEO.MODULAR.01; QEC.SEAM.01",
      "scripts": [
        "v238_modular_lindblad_dynamics.py"
      ],
      "script": "v238_modular_lindblad_dynamics.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 324
    },
    {
      "id": "DYN.KMS.01",
      "claim": "KMS / THERMAL TIME (2026-06-17): the modular flow of v238 makes the seam state a KMS (thermal) state at the canonical inverse temperature beta = 1, and the geometric (Unruh/Bisognano-Wichmann) normalisation fixes the physical seam temperature via the seam unit 2 pi = 1/(4 c3). Tomita-Takesaki: a cyclic-separating state is automatically KMS at beta = 1 for its own modular flow. (1) [E] KMS at beta=1: omega(A sigma_i(B)) = omega(B A) for random Hermitian A,B (omega(X)=Tr(rho X), rho=e^{-Lambda_Sigma}/Z), so modular time (v238) is a thermal time (Connes-Rovelli) and the seam state is the Gibbs state of the modular Hamiltonian. (2) [E] NEG: the same identity at beta=1/2 fails -- Tomita-Takesaki pins beta=1, no tunable knob. (3) [E] the seam unit: beta_phys = 2 pi = 1/(4 c3) exactly (v58; 1/4=1/|mu4|), so the modular KMS temperature is the horizon temperature T_H = kappa/(2 pi) = c3/M (v57/v208) -- thermal time = horizon time. (4) [E] one object with v238: the same Lambda_Sigma generates sigma_t and rho=e^{-Lambda_Sigma}/Z; the recovery generator L's stationary state is the beta->0 democratic point. (5) [O] residual: that this beta=1 flow is the geometric boost (T_H physical) is the QGEO.SYM.01/BW premise (HOR.BHTHERMO.01 [P]) -- a target, not a closure",
      "status": "Numerical (finite-block) [E]/[O]",
      "canonicalStatus": "[E] KMS at beta=1 (modular flow is thermal); seam unit 2 pi = 1/(4 c3) = horizon temperature T_H = c3/M; one object with v238. [O] the geometric-boost identification = QGEO.SYM.01 -- a target, not a closure",
      "dependencies": "DYN.SEMIGROUP.01; HOR.BHTHERMO.01",
      "scripts": [
        "v239_kms_thermal_time.py"
      ],
      "script": "v239_kms_thermal_time.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 325
    },
    {
      "id": "DYN.GNS.01",
      "claim": "GNS / OS RECONSTRUCTION (2026-06-17, finite toy): from the discrete seam data (a *-algebra M + the state omega) one reconstructs a Hilbert space, a cyclic vector, the modular flow (v238/v239) AND a POSITIVE Hamiltonian -- so two QFT primitives (the Hilbert space and a positive Hamiltonian) are OUTPUTS of the seam data, not extra inputs. (1) [E] GNS: <A,B>=Tr(rho A^dag B) is Hermitian and positive-definite (rho>0), dim n^2; the identity is cyclic (pi(A)1 = A spans M) and separating. (2) [E] modular flow sigma_t(x)=rho^{it} x rho^{-it} is a state-preserving *-automorphism with generator i[log rho,.] = Lambda_Sigma (v238); the modular operator Delta:X|->rho X rho^{-1} is positive. (3) [E] OS Hamiltonian H_OS = -log T (T the v221 recovery transfer) is POSITIVE: eigenvalues {0, 6 log(3/2), 6 log 3} >= 0, ground state = the stationary mode, gap = 6 log(3/2) = the v64 mass gap -- OS reconstruction delivers v64's 'H >= 0 + mass gap'. (4) [E] H_OS = -L: the OS Hamiltonian is MINUS the v238 dissipative generator -- the recovery semigroup e^{L tau} IS the Euclidean evolution e^{-H_OS tau}; two Hamiltonians from one seam (modular thermal-time + OS positive-energy). (5) [O] residual: the geometric/operator-level identification = QGEO.SYM.01/BW and the full reconstruction = the standard OS theorem under the named RP/gap hypotheses (Layer 2) -- a target, not a closure",
      "status": "Numerical (finite-block) [E]/[O]",
      "canonicalStatus": "[E] GNS Hilbert space + cyclic/separating vector from (M,omega); modular flow = v238; OS Hamiltonian H_OS = -log T >= 0 with gap = v64 mass gap, and H_OS = -L. [O] the geometric/operator-level OS step = QGEO.SYM.01 + Layer-2 RP/gap -- a target, not a closure",
      "dependencies": "DYN.SEMIGROUP.01; QEC.SEAM.01; QG.GAPDEC.01",
      "scripts": [
        "v240_gns_os_reconstruction.py"
      ],
      "script": "v240_gns_os_reconstruction.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 326
    },
    {
      "id": "DHR.SECTORS.01",
      "claim": "PARTICLES = DHR SUPERSELECTION SECTORS (2026-06-17): the discriminant/anyon content of the carrier and the condensation tower (v235/v237) made explicit as an abelian modular tensor category (MTC) -- the particle-spectrum layer of the emergent QFT. The carrier discriminant form is the Lean-verified glue form (FORM.GLUE.01): on Z4 x Z4, q(x,y)=(5x^2+3y^2)/8 mod 1, bilinear B((x,y),(x',y'))=(5xx'+3yy')/4 mod 1, topological spin theta(a)=e^{2 pi i q(a)}. (1) [E] 16 sectors = Z4 x Z4 (= det K_carrier = dim S^+ = 16, v235); fusion = abelian group law (closed, associative, vacuum (0,0), inverses). (2) [E] spin/condensability: bosonic <=> theta=1 <=> q in Z <=> 5x^2+3y^2=0 mod 8 -- exactly the six isotropic elements {(0,0),(1,1),(1,3),(2,2),(3,1),(3,3)}. (3) [E] condensation tower = v235/v237: condensing the Lagrangian H1=<(1,1)> (order 4, self-dual H1^perp=H1) leaves |H^perp/H|=1 sector = (E8)_1 (det 1, holomorphic); the order-2 isotropic <(2,2)> leaves 4 sectors = SO(16)_1 = D8 (det 4); so carrier 16 -> D8 4 -> E8 1. (4) [E] NEG: a non-isotropic anyon (1,0), q=5/8, theta != 1, is confined; <(1,0)> contains non-isotropic elements so cannot be a condensation algebra -- the tower is forced by isotropy. (5) [C] identifying sectors / deconfined anyons with SM particle multiplets is the interpretation; the MTC data (fusion + spins + tower) is exact",
      "status": "Numerical (finite MTC) [E]/[C]",
      "canonicalStatus": "[E] 16-sector abelian MTC from the glue form, fusion = group law, condensation tower 16->4->1 = v235/v237 (Lagrangian -> E8, order-2 -> D8); [C] sector<->SM-multiplet identification -- particles = stable defect sectors of the seam code, made precise",
      "dependencies": "GATE.HOLO.02; GATE.HOLO.03; FORM.GLUE.01",
      "scripts": [
        "v241_dhr_sectors.py"
      ],
      "script": "v241_dhr_sectors.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 327
    },
    {
      "id": "DHR.MODULAR.01",
      "claim": "SPIN-STATISTICS + MODULAR (S,T) DATA (2026-06-17): the exchange statistics, modular S/T matrices, Verlinde fusion and Gauss-Milgram central charge of the carrier MTC -- the statistics+kinematics layer on top of v241. Same abelian MTC: Z4 x Z4, q(x,y)=(5x^2+3y^2)/8 mod 1 (the Lean glue form FORM.GLUE.01), B(a,b)=q(a+b)-q(a)-q(b), theta(a)=e^{2 pi i q(a)}. (1) [E] spin-statistics: T=diag(theta_a) splits the 16 anyons into 6 BOSONS (theta=1, the condensable/isotropic set), 2 FERMIONS (theta=-1) = {(0,2),(2,0)}, and 8 genuine anyons. (2) [E] modular S_ab=(1/sqrt 16) e^{-2 pi i B(a,b)} is unitary, symmetric, S^2=C (charge conjugation, C_ab=delta(b,-a)), S^4=I. (3) [E] Verlinde N_ab^c = sum_x S_ax S_bx conj(S_cx)/S_0x = delta(c,a+b) -- the fusion ring is RECOVERED from the modular data, not posited. (4) [E] Gauss-Milgram (1/sqrt 16) sum_a theta_a = e^{2 pi i c/8} with c=8=g_car+N_fam -- the anyon data already encodes the chiral central charge of the (E8)_1 net (v156/v175); statistics and c=8 are one fact. (5) [C] the boson/fermion split is the kinematic skeleton of the matter content; mapping sectors to physical fermion multiplets is the interpretation",
      "status": "Numerical (finite MTC) [E]/[C]",
      "canonicalStatus": "[E] T=diag(theta) spin-statistics (6 bosons, 2 fermions, 8 anyons); modular S unitary/symmetric, S^2=C, S^4=I; Verlinde reproduces the group-law fusion; Gauss-Milgram gives c=8=g_car+N_fam. [C] the boson/fermion-to-field map is the interpretation",
      "dependencies": "DHR.SECTORS.01; GATE.HOLO.02",
      "scripts": [
        "v242_spin_statistics_modular.py"
      ],
      "script": "v242_spin_statistics_modular.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 328
    },
    {
      "id": "DHR.SMATRIX.01",
      "claim": "SCATTERING: HAAG-RUELLE SKELETON + BRAIDING S-MATRIX (2026-06-17): the S-matrix layer of the emergent QFT. The seam net has a mass gap (v64/v238), so Haag-Ruelle scattering theory constructs asymptotic in/out states; and for the abelian anyon MTC (v241/v242) the 2-particle S-matrix is the monodromy (double braiding), a pure phase. Same MTC: Z4 x Z4, q=(5x^2+3y^2)/8, theta=e^{2 pi i q}, M(a,b)=theta(a+b)/(theta(a)theta(b))=e^{2 pi i B(a,b)}. (1) [E] mass gap Delta=6 log(3/2)>0 (v64/v238) is the verified input; HR then constructs asymptotic states (standard gapped-net theorem). (2) [E] 2-particle S = monodromy: |M|=1 (unitary), diagonal in the conserved anyon labels -> NO particle production, factorised (integrable) scattering. (3) [E] Yang-Baxter/factorisation: M is a bicharacter (M(a+a',b)=M(a,b)M(a',b)), checked on all triples -> multi-particle factorisation. (4) [E] crossing+symmetry: M(a,b)=M(b,a) and M(a,-b)=conj(M(a,b)). (5) [E] holomorphic point: on the condensed (E8)_1 (single deconfined sector, v241/v235/v237) M=1, trivial braiding -- a closed anomaly-free bulk (det 1/SRE); scattering non-trivial only in the un-condensed matter sectors. (6) [O] residual: the 4d S-matrix and physical cross-sections are the open holographic uplift (Phase 3 / the spectral-action route) -- this exhibits the EXACT 2d topological scattering data (braiding = S-matrix) with the mass gap from v238",
      "status": "Numerical (finite MTC + cited HR) [E]/[O]",
      "canonicalStatus": "[E] mass gap -> HR asymptotic states; 2-particle S = monodromy (unitary, diagonal, factorised); Yang-Baxter bicharacter; crossing; trivial S on condensed E8. [O] the 4d S-matrix / cross-sections = the open holographic uplift",
      "dependencies": "DHR.MODULAR.01; DYN.SEMIGROUP.01; v64",
      "scripts": [
        "v243_haag_ruelle_braiding.py"
      ],
      "script": "v243_haag_ruelle_braiding.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 329
    },
    {
      "id": "CONTRACT.QFT4D.01",
      "claim": "4D LAGRANGIAN VIA THE SPECTRAL ACTION (2026-06-17): the honest CONTRACT for the Phase-3 structural gap (the 2d chiral seam net v156/v175 + its dynamics v238-v243 -> the physical 4d theory). The natural TFPT bridge is NOT a holographic guess but Connes' spectral action: a SINGLE Dirac operator whose spectral action Tr f(D/Lambda) yields gravity AND the matter (gauge+Higgs) Lagrangian. The gravity half is verified (v36: R+R^2); this types the matter half as a contract and checks the arithmetic pieces. (1) [E] carrier spinor = one generation: carrier D5(+)A3 = SO(10)xSO(6) (=SO(16) grading, v156); the SO(10) half-spinor = dim S^+ = 2^(g_car-1) = 16 = one SM generation (15 Weyl fermions + nu_R), N_fam = 3 -- the carrier carries the SM matter representation. (2) [E] SM gauge group inside the carrier: SU(3)xSU(2)xU(1) (rank 4, dim 12) in SU(5) (rank 4, dim 24) in SO(10)=D5 (rank 5, dim 45); color SU(3) also in A3=SU(4) (Pati-Salam, dim 15); rank(D5)+rank(A3)=8=rank E8. (3) [E] one Dirac operator -> gravity + matter: the Seeley-DeWitt expansion gives a2~R (Einstein-Hilbert) and a4~(R^2, Weyl^2, F_{mu nu}F^{mu nu} Yang-Mills, |D phi|^2 Higgs); gravity half machine-verified (v36), Yang-Mills a4 standard Gilkey. (4) [X] falsifiable hooks: spectral-action boundary relations g3=g2=sqrt(5/3)g1 and a Higgs-quartic relation (Chamseddine-Connes-Marcolli). (5) [O] CONTRACT.QFT4D.01 obligations: (i) the seam induces the SM Dirac operator D; (ii) the finite NCG geometry = the carrier algebra (FORM.P2.02); (iii) the spectral action of D reproduces the measured SM couplings under the v36 gravity normalisation -> then the 4d SM+gravity Lagrangian is an OUTPUT. The genuine Phase-3 target, NOT closed",
      "status": "Contract [O] (with [E] arithmetic + [X] hooks)",
      "canonicalStatus": "[E] carrier half-spinor = one SM generation (16), SM gauge group embeds in the carrier, one Dirac operator gives gravity (v36) + matter (Gilkey); [X] unification + Higgs hooks; [O] CONTRACT.QFT4D.01: the spectral action of the seam Dirac operator = 4d SM+gravity Lagrangian -- the Phase-3 target, not closed",
      "dependencies": "FORM.P2.02; QG.G2.01; DHR.SECTORS.01",
      "scripts": [
        "v244_spectral_action_contract.py"
      ],
      "script": "v244_spectral_action_contract.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 330
    },
    {
      "id": "QFT4D.MATTER.01",
      "claim": "SPECTRAL-ACTION MATTER CONTENT (2026-06-17): the computable core of CONTRACT.QFT4D.01 (v244) -- the NCG matter representation, discharging the representation-theory + tree-relation core to [E] and leaving only the seam->Dirac realisation and the measured-coupling reproduction as the open residual. The carrier half-spinor is the SO(10) 16 (dim S^+ = 2^(g_car-1) = 16). One generation = SO(10) 16 = SU(5)(10+5bar+1) = SM: Q=(3,2)_{1/6}, u^c=(3bar,1)_{-2/3}, e^c=(1,1)_{1}, L=(1,2)_{-1/2}, d^c=(3bar,1)_{1/3}, nu^c=(1,1)_{0}. (1) [E] 16 = one generation: SM multiplet dims sum to 6+3+1+2+3+1 = 16 = dim S^+, N_fam=3 (15 SM Weyl fermions + a right-handed neutrino). (2) [E] electric charges Q=T3+Y are exactly the SM values (quarks +2/3,-1/3; anti-quarks -2/3,+1/3; leptons 0,-1; e^c +1; nu^c 0) -- the hypercharges are forced by the SO(10) embedding. (3) [E] anomaly-free: all four gauge anomalies cancel exactly over the 16: sum Y = 0, sum Y^3 = 0, [SU(2)]^2 U(1) = 0, [SU(3)]^2 U(1) = 0. (4) [E] spectral-action unification: sin^2 th_W = (sum T3^2)/(sum Q^2) = 3/8 at the spectral scale, i.e. g3=g2=sqrt(5/3)g1 (Chamseddine-Connes-Marcolli). (5) [E] Higgs = the (1,2)_{1/2} off-diagonal part of the finite Dirac operator (inner fluctuation), one SM doublet. (6) [O] residual: (i) the seam induces the Dirac operator D and (iii) the spectral action reproduces the MEASURED couplings after RG running -- still a contract, now with its rep-theory core verified",
      "status": "Numerical (exact rep theory) [E]/[O]",
      "canonicalStatus": "[E] SO(10) 16 = one anomaly-free SM generation with correct hypercharges/charges; sin^2 th_W = 3/8 (g3=g2=sqrt(5/3)g1); Higgs = (1,2)_{1/2} inner fluctuation. [O] the seam->Dirac realisation + measured-coupling reproduction stay the open contract",
      "dependencies": "CONTRACT.QFT4D.01; FORM.P2.02",
      "scripts": [
        "v245_spectral_matter_content.py"
      ],
      "script": "v245_spectral_matter_content.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 331
    },
    {
      "id": "QFT4D.RGTEST.01",
      "claim": "UNIFICATION DATA CROSS-CHECK (2026-06-17): the honest data-confrontation kill-test for the spectral-action prediction (CONTRACT.QFT4D.01, v244/v245). Does the boundary condition g1=g2=g3, sin^2 th_W=3/8 at the unification scale survive contact with the MEASURED Standard-Model couplings? Answer: NOT in the plain SM. Inputs (PDG, GUT-normalised, at M_Z; consistent with the PyR@TE cross-check v159): alpha_i^-1 = (59.01, 29.59, 8.47), SM 1-loop betas b=(41/10,-19/6,-7), sin^2 th_W(M_Z)=0.23122. (1) [E] the prediction: at Lambda, alpha1=alpha2=alpha3 and sin^2 th_W=3/8. (2) [X] tension: running the measured couplings up 1-loop, the three do NOT meet at one scale -- pairwise crossings a1=a2 ~1.0e13, a1=a3 ~2.4e14, a2=a3 ~9.9e16 GeV (~4 decades spread), residual spread at 2e16 GeV Delta(alpha^-1) ~ 8.8. (3) [X] sin^2 th_W: predicted 3/8 vs measured 0.23122 (the known few-percent non-SUSY shortfall). (4) [X] two-loop also misses: an RK4 2-loop run gives minimal spread Delta(alpha^-1) ~ 3.2 at mu ~ 1.7e14 GeV (never a common point), ~8.7 at 2e16. (5) [O] NO rescue in the current theory: TFPT has SM gauge content and no new states (tfpt_3 'Dark matter = no new state'), and the E8 cascade is an arithmetic spine (D_n=60-2n, v5), NOT a threshold tower -- there is no admissible gauge-threshold source; closing the spread would require EITHER new gauge-charged matter (a new postulate, contradicting 'no new state') OR a non-GUT/boundary reading of the spectral-action cutoff. The spectral-action 4d-GUT route is in genuine tension -- the SAME status as the Connes-Chamseddine NCG Standard Model -- likely falsified unless the theory is extended (the AQFT skeleton + DHR + SM rep-core survive as an emergent boundary QFT). Typed [X]/[O], NEVER [E]-as-success; the tension IS the result",
      "status": "Data confrontation (kill-test) [X]/[O]",
      "canonicalStatus": "[X] plain-SM run misses unification at 1- and 2-loop (1-loop spread ~9; 2-loop min spread ~3.2 at 1.7e14) and sin^2 th_W 3/8 vs 0.231; [O] no admissible threshold source (SM content, no new state; the E8 spine is arithmetic, not a tower) -- spectral-action 4d-GUT route in tension like NCG-SM, likely falsified unless extended; not a confirmation",
      "dependencies": "QFT4D.MATTER.01; CONTRACT.QFT4D.01",
      "scripts": [
        "v246_unification_data_crosscheck.py"
      ],
      "script": "v246_unification_data_crosscheck.py",
      "location": "tfpt_research_contracts",
      "externalData": "PDG gauge couplings + sin^2 thetaW(M_Z)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 332
    },
    {
      "id": "PS.E8BRANCH.01",
      "claim": "E8 HULL FORBIDS THE 126 (2026-06-17): the E8 audit hull supplies exactly {1,10,16,45} for SO(10) -- turning the scalaron-match '16-Higgs over 126-Higgs' selection into an algebraic statement. The carrier D5(+)A3 = SO(10)xSU(4) is a rank-8 subalgebra of E8 via SO(16); the adjoint 248 branches as (45,1)+(1,15)+(10,6)+(16,4)+(16bar,4bar) = 45+15+60+64+64 = 248 (SO(16) split: 120 adjoint + 128 spinor = spinor_SO10 (x) spinor_SO6). (1) [E] ranks 5+3=8=rank E8. (2) [E] dims sum to 248. (3) [E] the SO(10) reps in the hull are exactly {1,10,16,45}; the 126 (dim 126) is ABSENT -> the renormalisable 126_H is algebraically forbidden if fields are E8-hull reps, and 10+16+45 is E8-supplied, not fitted. (4) [E] MULTIPLICITY CAVEAT: only ONE (45,1) and ONE (1,15) in the 248 -> the minimal E8-allowed content (one 45) is proton-MARGINAL, a sharp falsifiable constraint. (5) [O] the field-content principle ('TFPT fields are E8-hull reps') + the NCG inner-fluctuation complement (v250).",
      "status": "Lattice/rep theory [E]",
      "canonicalStatus": "[E] E8->SO(10)xSU(4) branching: SO(10) content {1,10,16,45}, 126 absent (126_H forbidden), only one 45/15 (proton-marginal); [O] the field-content principle",
      "dependencies": "E8.GLU.01; QFT4D.MATTER.01",
      "scripts": [
        "v247_e8_branching_no126.py"
      ],
      "script": "v247_e8_branching_no126.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 333
    },
    {
      "id": "PS.ALGEBRA.01",
      "claim": "PATI-SALAM NCG ALGEBRA + EXACT SM MATCHING (2026-06-17): the finite algebra A_F = H_L (+) H_R (+) M_4(C) and its consequences (Reviewer-A Theorem 2). (1) [E] unimodular gauge group SU(2)_L x SU(2)_R x SU(4)_c (ranks 1+1+3=5=rank SO(10), dims 3+3+15=21). (2) [E] one generation 16 = (4,2,1)+(4bar,1,2) = 8+8. (3) [E] Y = T_3R + (B-L)/2 reproduces the EXACT SM hypercharges for all six multiplets (Q,u^c,d^c,L,e^c,nu^c), and Q = T_3L + Y the exact electric charges {2/3,-1/3,0,-1,1,0}; B-L is the SU(4)_c generator (lepton = 4th colour). (4) [E] coupling matching a1^-1=(3/5)a2R^-1+(2/5)a4^-1 (coeffs sum to 1, so a1=a_GUT at the SO(10) point), sin^2 thetaW=3/8 (=v245). (5) [O] that A_F is the UNIQUE real even algebra generated by seam+sheet-Z2+mu4 (so 'carrier gauged' is a theorem, not a fork). (6) [O] CAVEAT: the PS colour SU(4)_c sits inside SO(10)=D5; it is NOT the carrier factor A3=SU(4) (= family structure, N_fam=3=rank A3) -- a precision the reviews blur.",
      "status": "Numerical (exact rep theory) [E]/[O]",
      "canonicalStatus": "[E] A_F -> SU(2)xSU(2)xSU(4) PS group, 16=(4,2,1)+(4bar,1,2), exact hypercharges/charges, matching a1^-1=(3/5)a2R^-1+(2/5)a4^-1, sin^2=3/8; [O] seam-reconstruction + the colour/family distinction",
      "dependencies": "QFT4D.MATTER.01; PS.E8BRANCH.01",
      "scripts": [
        "v248_ps_algebra.py"
      ],
      "script": "v248_ps_algebra.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 334
    },
    {
      "id": "PS.RGTEST.01",
      "claim": "CARRIER-NATIVE PATI-SALAM/SO(10) UNIFICATION KILL-TEST + NEGATIVE CONTROLS (2026-06-17): the suite claim-check of experiments/gauge-unification. Two-step M_Z->(SM)->M_PS->(PS)->M_GUT (SO(10): a4=a2L=a2R), matching a4=a3, a2L=a2, a2R^-1=(5/3)a1^-1-(2/3)a3^-1. (1) [E] matching holds and a valid 1-loop unification solution exists (M_PS < M_GUT < M_Pl). (2) [X] NEG: pure SM running does NOT unify (residual spread ~8.8 at 2e16 GeV, the universal non-SUSY gap). (3) [C] SCALE COINCIDENCE: the unifying M_PS lands on the scalaron scale M_s=c3^(7/2)Mbar -- M_PS/M_s = 1.0-1.7 at 1-loop (experiment 2-loop shrinks to ~1.0-1.2); two independent TFPT scales (gauge unification + gravitational scalaron) coincide. (4) [X] NEG: 126-type (10,1,3) content (large SU(2)_R, b2R>0) yields NO scalaron-matched solution (1-loop unphysical / 2-loop ~33x) -- match lost; 126 also E8-forbidden (v247). (5) [X] NEG: proton decay selects content -- minimal 16-Higgs M_GUT~2e15 -> tau_p < Super-K (2.4e34 yr) EXCLUDED; +(15,1,1) [E8-allowed 45] raises M_GUT proton-safer. (6) [O] residuals: the gauging fork, the exact kappa in M_PS=kappa M_s (now ~1.0-1.2), the one-45 E8 marginality (v247).",
      "status": "Data confrontation (kill-test) [X]/[O]",
      "canonicalStatus": "[E] matching+unification-exists; [X] SM-only fails, 126 breaks the match, proton decay selects content; [C] M_PS=scalaron coincidence; [O] gauging fork + exact kappa + one-45 marginality",
      "dependencies": "PS.ALGEBRA.01; QFT4D.RGTEST.01",
      "scripts": [
        "v249_ps_unification.py"
      ],
      "script": "v249_ps_unification.py",
      "location": "tfpt_research_contracts",
      "externalData": "PDG gauge couplings",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 335
    },
    {
      "id": "CONTRACT.QFT4D.DIRAC.01",
      "claim": "THE SPECTRAL-TRIPLE CONTRACT (2026-06-17): the object that would turn 'carrier is gauged' from a fork into a theorem (Reviewer-A Theorem 3 + the sigma=scalaron identification). Target: a real even almost-commutative triple (A,H,D,J,gamma) with A_F = PS algebra (v248), H_F = 3x16, D_F = D_M (x) 1 + gamma_5 (x) D_F(R,L,phi0,Majorana). (1) [E] H_F = N_fam x dim S^+ = 3x16 = 48 (three generations of the SO(10) spinor); A_F acts on it. (2) [E] the charged-fermion Yukawas are ALREADY fixed (FLAV.QUARK.01/v18), so the open test is NCG CONSISTENCY of D_F, not 'nice Yukawas' (Reviewer A). (3) [O] the 7 NCG obligations: reality J, grading gamma, KO-dimension (6 mod 8), the FIRST-ORDER CONDITION (the historic SO(10)-NCG obstruction), orientability, Poincare duality, no-junk forms. (4) [O] sigma=SCALARON HYPOTHESIS: the Chamseddine-Connes-Mukhanov singlet sigma (rescues first-order + gives nu_R Majorana) is conjectured to BE the R+R^2 scalaron (supported by M_PS=kappa M_s, v249); CAVEAT: the scalaron<->nu_R nu_R coupling is NOT automatic and must be proven. (5) [O] inner fluctuations Omega^1_D -> Higgs content 10+16+45 not 126 (NCG complement of v247). (6) [C] verdict: reduces the '4d QFT' residual to ONE constructive object (the triple) + the finite NCG checks; gauge group (v248), matter rep (v245), Yukawas (v18) and unification corridor (v249) already in place -- what is missing is D_F + its axioms.",
      "status": "Contract [O] (with [E] finite facts)",
      "canonicalStatus": "[E] H_F=3x16=48, Yukawas already fixed (test is NCG consistency); [O] the 7 NCG axioms (first-order condition the hard one), sigma=scalaron, inner-fluctuation Higgs content -- reduces 4d QFT to one spectral triple",
      "dependencies": "PS.ALGEBRA.01; CONTRACT.QFT4D.01",
      "scripts": [
        "v250_dirac_triple_contract.py"
      ],
      "script": "v250_dirac_triple_contract.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 336
    },
    {
      "id": "PS.DIRAC.01",
      "claim": "FIRST-ORDER CONDITION VIOLATED BY THE MAJORANA/sigma (2026-06-17): the Chamseddine-Connes-van Suijlekom 'beyond first order' mechanism exhibited on a faithful minimal model -- advances CONTRACT.QFT4D.DIRAC.01 (v250) from a contract to an exhibited mechanism. HONEST CORRECTION to the external review: the sigma field does NOT 'rescue' the first-order condition; the opposite is true (CCvS 2013) -- one DROPS the first-order condition, and the inner fluctuations of D THEN generate the fields that promote the SM gauge group to Pati-Salam SU(2)xSU(2)xSU(4) plus a singlet sigma (which gives the nu_R Majorana mass and drives the breaking). Model: H = H_F (+) H_F^c, H_F=C^3 (nu_L,nu_R,e_R); pi(a)=diag(rho(a),0) on particles; J = swap.conj so J pi(b) J^-1 = diag(0,conj rho(b)); D=[[D_p,M],[M^dag,conj D_p]] with D_p the Yukawa block and M the Majorana (nu_R nu_R^c = sigma) block. (1) [E] order-zero [pi(a),J pi(b) J^-1]=0. (2) [E] real-even: J^2=+1, gamma^2=1, D gamma=-gamma D (KO-dim 6 signs; physical SM triple is KO-6). (3) [E] first order HOLDS for the Yukawa block (M=0): [[D,pi(a)],J pi(b) J^-1]=0. (4) [E] first order VIOLATED with M!=0, supported EXACTLY on the off-diagonal (nu_R nu_R^c) Majorana blocks -- the sigma is the unique first-order violator; the controlled violation IS the Pati-Salam mechanism. (5) [C] sigma=scalaron (M_PS=kappa M_s, v249); physical identification [O] (scalaron<->nu_R nu_R coupling to be proven). (6) [O] the full 96-dim SM/PS finite triple + spectral-action expansion remain the open contract (v250).",
      "status": "Numerical (NCG mechanism) [E]/[C]/[O]",
      "canonicalStatus": "[E] order-zero + real-even (J^2=1,KO-6 signs,D odd) + first order holds(Yukawa)/violated(Majorana, localized to sigma); the CCvS beyond-first-order Pati-Salam mechanism exhibited; [C] sigma=scalaron; [O] full 96-dim triple + spectral action",
      "dependencies": "CONTRACT.QFT4D.DIRAC.01; PS.ALGEBRA.01",
      "scripts": [
        "v251_first_order_sigma.py"
      ],
      "script": "v251_first_order_sigma.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 337
    },
    {
      "id": "PS.DIRAC.02",
      "claim": "FULL 96-DIM FINITE SPECTRAL TRIPLE BUILT + NCG AXIOMS VERIFIED (2026-06-17): advances CONTRACT.QFT4D.DIRAC.01 (v250) from a contract listing 7 open obligations to a concrete object on which 5 are discharged to [E]. H_F = H_particle (+) H_antiparticle = particle/antiparticle doubling of three SO(10) 16-plets, dim 96 = 2 x Omega_adm. One generation = nu_L,nu_R,e_L,e_R (colour singlets) + (u,d)x(L,R)x3 colours = 16. D_F = [[S,T],[T^dag,conj S]] with S the Dirac/Yukawa block and T the Majorana (nu_R nu_R^c = sigma). J = (particle<->antiparticle swap).conj; gamma = chirality (flipped on antiparticles). A_F = C(+)H(+)M_3(C) as a bimodule: particles via (lambda,q) colour-trivial, antiparticles via (lambda,m) weak-trivial. (1) [E] dim 96. (2) [E] D_F=D_F^dag and D_F gamma=-gamma D_F. (3) [E] reality J^2=+1, JD=DJ. (4) [E] gamma^2=1, [gamma,pi(a)]=0, J gamma=-gamma J. (5) [E] KO-dimension 6 mod 8 (Chamseddine-Connes-Marcolli). (6) [E] order-zero [pi(a),J pi(b) J^-1]=0 for the SM algebra. (7) [E] first-order condition HOLDS for the SM algebra C(+)H(+)M_3 INCLUDING the full Majorana term (van Suijlekom). (8) [E] first-order VIOLATED for the Pati-Salam algebra (an SU(2)_R pairing nu_R with e_R) EXACTLY by the Majorana, with the violation supported exactly on the right-handed lepton doublet {nu_R,e_R} -- the Chamseddine-Connes-van Suijlekom beyond-first-order mechanism (SM -> Pati-Salam + sigma, cf v251) now on the full 96-dim triple. (9) [E] eigenvalues of D_F reproduce the fermion spectrum: charged Dirac eigenvalues = +/- m_f (all 9 charged masses) and the neutrino block gives the type-I seesaw m_light = m_D^2/M_R. (10) [C] Poincare duality: the intersection (multiplicity) form is non-degenerate (det!=0). (11) [O] orientability (Hochschild cycle), no-junk 1-forms and the spectral-action heat-kernel expansion remain open.",
      "status": "Numerical (NCG axioms) [E]/[C]/[O]",
      "canonicalStatus": "[E] dim 96 + Hermitian/odd + reality J + grading + KO-6 + order-zero + first-order holds(SM)/violated(PS, localized to nu_R,e_R) + spectrum=masses+seesaw; [C] Poincare duality; [O] orientability + no-junk + spectral action; discharges 5 of the 7 v250 obligations to [E]",
      "dependencies": "CONTRACT.QFT4D.DIRAC.01; PS.DIRAC.01; PS.ALGEBRA.01",
      "scripts": [
        "v252_full_finite_triple.py"
      ],
      "script": "v252_full_finite_triple.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 338
    },
    {
      "id": "PS.KAPPA.01",
      "claim": "kappa IN M_PS = kappa M_s PINNED TO O(1) ~ 1 (2026-06-17): discharges the open residual 6(b) of v249 ('exact kappa, now ~1.0-1.2, not pinned') as far as is honest. The scalaron scale M_s = c3^{7/2} Mbar ~ 3.06e13 GeV is exact from {c3=1/(8pi), Mbar}. kappa := M_PS/M_s with M_PS the RG output of the carrier-native PS two-step (v249). HEAT-KERNEL ORIGIN: in the spectral action S = Tr f(D/Lambda) the gauge kinetic terms (=> unification scale), the Einstein-Hilbert term and the R^2 term (=> scalaron) all descend from ONE cutoff Lambda and the SAME moments f_0,f_2,f_4 of one cutoff function (Gilkey/Seeley-DeWitt; Chamseddine-Connes); hence both M_PS and M_s are Lambda x dimensionless heat-kernel factor and kappa = sqrt((f_2/f_0) c_PS/c_grav), a ratio of traces over the 96-dim H_F (v252) -- Lambda cancels, kappa is scale-free. (1) [E] M_s exact. (2) [E] kappa pinned by RG: scanning PDG coupling errors x the two E8-allowed contents gives kappa in a tight O(1) interval ~1.3 (1-loop [1.0,1.7], experiment 2-loop [1.0,1.2]) -- pinned, not free. (3) [C] heat-kernel origin: O(1) cutoff moments give O(1) kappa; the RG interval sits inside the structural band [0.3,3]. (4) [X] negative control: 126-type content drives kappa out of the O(1) window (content-selective; 126 E8-forbidden anyway, v247). (5) [O] the exact scheme-independent decimal needs f_2/f_0 fixed; c3 is the candidate normalisation but the f-fixing is unproven.",
      "status": "Numerical (RG interval + heat-kernel) [E]/[C]/[X]/[O]",
      "canonicalStatus": "[E] M_s exact + kappa pinned to O(1)~1.3 by RG interval scan; [C] heat-kernel origin kappa=sqrt((f2/f0)c_PS/c_grav) scale-free; [X] 126 neg control; [O] exact decimal needs cutoff moments fixed",
      "dependencies": "PS.RGTEST.01; PS.DIRAC.02",
      "scripts": [
        "v253_kappa_scalaron_ps.py"
      ],
      "script": "v253_kappa_scalaron_ps.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 339
    },
    {
      "id": "PS.NCG.FLUCT.01",
      "claim": "INNER FLUCTUATIONS Omega^1_D -> SM HIGGS DOUBLET (no 126, no junk) + sigma (2026-06-17): closes the open obligation v250 #5 (Higgs content, NOT 126), the NO-JUNK obligation, and fixes the sigma quantum numbers (v250 #4 rep-theory half). Inner fluctuation D -> D + A + eps' J A J^-1 with A = sum pi(a)[D_F,pi(b)] in Omega^1_D = span{pi(a)[D_F,pi(b)]}. Because D_F is purely chirality-odd (Yukawa L<->R + Majorana), every 1-form is odd, so the FINITE fluctuations are SCALARS (gauge bosons come from the continuum [D_M,a]). (1) [E] Omega^1_{D_F} purely chirality-odd, gauge part 0. (2) [E] NO JUNK / NO 126: for the SM algebra C(+)H(+)M_3 every 1-form has support exactly inside the Higgs-doublet pattern (chirality-odd, colour-diagonal, no quark<->lepton mixing) = the single SM Higgs doublet (1,2)_{1/2}; no colour/weak triplet, no (10,1,3)/126-type scalar (0 junk entries, machine-checked). (3) [E] sigma NEEDS BEYOND FIRST ORDER: with the SM algebra the sigma (nu_R nu_R^c Majorana) direction is NOT generated by Omega^1_D (consistent with first order HOLDING, v252) but with the Pati-Salam algebra it IS generated -- sigma is precisely the Chamseddine-Connes-van Suijlekom beyond-first-order scalar (v251/v252). (4) [E] sigma quantum numbers: SM singlet (1,1,0) with B-L=2 (nu_R has B-L=-1, so sigma*(nu_R nu_R) is gauge invariant only for B-L(sigma)=+2), giving the nu_R Majorana mass and breaking B-L (the PS (4,1,2)->SM step). (5) [C] sigma=scalaron: sigma is a real SM-singlet scalar at the B-L/PS scale, consistent in quantum numbers and scale with the R+R^2 scalaron (M_PS=kappa M_s, v249/v253); the gravitational coupling proper remains [C]/[O].",
      "status": "Numerical (NCG inner fluctuations) [E]/[C]",
      "canonicalStatus": "[E] finite fluctuations = SM Higgs doublet (no junk/no 126) + sigma beyond first order + sigma B-L=2 singlet; [C] sigma=scalaron; closes v250 #5 + no-junk obligation + sigma QN",
      "dependencies": "PS.DIRAC.02; PS.DIRAC.01",
      "scripts": [
        "v254_inner_fluctuations.py"
      ],
      "script": "v254_inner_fluctuations.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 340
    },
    {
      "id": "PS.SPECACT.01",
      "claim": "SPECTRAL-ACTION HEAT-KERNEL EXPANSION (a0,a2,a4) + EXPLICIT kappa(f2/f0) (2026-06-17): closes the open obligation v252 #11 (spectral-action expansion) and makes the kappa-dependence on the cutoff moments explicit (v253 #5). The bosonic spectral action S = Tr f(D/Lambda) has the Seeley-DeWitt expansion S ~ 2 f4 Lambda^4 a0 + 2 f2 Lambda^2 a2 + f0 a4 (f4,f2 moments, f0=f(0)). (1) [E] expansion structure: a0=cosmological (N int sqrt g), a2=Einstein-Hilbert + Higgs mass, a4=Yang-Mills + Weyl^2 + R^2 + Higgs quartic + R|H|^2, with N=dim H_F=96 (Gilkey/Chamseddine-Connes). (2) [E] trace inputs a=Tr(Y+Y)=2.96, b=Tr((Y+Y)^2)=2.92, b/a^2=0.333=1/3 (top-Yukawa dominated, the Chamseddine-Connes value). (3) [E] gauge normalisation: the a4 Yang-Mills term fixes g3=g2=sqrt(5/3)g1, sin^2 thetaW=(sum T3^2)/(sum Q^2)=3/8 (cross-check v245). (4) [C] Higgs quartic: lambda(Lambda) ~ g^2 b/a^2 = g^2/3; with the sigma ingredient present (v254) this is the famous ~125 GeV (vs ~170 GeV without sigma) Higgs-mass prediction after RG running (m_H vs 125.25 GeV); exact coefficient + running cited. (5) [E] scalaron from R^2: the gravitational a4 carries a Weyl^2 + R^2 term with coefficient ~ f0 N; the R^2 spin-0 mode IS the Starobinsky scalaron, M_s its mass (R+R^2 gravity emerges from the spectral action). (6) [C] kappa made explicit: a2 carries f2 (Einstein-Hilbert), a4 carries f0 (R^2/scalaron + gauge), so kappa=M_PS/M_s=sqrt((f2/f0) c_PS/c_grav) is an explicit ratio of the two moments x traces over H_F; the RG value kappa~1.3 (v253) fixes f2/f0 to O(1). (7) [O] the exact kappa decimal and exact lambda(Lambda) need the cutoff function f fixed -- the last genuinely open input.",
      "status": "Numerical (spectral action) [E]/[C]/[O]",
      "canonicalStatus": "[E] Seeley-DeWitt expansion structure + trace inputs (b/a^2=1/3) + sin^2 thetaW=3/8 + scalaron from R^2; [C] Higgs quartic ~125 GeV + explicit kappa(f2/f0); [O] cutoff moments; closes v252 spectral-action obligation",
      "dependencies": "PS.DIRAC.02; PS.KAPPA.01; CONTRACT.QFT4D.01",
      "scripts": [
        "v255_spectral_action_expansion.py"
      ],
      "script": "v255_spectral_action_expansion.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 341
    },
    {
      "id": "PS.NCG.ORIENT.01",
      "claim": "ORIENTABILITY + POINCARE DUALITY -- HONEST STRUCTURAL TREATMENT (2026-06-17): the two NCG obligations of v250 that do NOT close to a clean [E] on the simplified v252 representation. NO faked proof. KO-dimension 6 forces the intersection form cap(p,q)=Tr(gamma pi(p) J pi(q) J^-1) to be ANTISYMMETRIC (from J gamma=-gamma J + order-zero + reality), so a skew form on the SM K-theory K_0(C(+)H(+)M_3)=Z^3 (odd rank) is necessarily degenerate -- a genuine structural fact. (1) [E] skew intersection form forced by KO-6, verified numerically on the explicit triple (v252). (2) [E] rank/corank: rank 2, corank 1, null direction ~ (1,1,-1) in (C,H,M_3). (3) [E] orientability local condition [gamma_F,pi(a)]=0 (v252); orientability proper is a degree-4 Hochschild cycle on the PRODUCT geometry M x F (metric dim 4), the finite factor (metric dim 0) only contributing gamma_F -- not a finite-only check. (4) [C] for the full hypercharge-faithful SM/PS spectral triple orientability and Poincare duality DO hold (van Suijlekom 'NCG and Particle Physics'; Chamseddine-Connes-Marcolli) -- the corank-1 of the naive skew form is resolved on the physical sublattice / proper K-theory pairing; established in the literature, not re-derived. (5) [O] a fully self-contained machine proof needs the COMPLETE hypercharge-faithful canonical bimodule; the v252 simplified rep faithfully verifies the five substantive axioms (reality, grading, KO-6, order-zero, first-order) but not these two skew/degree-4 conditions. Honestly open.",
      "status": "Numerical (NCG; honest status) [E]/[C]/[O]",
      "canonicalStatus": "[E] KO-6 skew intersection form + rank2/corank1/null(1,1,-1) + orientability local condition; [C] canonical triple satisfies both (literature); [O] self-contained machine proof needs full canonical bimodule -- no faked closure",
      "dependencies": "PS.DIRAC.02",
      "scripts": [
        "v256_orientability_poincare.py"
      ],
      "script": "v256_orientability_poincare.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 342
    },
    {
      "id": "PS.NCG.ORIENT.02",
      "claim": "ORIENTABILITY + POINCARE DUALITY -- HONEST LITERATURE-CONFIRMED RESOLUTION (2026-06-17): corrects the slightly-too-generous typing of v256. The published result (Cacic-Stephan 'Moduli Spaces of Dirac Operators for Finite Spectral Triples' arXiv:0902.2068; Chamseddine-Connes-Marcolli CCM07; Standard-Model-vacuum arXiv:hep-th/0601192) is that the CCM Standard-Model finite triple (KO-dimension 6, with three right-handed neutrinos -- exactly the TFPT case for the seesaw) GENUINELY FAILS strict orientability and strict Poincare duality, and instead satisfies the physically correct WEAKENED versions: quasi-orientability and modified Poincare duality. This is NOT a TFPT defect -- it is a known property of the whole NCG-Standard-Model model class. Proven on the explicit v252 triple: (1) [E] strict orientability FAILS: gamma is NOT in span{lambda(a) rho(b)} (degree-0 Hochschild image; rel residual 0.354) -- non-orientable, matching CCM/Cacic-Stephan. (2) [E] QUASI-orientability HOLDS: the (left,right) A-irrep boxes are DISJOINT by chirality (left-handed = H weak-doublet, right-handed = C singlet), so L^LR(H^even,H^odd)={0} -- the weakened orientability (Paschke-Sitarz/Krajewski formalism) that the CCM SM satisfies. (3) [E] strict Poincare duality FAILS: KO-6 forces the skew (antisymmetric) intersection form, corank 1 (v256) -- per the SM-vacuum analysis exactly the case of EQUAL left/right neutrino numbers. (4) [C] modified Poincare duality holds (CCM); Stephan's alternative finite triple (St06) restores STRICT Poincare duality with identical physical content. (5) [E] correction of v256: 'canonical triple satisfies both' is replaced by this precise statement (strict FAIL, weakened HOLD). Honest -- no faked closure.",
      "status": "Numerical (NCG; honest status) [E]/[C]",
      "canonicalStatus": "[E] strict orientability + strict Poincare duality FAIL (KO-6, 3 nu_R -- known model-class feature) + quasi-orientability HOLDS (boxes disjoint by chirality); [C] modified Poincare duality / Stephan PD-restoring triple; corrects v256 over-claim -- no faked closure",
      "dependencies": "PS.NCG.ORIENT.01; PS.DIRAC.02",
      "scripts": [
        "v257_quasiorient_modpd.py"
      ],
      "script": "v257_quasiorient_modpd.py",
      "location": "tfpt_research_contracts",
      "externalData": "arXiv:0902.2068 (Cacic-Stephan); arXiv:hep-th/0601192 (SM vacuum); CCM07",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 343
    },
    {
      "id": "PS.DIRAC.03",
      "claim": "DIRAC AS MODULAR/COVARIANCE INDUCTION (2026-06-17): the finite Dirac operator D_F (v252) is the modular/covariance induction of the boundary seam quasi-free (KMS, beta=1) state restricted to the carrier, not an independent posit -- closes the 'posited Yukawa vs derived operator' gap of v250/v252 (proposal Q2: D_F as Kasparov product, covariance formula as the finite implementation). Quasi-free/modular dictionary: one-particle modular Hamiltonian H=log((1-C)C^-1) (Araki; Casini-Huerta), inverse to the KMS occupation C=(1+e^H)^-1. The v18 Yukawas are the readout of C_Sigma=P_F C_Sigma P_F, not an extra input.",
      "status": "Numerical (NCG/modular) [E]/[C]",
      "canonicalStatus": "[E] inversion identity log((1-C)C^-1)=H exactly + C_F a positive contraction + chirality-odd (Dirac-type) + unitary-equivalent to v252 (same mass spectrum) + Majorana=off-diagonal covariance (seesaw m_D^2/M_R); [C] C_F=P_F C_Sigma P_F so D_F is the KK shadow of the boundary geometry (rests on the open seam realisation).",
      "dependencies": "PS.DIRAC.02; DYN.KMS.01; CONTRACT.QFT4D.DIRAC.01",
      "scripts": [
        "v258_dirac_covariance_induction.py"
      ],
      "script": "v258_dirac_covariance_induction.py",
      "location": "tfpt_research_contracts",
      "externalData": "Araki quasi-free states; Casini-Huerta modular Hamiltonian",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 344
    },
    {
      "id": "PS.SPECACT.02",
      "claim": "MODULAR/KMS SPECTRAL-ACTION CUTOFF (2026-06-17): the spectral-action cutoff f is the seam's own modular/KMS weight (beta=1 by Tomita-Takesaki/BW, v239; 2pi=1/(4c3), v58), so the heat-kernel cutoff f(u)=e^{-u} is derived not chosen and the moment ratio f_2/f_0=1 exactly -- reducing the last open scheme freedom of PS.KAPPA.01/PS.SPECACT.01 from a free function to a finite-triple trace ratio kappa=sqrt(c_PS/c_grav) (proposal Q4: the cutoff is modular, not chosen).",
      "status": "Numerical (spectral action) [E]/[C]",
      "canonicalStatus": "[E] KMS cutoff moments f_0=f_2=f_4=1 so f_2/f_0=f_4/f_2=1 (sympy) + seam-fixed beta=1 + non-vacuous neg control (Gaussian cutoff gives sqrt(pi)/2 != 1) + kappa=sqrt(c_PS/c_grav) (no cutoff function left); [C] RG kappa~1.3 pins c_PS/c_grav~1.7 (O(1) band [0.3,3]) and lambda(Lambda) likewise a finite-triple number -- the [O] free scheme becomes a finite computation over the built triple.",
      "dependencies": "PS.SPECACT.01; PS.KAPPA.01; DYN.KMS.01",
      "scripts": [
        "v259_modular_cutoff_kappa.py"
      ],
      "script": "v259_modular_cutoff_kappa.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 345
    },
    {
      "id": "ARCH.K3.01",
      "claim": "KUMMER/K3 UNIFICATION OF SEAM + CARRIER-16 + E8 (2026-06-17): one smooth Kummer/K3 surface carries the seam (pillowcase = the elliptic-involution quotient of the square torus tau=i, j=1728, v214), the carrier-16 (the 16 Kummer nodes = dim S+, v197) and the E8 lattice (the H^2(K3) intersection form U^3(+)E8(-1)^2), so the v1 glue D5(+)A3 + mu4 => E8 is the lattice shadow of a single geometric object (the K3/Kummer proposal). A unification, not a closure.",
      "status": "Lattice (K3/Kummer) [E]/[C]",
      "canonicalStatus": "[E] E8 Cartan even/det1/pos-def + K3 lattice rank22/det-1/even/signature(3,19) with E8(-1)^2 a summand + seam=T^2/(z->-z) 4 fixed points (pillowcase, orbifold Euler char 0) + carrier-16=Kummer nodes |A[2]|=2^4=16=dim S+=4x4 + honesty (the rank-16 Kummer lattice is NOT literally E8(-1)^2, distinct sublattices of H^2(K3)); [C] two faces of E8 in K3 (local du Val C^2/2I v232/v236; global H^2 summand) + the carrier<->node geometric unification.",
      "dependencies": "E8.GLU.01; QGEO.PILLOW.01; ARCH.RRCAR.02",
      "scripts": [
        "v260_k3_kummer_unification.py"
      ],
      "script": "v260_k3_kummer_unification.py",
      "location": "tfpt_research_contracts",
      "externalData": "standard K3 lattice theory (U^3(+)E8(-1)^2)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 346
    },
    {
      "id": "QFT.MSC.01",
      "claim": "MODULAR SPECTRAL CLOSURE (2026-06-17): the assembly/reduction certificate turning the whole QFT layer (v238-v260) into ONE relative object TFPT_QFT=(A_Sigma,omega_Sigma,Delta_Sigma,rho,A_F,H_F,D_F,J,gamma,S_rel) reduced to ONE open premise. Cross-validates the invariants that must agree across the round (index=marks=fixed points=glue order=4; carrier-16 across spinor/lattice/triple; one gap 6log(3/2) static+dynamic), re-derives the three closures (v258 Kasparov, v259 KMS cutoff, v260 K3), and certifies the relative-object manifest. The Modular Spectral Universality master statement: the unique RP holomorphic index-4 KMS seam state induces via Kasparov the 96-dim KO-6 triple whose inner fluctuations give the gauged Pati-Salam spectral action -- so Dirac, cutoff, gauging, glue and orientability are facets of one seam state, complete modulo QGEO.SYM.01.",
      "status": "Numerical (assembly certificate) [E]/[C]/[O]",
      "canonicalStatus": "[E] one index=marks=fixed points=glue order=4 + one carrier-16 (spinor/lattice/triple) + one gap 6log(3/2) (OS/recovery/Lindblad) + the three closures re-derived (Kasparov/KMS cutoff/K3) + the 10-component relative-object manifest; [O] the whole QFT layer reduces to one premise QGEO.SYM.01 (seam realisation), ambient QG QG.AMB.01 separate; [C] the Modular Spectral Universality master statement holds modulo QGEO.SYM.01.",
      "dependencies": "PS.DIRAC.03; PS.SPECACT.02; ARCH.K3.01; PS.DIRAC.02; GATE.METRIC.06; GATE.METRIC.07; QGEO.MARKS.03; DYN.KMS.01; DYN.GNS.01",
      "scripts": [
        "v261_modular_spectral_closure.py"
      ],
      "script": "v261_modular_spectral_closure.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 347
    },
    {
      "id": "FR.MPME.02",
      "claim": "F_QCD TRANSFER REPRODUCTION OF m_p/m_e (2026-06-18): the contract-only F_transfer/qcd_matching_mp_me pipeline implemented as a real solver. 2-loop alpha_s(M_Z)=0.1179 run with n_f thresholds (carrier b3=-7) gives alpha_s(2 GeV)~0.30 and Lambda^(3)_MSbar~0.33 GeV; with the lattice bridge C_p=m_p/Lambda (external O(1)) and the closed m_e, m_p/m_e~1824 vs measured 1836.15 within ~7% external budget. NOT a compiler power -- a [C] transfer reproduction; firewall forbids a compiler integer (SU(9) absent from D5(+)A3). Advances FR.MPME.01.",
      "status": "Numerical (transfer reproduction) [C]",
      "canonicalStatus": "[E] b3=-7 carrier output; [N] 2-loop alpha_s running reproduces PDG alpha_s(2 GeV)~0.30; [C] m_p/m_e~1824 reproduced within ~7% external budget (lattice C_p + alpha_s); [X] firewall -- m_p/m_e not an admissible compiler integer, stays [C].",
      "dependencies": "FR.MPME.01; QCD.LAMBDA.01; CONTRACT.F.01",
      "scripts": [
        "v262_fqcd_mp_me.py"
      ],
      "script": "v262_fqcd_mp_me.py",
      "location": "tfpt_4_frontier",
      "externalData": "alpha_s(M_Z) PDG; lattice m_p/Lambda_MSbar O(1)",
      "supersedes": "FR.MPME.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 348
    },
    {
      "id": "FLAV.PMNS.02",
      "claim": "M_nu AS THE TYPE-I SEESAW OF D_F (2026-06-18): the light neutrino matrix is the seesaw of the finite Dirac operator's own blocks, M_nu=-m_D M_R^{-1} m_D^T, advancing the v9 solar-angle texture from a posited matrix to a seesaw object. A mu-tau-symmetric (m_D,M_R) gives a mu-tau-symmetric M_nu reproducing theta23=45deg and sin^2 theta12=1/3-phi0/2 under the seam misalignment. theta13=0 in the leading texture, the CP magnitude (240deg) is the mu6/triality phase (not from M_nu), and M_R is the free seesaw scale -- an operator advance, NOT a closure of full PMNS dynamics.",
      "status": "Numerical (seesaw operator) [E]/[C]/[O]",
      "canonicalStatus": "[E] M_nu=-m_D M_R^{-1} m_D^T symmetric + seesaw-suppressed (output of D_F) + mu-tau-symmetric image; [C] reproduces v9 angles (theta23=45, sin^2 th12=1/3-phi0/2) under the seam lemma; [O] theta13=0 in the leading texture, CP magnitude from mu6/triality (v231/v233), absolute nu-mass scale (M_R) open. Operator advance, not a closure.",
      "dependencies": "FLAV.TH12.01; PS.DIRAC.02; FLAV.SOLAR.01; CP.TRIALITY.01",
      "scripts": [
        "v263_mnu_from_df.py"
      ],
      "script": "v263_mnu_from_df.py",
      "location": "tfpt_2_standard_model",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 349
    },
    {
      "id": "QGEO.ENERGY.03",
      "claim": "RAW-SEAM => MARK-LOCALITY REDUCTION (2026-06-18): composes v216+v201+v198 on the canonical pillowcase metric to make the converse implication explicit and isolate the bedrock. The flat pillowcase has curvature only at the 4 marks (Gauss-Bonnet), so the DtN sub-principal symbol is a mu4-orbit sum, Z4-invariant (Fourier on 4Z), block-diagonal, [rho,M_f]=0; with |k| commuting (v198) [rho,Lambda]=0 => omega o rho=omega ON THIS METRIC. Reduces the whole bedrock to the single statement 'the raw RP seam carries the uniformising flat pillowcase metric' (= QGEO.SYM.01). A reduction/sharpening, NOT a closure -- the converse (raw RP forces the metric) stays the irreducible postulate.",
      "status": "Geometric premise [O] (reduction, not closure)",
      "canonicalStatus": "[E] Gauss-Bonnet 4 marks + block-diagonal/state-invariance chain on the pillowcase; [N] Z4-Fourier support (4Z) of the pillowcase curvature; [O] the metric-realisation premise (QGEO.SYM.01) stays open -- a reduction to one metric statement, NOT a closure (NEG control: curvature off the mu4 orbit breaks the chain).",
      "dependencies": "QGEO.SYM.01; QGEO.SUBPRIN.01; QGEO.MODULAR.01; QGEO.MARKS.03; QGEO.PILLOW.01",
      "scripts": [
        "v264_raw_seam_marklocal.py"
      ],
      "script": "v264_raw_seam_marklocal.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 350
    },
    {
      "id": "QFT4D.FORK.01",
      "claim": "4D-QFT FORK FROZEN AS A BRANCH POLICY (2026-06-18): boundary-only is the canonical 4D reading; carrier-native Pati-Salam is a falsifiable UV branch; SM-only 4D-GUT unification is killed. A frozen decision tree (qft4d_fork_freeze.json) with explicit promotion/demotion rules, not an open ambiguity.",
      "status": "Branch policy (frozen) [C]/[X]",
      "canonicalStatus": "[C] A canonical (boundary only) + B conditional PS UV branch; [X] SM-only GUT killed; [E] wording guard + decision JSON written. The fork is a frozen decision tree, not an ambiguity.",
      "dependencies": "QFT4D.RGTEST.01; PS.RGTEST.01; PS.E8BRANCH.01; QFT.MSC.01",
      "scripts": [
        "v265_qft4d_fork_freeze.py"
      ],
      "script": "v265_qft4d_fork_freeze.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 351
    },
    {
      "id": "QFT4D.BOUNDARY.DEFAULT.01",
      "claim": "CANONICAL 4D READING = BOUNDARY ONLY (2026-06-18): TFPT delivers the boundary QFT, the relative spectral action and the SM readout; a 4D-GUT is not claimed by default (E8 is the audit hull, not an unbroken 4D gauge group).",
      "status": "Branch policy (canonical) [C]",
      "canonicalStatus": "[C] boundary-only is the default 4D reading; the MSC closes the relative boundary object modulo QGEO.SYM.01, ambient QG separate.",
      "dependencies": "QFT.MSC.01; QFT4D.FORK.01",
      "scripts": [
        "v265_qft4d_fork_freeze.py"
      ],
      "script": "v265_qft4d_fork_freeze.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 352
    },
    {
      "id": "QFT4D.SMONLYGUT.01",
      "claim": "SM-ONLY 4D-GUT UNIFICATION KILLED (2026-06-18): 1-loop SM running with b=(41/10,-19/6,-7) does not unify -- at the alpha1=alpha2 scale alpha3^-1 is off by ~5.6, spread ~8.8 at 2e16 GeV. The canonical expectation, not a failure (re-derives v246).",
      "status": "Kill-test [X]",
      "canonicalStatus": "[X] the plain SM couplings do not meet (recomputed); E8 is the audit hull, not a 4D gauge group, so non-unification is expected.",
      "dependencies": "QFT4D.RGTEST.01",
      "scripts": [
        "v265_qft4d_fork_freeze.py"
      ],
      "script": "v265_qft4d_fork_freeze.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 353
    },
    {
      "id": "PS.UVBRANCH.01",
      "claim": "PATI-SALAM AS A CONDITIONAL UV BRANCH (2026-06-18): carrier-native Pati-Salam is permitted as a falsifiable UV completion, NOT the default -- E8-allowed content {1,10,16,45} (no 126), sin^2 thetaW=3/8, kappa O(1) band; carrier gauging stays the contract CONTRACT.QFT4D.DIRAC.01.",
      "status": "Branch policy (conditional) [C]",
      "canonicalStatus": "[C] PS is a conditional UV candidate (E8 content + sin^2 thetaW=3/8 + kappa band); not promoted to default; carrier gauging stays [O].",
      "dependencies": "PS.E8BRANCH.01; PS.ALGEBRA.01; PS.RGTEST.01; QFT4D.FORK.01",
      "scripts": [
        "v265_qft4d_fork_freeze.py"
      ],
      "script": "v265_qft4d_fork_freeze.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 354
    },
    {
      "id": "PS.THRESHOLD.01",
      "claim": "PATI-SALAM THRESHOLD MODEL EXPLICIT (2026-06-18): the PS UV branch is admissible only with an explicit two-step threshold model meeting the scalaron scale (M_PS=kappa M_s, kappa in [1.0,1.7]); the detailed reps-at-scales model is the open part (v266 target).",
      "status": "Branch policy (threshold) [C]/[O]",
      "canonicalStatus": "[C] kappa band met; [O] the explicit reps-at-scales threshold model is the open part (deferred to v266).",
      "dependencies": "PS.RGTEST.01; PS.KAPPA.01; QFT4D.FORK.01",
      "scripts": [
        "v265_qft4d_fork_freeze.py"
      ],
      "script": "v265_qft4d_fork_freeze.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 355
    },
    {
      "id": "PS.PROTON.KILL.01",
      "claim": "PROTON-DECAY SAFETY RULE (2026-06-18): no discovery tau_p is predicted; PS is killed if its thresholds need forbidden reps, leave the kappa band, or predict tau_p below the current bound; until a sharp window is published proton decay is a safety constraint, not a discovery.",
      "status": "Kill-test rule [X]",
      "canonicalStatus": "[X] proton-decay is a frozen safety constraint/kill rule for the PS branch, not a discovery prediction (no fake tau_p window).",
      "dependencies": "PS.RGTEST.01; QFT4D.FORK.01",
      "scripts": [
        "v265_qft4d_fork_freeze.py"
      ],
      "script": "v265_qft4d_fork_freeze.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 356
    },
    {
      "id": "QFT4D.NO_OVERCLAIM.01",
      "claim": "4D OVER-CLAIM WORDING GUARD (2026-06-18): bare 4D over-claims ('TFPT solves 4D QFT', 'SM couplings unify', '4D GUT prediction', 'unbroken E8 gauge') are absent from the active docs, and the scoped fork-policy phrases ('boundary only is canonical', 'SM-only ... killed', 'not claimed by default') are present.",
      "status": "Formal/prose guard [E]",
      "canonicalStatus": "[E] prose sentinel over the active .tex: forbidden bare over-claims absent (6/6) + required scoped policy phrases present (3/3).",
      "dependencies": "QFT4D.FORK.01",
      "scripts": [
        "v265_qft4d_fork_freeze.py"
      ],
      "script": "v265_qft4d_fork_freeze.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 357
    },
    {
      "id": "PS.THRESHOLD.02",
      "claim": "EXPLICIT PATI-SALAM THRESHOLD MODEL (2026-06-18): the two-step carrier-native PS chain made explicit for the two E8-allowed contents -- minimal 16-Higgs (M_PS~4.2e13=scalaron, M_GUT~2.4e15) and +(15,1,1) (M_GUT~5.9e15) -- realising the PS.THRESHOLD.01 policy with concrete reps-at-scales and M_PS on the scalaron scale. Decides branch B together with PS.PROTON.02.",
      "status": "Numerical (two-step RG) [E]/[C]",
      "canonicalStatus": "[E] explicit two-step solutions (M_PS,M_GUT,alpha_GUT) for both E8-allowed contents, M_PS on the scalaron scale; [C] realises the threshold policy; selects the content via proton decay (PS.PROTON.02).",
      "dependencies": "PS.THRESHOLD.01; PS.RGTEST.01; PS.KAPPA.01",
      "scripts": [
        "v266_ps_threshold_proton.py"
      ],
      "script": "v266_ps_threshold_proton.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 358
    },
    {
      "id": "PS.PROTON.02",
      "claim": "PROTON-DECAY WINDOW DECIDES BRANCH B (2026-06-18): d=6 tau_p(p->e+pi0) over the explicit thresholds -- minimal 16-Higgs tau_p~4.4e33 yr < Super-K (2.4e34) => KILLED even at the high hadronic band; +(15,1,1) [E8-allowed single 45] tau_p~1.6e35 yr > Super-K => survives now, but sits AT the Hyper-K reach (~1.4e35) => decisively falsifiable within the decade. Branch B is content-selected (must be +(15,1,1)) and Hyper-K-decisive; A (boundary-only) stays canonical. Marginal (one 45 in the hull, v247) + O(3) tau_p uncertainty.",
      "status": "Kill-test (proton decay) [X]/[C]",
      "canonicalStatus": "[X] minimal content killed by Super-K + Hyper-K decisive for the survivor; [C] +(15,1,1) survives now and is the only E8-allowed proton-safe content; [O] marginal (one 45) + O(3) hadronic uncertainty. Promotes branch B to a content-selected, dated-kill-test UV branch (still not the default).",
      "dependencies": "PS.THRESHOLD.02; PS.PROTON.KILL.01; PS.E8BRANCH.01; QFT4D.FORK.01",
      "scripts": [
        "v266_ps_threshold_proton.py"
      ],
      "script": "v266_ps_threshold_proton.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 359
    },
    {
      "id": "QGEO.SYM.02",
      "claim": "QGEO RIGIDITY / MINIMAL-AXIOM FORM (2026-06-18): the bedrock QGEO.SYM.01 is sharpened to its minimal form -- the single bare statement 'the raw RP seam admits the carrier mu4 clock as a conformal symmetry permuting its 4 marks' (= omega o rho = omega) FORCES everything: square config (cross-ratio 2 <=> j=1728 <=> order-4 CM auto), the unique flat pillowcase metric (Troyanov, v214), mark-local Z4-Fourier DtN (v201/v264), [rho,Lambda]=0 and omega o rho=omega (v198). Neg controls: generic config j!=1728 (Z2 only), hexagonal j=0 (Z6, order 6). Arithmetic rigidity (Belyi/dessins): {1,i,-1,-i} over Q, j=1728 CM by Z[i]. Closes the REDUCTION (axiom => everything), NOT the axiom -- the bare order-4 symmetry stays the one foundational postulate.",
      "status": "Geometric premise [E]/[C]/[O] (rigidity; minimal axiom)",
      "canonicalStatus": "[E] order-4 symmetry forces cross-ratio 2 / j=1728 / the pillowcase / mark-local DtN / omega o rho=omega (the whole chain) + neg controls (generic Z2, hex Z6); [C] arithmetic/Galois rigidity (j=1728 CM, {1,i,-1,-i} over Q); [O] the bare order-4 symmetry (QGEO.SYM.01) stays the one foundational postulate -- the reduction is closed, the axiom is not.",
      "dependencies": "QGEO.SYM.01; QGEO.PILLOW.01; QGEO.ENERGY.03; QGEO.RIGID.01; QGEO.MODULAR.01",
      "scripts": [
        "v267_qgeo_rigidity_minimal_axiom.py"
      ],
      "script": "v267_qgeo_rigidity_minimal_axiom.py",
      "location": "tfpt_research_contracts",
      "externalData": "Troyanov uniformisation; Belyi/Grothendieck dessins d'enfants",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 360
    },
    {
      "id": "FLAV.TH13.01",
      "claim": "REACTOR ANGLE EXPONENT = CARRIER HYPERCHARGE TRACE (2026-06-18): the theta_13 value sin^2 theta_13 = phi0 e^{-5/6} = 0.0231 (v16) has its EXPONENT origin closed exactly -- gamma = tr_E Y^2 = 5/6 over the 5-slot carrier hypercharge Y=diag(-1/3,-1/3,-1/3,1/2,1/2) (anomaly-free roots of 6Y^2-Y-1=0), complement 1/6 the neutrino ratio. theta_13 is confirmed a SEPARATE carrier-trace channel: the mu-tau-breaking eps=3phi0/4 (which gives theta_12) induces only ~1.6e-3 << 0.023, correcting the tempting 'theta_13 from M_nu' route (v263). CP magnitude (delta_PMNS=240 deg, mu6/triality) and absolute nu-mass scale (M_R) remain the open PMNS items.",
      "status": "Numerical/identity (flavor readout) [E]/[C]/[O]",
      "canonicalStatus": "[E] gamma=5/6=tr_E Y^2 (exponent origin) + own-channel separation (eps^2 ~ 1.6e-3 != 0.023); [C] value 0.0231 vs ~0.0222; [O] CP magnitude + absolute scale open. Closes the theta_13 exponent origin, not full PMNS dynamics.",
      "dependencies": "FLAV.TH12.01; FLAV.SOLAR.01; FLAV.PMNS.02; CAR.SM.01",
      "scripts": [
        "v268_theta13_carrier_trace.py"
      ],
      "script": "v268_theta13_carrier_trace.py",
      "location": "tfpt_2_standard_model",
      "externalData": "NuFIT 6.0 / PDG sin^2 theta_13 ~ 0.0222",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 361
    },
    {
      "id": "QFT4D.SPERT.01",
      "claim": "PERTURBATIVE 4D S-MATRIX (S_pert) AS AN EPSTEIN-GLASER / pAQFT SKELETON (2026-06-18): the honest middle layer of the three-layer S-matrix -- S_top (2d DHR braiding monodromy |M|=1, statistics, v243), S_pert (4d EG/BV S-matrix of the Connes spectral action), S_phys (LSZ on the OS-reconstructed admissible Wightman functions, v240). The spectral-action a_4 interaction is power-counting renormalizable (all operators mass dim <= 4) with a finite counterterm basis (7 marginal + 3 relevant), so by the Epstein-Glaser theorem the perturbative S-matrix S(g)=T exp(i int g L_int) exists order by order (causal factorisation + finite local extension; no path integral, no UV divergences). Gapped admissible sector (Delta=6log(3/2)) gives an IR-safe adiabatic limit. PERTURBATIVE only: the nonperturbative ambient measure QG.AMB.01 stays open; the canonical 4D reading is boundary-only (v265).",
      "status": "Contract [C]/[O] (perturbative skeleton)",
      "canonicalStatus": "[E] three-layer typing + power-counting renormalizable (dim<=4) + finite counterterm basis (7+3); [C] Epstein-Glaser order-by-order existence (cited theorem applied) + adiabatic/gap link to S_phys; [O] perturbative only -- ambient measure QG.AMB.01 stays open. A contract/skeleton, not a nonperturbative closure.",
      "dependencies": "CONTRACT.QFT4D.01; DHR.SMATRIX.01; DYN.GNS.01; PS.SPECACT.01; QFT4D.FORK.01",
      "scripts": [
        "v269_spert_paqft_skeleton.py"
      ],
      "script": "v269_spert_paqft_skeleton.py",
      "location": "tfpt_research_contracts",
      "externalData": "Epstein-Glaser causal perturbation theory; Brunetti-Fredenhagen pAQFT",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 362
    },
    {
      "id": "FLAV.PMNS.03",
      "claim": "COMPLETE COMPLEX PMNS MATRIX + LEPTONIC JARLSKOG J_PMNS (CP STRENGTH) (2026-06-18): the four TFPT channels (sin^2 th12 = 1/3 - phi0/2 v9/v263, sin^2 th23 = 1/2 v9/v263, sin^2 th13 = phi0 e^{-5/6} v16/v268, delta = 4 pi/3 = 240 deg v231/v233) assemble into one exactly-unitary complex PMNS matrix U = R23 U13(delta) R12, from which the Jarlskog CP strength J_PMNS = Im(U_e1 U_mu2 U*_e2 U*_mu1) = s12 c12 s23 c23 s13 c13^2 sin(delta) = -0.02965 (J_max = 0.03424) is a DERIVED number (the leptonic analogue of the CKM J, v88). Data: J_max vs NuFIT 5.2 J_max 0.0332 (~3%), J vs best fit -0.026 (delta_data ~232 deg within ~1 sigma), the negative CP-violating sign reproduced. Closes the CP-STRENGTH readout; delta itself is the mu6/triality phase (assembled, not from M_nu/D_F) and the absolute neutrino-mass scale (Sigma m_nu, M_R) stays open.",
      "status": "Readout [C] (CP strength); [O] phase origin + absolute scale",
      "canonicalStatus": "[E] the three angle channels + the exact unitary assembly; [C] the derived Jarlskog CP strength + its NuFIT confrontation; [O] the phase's operator origin (delta = mu6/triality, not M_nu) + the absolute scale. A PMNS assembly + CP-strength readout, not a full operator closure.",
      "dependencies": "FLAV.TH12.01; FLAV.PMNS.02; FLAV.TH13.01; CP.MU6.01; CP.TRIALITY.01",
      "scripts": [
        "v270_pmns_jarlskog_assembly.py"
      ],
      "script": "v270_pmns_jarlskog_assembly.py",
      "location": "tfpt_2_standard_model",
      "externalData": "NuFIT 5.2 (2022) PMNS global fit; PDG 2024",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 363
    },
    {
      "id": "QFT4D.SPERT.02",
      "claim": "CONCRETE EPSTEIN-GLASER ONE-LOOP QUARTIC RENORMALIZATION (2026-06-18): takes the v269 S_pert pAQFT skeleton from 'exists' to an actual EG number (no path integral). [E] T_1=:L_int: needs no extension; the first EG renormalization is the order-2 product T_2. [E] the massless Feynman propagator has Steinmann scaling degree sd=2 in d=4; the one-loop bubble (two propagators) has sd=4=d, so the UV singular order omega=sd-d=0. [E] omega=0 => the EG extension freedom is C(omega+d,d)=1 local constant -- EXACTLY one logarithmic counterterm (renormalizable, finite, no infinite tower; matches v269). [E] the loop factor Omega_3/(2(2pi)^4)=(2pi^2)/(2(2pi)^4)=1/(16 pi^2) exactly -- the same 1/(16 pi^2) that normalises the spectral-action a_4 geometric quartic. [C] the phi^4 one-loop beta_lambda = 3 lambda^2/(16 pi^2) (symmetry factor 3 = s,t,u channels), EG initial condition lambda_Phi(UV)=1/(16 pi^2). [O] order-2/one-loop, one diagram class; all-order EG + nonperturbative measure QG.AMB.01 stay open. Exact core mirrored in Wolfram.",
      "status": "Numerical/Identity [E] (one-loop EG core) + [C] (beta + IC) + [O] (all-order)",
      "canonicalStatus": "[E] order structure + scaling degree (sd=4=d) + one log counterterm (omega=0) + loop factor 1/(16 pi^2) (exact, Wolfram-mirrored); [C] the phi^4 one-loop beta + the spectral-action initial condition; [O] order-2/one-loop only, all-order + nonperturbative stay open. Turns the v269 skeleton into a concrete one-loop number.",
      "dependencies": "QFT4D.SPERT.01; CONTRACT.QFT4D.01; PS.SPECACT.01; HIGGS.FREESEAM.01",
      "scripts": [
        "v271_eg_oneloop_quartic.py"
      ],
      "script": "v271_eg_oneloop_quartic.py",
      "location": "tfpt_research_contracts",
      "externalData": "Epstein-Glaser causal perturbation theory; Steinmann scaling degree",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 364
    },
    {
      "id": "FLAV.NUSCALE.01",
      "claim": "ABSOLUTE NEUTRINO-MASS SCALE TYPED HONESTLY (2026-06-18): does NOT fabricate a Sigma m_nu number (would contradict the v184 firewall); closes the PMNS bookkeeping instead. [E] the absolute scale reduces to ONE seesaw ratio m_3 = (y_nu v_EW)^2/M_R -- one irreducible UV input like v_geo (No-Unit, v153); the ratios (m2/m3, ordering, PMNS angles v270) are pinned. [N] the observed m_3 = sqrt(dm2_atm) = 0.050 eV with the TFPT PS->GUT scale window M_R in [M_PS=4.2e13, M_GUT=2.4e15] GeV (v249) needs a third-generation Dirac Yukawa y_nu in [0.26, 2.0] -- O(1), Yukawa-natural, no tuning. [O] the y_nu=1 seesaw scale M_R=v_EW^2/m_3=6.1e14 GeV has log_c3(M_R/Mbar)=2.57 (non-integer), not a TFPT compiler output; (y_nu,M_R) trade off so the absolute Sigma m_nu stays one open UV input. [C] the normal-ordering floor Sigma m_nu = m_2+m_3 = 0.0586 eV is consistent with Planck+BAO (<0.12 eV) and DESI 2024 (~0.072 eV). [X] a cosmological Sigma < 0.0586 eV excludes the NO m1~0 spectrum. UPDATE 2026-07-14 (v481/FLAV.NUSCALE.02): under the CARRIER normalisation y_nu = y_t (minimal 16.16.10 Yukawa sector, [P]) the (y_nu, M_R) trade-off collapses to M_R alone; the required M_R = 9.3e13 GeV sits INSIDE the PS window at log_c3(Mbar/M_R) = 3.15; the integer rung c3^3 Mbar is DECLINED at 1-loop (m_3 40% low). The absolute scale stays [O]; candidate class, frozen record untouched.",
      "status": "Bookkeeping [E] + window [N]/[C] + [O] absolute scale + [X] kill",
      "canonicalStatus": "[E] the absolute nu-scale = one seesaw ratio (parallels v_geo); [N]/[C] the TFPT PS->GUT window is consistent with the observed spectrum at O(1) Dirac Yukawa + the cosmological floor; [O] the absolute value stays one open UV input (M_R not a clean Mbar power, v184); [X] the NO floor is the kill test. Closes the PMNS bookkeeping, not the absolute number.",
      "dependencies": "FLAV.PMNS.02; FLAV.PMNS.03; PS.RGTEST.01; FR.MPME.02; ANCHOR.VGEO.02",
      "scripts": [
        "v272_nu_mass_scale.py"
      ],
      "script": "v272_nu_mass_scale.py",
      "location": "tfpt_2_standard_model",
      "externalData": "NuFIT 5.2 dm2; Planck 2018 + DESI 2024 Sigma m_nu bound",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 365
    },
    {
      "id": "QFT4D.SPERT.03",
      "claim": "EG ONE-LOOP GAUGE SELF-ENERGY -> (b1,b2,b3) = (41/10,-19/6,-7) (2026-06-18): takes v271's quartic mechanism to the gauge sector. [E] the one-loop gauge 2-point function is the order-2 EG product of two currents, scaling degree sd=4=d so omega=sd-d=0 -> exactly one local counterterm per gauge coupling (wavefunction/coupling renormalization), the SAME marginal mechanism and loop factor 1/(16 pi^2) as the quartic. [E] the one-loop coefficients from the explicit carrier/SM content (b_i = -(11/3)C2(G) + (2/3) sum_f T(R) + (1/3) sum_s T(R)): b3 = -(11/3)(3)+(2/3)(6) = -7; b2 = -(11/3)(2)+(2/3)(6)+(1/3)(1/2) = -19/6; b1 (GUT norm) = (3/5)[(2/3)(10)+(1/3)(1/2)] = 41/10 -- the same 41 as the carrier algebra 10 b1 = g_car 2^{g_car-2}+1 (v159/CAR.SM.01). [C] the physical RG directions (b3<0 QCD asymptotic freedom, b2<0, b1>0 U(1) Landau) are EG-derived and feed the unification cross-check (v246/v249). [O] order-2/one-loop; the two-loop gauge matrix (v159, PyR@TE) and the all-order EG construction stay open. Exact b_i mirrored in Wolfram.",
      "status": "Identity [E] (EG one-loop b_i, exact) + [C] (RG directions) + [O] (all-order)",
      "canonicalStatus": "[E] the EG gauge self-energy scaling-degree mechanism + the exact one-loop b_i=(41/10,-19/6,-7) from the content (Wolfram-mirrored); [C] the physical RG directions; [O] two-loop + all-order. The gauge analogue of v271's quartic.",
      "dependencies": "QFT4D.SPERT.02; QFT4D.SPERT.01; EM.BUDGET.02; CAR.SM.01",
      "scripts": [
        "v273_eg_gauge_running.py"
      ],
      "script": "v273_eg_gauge_running.py",
      "location": "tfpt_research_contracts",
      "externalData": "Epstein-Glaser causal perturbation theory; standard one-loop gauge beta",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 366
    },
    {
      "id": "SCALE.OVERDET.01",
      "claim": "THE SINGLE MASS ANCHOR IS OVER-DETERMINED; NO DERIVABLE ABSOLUTE SCALE IS A THEOREM (2026-06-18): sharpens the No-Unit theorem (v153). [E] one anchor: every dimensionful output is a pure number x M_bar^d (rho_Lambda/M_bar^4 = (3/4 pi^2) e^{-2 alpha^-1} = 7.13e-121 v60/v7, M_scal/M_bar = c3^{7/2} v7, the whole mass spectrum); the absolute neutrino scale (v272) and v_geo are the SAME anchor. [N] OVER-DETERMINATION: two independent routes to M_bar agree -- Route 1 (gravity) M_bar = (8 pi G)^{-1/2} = 2.4353e18 GeV; Route 2 (cosmology) invert the compiler prediction rho_Lambda/M_bar^4 with the measured rho_Lambda^{1/4} = 2.24 meV -> M_bar = 2.438e18 GeV, agreeing to 0.11%. [E] the anchor is the gravitational unit (G), the one measured dimensionful input of all physics (modern SI fixes hbar, c exactly). [E] THEOREM not gap: the seam is a conformal c=8 chiral CFT (v157/v158), scale-invariant by construction so provably no intrinsic scale; the scale is gravitational, entering only through the ambient measure (QG.AMB.01). [A] TFPT is scale-complete up to one over-determined anchor (the Planck mass) -- maximal predictivity for a dimensionless compiler, not a missing derivation.",
      "status": "Identity [E] + over-determination [N] + re-typing [A]",
      "canonicalStatus": "[E] one anchor + the conformal-seam reason; [N] the two-route over-determination (gravity vs dark energy, 0.11%); [A] the re-typing of v_geo to an over-determined anchor = the Planck mass. Sharpens v153/v78.",
      "dependencies": "ANCHOR.VGEO.02; COSMO.LAM.01; QG.AMB.01; FLAV.NUSCALE.01",
      "scripts": [
        "v274_scale_overdetermination.py"
      ],
      "script": "v274_scale_overdetermination.py",
      "location": "tfpt_4_frontier",
      "externalData": "Planck 2018 dark-energy density; CODATA G / reduced Planck mass",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 367
    },
    {
      "id": "QGAMB.ROADMAP.01",
      "claim": "OBLIGATION ROADMAP FOR QG.AMB.01 (the ambient nonperturbative QG measure), CONSOLIDATED INTO ONE MODULE (2026-06-18; QG.AMB.01 since reclassified as a DECOUPLED general QG problem, NOT a TFPT gate, v335): the (now decoupled, non-TFPT) general QG problem had no module of its own; this gives it one without claiming closure. [E] Tier A gap decoupling: Delta_eff = 6 log(3/2) - 2*248 c3^2 = 2.433 - 0.785 = 1.648 > 0 (v36/v76), so every testable low-energy readout (masses, mixings, alpha^-1, R+R^2) is independent of the un-built ambient measure -- QG.AMB.01 is gap-decoupled, not a blocker. [E]/[P] Tier B boundary-net reduction: the ambient bulk measure reduces to identifying the seam-Calderon boundary measure with the holomorphic (E8)_1 conformal net (c = 248/31 = 8 = rank E8; v77/GATE.METRIC.03), the gap giving clustering -> tightness; residual = the constructive seam-Calderon -> (E8)_1 map + bulk reconstruction (REDTEAM.A.01: reduced, not closed). [C] the perturbative layer is in hand (S_pert EG-constructible v269 + one-loop numbers v271/v273). [O] the nonperturbative interacting ambient measure (the bulk path integral / projective limit) is NOT constructed -- the one genuine structural frontier. [X] QG.AMB.01 closes IFF the seam net is identified with (E8)_1 AND the bulk reconstruction is proved; a proof that the seam net is NOT (E8)_1 or that the gap fails (Delta_eff<=0) kills the reduction.",
      "status": "Roadmap [E]/[C] + [O] open core + [X] promotion/kill",
      "canonicalStatus": "[E] Tier A decoupling + the c=8 boundary-net facts; [C] the perturbative layer; [O] the nonperturbative ambient measure stays open; [X] the promotion/kill condition. A consolidation/roadmap that gives the QG.AMB.01 gate a module -- it does not close it.",
      "dependencies": "QG.AMB.01; QG.GAPDEC.01; GATE.METRIC.03; QFT4D.SPERT.01",
      "scripts": [
        "v275_qgamb_roadmap.py"
      ],
      "script": "v275_qgamb_roadmap.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 368
    },
    {
      "id": "REDTEAM.F.01",
      "claim": "RED TEAM Target F -- adversarial audit of the perturbative 4D-QFT + scale round (v269-v275), published in release 5.2. TWO attacks land and force a narrowing (now folded into v269/v274); the rest survive as typed. (1) GRAVITY S_pert NON-UNITARY: power-counting renormalizability is not unitarity -- the spectral-action R^2/Weyl^2 sector gives a 4-derivative propagator 1/(p^2(p^2+M^2)) whose massive spin-2 mode has a NEGATIVE residue (the Stelle ghost), so S_pert is unitary for the matter+gauge sector only; the gravity perturbative S-matrix carries the ghost, which is exactly why the nonperturbative QG.AMB.01 (asymptotic safety v207 / boundary (E8)_1 net) is the real frontier, not a new gap. (2) OVER-DETERMINATION CONDITIONAL: v274 Route-2 inverts rho_Lambda/M_bar^4=(3/4pi^2)e^{-2 alpha^-1} = LAMBDA.BRANCH.01 [C], so the 0.11% gravity-vs-cosmology agreement is conditional on the Lambda-branch (the gravity route is unconditional). Already-honest: J_PMNS (v270) inherits delta's [C] provenance (assembled mu6/triality phase, not an M_nu output); the nu-scale (v272) is [O] with the NO floor Sigma m_nu=0.0586 eV as the one kill test. FIREWALL HOLDS: no positive-claim doc carries an unscoped overclaim (the red-team paper itself is excluded as it quotes forbidden phrases to forbid them). Verdict: SURVIVES (narrowed).",
      "status": "Red-team [P] (survives, narrowed)",
      "canonicalStatus": "Red-team verdict: the v269-v275 round survives as typed; two narrowings folded into v269 (gravity-sector Stelle ghost -> S_pert unitary for matter+gauge only) and v274 (over-determination conditional on the Lambda-branch). The perturbative gravity ghost IS the QG.AMB.01 frontier.",
      "dependencies": "QFT4D.SPERT.01; QFT4D.SPERT.02; QFT4D.SPERT.03; FLAV.PMNS.03; FLAV.NUSCALE.01; SCALE.OVERDET.01; QGAMB.ROADMAP.01",
      "scripts": [
        "redteam/rt_F_qft4d.py"
      ],
      "script": "redteam/rt_F_qft4d.py",
      "location": "tfpt_5_redteam",
      "externalData": "Stelle 1977 (higher-derivative gravity ghost)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 369
    },
    {
      "id": "QGEO.SYM.03",
      "claim": "FLAT PILLOWCASE CLOSES THE MODULAR COMMUTATOR TO ALL ORDERS (2026-06-18): collapses the bedrock QGEO.SYM.01 to ONE geometric postulate (does NOT prove the premise). For a quasi-free seam state QGEO.SYM.01 = (omega o rho = omega) <=> [rho,C]=0 <=> [rho,H]=0 (C=(1+e^H)^-1, v258). [E] ISOMETRY=>COMMUTATOR: if H=f(Delta_flat) (the DtN of a flat metric, no curvature corrections) and the order-4 deck rho (z->iz) is a flat-tau=i isometry, then [rho,Delta]=0 => [rho,H]=[rho,f(Delta)]=0 EXACTLY to all orders -- verified numerically on the discretised square torus (rho^4=I, [rho,Delta]=0, [rho,sqrt(-Delta)]=0; a generic non-isometry permutation fails). [E] this UPGRADES the leading-order results v198 (principal symbol |k| commutes) and v201 (subprincipal block-diagonal) to the FULL operator H. [C] tau=i is forced by the order-4 requirement (cross-ratio 2 <=> j=1728, the unique order-4 CM by Z[i]; hexagonal tau=omega has j=0, order-6, excluded; v214/v267). [O] QGEO.SYM.01 therefore collapses to the single geometric statement 'the raw RP seam state is the flat tau=i pillowcase quasi-free state (H=sqrt(-Delta_flat))' -- everything downstream [E]; this one statement is the bedrock, needs a human constructive-QFT proof or stays an honest axiom. Sharpest non-circular form.",
      "status": "Reduction [E] (flat => commutator all orders) + [C] (tau=i forced) + [O] (one geometric postulate)",
      "canonicalStatus": "[E] the isometry=>commutator closure (numerically verified, all orders) + the all-orders upgrade of v198/v201; [C] tau=i forced by order-4; [O] the one geometric postulate (raw seam = flat tau=i pillowcase) stays open. Narrows QGEO.SYM.01 to a single flat-geometry statement; does NOT close it.",
      "dependencies": "QGEO.SYM.01; QGEO.SYM.02; QGEO.MODULAR.01; QGEO.SUBPRIN.01; QGEO.PILLOW.01; PS.DIRAC.03",
      "scripts": [
        "v276_qgeo_flat_closes_commutator.py"
      ],
      "script": "v276_qgeo_flat_closes_commutator.py",
      "location": "tfpt_research_contracts",
      "externalData": "Stone/spectral theorem (isometry commutes with functions of the Laplacian)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 370
    },
    {
      "id": "QGAMB.TIERB.01",
      "claim": "SEAM-CALDERON -> (E8)_1 MATCHING CERTIFICATE (2026-06-18): the concrete Tier-B step of QG.AMB.01 (v275) -- computes the full (E8)_1 matching target and pins the residual to ONE bit (does NOT close it; the rt_A 'reduced, not closed' verdict stands). [E] c=8 (Sugawara, c((E8)_1)=248/31=8=g_car+N_fam), matching the seam (v77), but c=8 alone does not pin the net. [E] THE DISCRIMINATOR IS HOLOMORPHY: det Cartan(E8)=1 (one primary/anyon, holomorphic) vs the c=8 counterexample det Cartan(D8=SO(16))=4 (four primaries 1,v,s,c, non-holomorphic) -- same c, different det, so the single discriminator is |det K|=1. [E] THE (E8)_1 CHARACTER: the unique holomorphic c=8 character E4/eta^8=j^{1/3}=q^{-1/3}(1+248q+4124q^2+...) has a SINGLE primary and exactly 248=dim E8 weight-1 currents (the affine E8 current algebra; 240 roots from Theta_E8=E4). [E] EQUIVALENCE CHAIN (v234/v235): holomorphic <=> det Cartan 1 <=> E8 the unique even unimodular rank-8 lattice (v83) <=> seam link a homology 3-sphere (|Gamma^ab|=1) <=> |det K|=1 (Chern-Simons) -- all Tier-B faces are ONE condition. [C] SEAM-SIDE MATCH: the seam is pinned to the flat tau=i pillowcase (v276) and one Kummer/K3 carries seam+carrier-16+E8 (v260), so the boundary chiral data matches (E8)_1 on every computable invariant. [O] RESIDUAL = the single holomorphy bit (det K=1 / mu-index 1 / single sector); reduced to one bit, NOT closed.",
      "status": "Reduction [E] (full (E8)_1 match + discriminator) + [C] (seam-side) + [O] (one holomorphy bit)",
      "canonicalStatus": "[E] the full (E8)_1 matching target (c=8, det 1, 248 currents, single primary) + the explicit D8 counterexample isolating holomorphy as THE discriminator; [C] the seam-side match via v276/v260; [O] the single holomorphy bit stays open. Reduces QG.AMB.01 Tier-B to one checkable condition; does NOT close it.",
      "dependencies": "QGAMB.ROADMAP.01; GATE.METRIC.03; REDTEAM.A.01; QGEO.SYM.03; ARCH.K3.01; GATE.HOLO.02",
      "scripts": [
        "v277_seam_calderon_e8_match.py"
      ],
      "script": "v277_seam_calderon_e8_match.py",
      "location": "tfpt_4_frontier",
      "externalData": "E8/D8 lattice theta functions; Frenkel-Kac-Segal lattice VOA",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 371
    },
    {
      "id": "QFT4D.SPERT.04",
      "claim": "S_pert -> S_phys LSZ BRIDGE + ONE-LOOP UNITARITY (2026-06-18): connects the EG perturbative S-matrix (v269/v271/v273) to the physical asymptotic S-matrix (S_phys = LSZ on the OS-reconstructed Wightman functions, v240); the three-layer S-matrix is now S_top (v243) | S_pert (v269) --LSZ--> S_phys (v240). [E] LSZ BRIDGE: S_phys matrix elements = (amputate external legs)(on-shell residue sqrt(Z)) of the connected time-ordered correlators, and the EG time-ordered products of S_pert ARE those correlators (the OS Wick-rotation of the seam Wightman functions), so the perturbative amplitudes feed S_phys. [C] MASS GAP: Delta=6log(3/2)>0 (v64/v240) gives isolated one-particle states, so Haag-Ruelle scattering theory constructs the in/out asymptotic states and LSZ is valid. [E] ONE-LOOP UNITARITY (optical theorem): the s-channel one-loop bubble B(s)=int_0^1 dx ln(m^2-x(1-x)s-i eps) has discontinuity Im B(s)/pi = x_+ - x_- = sqrt(1-4m^2/s) for s>4m^2 EXACTLY -- the two-body phase-space factor -- so 2 Im M = sum_int dPi |M|^2 (cutting rule) holds at one loop: S_pert is unitary order by order for the matter+gauge sector. [E] THRESHOLD: Im B=0 for s<4m^2, turning on exactly at the physical two-particle threshold s=4m^2 (no spurious thresholds). [O] GRAVITY CAVEAT (rt_F): the R^2/Weyl^2 gravity sector has a negative-residue massive spin-2 mode (the Stelle ghost), so the gravity perturbative S-matrix is NOT unitary -- S_phys for gravity is the nonperturbative QG.AMB.01 frontier. [C] FIELD-STRENGTH Z: the EG scheme gives finite Z so the LSZ reduction is well-defined.",
      "status": "Bridge [E] (LSZ structure + one-loop optical theorem) + [C] (gap/Z) + [O] (gravity ghost)",
      "canonicalStatus": "[E] the LSZ bridge structure + the one-loop optical theorem (exact: cut = two-body phase space) + the threshold analyticity; [C] the gap->asymptotic-states link + finite Z; [O] the gravity ghost (-> QG.AMB.01). Connects S_pert to S_phys and verifies one-loop unitarity for matter+gauge.",
      "dependencies": "QFT4D.SPERT.01; QFT4D.SPERT.02; DYN.GNS.01; DHR.SMATRIX.01; REDTEAM.F.01",
      "scripts": [
        "v278_lsz_bridge_unitarity.py"
      ],
      "script": "v278_lsz_bridge_unitarity.py",
      "location": "tfpt_research_contracts",
      "externalData": "Lehmann-Symanzik-Zimmermann reduction; Haag-Ruelle scattering theory",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 372
    },
    {
      "id": "QGEO.OBLIG.01",
      "claim": "QGEO.SYM.01 AS ONE PRECISE CONSTRUCTIVE-QFT LEMMA + PROOF-TREE COMPLETENESS (2026-06-18): the formal write-up of the one human math obligation (does NOT close it). LEMMA (constructive-QFT form): let Sigma = P^1 minus mu_4 carry the free c=8 RP/OS quasi-free state omega_Sigma with DtN Lambda_Sigma; PREMISE (the one open input, QGEO.SYM.01): the raw seam carries the FLAT tau=i pillowcase metric g_flat (the Troyanov-unique constant-curvature representative, cone angles pi, of the conformal class fixed by the order-4 deck), so Lambda_Sigma = sqrt(-Delta_{g_flat}); CONCLUSION: omega_Sigma o rho = omega_Sigma (the carrier clock is a symmetry of the seam STATE) -> mark-locality -> metric-sector inclusion -> E8. [E] PROOF-TREE COMPLETENESS: the reduction DAG from 'free RP seam' (v155/v175 -> marks v195/v216 -> conformal class tau=i v214/v267 -> Troyanov flat -> commutator all orders v276/Lean -> state-invariance v198/v201/SeamDeckClosure -> mark-locality v210/v264 -> E8 v154/v277) has EXACTLY ONE un-discharged leaf: 'the raw seam state IS the flat g_flat state' (Lambda_Sigma = sqrt(-Delta_flat)). [E] the GEOMETRIC SIDE is fixed (flat DtN diagonal in Fourier modes, rho commutes with any spectral function to all orders, Lean FORM.QGEO.04), so the obligation is STATE-IDENTIFICATION not geometric ambiguity. [C] two proof routes: (i) RP-state uniqueness on the flat tau=i orbifold, (ii) Troyanov + OS reconstruction. [O] one constructive-QFT statement closes QGEO.SYM.01.",
      "status": "Obligation lemma [E] (proof-tree complete) + [C] (routes) + [O] (one residual)",
      "canonicalStatus": "[E] the precise lemma + proof-tree completeness (exactly one open leaf) + the fixed geometric side; [C] the two proof routes; [F] the all-orders step in Lean; [O] the one constructive-QFT residual. The formal obligation write-up; does NOT close QGEO.SYM.01.",
      "dependencies": "QGEO.SYM.01; QGEO.SYM.02; QGEO.SYM.03; FORM.QGEO.04; QGEO.MARKLOCAL.01; QGEO.PILLOW.01",
      "scripts": [
        "v279_qgeo_obligation_lemma.py"
      ],
      "script": "v279_qgeo_obligation_lemma.py",
      "location": "tfpt_research_contracts",
      "externalData": "Troyanov (prescribed-curvature cone metrics); Osterwalder-Schrader reconstruction",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 373
    },
    {
      "id": "FORM.QGEO.04",
      "claim": "FLAT-METRIC ALL-ORDERS CLOSURE FORMALISED IN LEAN (2026-06-18): TfptCarrier/SeamDeckClosure.lean (theorems diagonal_commutesClock, flat_all_orders_clock) machine-proves the v276 all-orders step -- any SPECTRAL FUNCTION g of the Fourier-diagonal flat Laplacian (specFun g d) commutes with the carrier clock rho=diag(i^n) (CommutesClock), i.e. given the flat tau=i metric the DtN modular Hamiltonian H=g(Delta_flat) is clock-invariant to ALL orders, not merely the principal (v198) and sub-principal (v201) orders. Upgrades the conditional block-diagonality (markLocal_blockDiagonal / SeamDeckPremise.clock_invariant) to the full operator. Only the standard axioms {propext, Classical.choice, Quot.sound}, no sorry; full lake build green (3356 jobs), listed in AxiomCheck.lean. The premise (the seam IS flat) stays the [O] QGEO.SYM.01 input. QA NOTE (2026-08-08, round 36, v878_ledger_integrity.py): claim_id RENAMED FORM.QGEO.03 -> FORM.QGEO.04 -- the id FORM.QGEO.03 was double-booked on 2026-06-18 (it had been assigned to the CohomologyGrading formalisation on 2026-06-15): a primary-key violation found by the round-36 ledger audit; every flat-closure reference (papers, scripts, registry, website) updated to FORM.QGEO.04; the CohomologyGrading row keeps FORM.QGEO.03. The claim content is UNCHANGED.",
      "status": "Formal [F] (Lean theorem, all-orders closure)",
      "canonicalStatus": "Formal [F]: the v276 flat-metric all-orders commutator closure is a Lean theorem (SeamDeckClosure.flat_all_orders_clock), standard axioms only, no sorry, full build green.",
      "dependencies": "FORM.QGEO.01; FORM.QGEO.02; QGEO.SYM.03",
      "scripts": [
        "lean4-carrier-rigidity"
      ],
      "script": "lean4-carrier-rigidity",
      "location": "tfpt_research_contracts",
      "externalData": "Lean 4 (leanprover/lean4 v4.29.1) + Mathlib",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 374
    },
    {
      "id": "QGEO.STEKLOV.01",
      "claim": "FLAT tau=i PILLOWCASE DtN EXPERIMENT (2026-06-18): a direct numerical experiment on the actual seam orbifold, realising the QGEO chain on the geometric side (does NOT prove the premise). [E] the pillowcase = the Z2-even sector of the flat torus T^2=C/(Z+iZ) modulo z->-z; the 4 cone points are the 2-torsion fixed points (deficit pi each, sum 4pi=2pi*2 Gauss-Bonnet). [E] the order-4 deck rho (z->iz) descends to the pillowcase ([rho,Z2]=[rho,P_even]=0) and permutes the 4 cone points as 2 fixed + 1 swapped pair (the order-4 mark structure). [E] the geometric chain is numerically EXACT: [rho,Delta]=0 => [rho,H]=0 (H=sqrt(-Delta)) => [rho,C]=0 (C=(1+e^H)^-1) => omega o rho = omega -- the v276/v279 all-orders closure realised on the real orbifold. [C] the Steklov/DtN is positive self-adjoint, fixed by the flat metric alone (numerical evidence for QGEO.OBLIG.01 route (i), RP-state uniqueness). [O] confirms the conclusion GIVEN the flat geometry; the premise (the raw seam state IS this flat pillowcase state) stays the one open QGEO.SYM.01 input.",
      "status": "Numerical experiment [E] (geometric chain) + [C] (RP-state) + [O] (premise)",
      "canonicalStatus": "[E] the pillowcase orbifold + order-4 orbit + the exact geometric chain omega o rho = omega numerically realised; [C] the RP-state is metric-determined (route-(i) evidence); [O] the premise stays open. A self-investigable numerical experiment on the real flat tau=i pillowcase.",
      "dependencies": "QGEO.SYM.01; QGEO.SYM.03; QGEO.OBLIG.01; QGEO.PILLOW.01; QGEO.MARKS.03",
      "scripts": [
        "v280_pillowcase_steklov.py"
      ],
      "script": "v280_pillowcase_steklov.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 375
    },
    {
      "id": "QGAMB.MODULAR.01",
      "claim": "MODULAR-DATA / ANYON-CONDENSATION LAYER OF THE Tier-B HOLOMORPHY BIT (2026-06-18): complements v277 (det+character) with the anyon/modular-data view. [E] #anyons = |det Gram|: E8 -> 1 (holomorphic, single sector), D8=SO(16)_1 -> 4 (anyons 1,v,s,c), carrier D5(+)A3 -> det(D5)*det(A3)=4*4=16. [E] the anyon-condensation tower D5(+)A3 (det 16) -> D8 (det 4) -> E8 (det 1), each step condensing a Lagrangian Z2 boson (|det|/4=2^2 per step), the Kitaev E8 quantum-Hall endpoint (v92/v235). [E] Gauss-Milgram for the D8 anyons (topological spins 0,1/2,1,1): (1/sqrt|A|) sum_a e^{2pi i h_a} = e^{2pi i c/8} recovers c=8 mod 8 (the chiral central charge = the modular anomaly). [E] E8 is the unique holomorphic c=8 net (1 anyon <=> det 1 <=> even unimodular rank-8 = E8); SO(16)_1 has the SAME c=8 but 4 anyons, so holomorphy (det 1) is THE discriminator (matches v277). [O] the open Tier-B bit: does the seam-Calderon net condense to the det-1 holomorphic endpoint (|det K|=1)? Reduced to one bit, not closed.",
      "status": "Reduction [E] (modular data + condensation tower + Gauss-Milgram) + [O] (one bit)",
      "canonicalStatus": "[E] #anyons=|det Gram| + the condensation tower 16->4->1 + Gauss-Milgram c=8 + E8 the unique holomorphic point; [O] the single holomorphy bit. The modular-data view of QG.AMB.01 Tier-B; complements v277. Exact core mirrored in Wolfram.",
      "dependencies": "QGAMB.TIERB.01; GATE.HOLO.02; QGAMB.ROADMAP.01; E8.GLU.01",
      "scripts": [
        "v281_holomorphy_modular_data.py"
      ],
      "script": "v281_holomorphy_modular_data.py",
      "location": "tfpt_4_frontier",
      "externalData": "Frenkel-Kac-Segal lattice VOA; Gauss-Milgram / anyon condensation",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 376
    },
    {
      "id": "QGAMB.UNIFY.01",
      "claim": "THE E8-AT-tau=i UNIFICATION: TWO OBLIGATIONS -> ONE OBJECT (2026-06-18): the candidate simplification. The (E8)_1 character as a modular function of the seam torus modulus tau is chi_E8(tau)=Theta_E8/eta^8=E4/eta^8=j(tau)^{1/3} (since j=E4^3/eta^24=(E4/eta^8)^3=chi_E8^3). [E] tau=i is the order-4 CM point (cross-ratio 2 <=> j=1728 <=> tau=i, unique order-4 CM by Z[i], v214/v267) -- the flat tau=i pillowcase geometry of QGEO.SYM.01. [E] chi_E8 = j^{1/3} (verified chi^3 = q*j as a q-series), so chi_E8(i)=1728^{1/3}=12 -- the (E8)_1 partition function at the SAME modulus tau=i, the holomorphy of QG.AMB.01. [E] ONE OBJECT, TWO FACES: tau=i is simultaneously the order-4 conformal CM point (QGEO.SYM.01, flat geometry) and the modulus at which the holomorphic (E8)_1 character is the seam partition function (QG.AMB.01, holomorphy). [C] OBLIGATION COUNT 2->1: prove the single statement 'the raw seam state is the (E8)_1 holomorphic CFT at tau=i' and BOTH QGEO.SYM.01 (flat tau=i, order-4 CM) AND QG.AMB.01 (E8 holomorphic, det K=1) follow -- the flat geometry IS the CM modulus, the holomorphy IS the E8 algebra. [O] the single unified premise stays open (it IS the seam-identification) but is now ONE object, not two separate premises -- a genuine reduction of the open count, not a closure.",
      "status": "Unification [E] (chi_E8=j^{1/3}, tau=i shared) + [C] (count 2->1) + [O] (one premise)",
      "canonicalStatus": "[E] tau=i the order-4 CM point + chi_E8=j^{1/3} (chi_E8(i)=12) + the one-object identification; [C] the obligation count drops 2->1; [O] the single unified premise stays open. The candidate simplification of the two math obligations. Exact core mirrored in Wolfram.",
      "dependencies": "QGEO.SYM.03; QGEO.OBLIG.01; QGAMB.TIERB.01; QGAMB.MODULAR.01; QGEO.PILLOW.01",
      "scripts": [
        "v282_e8_tau_i_unification.py"
      ],
      "script": "v282_e8_tau_i_unification.py",
      "location": "tfpt_research_contracts",
      "externalData": "E8 theta / Klein j-function CM values (j(i)=1728)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 377
    },
    {
      "id": "PRED.LIVE.01",
      "claim": "LIVE KILL-TEST SCORECARD OF THE RECENT ROUND'S COMPUTABLE PREDICTIONS (2026-06-18): a data-confrontation consolidation (confronts, does not add physics), each row tagged with the decisive near-term experiment. [N] PMNS angles: sin^2 th12=1/3-phi0/2=0.30675 vs NuFIT 0.303(12) [+0.31 sigma]; sin^2 th13=phi0 e^{-5/6}=0.02311 vs 0.02225(59) [+1.45 sigma]; sin^2 th23=0.5 maximal (octant-ambiguous) -- decisive JUNO/Daya-Bay/DUNE. [C] leptonic CP: J_PMNS=-0.0297 (J_max 0.0342 vs NuFIT 0.0332 ~3%), delta=240 vs NuFIT ~232(30) -- decisive DUNE/Hyper-K. [X] Sigma m_nu NO floor=0.0586 eV vs Planck+BAO<0.12, DESI~<0.072 (consistent at the edge; Sigma<0.0586 excludes the NO m1~0 spectrum) -- decisive DESI/CMB-S4. [X] proton decay tau_p(p->e+pi0)=1.55e35 yr for the E8-allowed +(15,1,1) content vs Super-K>2.4e34 (safe); minimal 16-Higgs 4.4e33 EXCLUDED -- decisive Hyper-Kamiokande (reach ~1.4e35). [N] closed readouts alpha^-1=137.0359992 (1.9 sigma known offset), Omega_b=(1-1/4pi)phi0=0.04894, birefringence beta=phi0/(4pi)=0.2424 deg -- decisive CMB birefringence. [N] all confronted rows consistent < 2 sigma; 2 sharp near-term kill tests (Sigma m_nu floor, proton decay).",
      "status": "Data confrontation [N]/[C] + [X] (two near-term kill tests)",
      "canonicalStatus": "[N]/[C] data confrontation (pulls vs current bounds) + [X] the two sharp near-term kill tests. A consolidation/scorecard, not new physics.",
      "dependencies": "FLAV.TH13.01; FLAV.PMNS.03; FLAV.NUSCALE.01; PS.PROTON.02; EM.FP.01",
      "scripts": [
        "v283_live_killtest_scorecard.py"
      ],
      "script": "v283_live_killtest_scorecard.py",
      "location": "tfpt_4_frontier",
      "externalData": "NuFIT 5.2/6.0; PDG 2024; Planck 2018 + DESI 2024; Super-K",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 378
    },
    {
      "id": "QGEO.ROUTEI.01",
      "claim": "ROUTE (i) (RP-STATE UNIQUENESS) LEMMA DECOMPOSITION (2026-06-18): turns Route (i) from one hard target into a 6-lemma chain with 5 lemmas discharged [E]/[F] and exactly one open lemma isolated (does NOT prove the open premise). L1 [E] RP data define the raw DtN intrinsically (v194 QGEO.DTN.01, OS on the quasi-free state). L2 [E] the four branch marks force the pillowcase conformal class (v195 QGEO.MARKS.02 Lefschetz/character + v216 QGEO.MARKS.03 n=2chi=4 + v214 QGEO.PILLOW.01). L3 [E] the flat orbifold Steklov/DtN operator = sqrt(-Delta_flat): the DtN principal symbol is |xi|=sqrt(-Delta_bdry) (Lee-Uhlmann) with no curvature correction for a flat metric (flat-strip DtN(k)/k->1; v280). L4 [E] the quasi-free covariance is determined by the DtN, C=(1+e^H)^-1 with H=log-DtN (v258 PS.DIRAC.03). L5 [E]/[F] mu4-invariance follows: [rho,Delta]=0 => [rho,H]=0 => [rho,C]=0 => omega o rho = omega (v276 QGEO.SYM.03 + Lean FORM.QGEO.04 + v280). TROYANOV REDUCTION [C]: 4 cone points of angle pi => chi_orb = 2-4(1-1/2) = 0 => the unique constant-curvature representative is flat = the tau=i square (j=1728), so the residual reduces to 'the raw seam is the constant-curvature geometric state'. THE ONE OPEN LEMMA [O]: the raw RP seam state is the constant-curvature/geometric state (Lambda = sqrt(-Delta_g) for constant-curvature g) -- the state-identification theory-selection step; proving it closes Route (i).",
      "status": "Lemma decomposition [E] (5/6 discharged) + [C] (Troyanov) + [O] (one open lemma)",
      "canonicalStatus": "[E] 5/6 Route-(i) lemmas discharged + the L3 flat-DtN verification; [C] the Troyanov reduction (constant curvature -> flat tau=i); [O] the one constant-curvature/geometric-state lemma. Decomposes Route (i) and isolates its single open lemma; does NOT close it.",
      "dependencies": "QGEO.SYM.01; QGEO.OBLIG.01; QGEO.DTN.01; QGEO.MARKS.03; PS.DIRAC.03; QGEO.SYM.03; QGEO.STEKLOV.01",
      "scripts": [
        "v284_route_i_rp_uniqueness.py"
      ],
      "script": "v284_route_i_rp_uniqueness.py",
      "location": "tfpt_research_contracts",
      "externalData": "Troyanov (constant-curvature cone metrics); Lee-Uhlmann (DtN symbol)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 379
    },
    {
      "id": "QGAMB.ROUTEII.01",
      "claim": "ROUTE (ii) (SEAM-NET CONDENSATION) LEMMA DECOMPOSITION + ROUTE-COINCIDENCE (2026-06-18): a 4-lemma chain with 3 discharged and the open lemma shown to COINCIDE with Route (i)'s (the v282 unification at the lemma level; does NOT prove the open premise). M1 [E] no nontrivial abelian sector <=> |det K|=1 <=> holomorphic (mu-index 1) (v234 GATE.HOLO.01: #(1-dim irreps)=|Gamma^ab|=|H1|=1 <=> E8; v235 GATE.HOLO.02 Chern-Simons det K=1). M2 [E] holomorphic c=8 <=> (E8)_1 the unique even unimodular rank-8 lattice; the c=8 counterexample SO(16)_1 has |det|=4 (v277 QGAMB.TIERB.01 + v281 QGAMB.MODULAR.01 #anyons=|det Gram|). M3 [E] the condensation tower D5(+)A3 (det 16) -> D8 (4) -> E8 (1) is the unique mu4-Lagrangian glue (v92 GATE.METRIC.07 + v281 + v235). THE DISCRIMINATOR [E]: det 1 (E8, holomorphic) vs det 4 (SO(16)) at the same c=8. ROUTE COINCIDENCE [E]: Route (i)'s L_open ('raw seam = constant-curvature flat tau=i state', v284) and Route (ii)'s M_open ('the free RP seam condenses the mu4-Lagrangian glue' = QGEO.SYM.01) are the SAME lemma -- both 'the raw seam IS the (E8)_1-at-tau=i object' (v282); proving EITHER route closes BOTH. THE UNIFIED OPEN LEMMA [O]: construct a canonical equivalence between the raw seam KMS/DtN state and the holomorphic (E8)_1 net at tau=i -- the state-identification step.",
      "status": "Lemma decomposition [E] (3/4 discharged + route coincidence) + [O] (unified lemma)",
      "canonicalStatus": "[E] 3/4 Route-(ii) lemmas discharged + the discriminator + the route-(i)/(ii) coincidence; [O] the unified open lemma (canonical seam<->(E8)_1 equivalence). Decomposes Route (ii), isolates its single open lemma, and shows it coincides with Route (i)'s -- confirming v282 at the lemma level. Does NOT close it.",
      "dependencies": "QGEO.SYM.01; GATE.HOLO.01; GATE.HOLO.02; QGAMB.TIERB.01; QGAMB.MODULAR.01; QGAMB.UNIFY.01; QGEO.ROUTEI.01",
      "scripts": [
        "v285_route_ii_seam_condensation.py"
      ],
      "script": "v285_route_ii_seam_condensation.py",
      "location": "tfpt_research_contracts",
      "externalData": "Frenkel-Kac-Segal lattice VOA; anyon condensation",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 380
    },
    {
      "id": "SEAM.EQUIV.01",
      "claim": "THE SEAM EQUIVALENCE CONTRACT (2026-06-18): concentrates the whole open structure on ONE named arrow (does NOT prove it). THE NAMED THEOREM: the raw reflection-positive seam state (gap, chirality, carrier clock) reconstructs canonically the SAME state as the holomorphic (E8)_1 boundary net at the tau=i pillowcase, respecting the DtN, KMS modular flow, mu4 deck, Q-system extension, det K=1 and the (E8)_1 character (short: the raw seam is (E8)_1 at tau=i). [E] four objects (RawRPSeam, FlatTauIPillowcaseDtN, HolomorphicE8Net, SREKitaevE8Phase). [E] six arrows: 2 closed (Route i: FlatDtN->omega o rho=omega, v276/v280/v284; Route ii: SRE->HolomorphicE8Net, v235/v277/v281), 1 conditional (FlatDtN<->HolomorphicE8Net at tau=i, v282), 2 open premises (RawRPSeam->FlatDtN and RawRPSeam->SRE, the SAME lemma per v284/v285), 1 Gral (RawRPSeam->HolomorphicE8Net = SEAM.EQUIV.01). [E] IMPORT FIREWALL (anti-circularity): no Route-i module (v276/v280/v284) imports any Route-ii module (v277/v281/v285) or vice versa, verified by scanning the sources -- the routes converge ONLY through this contract, killing the 'E8 smuggled into the geometry and pulled back out' attack. [E] exact-label taxonomy {Formal Exact (Lean), Numerical Exact (reproducible experiment), Conditional Exact (standard theorem under a stated premise), Open Selection (theory-selection premise)}. [O] the one open arrow: free Gaussian RP bulk => holomorphic single-sector boundary net. Writes seam_equivalence.json. NOT closed by this: v_geo (metrology primitive), F_transfer (downstream), QG.AMB.01 (nonperturbative ambient measure). ROUTE SPLIT (2026-07-22; ID restructuring after the external review, NO status change in substance): the target statement now carries two named routes -- SEAM.EQUIV.MMST.01 (the conditional scaling-limit route: closed modulo cited existence theorems) and SEAM.EQUIV.TWISTOR.01 (the open Costello-Li construction). PARENT LOGIC (the claim text of this row): closed if MMST.01 OR TWISTOR.01 closes; currently: MMST conditional-closed modulo cited theorems, TWISTOR open => parent stays [O] as an unconditional claim.  UPDATED 2026-08-27 (v973): the route narrowing -- the MMST route's cited-theorem coverage is carved into COVERED (bilinear half) vs NOT-COVERED (the named N1-N4 list, headed by the mu4/GSO net extension), the twistor route's declared measure is unique on the computable sector up to exactly ONE named flat direction, and the two routes provably share the executable OS skeleton 5/5 with all divergences control-typed (NEW row SEAM.EQUIV.SKELETON.01 [E]) -- the open structure of the parent is now concentrated on the SAME named extension question from both sides (N1/N4 on the MMST side, the global quantisation + OS bridge on the twistor side), with the executable levels verified to agree. Parent logic unchanged: closed if MMST.01 OR TWISTOR.01 closes; both stay [O]; parent stays [O].",
      "status": "Contract [E] (typing + firewall + taxonomy) + [O] (the named theorem)",
      "canonicalStatus": "[E] the four-object/six-arrow typing + the import firewall (routes non-circular) + the exact-label taxonomy + the single-open-arrow certificate; [O] the named Seam Equivalence Theorem stays open. A contract concentrating the open structure on one arrow; does NOT prove it. Route split 2026-07-22: closed if SEAM.EQUIV.MMST.01 OR SEAM.EQUIV.TWISTOR.01 closes; currently MMST conditional-closed modulo cited theorems, TWISTOR open => parent stays [O] as an unconditional claim.",
      "dependencies": "QGEO.SYM.01; QGEO.OBLIG.01; QGAMB.UNIFY.01; QGEO.ROUTEI.01; QGAMB.ROUTEII.01; SEAM.EQUIV.MMST.01; SEAM.EQUIV.TWISTOR.01",
      "scripts": [
        "v286_seam_equivalence_contract.py"
      ],
      "script": "v286_seam_equivalence_contract.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 381
    },
    {
      "id": "SEAM.EQUIV.A01",
      "claim": "ROUTE A (AQFT) DEPENDENCY CHECKER (2026-06-18): for the open Gral arrow 'free Gaussian RP bulk => holomorphic single-sector boundary net', types the 5-lemma AQFT chain and isolates the ONE missing standard theorem (does NOT prove the arrow). L1 [E] RawRPSeam --OS--> CAR/quasi-free boundary system (v155 quasi-free boundary, v160 Gaussianity from the Pfaffian, v240 GNS/OS, v175 net existence + full-cone RP). L2 [E] gap + chirality => gapped chiral bulk (gap Delta=6log(3/2)>0 v64; c=8 chiral net v156/v157). L3 [O] invertible/SRE gapped Gaussian bulk => single-sector boundary (mu-index 1) -- THE ONE MISSING STANDARD THEOREM: an invertible (SRE) gapped bulk has no bulk topological order, hence a single-sector boundary net; the AQFT form of det K=1, to be imported/proved externally. L4 [E] mu-index 1 + c=8 => holomorphic (v234 holomorphy<=>1 sector). L5 [E] holomorphic c=8 => (E8)_1 (v83/v277/v281, unique even unimodular rank-8; SO(16) the det-4 non-holomorphic counterexample). VERDICT: 4/5 lemmas discharged, so Route A reduces SEAM.EQUIV.01 to ONE cited standard theorem (invertibility => single-sector), not a diffuse programme.",
      "status": "Dependency checker [E] (4/5 discharged) + [O] (the one external theorem)",
      "canonicalStatus": "[E] the AQFT chain typing + the four discharged steps (L1/L2/L4/L5) + the single-missing-theorem isolation (L3); [O] the one external theorem 'invertible Gaussian bulk => single-sector boundary'. A dependency checker, not a proof.",
      "dependencies": "SEAM.EQUIV.01; QFT.NETRP.01; GATE.HOLO.01; QGAMB.TIERB.01; QGAMB.ROUTEII.01",
      "scripts": [
        "v287_free_rp_bulk_to_holomorphic_boundary.py"
      ],
      "script": "v287_free_rp_bulk_to_holomorphic_boundary.py",
      "location": "tfpt_research_contracts",
      "externalData": "Doplicher-Haag-Roberts / invertible-phase (SRE) boundary theory",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 382
    },
    {
      "id": "SEAM.EQUIV.B01",
      "claim": "ROUTE B (DtN) FULL-L^2 SUB-PRINCIPAL Z4 BLOCK-DIAGONALITY (2026-06-18): proves (numerically, on the full Toeplitz operator) the advisor's 'probably provable' step, lifting v201/v284 from the finite H^1 block to the full L^2 operator. Setup: the seam DtN Lambda = |D_theta| + M_f with <n|M_f|n'> = f_{n-n'} (boundary curvature f), rho = diag(i^n). [E] CHARACTER ORTHOGONALITY: a mark-sourced curvature f(theta)=sum_{j<4} g(theta - j pi/2) has Fourier coefficient f_m = g_m sum_{j<4}((-i)^m)^j = 4 g_m [4|m], so support ONLY on modes m==0 (mod 4) (verified nonzero modes {0,+-4,+-8,+-12}). [E] FULL-L^2 BLOCK-DIAGONALITY: on the full mode range -50..50 the Toeplitz M_f connects only equal clock-characters, so Lambda commutes with rho, ||[rho,Lambda]||~2e-15 -- the all-orders statement on L^2, lifting the finite block (v201/v284). [E] NEGATIVE CONTROL: a generic (non-mark-local) curvature gives ||[rho,Lambda]||=4.44>0. [E] LEAN CONSISTENCY: the operator form of SeamDeckClosure.geom_sum_fourth_root + markLocal_blockDiagonal. [O] THE SHARPER RESIDUAL: why is the raw seam sub-principal term mark-local (a sum of mu4-orbit sources)? -- the curvature-locality half of SEAM.EQUIV.01.",
      "status": "Reduction [E] (full-L^2 lift, lifts v201/v284) + [O] (mark-locality residual)",
      "canonicalStatus": "[E] character orthogonality + the full-L^2 Z4 block-diagonality (lifts v201/v284 to the full operator) + the negative control + Lean consistency; [O] the one sharper residual (raw sub-principal mark-locality). Real progress on Route B; does NOT close the premise.",
      "dependencies": "SEAM.EQUIV.01; QGEO.SUBPRIN.01; QGEO.ROUTEI.01; FORM.QGEO.04; QGEO.MODULAR.01",
      "scripts": [
        "v288_full_l2_subprincipal_z4.py"
      ],
      "script": "v288_full_l2_subprincipal_z4.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 383
    },
    {
      "id": "MARKLOCAL.RAW.01",
      "claim": "RAW MARK-LOCALITY DECOMPOSITION (2026-06-18): decompose the ONE remaining Route-B residual after v288 ('why is the raw seam sub-principal term mark-local?') into an explicit 5-lemma chain, discharge the four standard/established lemmas and isolate the single open analytic lemma (Flat-Away). The best closing lever for Route B; does NOT close mark-locality. CHAIN (raw RP seam => mark-local sub-principal term): L1 [C] CONIC PARAMETRIX -- the Steklov/DtN of a conic surface splits as Lambda = |D_theta| (principal) + M_f (sub-principal = multiplication by boundary curvature f) + delta-sources at the cone points (b-calculus/conic parametrix, Melrose; imported). L2 [O] FLAT-AWAY -- THE ONE OPEN ANALYTIC LEMMA: reflection positivity + the mass gap + the four branch marks force the seam curvature to VANISH away from the four marks (the flat-pillowcase realisation, the curvature-locality heart of SEAM.EQUIV.01). L3 [E] ORBIT-SOURCE -- the four marks form a single mu4 orbit {0,pi/2,pi,3pi/2} (v195 Lefschetz/character marks + order-4 clock; v216 Gauss-Bonnet 4 cone points deficit pi). L4 [E] FOURIER-SUPPORT -- a curvature sourced on the mu4 orbit has discrete Fourier support ONLY on modes m==0 mod 4 (v264/v288; verified by FFT, with a non-orbit negative control showing the ORBIT structure, not just '4 marks', gives mark-locality). L5 [E] FULL-OPERATOR -- hence M_f is Z4 block-diagonal and [rho,Lambda]=0 on the full L^2 (v288), so omega o rho = omega (Tomita-Takesaki, v198). VERDICT: 4/5 discharged (L1 conditional-by-literature, L3/L4/L5 exact); Route B reduces to exactly L2 (Flat-Away) -- if L2 falls, Route B closes and SEAM.EQUIV.01 needs only the holomorphy half (Route A).",
      "status": "Decomposition [E] (chain + L3/L4/L5) + [C] (L1 literature) + [O] (L2 Flat-Away)",
      "canonicalStatus": "[E] the chain typing + L3 (orbit) + L4 (Fourier support) + L5 (full operator); [C] L1 (conic parametrix, literature); [O] L2 (Flat-Away), the single open analytic lemma. A finer decomposition of the v288 residual, NOT a closure.",
      "dependencies": "SEAM.EQUIV.B01; SEAM.EQUIV.01; QGEO.ENERGY.03; QGEO.SUBPRIN.01; FORM.QGEO.04",
      "scripts": [
        "v289_marklocal_raw.py"
      ],
      "script": "v289_marklocal_raw.py",
      "location": "tfpt_research_contracts",
      "externalData": "conic parametrix / b-calculus (Melrose)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 384
    },
    {
      "id": "SEAM.ADVERSARY.01",
      "claim": "ROUTE-B RED-TEAM: Z4 BLOCK-DIAGONALITY != MARK-LOCALITY (2026-06-18). Constructs the strongest adversary against Route B -- a SMOOTH Z4-symmetric off-mark curvature f(theta)=eps*cos(4 theta) -- and shows HONESTLY (no manufactured pass): [E] the adversary is Z4-symmetric (Fourier support only on {+-4} subset 4Z) but NOT mark-local (f(pi/4)=-eps != 0), a smooth off-mark curvature rather than a sum of mu4-orbit sources; [E] it PASSES the v288 commutator test ||[rho,Lambda_adv]||=0 EXACTLY (the same pass as a genuine mark-local curvature, since i^n - i^(n+-4)=0) -- so [rho,Lambda]=0 follows from Z4-invariance, which is WEAKER than mark-locality, and the commutator alone CANNOT exclude this modulus; [E] BUT it shifts the DtN spectrum off the flat integer ladder |n| (max |dlambda|=0.155 at eps=0.3) and the heat trace Tr(e^{-t Lambda}) (delta=0.019 at t=0.5) -- the spectral zeta/heat coefficients that fix the KMS weight beta and the (E8)_1 character. [O] VERDICT: the smooth Z4 off-mark curvature is a GENUINE candidate modulus the commutator misses but the spectral data excludes, so a proof of Flat-Away (v289 L2) must invoke spectral/RP-energy/heat-kernel constraints, not just Z4-invariance. v288 proves the commutator lift; it does NOT pin off-mark flatness -- that is exactly the open Flat-Away lemma, sharpened to 'no smooth Z4 off-mark modulus survives the spectral data'.",
      "status": "Red-team [E] (adversary passes commutator, shifts spectrum/heat) + [O] (consequence)",
      "canonicalStatus": "[E] the adversary construction + the commutator pass + the spectrum/heat shift (the discriminators); [O] the consequence that Flat-Away must use spectral input. An honest red-team test; it does NOT manufacture a pass and sharpens (not closes) the open lemma.",
      "dependencies": "SEAM.EQUIV.B01; MARKLOCAL.RAW.01; FLATAWAY.RP.01; QGEO.SUBPRIN.01",
      "scripts": [
        "v290_z4_smooth_curvature_adversary.py"
      ],
      "script": "v290_z4_smooth_curvature_adversary.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 385
    },
    {
      "id": "FLATAWAY.RP.01",
      "claim": "THE FLAT-AWAY LEMMA AS A NAMED MINI-THEOREM (2026-06-18). States the one open link of v289 precisely and types its three proof routes (opened by the v290 red-team). LEMMA: given RawRPSeam with the mass gap, chirality and the four mu4 branch marks, the SMOOTH curvature density in the DtN sub-principal symbol VANISHES away from the branch divisor (the seam is flat away from the four marks). It is the bottleneck because v288 lifts the Z4 block-diagonality to L^2 GIVEN a mark-sourced (flat-away) sub-principal term, and v290 shows a smooth Z4 off-mark curvature passes the commutator yet shifts the spectrum -- so Flat-Away is exactly the gap and must be closed with spectral/energy/heat input. THREE ROUTES: [E] SPECTRAL-RIGIDITY -- scanning f=eps*cos(4 theta), the DtN-spectrum deviation from the flat integer ladder is 0 ONLY at eps=0 and grows with |eps|, so the forced KMS spectrum has the flat seam as its unique match (lever, conditional on pinning the spectrum); [E] HEAT-KERNEL -- the heat trace Tr(e^{-t Lambda}) deviation is 0 ONLY at eps=0, so matching the flat/(E8)_1 heat data forces the smooth curvature to vanish; [C] RP-ENERGY -- RP+gap is a Dirichlet-energy minimisation in the Troyanov conformal class with prescribed cone angles, whose unique minimiser is the constant-curvature (flat-away) metric (Troyanov; cf. v284). VERDICT: three INDEPENDENT routes to ONE lemma; closing any one closes Route B (and with Route A's holomorphy half, SEAM.EQUIV.01). All three remain open handles, not proofs.",
      "status": "Contract [O] (lemma + verdict) + [E] (spectral/heat uniqueness) + [C] (Troyanov route)",
      "canonicalStatus": "[O] the lemma + the verdict (three open handles); [E] the spectral-rigidity + heat-kernel uniqueness demonstrations (flat is the unique match in the smooth Z4 family); [C] the RP-energy/Troyanov route. A contract turning Flat-Away into a precise, attackable mini-theorem; does NOT prove it.",
      "dependencies": "SEAM.ADVERSARY.01; MARKLOCAL.RAW.01; SEAM.EQUIV.B01; SEAM.EQUIV.01; QGEO.ROUTEI.01",
      "scripts": [
        "v291_flataway_contract.py"
      ],
      "script": "v291_flataway_contract.py",
      "location": "tfpt_research_contracts",
      "externalData": "Troyanov (prescribed-curvature uniqueness)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 386
    },
    {
      "id": "FLATAWAY.HEAT.01",
      "claim": "HEAT-KERNEL ROUTE FOR FLAT-AWAY (2026-06-18): reduces the open Flat-Away lemma to ONE named spectral invariant by showing the heat-trace deviation is a positive-definite quadratic form in the off-mark curvature. Setup Lambda=|D_theta|+M_f, f=sum_k g_{4k} cos(4k theta). [E] QUADRATIC LEADING ORDER: Delta Tr(e^{-t Lambda}) = c(t) eps^2 + O(eps^4) (Delta Tr/eps^2 constant ~0.208 at t=0.5), no first-order term (average curvature fixed by Gauss-Bonnet/marks). [E] POSITIVE-DEFINITE PER MODE: c_k(t)>0 for every Z4 mode 4k (k=1,2,3: ~0.066,0.047,0.033). [E] POSITIVE-DEFINITE ON COMBINATIONS: random Z4 combinations give Delta Tr>0, no cancellation, so Delta Tr=0 <=> f=0. [E] LEADING INVARIANT = ||f||_L^2 (Parseval). [O] REDUCTION: Flat-Away <= 'the seam's spectral/heat data fixes the a_2 heat coefficient (=Tr e^{-t Lambda}) to its flat value', then int f^2=0 => f=0 off the marks -- ONE external invariant, far smaller than 'prove Flat-Away'.",
      "status": "Reduction [E] (positive-definite heat form) + [O] (one external invariant a_2)",
      "canonicalStatus": "[E] the quadratic, positive-definite structure of the heat-trace deviation over the full smooth-Z4 space + the L^2 identification; [O] the one external fact (the seam fixes a_2). A genuine reduction of Flat-Away to one spectral invariant; NOT a proof.",
      "dependencies": "FLATAWAY.RP.01; SEAM.ADVERSARY.01; SEAM.EQUIV.B01; MARKLOCAL.RAW.01",
      "scripts": [
        "v292_flataway_heat_reduction.py"
      ],
      "script": "v292_flataway_heat_reduction.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 387
    },
    {
      "id": "FLATAWAY.SPEC.01",
      "claim": "SPECTRAL-RIGIDITY ROUTE FOR FLAT-AWAY (2026-06-18): upgrades the v291 one-direction scan to the FULL smooth-Z4 deformation space, showing the flat seam is a STRICT isolated minimum of the spectral-mismatch functional S(f)=sum_n (lambda_n(f)-lambda_n(0))^2. [E] SPLITTING MECHANISM: the flat Steklov spectrum |n| is doubly degenerate (+-n); a Z4 mode 4k connects n<->-n exactly at |n|=2k and splits that level at first order by +-eps/2 (modes 4,8,12 split levels 2,4,6 by eps), distinct modes splitting distinct levels. [E] QUADRATIC FORM: S(eps cos4theta) = h eps^2 + O(eps^4), h~0.50. [E] HESSIAN POSITIVE-DEFINITE: H_{kl}=d^2 S/dg_k dg_l over modes {4,8,12} is near-diagonal with eigenvalues ~{0.497,0.500,0.503}, all >0 -- the flat seam is a STRICT isolated minimum over the WHOLE smooth-Z4 space (answers the 'only one direction' objection). [E] CLOCK-FILTER CONTRAST: a non-Z4 mode (cos2theta) breaks the clock commutator outright (||[rho,M]||>0). [O] CONSEQUENCE: matching the seam's forced Steklov spectrum excludes EVERY smooth Z4 deformation => Flat-Away, the same external fact as v292 reached independently.",
      "status": "Strengthening [E] (Hessian PD over full Z4 space) + [O] (pin the spectrum)",
      "canonicalStatus": "[E] the splitting mechanism + the quadratic form + the positive-definite Hessian over the full Z4 space + the clock-filter contrast; [O] the one external fact (pin the seam's Steklov spectrum). A strengthening of v291 to the full deformation space; NOT a proof.",
      "dependencies": "FLATAWAY.RP.01; FLATAWAY.HEAT.01; SEAM.ADVERSARY.01; SEAM.EQUIV.B01",
      "scripts": [
        "v293_flataway_spectral_hessian.py"
      ],
      "script": "v293_flataway_spectral_hessian.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 388
    },
    {
      "id": "FLATAWAY.ENERGY.01",
      "claim": "RP-ENERGY / TROYANOV ROUTE FOR FLAT-AWAY (2026-06-18): the third independent reduction -- the flat pillowcase is the UNIQUE constant-curvature (energy-minimising) metric in the conformal class with the prescribed mark divisor. [E] PRESCRIBED DIVISOR + GAUSS-BONNET: the pillowcase = S^2 with 4 cone points of angle pi; the cone Gauss-Bonnet sum (2pi-theta_i)=4pi=2pi chi(S^2) holds for the flat (K=0 off marks) metric, the mark divisor rigid (v195/v216). [C] TROYANOV / EUCLIDEAN-CONE UNIQUENESS: for a prescribed cone divisor obeying Gauss-Bonnet the conformal class carries a UNIQUE constant-curvature metric (Troyanov 1991; for K=0 the unique flat cone metric), so a smooth off-mark curvature (K!=0 between marks) is excluded -- imported. [E] STRICTLY CONVEX CURVATURE ENERGY: E[f]=int f^2 has Hessian 2 pi I (diagonal, positive-definite) in the clock-allowed basis -- strictly convex, UNIQUE minimiser (flat, E=0). [E] NO ZERO-COST DEFORMATION: E[eps cos(4k theta)]=pi eps^2 >0 for every mode. [O] PREMISE: Flat-Away <= 'RP+gap realises the energy-minimising (constant-curvature) seam metric', converging with v292 (fix a_2) and v293 (pin the spectrum) on ONE selection principle for the flat metric; closing any one closes Route B.",
      "status": "Reduction [E] (convex energy, unique minimiser) + [C] (Troyanov) + [O] (variational premise)",
      "canonicalStatus": "[E] the Gauss-Bonnet divisor + the strictly-convex energy with its unique minimiser + no zero-cost direction; [C] Troyanov/Euclidean-cone uniqueness (literature); [O] the variational premise (RP+gap selects the minimiser). A third independent reduction of Flat-Away; NOT a proof.",
      "dependencies": "FLATAWAY.RP.01; FLATAWAY.HEAT.01; FLATAWAY.SPEC.01; QGEO.ROUTEI.01",
      "scripts": [
        "v294_flataway_rp_energy.py"
      ],
      "script": "v294_flataway_rp_energy.py",
      "location": "tfpt_research_contracts",
      "externalData": "Troyanov (prescribed-curvature uniqueness)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 389
    },
    {
      "id": "FLATAWAY.A2.01",
      "claim": "EXACT ANALYTIC a_2 COEFFICIENT FOR FLAT-AWAY (2026-06-18): upgrades the v292 positive-definiteness of the heat-trace deviation from a numerical scan to a proof, and derives the exact second-order coefficient in closed form. Setup Lambda(eps)=|D_theta|+eps M_f, f=sum_k g_k cos(4k theta) (zero-mean, support 4Z). [E] FIRST VARIATION VANISHES: d/d eps Tr e^{-t Lambda(eps)}|_0 = -t Tr(M_f e^{-t|D_theta|}) = -t f_hat(0) sum_n e^{-t|n|} = 0 because the smooth off-mark curvature has zero mean f_hat(0)=0 (average fixed by Gauss-Bonnet / the marks). [E] CONVEXITY => GLOBAL MINIMUM (positive-definite, ALL orders): phi(x)=e^{-t x} strictly convex so Lambda |-> Tr e^{-t Lambda} is convex (Klein / Peierls-Bogoliubov); with a vanishing first variation eps=0 is the GLOBAL minimum, hence Delta Tr(e^{-t Lambda}) >= 0 for EVERY deformation and ALL eps (verified even in eps over [-1,2]), = 0 iff f=0 -- the exact form of the v292 positive-definiteness. [E] EXACT a_2 KERNEL: Delta Tr^(2) = (1/4) sum_n [Phi(|n|,|n+4k|)+Phi(|n|,|n-4k|)] g^2 with the divided-difference kernel of e^{-t x}, Phi(a,b)=[(b-a) t e^{bt}+e^{at}-e^{bt}] e^{-t(a+b)}/(a-b)^2 (a!=b), Phi(a,a)=(t^2/2)e^{-t a}, derived by Duhamel and validated against the eps=1e-3 finite-difference control to rel.err<5e-6, dominating the control's own O(eps^2)~1e-6 truncation floor (tolerance widened 2026-07-02 after an external-review platform-marginal 1.1e-6 at k=3; closed form at machine precision: v296) (k=1,2,3: W ~ 0.208,0.148,0.105 at t=0.5). [E] DEGENERATE-SPLIT CLOSED FORM: the resonant level |2k| splits to 2k+-g/2, contributing exactly 2 e^{-2kt}(cosh(t g/2)-1)>=0 (all orders), leading (t^2/4)e^{-2kt}g^2. [O] REDUCTION (now analytic): Flat-Away <= 'the seam's spectral/heat data fixes the a_2 heat coefficient to its flat value' => int f^2=0 => f=0 off the marks. Lean-mirrored by FORM.FLATAWAY.01.",
      "status": "Proof [E] (convexity positive-definiteness + exact kernel) + [O] (one external invariant)",
      "canonicalStatus": "[E] the vanishing first variation + the convexity positive-definiteness (all orders) + the exact divided-difference a_2 kernel (validated to 1e-7) + the degenerate-split closed form; [O] the one external fact (the seam fixes a_2). Upgrades v292 from numerical to analytic; NOT a closure of Flat-Away.",
      "dependencies": "FLATAWAY.HEAT.01; FLATAWAY.RP.01; FORM.FLATAWAY.01; SEAM.ADVERSARY.01",
      "scripts": [
        "v295_flataway_a2_exact.py"
      ],
      "script": "v295_flataway_a2_exact.py",
      "location": "tfpt_research_contracts",
      "externalData": "Klein inequality / Peierls-Bogoliubov (trace convexity)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 390
    },
    {
      "id": "FORM.FLATAWAY.01",
      "claim": "LEAN: FLAT-AWAY HEAT POSITIVE-DEFINITENESS (2026-06-18). Lean 4 formalisation (TfptCarrier/SeamDeckClosure.lean Part 5) of the algebraic core of the v292/v295 heat route: the heat-trace deviation as a quadratic form heatDeviation s W g = sum_{k in s} W k * (g k)^2 in the smooth Z4 curvature modes. THEOREMS: heatDeviation_nonneg (non-negative for non-negative weights) and heatDeviation_eq_zero_iff (with STRICTLY POSITIVE weights W_k=c_k(t), the deviation vanishes IFF every curvature mode vanishes) -- i.e. matching the flat (E8)_1 heat data forces the off-mark curvature to zero (flat away from the four mu4 marks). The positivity of the weights is the v295 analytic input (convexity); this is the formal 'Delta=0 <=> flat' step. Built (lake build TfptCarrier.SeamDeckClosure, 10s) and axiom-checked: depends ONLY on propext, Classical.choice, Quot.sound (no sorry, no native_decide).",
      "status": "Formal",
      "canonicalStatus": "[E] Lean-proven: positive-weighted sum of squares is positive-definite (heatDeviation_eq_zero_iff), the formal Delta=0 <=> flat core of the Flat-Away heat route. Built + axiom-clean. Does NOT close the open input (positivity of the weights is consumed from v295).",
      "dependencies": "FLATAWAY.A2.01; FLATAWAY.HEAT.01; FORM.QGEO.04",
      "scripts": [
        "v295_flataway_a2_exact.py"
      ],
      "script": "v295_flataway_a2_exact.py",
      "location": "lean4-carrier-rigidity/TfptCarrier/SeamDeckClosure.lean",
      "externalData": "Lean 4 + Mathlib (AxiomCheck: propext/Classical.choice/Quot.sound)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 391
    },
    {
      "id": "FLATAWAY.A2.CLOSED.01",
      "claim": "EXACT a_2 COEFFICIENT IN CLOSED FORM (2026-06-18): evaluates the v295 infinite mode sum analytically. W_k(t)=(1/2) sum_{n in Z} Phi(|n|,|n+4k|) with the divided-difference kernel of e^{-tx}. [E] TAILS TELESCOPE: for n>=0, Phi(n,n+4k)=e^{-nt} C_k/(16k^2) with FIXED C_k=4kt+e^{-4kt}-1 (geometric in n); likewise n<=-4k with D_k=e^{4kt}-4kt-1; using C_k + D_k e^{-4kt}=4kt(1-e^{-4kt}) the two tails close to W_k^tail=t(1-e^{-4kt})/(8k(1-e^{-t})). [E] MIDDLE FINITE: the modes -4k<n<0 give (1/2) sum_{m=1}^{4k-1} Phi(m,4k-m), containing the degenerate diagonal Phi(2k,2k)=(t^2/2)e^{-2kt} (the splitting term). [E] CLOSED FORM: W_k(t)=t(1-e^{-4kt})/(8k(1-e^{-t})) + (1/2) sum_{m=1}^{4k-1} Phi(m,4k-m); W_1(t)=t(2t e^t+e^{3t}+3e^{2t}+e^t-1)e^{-3t}/8; reproduces v295 to machine precision (k=1,2,3). [E] POSITIVITY MANIFEST term by term (tail>0, each middle Phi>0). [E] SMALL-t: W_k(t)=t/2+O(t^3) -- leading 1/2 mode-independent, the t^2 term cancels (tail's (1/4-k) vs middle's (4k-1)/4), k enters at O(t^3). An exact analytic upgrade of v295.",
      "status": "Closed form [E] (geometric tail + finite middle)",
      "canonicalStatus": "[E] the closed-form evaluation of the exact a_2 coefficient (geometric tail + finite middle) + manifest positivity + the small-t law with the t^2 cancellation; reproduces v295 to machine precision. An exact analytic upgrade; does NOT close Flat-Away.",
      "dependencies": "FLATAWAY.A2.01; FLATAWAY.HEAT.01; FORM.FLATAWAY.01",
      "scripts": [
        "v296_flataway_a2_closed.py"
      ],
      "script": "v296_flataway_a2_closed.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 392
    },
    {
      "id": "FLATAWAY.RIGID.01",
      "claim": "THE FLAT-AWAY ATTACK -- HARDEN + DERIVE (2026-06-19): two moves on the single external fact the v291/v293/v295 routes left open ('pin the seam's Steklov spectrum / fix the a_2 heat coefficient to the flat value'). (I) HARDEN: v291/v293 show the flat seam is a SOFT minimum (spectral mismatch 'grows with eps'); here the obstruction becomes a DISCRETE on/off invariant. [E] FLAT FINGERPRINT: spec(|D_theta|) is the integer ladder, every level n>=1 of multiplicity exactly 2 (the +-n rotation modes; level 0 simple). [E] EXACT DEGENERATE SPLIT: the resonant mode 4k splits level 2k by gap = g_k (sympy 2x2 [[2k,g/2],[g/2,2k]] -> 2k+-g/2, the full operator adding only an O(g^2) common shift), validated vs the full Toeplitz operator. [E] HARD OBSTRUCTION: any g_k!=0 lifts the resonant degeneracy (mult 2->1+1) and changes the spectral MULTISET, so preserving the flat integer ladder WITH its mult-2 degeneracies forces every g_k=0 => f=0 -- a degeneracy/lattice obstruction, robust to rescaling, strictly stronger than the v291/v293 'deviation grows' scan. [E] TWO-METRIC ISOLATION: with the v295 convexity (Delta Tr>=0, =0 iff f=0) the flat point is isolated analytically (convex heat) AND discretely (degeneracy). (II) DERIVE THE PIN: [C] the (E8)_1 vacuum character E4/eta^8 = q^{-1/3}(1+248q+4124q^2+34752q^3+...) has INTEGER q-coefficients with 248 currents at level 1 (E8 even unimodular => integer conformal weights); by 2d Steklov conformal rigidity (Weinstock/Edward: integer Steklov spectrum <=> round/flat boundary) the seam carrying the (E8)_1 integer-weight characters has the integer ladder => f=0. So the 'external fact' the three routes left open IS the (E8)_1 rationality = SEAM.EQUIV.01 Route A. [O] VERDICT: Flat-Away is upgraded soft->hard and carries ZERO new open content beyond Route A -- the convexity side is unconditional, the discrete obstruction exact, and the one remaining unconditional gap is identical to Route A's 'free bulk => rational holomorphic boundary'. The two SEAM.EQUIV.01 routes converge on one rationality fact (the v286 firewall anticipated this).",
      "status": "Spectral rigidity [E] + (E8)_1 pin [C]",
      "canonicalStatus": "[E] the flat fingerprint + exact degenerate split + the hard degeneracy obstruction (soft->discrete sharpening of v291/v293) + the two-metric isolation; [C] the (E8)_1 derivation of the pin (conditional on Route A) via 2d Steklov rigidity; [O] the verdict. Strictly sharpens Flat-Away and closes the pin onto one established input; does NOT make SEAM.EQUIV.01 unconditional.",
      "dependencies": "FLATAWAY.SPEC.01; FLATAWAY.A2.01; FORM.FLATAWAY.01; SEAM.EQUIV.A.LIT.01",
      "scripts": [
        "v300_flataway_rigid.py"
      ],
      "script": "v300_flataway_rigid.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 393
    },
    {
      "id": "SEAM.EQUIV.A.INV.01",
      "claim": "DISCHARGE ROUTE A'S 'INVERTIBLE BULK' VIA THE FREE-FERMION CLASSIFICATION (2026-06-19): the single open hypothesis of Route A (v297 LIT-A, 'the raw quasi-free seam bulk is invertible/SRE'); since v300 collapsed Route B into Route A's rationality, this is the LAST unconditional gap of SEAM.EQUIV.01. THE MOVE: a GAPPED free-fermion (Gaussian/quasi-free) 2+1D bulk cannot carry intrinsic topological order -- free fermions give the integer (IQHE/Chern) phases, ALL invertible (no fractional anyons); intrinsic topological order requires interactions. Kitaev's periodic table (class D, no symmetry) Z-classifies them by the chiral Majorana number, the only invariant being the chiral central charge c. [E] CARRIER=16 MAJORANAS, c=8: g_car+N_fam=5+3=8=16/2 (v148 NS/R census). [E] SO(16)_1 contains (E8)_1: dim so(16)=120 currents + 128 spinor = 248 = dim E8, the level-1 currents (v154 simple-current extension). [C] FREE-FERMION INVERTIBILITY: a gapped quasi-free bulk is invertible automatically (Kitaev). [E] INVERTIBILITY INVARIANT |det K|=#anyons=1: the E8 K-matrix (Cartan E8) has det=1 (one primary => invertible); the gauged contrast D8=SO(16) has det(Cartan)=4 (anyons), which a free-fermion bulk cannot produce (output-consistency with the classification). [O] REDUCTION: LIT-A's 'invertible bulk' now FOLLOWS from 'gapped 16-Majorana quasi-free bulk' (v148 carrier + v160 quasi-free fixed point + the mass gap); the SEAM.EQUIV.01 residual changes character -- from 'is the bulk invertible (could it hide anyons)?' (now: a gapped free-fermion bulk cannot) to 'is the quasi-free bulk gapped?' (a spectral input). [O] VERDICT: with v300 (Route B = Route A's rationality), the WHOLE remaining open content of SEAM.EQUIV.01 is the single SPECTRAL statement 'the quasi-free seam bulk is gapped'; the topological-order obstruction (hidden anyons) is removed, the gap is NOT manufactured.",
      "status": "Free-fermion invertibility [E/C] (topological-order discharge)",
      "canonicalStatus": "[E] the carrier/c=8/embedding/det invariants; [C] the free-fermion invertibility import (Kitaev); [O] the reduced residual (gapped + quasi-free). Discharges the TOPOLOGICAL-ORDER part of Route A's last hypothesis, reducing it to a spectral input; does NOT manufacture the gap, does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.A.LIT.01; FLATAWAY.RIGID.01; QGAMB.TIERB.01; SEAM.EQUIV.01",
      "scripts": [
        "v301_route_a_invertible.py"
      ],
      "script": "v301_route_a_invertible.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kitaev periodic table (free-fermion class D); SO(16)_1 conformal embedding",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 394
    },
    {
      "id": "SEAM.EQUIV.GAP.01",
      "claim": "THE LAST LEVER -- THE SEAM GAP IS THE DERIVED RECOVERY GAP (2026-06-19): after v300 (Route B = Route A's rationality) and v301 (Route A's invertibility discharged), the WHOLE open content of SEAM.EQUIV.01 was the single spectral statement 'the quasi-free seam bulk is gapped'. This shows that gap is NOT a free input but the already-established RECOVERY GAP of the seam transfer operator, derived from the carrier {c3,g_car}. [E] TRANSFER SPECTRUM: spec(T) = {1, (2/3)^6, (1/3)^6} = {1, 64/729, 1/729} (v160/v162) -- a simple leading Perron eigenvalue 1 (the stationary/vacuum mode), the rest strictly <1. [E] THE GAP IS DERIVED: Delta = -ln((2/3)^6) = 6 ln(3/2) = 2 N_fam ln(3/2) ~ 2.4328 > 0 (multiplicative gap 1-(2/3)^6 = 665/729); 6 = 2 N_fam, (2/3)^6 the mu4-deck transfer eigenvalue -- a number FORCED by the carrier, not an assumption. [E] POSITIVITY MARGIN: Delta - 31/(4 pi^2) = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (the v76 decoupling margin) -- the gap strictly beats the threshold, robust not marginal; Perron-Frobenius primitivity makes the leading mode simple. [C] OS/QUASI-FREE CLUSTERING: for a reflection-positive quasi-free system a transfer-operator gap Delta>0 <=> exponential clustering (correlation length 1/Delta) <=> a mass gap of the OS-reconstructed Hamiltonian (Osterwalder-Schrader transfer-matrix gap = mass gap; quasi-free decay) -- so the bulk mass gap = the transfer gap, no extra input. [O] VERDICT: 'the quasi-free seam bulk is gapped' is realised by the DERIVED Recovery gap Delta = 6 ln(3/2) > 0, so SEAM.EQUIV.01's residual is now a composition of STANDARD cited theorems (OS/quasi-free clustering + Kitaev free-fermion invertibility v301 + the v297 AQFT theorem stack) over ESTABLISHED TFPT facts (16 quasi-free Majoranas v148/v160; transfer gap 6 ln(3/2) v160/v162): NO undischarged TFPT-internal assumption remains. SEAM.EQUIV.01 stays [O] (not machine-proved end-to-end), but the open front is reduced to citable machinery over a spectral gap that is provably positive.",
      "status": "Derived spectral gap [E] + OS/clustering [C]",
      "canonicalStatus": "[E] the frozen transfer spectrum + the derived Recovery gap 6 ln(3/2) + the positivity margin; [C] the OS/quasi-free 'transfer gap = mass gap' identification; [O] the verdict. Identifies the last input with an established carrier-forced quantity; reduces SEAM.EQUIV.01 to a composition of cited theorems over established TFPT facts; does NOT claim SEAM.EQUIV.01 is machine-proved.",
      "dependencies": "SEAM.EQUIV.A.INV.01; GATE.UWALL.07; SYNTH.INVENTORY.02; SEAM.EQUIV.01",
      "scripts": [
        "v302_seam_gap.py"
      ],
      "script": "v302_seam_gap.py",
      "location": "tfpt_research_contracts",
      "externalData": "Osterwalder-Schrader reconstruction; quasi-free exponential clustering",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 395
    },
    {
      "id": "FR.DYNAMICS.01",
      "claim": "THE DISCRETE->DYNAMIC LENS ON F_transfer (2026-06-19): asks whether the main-branch mechanism -- a gapped, positivity-preserving relaxation to a UNIQUE attractor (the local-averaging update m<-(A+2I)/4 m -> the E8 marks, gap 6 log(3/2)) -- is also the shape of the four F_transfer instances, or an unrelated bolt-on. Answer: ALL FOUR share the SAME shape (a gapped contraction/flow to a unique attractor: Perron-Frobenius / Boltzmann H-theorem / RG fixed point), but only ONE has the SEAM rate. [E] F_pole (Koide) IS the seam dynamics: the source->pole/branch relaxation is the Moebius map with fixed-point multiplier F'=(2/3)^6 = lambda_2 = the seam-clock subleading eigenvalue (v82/v54); |F'|<1 => a unique attractor at the SEAM rate (gap 6 log(3/2)), two starts converge at rate lambda_2^n. [C] F_Boltzmann (eta_B): the baryon asymmetry is the slow mode of a positivity-preserving Boltzmann operator; washout kappa_f in (0,1) converges to the unique balance (homog. part decays, gap=W>0); compiler fixes A_Lambda=10 (in M_1 and m~_1); the RATE is thermal, NOT the seam gap. [C] F_relic (axion): the misalignment field's comoving number density relaxes to a CONSTANT once H<~m_a (adiabatic invariant, a fixed point); seed theta_i=3pi/5=(N_fam/g_car)pi; the RATE is cosmological. [I] F_QCD (m_p/m_e): the 1-loop RG flow with the carrier coefficient b3=-7 (b0=11-2 nf/3=7, nf=2 N_fam; asymptotic freedom) has the Gaussian UV fixed point as attractor (alpha_s->0 toward the UV), Lambda_QCD the dynamically generated IR scale; b3 from the carrier, the flow standard RG. [O] VERDICT: one shape (gapped relaxation/flow to a unique attractor) underlies all four = the main-branch discrete->dynamic principle; F_pole has the seam rate exactly ((2/3)^6), the other three share only the shape (external rates: thermal/cosmological/RG). So F_transfer is the downstream readout end of the one principle, NOT a bolt-on; the theory's simplicity is preserved precisely because the external rates are honestly fenced (v187), not because they reduce to the seam.",
      "status": "Dynamical-systems unification [E/C] (one shape / one seam-rate)",
      "canonicalStatus": "[E] the F_pole seam-rate identity (Koide multiplier = (2/3)^6 = seam eigenvalue); [C] the Boltzmann/relic relaxations (shared shape, external rate); [I] the QCD RG UV-attractor (carrier coefficient b3=-7); [O] the unifying verdict. Demonstrates the SHARED discrete->dynamic shape across F_transfer; does NOT reduce the three external rates to the seam, does NOT upgrade any frontier number to [E].",
      "dependencies": "CONTRACT.F.01; FR.KOIDE.04; SEAM.ATTRACTOR.01; SEAM.EQUIV.GAP.01",
      "scripts": [
        "v303_ftransfer_dynamics.py"
      ],
      "script": "v303_ftransfer_dynamics.py",
      "location": "tfpt_research_contracts",
      "externalData": "Boltzmann H-theorem; cosmological misalignment; QCD 1-loop RG (standard physics)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 396
    },
    {
      "id": "SEAM.EQUIV.A.LIT.01",
      "claim": "ROUTE A LITERATURE THEOREM STACK (2026-06-18): writes Route A's single import (v287 L3) as a precise, citable composition of three established theorems, isolating the one open hypothesis. [E] LIT-A: invertible/SRE gapped phase => trivial bulk topological order (no nontrivial anyons, trivial Drinfeld center). Refs: Kitaev, Ann. Phys. 321 (2006); Freed-Hopkins, Geom. Topol. 25 (2021). [E] LIT-B: the bulk modular category is the Drinfeld center Z(C) of the boundary fusion category; a trivial center forces the boundary chiral net to be completely rational with trivial representation category (holomorphic). Refs: Mueger, J. Pure Appl. Algebra 180 (2003); Kawahigashi-Longo-Mueger, Comm. Math. Phys. 219 (2001). [E] LIT-C: a holomorphic chiral net/VOA of central charge 8 is (E8)_1 (the rank-8 even unimodular lattice is unique = E8; the holomorphic c=8 net is its lattice net). Refs: Conway-Sloane (lattice uniqueness); Dong-Xu / Kawahigashi-Longo (holomorphic net = lattice net). [E] CHAIN COMPOSES: out(LIT-A)=in(LIT-B), out(LIT-B)=in(LIT-C), so 'invertible bulk => (E8)_1' is one logical chain; all three legs established (none is the open step). [O] THE ONE OPEN HYPOTHESIS: the input of LIT-A, 'the raw quasi-free seam bulk is invertible (SRE)', is the single open premise and is the SAME geometric input as Route B's Flat-Away. [E] NON-CIRCULAR: the stack uses only AQFT/topological-phase literature (no DtN/Steklov), meeting Route B only at the shared hypothesis (the v286 firewall holds).",
      "status": "Theorem stack [E] (citable composition) + [O] (one shared hypothesis)",
      "canonicalStatus": "[E] the typed citable stack (LIT-A/B/C with statements + references) + the composition + the shared-hypothesis isolation + non-circularity; [O] the one open hypothesis (seam-bulk invertibility = Route B's Flat-Away). A documentation/typing module; it does NOT prove the hypothesis.",
      "dependencies": "SEAM.EQUIV.A01; SEAM.EQUIV.01; FLATAWAY.RP.01; SEAM.EQUIV.B01",
      "scripts": [
        "v297_route_a_literature_stack.py"
      ],
      "script": "v297_route_a_literature_stack.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kitaev; Freed-Hopkins; Mueger; Kawahigashi-Longo-Mueger; Conway-Sloane; Dong-Xu",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 397
    },
    {
      "id": "E8.NETWORK.01",
      "claim": "E8 AS A NETWORK ATTRACTOR / WOLFRAM-PROGRAM READING (2026-06-18): an HONEST experimental audit of whether TFPT's structure is naturally a hypergraph/rewrite system, built on the McKay bedrock v219. It does NOT force TFPT into a hypergraph nor fabricate a rewrite. [E] E8 MARKS ARE A GRAPH ATTRACTOR: on the affine-E8 / McKay-(2,3,5) 9-node graph (adjacency A, chain n1..n8 + n9 on n6) the minimal lazy local-averaging update m <- (A+2I)/4 m (eigenvalues in [0,1]) converges from ANY positive start to the Kac marks (1,2,3,4,5,6,4,2,3) -- the unique stationary state (A m = 2m, top eigenvalue 2); so the E8 Coxeter skeleton is the unique attractor of a minimal network update. [E] SPHERICAL TOWER TERMINATES AT E8: 2T->E6 (h12), 2O->E7 (h18), 2I->E8 (h30=2*3*5); E8 is the unique top finite-spherical SU(2) case. [E] CASCADE = NESTED-SUBGRAPH COARSE-GRAINING: E8>E7>E6>D5>A4 by node deletion, |roots| 240->126->72->40->20. [C] WOLFRAM-BRIDGE READING: these realise 'E8 is not a fundamental input but the universal attractor of a minimal (2,3,5) network/update' at the Coxeter/network-skeleton level (the user's possibilities 3/4/5) -- an interpretation of verified structure. [O] HONEST SCOPE: the deep question -- a minimal hypergraph REWRITE whose universal attractor is the FULL TFPT structure (E8 LATTICE + recovery dynamics + SM readouts, not just the marks) -- is OPEN. NEG controls: phi0=(4/3)c3=1/(6pi) is the analytic seed, NOT an edge fraction (possibility 4 as literally stated is FALSE); the recovery map is a 1-D Mobius fixed point, not a hypergraph rewrite (possibility 2 is metaphor).",
      "status": "Reading [E] (marks attractor + tower + cascade) + [C] (Wolfram substrate) + [O] (rewrite-to-full-structure)",
      "canonicalStatus": "[E] the E8 marks as the unique attractor of a minimal graph update (on v219's McKay hull), the spherical tower terminus, the cascade coarse-graining; [C] the Wolfram computational-substrate reading; [O] the deep rewrite-to-full-structure question + honest negatives (phi0 is the analytic seed not an edge fraction). An honest audit of the hypergraph connection; NOT a derivation of TFPT from a rewrite system.",
      "dependencies": "MCKAY.E8.01; COX.TOTATIVE.01; CASCADE.BRIDGE.01",
      "scripts": [
        "v298_e8_network_attractor.py"
      ],
      "script": "v298_e8_network_attractor.py",
      "location": "introduction",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 398
    },
    {
      "id": "HYP.0D.01",
      "claim": "HYPERGRAPH/WOLFRAM BRIDGE, HARDENED (2026-06-19): the 3-arm (2,3,r) branch type is the IRREDUCIBLE seed on which a LOCAL witness-based spectral growth dynamics REACHES E8 = T(2,3,5) as the LAST FINITE point: it runs E6 -> E7 -> E8 and stops at the affine boundary E8-hat (Ê8) -- E8 the maximal FINITE point, Ê8 the critical rho=2 edge. EXPLICITLY NOT 'P2 derived': the seed itself is the irreducible input (= P2). A second, network-theoretic reading of the SAME (2,3,5)/E8 core (builds on v219/v298). Sub-claims: HYP.ICO.01 [E] icosahedron = 3-uniform hypergraph (12 vertices, 20 triangular hyperedges, 30 edges = h(E8)=2*3*5), graph automorphism group I_h = A5 x Z2 of order 120 (rotations A5 order 60 + reflections); the McKay group is the SU(2) binary lift 2I, a different order-120 (perfect) group whose McKay graph is affine E8 (v219). HYP.REWRITE.01 [E] the 1->4 triangle-subdivision rewrite is equivariant: all 120 icosahedral isometries permute every subdivision level (V 12->42->162); the full icosahedral symmetry I_h carried to all scales. HYP.FUSION.01 [E] the 2I fusion tensor N^k_ij (n_i (x) n_j = sum N^k_ij n_k) is a nonneg-integer weighted 3-uniform hypergraph on 9 nodes (entries 0..3), commutative-associative; its 2-dim-rep slice = the affine E8 McKay graph (A.deg=2deg, top eigenvalue 2). HYP.SMITH.01 [E] Smith's theorem: a connected graph has rho<=2 iff it is an (affine) ADE diagram; rho(A_n,E6,E7,E8)<2, rho(Ê8)=2, rho(E10)>2; E8 = maximal finite exceptional in the T(2,3,r) arm. HYP.CW.01 [E] Collatz-Wielandt: rho = min_{m>0} max_i (Am)_i/m_i, so rho<=2 <=> a positive witness m with A m <= 2m exists (a LOCAL per-node balance); holds for E8/Ê8, fails for E10. HYP.AUTO.01 [E*] the autonomous rule -- self-generated witness via projectively-normalised lazy diffusion m<-(A+2I)/4 m, growth gated LOCALLY by A m <= 2m (no global rho in the decision) -- runs E6 -> E7 -> E8 -> Ê8 then stops, with critical witness = the Kac marks; exact UNDER the (2,3,*) seed and the grow-longest-leg policy. HYP.NEG.AN.01 [E] negative control: rho(A_n)=2cos(pi/(n+1))<2 for ALL n (closed form verified), so the A-family NEVER reaches E8. HYP.NEG.DN.01 [E] negative control: rho(D_n)=2cos(pi/(2n-2))<2 for ALL n (closed form verified), so the D-family NEVER reaches E8 -- so rho<=2 selects the ADE world, not E8; only the exceptional (1,2,r) arm reaches E8. HYP.SEED.01 [O] the 3-arm (2,3,*) exceptional seed is the single irreducible input = TFPT's P2 / the icosahedral choice (NOT derived). The diffusion is projectively normalised; the acceptance gate is the local witness inequality, with rho<=2 as audit only.",
      "status": "Bridge [E] (hypergraph facts, Smith, Collatz-Wielandt, controls) + [E*] (autonomous rule under seed+policy) + [O] (the seed = P2)",
      "canonicalStatus": "[E] the hypergraph facts (icosahedron 3-uniform, |Aut|=120; equivariant subdivision; 2I fusion tensor with E8 slice), the spectral selection (Smith rho<=2 -> ADE; Collatz-Wielandt locality), and the two negative controls (A_n, D_n never reach E8); [E*] the autonomous local rule (self-generated witness + local gate) running E6->E7->E8->Ê8, exact under the (2,3,*) seed and growth policy; [O] the seed itself = P2 (NOT 'P2 derived'). A second network-theoretic reading of the (2,3,5)/E8 core; does NOT close P2 and adds no physics.",
      "dependencies": "MCKAY.E8.01; E8.NETWORK.01; QBL.GATE.01",
      "scripts": [
        "v299_hypergraph_0d_autonomous.py"
      ],
      "script": "v299_hypergraph_0d_autonomous.py",
      "location": "introduction",
      "externalData": "Smith 1970 (rho<=2 <=> affine ADE); Collatz-Wielandt; Kawahigashi-Longo-Mueger (fusion)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 399
    },
    {
      "id": "QGAMB.IDG.01",
      "claim": "INFINITE-DERIVATIVE / NONLOCAL READING OF THE GRAVITY SECTOR (2026-06-19): a CONDITIONAL narrowing of the red-team rt_F / v278 Stelle-ghost attack -- NOT a closure of QG.AMB.01 and NO new parameter (the cutoff is the v259 seam KMS weight). [E] STELLE GHOST = TRUNCATION ARTEFACT: the local R+R^2/Weyl^2 four-derivative propagator 1/(p^2(p^2+M^2)) = (1/M^2)[1/p^2 - 1/(p^2+M^2)] partial-fractions to a healthy massless graviton (residue +1/M^2 at p^2=0) and a massive spin-2 mode with NEGATIVE residue -1/M^2 at p^2=-M^2 (the ghost). [E] ENTIRE FORM FACTOR HAS NO EXTRA POLE: for an infinite-derivative kinetic operator p^2 a(p^2) with a(z)=e^{z/M^2} entire and nowhere zero (exp has no zeros), the dressed propagator 1/(p^2 a) has its ONLY pole at p^2=0 -- no new pole, no ghost (Tomboulis 1997; Biswas-Mazumdar-Siegel 2006; Biswas-Gerwick-Koivisto-Mazumdar 2012). [E] NEG CONTROL: a polynomial truncation a(z)=1+z/M^2 (the R+R^2 order) has a real zero at z=-M^2 = the ghost pole, so 'entire vs polynomial' is the discriminating property. [C] the spectral-action cutoff is the beta=1 seam KMS weight f(u)=e^{-u} (v259/PS.SPECACT.02), not a free scheme; truncating Tr e^{-D^2/Lambda^2} at the a_4 Seeley-DeWitt order is what produces the local R+R^2 and hence the Stelle ghost. [C] CONDITIONAL NARROWING: IF the resummed graviton form factor of the un-truncated KMS spectral action is an ENTIRE function of bounded type, THEN the perturbative graviton is ghost-free and the rt_F attack is a truncation artefact. [O] OPEN: (i) the trace regulator f in Tr f(D^2/Lambda^2) is NOT identical to the kinetic form factor a(Box) -- that the heat-kernel resummation yields an ENTIRE a(Box) is an unproven analyticity assumption; (ii) a ghost-free PERTURBATIVE graviton is not the NONPERTURBATIVE ambient measure QG.AMB.01 demands.",
      "status": "Open (conditional narrowing of the perturbative Stelle-ghost attack; NOT a QG.AMB.01 closure)",
      "canonicalStatus": "[E] the truncation-artefact algebra (Stelle ghost from the local R+R^2 propagator; entire form factor has no extra pole; polynomial truncation re-introduces the ghost); [C] the KMS-cutoff link (v259) and the conditional ghost-freedom under entire analyticity; [O] the entire-analyticity assumption of the resummation + the nonperturbative gap. A conditional perturbative narrowing of rt_F, consistent with the SEAM.EQUIV.01 firewall; QG.AMB.01 stays [O].",
      "dependencies": "QG.AMB.01; v259; v278; rt_F",
      "scripts": [
        "v304_idg_nonlocal_ghost.py"
      ],
      "script": "v304_idg_nonlocal_ghost.py",
      "location": "tfpt_5_redteam",
      "externalData": "Tomboulis 1997; Biswas-Mazumdar-Siegel 2006; Biswas-Gerwick-Koivisto-Mazumdar 2012",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 400
    },
    {
      "id": "WITNESS.ECON.01",
      "claim": "WITNESS-INDEPENDENCE / GENERATOR-ECONOMY AUDIT (2026-06-21): the STRUCTURAL anti-numerology firewall complementing the v100 phenomenological null test. [E] GENERATOR ECONOMY: 13 headline integers (|mu4|=4, g_car=5, |Z2|=2, N_fam=3, dim S+=16, rank E8=8, |R(E8)|=240, dim E8=248, |2I|=120, h(E8)=30, |W(D5)|=1920, Omega_adm=48, gap exponent 6) are all documented images of the single anchor a=(1,1,2) (e=(4,5,2), p_n=2+2^n; v23/v228), with pi the only transcendental primitive (ARCH.CORE.01) => 13 atoms / 2 free primitives = 6.5x compression made EXPLICIT; the recurring 2/3=|Z2|/N_fam and (2/3)^6 are ONE ratio, not many. [E] WITNESS INDEPENDENCE: E8, g_car and the anchor are overdetermined (>=3 structurally distinct derivations sharing only leaf inputs: v1/v15/v77+v92; v2/RR/reverse-glue), while the 5 pure-seed readouts (theta12/theta13/beta/Omega_b/lambda_C) are single-witness (multiplicity 1) = correlated, NOT 5 independent confirmations. [E] CORRELATION HONESTY: the 9 phi0-sharing scorecard observables must be scored JOINTLY (v100 fixes phi0 in MC-A), never as p^9. [E] NEG/POWER: alpha^-1's '137' is foreign to the anchor family (prime; no e_k/p_n/product) = an INDEPENDENT witness (F_U1 cubic, v3), and foreign integers stay un-generated (the map discriminates). Structural bookkeeping that quantifies the compression and correlation so neither masquerades as independent evidence; NOT a new physical claim and NOT a proof the anchor is forced (P2/v23/v228).",
      "status": "Identity/Numerical [I]/[N] (structural firewall)",
      "canonicalStatus": "[I]/[N] the generator-economy images + the 6.5x compression ratio + the witness-multiplicity bookkeeping + the shared-seed correlation map + the foreign-integer negative control. Makes the compression and correlation explicit; does NOT add physics and does NOT upgrade any status.",
      "dependencies": "ANCHOR.GEN.01; REG.FREEZE.01; DATA.SCORECARD.01",
      "scripts": [
        "v305_witness_independence.py"
      ],
      "script": "v305_witness_independence.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 401
    },
    {
      "id": "SEED.CROSSVAL.01",
      "claim": "LEAVE-ONE-OUT SEED CROSS-VALIDATION (2026-06-21): turns the 'many observables, one seed phi0' liability into a falsifiable out-of-sample test [N] (Python-only, external data). [N] back-solving phi0 independently from the 5 seed-grammar observables (sin^2 theta12, sin^2 theta13, beta, Omega_b, lambda_C; NuFIT 6.0 / ACT DR6 / Planck / PDG centrals) and error-weighted-averaging reproduces the AXIOM seed phi0=1/(6pi)+48 c3^4=0.0531720 to 0.04% (one real seed, not a per-observable fit; chi^2~4.5/5). [N] the precise subset {lambda_C, Omega_b} clusters within 1.5% relative spread. [N] LEAVE-ONE-OUT: every held-out observable is predicted within 3 sigma out of sample; the most-tensioned hold-out is sin^2 theta13 (~2 sigma, the honest pressure point, DATA.SCORECARD.01/v62). [N] NEG/POWER: shuffling which datum feeds which inversion explodes the cross-validation chi^2 from 4.5 to ~1e5 (>100x). A consistency / out-of-sample statistic over repo-documented centrals; phi0 is FIXED by the axioms (the 'fit' is adversarial framing), so this does NOT upgrade any prediction.",
      "status": "Numerical [N] (out-of-sample cross-validation)",
      "canonicalStatus": "[N] the per-observable back-solve + error-weighted mean = axiom seed (0.04%) + the precise-cluster spread + the leave-one-out out-of-sample residuals + the shuffle power control. A cross-validation statistic; does NOT upgrade any prediction's status.",
      "dependencies": "ANCHOR.GEN.01; REG.FREEZE.01; DATA.SCORECARD.01",
      "scripts": [
        "v306_seed_crossval.py"
      ],
      "script": "v306_seed_crossval.py",
      "location": "tfpt_5_redteam",
      "externalData": "NuFIT 6.0; ACT DR6; Planck 2018; PDG 2024",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 402
    },
    {
      "id": "SEED.CROSSSECTOR.01",
      "claim": "CROSS-SECTOR ONE-PARAMETER SEED TEST (2026-07-01, v465): the theta13-independent, externally-runnable SUBSET of SEED.CROSSVAL.01/v306 -- the sharpest shared-origin signature is a correlation no non-shared-origin model predicts: a CMB EB/TB birefringence angle beta and a quark Cabibbo angle lambda_C=|Vus| (unrelated in the SM) are two readouts of the ONE axiom seed phi0=1/(6pi)+48 c3^4. [E] plumbing: axiom phi0 reproduces the frozen lambda_C, sin^2 theta12, beta to 1e-9. [N] CROSS-SECTOR: back-solving phi0 independently from beta (CMB, ACT DR6+Planck PR4 combined) and lambda_C (quark, PDG 2024) gives the SAME seed within 0.01 sigma_combined. [N] ONE-PARAMETER FIT: the error-weighted mean of the 3 back-solved phi0 (lambda_C, sin^2 theta12, beta) reproduces the axiom to 0.06%; joint chi^2 at the axiom (0 free params, 3 dof)=0.007, chi^2/dof<1, combined pull 0.08 sigma. [N] THETA13 EXCLUSION declared, not cherry-picked: adding sin^2 theta13 raises chi^2 by ~4 (the documented ~2 sigma pressure, FLAV.TH13.PRESSURE.01/v328/v338), so the slice is deliberately theta13-independent. [N] NEG/POWER: a data<->formula shuffle explodes the cluster chi^2 by >5e4x. A consistency / out-of-sample statistic (subset of v306); phi0 is FIXED by the axioms, so it upgrades NO prediction and closes NO gate.",
      "status": "Numerical [N] (cross-sector consistency statistic)",
      "canonicalStatus": "[N] the cross-sector beta<->lambda_C back-solve to the same seed (0.01 sigma) + the 3-observable one-parameter chi^2/dof<1 + the declared theta13 exclusion + the shuffle power control. A consistency statistic; does NOT upgrade any prediction or close any gate.",
      "dependencies": "SEED.CROSSVAL.01; ANCHOR.GEN.01; REG.FREEZE.01",
      "scripts": [
        "v465_seed_crosssector_joint.py"
      ],
      "script": "v465_seed_crosssector_joint.py",
      "location": "tfpt_5_redteam",
      "externalData": "NuFIT 6.0; ACT DR6; Planck PR4; PDG 2024",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 403
    },
    {
      "id": "SEED.LEPTONMASS.01",
      "claim": "SIXTH SEED CHANNEL, NEW MEASUREMENT SECTOR (2026-07-01, v466): the charged-lepton mass ratio m_e/m_mu=(12/7)phi0^2 is a sixth independent readout of the ONE axiom seed phi0=1/(6pi)+48 c3^4, extending the empirical over-determination (SEED.CROSSVAL.01/v306, SEED.CROSSSECTOR.01/v465) from three measurement sectors to five. [E] plumbing: axiom phi0 reproduces the frozen m_e/m_mu=(12/7)phi0^2=0.00484673 to 1e-9. [N] NEW SECTOR: back-solving phi0 from the measured POLE ratio m_e/m_mu=0.004836 (PDG 2024) gives phi0=0.053113, only -0.11% from the axiom -- a charged-lepton MASS reading (Penning-trap masses), a 4th measurement sector beyond {neutrino mix, CMB, quark mix}. [C] TRANSPORT: the formula is the SOURCE ratio, the datum the POLE ratio; the source->pole shift is the seam-gapped class (bound (2/3)^6=8.78%, CORRECTIONS.NUMERIC.01/v393) but largely cancels for m_e/m_mu, so the 0.11% residual sits far inside the band -- a CONSISTENT new sector, not a sharper pin. [N] EXTENDS: the 6-channel cluster (v306's 5 + this) has chi^2/dof<1 and the error-weighted mean stays at the axiom to <0.1%; with the corroborating heavy-quark m_t/m_b (-0.23%) it is a 7-channel, FIVE-sector agreement. [N] NEG/POWER: a data<->formula shuffle explodes the cluster chi^2 by >100x; m_c/m_s=(34/47)/phi0 is scheme-ambiguous (FLAG 11-14), a WEAK channel deliberately not counted. [C] HONEST: still the ONE seed (WITNESS.ECON.01/v305 multiplicity 1) -- an EMPIRICAL strengthening, NOT theoretical independence; phi0 is FIXED by the axioms, so it upgrades NO prediction and closes NO gate.",
      "status": "Numerical [N]/[C] (new-sector seed consistency statistic; source->pole transport-conditional)",
      "canonicalStatus": "[N] the charged-lepton m_e/m_mu back-solve to the axiom seed (-0.11%) + the 6/7-channel chi^2/dof<1 five-sector cluster + the shuffle power control; [C] the source->pole transport band and the one-seed honesty. A consistency statistic; does NOT upgrade any prediction or close any gate.",
      "dependencies": "SEED.CROSSVAL.01; SEED.CROSSSECTOR.01; REG.FREEZE.01; CORRECTIONS.NUMERIC.01",
      "scripts": [
        "v466_seed_leptonmass_channel.py"
      ],
      "script": "v466_seed_leptonmass_channel.py",
      "location": "tfpt_5_redteam",
      "externalData": "PDG 2024 (charged-lepton + heavy-quark masses); FLAG (m_c/m_s)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 404
    },
    {
      "id": "DATA.WATCHDOG.01",
      "claim": "DATA WATCHDOG / DECISION PIPELINE (2026-06-21): the operational decision layer over the frozen registry (REG.FREEZE.01/v84) [N] (Python-only, external data). [N] reads predictions_frozen.json and classifies all 21 entries (16 predictions + 2 assigned textures + 2 conditional bands + 1 F_transfer bridge: rare kaon) PASS/WATCH/TENSION/KILL by live n_sigma vs repo-documented current bests (CODATA 2022, NuFIT 6.0, ACT DR6, Planck 2018, PDG 2024, BK18, NA62 2016-2024); ratio observables judged against the conservative v18/v100 scheme window. [N] DATA PROVENANCE (2026-06-22 hardening): every measured/bridge source carries a release date (SOURCE_DATES) so the board cannot silently drift onto stale numbers; [N] RARE-KAON BRIDGE: BR(K+)=9.45e-11 lands on NA62 2016-2024 (9.6 +1.9/-1.8)e-11 at +0.08 sigma (was ~1.2 sigma vs the 2016-2022 13.0e-11), an F_transfer bridge readout (not a unique TFPT-vs-SM discriminator). [N] THEORY ALIVE: no decidable prediction in KILL (|n_sigma|<3). [N] POWER + PRESSURE POINT: watch items alpha^-1 (1.9 sigma), s23 (1.8 sigma), delta_CKM (1.0 sigma) and the most-tensioned CORE prediction sin^2 theta13 (2.0 sigma) = the honest current pressure point (DATA.SCORECARD.01/v62). [N] JUNO #1 KILL TEST: the prediction-of-record sin^2 theta12=0.306747 currently compatible (0.02 sigma), the fastest falsifier. [N] FRONTS: r in [0.0033,0.0048] below BK18 0.036 and the kill 0.01 (allowed); n_s band overlaps Planck; w=-1 flagged WATCHDOG (DESI dynamical-DE = the most dangerous negative front). A decision harness over repo-documented data, not a new claim.",
      "status": "Numerical/data [N] (decision pipeline)",
      "canonicalStatus": "[N] the registry-wide classification + theory-alive (no KILL) + the watch/pressure-point identification (sin^2 theta13 ~2 sigma) + the JUNO kill test + the r/n_s/w fronts. An operational decision layer over REG.FREEZE.01; does NOT change any prediction's status.",
      "dependencies": "REG.FREEZE.01; DATA.SCORECARD.01",
      "scripts": [
        "v307_data_watchdog.py"
      ],
      "script": "v307_data_watchdog.py",
      "location": "origin_theory",
      "externalData": "CODATA 2022; NuFIT 6.0; ACT DR6; Planck 2018; PDG 2024; BK18",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 405
    },
    {
      "id": "SEAM.EQUIV.CHAIN.01",
      "claim": "SEAM.EQUIV.01 COMPOSITION CERTIFICATE (2026-06-21, TOE attack point 1): assembles the v297 literature stack into ONE well-typed logical chain gap>0 -> invertible/SRE (Kitaev) -> holomorphic c=8 (Mueger/KLM) -> (E8)_1 (Conway-Sloane), out(i)=in(i+1). [E] arithmetic discriminators exact (sympy): |det Cartan(E8)|=1 (one anyon/primary/invertible/holomorphic) vs |det Cartan(D8=SO(16))|=4 (the same-c rival, NON-holomorphic, excluded); carrier |det D5|*|det A3|=4*4=16 => anyon-condensation tower 16->4->1; c=g_car+N_fam=8. [E] the input gap is the DERIVED recovery gap 6 ln(3/2)>0 with the v76 margin>0. [O] the only undischarged TFPT-internal premise is the cited OS step (OS-reconstructed bulk gap = transfer gap). An assembly/reduction certificate (like v261/v297) + a Lean target; does NOT machine-prove SEAM.EQUIV.01 end-to-end.",
      "status": "Reduction/certificate [E]/[O] (composition)",
      "canonicalStatus": "[E] the exact Cartan-det discriminators + the well-typed four-link composition + the derived-gap input; [O] the cited OS step + the end-to-end machine proof (Lean target). Pins the keystone's open content to one named step; NOT a closure of SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.GAP.01; v297; v301; v302",
      "scripts": [
        "v308_seam_equiv_chain.py"
      ],
      "script": "v308_seam_equiv_chain.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 406
    },
    {
      "id": "QGEO.MODHAM.01",
      "claim": "BEDROCK omega o rho = omega VIA AN EXPLICIT MODULAR HAMILTONIAN (2026-06-21, TOE attack point 2): the next concrete step on the open bedrock QGEO.SYM.01. [E] NON-CIRCULAR upstream: marks=2 chi(S^2)=4 (Gauss-Bonnet, v216) and the order-4 clock z->iz from cross-ratio 2 / j=1728 (v214/v267), so mu4 is DERIVED, not assumed (rho^4=I). [E] for the thermal quasi-free covariance C=1/(1+e^{H1}) the one-particle modular Hamiltonian K=log((1-C)/C) recovered from C round-trips to the seam operator H1 (machine precision). [E] rho C rho^dag=C AND [rho,K]=0 <=> omega o rho = omega, because mu4 curvature couples only modes n-n' in 4Z (i^{n-n'}=1). [E] NEG: a non-mu4 (period-2/Z2) curvature breaks the invariance. [O] residual: the RAW RP-collar C is intrinsically this clock-invariant thermal quasi-free state (Bisognano-Wichmann intrinsicality). Sharpens v198/v210; does NOT close QGEO.SYM.01.",
      "status": "Reduction [E]/[O] (modular bedrock)",
      "canonicalStatus": "[E] the non-circular marks+clock assembly + the explicit modular-Hamiltonian round-trip + [rho,K]=0 state invariance + the Z2 negative control; [O] the Bisognano-Wichmann intrinsicality of the raw collar. The sharpest single open theorem made explicit; NOT a closure.",
      "dependencies": "QGEO.SYM.01; v198; v210; v214; v216",
      "scripts": [
        "v309_modular_bedrock.py"
      ],
      "script": "v309_modular_bedrock.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 407
    },
    {
      "id": "CAR.SM.ANOM.01",
      "claim": "THE CARRIER HALF-SPINOR IS ONE ANOMALY-FREE SM GENERATION + THE SM RG (2026-06-21, TOE attack point 3): connects the discrete compiler to constructive 4D QFT, closing the spectrum+anomaly+beta sub-question exactly. [E] 16=Lambda^even(C^5)=1+10+5; SU(5) c SO(10): 10+5bar+1 = one SM generation (Q,u^c,d^c,L,e^c,nu^c), 16 Weyl states. [E] ANOMALY-FREE: sum Y=0, sum Y^3=0, [SU2]^2U1=0, [SU3]^2U1=0 (exact Fraction arithmetic). [E] one-loop RG from the carrier content (3 gen + 1 Higgs): (b1,b2,b3)=(41/10,-19/6,-7); b1 numerator 41 = the EM-budget integer (v3/v159), b3=-(11-2 n_f/3) with n_f=2 N_fam. [E] NEG: dropping nu^c leaves an incomplete 15!=16 multiplet; a 4th family shifts b1 off 41/10 (N_fam=3 pins the RG). [C]/[O] the full Epstein-Glaser interacting S-matrix (v269/v271/v273) and the Yukawa sector stay open. Closes the spectrum/anomaly/beta layer; not the interacting S-matrix.",
      "status": "Identity [E] (spectrum+anomaly+RG) + [C]/[O] (S-matrix)",
      "canonicalStatus": "[E] the 16=1+10+5 decomposition + the four exact anomaly cancellations + the one-loop (b1,b2,b3)=(41/10,-19/6,-7) + the N_fam=3 RG-pinning negative controls; [C]/[O] the full EG interacting S-matrix + Yukawa. A genuine closure of the carrier->anomaly-free-SM+RG sub-question; does NOT close the interacting theory.",
      "dependencies": "CAR.SM.01; QFT4D.SPERT.03; v273; v159",
      "scripts": [
        "v310_carrier_sm_anomaly.py"
      ],
      "script": "v310_carrier_sm_anomaly.py",
      "location": "tfpt_research_contracts",
      "externalData": "standard GUT representation theory",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 408
    },
    {
      "id": "QGAMB.CLUSTER.01",
      "claim": "THE PHYSICAL QG MEASURE AS A GAP-DECOUPLED, CLUSTERING OBJECT (2026-06-21, TOE attack point 4): the honest statement in place of an over-claimed 4D measure for QG.AMB.01/G_metric. [E] the seam transfer spectrum {1,(2/3)^6,(1/3)^6} gives geometric clustering of the connected correlator at rate (2/3)^6, correlation length xi=1/Delta=1/(6 ln(3/2)). [E] finite susceptibility sum (2/3)^{6n}=1/(1-(2/3)^6)=729/665<inf => a well-defined admissible-sector thermodynamic limit (unique vacuum). [E] gap-decoupling (v76): Delta-31/(4pi^2)~1.648>0, the gap survives metric dressing so the physical sector is DECOUPLED from the un-built ambient. [E] NEG: an ungapped transfer (2nd eig->1) sends xi,chi->inf (no clustering). [C] spectral-action dictionary a2=EH, a4=R^2/72 (v36) + KMS beta=1 (v259). [O] the FULL ambient measure QG.AMB.01 is not built -- gap-decoupled and inert, so the physical sector does not need it.",
      "status": "Reduction [E]/[C]/[O] (gap-decoupled measure)",
      "canonicalStatus": "[E] the geometric clustering at rate (2/3)^6 + finite correlation length/susceptibility + the v76 decoupling margin + the ungapped negative control; [C] the spectral-action low-energy dictionary; [O] the full ambient measure (gap-decoupled, inert). Shows the physical sector is a convergent object; does NOT construct QG.AMB.01.",
      "dependencies": "QG.AMB.01; GATE.METRIC.01; QG.GAPDEC.01; v36; v76; v259",
      "scripts": [
        "v311_gap_decoupled_measure.py"
      ],
      "script": "v311_gap_decoupled_measure.py",
      "location": "tfpt_research_contracts",
      "externalData": "Osterwalder-Schrader / Chamseddine-Connes",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 409
    },
    {
      "id": "HYP.INJECT.01",
      "claim": "THE HYPERGRAPH SUBSTRATE SHARPENED -- WHAT A REWRITE MUST INJECT (2026-06-21, TOE attack point 5): turns the open rewrite-to-full-structure question into a precise list. [E] the affine-E8 Kac marks (1,2,3,4,5,6,4,2,3) are the Perron eigenvector (A.marks=2 marks, top eigenvalue 2) -- the Coxeter skeleton is graph-spectral (v298/v299). [E] the subleading eigenvalue is exactly 2cos(pi/5)=golden ratio phi (the icosahedral (2,3,5)/5-fold signature; spectrum {+-2,+-phi,+-1,+-1/phi,0}). [E] HONEST NEGATIVE: the recovery rate (2/3)^6 is NOT any eigenvalue of A (the lazy-update rate (phi+2)/4=0.9045 != 2/3), so the family-recovery dynamics (gap 6 ln(3/2)) is carried by the cusp weights {0,1/3,2/3}, NOT the adjacency. [E] HONEST NEGATIVE: phi0=1/(6pi)+48c3^4 is analytic, not a rational edge fraction (v298). [O] CONCLUSION: a full-structure rewrite must inject exactly two non-graph-spectral data -- the cusp weight 2/3=|Z2|/N_fam and the analytic seed phi0; the open question is precisely delimited, not closed (NOT a derivation of P1/P2).",
      "status": "Reduction [E]/[O] (substrate sharpening)",
      "canonicalStatus": "[E] the marks-as-Perron + the golden-ratio subleading eigenvalue + the two honest negatives (recovery rate not graph-spectral; seed not an edge fraction); [O] the rewrite-to-full-structure question (now delimited to two injectables). Sharpens v298/v299; does NOT derive P1/P2 from a rewrite.",
      "dependencies": "E8.NETWORK.01; HYP.0D.01; v219; v298; v299",
      "scripts": [
        "v312_hypergraph_substrate.py"
      ],
      "script": "v312_hypergraph_substrate.py",
      "location": "introduction",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 410
    },
    {
      "id": "GOLD.ATOMS.01",
      "claim": "THE GOLDEN RATIO IS THE g_car=5 SIGNATURE -- THE (2,3,5) ATOMS ARE THE NETWORK SPECTRAL ANGLES (2026-06-21, v312 follow-up made exact). [E] EXACT: the affine-E8 adjacency charpoly factors x(x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1); the golden minimal polynomial x^2-x-1 divides it, so phi=2cos(pi/5)=(1+sqrt5)/2 is an EXACT algebraic eigenvalue (not a numerical near-value). [E] ATOMS = SPECTRAL ANGLES: the non-trivial eigenvalues are 2cos(pi/k) for exactly k in {2,3,5}={|Z2|,N_fam,g_car}: 2cos(pi/2)=0, 2cos(pi/3)=1, 2cos(pi/5)=phi, plus the second 5-fold harmonic 2cos(2pi/5)=1/phi -- the golden ratio is SPECIFICALLY the g_car=5 (5-fold) signature. [E] NEW WITNESS for g_car=5: the 5-fold spectral angle pi/5, structurally distinct from Pascal (v2)/RR (v228)/reverse glue (raises the v305 multiplicity). [E] ICOSAHEDRAL SELECTION: 1/|Z2|+1/N_fam+1/g_car=31/30>1 (spherical => finite 2I => E8); numerator 31=2^g_car-1=248/8=1+h(E8) (the v63/v76 gap-margin numerator), denominator 30=h(E8)=|Z2|*N_fam*g_car. HONEST anti-numerology (v305): the four readings of 31 are ONE anchor-derived quantity, a unification/compression NOT new independent evidence. [E] HONEST NULL: phi is NOT a TFPT phenomenological readout (structural lattice/network quantity, kept out of the data grammar). [C] DIRECTION: phi=2cos(pi/5) is the E8-quasicrystal (Elser-Sloane cut-and-project) ratio -> a candidate aperiodic substrate for the v312 question.",
      "status": "Identity/Lattice [E] (spectral) + [C] (quasicrystal direction)",
      "canonicalStatus": "[E] the exact golden-min-poly factorisation + the (2,3,5)=2cos(pi/k) spectral-angle identification + the new g_car spectral witness + the 31/30 icosahedral-selection identity (with the honest anti-numerology framing) + the phi-not-a-readout null; [C] the E8-quasicrystal substrate direction. A new structural reading; does NOT add a physical readout and does NOT close the substrate question.",
      "dependencies": "MCKAY.E8.01; HYP.INJECT.01; v219; v312; v63",
      "scripts": [
        "v313_golden_atoms_spectral.py"
      ],
      "script": "v313_golden_atoms_spectral.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 411
    },
    {
      "id": "COX.COUPLE.01",
      "claim": "COUPLE OR FACTORIZE? THE ORDER-30 COXETER SECTORS COUPLE AS A CYCLOTOMIC FIELD (2026-06-21, v314 follow-up). The v314 question -- does 30=g_car*(2 N_fam)=5*6 couple the carrier (5-fold, golden, Q(sqrt5)) and family (6=2*3, rational) sectors, or just factorize? [E] GROUP FACTORIZES: gcd(5,6)=1 => Z/30 = Z/5 x Z/6 (CRT, direct product); order-5 subgroup <c^6>, order-6 <c^5> -- group-level decoupled (consistent with the v314 field obstruction). [E] EXPONENTS FACTOR: the E8 exponents = totatives(30) split as totatives(5) x totatives(6); phi(30)=phi(5)phi(6)=4*2=8=rank E8; (m mod5, m mod6) is a bijection onto {1,2,3,4}x{1,5}. [E] FIELD COUPLES: [Q(zeta30):Q]=phi(30)=8=rank E8; Q(zeta30) is the COMPOSITUM of the carrier field Q(zeta5) (contains the golden sqrt5) and the family field Q(zeta6)=Q(zeta3) (contains zeta3). [E] GALOIS = mu4 x Z2: (Z/30)^x=(Z/5)^x x (Z/3)^x=Z/4 x Z/2 (order 8, max element order 4, NOT cyclic Z/8); mu4=(Z/5)^x is the CARRIER Galois, Z2=(Z/3)^x the FAMILY Galois -- so the P2 seam clock mu4 IS the Galois group of the carrier's golden field (refines v223). [E] CARRIER GENERATOR: sqrt5=phi+1/phi=2 phi-1 (the sum of the two golden network eigenvalues) = the quadratic Gauss sum zeta5-zeta5^2-zeta5^3+zeta5^4 in Q(zeta5), g^2=5. [C] VERDICT: 30=5*6 does NOT dynamically entangle the sectors (direct product + the v314 field obstruction) -- it couples them as the cyclotomic compositum Q(zeta30), Galois mu4 x Z2, degree rank E8; the carrier (sqrt5) and family (zeta3) are conjugate sub-data of ONE cyclotomic field, bridged by GALOIS not by a dynamical map.",
      "status": "Identity/Lattice [E] (cyclotomic Galois) + [C] (verdict)",
      "canonicalStatus": "[E] the CRT direct-product factorisation + the exponent/totative factorisation + the Q(zeta30) compositum + the Galois=mu4 x Z2 identification (seam clock = carrier-field Galois) + the Gauss-sum carrier generator; [C] the couple-vs-factorize verdict. Answers the v314 coupling question exactly (Galois compositum, not a dynamical map); does NOT add a physical readout.",
      "dependencies": "RATE.TRANSLATE.01; COX.TOTATIVE.01; v314; v223; v55",
      "scripts": [
        "v315_coxeter_coupling.py"
      ],
      "script": "v315_coxeter_coupling.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 412
    },
    {
      "id": "GALOIS.READOUT.01",
      "claim": "DOES THE GALOIS mu4 x Z2 (v315) ORGANIZE THE PHYSICAL READOUTS? YES -- THE CP PHASES (2026-06-21, v315 follow-up to phenomenology). [E] FAMILY FACTOR: the hexagonal CP unit rho=zeta_6=zeta_30^5 is the order-6=2 N_fam factor c^5 of v315 (5=30/6); the carrier is zeta_5=zeta_30^6 (order 5=g_car, c^6) -- the CP phase lives in the FAMILY factor, not the carrier. [C] CP PHASES (inherits v231/v233): delta_CKM,lead=pi/3=arg(zeta_6), delta_PMNS=4pi/3=arg(zeta_6^4) (the power 4=|mu4|, the seam clock). [E] SHEET FLIP: zeta_6^4=-zeta_6 (rho^3=-1), so delta_PMNS=delta_CKM+pi (the |Z2| sheet); mu6=mu3 x mu2 (family x sheet), zeta_6=-zeta_3^2. [E] GALOIS Z2 = CP CONJUGATION: Gal(Q(zeta_6)/Q)=(Z/6)^x={1,5}=Z2, sigma: zeta_6->zeta_6^5=conj(zeta_6) (delta->-delta) -- the FAMILY Galois factor of mu4 x Z2 IS the physical CP-conjugation symmetry. [E] CARRIER STATIC: zeta_5=zeta_30^6 (the golden sqrt5) carries NO phase readout -- mu4=(Z/5)^x organizes the lattice/spectrum, not a CP phase. [E] HONEST SEPARATION: the magnitudes (sin^2 theta12=1/3-phi0/2, phi0=1/(6pi)+48c3^4 transcendental) are NOT in any Q(zeta_n) -- Galois organizes PHASES (cyclotomic), the magnitudes are the analytic seed. [C] VERDICT: the v315 Galois coupling reaches SM phenomenology through the FAMILY (order-6) factor -- the CP phases are its cyclotomic data (rho=zeta_6=zeta_30^5, mu6=mu3 x mu2, Gal Z2=CP conjugation); the carrier (mu4/golden) factor is static; the magnitudes are the analytic seed (v314 field split).",
      "status": "Identity/Lattice [E] (cyclotomic CP) + [C] (physical identification)",
      "canonicalStatus": "[E] the family-factor placement of the CP unit zeta_6=zeta_30^5 + the sheet flip + the Galois-Z2 = CP-conjugation identification + the carrier-static fact + the magnitude/seed separation; [C] the identification with the physical delta_CKM/delta_PMNS (inherits v231/v233). Bridges the v315 Galois coupling to SM phenomenology (the phase sector); does NOT upgrade the CP-phase typing.",
      "dependencies": "COX.COUPLE.01; CP.TRIALITY.01; v315; v231; v233",
      "scripts": [
        "v316_galois_readout.py"
      ],
      "script": "v316_galois_readout.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 413
    },
    {
      "id": "GALOIS.FAMILY.01",
      "claim": "ARE THE 3 GENERATIONS A mu3/GALOIS ORBIT? YES (mu3), REFINED 1+2 BY GALOIS (2026-06-21, v316 follow-up to the generation structure). [E] GENERATIONS = CUBE ROOTS: the 3 cusp weights {0,1/3,2/3} (Spec(Q+)=3a+1={1,2,3}=A3 exponents) map under w->e^{2 pi i w} to the 3rd roots of unity {1,zeta_3,zeta_3^2}. [E] mu3 ORBIT (transitive): the cyclic family symmetry w->w+1/3 mod 1 permutes them cyclically -- ONE transitive Z/3=mu3 orbit (the family IS the cube-root orbit). [E] RECOVERY: the transfer eigenvalues (1-w)^6={1,(2/3)^6,(1/3)^6} attach to the weights; the attractor (eigenvalue 1, the stationary law, v56) is the w=0 weight. [E] GALOIS REFINEMENT 1+2: Gal(Q(zeta3)/Q)=Z2, sigma: zeta_3->zeta_3^2 acts on weights by w->2w mod 1, FIXING w=0 (rational) and SWAPPING w=1/3<->w=2/3 -- so the Galois orbit is 1 (fixed) + 2 (pair), NOT transitive. [E] FIXED = ATTRACTOR: the Galois-fixed w=0 <-> transfer eigenvalue 1 (the law); the Galois-conjugate pair w=1/3,2/3 <-> the decaying modes (2/3)^6,(1/3)^6 (the hierarchy) -- selection = the rational generation. [E] SAME Z2 AS CP CONJUGATION (v316): sigma: zeta_3->zeta_3^2=conj(zeta_3). [E] COUNTS: N_fam=3=|mu3|=#weights=|Spec(Q+)| (A3 exponents {1,2,3}). [C] VERDICT: the 3 generations ARE a single mu3 (cyclic family) orbit {1,zeta_3,zeta_3^2}; the Galois group refines it to 1 (rational=the recovery attractor) + 2 (conjugate pair=the hierarchy), that Z2 = the v316 CP conjugation; the family STRUCTURE is arithmetic, the mass MAGNITUDES stay the analytic seed (NOT derived).",
      "status": "Identity/Lattice [E] (mu3+Galois orbit) + [C] (verdict)",
      "canonicalStatus": "[E] the cusp-weights = cube-roots identification + the transitive mu3 orbit + the recovery-eigenvalue attachment + the Galois 1+2 refinement (fixed=attractor, pair=hierarchy) + the same-Z2-as-CP-conjugation + the counts; [C] the verdict. Shows the family STRUCTURE is the same arithmetic (mu3 orbit, Galois-refined) as the CP phase; does NOT derive the mass magnitudes (analytic seed).",
      "dependencies": "GALOIS.READOUT.01; COX.COUPLE.01; v316; v315; v56",
      "scripts": [
        "v317_galois_family.py"
      ],
      "script": "v317_galois_family.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 414
    },
    {
      "id": "ARITH.CAPSTONE.01",
      "claim": "THE CAPSTONE: THE SM STRUCTURAL SECTOR IS Q(zeta30) + GALOIS mu4 x Z2, COMPLETE INPUT {a, pi, v_geo} (2026-06-21, synthesis of the v313-v317 arc). [E] ARITHMETIC SKELETON: the SM STRUCTURAL sector (3 generations, 2 CP phases, the orbit/hierarchy ordering) lives in the cyclotomic field Q(zeta30): [Q(zeta30):Q]=phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=h(E8), Galois (Z/30)^x=mu4 x Z2 (order 8) -- carrier mu4=(Z/5)^x (golden sqrt5, v313/v315), family/sheet Z2=(Z/3)^x (zeta3=CP conjugation, v316); the 3 generations = the mu3 cube-root orbit (v317), the 2 CP phases = the zeta6 family-factor data (v316). [E] RIGIDITY: forced by the anchor atoms {|Z2|,N_fam,g_car}={2,3,5}=elementary-symmetric data of a=(1,1,2); 30=2*3*5 -- no freedom in the skeleton. [E] MAGNITUDE REDUCTION: the seed phi0=(|mu4|/N_fam)c3+Omega_adm c3^4=(4/3)c3+48 c3^4 (c3=1/(8pi)) is a PURE function of pi (no free parameters), coefficients anchor-derived (4/3=|mu4|/N_fam, 48=Omega_adm=N_fam*dim_Splus); so the magnitudes (sin^2 theta12=1/3-phi0/2) are {a,pi}-determined, transcendental (NOT cyclotomic) but NOT free. [E] FREE INVENTORY: dimensionless free parameters = 0 (everything is {a,pi}); pi the unique transcendental primitive (ARCH.CORE.01/v53); v_geo the unique dimensionful input (No-Unit Theorem v153/v78) -- complete input = {a, pi, v_geo}. [O] HONEST RESIDUAL: this rigidity holds GIVEN the axioms; it does NOT prove the raw seam REALIZES the structure -- the open bedrock QGEO.SYM.01/SEAM.EQUIV.01 stays [O]. VERDICT: SM structural sector = Q(zeta30)+Galois mu4 x Z2 (rigid, anchor-forced, degree=rank E8); magnitudes = {a,pi}-transcendental (phi0); the only genuinely free datum is v_geo; dimensionless free parameters: 0.",
      "status": "Synthesis/inventory [E] (arithmetic + {a;pi} reduction) + [O] (seam realisation)",
      "canonicalStatus": "[E] the Q(zeta30)/Galois=mu4 x Z2 skeleton (degree=rank E8) + the anchor-rigidity + the phi0={a,pi} reduction (pure pi, anchor-derived coefficients) + the 0-dimensionless-free-parameter inventory ({a,pi,v_geo}); [O] the seam realisation. A synthesis/inventory capstone of the v313-v317 arc; does NOT close the open bedrock and does NOT add a physical readout.",
      "dependencies": "GALOIS.FAMILY.01; GALOIS.READOUT.01; COX.COUPLE.01; ARCH.CORE.01; NOUNIT.01; v313; v314; v315; v316; v317",
      "scripts": [
        "v318_arithmetic_capstone.py"
      ],
      "script": "v318_arithmetic_capstone.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 415
    },
    {
      "id": "RATE.TRANSLATE.01",
      "claim": "DO THE DISCRETE-KERNEL AND DYNAMIC RATES TRANSLATE? AN EXACT NUMBER-FIELD SPLIT (2026-06-21, v312/v313 follow-up). [E] FIELD SPLIT: the affine-E8 adjacency spectrum is the rational part {0,+-1,+-2} in Q plus the golden part {+-phi,+-1/phi} in Q(sqrt5) (min poly x^2-x-1, discriminant 5); the dynamic transfer spectrum {1,(2/3)^6,(1/3)^6} is ENTIRELY rational (in Q). [E] TRANSLATION (the (2,3)-fold/family sector): the rational discrete angles 2cos(pi/2)=0 (|Z2|=2) and 2cos(pi/3)=1 (N_fam=3) are the SAME atoms that build the recovery rate (2/3)^6=(|Z2|/N_fam)^(2 N_fam) in Q -- same field, same atoms, so the family/sheet rates DO translate. [E] NO-TRANSLATION (the 5-fold/carrier sector): phi in Q(sqrt5) minus Q has NO rational dynamic image (2/3 is neither an adjacency nor a lazy-update eigenvalue; heat-kernel t=Delta/(2-phi)~6.37 not clean) -- the carrier 5-fold is STATIC-only (builds the lattice/spectrum, not a dynamic rate). [E] SHARED CLOCK: both are facets of the order-30 Coxeter element, 30=h(E8)=g_car*(2 N_fam)=5*6; the golden in the order-5 factor, the recovery exponent 6=2 N_fam=30/g_car in the order-6=2*3 factor. [E] QUASICRYSTAL: E8=H4+phi H4 (240=2*120, 120=|2I|=|R+(E8)|), so the phi cut-and-project carries the 5-fold carrier (Q(sqrt5)) but the cusp weights {0,1/3,2/3} are RATIONAL (3-fold), a separate parabolic datum -- the substrate carries the carrier, NOT the family dynamics (confirms v313/v312). [C]/[O] VERDICT: a static<->dynamic translation functor exists ONLY on the rational (2,3-fold/family) sector; the carrier 5-fold is field-obstructed (static-only), the bridge holding both is the order-30 Coxeter element.",
      "status": "Identity/Lattice [E] (number-field split) + [C]/[O] (functor verdict)",
      "canonicalStatus": "[E] the Q vs Q(sqrt5) field split of the discrete spectrum + the dynamic-in-Q fact + the (2,3)-sector translation + the 5-fold field obstruction (heat-kernel + eigenvalue negatives) + the order-30 shared clock + the quasicrystal counts; [C]/[O] the translation-functor verdict. Answers the discrete<->dynamic translation question exactly (a field split); does NOT close the substrate question.",
      "dependencies": "GOLD.ATOMS.01; SEAM.EQUIV.GAP.01; v313; v54; v56",
      "scripts": [
        "v314_rate_translation.py"
      ],
      "script": "v314_rate_translation.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 416
    },
    {
      "id": "TRANSLATE.CLOCK.01",
      "claim": "THE TRANSLATION CLOCK: THE DISCRETE<->DYNAMIC BRIDGE IS THE ORDER-30 COXETER ELEMENT, A CLOCK WITH TWO COPRIME HANDS 5 x 6 (2026-06-22, synthesis of the v313-v318 arc + v124/v55, answering 'is the static<->dynamic translation a clock running 0,1,2,3,4,5 or 1,2,3,4,5?'). [E] TRANSLATION IS A CLOCK: the only object holding both the static carrier (Q(sqrt5)) and the dynamic family (rational) data is the order-30 Coxeter element (v314); 30=h(E8)=g_car*(2 N_fam)=5*6, gcd(5,6)=1 => Z/30=Z/5 x Z/6 (two coprime hands, v315). [E] DYNAMIC HAND RUNS 0..5: the dynamic factor is Z/6=2 N_fam (ticks {0,1,2,3,4,5}); the recovery exponent is EXACTLY 6=|Z/6|=2 N_fam, rate (2/3)^6=(|Z2|/N_fam)^(2 N_fam); the order-6 generator is c^5 (zeta_6=zeta_30^5, the hexagonal CP unit, v316). [E] POSITION 0=THE CONSERVED LAW, 1..=THE LIVE PHASES: the resummed translation clock rate(n)=-p2 ln(1-n/N_fam) (v124) has rate(0)=0 (the attractor/stationary law, eigenvalue 1, v56) and live modes rate(1)=6 ln(3/2), spectrum {1,(2/3)^6,(1/3)^6}; the E8 live phases (Coxeter exponents=totatives of 30) start at 1 ({1,7,...,29}), the 0 phase is NOT a totative -- so '0,1,2,3,4,5' is the law-inclusive labelling, '1,2,3,4,5' the live-only labelling. [E] STATIC HAND RUNS 1..5: the static factor is Z/5=g_car (ticks {1,2,3,4,5}); golden field Q(sqrt5) (phi=2cos(pi/5), min poly x^2-x-1, disc 5), field-obstructed (no rational dynamic image, v314) -- static-only; zeta_5=zeta_30^6 carries no phase (v316). [E] ONE FULL TURN=30: lcm(5,6)=30=h(E8)=the full Coxeter period; rank E8=8=phi(30)=#live phases pairing m+(30-m)=30 into |mu4|=4 invariant planes (v55). [C] VERDICT: the discrete->dynamic translation IS the order-30 clock = (static 5-ring) x (dynamic 6-ring); the dynamic hand {0,...,5} carries the rate (exponent 6=2 N_fam), the static hand {1,...,5} the golden carrier (no rate); '0 vs 1' = law-inclusive vs live-only.",
      "status": "Synthesis/Identity [E] (clock factorization + hand arithmetic) + [C] (one-clock reading)",
      "canonicalStatus": "[E] the order-30 = 5x6 coprime-hand factorization + the dynamic-hand arithmetic (Z/6=2 N_fam, exponent 6, rate (2/3)^6, zeta_6=zeta_30^5) + the law-vs-live (0 vs 1) reading (rate(0)=0 the attractor, totatives start at 1) + the static-hand golden field-obstruction (Z/5=g_car, no rate) + the one-full-turn=30 Coxeter period; [C] the 'the bridge IS one clock' interpretation. A synthesis of the v313-v318 arc + v124/v55 into the clock picture; does NOT add a physical readout and does NOT close the open bedrock.",
      "dependencies": "RATE.TRANSLATE.01; COX.COUPLE.01; ARITH.CAPSTONE.01; v314; v315; v124; v55",
      "scripts": [
        "v319_translation_clock.py"
      ],
      "script": "v319_translation_clock.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 417
    },
    {
      "id": "GALOIS.CP.PREDICT.01",
      "claim": "A NEW FALSIFIABLE PREDICTION FROM THE GALOIS STRUCTURE: THE TWO CP PHASES ARE LOCKED, delta_PMNS = delta_CKM,lead + pi (2026-06-22, the v316 follow-up turned testable). [E] ONE UNIT TWO POWERS: both CP phases are powers of one hexagonal unit rho=zeta_6 in the family Galois factor (v316): delta_CKM,lead=arg(rho)=pi/3 (60 deg), delta_PMNS=arg(rho^4)=4pi/3 (240 deg), rho^4=-rho. [E] FORCED RELATION: delta_PMNS-delta_CKM,lead=pi exactly (the Z2 sheet flip rho^3=-1), equivalently delta_PMNS=4*delta_CKM,lead -- the quark and lepton leading CP phases are NOT independent (one is fixed by the other). [C] PREDICTION OF RECORD: delta_PMNS,lead=240 deg, forced from delta_CKM,lead=60 deg + the sheet flip (180 deg) -- upgrades delta_PMNS from an assigned texture to a Galois-forced relation to the measured quark phase. [N] CURRENTLY COMPATIBLE: 240 deg lies within the broad current global-fit range for the PMNS Dirac phase (NuFIT 6.0, normal ordering); decisive test = DUNE/Hyper-K/JUNO. [X] KILL TEST: a measured delta_PMNS robustly incompatible with delta_CKM,lead+pi=240 deg (beyond the sub-leading budget, at >3 sigma) falsifies the Galois-CP organization (v316/v317). A genuine NEW falsifiable cross-prediction from the v313-v319 arithmetic arc; the leading-order/holonomy reading is [C] like delta_CKM, the relation itself [E], the kill test [X]; does NOT derive the magnitudes.",
      "status": "Identity [E] (Galois relation) + [C] (prediction) + [X] (kill test)",
      "canonicalStatus": "[E] the exact Galois relation delta_PMNS=delta_CKM,lead+pi (rho^4=-rho, both powers of zeta_6) + the non-triviality; [C] the leading-order prediction delta_PMNS,lead=240 deg; [N] current compatibility; [X] the pre-registered kill test (DUNE/Hyper-K/JUNO). A NEW falsifiable cross-prediction from the Galois arc; does NOT derive magnitudes and does NOT upgrade the CP-phase typing beyond the relation.",
      "dependencies": "GALOIS.READOUT.01; ARITH.CAPSTONE.01; v316; v231; v233",
      "scripts": [
        "v320_galois_cp_relation.py"
      ],
      "script": "v320_galois_cp_relation.py",
      "location": "origin_theory",
      "externalData": "NuFIT 6.0 (PMNS delta_CP)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 418
    },
    {
      "id": "KILL.BOARD.01",
      "claim": "THE FORWARD KILL-TEST BOARD: THE DECISIVE UPCOMING MEASUREMENTS, RANKED (2026-06-22, forward companion to v307's current-data board). [N] assembles the decisive NEXT measurements, each LOCKED to a freeze_file.csv kill row (so the test and the frozen signal cannot drift apart): #1 JUNO sin^2 theta12=0.306747 (fastest, sub-percent), #2 CMB-S4/LiteBIRD r in [0.0033,0.0048] (kill robust r>0.01), #3 DESI/Euclid w=-1 (kill robust w!=-1), #4 DUNE/Hyper-K delta_PMNS=240deg=delta_CKM,lead+pi (the NEW v320 Galois-CP relation; kill if away from 240 deg at >3 sigma), #5 DUNE/LEGEND/nEXO normal ordering + small m_betabeta, #6 PSI n2EDM theta_eff=0, #7 JUNO sin^2 theta13=0.0231 (now ~2 sigma), #8 LiteBIRD/Simons beta=0.2424 deg. [N] COVERAGE: every entry backed by a freeze_file kill row; the v320 CP relation is on the board; JUNO theta12 rank 1; >=7 decisive tests each with a concrete kill threshold. A forward decision ledger (timelines/reach are the publicly-stated experimental targets, not TFPT outputs); complements v307; no new physics.",
      "status": "Numerical/data [N] (forward decision ledger)",
      "canonicalStatus": "[N] the ranked forward board (each kill test locked to a freeze_file row) + the v320 CP-relation entry + the JUNO-rank-1 ordering + the power check. A forward decision ledger complementing v307; does NOT add a physical readout.",
      "dependencies": "DATA.WATCHDOG.01; GALOIS.CP.PREDICT.01; REG.FREEZE.01",
      "scripts": [
        "v321_killtest_board.py"
      ],
      "script": "v321_killtest_board.py",
      "location": "origin_theory",
      "externalData": "JUNO; CMB-S4; LiteBIRD; DESI; Euclid; DUNE; Hyper-K; LEGEND; nEXO; PSI n2EDM",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 419
    },
    {
      "id": "GALOIS.CP.SHARPEN.01",
      "claim": "THE v320 CP-LOCK PREDICTION SHARPENED INTO A QUANTITATIVE PRE-REGISTERED KILL TEST (2026-06-22). [E] DISCRETE LADDER: the leading CP phase sits on the six hexagonal nodes arg(rho^k)=k*60 deg; delta_CKM,lead=60 (node 1), delta_PMNS,lead=240 (node 4). [E] MULTIPLIER = DECK ORDER: delta_PMNS,lead=|mu4|*delta_CKM,lead=4*(pi/3)=4pi/3=delta_CKM,lead+pi (rho^4=-rho) -- the node selector IS the seam clock order |mu4|=4. [C] SUB-LEADING BUDGET: delta_CKM,full=pi/3+3 lambda^2 (~68.7 deg); the lepton sub-leading is a SEPARATE open holonomy term bounded by ~3 lambda^2~8.7 deg => band 240 +- ~9 deg. [N] CURRENT TENSION: vs NuFIT 6.0 NO (delta_CP=212 +26/-41 deg) the leading 240 deg sits at +1.08 sigma, band lower edge ~231 deg at +0.74 sigma -- compatible today. [X] SHARPENED KILL: the nearest wrong node (180/300) is 60 deg away, far beyond the budget; DUNE/Hyper-K at ~5-15 deg discriminate the node, >3 sigma outside kills the lock. [E] DISCRIMINATION POWER: an anarchic delta_CP hits the band with p~5% and node 4 (not 1/3/5) 1-in-6. Sharpens v320; no magnitudes derived.",
      "status": "Identity [E] (node+multiplier) + [C] (budget) + [N] (tension) + [X] (kill test)",
      "canonicalStatus": "[E] the discrete hexagonal node + the |mu4| multiplier + the discrimination geometry; [C] the sub-leading budget (the lepton holonomy term is only bounded by the quark analog, not closed); [N] the +1.08 sigma current tension; [X] the sharpened pre-registered kill test (node discrimination). Sharpens v320; does NOT derive magnitudes or upgrade the CP typing.",
      "dependencies": "GALOIS.CP.PREDICT.01; GALOIS.READOUT.01; v320; v316",
      "scripts": [
        "v322_cp_lock_sharpened.py"
      ],
      "script": "v322_cp_lock_sharpened.py",
      "location": "origin_theory",
      "externalData": "NuFIT 6.0 (PMNS delta_CP)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 420
    },
    {
      "id": "QGEO.BW.01",
      "claim": "THE KEYSTONE PUSHED ONE STEP -- BISOGNANO-WICHMANN MAKES THE SEAM MODULAR FLOW GEOMETRIC (2026-06-22): the mu4 deck invariance is NOT an independent axiom. [E] CLOCK = GEOMETRIC ROTATION: rho=diag(i^n)=exp(i(pi/2)L), L=diag(n) the seam rotation generator; rho^4=exp(i2pi L)=I -- mu4 in U(1)_rot (quarter turns). [E] BW: a rotation-covariant covariance C=f(L) gives K=log((1-C)/C)=g(L), so [K,L]=0 (the modular flow commutes with all rotations) AND [rho,K]=[rho,C]=0 -- the mu4 clock is a modular symmetry FOR FREE (no fine-tuned curvature, unlike v309). [E] DISCRIMINATOR vs v309: a period-4 curvature preserves the clock ([rho,K]=0) but its modular flow is NOT geometric ([K,L]!=0) -- BW is strictly stronger (pins the whole rotation group, not just mu4). [E] NEG: a non-rotation-covariant C breaks [rho,K]!=0. [C] LINKED BEDROCK: GIVEN the seam is the (E8)_1 chiral net (v308/SEAM.EQUIV.01), Bisognano-Wichmann/Hislop-Longo make the vacuum modular flow geometric => QGEO.SYM.01 (omega o rho=omega) is DOWNSTREAM of SEAM.EQUIV.01 + a rotation-invariant vacuum, NOT an independent axiom (two open items collapse toward one). [O] residual: the raw collar vacuum is rotation-invariant (cleaner than v309's bespoke thermal state, shared with v308). Sharpens v309/v198; does NOT close QGEO.SYM.01.",
      "status": "Reduction [E]/[C]/[O] (BW geometric modular flow)",
      "canonicalStatus": "[E] the finite-model BW construction (rotation-covariant C => geometric modular flow + a free clock symmetry) + the v309 discriminator (period-4 preserves the clock but is not geometric); [C] the linkage to v308 via the cited BW/Hislop-Longo theorem (QGEO.SYM.01 downstream of SEAM.EQUIV.01); [O] rotation-invariance of the raw seam. Sharpens v309; does NOT close the bedrock.",
      "dependencies": "QGEO.SYM.01; QGEO.MODHAM.01; SEAM.EQUIV.CHAIN.01; v308; v309; v198",
      "scripts": [
        "v323_bw_geometric_modular.py"
      ],
      "script": "v323_bw_geometric_modular.py",
      "location": "origin_theory",
      "externalData": "Bisognano-Wichmann / Hislop-Longo (cited)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 421
    },
    {
      "id": "HYP.FIBER.01",
      "claim": "THE MINIMAL HYPERGRAPH SUBSTRATE THAT WORKS -- A (2,3,5)-NETWORK FIBERED BY A 3-FOLD CUSP CHANNEL (2026-06-22, the constructive v312 follow-up): reproduces BOTH the E8 skeleton AND the recovery gap. [E] NETWORK FACTOR (carrier/5-fold): T_net=(A+2I)/4 fixes the Kac marks (eigenvalue 1 = E8 skeleton), subleading (phi+2)/4 (golden); (2/3)^6 NOT in its spectrum (the v312 negative). [E] CUSP FIBER (family/3-fold): T_cusp=diag((1-w)^{2N_fam}), w in {0,1/3,2/3} (N_fam=3, v317), spectrum {1,(2/3)^6,(1/3)^6}, recovery gap at w=1/3 (v76/v221). [E] FIBERED SUBSTRATE CARRIES BOTH: T=T_net (x) T_cusp has top eigenvalue 1 (eigenvector marks (x) w=0 = E8 skeleton x democratic cusp) AND (2/3)^6 as a genuine eigenvalue (network attractor x cusp subleading). [E] HONEST INJECTION (v312): without the fiber (T_net (x) I_3) there is NO (2/3)^6 -- the cusp datum is injected, not graph-spectral; substrate = network x cusp = the v315 split 30=g_car*(2 N_fam)=5*6, the gap living entirely in the family factor. [E] MINIMALITY: the smallest fiber is the 3-node cusp (=N_fam); a 2-node fiber cannot hold 3 weights -> minimal substrate = (2,3,5)-net x 3-cusp = order-30 (v319 clock). [C] VERDICT: a CONSISTENT fibered realization (carrier x family) matching the arithmetic 5x6 split, but the cusp weight stays injected (v312) -- not a derivation of 2/3 from a pure rewrite. Sharpens v312; the open rewrite question is unchanged.",
      "status": "Construction [E]/[C] (fibered substrate)",
      "canonicalStatus": "[E] the fibration construction (T_net (x) T_cusp carries the skeleton + the gap) + the v312 negative (no gap without the fiber); [C] the 'minimal substrate = carrier x family' reading matching the v315 5x6 split; [O] deriving the cusp weight from a pure rewrite stays open. Sharpens v312; not a closure.",
      "dependencies": "HYP.INJECT.01; COX.COUPLE.01; GALOIS.FAMILY.01; v312; v315; v317; v319; v221; v76",
      "scripts": [
        "v324_hypergraph_fiber.py"
      ],
      "script": "v324_hypergraph_fiber.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 422
    },
    {
      "id": "FORM.SEAMEQUIV.01",
      "claim": "LEAN: SEAM.EQUIV.01 COMPOSITION CHAIN (2026-06-22). Lean 4 formalisation (TfptCarrier/SeamEquivChain.lean) of the v308 closing chain: the four nodes GapPositive -> InvertibleSRE -> HolomorphicC8 -> SeamIsE8 assembled as a well-typed composition theorem seamEquivChain, with the three cited literature steps (Kitaev free-fermion invertibility; Mueger / Kawahigashi-Longo-Mueger holomorphy; Conway-Sloane E8 uniqueness) declared as named axioms and the carrier recovery gap (the OS step) as the single open input. #print axioms seamEquivChain confirms the residual is EXACTLY those named steps + recovery_gap (no hidden assumption). Plus a decide-proven discriminator: primariesE8=1 != primariesD8=4 (|det Cartan| distinguishes the holomorphic E8 from the same-central-charge rival D8=SO(16)). Built end-to-end (lake build, 3357 jobs) on Lean v4.29.1 + Mathlib. The VALUE is the machine-checked composition + the axiom audit (which pins the keystone residual to the named cited theorems + the one OS input), NOT a from-scratch proof of the cited steps -- so it does NOT close SEAM.EQUIV.01.",
      "status": "Formal",
      "canonicalStatus": "[E] Lean-built: the well-typed composition seamEquivChain + the axiom audit (residual = the named cited theorems + the one OS input) + the decide-proven discriminator (1 != 4). A formal audit-contract skeleton of the v308 keystone chain; the three literature steps are named axioms (cited, not re-proved), so it MACHINE-PINS the residual but does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.CHAIN.01; SEAM.EQUIV.01; SEAM.EQUIV.GAP.01",
      "scripts": [
        "v308_seam_equiv_chain.py"
      ],
      "script": "v308_seam_equiv_chain.py",
      "location": "lean4-carrier-rigidity/TfptCarrier/SeamEquivChain.lean",
      "externalData": "Lean 4 + Mathlib (v4.29.1; #print axioms = named cited steps + OS gap)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 423
    },
    {
      "id": "QGEO.KEYSTONE.01",
      "claim": "THE SEAM KEYSTONE AS ONE CITABLE THEOREM (2026-06-22): the raw seam state is the flat tau=i pillowcase state, with EXACTLY ONE open lemma. Assembles the keystone pieces into one statement -- IF (raw RP-collar seam state = flat tau=i pillowcase / rotation-invariant state) THEN 4 square marks + mu4 deck + omega o rho=omega + holomorphic c=8 => (E8)_1, with the carrier recovery gap as input. [E] FOUR SQUARE MARKS: n=2chi(S^2)=4 (Gauss-Bonnet, v216); cross-ratio 2 => j=1728 (order-4 CM, v214/v267). [E] FLAT PILLOWCASE: S^2(2,2,2,2), chi_orb=2-4(1-1/2)=0 => flat tau=i (Troyanov). [E] MU4 FINGERPRINT: DtN sub-principal at m=0 mod 4 (v201/v280). [E] HOLOMORPHY SELECTOR: det Cartan(E8)=1 vs D8=4 (v83/v143/v308). [E] INPUT GAP: Delta=6 ln(3/2)>0 (v76/v302). [O] ONE OPEN LEMMA: the raw collar state IS the flat tau=i pillowcase / rotation-invariant state, SHARED by both bedrock IDs (QGEO.SYM.01, SEAM.EQUIV.01; v323), machine-pinned in Lean (FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). An assembly/reduction certificate (like v176/v261), NOT a closure.",
      "status": "Synthesis/reduction [E]/[O] (keystone-as-one-theorem)",
      "canonicalStatus": "[E] the assembled keystone theorem with all exact pieces machine-checked (marks=4, j=1728, chi_orb=0, det Cartan E8=1, gap>0, mod-4 fingerprint); [O] the single open lemma (raw seam = flat tau=i pillowcase / rotation-invariant state) shared by both bedrock IDs and Lean-pinned. An assembly certificate (like v176/v261), does NOT close the bedrock.",
      "dependencies": "QGEO.SYM.01; SEAM.EQUIV.01; QGEO.BW.01; v214; v216; v280; v308; v323; v76",
      "scripts": [
        "v325_pillowcase_keystone.py"
      ],
      "script": "v325_pillowcase_keystone.py",
      "location": "origin_theory",
      "externalData": "Troyanov uniformisation (cited)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 424
    },
    {
      "id": "FR.SUITE.01",
      "claim": "THE F_transfer SOLVER SUITE (2026-06-22): the four transfer readouts as ONE runnable harness, each a gapped relaxation to a unique attractor, with values + uncertainties + honest typing (productises FR.DYNAMICS.01/v303). [E] F_pole (Koide): Moebius contraction to Q*=2/3 with multiplier (2/3)^6 = the seam-clock subleading eigenvalue, unique attractor at the SEAM rate. [C] F_Boltzmann (eta_B): washout contraction to the balance 6.1e-11, monotone (H-theorem), external thermal rate. [C] F_relic (axion): the comoving number N=E/omega is the adiabatic invariant, the relic freezes (seed theta_i=3pi/5), cosmological rate. [I] F_QCD (m_p/m_e): 1-loop RG with carrier b0=7 flows to the Gaussian UV fixed point, Lambda_QCD generated; standard RG. [N] F_RareKaon (bridge): BR(K+)=9.45e-11 on NA62 2016-2024 (v202). [O] SHARED SHAPE: all 5 are gapped relaxations to a unique attractor; only F_pole has the seam rate (2/3)^6, the other 4 share the SHAPE with external rates honestly fenced (v187/v303) -- the suite makes the frontier measurable WITHOUT claiming the external rates.",
      "status": "Numerical/synthesis [E]/[C]/[I] (F_transfer solver suite)",
      "canonicalStatus": "[E] the F_pole seam-rate contraction; [C] the Boltzmann/relic relaxations (shape, external rates); [I] the QCD RG flow; [N] the rare-kaon bridge. A productisation/solver harness of v303, NOT a new claim; the external rates stay honestly fenced.",
      "dependencies": "FR.DYNAMICS.01; v303; v82; v184; v185; v202",
      "scripts": [
        "v326_ftransfer_suite.py"
      ],
      "script": "v326_ftransfer_suite.py",
      "location": "origin_theory",
      "externalData": "NuFIT/PDG/NA62 (data targets)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 425
    },
    {
      "id": "HYP.REWRITE.01",
      "claim": "DERIVING THE CUSP WEIGHT 2/3 FROM A MINIMAL REWRITE (2026-06-22, sharpening HYP.FIBER.01/v324 + HYP.INJECT.01/v312). A minimal local rule -- a family channel with N_fam=3 slots, one absorbing attractor (w=0) and |Z2|=2 surviving (the Z2 pair) -- supplies the cusp fiber. [E] MINIMAL REWRITE: M=[[1,0,0],[0,1/3,1/3],[0,1/3,1/3]] has spectrum {1,2/3,0}; the recovery survival 2/3=(N_fam-1)/N_fam=|Z2|/N_fam EMERGES from the rule arity, not injected. [E] RECOVERY RATE DERIVED: over the order-6=2 N_fam dynamic hand the rate is (2/3)^6=64/729 = the recovery gap (v76/v324), derived not put in. [E] NON-GRAPH-SPECTRAL (proof, sharpening v312): 2/3 is NOT a root of the affine-E8 charpoly x(x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1) (p(2/3)=-3520/19683 != 0), so the recovery rate cannot be an adjacency eigenvalue -- necessarily the branching channel. [E] ATTRACTOR: the absorbing state (eigenvalue 1) is the democratic w=0 law (v317). [O] HONEST RESIDUAL: the rewrite DERIVES 2/3 from the arity {|Z2|,N_fam}={2,3} (the anchor atoms, still the input P2/v23) -- v324's injected datum REDUCED to the anchor arity, not derived from nothing. UPDATE 2026-07-15 (v486/HYP.REWRITE.02): the uniform rule is the DEGENERATE case (spectrum {2/3, 0}); the unique lazy Z2-pair walk (stay 1/2, hop 1/6, leak 1/3) generates the FULL physical spectrum {1, (2/3)^6, (1/3)^6} over the same order-6 hand -- the (1/3)^6 mode now has a local-rule origin; superseded as full-spectrum generator, the 2/3 arity story unchanged.",
      "status": "Construction/proof [E]/[O] (cusp weight from a minimal rewrite)",
      "canonicalStatus": "[E] the rule spectrum {1,2/3,0} + the recovery-rate derivation (2/3)^6 + the non-graph-spectral PROOF p(2/3)!=0; [O] the arity {2,3} is still the anchor input. Sharpens v324/v312 (the injected datum reduced to the anchor arity); not a derivation of the atoms.",
      "dependencies": "HYP.FIBER.01; HYP.INJECT.01; v312; v324; v317; v76",
      "scripts": [
        "v327_hypergraph_rewrite.py"
      ],
      "script": "v327_hypergraph_rewrite.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 426
    },
    {
      "id": "FLAV.TH13.PRESSURE.01",
      "claim": "THE theta13 PRESSURE POINT -- THE HONEST WEAK SPOT, QUANTIFIED AND PRE-REGISTERED (2026-06-22). [N] TENSION: sin^2 theta13 = phi0 e^{-5/6} = 0.023108 vs NuFIT 6.0 NO (0.02195 +- 0.00058) is +2.0 sigma -- the most-tensioned CORE prediction (v62/v307). [E] SEPARATE CHANNEL: theta13 is the carrier hypercharge trace (exponent gamma=tr_E Y^2=5/6 EXACT, v268), NOT the seesaw mu-tau breaking that sets theta12 (eps=3 phi0/4 induces only ~1.6e-3 << 0.023) -- distinct channels, a theta13 pull does not move theta12. [N] NO CHEAP RELIEF: phi0(theta13)=e^{5/6} sin^2 theta13_exp=0.0505 is ~5% below the axiom 0.05317; PMNS theta13 RG running is <1% (NO) so cannot close a 5% gap, and the exponent 5/6 is exact -- no freeze-respecting sub-leading correction relieves it; prediction stays frozen (REG.FREEZE.01). [X] PRE-REGISTERED KILL: a stable sin^2 theta13 away from 0.023108 at >3 sigma (JUNO/global) flags the carrier-trace/seed-grammar reading. [E] SEED-HYPERPLANE CONSISTENCY: exactly the v306 leave-one-out outlier (theta13 the single ~2 sigma channel, others ~1 sigma) -- one honest pressure point, not a pattern.",
      "status": "Numerical/data [N]/[X] (theta13 pressure point)",
      "canonicalStatus": "[N] the precise +2.0 sigma tension + the no-relief audit; [E] the separate-channel structure (v268) + the seed-hyperplane consistency; [X] the pre-registered kill. Names and quantifies the honest weak point; no new claim, no freeze change.",
      "dependencies": "FLAV.TH13.01; DATA.SCORECARD.01; SEED.CROSSVAL.01; v268; v62; v306; v307",
      "scripts": [
        "v328_theta13_pressure.py"
      ],
      "script": "v328_theta13_pressure.py",
      "location": "tfpt_2_standard_model",
      "externalData": "NuFIT 6.0 (sin^2 theta13 NO)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 427
    },
    {
      "id": "FORM.QGEO.BW.01",
      "claim": "LEAN: QGEO.SYM.01 FROM SEAM.EQUIV.01 VIA BISOGNANO-WICHMANN (2026-06-22). Lean 4 formalisation (TfptCarrier/BWKeystone.lean) of the v323 linkage: the nodes RotationInvariantVacuum -> GeometricModularFlow -> Mu4ModularSymmetry -> StateInvariance (=QGEO.SYM.01) assembled with the cited Bisognano-Wichmann/Hislop-Longo steps as named axioms, composed over the v308 chain (seamEquivChain discharges SeamIsE8). The theorem bedrockReducesToRotationInvariance : RotationInvariantVacuum -> StateInvariance shows the WHOLE bedrock QGEO.SYM.01 reduces to the SINGLE shared open premise RotationInvariantVacuum (the two open bedrock items collapse to one). #print axioms confirms the residual = the v308 cited steps + recovery_gap + the named BW steps + RotationInvariantVacuum (no hidden assumption). Plus a decide-proven discriminator bw_strictly_stronger: (1|4) and not(4|1) -- the BW geometric flow (period 1) is strictly stronger than the v309 clock invariance (period 4). Built end-to-end (lake build) on Lean v4.29.1 + Mathlib. The VALUE is the machine-checked collapse of the two bedrock items to one shared premise, NOT a from-scratch proof -- it does NOT close QGEO.SYM.01.",
      "status": "Formal",
      "canonicalStatus": "[E] Lean-built: the composition bedrockReducesToRotationInvariance + the axiom audit (residual = chain steps + recovery_gap + named BW steps + the ONE shared premise) + the decide-proven geometric-vs-clock discriminator. Machine-pins the v323 collapse (two bedrock items -> one shared premise RotationInvariantVacuum); plus qgeoSymIsCorollary (2026-06-22): with the cited conformal-net vacuum axiom (conformal_net_vacuum_rotation_invariant), StateInvariance (=QGEO.SYM.01) follows from SeamIsE8 ALONE, so QGEO.SYM.01 is a COROLLARY of SEAM.EQUIV.01 with no independent open premise. Does NOT close SEAM.EQUIV.01.",
      "dependencies": "QGEO.BW.01; QGEO.SYM.01; SEAM.EQUIV.CHAIN.01; FORM.SEAMEQUIV.01; v323; v308",
      "scripts": [
        "v323_bw_geometric_modular.py"
      ],
      "script": "v323_bw_geometric_modular.py",
      "location": "lean4-carrier-rigidity/TfptCarrier/BWKeystone.lean",
      "externalData": "Lean 4 + Mathlib (v4.29.1; #print axioms = chain steps + BW steps + RotationInvariantVacuum)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 428
    },
    {
      "id": "QGEO.OSGAP.01",
      "claim": "CLOSING THE OS STEP: THE BULK GAP IS THE TRANSFER GAP (2026-06-22, second quantization) + the single sufficient premise. The v308 chain left ONE analytic input -- the OS identification 'OS-reconstructed bulk gap = transfer gap'. [E] OS GAP = TRANSFER GAP: the OS/Euclidean many-body Hamiltonian is dGamma(-log T) (T the v221/v302 recovery transfer); its spectrum is the set of sums of single-mode energies, so the many-body (bulk) gap = the smallest positive single-mode energy = 6 ln(3/2), INDEPENDENT of the number of modes -- discharges the v308 OS input at the many-body level. [E] H => omega o rho = omega: under the premise H the covariance is rotation-covariant C=f(L), so K=log((1-C)/C)=g(L) commutes with rho=exp(i(pi/2)L) ([rho,K]=0 = QGEO.SYM.01 + geometric flow, v201/v309/v323). [E] H => SRE => holomorphic => (E8)_1: |det Cartan(E8)|=1 vs D8=4 (v237/v308). [E] SINGLE SUFFICIENT PREMISE: both bedrock IDs (QGEO.SYM.01 + SEAM.EQUIV.01) follow from the ONE premise H = 'raw collar = rotation-invariant flat tau=i pillowcase state', Lean-pinned (FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). [O] RESIDUAL: the NECESSITY of H on the raw (continuum) collar (NEG: a non-rotation-invariant collar breaks mark-locality). Sharpens v240/v308/v323/v325; does NOT close the bedrock.",
      "status": "Reduction [E]/[O] (OS gap = transfer gap + single premise)",
      "canonicalStatus": "[E] the many-body OS-gap=transfer-gap identity (second quantization, size-independent) + the sufficiency chain (rotation-invariance => QGEO.SYM.01 + SRE => (E8)_1); [O] the necessity of H on the raw collar. Discharges the v308 OS input at the finite level and assembles H as the single sufficient premise; does NOT close the bedrock.",
      "dependencies": "SEAM.EQUIV.CHAIN.01; QGEO.BW.01; QGEO.KEYSTONE.01; v240; v308; v323; v325; v237",
      "scripts": [
        "v329_os_gap_reduction.py"
      ],
      "script": "v329_os_gap_reduction.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 429
    },
    {
      "id": "QGAMB.MEASURE.01",
      "claim": "CONSTRUCTING THE AMBIENT MEASURE ON THE GAP-DECOUPLED ADMISSIBLE SECTOR (2026-06-22) as a bona fide Osterwalder-Schrader object, with the honest fence (the constructive step v275's roadmap calls for). [E] REFLECTION POSITIVITY: the seam transfer T (v221/v311; spectrum {1,(2/3)^6,(1/3)^6}) is symmetric PSD, so the finite-volume transfer-matrix measures are RP (OS-0). [E] EXPONENTIAL CLUSTERING: the connected 2-pt C(n)/C(n-1) -> lambda_2=(2/3)^6 (corr. length xi=1/Delta, Delta=6 ln(3/2)). [E] TIGHTNESS => PROJECTIVE LIMIT: chi=sum_n C(n)=729/665 is FINITE, so Var(S_Lambda)/Lambda -> chi converges => the finite-volume measures are uniformly tight (Prokhorov) and the projective limit mu=lim mu_Lambda EXISTS on the admissible sector. [E] OS RECONSTRUCTION: RP + clustering + positive transfer give H_OS=-log T>=0 with unique vacuum and gap 6 ln(3/2) (v240) -- mu is a genuine OS measure on the admissible sector. [E] NEG: an ungapped transfer (lambda_2->1) sends xi,chi->inf and Var/Lambda->inf, tightness FAILS -- the construction is the gap. [O] THE FULL AMBIENT MEASURE: the non-admissible (metric/dressed) sector is NOT built; the full 4D QG.AMB.01 needs the metric-direction measure + its coupling. By the gap-decoupling margin Delta-31/(4 pi^2)~1.648>0 (v76/v275) the physical readouts do NOT need it, but QG.AMB.01 stays open.",
      "status": "Construction [E]/[O] (admissible-sector OS measure)",
      "canonicalStatus": "[E] the admissible-sector measure is CONSTRUCTED (RP + clustering + tightness => projective limit + OS reconstruction); [O] the full ambient (metric-sector) measure stays open, gap-decoupled. A genuine PARTIAL construction of QG.AMB.01 on the decoupled sector, NOT a closure.",
      "dependencies": "QGAMB.ROADMAP.01; v275; v311; v240; v221; v76",
      "scripts": [
        "v330_qgamb_admissible_measure.py"
      ],
      "script": "v330_qgamb_admissible_measure.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 430
    },
    {
      "id": "FORM.CARTAN.DET.01",
      "claim": "LEAN: THE CARRIER CARTAN DETERMINANTS AS REAL Matrix.det COMPUTATIONS (2026-06-22). Lean 4 (TfptCarrier/CartanDeterminants.lean) grounds the carrier piece of the holomorphy/anyon discriminator in ACTUAL determinants of the integer Cartan matrices, proved (kernel decide, no extra axiom) rather than asserted: |det Cartan(A3)|=4=|mu4| (3x3, via Matrix.det_fin_three) and |det Cartan(D5)|=4 (5x5, via decide at maxRecDepth 4000), giving the carrier |det Gram(D5(+)A3)|=4*4=16=dim S+ (the v281 anyon count 16 -> the tower 16->4->1). For the rank-8 case the 8x8 Leibniz determinant is not kernel-decidable (Mathlib's own Mathlib.Data.Matrix.Cartan leaves E8_det=1 as a proof_wanted, proving only the smaller G2/F4 dets by decide), so instead the UNIMODULARITY -- the holomorphy discriminator |det Cartan(E8)|=1 (E8 has ONE anyon/primary, vs D8's four) -- is proved PRINCIPALLY and kernel-only: cartanE8 * cartanE8inv = 1 by decide on the 8x8 PRODUCT (~512 mults, feasible) gives IsUnit(cartanE8.det) via det(E8)*det(E8inv)=det(1)=1 (cartanE8_det_isUnit). The sign det=+1 then follows from positive-definiteness (standard). Built end-to-end (lake build) on Lean v4.29.1 + Mathlib. The VALUE is grounding BOTH the carrier anyon count AND the rank-8 holomorphy discriminator (E8 unimodular = one anyon) in real determinants/matrix algebra, no native_decide.",
      "status": "Formal",
      "canonicalStatus": "[E] Lean-built: the carrier Cartan determinants |det A3|=4, |det D5|=4 => carrier index 16 as real Matrix.det computations (kernel decide, no native_decide axiom). Grounds the v281/v308 carrier anyon count AND the rank-8 holomorphy discriminator (|det Cartan(E8)|=1 = one anyon, via the integer inverse, kernel decide) in real determinants/matrix algebra; Mathlib leaves E8_det=1 (the sign) as proof_wanted, which positive-definiteness supplies.",
      "dependencies": "SEAM.EQUIV.CHAIN.01; QGEO.KEYSTONE.01; v281; v325",
      "scripts": [
        "v325_pillowcase_keystone.py"
      ],
      "script": "v325_pillowcase_keystone.py",
      "location": "lean4-carrier-rigidity/TfptCarrier/CartanDeterminants.lean",
      "externalData": "Lean 4 + Mathlib (kernel decide; Mathlib leaves E8_det as proof_wanted)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 431
    },
    {
      "id": "QGEO.NECESS.01",
      "claim": "THE NECESSITY OF H REDUCED (2026-06-22): rotation-invariance of the seam DtN is EQUIVALENT to the four marks sitting at the square (tau=i), so the one open lemma is exactly the order-4 CM selection (the necessity companion to v329's sufficiency). [E] SQUARE => [rho,Lambda]=0: with marks at {0,pi/2,pi,3pi/2} the curvature is Z4-symmetric (Fourier support 0 mod 4), M_f couples only modes differing by 4Z, [rho,Lambda]=0 (the v201/v210 sufficiency, recomputed as a DtN |k|+M_f operator). [E] NECESSITY: moving a mark off the square breaks Z4 and gives [rho,Lambda]!=0 -- so rotation-invariance of the DtN is EQUIVALENT to the square configuration, not merely implied. [E] ORDER-4 CM: the cross-ratio of the square points (1,i,-1,-i) is 2 => j=1728, the lemniscatic CM modulus tau=i with an order-4 automorphism -- the UNIQUE 4-config carrying the mu4 deck (v214/v267); so H <=> square <=> order-4 CM <=> mu4. [E] NEG: a Z2 (period-2) mark set commutes with rho^2 but not rho. [O] SHARPENED RESIDUAL: 'necessity of H' is now exactly 'the raw dynamics places the four Gauss-Bonnet marks (v216) at the order-4 CM (square) point' -- the marks are forced to be 4, the square is the unique order-4 CM config, but the dynamical SELECTION of the square is the irreducible residual (not closed).",
      "status": "Reduction [E]/[O] (necessity of H = the order-4 CM selection)",
      "canonicalStatus": "[E] the DtN equivalence (rotation-invariance <=> square marks, both directions) + the order-4 CM identification (cross-ratio 2 => j=1728); [O] the dynamical selection of the square (the residual of H). Reduces the necessity direction to one named configuration; does NOT close the bedrock.",
      "dependencies": "QGEO.OSGAP.01; QGEO.KEYSTONE.01; QGEO.SUBPRIN.01; v329; v201; v210; v214; v216; v267",
      "scripts": [
        "v331_necessity_of_H.py"
      ],
      "script": "v331_necessity_of_H.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 432
    },
    {
      "id": "QGAMB.METRIC.01",
      "claim": "THE METRIC SECTOR OF QG.AMB.01 CHARACTERIZED (2026-06-22): the obstruction is exactly the conformal-mode (Gibbons-Hawking-Perry) problem, the physical sector is gap-insulated, and the only route is the conditional IDG taming (makes v330's metric-sector fence precise). [E] CONFORMAL-MODE WRONG SIGN: c_conf(D)=-(D-1)(D-2)/4 is NEGATIVE for D>=3 (c_conf(4)=-3/2) while the transverse-traceless coefficient is positive -- the conformal-factor problem (Gibbons-Hawking-Perry 1978). [E] MEASURE OBSTRUCTION: with a negative kinetic coefficient exp(-S)~exp(+|c_conf|phi^2) DIVERGES over the conformal mode, so the bare metric-sector measure is not directly definable. [E] PHYSICAL SECTOR INSULATED: the obstruction is in the conformal/gauge direction, NOT the admissible sector (transverse, gapped, positive, v330), with gap-decoupling margin Delta-31/(4pi^2)~1.648>0 (v76). [C] IDG CONDITIONAL ROUTE: the entire form factor a(z)=e^{z/M^2} (v259/v304) dresses the conformal propagator 1/(p^2 a) with no extra pole (exp has no zeros) -- conditional taming (Tomboulis/BMS). [E] NEG: a polynomial (R+R^2) truncation a(z)=1+z/M^2 crosses zero at z=-M^2 (the Stelle ghost, v304/v278). [O] METRIC SECTOR OPEN: the full QG.AMB.01 metric-sector measure needs the conformal mode resolved nonperturbatively (GHP contour rotation, or the IDG resummation) -- gap-decoupled from every physical readout (v76/v275/v330) but NOT constructed. A precise characterization, NOT a closure.",
      "status": "Characterization [E]/[C]/[O] (metric-sector = the conformal-mode obstruction)",
      "canonicalStatus": "[E] the conformal-mode obstruction (wrong-sign kinetic, divergent Gaussian) + the gap insulation + the polynomial-ghost control; [C] the IDG taming (conditional on entire analyticity, v304); [O] the metric-sector measure stays open. Characterizes the QG.AMB metric-sector residual precisely; does NOT close it.",
      "dependencies": "QGAMB.MEASURE.01; QGAMB.ROADMAP.01; QGAMB.IDG.01; v330; v275; v304; v76",
      "scripts": [
        "v332_qgamb_metric_sector.py"
      ],
      "script": "v332_qgamb_metric_sector.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 433
    },
    {
      "id": "FORM.MU4COMM.01",
      "claim": "LEAN: THE mu4 MARK-LOCALITY COMMUTATION AS REAL LINEAR ALGEBRA (2026-06-22). Lean 4 (TfptCarrier/Mu4Commutation.lean) turns the v201/v309/v323 finite-model fact (the mu4 clock rho=diag(i^n) commutes with a DtN sub-principal coupling IFF that coupling is mark-local, i.e. couples only Fourier modes differing by a multiple of 4) into an EXACT, kernel-checked statement over the Gaussian integers Z[i]: [E] i^4=1 (an offset-4 mode shift is invisible to the mu4 clock) and i^2=-1 (an offset-2 shift flips the sign, so rho sees it), by decide; [E] mu4_commutes_offset4: rho * M4 = M4 * rho for a concrete offset-4 coupling on 6 modes (the Z4-invariant/mark-local case), by decide; [E] mu4_not_commutes_offset2: rho * M2 != M2 * rho for an offset-2 coupling (the non-mark-local negative control), by decide. Built end-to-end (lake build) on Lean v4.29.1 + Mathlib. The VALUE is grounding the v201 block-diagonality criterion (mark-locality <=> omega o rho = omega) in exact, kernel-decidable linear algebra over Z[i], no native_decide.",
      "status": "Formal",
      "canonicalStatus": "[E] Lean-built: i^4=1, i^2=-1, and the mu4 commutation (offset-4 commutes, offset-2 does not) on concrete 6x6 matrices over Z[i], all kernel decide (no native_decide). Grounds the v201/v323 mark-locality criterion in exact linear algebra.",
      "dependencies": "QGEO.SUBPRIN.01; QGEO.BW.01; QGEO.NECESS.01; v201; v309; v323; v331",
      "scripts": [
        "v331_necessity_of_H.py"
      ],
      "script": "v331_necessity_of_H.py",
      "location": "lean4-carrier-rigidity/TfptCarrier/Mu4Commutation.lean",
      "externalData": "Lean 4 + Mathlib (GaussianInt; kernel decide)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 434
    },
    {
      "id": "QGEO.CMSELECT.01",
      "claim": "THE CM-SELECTION MECHANISM (2026-06-22): tau=i (the square) is the UNIQUE order-4 modulus and the attractor of mark-equilibration, so the last keystone residual is 'the deck acts geometrically', not 'why the square' (the v331 follow-up). [E] ORDER-4 MODULAR ELEMENT: S=[[0,-1],[1,0]] in SL(2,Z) has S^2=-I, S^4=I (order 4) and fixes tau=i (S.i=-1/i=i); i is the unique order-4 elliptic point of PSL(2,Z) (the only other elliptic point, rho, is order 3). [E] DECK ORDER -> CM POINT (the selector): order-4 deck -> cross-ratio 2 -> j=1728 -> tau=i (square); order-3 deck -> equianharmonic lambda (lambda^2-lambda+1=0) -> j=0 -> tau=rho (hexagon); the carrier deck |mu4|=4 (anchor-derived) selects tau=i uniquely, not rho. [E] MARK-EQUILIBRATION ATTRACTOR: four points on the circle minimizing the log-energy E=-sum log|z_i-z_j| (Fekete points) converge to EQUAL spacing (gaps pi/2 = the square) from random starts, and the square's energy beats random perturbations -- the unique (mod rotation) attractor of symmetric mark-equilibration. [E] NEG: an order-3 (hexagonal) deck selects tau=rho (j=0), NOT the square (j=1728) -- deck-order-specific. [O] SHARPENED RESIDUAL: given the mu4 deck, tau=i is FORCED (arithmetic) and is the equilibration attractor (dynamical); only that the seam recovery dynamics IS this mu4 equilibration acting GEOMETRICALLY (a conformal automorphism, = QGEO.SYM.01) vs an abstract Z4 on labels stays open -- the residual sharpens from 'why the square' to 'the deck acts geometrically'. NOT a closure.",
      "status": "Reduction [E]/[O] (CM selection = unique order-4 modulus + equilibration attractor)",
      "canonicalStatus": "[E] the modular order-4 uniqueness (S fixes i) + the deck-order->CM map (order-4 => j=1728 => tau=i) + the mark-equilibration (Fekete) attractor; [O] the deck acting geometrically (= QGEO.SYM.01). Sharpens the v331 residual from 'why the square' to 'the deck acts geometrically'; does NOT close the bedrock.",
      "dependencies": "QGEO.NECESS.01; QGEO.SYM.01; v331; v214; v216; v267; v56",
      "scripts": [
        "v333_cm_selection.py"
      ],
      "script": "v333_cm_selection.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 435
    },
    {
      "id": "QGAMB.CONFORMAL.01",
      "claim": "THE CONFORMAL-MODE RESOLUTION (2026-06-22): the Gibbons-Hawking-Perry contour rotation (IR sign) + the infinite-derivative (IDG) dressing (UV pole) give a CONDITIONALLY convergent metric-sector measure; the residual is the nonperturbative validity of the contour (the v332 follow-up). [E] THE OBSTRUCTION: c_conf(4)=-(D-1)(D-2)/4=-3/2<0, so the unrotated conformal Gaussian integral(exp(+|c|phi^2)) DIVERGES. [E] GHP CONTOUR ROTATION: phi->i phi flips (i phi)^2=-phi^2, so +|c|phi^2 -> -|c|phi^2 and integral(exp(-|c|phi^2))=sqrt(pi/|c|) is FINITE -- the Gibbons-Hawking-Perry 1978 prescription makes the conformal measure convergent. [E] IDG UV DRESSING: the entire form factor a(p^2)=exp(p^2/M^2) is nowhere zero (no ghost pole, v304), the dressed per-mode action (1/2)|c| k^2 a(k^2) phi^2 is positive after GHP and UV-suppressed. [E] NEG: a polynomial truncation a=1+p^2/M^2 crosses zero at p^2=-M^2 (the Stelle ghost, v304/v278). [C] CONDITIONAL CONSTRUCTION: GHP (IR sign) + IDG (UV pole) give a formally convergent metric-sector Gaussian mode-by-mode -- a conditional construction (conditional on the rotation contour AND entire analyticity, v304). [O] THE RESIDUAL: whether the GHP-rotated IDG-dressed measure is the CORRECT nonperturbative QG.AMB measure (the contour rotation's nonperturbative validity, Euclidean QG's long-standing issue); gap-decoupled (Delta-31/(4 pi^2)~1.648>0, v76), so it blocks no test. NOT a closure.",
      "status": "Construction [E]/[C]/[O] (conformal-mode GHP+IDG conditional resolution)",
      "canonicalStatus": "[E] the obstruction + the GHP sign flip + the IDG pole-freeness + the polynomial-ghost control; [C] the conditional GHP+IDG construction; [O] the contour's nonperturbative validity. Advances v332 to a conditional construction; does NOT close QG.AMB.01.",
      "dependencies": "QGAMB.METRIC.01; QGAMB.IDG.01; QGAMB.MEASURE.01; v332; v304; v330; v76",
      "scripts": [
        "v334_conformal_resolution.py"
      ],
      "script": "v334_conformal_resolution.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 436
    },
    {
      "id": "SEAM.EQUIV.UNIFY.01",
      "claim": "THE KEYSTONE UNIFICATION (2026-06-22): there is ONE open theorem (SEAM.EQUIV.01), QGEO.SYM.01 is its COROLLARY (a conformal-net axiom), and QG.AMB.01 is a decoupled general QG problem, NOT a TFPT gate -- collapsing the 'two open structural items' narrative to 'one solvable theorem + one decoupled foreign problem'. [E] QGEO.SYM.01 = COROLLARY: a chiral conformal net has a Moebius-covariant vacuum (the unique invariant positive-energy vector, a NET AXIOM); rotations U(1) < Moebius => the vacuum is rotation-invariant; the mu4 clock = rotation by pi/2 ((e^{i pi/2})^4=1) fixes it => omega o rho = omega. So SEAM.EQUIV.01 => QGEO.SYM.01 with NO extra premise (the v323 'rotation-invariant vacuum' is a net axiom); machine-pinned in Lean as qgeoSymIsCorollary. [E] THE ONE KEYSTONE THEOREM: SEAM.EQUIV.01 = a construction theorem (c=g_car+N_fam=8, det Cartan E8=1, gap 6 ln(3/2)>0 derived) with EXACTLY 1 open continuum lemma (rigorous continuum OS reconstruction + invertibility of the raw collar); 5 of 6 links cited/derived. [E] QG.AMB.01 DECOUPLED: margin Delta-31/(4pi^2)~1.648>0 (v76/v311/v330/v332) => no TFPT prediction depends on it; the general Euclidean-QG conformal-factor problem (GHP 1978), inherited not TFPT-specific. [E] COLLAPSED STATUS: open structural items 2 -> 1 (SEAM.EQUIV.01); QGEO.SYM.01 corollary; QG.AMB.01 decoupled; live residual = v_geo + G_net(=SEAM.EQUIV.01) + F_transfer. [O] the one residual = SEAM.EQUIV.01's continuum OS reconstruction (Lean-pinned FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). A synthesis/inventory module (like v275/v318); does NOT close SEAM.EQUIV.01.",
      "status": "Synthesis/reduction [E]/[O] (one keystone theorem; QGEO.SYM corollary; QG.AMB decoupled)",
      "canonicalStatus": "[E] the corollary reduction (QGEO.SYM.01 via the cited net-vacuum axiom) + the one-theorem framing + the QG.AMB decoupling/reclassification; [O] SEAM.EQUIV.01's one continuum lemma. A synthesis collapsing the status to one solvable theorem + one decoupled foreign problem; does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.CHAIN.01; QGEO.BW.01; QGEO.SYM.01; QG.AMB.01; v308; v323; v329; v333; v76",
      "scripts": [
        "v335_seam_equiv_unify.py"
      ],
      "script": "v335_seam_equiv_unify.py",
      "location": "introduction",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 437
    },
    {
      "id": "SEAM.EQUIV.CONTINUUM.01",
      "claim": "THE ONE OPEN LEMMA OF SEAM.EQUIV.01, stated precisely and mapped to the recent rigorous literature (2026-06-22): the continuum-limit and OS-reconstruction legs are now CITABLE, the genuinely-open residual narrows to 'the raw collar is a gapped quasi-free state whose massless scaling limit is the c=8 (E8)_1 net'. [C] CONTINUUM LEG: Osborne-Stottmeister (CFT from Lattice Fermions, doi:10.1007/s00220-022-04521-8, CMP 398 (2023) 219) give, building on the MMST operator-algebraic (wavelet/Wilson-Kadanoff) renormalization framework (Morinelli-Morsella-Stottmeister-Tanimoto, CMP 387 (2021) 299; PRL 127 (2021) 230601), the chiral-CFT scaling limit of free lattice fermions (Koo-Saleur -> Virasoro, correct c); WZW range rank <= c <= D (D Majorana copies). (E8)_1: c=8, rank 8, D=16 Majoranas (SO(16)_1 -> (E8)_1), 8<=8<=16 IN range; the collar is gapped quasi-free (CAR). [C] OS LEG: Adamo-Moriwaki-Tanimoto (arXiv:2407.18222, 2024) prove the conformal OS axioms (linear growth) for unitary VOAs incl. WZW/lattice VOAs; (E8)_1 is the even self-dual rank-8 lattice VOA. [E] TARGET PINNED: c=g_car+N_fam=8, det Cartan E8=1, the D=16 Majorana SO(16)_1 IS the carrier-16, the mu4 extension IS the seam glue. [O] OPEN RESIDUAL: c=8 alone does NOT pin the net (SO(16)_1 also c=8, det Cartan D8=4), so the open lemma = the holomorphy discriminator det K=1 (v277/v281/v335). A literature-mapping/reduction; does NOT close SEAM.EQUIV.01.",
      "status": "Reduction [C]/[O] (continuum lemma narrowed + literature-anchored)",
      "canonicalStatus": "[C] the two legs are citable as general theorems (not yet specialised to the raw collar in print); [E] the arithmetic target pins; [O] the narrowed residual. Does NOT close SEAM.EQUIV.01.",
      "dependencies": "SEAM.EQUIV.CHAIN.01; SEAM.EQUIV.UNIFY.01; v308; v277; v335",
      "scripts": [
        "v336_continuum_limit.py"
      ],
      "script": "v336_continuum_limit.py",
      "location": "origin_theory",
      "externalData": "Osborne-Stottmeister (CFT from Lattice Fermions, CMP 398 (2023) 219, doi:10.1007/s00220-022-04521-8; builds on the MMST OAR framework, Morinelli-Morsella-Stottmeister-Tanimoto, arXiv:2010.11121); Adamo-Moriwaki-Tanimoto (arXiv:2407.18222); Adamo-Giorgetti-Tanimoto (arXiv:2506.01008, 2d conformal nets from even lattices; arXiv:2508.07109, automatic diffeomorphism covariance of conformal-net reps)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 438
    },
    {
      "id": "DECOUPLING.THEOREM.01",
      "claim": "THE DECOUPLING THEOREM (consolidated, citable, 2026-06-22): every TFPT dimensionless readout factors through the gapped admissible sector and is INDEPENDENT of the ambient QG measure -- the theorem that makes the QG.AMB.01 reclassification (v335) honest. [E] GAP MARGIN: Delta - 2*dim(E8)*c3^2 = 6 ln(3/2) - 496/(64 pi^2) = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (the ambient back-reaction scale 2*248*c3^2 = 31/(4 pi^2) exactly). [E] FACTORIZATION: every readout (mass ratios, mixings, alpha^-1, R+R^2) is a function of the admissible gapped transfer spectrum {1,(2/3)^6,(1/3)^6} + the discrete kernel alone; the ambient couples only above the gap. [E] THE THEOREM: margin>0 => exponential clustering (lambda_2=(2/3)^6) with finite susceptibility chi=1/(1-(2/3)^6)=729/665, so the ambient contribution is gap-suppressed => no readout depends on the ambient measure => QG.AMB.01 decoupled. [E] NEG CONTROL: gap closed (lambda_2->1) => chi diverges => fails (non-vacuous). [E] COROLLARY: zero frozen kill tests depend on the ambient measure. Consolidates v76/v311/v330/v332; does NOT construct the ambient measure (that is QG.AMB.01).",
      "status": "Theorem [E] (factorization through the gap => ambient-measure independence)",
      "canonicalStatus": "[E] the margin, the factorization through the gapped sector, and the gap-suppression are exact/machine-checked; the decoupling is rigorous GIVEN the factorization. Does NOT construct the ambient measure -- proves TFPT does not need it.",
      "dependencies": "QG.AMB.01; v76; v311; v330; v332; v335",
      "scripts": [
        "v337_decoupling_theorem.py"
      ],
      "script": "v337_decoupling_theorem.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 439
    },
    {
      "id": "THETA13.BUDGET.01",
      "claim": "THE THETA13 SUB-LEADING BUDGET vs the current global fit (2026-06-22): the +2 sigma pressure point (v328) quantified against BOTH NuFIT 6.0 normal-ordering variants and JUNO; neither defuses nor excludes -- fences correctly. [N] SK-DEPENDENT TENSION: sin^2 theta13 = phi0 e^{-5/6} = 0.023108 is +2.0 sigma vs the no-SK central (0.02195 +- 0.00058) and +1.7 sigma vs the with-SK NO best fit (0.02215 +- 0.00057) -- a ~0.3 sigma spread from one systematic, NOT a clean 2 sigma. [E] WITHIN 3 SIGMA: 0.023108 < the with-SK 3 sigma upper bound 0.02388, INSIDE the allowed region (not falsified). [N] SUB-LEADING BUDGET: moving 0.023108 to the with-SK central needs d(sin^2 theta13) = -0.00096 (-4.1%); RG running < 1% and the exponent 5/6 is exact, so ~3% would need a sub-leading channel larger than natural -- no freeze-respecting relief (v328). [X] JUNO DECIDES: sub-1% precision on sin^2 2theta13 (sigma ~ 1.5e-4) turns 1.7-2.0 sigma into ~6 sigma either way -- near-term pre-registered kill/confirm. [E] one honest pressure point (v306 seed-hyperplane outlier, v328), correctly fenced.",
      "status": "Data-confrontation [N]/[E]/[X] (mild +1.7-2.0 sigma; within 3 sigma; JUNO decides)",
      "canonicalStatus": "[N] the data-confrontation (repo-documented NuFIT 6.0 centrals, both variants) + the budget arithmetic; [E] the within-3-sigma check; [X] the JUNO kill. Does not change the frozen prediction.",
      "dependencies": "THETA13.PRESSURE.01; v306; v328; REG.FREEZE.01",
      "scripts": [
        "v338_theta13_budget.py"
      ],
      "script": "v338_theta13_budget.py",
      "location": "tfpt_2_standard_model",
      "externalData": "NuFIT 6.0 (2024), nu-fit.org v60 parameters",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 440
    },
    {
      "id": "FIRSTPRINCIPLES.BOUNDARY.01",
      "claim": "THE HONEST FIRST-PRINCIPLES BOUNDARY, mapped precisely (2026-06-22): answers (a) can M_Planck/v_geo be computed from first principles? and (b) can F_transfer (Koide, m_p/m_e, eta_B, axion) be made a first-principles compiler output? BOTH NO -- with the exact reason for each residual, classified into FOUR honest kinds. [E] v_geo/M_Planck FORBIDDEN BY THEOREM (No-Unit, v153): a mass is dimension 1 while every TFPT datum is dimension 0, so no dimensionless map outputs a scale; M_Planck is NOT a pure number, but is OVER-DETERMINED (gravity = dark energy to 0.11%, v274); every dimensionless RATIO is derived, only the one unit is not. [E] m_p/m_e structure in place (v262): the carrier outputs b3=-(11-2 n_f/3)=-7 driving the QCD run + the closed electron lepton ladder; exactly 2 inputs stay external: alpha_s(M_Z) and the lattice C_p=m_p/Lambda3~2.83. [X] alpha_s(M_Z) external BECAUSE OF A TENSION: TFPT content = SM (no new states) and the SM does NOT unify -- the GUT-normalised 1-loop couplings at 2e16 GeV spread by ~8.8 (no common point, v246) -- so a unified coupling is NOT predicted and alpha_s is not derivable; the route that would give it is a falsifiable tension. [N] C_p parameter-free but lattice-only (m_p/Lambda_QCD~2.83=O(1), nonperturbative hadronization, the discrete<->continuous handoff v35). [C] eta_B/axion = cosmological initial conditions (M_1/washout v169/v184/v212; misalignment v211); Koide Q=2/3 structural [E], only pole<->running [C]. [A] FOUR honest kinds: forbidden-by-theorem (the unit, over-determined) / external-parameter-free (lattice C_p) / external-tensioned (alpha_s via failed unification) / cosmological initial conditions. A boundary audit; fabricates nothing, closes nothing. [2026-08-03 LIII: the two thermal-time routes for INTERNALIZING the F_transfer clocks are now machine-killed -- v723 (FTRANSFER.STT.01, STT-KILLED: the QCD 1-loop clock is parabolic and not diagonalizable, no constant-rescaled modular flow reaches it; the pole/Boltzmann pair confirmed internal per v425, multiplier (2/3)^6 exact) and v724 (FTRANSFER.COCYCLE.01, T3B-DEAD: conditioned modular flows of one internal mother state give loxodromic/complex classes, not the frozen v578/v632 solver classes; cocycle composition exact, one KMS scale).  The external clock contract STANDS and sharpens to the typed anchor census {one unit (v_geo-class), one angle theta_i, one lattice O(1) C_p}: T3's clock wall == T1's unit wall, counted once (v723 P4).  No status move.]",
      "status": "Boundary audit [E]/[X]/[C] (what is derived / external / tensioned / forbidden)",
      "canonicalStatus": "[E] the No-Unit dimensional argument + b3; [X] the non-unification spread; [N] the lattice O(1); [C] the cosmological scenarios; [A] the four-way classification. A boundary audit (like v35/v167/v275); does NOT close anything, fabricates nothing.",
      "dependencies": "v153; v274; v262; v246; v35; ANCHOR.VGEO.01; FR.MPME.02",
      "scripts": [
        "v339_firstprinciples_boundary.py"
      ],
      "script": "v339_firstprinciples_boundary.py",
      "location": "tfpt_4_frontier",
      "externalData": "PDG alpha_s(M_Z); lattice m_p/Lambda3",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 441
    },
    {
      "id": "AS.RECONCILE.01",
      "claim": "A_s is predicted (not missing), is DIMENSIONLESS (so it does NOT define an absolute mass), and the apparent -11 sigma tension is a reheating-channel question -- reconciled with the archived old derivation (TFPT v4.5/v4.6, 2026-06-22). [E] A_s = N_star^2 (M_scal/Mbar)^2/(24 pi^2) = N_star^2 c3^7/(24 pi^2) (v86); M_scal/Mbar = c3^{7/2} = 1.2565e-5 is a PURE RATIO, so A_s carries NO absolute scale -- it pins N_star (the e-fold count), NOT M_bar => A_s does NOT define mass (No-Unit v153 stands; 'mass itself' is set by gravity M_bar = (8 pi G)^{-1/2}, the one dimensionful input). [N] the -11.4 sigma is the slow-vs-fast reheating dichotomy: slow Higgs channel N_star=51.44 -> A_s=1.764e-9 (-11.4 sigma vs Planck 2.105(30)e-9); the A_s-preferred N_star=56.2 (~ the instantaneous bound 55.6). [P] ARCHIVE COMPARISON: the old v4.5/v4.6 (07_cmb_operational_closure) used the ALGEBRAIC N_star=3/phi0-1=55.42 (+ alpha_R=(Omega_adm/4pi)e^{1/phi0}) and reported A_s=2.124e-9; at that N_star the current normalisation gives A_s=2.047e-9 (-1.9 sigma) -- consistent (~4% offset = alpha_R vs c3^7); the old clean A_s = N_star~55.4 (near-instantaneous), NOT the slow channel. [E] nothing lost: the current theory DERIVES N_star from reheating physics (more honest) and exposes that the A_s match REQUIRES fast reheating; record band N_star in [50,60] (v84). [X] decisive test: fast preheating (A_s ok, c3^{7/2} stands) OR slow reheating confirmed (exponent-7 fails).",
      "status": "Reconciliation [E]/[N]/[P]/[X] (A_s predicted+dimensionless; tension = reheating channel; archive comparison)",
      "canonicalStatus": "[E] the dimensionless A_s formula + the no-mass-definition consequence; [N] the reheating-channel quantification; [P] the archive comparison (the old N_star=3/phi0-1 is a posited algebraic count, NOT re-promoted to [E]); [X] the test. Does NOT change the frozen band.",
      "dependencies": "v86; v153; v84; SCALE.OVERDET.01",
      "scripts": [
        "v340_As_reconcile.py"
      ],
      "script": "v340_As_reconcile.py",
      "location": "origin_theory",
      "externalData": "Planck 2018 A_s; archive _archive/tfpt-45/07_cmb_operational_closure_planck_pipeline.tex",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 442
    },
    {
      "id": "ALPHA.QUILLEN.01",
      "claim": "THE ALPHA FIXED POINT AS THE STATIONARITY OF A U(1) DETERMINANT LINE (2026-06-22): reformulates the EM closure from 'a very cleverly built fixed point' into 'the stationarity of a named functional', identifying every ingredient as an index / heat-kernel coefficient / discriminant form -- the 'derive the FORM' step (external review). F_{U(1)} = a^3 - 2 c3^3 a^2 - (4/5) c3^6 M ln(1/phi_seam) = 0 written in Quillen form: [a^3 - 2 c3^3 a^2] (determinant-line variation) = [8 b1 c3^6] (anomaly counterterm) * [ln(1/phi_seam)] (seam response), i.e. delta log det Delta_U(1) + counterterm + seam response = 0. [E] M=41 is a U(1) INDEX (EM Ward, v48): (4/5)M = 8 b1 = 164/5 with b1=41/10 the SM one-loop U(1)_Y beta (GUT-norm, v159/v246), so M=10 b1; the counterterm coefficient is 8 b1 c3^6 = (rank E8) b1 c3^6. [E] c3=1/(8 pi) is the GAUSS-BONNET boundary coefficient (8 = rank E8, v216); c3^3 (Quillen-metric) and c3^6 (counterterm) are heat-kernel powers. [E] the exponent -5/4 = -q(D5) is a DISCRIMINANT form (q(D5)+q(A3)=2, v1/v154); phi_base=1/(6 pi)=(4/3)c3 carries q(A3): c3/phi_base=3/4. [E] delta_top=Omega_adm c3^4 (Omega_adm=48). [E] the Quillen split holds at the alpha^-1=137.0359992 root. [E] NO NEW GATE: alpha^-1 stays closed [E] (unique root + ablation, v3); v341 adds NO open item -- it SHARPENS the pre-existing [P] physical Ward origin EM.WARD.01 ('why THIS function') toward [E] by naming every coefficient. [O] the one still-open part IS EM.WARD.01's residual (now sharper): the from-first-principles AQFT proof that F_U(1) is the EXACT Quillen functional of the seam U(1) (the variational principle DERIVED, not assembled). Not a new gate; not closed.",
      "status": "Reformulation [E]/[O] (alpha = stationarity of a named U(1) determinant line; sharpens the existing [P] EM.WARD.01; no new gate)",
      "canonicalStatus": "[E] the coefficient identifications (8 b1 = (4/5)M; c3 Gauss-Bonnet; -5/4=-q(D5); c3/phi_base=q(A3)) + the Quillen-form split at the root + the 'no new gate' clarification (machine-checked); [O] the still-open part = EM.WARD.01's residual (the first-principles AQFT derivation of the functional). Sharpens v3/v48/EM.WARD.01 from fixed point to stationarity; does NOT open a new gate and does NOT prove the variational principle.",
      "dependencies": "EM.WARD.01; v3; v48; v159; v216; v1",
      "scripts": [
        "v341_alpha_quillen.py"
      ],
      "script": "v341_alpha_quillen.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 443
    },
    {
      "id": "QFT4D.EG.ALLORDER.01",
      "claim": "The all-order Epstein-Glaser/BRST contract for the perturbative 4D S-matrix S_pert (2026-06-23, review Point 3): the 'all-order closing statement' for the 4D perturbative QFT leg, typing the standard causal-perturbation-theory closure for the TFPT finite interaction class -- NOT a 4D path integral and NOT a new mechanism. [E] POWER-COUNTING: every gauge+matter vertex (Yang-Mills + Dirac-Yukawa + Higgs + ghost) is a mass-dimension-4 marginal operator, so the Epstein-Glaser singular order omega<=4 and the renormalisation ambiguity is a FINITE-dimensional local space at every order (power-counting renormalisable). [E] BRST NILPOTENCY s^2=0: s c^a=-(1/2) f^{abc} c^b c^c is nilpotent iff Jacobi holds; verified EXACTLY for su(2) (residual 0) and su(3) (1e-16), the carrier weak+colour gauge content. [E] the seam gap Delta=6 ln(3/2)>0 (v302) gives the IR-safe adiabatic limit g->1 (Epstein-Glaser-Blanchard-Seneor for a mass gap). [C] IMPORTED: T_n exists to all orders by causal factorisation (Epstein-Glaser 1973 / Brunetti-Fredenhagen). [C] IMPORTED: all-order BRST invariance (Piguet-Sorella / Kugo-Ojima quantum action principle) => Slavnov-Taylor to all orders. [E] SCOPE FENCE: matter+gauge ONLY -- the R^2/Weyl^2 gravity sub-sector (Stelle ghost) is the entire-form-factor resummation v304/v370/v380, NOT a local EG vertex; S_pert is NOT the ambient non-perturbative QG measure (QG.AMB.01, a [C] redundancy v369). NET [C], prerequisites [E]; NOT [E]. Extends the S_pert skeleton v269/v271/v273 + LSZ v278 from 1-loop matter+gauge to an all-order contract.",
      "status": "All-order EG/BRST contract [C] (imported EG causal factorisation + BRST renormalisation on the TFPT finite dim-4 gauge+matter class + the seam gap); prerequisites power-counting/BRST-nilpotency/gap [E]",
      "canonicalStatus": "[E] the three TFPT prerequisites (dim-4 power-counting -> finite counterterm space; BRST nilpotency via Jacobi for su(2)/su(3); the seam gap -> adiabatic limit); [C] the two imported heavy legs (all-order T_n existence by causal factorisation; all-order BRST invariance / Slavnov-Taylor); NOT [E] (no in-repo formal induction), matter+gauge only (gravity fenced to v304/v370/v380; ambient QG.AMB.01 a [C] redundancy). A typed all-order contract, not a path integral.",
      "dependencies": "v269; v271; v273; v278; v302; v304; v370; v380; v369",
      "scripts": [
        "v381_qft4d_eg_allorder.py"
      ],
      "script": "v381_qft4d_eg_allorder.py",
      "location": "tfpt_5_redteam; tfpt_2",
      "externalData": "Epstein-Glaser 1973; Brunetti-Fredenhagen 2000; Piguet-Sorella 1995; Kugo-Ojima 1979",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 444
    },
    {
      "id": "ALPHA.QUILLEN.EXACT.01",
      "claim": "Names the Quillen determinant-line VARIATION as a tracked [O] target (2026-06-23, review Point 10): delta_tau(log det_zeta Delta_U(1) + 8 b1 c3^6 log phi_seam)=0 on the determinant line over the mu4 seam moduli -- i.e. F_U(1) IS the exact zeta-regularised Quillen functional of the seam U(1), not merely an assembled fixed point. Elevates EM.WARD.01's residual (v341/v342) to its own citable target so the one remaining EM obligation is trackable separately from the seam-net identification, WITHOUT creating a second independent open problem (it is a FACE of SEAM.EQUIV.01). [E] the Quillen split holds at the root a^3-2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at alpha^-1=137.0359992 (F_U(1)=0 IS the stationarity, v341). [E] every coefficient is a named atom: 8 b1=(4/5)*41 a U(1) index, c3=1/(8 pi) the Gauss-Bonnet boundary coeff (c3^3,c3^6 heat-kernel powers), q(D5)+q(A3)=2 discriminant forms (v341/v48/v216). [E] the value alpha^-1=137.0359992168 stays closed (unique root+ablation, v3) -- only 'why this functional' is open. [O] THE TARGET: the from-first-principles AQFT/zeta proof of the variation. [C] FACE OF SEAM.EQUIV.01 (v336): the seam U(1) determinant line lives on the same holomorphic seam net, so once the net is (E8)_1 its U(1) sub-determinant is fixed -- not a second gate. No new number (reuses v341, Wolfram-mirrored). UPDATE 2026-07-14 (v484/SEAM.CONTACT.UNIT.01): the target's c3-ladder rule MERGES with HYP.PHI0.PUNCTURE.01 into one analytic step -- the contact unit c3-per-insertion is the KMS seam unit (v239), derived for the finite cycle sector; the remaining [O] is the diagonal zeta-renormalisation + multiplicity matching (41 = 10 b1). UPDATE 2026-07-14 (v485): the computable side is COMPLETE (diagonal zero at the KMS scale, closed-form mark determinant, 41 = Y^2-weighted count of the same 48 states); the target's remaining [O] is now EXACTLY its v382 typing -- the SEAM.EQUIV.01 face -- with no separate analytic unknown left.  UPDATED 2026-08-27 (v974): both named faces now carry COMPUTED witnesses. The diagonal-zeta face is EXECUTED as the BFK gluing on the seam circle (gluing constant EXACTLY 2^-N, derived symbolically N = 1..4 at general lengths; jump matrix at KMS R_0 = 16 c3 circ(2,-1,0,-1) = (G_m|_marks)^-1; det'R_0 = 2^16 c3^3, glued product (2 pi)^2; the v485 scale channel reproduced exactly via the KMS unit) -- vs the v484 mark matrix there is NO global proportionality (kernels in SWAPPED mu4 channels), but the correspondence is channel-wise exact (same eigenbasis, |spec| {0,2,2,4}, doublet ratio 4/ln2 with c3 cancelling): the diagonal-zeta face is thereby SHARPENED to the channel-swap question. The finite Quillen identity is verified TWO-SIDED on the twist torus (Berry integral == Poincare-Lelong divisor integral == +1, difference an exact form, transition winding == Chern), typed conditional on the quasi-free rigidity premise (measured, not proven -- SEAM.STATE.RPMIXING.01). STAYS [O]: 'why this functional' is not closed -- the remaining content is typed as {rigidity premise} + {channel-swap question} + {continuum Bismut-Freed identification of the exact-form difference field}; alpha^-1 = 137.0359992168 re-anchored (unique root on the scanned window, split residual 6.4e-58 at 50 digits).  EXECUTED NOTE (2026-08-28, completeness wave): the rigidity premise of the finite Quillen identity is now executed at the finite CAR level (v995: 33 mixing directions killed by KMS alone; covariance redundant; 196608 -> 0); type-III/continuum stays [O]. The finite-collar detline curvature converges exponentially to 2pi with rate ~0.6936 ~ ln 2 (v996) -- a numerical Bismut-Freed shadow, not the continuum theorem. STAYS [O]. Python-only / Wolfram mirror deferred.  RELATIVE-DETERMINANT NOTE (2026-08-29, review wave 4): the correct object is the RELATIVE determinant Q(A) = log[det' Delta_A / det' Delta_{A0}] + 8 b1 c3^6 log phi(A); the BFK constant 2^{-4} is A/tau/alpha-INDEPENDENT at fixed cut number, so delta log 2^{-4} = 0 -- no physical compensation needed (consistent with COMPENSATION_INTERNAL + the reviewer's correction).  Absolute normalization is fixed by the reference A0.  Contract stays [O].  LATE EVENING (2026-08-30, v1014): finite restriction isomorphism verified (c_phase constancy 3.7e-16, conjugation covariance exact, C = W_bulk = W_seam = +1 / mirror -1); orientation fixes the conjugate branch (2 -> 1); the residual constant phase is fixed by the A0 reference normalization -- recorded.  Continuum Bismut-Freed stays [O].  Contract stays [O].",
      "status": "Named target [E]/[O] (the Quillen determinant-line variation-exactness; a face of SEAM.EQUIV.01, elevates EM.WARD.01's residual to a citable target; no new gate; 2026-08-28: the v974 CHANNEL-SWAP face resolved at the finite level by mu4-character grading, v985; completeness wave: finite KMS rigidity-premise [E] v995, ln-2-rate detline shadow v996 -- continuum Bismut-Freed stays [O])",
      "canonicalStatus": "[E] the Quillen split at the root + named-atom coefficients + the value alpha^-1=137.0359992 (reused from v3/v48/v341, machine-checked); [O] the from-first-principles exactness proof (the named target); [C] it is a face of SEAM.EQUIV.01, not a second independent gate. A typing/registration of the pre-existing EM.WARD.01 residual, reusing v341's (Wolfram-mirrored) identities; introduces no new number. UPDATE (2026-08-28, external master-object review wave 2, v985): the channel-swap face is resolved at the FINITE level -- the deck swap r -> r+2 maps the jump |spectrum| [0,2,4,2] onto the mark pattern |[4,-2,0,-2]| exactly (odd swap fails), the graded det' difference is c3-free = log(ln2/4), and the channelwise unit ratio is uniform 4/ln2; finite KMS rigidity-premise executed [E] (v995: 33 directions killed; 196608 -> 0); ln-2-rate detline shadow [E-measured] (v996, numerical not theorem); continuum Bismut-Freed, type-III KMS uniqueness and the from-first-principles exactness of F_U(1) stay [O] (BFK 2^{-4} needs no physical compensation: relative det, 2026-08-29).  | 2026-08-29: relative det Q(A)=log[det'Delta_A/det'Delta_{A0}]+8 b1 c3^6 log phi(A); BFK 2^{-4} A/tau/alpha-independent at fixed cut number (delta log 2^{-4}=0, no physical compensation).  Stays [O].  WAVE-5 (2026-08-29): rigidity premise of SEAM.MMST.TYPEIII.CHARGED.01; no marker move. | LATE EVENING (2026-08-30 v1014): finite restriction isomorphism verified; orientation fixes the conjugate branch; the residual constant phase is fixed by the A0 reference normalization -- recorded.  Stays [O].",
      "dependencies": "EM.WARD.01; SEAM.EQUIV.01; v341; v342; v3; v48; v216; v336",
      "scripts": [
        "v382_alpha_quillen_exact.py",
        "v985_quillen_channel_swap.py",
        "v995_kms_subnet_rigidity.py",
        "v996_gauge_grammar_kill.py",
        "v1014_bridge_refinements.py"
      ],
      "script": "v382_alpha_quillen_exact.py; v985_quillen_channel_swap.py; v995_kms_subnet_rigidity.py; v996_gauge_grammar_kill.py; v1014_bridge_refinements.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 445
    },
    {
      "id": "DYNAMICS.UNIVERSAL.01",
      "claim": "The UNIVERSAL spectral-gap / Perron-Frobenius principle (2026-06-23, bird's-eye synthesis): every TFPT sector is the SAME structural object -- a gapped operator with a UNIQUE leading attractor (the physics) and a spectral gap (the reason there is no free parameter). Extends v303 (which named this only for F_transfer) to the sectors made parameter-free AFTER it. [E] FLAVOR (computed): cusp transfer spectrum {1,(2/3)^6,(1/3)^6}, simple Perron leading 1, gap 6 ln(3/2)>0 (v56/v82/v240). [E] DISCRETE COMPILER (computed): affine-E8 adjacency Perron 2 (Kac marks, v312/v313), subleading 2cos(pi/5)=phi, gap 2-phi=1/phi^2>0. [E] DECOUPLING MARGIN (computed): Delta_eff=6 ln(3/2)-31/(4 pi^2)~1.648>0 (v76/v337). [E] TWO NUMBER-FIELD FACETS (v314): golden phi in Q(sqrt5) (carrier-5, static) and (2/3)^6 in Q (family-3, dynamic) are the prime-5 and prime-3 facets of the order-30 Coxeter clock 30=2*3*5=h(E8) -- a factorisation, NOT a forced equality. [C] OTHER SECTORS = SAME STRUCTURE: gravity=deltaS=0 equilibrium (v358/v359), QG=Gaussian saddle (v365), QFT=stable holomorphic c=8 RG fixed point (v157/v158), all-order S-matrix=adiabatic limit (v381), recovery=Petz fixed point (2/3)^6 (v221), F_transfer (v303), tau=i (v333). [C] META-THEOREM: gapped => unique attractor => parameter-free, theory-wide -- 'parameter-freeness is a theorem' (v56) is ONE spectral-gap statement wearing many hats. Anti-numerology: no new number.",
      "status": "Universal synthesis [E]/[C] (every sector = a gapped operator with a unique attractor + spectral gap => parameter-free; extends v303 theory-wide; 3 gaps computed, the cross-sector identification [C])",
      "canonicalStatus": "[E] the three computed gaps (flavor 6 ln(3/2); discrete 2-phi=1/phi^2; decoupling 1.648) + the two number-field facets (v314); [C] the cross-sector identification (gravity/QG/QFT/recovery/F_transfer) + the meta-theorem 'parameter-freeness = spectral gap, theory-wide'. A synthesis (like v303/v337/v353), reuses established gaps, introduces no new number.",
      "dependencies": "v56; v82; v303; v312; v313; v314; v337; v358; v359; v365; v381; v221; v157; v158",
      "scripts": [
        "v383_dynamics_universal.py"
      ],
      "script": "v383_dynamics_universal.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 446
    },
    {
      "id": "RESIDUAL.CERTIFICATION.01",
      "claim": "Historical compiler-residual audit (2026-06-23; scope corrected 2026-09-05): v384 classifies the then-listed compiler residual into three external kinds, (A) external math proof, (B) theorem-forbidden, and (C) external physics. It does not classify the physical completeness gates T1-T8 and does not imply zero open internal TFPT mechanisms. [E] LEDGER CI: every claim in that historical residual list is present in status_ledger.csv. [E] NO-UNIT: a dimensionless compiler cannot derive the absolute scale v_geo. [E] CHECKED GAP-DECOUPLING: Delta_eff=6 ln(3/2)-31/(4 pi^2)~1.648>0 for the audited readouts. [E] HISTORICAL CLASSIFICATION: SEAM.EQUIV.01 and ALPHA.QUILLEN.EXACT.01 -> A; QG.AMB.01 -> A; v_geo -> B; F_transfer -> C. [O] CURRENT TOE FENCE: TFPT.TOE.COMPLETE.01 and every T1-T8 gate remain open; missing dynamics and continuum mechanisms, including a shared 3+1D parent satisfying T3-T8, are outside v384's scope.",
      "status": "Historical compiler-residual audit [E]; explicitly not a TOE-completeness audit. TFPT.TOE.COMPLETE.01 and all T1-T8 remain Open [O].",
      "canonicalStatus": "[E] ledger CI, No-Unit dimensionlessness, checked gap-decoupling margin and three-way classification within the historical compiler residual only. [O] no statement about T1-T8 closure; the audit closes nothing and introduces no number.",
      "dependencies": "SEAM.EQUIV.01; ALPHA.QUILLEN.EXACT.01; QG.AMB.01; ANCHOR.VGEO.01; FR.POLE.SOLVE.01; TFPT.TOE.COMPLETE.01; v153; v336; v369; v337; v187",
      "scripts": [
        "v384_residual_certification.py"
      ],
      "script": "v384_residual_certification.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 447
    },
    {
      "id": "UVBRANCH.KILLTEST.01",
      "claim": "The optional carrier-Pati-Salam UV branch on the ALL-ORDER footing -- a single citable kill-test surface with the proton-decay SAFETY hierarchy computed (2026-06-24). A consolidation: does NOT re-run the RGE (v246 did 1+2-loop) nor re-derive the PS scale (v249/v253); it ADDS the all-order connection (v381) and the computed proton-safety margin (turning v265's 'no fake tau_p window' policy into a number). [E] CARRIER CONTENT = SM, ALL-ORDER: b=(41/10,-19/6,-7) (v159/v246), no new gauge-charged states (tfpt_3), S_pert closes to all orders (v381) => the SM-non-unification (v246) is all-order-robust. [X] PLAIN-SM 4D-GUT KILLED: sin^2 thW=3/8 vs measured 0.231, 1+2-loop couplings never meet (v246). [C] OPTIONAL CARRIER-PS BRANCH: M_PS=kappa*M_scal, M_scal=c3^{7/2}Mbar~3.06e13 GeV, kappa in [1.0,1.2] (2-loop v253), content {1,10,16,45} no 126 (v247/v248), NOT the default (v265 fork). [E] PROTON-DECAY SAFETY HIERARCHY: minimal PS gauge bosons are SU(4) leptoquarks (B-L-conserving q<->l = rare LFV K_L->mu e), NOT the p->e+ pi0 dim-6 operator (needs an SO(10)/SU(5) diquark, absent); the binding bound M_PS>~1e6 GeV is cleared by M_PS/M_bound~1e7>>1 => proton-safe. [X] FROZEN KILL TESTS (v265): the branch dies if kappa leaves O(1) / a forbidden rep (126) is required / a new gauge-charged state appears / a rare-LFV signal appears below the M_PS bound; NO discovery tau_p window.",
      "status": "UV-branch kill-test surface [E]/[X]/[C] (carrier-PS on the all-order footing; proton-SAFE hierarchy computed; plain-SM GUT killed; no fake tau_p)",
      "canonicalStatus": "[E] the all-order-robustness link (v381) + the proton-safety scale hierarchy (M_PS/M_bound~1e7); [X] the plain-SM-GUT kill + the frozen kill-test surface; [C] the optional PS branch. A consolidation (like v265/v337/v384); does NOT re-run the RGE or claim a tau_p discovery; no new number.",
      "dependencies": "v159; v246; v247; v248; v249; v253; v265; v381",
      "scripts": [
        "v385_uvbranch_killtest.py"
      ],
      "script": "v385_uvbranch_killtest.py",
      "location": "tfpt_5_redteam",
      "externalData": "PDG K_L->mu e bound; Super-K proton decay",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 448
    },
    {
      "id": "GRAV.AMPLITUDE.01",
      "claim": "The entire-form-factor graviton-exchange AMPLITUDE is finite, UV-softened and tree-unitary (2026-06-24) -- perturbative gravity as an explicit scattering problem, extending v304/v370/v380 from the propagator to the amplitude. The dressed spin-2 propagator P(p^2)=e^{-p^2/M^2}/p^2=1/(p^2 a) with a(u)=e^u (u=p^2/M^2), M=M_scal. [E] ONE POLE, GHOST-FREE: a=e^u entire and nowhere zero (v380) => the only pole is p^2=0 with residue lim p^2 P=e^0=1>0 (one healthy massless graviton). [E] UV-SOFTENED: the dressed/GR ratio e^{-u} at u=1,5,10,20 = 0.37,6.7e-3,4.5e-5,2.1e-9 -- suppressed faster than any power (infinite-derivative improvement), monotone->0. [E] LOW-ENERGY = GR + CALCULABLE CORRECTION: e^{-u}=1-u+u^2/2-...; the leading deviation -u=-p^2/M^2 (M=M_scal=c3^{7/2}Mbar~3.06e13 GeV, v253). [C] TREE UNITARITY: a single positive-residue pole, no extra poles / negative-norm states (entire zero-free form factor) => Im A>=0, consistent with v370 (spin-2) + v380 (entire). [C] AMPLITUDE STRUCTURE: A=(kappa^2/2)(T.P2.T')e^{-p^2/M^2}/p^2, GR at low energy, softened in the UV (P2 = Barnes-Rivers spin-2 projector, v370); perturbative-only (QG.AMB.01 stays a [C] redundancy, v369).",
      "status": "Perturbative graviton amplitude [E]/[C] (entire-form-factor exchange: finite, ghost-free, UV-softened, tree-unitary; perturbative-only)",
      "canonicalStatus": "[E] the single positive-residue pole + the UV softening (e^{-u} faster than any power) + the GR limit and first correction; [C] the tree unitarity and the amplitude structure (perturbative-only). Extends v304/v370/v380 from propagator to amplitude; does NOT touch the non-perturbative measure (QG.AMB.01); the scale M=M_scal (v253) and the form factor = the seam KMS cutoff (v259), no new number.",
      "dependencies": "v259; v304; v370; v380; v253; v369",
      "scripts": [
        "v386_grav_amplitude.py"
      ],
      "script": "v386_grav_amplitude.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 449
    },
    {
      "id": "CORRECTIONS.GAP.01",
      "claim": "The practical harvest of v383 (2026-06-24): the same spectral gap that makes each sector parameter-free ALSO sets the SIZE of its first correction -- for a gapped operator the deviation from the attractor decays at the ratio lambda_2/lambda_1 per step, so that ratio IS the leading-correction size. [E] FLAVOR: lambda_1=1, lambda_2=(2/3)^6=64/729 => first Koide correction ~0.0878 (v56/v82). [E] RECOVERY: the seam QECC Petz rate is the same (2/3)^6 (v221). [E] QG DECOUPLING: the ambient correction is bounded by chi=1/(1-(2/3)^6)=729/665 (finite, v337). [E] DISCRETE COMPILER: the main-branch update T_net=(A+2I)/4 fixes the Kac marks (lambda_1=1) and has subleading lambda_2=(phi+2)/4~0.9045 -- the compiler transient decays at the golden rate (carrier-5 facet, v303/v312). [E] UNIFIED: correction_n~(lambda_2/lambda_1)^n everywhere; the rates split into the (2/3)^6 family (flavor/recovery/QG, family-3 dynamic, in Q) and the golden family (the compiler, carrier-5 static, in Q(sqrt5)) -- the two number-field facets of v314/v383. [C] the correction calculus: the gap does double duty -- it is both WHY there is no free parameter and HOW BIG the first correction is, so the magnitude of sub-leading effects across TFPT is organised, not free.",
      "status": "Correction calculus [E]/[C] (each sector's gap sets its first-correction size; (2/3)^6 family vs golden family; the harvest of v383)",
      "canonicalStatus": "[E] the four computed correction rates + the unified (lambda_2/lambda_1)^n form + the two-facet split (Q vs Q(sqrt5)); [C] the correction-calculus reading. A harvest/application of v383 (as v303/v337 are of v56); reuses established eigenvalues, introduces no new number.",
      "dependencies": "v56; v82; v221; v303; v312; v314; v337; v383",
      "scripts": [
        "v387_corrections_gap.py"
      ],
      "script": "v387_corrections_gap.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 450
    },
    {
      "id": "CORRECTIONS.BUDGET.01",
      "claim": "The gap correction (v387) is a TYPED budget, NOT a uniform band on every prediction (2026-06-24): the (lambda_2/lambda_1)^n size is a property of the distance from a gapped operator's leading eigenvector, so it is only defined where a subleading lambda_2 exists. The prediction surface splits into FOUR correction classes (the verdict already implied by v303: of the four F_transfer instances only F_pole carries the seam rate). [E] SEAM-GAPPED class: Koide (F_pole, v303), horizon recovery (v221) and the QG-decoupling bound (capped by chi=1/(1-(2/3)^6)=729/665, v337) carry lambda_2=(2/3)^6=64/729; the discrete compiler decays at the golden (phi+2)/4 (carrier-5 facet, v312) -- the two number-field facets (v314/v383/v387). [E] KOIDE FIRST CORRECTION (door #2): F_pole is the ONE seam-gapped prediction, per-step Moebius rate (2/3)^6~0.0878 (v371), while the source quotient Q_src=0.66446 sits 0.33% below Q*=2/3 -- the rate is the suppression SCALE of the sub-leading texture, NOT the residual itself (an honest distinction). [E] EXACT-IDENTITY class: det R=8, N_Phi=1, theta_eff=0 are exact, sin^2 theta13=phi0 e^{-5/6} a closed form -- NO lambda_2 => correction 0 STRUCTURALLY (a (2/3)^6 band would contradict the [E]-identity status). [E] EXTERNAL-RATE class: eta_B (thermal washout), axion (cosmological freeze), m_p/m_e (RG) share only the gapped SHAPE with external rates, firewalled (v187). [E] FIXED-POINT class: the sub-leading texture is already explicit in the closed form (sin^2 theta12: eps=(3/4)phi0=c3+36 c3^4 EXACTLY). [C] THE BUDGET: a single uniform band on every prediction is WRONG; the size is keyed to the MECHANISM class. A typing of v387 (as v384 is of v383), no new number.",
      "status": "Typed correction budget [E]/[C] (the gap correction is keyed to the mechanism class, not a uniform band; seam-gapped=(2/3)^6/golden, exact-identity=0, external-rate=external, fixed-point=explicit texture)",
      "canonicalStatus": "[E] the four-class classification + the seam (2/3)^6 and golden (phi+2)/4 rates + the exact-identity zero-correction facts (det R=8, N_Phi=1, theta_eff=0) + the explicit theta12 texture (eps=c3+36 c3^4); [C] the typed-correction-budget reading. A typing of v387 (as v384 is of v383); reuses established eigenvalues and closed forms, introduces no new number.",
      "dependencies": "v303; v371; v383; v387; v314; v337; v221; v312; v187",
      "scripts": [
        "v388_corrections_budget.py"
      ],
      "script": "v388_corrections_budget.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 451
    },
    {
      "id": "GRAV.LOOP.FINITE.01",
      "claim": "The entire-form-factor graviton is UV-finite at the LOOP level by power counting (2026-06-24) -- the honest next step past v386 (which established the tree amplitude), extending v304/v370/v380/v386 from the propagator to the superficial degree of divergence of loop graphs. HONEST SCOPE: this is the standard infinite-derivative/nonlocal-gravity finiteness statement (Tomboulis; Biswas-Mazumdar-Siegel; Modesto); it is NOT a full proof of perturbative quantum gravity. [E] SUPER-POLYNOMIAL FALLOFF: for every polynomial degree k, p^k e^{-p^2/M^2}->0 as p->inf (k=2..20), so the dressed propagator cannot leave a power-counting divergence. [E] ONE DRESSED LINE KILLS A QUADRATIC DIVERGENCE: int_{p0}^Lambda p^3(e^{-p^2/M^2}/p^2)dp -> (M^2/2)e^{-p0^2/M^2} (converged at 10M/20M), while the GR int p^3/p^2 dp grows quadratically. [E] SUPERFICIAL DEGREE -> -infinity: GR has a fixed degree D=4L+2V-2I per L-loop graph; the entire form factor multiplies the integrand by e^{-(#internal)p^2/M^2}, so the effective degree is -infinity at every loop order L (dressed cutoff-ratio ->1 vs GR ->4). [E] the regulator is the SEAM scale M=M_scal=c3^{7/2}Mbar~3.06e13 GeV (v253) with the seam KMS form factor e^{-u} (v259), a derived scale, not a Pauli-Villars dial. [C]/[O] LOOP UNITARITY is the subtle OPEN point: tree unitarity holds (single positive-residue pole, v386), but unitarity of a nonlocal form factor at loop level (Lorentzian continuation/macro-causality, Pius-Sen/Briscese-Modesto) is genuinely subtle and still debated -- NOT claimed solved.",
      "status": "Perturbative graviton loop UV-finiteness [E] (power counting; super-polynomial falloff, degree -> -infinity at every loop order, seam-regulated) + loop unitarity [C]/[O] (the honest open point, not claimed solved)",
      "canonicalStatus": "[E] the Euclidean power-counting UV-finiteness at every loop order (super-polynomial falloff; one dressed line turns a quadratic divergence into a finite integral; the superficial degree -> -infinity), regulated by the derived seam scale M_scal (v253/v259); [C]/[O] full loop UNITARITY of the nonlocal form factor (the Lorentzian continuation / macro-causality, the genuinely open point), NOT claimed solved. Extends v386 from tree to loop FINITENESS only; introduces no new number.",
      "dependencies": "v253; v259; v304; v370; v380; v386",
      "scripts": [
        "v389_grav_loop_finiteness.py"
      ],
      "script": "v389_grav_loop_finiteness.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 452
    },
    {
      "id": "COXETER.PRIME2.01",
      "claim": "The prime-2 facet of the order-30 Coxeter clock, completing v383 (2026-06-24). The clock 30=2*3*5=h(E8) had two named number-field facets (the golden phi in Q(sqrt5), prime-5/carrier; (2/3)^6 in Q, prime-3/family); this names the prime-2 facet and identifies it with structure ALREADY in the theory: the Gaussian field Q(i), i.e. the CM point tau=i and the order-4 mu4=Z/4 square seam deck (v282/v288/v333). [E] THE CLOCK FACTORS INTO THREE PRIMES: h(E8)=30=2*3*5 (exactly three prime factors). [E] ONE QUADRATIC FACET FIELD PER PRIME, EACH RAMIFIED AT ITS PRIME: disc Q(i)=-4 (2 ramifies), disc Q(sqrt-3)=-3 (3 ramifies), disc Q(sqrt5)=5 (5 ramifies); ramified primes {2,3,5}, product 2*3*5=30=h(E8). [E] THE PRIME-2 (GAUSSIAN) FACET = the Z/4 seam deck: min poly of i is x^2+1, tau=i has j(i)=1728=12^3 and is fixed by the modular S of order 4=2^2 (S^2=-I); |mu4|=4=2^2 (v288) -- the static, even, seam-orientation facet (previously unnamed). [E] THE PRIME-5 (GOLDEN) FACET (v383): phi=(1+sqrt5)/2, min poly x^2-x-1, 2cos(pi/5)=phi (pentagon), carrier-5. [E] THE PRIME-3 (FAMILY) FACET (v383/v220): the dynamic rational (2/3)^6 in Q (family-3) with the Eisenstein zeta_3 (j=0, delta_CKM=pi/3, hexagonal CP) the prime-3 CM point. [C] COMPLETION: the three primes of the Coxeter clock now each carry a number-field facet -- Gaussian(2)/Eisenstein(3)/golden(5)=square/hexagon/pentagon=seam/CP/carrier; the prime-2 facet was the previously-unnamed one, and the prime set {2,3,5} is forced by h(E8)=30, not chosen.",
      "status": "Prime-2 facet of the Coxeter clock [E]/[C] (h(E8)=30=2*3*5; one quadratic facet field per prime, ramified at it; the prime-2/Gaussian/Z4-seam facet completes v383's golden and family facets)",
      "canonicalStatus": "[E] the exact arithmetic (h(E8)=30=2*3*5; discriminants -4,-3,5; ramified-prime set {2,3,5} with product 30; j(i)=1728=12^3; 2cos(pi/5)=phi); [C] the completion/identification reading. A synthesis (like v383); reuses the three smallest CM/real-quadratic fields, tau=i (v282) and the Z/4 deck (v288), introduces no new number.",
      "dependencies": "v220; v282; v288; v314; v333; v383; v387",
      "scripts": [
        "v390_prime2_facet.py"
      ],
      "script": "v390_prime2_facet.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 453
    },
    {
      "id": "FORM.SPECTRALGAP.01",
      "claim": "LEAN 4 FORMALISATION of the SCALAR CORE of the universal spectral-gap principle (door #4, 2026-06-24) -- new module TfptCarrier/SpectralGapAttractor.lean, lake build OK (target TfptCarrier.SpectralGapAttractor; aggregator TfptCarrier rebuilt, 3364 jobs), no sorry, only the three standard kernel axioms. It formalises the meta-theorem of v303/v383/v387 at the one-dimensional reduction: the affine gap contraction iter(r,x*,x0) with multiplier r=lambda_2/lambda_1. [F] iter_sub: the deviation has the closed form iter_n - x* = r^n (x0 - x*) (general n, induction). [F] iter_tendsto: 0<=r<1 (the spectral gap) => iter_n -> x* (the unique attractor), via Mathlib tendsto_pow_atTop_nhds_zero_of_lt_one. [F] starts_agree: two starts a,b differ by r^n (a-b) -> 0, so the attractor is INDEPENDENT of the initial state (parameter-freeness). [F] no_gap_no_forget: r=1 (no gap) => iter_n - x* = x0 - x* is CONSTANT (the start is a free parameter) -- the gap r<1 is exactly what removes it. [F] seam_rate_lt_one: (2/3)^6 in [0,1) (prime-3/family facet); golden_rate_lt_one: (5+sqrt5)/8=(phi+2)/4 in [0,1) (prime-5/carrier facet). FINITE-DIM EXTENSION (the multi-dimensional case, reduced to the scalar core): component_tendsto (a non-dominant eigencomponent |r|<1 decays r^n c -> 0); deviation_tendsto (with the dominant eigenvalue normalised to 1, the deviation vector -- all non-dominant components -- tends to 0, so the iterate converges to the dominant eigendirection = the unique attractor, for any start); deviation_unique (two starts give deviation vectors both -> 0 => the selected eigendirection is independent of the start = parameter-freeness); flavor_deviation_tendsto (concrete instance on the cusp-transfer spectrum {1,(2/3)^6,(1/3)^6}, v56/v82/v383). HONEST SCOPE: the scalar reduction PLUS the finite-dimensional diagonalisable case; the general non-diagonalisable / infinite-dimensional Perron-Frobenius statement stays the cited theorem.",
      "status": "Formal [F] (Lean kernel; no extra axioms) -- the scalar core + the finite-dimensional diagonalisable case of the spectral-gap meta-theorem (v383): gap => unique attractor + geometric decay => parameter-free",
      "canonicalStatus": "Formal: iter_sub (closed form r^n), iter_tendsto (gap => attractor), starts_agree (parameter-freeness), no_gap_no_forget (gap is essential), seam/golden rates in [0,1); PLUS the finite-dim case (component_tendsto, deviation_tendsto = convergence to the dominant eigendirection, deviation_unique = parameter-free, flavor_deviation_tendsto on {1,(2/3)^6,(1/3)^6}); the general non-diagonalisable / infinite-dim PF stays cited",
      "dependencies": "DYNAMICS.UNIVERSAL.01; CORRECTIONS.GAP.01; FR.DYNAMICS.01",
      "scripts": [
        "lean4-carrier-rigidity (SpectralGapAttractor.lean)"
      ],
      "script": "lean4-carrier-rigidity (SpectralGapAttractor.lean)",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 454
    },
    {
      "id": "ALPHA.QUILLEN.PROGRESS.01",
      "claim": "An HONEST attempt at the external target ALPHA.QUILLEN.EXACT.01 (door #5, 2026-06-24): it REDUCES the open step to a sharper named sub-target via a solvable model, but does NOT close it -- the status of ALPHA.QUILLEN.EXACT.01 stays [O] (external math, type A, v384). [E] SOLVABLE MODEL: det_zeta(-d^2/dtau^2 + m^2) on a circle of circumference beta = (2 sinh(beta m/2))^2 (textbook); its m-variation is the closed form beta*coth(beta m/2) = 2/m + (beta^2/6)m - (beta^4/360)m^3 + ... an exact heat-kernel/Laurent series (the m^3 coefficient -beta^4/360 = -1/360 verified). [E] ZETA-DET VARIATION = HEAT-KERNEL DATA: the variation is governed entirely by closed heat-kernel coefficients, so a zeta-determinant's coupling-variation is STRUCTURED anomaly data, not a free constant -- hence v382's counterterm 8 b1 c3^6 (a Gauss-Bonnet heat-kernel power, c3=1/(8 pi)) is the RIGHT KIND of object. [E] THE v382 SPLIT re-verified at alpha^-1=137.0359992 (8 b1=(4/5)*41 a U(1) index, c3^3/c3^6 heat-kernel powers). [E] THE REDUCTION: the open step now reduces to the named sub-target 'the seam U(1) operator's Seeley-DeWitt coefficient = 8 b1 c3^6 (with a_0,a_1 vanishing on the mu_4 seam moduli)' -- a heat-kernel computation, sharper than 'prove the functional identity'. [O] NOT CLOSED: computing the actual seam-operator zeta-determinant (not the toy) and showing its variation = the cubic is the from-first-principles AQFT/zeta proof (stays [O]); SEAM.EQUIV.01 continuum existence (MMST, v336) likewise stays the other external [O]; the value alpha^-1 stays [E].",
      "status": "Honest attempt / reduction [E]/[O] (a solvable-model zeta-det mechanism + the reduction of ALPHA.QUILLEN.EXACT.01 to the named sub-target 'seam a_2 = 8 b1 c3^6'; the headline target stays OPEN, not closed)",
      "canonicalStatus": "[E] the solvable model (det_zeta(-d^2+m^2) on a circle and its closed-form heat-kernel variation), the heat-kernel-data mechanism (so 8 b1 c3^6 is the right kind of object), the re-verified v382 split, and the reduction to a named sub-target; [O] the headline ALPHA.QUILLEN.EXACT.01 stays open (NOT closed here) and SEAM.EQUIV.01 continuum existence stays the other external [O]. An honest attempt that reduces, does not close; no new number, alpha^-1 stays [E].",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; SEAM.EQUIV.01; EM.WARD.01; v336; v341; v382",
      "scripts": [
        "v391_alpha_quillen_progress.py"
      ],
      "script": "v391_alpha_quillen_progress.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 455
    },
    {
      "id": "ALPHA.QUILLEN.PROGRESS.02",
      "claim": "A SECOND HONEST step on the external target ALPHA.QUILLEN.EXACT.01 (2026-06-27, v433): it GROUNDS and CONNECTS, it does NOT close it -- ALPHA.QUILLEN.EXACT.01 stays [O] (external math, type A, v384), alpha^-1 stays [E]. [E] SOLVABLE 4D MODEL REACHES a_4: the flat Delta=-d^2+m^2 on T^4 (vol V) has Tr e^{-tD}=V/(4 pi)^2 t^-2 e^{-t m^2}, Gilkey coeffs a_0=V/(4 pi)^2, a_2=-V m^2/(4 pi)^2, a_4=V m^4/(2(4 pi)^2); the t^0/a_4 conformal-anomaly order is NON-ZERO -- the order v391's 1D circle toy (a_0,a_1 only) could not reach. [E] IT IS THE ORDER v342 USES: v342 grounds the quadratic term in the Gilkey a_4 gauge-curvature coefficient 30/360=1/12 (Omega^2/12, Gilkey Thm 4.8.16); v433 exhibits that a_4 is the operative, computable order -- unifying v391 (a zeta-det variation IS heat-kernel data) with v342 (the seam term IS the Gilkey a_4). [E] MEASURE IS c3-BUILT: (4 pi)^-2=1/(16 pi^2)=(2 c3)^2=4 c3^2 in the seam 2D-boundary normalisation c3=(1/2)(4 pi)^-1. [E] the v382 split re-verified at alpha^-1=137.0359992. [O] NOT CLOSED: the actual seam U(1) zeta-det a_4 coefficient (=8 b1 c3^6) on the mu_4 seam moduli and the origin of the cubic a^3 Maxwell/Chern moment stay open (external math, type A, v384); the exact seam F-normalisation fixing the coefficient VALUE stays [C] (v342); SEAM.EQUIV.01 continuum existence (MMST, v336) stays the other external [O]; alpha^-1 stays [E].",
      "status": "Second honest step / grounding [E]/[O] (a solvable 4D model exhibiting the nonzero a_4 conformal-anomaly order + the v391<->v342 connection; the headline target stays OPEN, not closed)",
      "canonicalStatus": "[E] the solvable 4D model reaching the nonzero a_4 (t^0) conformal-anomaly order (the order v391's 1D circle could not reach and the very order v342 grounds via Gilkey 30/360=1/12), the v391<->v342 unification (a zeta-det variation IS heat-kernel data AT the a_4 order), the measure identity (4 pi)^-2=4 c3^2, and the re-verified v382 split; [O] the headline ALPHA.QUILLEN.EXACT.01 stays open (the actual seam a_4=8 b1 c3^6 and the cubic a^3 Chern term, v384), and the exact seam F-normalisation fixing the coefficient value stays [C] (v342). A second honest step that grounds/connects, does not close; no new number, alpha^-1 stays [E].",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.PROGRESS.01; SEAM.EQUIV.01; EM.WARD.01; v336; v341; v342; v382; v391",
      "scripts": [
        "v433_alpha_quillen_heatkernel.py"
      ],
      "script": "v433_alpha_quillen_heatkernel.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 456
    },
    {
      "id": "ALPHA.QUILLEN.PROGRESS.03",
      "claim": "A THIRD HONEST step on the external target ALPHA.QUILLEN.EXACT.01 (2026-06-28, v434): it settles the status of the three residuals named after v433 and shows they are NOT three independent open problems -- the target stays [O], alpha^-1 stays [E]. [E] b1 IS THE U(1) a_4 HEAT-KERNEL COEFFICIENT (the beta-function = a_4 theorem, Vassilevich): computed from the carrier hypercharge content with the standard spin weights (2/3 per Weyl fermion, 1/3 per complex scalar), sum (2/3)Y_f^2 + (1/3)Y_s^2 = 41/6 (SM normalisation) and (3/5)*that = 41/10 = b1 (GUT) -- so the b1 in the counterterm 8 b1 c3^6 is the a_4 heat-kernel coefficient of the carrier U(1)_Y content (same number as the beta, v159/v246), the SAME KIND of object as the alpha^2 Calderon a_4 (v342), NOT a fitted multiplicity. [E] the geometry factors off: 8 b1 c3^6 = b1*(rank E8)*c3^6 (rank E8=8 the chi=2 seam boundary, v216; c3^6 the six boundary insertions, v342). [C] RESIDUAL (1) 'exact a_4 = 8 b1 c3^6 on the mu_4 seam moduli' == RESIDUAL (3) 'the seam F-normalisation': with b1 and the c3-powers heat-kernel-forced, the only freedom left in the exact a_4 value is how the gauge curvature couples to the boundary measure -- ONE [C] obligation, not two. [E] RESIDUAL (2) located (not closed): a^3 - 2 c3^3 a^2 = a^2 (a - 2 c3^3), with a^2 the Gilkey gauge-curvature (Quillen/Chern) density (v342), so the leading a^3 = a*a^2 is the first MOMENT of the Chern density -- a topological level at boundary order c3^0, not a Seeley-DeWitt curvature integral; its from-first-principles (Chern-Simons-type) origin stays [O]. [E] NET REDUCTION: the EM-Ward residual is ONE [C] (residuals 1&3) + ONE [O] (residual 2), not three independent problems. [O] NOT CLOSED: the alpha^3 Chern level + the from-first-principles proof are external math (type A, v384); the seam F-normalisation stays [C]; SEAM.EQUIV.01 continuum existence (MMST, v336) stays the other external [O]; alpha^-1 stays [E].",
      "status": "Third honest step / grounding+reduction [E]/[C]/[O] (b1 = the U(1) a_4 heat-kernel coefficient via beta=a_4; residuals 1&3 collapse to one seam F-normalisation [C]; residual 2 the alpha^3 Chern level located but [O]; the headline target stays OPEN)",
      "canonicalStatus": "[E] b1 = 41/6 (SM) = 41/10 (GUT) is the U(1)_Y a_4 heat-kernel coefficient of the carrier content (beta=a_4 theorem), grounding the counterterm's matter factor; the geometry factors as 8 b1 c3^6 = b1*(rank E8)*c3^6; the cubic factors as a^2(a-2c3^3) (a^3 = the first moment of the Chern density, a level). [C] residual (1) 'exact a_4 = 8 b1 c3^6' == residual (3) 'seam F-normalisation' (ONE obligation). [O] residual (2) the alpha^3 Chern level + ALPHA.QUILLEN.EXACT.01 stay open. A third honest step that grounds, reduces (3 residuals -> 1[C]+1[O]) and locates, it does not close; no new number, alpha^-1 stays [E].",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.PROGRESS.01; ALPHA.QUILLEN.PROGRESS.02; EM.WARD.01; EM.WARD.02; v159; v216; v246; v341; v342; v382; v433",
      "scripts": [
        "v434_alpha_quillen_betafunction.py"
      ],
      "script": "v434_alpha_quillen_betafunction.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 457
    },
    {
      "id": "ALPHA.QUILLEN.PROGRESS.04",
      "claim": "A FOURTH HONEST step on the external target ALPHA.QUILLEN.EXACT.01 (2026-06-29, v435): it attacks the SINGLE remaining [O] left after v434 -- residual (2), the cubic alpha^3 Chern/Maxwell moment -- and SHARPENS it with a verifiable test plus a conditional typing, it does NOT close it; the target stays [O], alpha^-1 stays [E]. [E] pi-POWER / METRIC-INDEPENDENCE TEST: the three EM-closure coefficients carry pi-powers exactly {0,3,6} = their c3-orders -- Maxwell k0=1 -> pi^0/c3^0, Calderon -2 c3^3 = -1/(256 pi^3) -> pi^3, transport -8 b1 c3^6 = -41/(327680 pi^6) -> pi^6 -- so the Maxwell a^3 is the UNIQUE pi^0/c3^0 (metric-INDEPENDENT) term. A heat-kernel boundary coefficient ALWAYS carries the (4 pi)^{-d/2} measure (c3=1/(8 pi)); a metric-independent term carries none -- the defining signature of a TOPOLOGICAL term (a Chern-Simons/eta level does not see the metric, a curvature integral does). So the test PARTITIONS the closure into ONE topological level (a^3) + TWO heat-kernel boundary terms, confirming v342/v434's 'a^3 is a level, not a Seeley-DeWitt curvature integral' typing from a TEST rather than a claim. [E] DUAL LADDER: coupling power (3-n) at c3-order 3n for n=0,1,2 (the n=2 transport carries log(1/phi) in place of a^1); a^3 is the n=0 rung -- maximal coupling power, ZERO boundary insertions, the only bulk/topological rung. [C] CONDITIONAL LEVEL QUANTISATION: IF a^3 is the seam U(1) Chern-Simons/eta boundary level (the EM1 reading, v48 -- a [P] input, NOT proven here), large-gauge invariance (Dirac quantisation of the U(1) level on the closed seam 3-boundary) forces the level into Z, and k0=1 is the UNIT (minimal nonzero) level -- a topological integer, NOT a tunable real; this converts 'the coefficient happens to be 1' into 'the coefficient is an integer level and 1 is the unit'. [O] NOT CLOSED: the EM1 lemma itself (the from-first-principles proof that d_tau log Z_Maxwell = a^3 IS the seam U(1) CS/eta level -- the genuine Chern-Simons-boundary computation) stays external math (type A, v384); the seam F-normalisation stays the one [C] (residuals 1&3, v434); SEAM.EQUIV.01 continuum existence (MMST, v336) stays the other external [O]; alpha^-1 = 137.0359992 stays [E].",
      "status": "Fourth honest step / metric-independence test + conditional level typing [E]/[C]/[O] (the pi-power test isolates a^3 as the unique pi^0/c3^0 metric-independent topological rung; conditional on the CS/eta reading the a^3 coefficient is an integer level, =1 the unit; the EM1 CS-boundary proof and the headline target stay OPEN)",
      "canonicalStatus": "[E] the pi-power/metric-independence test isolates a^3 as the UNIQUE pi^0/c3^0 (metric-independent) term -- the signature of a topological level, NOT a Seeley-DeWitt curvature integral (the other two carry pi^3, pi^6); plus the dual ladder (coupling 3-n at c3-order 3n). [C] conditional on the CS/eta reading (EM1, v48) the a^3 coefficient is an integer level, =1 the unit level, not a free real. [O] the EM1 CS-boundary derivation + ALPHA.QUILLEN.EXACT.01 stay open; seam F-norm stays the one [C] (v434); alpha^-1 stays [E]. A fourth honest step that sharpens the only remaining [O]; no new number, Python-only.",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.PROGRESS.01; ALPHA.QUILLEN.PROGRESS.02; ALPHA.QUILLEN.PROGRESS.03; EM.WARD.01; EM.WARD.02; v48; v342; v382; v384; v433; v434",
      "scripts": [
        "v435_alpha_quillen_chernlevel.py"
      ],
      "script": "v435_alpha_quillen_chernlevel.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 458
    },
    {
      "id": "SEAM.S3.SCALINGLIMIT.01",
      "claim": "An HONEST attack on the one open lemma of SEAM.EQUIV.01 -- the continuum scaling-limit existence (2026-06-24). It does NOT close v336 (the abstract existence is the cited MMST theorem); it STRENGTHENS the case with the existence-relevant observable v376 did not test: that the finite-size data CONVERGE (a divergent/drifting sequence would have no continuum limit). [E] MULTI-SIZE CONVERGENCE: the Calabrese-Cardy correlation-matrix EE slope c1(L) at L=66,130,258,386,514 is a monotone sequence approaching 1, with |c1(L)-1| strictly decreasing from 4.6e-4 to 7.1e-6. [E] RICHARDSON EXTRAPOLATION: a 1/L fit gives c1(inf)~0.9999, so the 16-Majorana (=8 complex mode) collar has c=8*c1(inf)~7.9993~8=g_car+N_fam -- the (E8)_1/SO(16)_1 continuum value reached IN the limit. [E] CAUCHY SIGNATURE: the successive differences |c1(L_{k+1})-c1(L_k)| are themselves strictly decreasing and tiny -- a numerical Cauchy sequence (necessary, not sufficient, for existence). [O] convergent finite-size DATA are strong evidence but NOT a proof that the continuum limit exists (the cited MMST theorem, v336), and do not pin (E8)_1 over the same-c SO(16)_1 (the holomorphy det K=1 discriminator, v377/v378). Strengthens the residual, does NOT close it -- SEAM.EQUIV.01 stays closed modulo the cited theorem.",
      "status": "Scaling-limit convergence evidence [E] (the finite-size central charge converges to c=8 and Richardson-extrapolates there -- the existence-relevant observable) + the abstract existence stays [O] (cited MMST v336); an honest attack that strengthens, does not close",
      "canonicalStatus": "[E] the multi-size convergence (monotone Cauchy-like c1(L)->1) + the Richardson extrapolation to c=8 + the decreasing successive differences (numerical evidence the scaling limit exists and gives c=8); [O] the abstract continuum existence (cited MMST v336) and the (E8)_1-vs-SO(16)_1 holomorphy discriminator (v377/v378) stay open. Extends v376 from the value (two sizes) to the convergence (five sizes); strengthens v336's residual, does not close it.",
      "dependencies": "SEAM.EQUIV.CONTINUUM.01; v336; v367; v368; v376; v377; v378",
      "scripts": [
        "v392_seam_s3_scalinglimit.py"
      ],
      "script": "v392_seam_s3_scalinglimit.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 459
    },
    {
      "id": "CORRECTIONS.NUMERIC.01",
      "claim": "The typed correction budget (v388) turned into ACTUAL first-correction magnitudes -- a computed theoretical-uncertainty surface (2026-06-24), the short-term testable harvest of v388: instead of merely TYPING each prediction's correction, this gives the NUMBER, so a parameter-free central value can be quoted as 'value +- computed first correction'. [E] FIXED-POINT band: the phi0-derived fixed points (theta12, theta13, lambda_C, beta_rad) carry the EXPLICIT phi0 puncture term as their first correction, dtop/phibase = 36 c3^4/c3 = 9/(128 pi^3) ~ 0.227% (exact, no fit); the SAME number is theta12's epsilon first-correction (eps=(3/4)phi0=c3+36 c3^4). [E] SEAM-GAPPED band: the seam-gapped class (Koide F_pole, recovery) has rate (2/3)^6=64/729 ~ 8.78%; the QG-decoupling bound is CAPPED by the susceptibility excess chi-1 = (2/3)^6/(1-(2/3)^6) = 64/665 ~ 9.62% (finite, v337). [E] EXACT-IDENTITY band = 0 STRUCTURALLY (det R=8, N_Phi=1, theta_eff=0 are exact -- a band would contradict [E]). [E] EXTERNAL-RATE band = external physics, quoted from the modules (m_p/m_e +-7.5% v374; n_s/r N* in [50,60]; eta_B ~1.07x v212; axion scenario), fenced by v187 -- NOT a seam band. [C] the assembled per-class numeric budget gives each prediction a 'central value +- computed first correction'.",
      "status": "Numeric correction budget [E]/[C] (the typed budget v388 as actual first-correction magnitudes: fixed-point 0.227%, seam-gapped 8.78%/QG-capped 9.62%, exact-identity 0, external quoted -- a computed theoretical-uncertainty surface)",
      "canonicalStatus": "[E] the four computed bands (fixed-point 9/(128 pi^3)~0.227%, seam-gapped (2/3)^6~8.78%, QG cap 64/665~9.62%, exact-identity 0) + the external quoted bands; [C] the assembled numeric budget. A harvest of v388 into testable numbers (theoretical error bars); reuses the phi0 puncture, (2/3)^6 and the chi cap, introduces no new number.",
      "dependencies": "CORRECTIONS.BUDGET.01; v388; v337; v374; v212; v187",
      "scripts": [
        "v393_corrections_numeric.py"
      ],
      "script": "v393_corrections_numeric.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 460
    },
    {
      "id": "COXETER.ATOMS.01",
      "claim": "The clockwork coherence (2026-06-24, bird's-eye): the three Coxeter primes ARE the three anchor atoms ARE the three number-field facets, and the prime-2 seam is the common dynamical factor. A re-reading tying v390 (facets) to the anchor a=(1,1,2) and v319 (5x6 clock). [E] ANCHOR ATOMS = COXETER PRIMES: a=(1,1,2) has e3=2=|Z2|, p0=3=N_fam, e2=5=g_car, product 2*3*5=30=h(E8) -- the clock's three primes ARE the anchor's three atoms. [E] ATOMS = FACET FIELDS (v390): 2->Q(i) (disc -4), 3->Q(sqrt-3) (disc -3), 5->Q(sqrt5) (disc 5), each ramified at its prime; {atoms}={primes}={fields}. [E] QG SUSCEPTIBILITY PURE {2,3}: chi=1/(1-(2/3)^6)=729/665=3^6/(3^6-2^6), (2/3)^6=2^6/3^6, exponent 6=2*N_fam. [E] PRIME-2 COMMON DYNAMIC FACTOR: 2/3=|Z2|/N_fam (prime-2/prime-3) and (phi+2)/4=(phi+2)/|mu4| with |mu4|=4=2^2, so both dynamic rates carry prime-2 (the seam involution). [E] 30=5x6=g_car*(2 N_fam) (v319). [C] the prime-2 seam is the universal dynamical engine; prime-3 (family) and prime-5 (carrier golden) the two attractors.",
      "status": "Clockwork coherence [E]/[C] (Coxeter primes = anchor atoms = facet fields; chi=3^6/(3^6-2^6); both dynamic rates carry prime-2; 30=5x6) -- a bird's-eye re-reading, no new number",
      "canonicalStatus": "[E] the exact arithmetic (atoms=primes=fields; chi=3^6/(3^6-2^6); 2/3=|Z2|/N_fam; 4=|mu4|=2^2; 30=5x6); [C] the seam-as-universal-dynamic-engine reading. A re-reading of established atoms/eigenvalues (v390/v319), introduces no new number.",
      "dependencies": "v314; v319; v383; v387; v390; v23; v53",
      "scripts": [
        "v394_coxeter_atoms.py"
      ],
      "script": "v394_coxeter_atoms.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 461
    },
    {
      "id": "HYP.COUPLED.01",
      "claim": "HYP.COUPLED.01: finite local carrier/family recursion F(m)=T_net m M^T, checked on all 27 basis inputs; full v486/v487 M has spectrum {1,2/3,1/3}. The joint Perron line is marks tensor e0. One hand is one network step and six family substeps, T_net tensor M^6, not T_micro^6. Exact even/odd conjugacy O^T M^6 O=diag(1,64/729,1/729) supplies the complete v324 readout. v299 growth is unchanged. P2 and phi0 remain inputs. No physical OS-transfer or local charged-field identification is claimed. Correction 2026-10-27: the former routine discarded its network update and its test did not call that routine; the former v327 matrix also erased the odd mode.",
      "status": "Coupled hypergraph rewrite [E]/[O] (one local rule unifies carrier x family; joint attractor marks x e0; recovery (2/3)^6 after one hand; v324 fiber emerges as cusp readout; v299 growth unchanged) -- mechanism unified, phi0 + seed remain",
      "canonicalStatus": "[E] finite local recursion on 27 cells, actual-function basis check, joint attractor, full three-mode basis conjugacy, and scheduled 1:6 hand; [O] P2 and phi0 inputs. No physical OS-transfer or charged-field derivation. Existing status retained; executable defect corrected 2026-10-27.",
      "dependencies": "HYP.0D.01; HYP.FIBER.01; HYP.REWRITE.01; HYP.REWRITE.02; HYP.REWRITE.03; v299; v324; v327; v486; v487",
      "scripts": [
        "v395_hypergraph_coupled.py"
      ],
      "script": "v395_hypergraph_coupled.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 462
    },
    {
      "id": "HYP.PHI0.01",
      "claim": "THE phi0 LEADING TERM FROM ICOSAHEDRAL COMBINATORICS (2026-06-24, sharpening v312/v395): the tree-level retained seed phibase = 1/(6 pi) has a forced combinatorial reading on the icosahedral hypergraph; the puncture 48 c3^4 stays analytic. [E] ICO COUNTS ANCHOR-FORCED: V=12, E=30=h(E8), F=20=g_car(g_car-1), |Aut|=120=4h=|2I| (all from {g_car,N_fam}={5,3}). [E] LEADING TERM EXACT: phibase = 1/(6 pi) = (4/3) c3 = F/(4 h pi) = F/(|Aut| pi) = (F/(g_car N_fam)) c3. [E] F/h = |Z2|/N_fam (hyperedge/Coxeter = v327 survival); phibase = (F/h)/(4 pi) = c3 (g_car-1)/N_fam. [E] GAUSS-BONNET: icosahedron vertex-deficit sum = 4 pi; c3 = 1/(|Z2| 4 pi) same boundary datum (v216/v342). [E] HONEST NEGATIVE: puncture dtop = 48 c3^4 not a rational hypergraph fraction; full phi0 still has analytic piece. [C] phibase is tree-level seed = (hyperedges)/(g_car N_fam) x c3, not an independent rewrite injection; puncture stays [O].",
      "status": "phi0 leading term [E]/[C]/[O] (phibase = F/(g_car N_fam) c3 = 1/(6 pi); F/h = |Z2|/N_fam; Gauss-Bonnet consistent; puncture 48 c3^4 still analytic) -- combinatorial reinterpretation, not full phi0 closure",
      "canonicalStatus": "[E] exact combinatorial identities + Gauss-Bonnet + puncture negative; [C] tree-level seed reading; [O] puncture term. Sharpens v312/v395; does not close full phi0.",
      "dependencies": "v312; v216; v342; v327; v395",
      "scripts": [
        "v396_phi0_icosahedral.py"
      ],
      "script": "v396_phi0_icosahedral.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 463
    },
    {
      "id": "HYP.PHI0.PUNCTURE.01",
      "claim": "NAMES THE phi0 PUNCTURE TERM AS A TRACKED [O] HEAT-KERNEL TARGET (2026-06-24, v408): dtop = Omega_adm c3^4 = 48 c3^4 = 3/(256 pi^4) IS the order-4 (|mu4|) local boundary Seeley-DeWitt contact term of the mu4 puncture divisor, summed over Omega_adm = N_fam dim S+ = 48 admissible fermion states -- the analytic complement of the icosahedral tree term phibase (v396, HYP.PHI0.01). Elevates v396's flagged puncture residual to its own citable target WITHOUT a new number (a FACE of the seam boundary analysis, not a new gate). [E] THE SPLIT IS EXACT: phi0 = phibase + dtop = 1/(6 pi) + 48 c3^4 (v396/ARCH.QUAD.01). [E] PUNCTURE COUNT = Omega_adm = N_fam dim S+ = 48, order = |mu4| = 4. [E] CLOSED FORM dtop = 3/(256 pi^4); c3 = 1/(|Z2| 4 pi) Gauss-Bonnet (v216/v342) so c3^4 is a fourth-order boundary heat-kernel power (the right KIND of object, cf. v391). [E] HONEST NEGATIVE: dtop not a rational hypergraph fraction (dtop/phibase < 1%). [O] THE TARGET: the from-first-principles heat-kernel proof that Omega_adm c3^4 IS the order-4 boundary contact term. [C] two engines, two terms: the hypergraph gives phibase, the boundary heat-kernel gives dtop -- a FACE of the seam boundary analysis (the c3 datum), not a second independent gate. No new number (reuses v396/ARCH.QUAD.01, Wolfram-mirrored under ARCH.QUAD.01/v37). UPDATE 2026-07-14 (v483/HYP.PHI0.GEOM.01): the geometric/equivariant side is EXACT (all twisted pillowcase traces t-independent rationals), so the target narrows to the ONE rule 'per-mark insertion weight = c3'; stays [O]. UPDATE 2026-07-14 (v484/SEAM.CONTACT.UNIT.01): the per-mark weight rule MERGES with ALPHA.QUILLEN.EXACT.01 into one analytic step -- the contact unit is the KMS seam unit (v239), derived for the finite cycle sector; the remaining [O] is the diagonal zeta-renormalisation + multiplicity matching (48 = Omega_adm). UPDATE 2026-07-14 (v485): the computable side is COMPLETE (the flat 48-count is the same state set as alpha's Y^2-weighted 41; diagonal zero at the KMS scale); the remaining [O] is the SEAM.EQUIV.01 face alone.",
      "status": "Named target [E]/[O] (the phi0 puncture 48 c3^4 as the order-4 boundary Seeley-DeWitt contact term of the mu4 puncture, summed over Omega_adm=N_fam dim S+=48; a face of the seam boundary analysis, elevates v396's puncture residual to a citable target; no new gate)",
      "canonicalStatus": "[E] the exact split phi0 = 1/(6 pi) + 48 c3^4, the puncture count Omega_adm=N_fam dim S+=48, the closed form 3/(256 pi^4), the hypergraph negative (reused from v396/ARCH.QUAD.01, machine-checked); [O] the from-first-principles heat-kernel-contact-term proof (the named target); [C] a face of the seam boundary analysis, not a second independent gate. A typing/registration of v396's puncture residual; introduces no new number.",
      "dependencies": "HYP.PHI0.01; ARCH.QUAD.01; v396; v312; v216; v342; v391",
      "scripts": [
        "v408_phi0_puncture_heatkernel.py"
      ],
      "script": "v408_phi0_puncture_heatkernel.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 464
    },
    {
      "id": "RES.COXETER.SYMMETRY.01",
      "claim": "CLOSED AS A STRUCTURAL LEMMA (2026-06-24, v409): prime-2 (the seam sheet involution) is NECESSARY but NOT autonomous, so no prime-2-only attractor exists -- resolving the v394 open question's falsifiable corollary. The facets are prime-2 (Q(i)/tau=i/Z4 seam deck, static) + prime-3 (Eisenstein zeta3 CP, static; (2/3)^6, dynamic) + prime-5 (golden, static; (phi+2)/4, dynamic). The prime-2 seam appears as the common FACTOR in BOTH dynamic rates (2/3=|Z2|/N_fam; (phi+2)/4=(phi+2)/|mu4|, |mu4|=2^2), NOT as a separate attractor. THE LEMMA (v409): the sheet involution sigma=diag(1,-1,-1) has spectrum {+1,-1} and projector (I+sigma)/2 spectrum {0,1} -- none in the open interval (0,1), so prime-2 supplies parity/projection, never a nonzero contraction rate; the genuine rates 2/3 and (1/3)^6 lie in (0,1) and carry prime-2 only as a FACTOR (2/3=|Z2|/N_fam needs prime-3; (phi+2)/4, |mu4|=2^2, needs the golden phi/prime-5; 30=5x6=g_car*(2 N_fam), carrier static, 2*3 dynamic). [E] So no prime-2-only attractor exists; [C] prime-2 is the universal dynamical parity factor, necessary in every sheet-paired transport, not autonomous -- dynamics begins only after coupling to family-3 or carrier-5 (mu4=2^2 the Galois bridge). This closes the falsifiable corollary and sharpens v383's 'every sector is the same gapped operator' towards 'the same SEAM-driven operator'; the full 'appears in every gap' universality stays the v383 reading.",
      "status": "Structural lemma [E]/[C] (prime-2 necessary, not autonomous: the sheet involution has spectrum {+-1} and projector {0,1}, none in (0,1), so no prime-2-only attractor; reuses established atoms, no new number) -- closes the v394 open-question corollary",
      "canonicalStatus": "[E] the sheet involution/projector spectra carry no contraction rate in (0,1), so no prime-2-only attractor exists; [C] prime-2 is the universal dynamical parity factor (necessary, not autonomous), dynamics needs coupling to family-3 or carrier-5. Closes the v394 open question's falsifiable corollary; the full 'appears in every gap' universality stays the v383 reading. No new number.",
      "dependencies": "COXETER.ATOMS.01; v314; v383; v390; v394",
      "scripts": [
        "v394_coxeter_atoms.py",
        "v409_coxeter_prime2_lemma.py"
      ],
      "script": "v394_coxeter_atoms.py; v409_coxeter_prime2_lemma.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 465
    },
    {
      "id": "FORM.COXETER.PRIME2.01",
      "claim": "LEAN 4 FORMALISATION of the RES.COXETER.SYMMETRY.01 falsifiable corollary (2026-06-24): new module TfptCarrier/CoxeterPrime2.lean machine-proves 'no prime-2-only attractor exists', upgrading the v409 Python check to a Lean theorem. [E] involution_eigenvalue: r*r=1 -> r in {1,-1}; idempotent_eigenvalue: r*r=r -> r in {0,1}; involution_not_contraction / idempotent_not_contraction: neither set meets the open interval (0,1); seam_rate_is_contraction: 2/3 in (0,1); no_prime2_only_attractor: the conjunction (parity + projection carry no contraction rate, while 2/3 does). [E] dynamical reading via the SpectralGapAttractor core: parity_eigenvalue_remembers_start (r=1 keeps the start = no attractor) vs seam_rate_has_attractor (r=2/3 converges to the attractor). Built clean (lake build, 3286 jobs); the new theorems are axiom-pure (#print axioms = propext, Classical.choice, Quot.sound only), and #check (AuditCheck) + signature-lock (AuditContract) pass. Strengthens v409/RES.COXETER.SYMMETRY.01 to the machine-proof tier.",
      "status": "Lean formalisation [E] (no prime-2-only attractor: involution eigenvalues +-1, projector {0,1}, none in (0,1), while 2/3 is a contraction rate; axiom-pure) -- machine-proves the v409 corollary",
      "canonicalStatus": "[E] the involution/projector eigenvalue lemmas + the no-contraction-rate corollary + the dynamical reading, machine-proved in TfptCarrier/CoxeterPrime2.lean (axiom-pure: only the three standard kernel axioms). Upgrades v409 to the Lean tier; no new number.",
      "dependencies": "RES.COXETER.SYMMETRY.01; COXETER.ATOMS.01; v394; v409",
      "scripts": [
        "v409_coxeter_prime2_lemma.py"
      ],
      "script": "v409_coxeter_prime2_lemma.py",
      "location": "tfpt_research_contracts",
      "externalData": "Lean 4 + Mathlib",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 466
    },
    {
      "id": "FR.CLOCK.PROBE.01",
      "claim": "A FALSIFIABLE probe of how external the 'external physics' really is (2026-06-24): do the frontier scheme-numbers land on the {2,3,5} clock (= the anchor atoms |Z2|,N_fam,g_car, v394) or not? Strict No-Free-Pattern discipline (v100): 'on the clock' means an EXACT match to a ratio/power of {2,3,5}/golden, NOT a wide-band fit. [E] eta_B DECUPLE: A_Lambda=10=|Z2|*g_car=2*5=p3(anchor) ON CLOCK (v212). [E] KOIDE RATE: the F_pole Moebius multiplier (2/3)^6=(|Z2|/N_fam)^6 ON CLOCK (v303/v82). [E] AXION SPINE ANGLE: the robust DM branch theta_i=pi*N_fam/g_car=3pi/5=108deg ON CLOCK (v211/v373). [E] PROTON C_p~2.83+-0.15 is NOT a discriminating clock value: within its +-5.3% band multiple distinct simple values fit (2^{3/2}=2.828, 17/6=2.833, 14/5=2.800), so a clock match is non-unique => by No-Free-Pattern C_p is EXTERNAL (firewall v187/v374 correct), NOT assigned a clock value. [C] VERDICT: 3 of the 4 frontier scheme-numbers land EXACTLY on the {2,3,5}/golden clock; the IRREDUCIBLE external residue shrinks to essentially the proton's QCD number C_p (pure non-perturbative QCD) -- 'external physics' was OVERSTATED.",
      "status": "Clock probe of the frontier [E]/[C] (3 of 4 'external' scheme-numbers -- A_Lambda, Koide rate, axion spine -- land EXACTLY on the {2,3,5} clock; only the proton C_p is genuinely external; a falsifiable anti-numerology test)",
      "canonicalStatus": "[E] the three exact clock matches (A_Lambda=2*5; Koide=(|Z2|/N_fam)^6; axion=pi*N_fam/g_car) + the proton C_p non-uniqueness (multiple simple values in its +-5.3% band => non-discriminating => external); [C] the verdict that the external residue shrinks to essentially C_p. A falsifiable probe (the answer could have been '0 on the clock'); refuses to assign C_p a clock value (anti-numerology); reuses established atoms/rates, introduces no new number.",
      "dependencies": "v100; v187; v211; v212; v303; v374; v394",
      "scripts": [
        "v397_external_clock_probe.py"
      ],
      "script": "v397_external_clock_probe.py",
      "location": "tfpt_4_frontier",
      "externalData": "PDG/lattice C_p, alpha_s",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 467
    },
    {
      "id": "SEAM.RIGIDITY.01",
      "claim": "THE SEAM STATE RIGIDITY THEOREM AS ONE NAMED TARGET (2026-06-24, Paper A / the G_net closure contract): consolidates the scattered state-invariance reductions (v177/v198/v199/v308/v309/v335) into a single citable theorem and HARDENS the finite-block evidence to a multi-mode sweep. Does NOT close SEAM.EQUIV.01 (the continuum scaling-limit existence stays the cited MMST/Adamo step, v336/v392). [E] RP-DEFINABILITY HINGE: OS reconstruction gives a canonical transfer operator T=e^{-H} from the RP state + reflection alone (v54) on a quasi-free state (v155), so Lambda_Sigma is RP-definable WITHOUT importing the mu4 normal form (non-circular). [E] GROUND-STATE REDUCTION: for a quasi-free RP state C=1/2(1+sgn H_1), so omega o rho=omega <=> [rho,H_1]=0 <=> H_1 block-diagonal in the four mu4-character classes. [E] BAND-LIMITED CORE (the hardening): [rho,|k|]=0 exactly and a character-block-diagonal sub-principal symbol commutes with rho while a single off-character entry breaks it, swept N=4..64 (beyond the 3-dim H^1 of v177/v199). [E] HOLOMORPHY=>(E8)_1: c=g_car+N_fam=8, |det Cartan(E8)|=1 (one primary) vs |det Cartan(D8)|=4, index 4=|mu4|, gap 6 ln(3/2)>0 -- the unique even-unimodular rank-8 lattice net (v83/v154/v308). [O] THE ONE RESIDUAL: [rho,Lambda_Sigma]=0 on the FULL L^2 <=> intrinsic Bisognano-Wichmann modular geometricity, the same [O] as v199/v309, now stated as the single theorem residual.",
      "status": "Seam State Rigidity contract+evidence [E]/[O] (Paper A: the G_net keystone as one named theorem -- RP-definability hinge + band-limited mu4-character core swept N=4..64 + holomorphy=>(E8)_1; the one residual = full-L^2 / intrinsic BW). NOT a closure of SEAM.EQUIV.01",
      "canonicalStatus": "[E] the RP-definability hinge (OS canonical transfer + quasi-free) + the band-limited character-block-diagonal core swept N=4..64 (hardens v177/v199 beyond H^1) + the holomorphy=>(E8)_1 arithmetic (det Cartan 1 vs 4, c=8, index 4); [O] the full-L^2 / intrinsic-Bisognano-Wichmann step (G_net's irreducible analytic residual). A contract+evidence consolidation (genre of v275/v343/v384/v392); NOT a closure of SEAM.EQUIV.01 (continuum=cited MMST v336); no new number.",
      "dependencies": "SEAM.EQUIV.01; v54; v155; v177; v198; v199; v308; v309; v335; v83; v154; v336",
      "scripts": [
        "v398_seam_state_rigidity.py"
      ],
      "script": "v398_seam_state_rigidity.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 468
    },
    {
      "id": "GRAVITY.COMPLETE.01",
      "claim": "GRAVITY-COMPLETE = BOUNDARY-COMPLETE + AMBIENT-REDUNDANCY (2026-06-24, Paper B / the gravity closure accounting): the explicit verdict combining the parameter-free LOCAL field equation (v358/v359) and the ambient REDUNDANCY (v369). Does NOT construct the ambient measure and does NOT close SEAM.EQUIV.01. [E] LOCAL EQUATION PARAMETER-FREE: G_ab+Lambda g_ab=(1/c3)T_ab with 1/c3=8pi (the four c3-roles consistent: Unruh 1/(2pi)=4c3, EH 1/(16pi)=c3/2, Einstein 8pi=1/c3, Bekenstein 1/4=1/|mu4|), Lambda prefactor (8pi)^2*48 c3^4=3/(4pi^2). [E] REDUNDANCY DISCRIMINATORS: holomorphic (E8)_1 => DHR=Vec (|det Cartan|=1 vs 4) + torus GSD=1, Petz recovery (2/3)^6, gap margin Delta_eff=6 ln(3/2)-31/(4pi^2)~1.648>0. [E] O_phys SUBSET A_Sigma: the five physically reachable gravitational readout classes (Newton 8pi, Lambda 3/(4pi^2), scalaron c3^7, horizon density 1/4=1/|mu4|, BH recovery (2/3)^6) are all boundary/seam/gap readouts -- the bulk ambient measure adds none. [C] THE VERDICT: Gravity-complete (TFPT sense) = Boundary-complete + Ambient-redundancy. [O] RESIDUAL (no new gate): SEAM.EQUIV.01 + intrinsic Bisognano-Wichmann (v369/v329).",
      "status": "Gravity-completeness accounting [E]/[C]/[O] (Paper B: the explicit verdict = local parameter-free equation + ambient redundancy; O_phys subset A_Sigma over 5 readout classes; residual = SEAM.EQUIV.01 + intrinsic BW, no new gate)",
      "canonicalStatus": "[E] the parameter-free local-equation coefficients (8pi=1/c3, Lambda prefactor 3/(4pi^2)) + the redundancy discriminators (det 1 vs 4, GSD=1, Petz (2/3)^6, margin 1.648) + the O_phys-subset-A_Sigma enumeration over 5 readout classes; [C] the completeness verdict; [O] SEAM.EQUIV.01 + intrinsic BW (no new gate). A completion-accounting module (genre of v384); no construction, no new number.",
      "dependencies": "GRAV.NONLINEAR.01; QGAMB.REDUNDANCY.01; QG.AMB.01; v358; v359; v369; v60; v36; v54",
      "scripts": [
        "v399_gravity_complete.py"
      ],
      "script": "v399_gravity_complete.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 469
    },
    {
      "id": "GRAV.NONLOCAL.01",
      "claim": "THE NONLOCAL SPECTRAL-GRAVITY ACTION; LOCAL R+R^2 IS ITS IR PROJECTION (2026-06-24, Paper C / the 4D gravity-action contract): consolidates v304/v370/v380/v386 and adds the IR-matching check. Does NOT prove the entire-analyticity of the resummed form factor (stays the cited heat-kernel resummation, the [O] of v304/v380); PERTURBATIVE only. [E] ENTIRE FORM FACTOR -> ONE POLE: the dressed spin-2 propagator 1/(p^2 a) with a(u)=e^u has its ONLY pole at p^2=0 (residue a(0)^{-1}=1>0, one healthy massless graviton, no Stelle ghost). [E] EVERY TRUNCATION CARRIES THE GHOST: T_1=1+u has the Stelle zero u=-1, and the nearest-zero modulus grows monotonically with order -> infinity, so 'entire'='do not truncate' (v380). [E] IR PROJECTION = R+R^2: a(u)=1+u+u^2/2+... reproduces the spectral-action R+R^2 structure (v28/v36) with the FIXED scale M_scal^2/Mbar^2=c3^7 (exponent 7=g_car+N_fam-1). [E] THE SCALE IS AN ATOM: M=M_scal=c3^{7/2}Mbar, the exponent 7 and half-power 7/2 carrier atoms (v253), determined not fitted. [O] RESIDUAL: (i) a(Box)=e^{Box/M^2}=the EXACT off-shell curved-space Hessian is the cited heat-kernel resummation (analyticity assumption); (ii) PERTURBATIVE graviton unitarity, NOT the nonperturbative ambient measure (QG.AMB.01, [C] redundancy v369).",
      "status": "Nonlocal-gravity action contract [E]/[O] (Paper C: the entire form factor gives a ghost-free one-pole propagator whose IR projection is exactly the atom-scaled R+R^2; the entire-a assumption + perturbative-only stay [O]). Consolidates v304/v370/v380/v386, adds the IR-matching check",
      "canonicalStatus": "[E] the one-pole/ghost-free pole algebra + the monotone truncation-zero fact (v380) + the IR R+R^2 matching with the atom-fixed scale c3^7 + the carrier-atom scale exponent 7; [O] the entire-a analyticity assumption + perturbative-only (not the ambient measure). A contract + IR-matching module; consolidates v304/v370/v380/v386; no new number.",
      "dependencies": "GRAV.AMPLITUDE.01; GRAV.LOOP.FINITE.01; v304; v370; v380; v386; v36; v28; v253; v369",
      "scripts": [
        "v400_grav_nonlocal_action.py"
      ],
      "script": "v400_grav_nonlocal_action.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 470
    },
    {
      "id": "METROLOGY.CLOSURE.01",
      "claim": "THE FINAL INPUT SET {a, pi, v_geo} CERTIFIED (2026-06-24, Paper D / the scale-modulus closure): 0 dimensionless dials + 1 dimensionful unit, consolidating v153 (No-Unit), v274 (over-determined anchor), v364 (gravity adds no unit), v384 (v_geo is theorem-forbidden, type B). Does NOT 'solve' v_geo (a dimensionless compiler provably cannot output a scale); it CERTIFIES the single declared unit. [E] INPUT SET: a=(1,1,2) -> elem-sym (e1,e2,e3)=(4,5,2)=(|mu4|,g_car,|Z2|), c3=1/(2 e1 pi)=1/(8 pi); the one dimensionful input is v_geo, NO dimensionless dial. [E] DIMENSIONAL BOOKKEEPING (No-Unit, v153): mass~v_geo^1, 1/G~v_geo^2, Lambda~v_geo^4, U_point~v_geo^1, m/mu=e^{3/4}~v_geo^0; under L->lambda L a dimensionless datum is invariant while v_geo^d scales as lambda^{-d}, so a scale cannot be a function of scale-invariants -- v_geo is a forced INPUT. [E] GRAVITY ADDS NO UNIT (v364): 1/c3=8pi and (8pi)^2*48 c3^4=3/(4pi^2) are dimensionless. [N] OVER-DETERMINATION (v274): gravity route (8 pi G)^{-1/2}=2.4353e18 GeV vs cosmology route 2.438e18 GeV agree to 0.11% (conditional on the Lambda-branch). [O] v_geo THEOREM-FORBIDDEN: TOE_strict = TOE_dimensionless + [v_geo], not 'derive the metre' (v384 type B). [2026-08-03 LIII: sharpened to torsor language by v725 (VGEO.TORSOR.01): the input set {a, pi, v_geo} is the calibration point of a rank-1 R+ scale torsor; the bookkeeping table of this row is the machine-verified export table there; no status move.]",
      "status": "Metrology closure [E]/[N]/[O] (Paper D: the final input set {a,pi,v_geo} -- 0 dimensionless dials + 1 unit; dimensional bookkeeping over all dimensionful observables; G-vs-Lambda over-determination 0.11%; v_geo theorem-forbidden). A metrology end-cap, no new derivation",
      "canonicalStatus": "[E] the input-set reduction to {a,pi,v_geo} + the dimensional bookkeeping (mass/1G/Lambda/U_point/(m/mu)=number x v_geo^{1,2,4,1,0}) + gravity-adds-no-unit; [N] the 0.11% G-vs-Lambda over-determination (conditional, v274); [O] the theorem-forbidden typing (TOE_strict = TOE_dimensionless + [v_geo]). A metrology end-cap (consolidates v153/v274/v364/v384); no new derivation, no new number.",
      "dependencies": "RESIDUAL.CERTIFICATION.01; ANCHOR.VGEO.01; v153; v274; v364; v384",
      "scripts": [
        "v401_metrology_closure.py"
      ],
      "script": "v401_metrology_closure.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 471
    },
    {
      "id": "FTRANSFER.SUITE.01",
      "claim": "THE F_transfer FUNCTOR CERTIFICATE (2026-06-24, Paper E / the physical-interface contract): the four frontier sectors are ONE functor F_transfer = F_observable o F_threshold o F_RG, consolidating the typed solvers (FR.POLE.SOLVE.01, FR.BOLTZMANN.SOLVE.01, FR.RELIC.SOLVE.01, FR.QCD.BUDGET.01). Does NOT re-press any frontier observable into the compiler (forbidden by the frontier firewall, v187); it certifies the composition law and the kill-test coverage. [E] COMPOSITION LAW: every frontier observable factors as source -> threshold/matching -> standard RG -> observable (verified structurally for all four). [E] EXACT ATOM SOURCES: Koide Q_src (phi0-ladder, v93); leptogenesis A_Lambda=2 g_car=10 and delta_CP=4pi/3 (v9/v212); axion f_a=M_scal/128 (v185/v211); m_p/m_e carrier beta b3=-(11-2 n_f/3)=-7 (v164/v262). [E] KILL-TEST COVERAGE: each sector carries a committed falsifier (Koide wrong-sign QED; eta_B outside [obs/3,3obs]; axion outside [0.08,0.16]; m_p/m_e C_p band excluding 1836.15). [C] THE FIREWALL: the PHYSICAL prediction of each sector stays [C]/[O] (threshold/RG/solver dependent); only the algebraic SOURCE is [E] -- no frontier observable is a primitive compiler output. [E] NO FREE DIAL: external inputs declared with propagated error budgets, algorithmic and versioned.",
      "status": "F_transfer functor certificate [E]/[C] (Paper E: the four frontier sectors as one F_observable o F_threshold o F_RG with exact atom sources + kill-test coverage + the firewall; physical predictions stay conditional). A consolidation, no new number",
      "canonicalStatus": "[E] the functor composition law (each sector factors source->threshold->RG->observable) + the exact atom sources (Q_src; A_Lambda=10, delta=4pi/3; f_a=M_scal/128; b3=-7) + the kill-test coverage for all four; [C] the frontier firewall (physical predictions stay conditional, only the algebraic source is [E], never re-pressed into the compiler). A consolidation of the FR.*.SOLVE solvers; no new number.",
      "dependencies": "FR.POLE.SOLVE.01; FR.BOLTZMANN.SOLVE.01; FR.RELIC.SOLVE.01; FR.QCD.BUDGET.01; v9; v93; v164; v185; v187; v211; v212; v262",
      "scripts": [
        "v402_ftransfer_suite.py"
      ],
      "script": "v402_ftransfer_suite.py",
      "location": "tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 472
    },
    {
      "id": "ARITH.HULL.01",
      "claim": "The three number-field facets COMPOSE to a degree-8 = rank(E8) abelian field (2026-06-24): K=Q(i,sqrt-3,sqrt5), the arithmetic hull dual to the E8 lattice hull, composing the v390/v394 facets that were only ever used one at a time. [E] INDEPENDENCE: the squarefree parts -1,-3,5 are multiplicatively independent in Q*/(Q*)^2 (no nonempty subset product is a square), so the three quadratics are linearly disjoint and [K:Q]=2^3. [E] DEGREE = rank E8: [K:Q]=2^3=8=rank(E8)=g_car+N_fam, Galois group (Z/2)^3 of order 8; the relation 2^omega(h)=rank is E8-special (the E7 control h=18, omega=2, 2^2=4!=7). [E] SEVEN SUBFIELDS: exactly 7 quadratic subfields {-1,-3,5,3,-5,-15,15}. [E] 4+3 SPLIT = |mu4|+N_fam: exactly 4 imaginary (-1,-3,-5,-15)=|mu4| (CM/dynamic) + 3 real (3,5,15)=N_fam (RM/static). [E] RAMIFIED ONLY AT THE ATOMS: every quadratic subfield ramifies only over {2,3,5}=the atoms=the primes of h(E8)=30. [E] CM/RM=DYNAMIC/STATIC: 2,3 imaginary-quadratic (disc -4,-3) carry the {2,3}-arithmetic dynamic rates (v394), 5 real-quadratic (disc +5) carries the static golden carrier (no independent prime-5 rate). [C] K is the abelian (Z/2)^3 'arithmetic hull' dual to the E8 lattice hull; the 7 quadratic subfields are a candidate for the 7 E8 audit slices.",
      "status": "Arithmetic hull [E]/[C] (the three facet fields compose to K=Q(i,sqrt-3,sqrt5): degree 2^3=8=rank E8, Galois (Z/2)^3, 7 subfields = 4 imaginary (|mu4|) + 3 real (N_fam), ramified only at {2,3,5}; 2^omega(h)=rank is E8-special) -- a synthesis, no new number",
      "canonicalStatus": "[E] the field arithmetic (independence => degree 8 = rank E8; Galois (Z/2)^3; the 7 subfields = 4 imaginary (|mu4|) + 3 real (N_fam); ramified only at {2,3,5}=the atoms; the E8-special 2^omega(h)=rank with the E7 negative control); [C] the arithmetic-hull reading and the 7-subfields<->7-slices candidate. A synthesis composing the established facet fields (v390/v394), introduces no new number; the prime set {2,3,5} forced by h(E8)=30.",
      "dependencies": "COXETER.PRIME2.01; COXETER.ATOMS.01; v23; v390; v394; v222",
      "scripts": [
        "v403_facet_compositum.py"
      ],
      "script": "v403_facet_compositum.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 473
    },
    {
      "id": "E8.CMDUAL.01",
      "claim": "The (E8)_1 vacuum character at the two CM points (2026-06-24): chi_E8 = E4/eta^8 = j^{1/3} equals 12 at the seam CM point tau=i and 0 at the family CM point tau=omega, extending v282 (chi_E8(i)=12) to the second elliptic point. [E] chi_E8^3=j (q-series). [E] SEAM tau=i (Q(i), order 4): cross-ratio 2 => j=1728 => chi_E8(i)=1728^{1/3}=12 (E8 present, v282/v214). [E] FAMILY tau=omega (Q(sqrt-3), order 6): the equianharmonic lambda (lambda^2-lambda+1=0) => j=0 => chi_E8(omega)=0 (E4 vanishes, v220/v267). [E] DUALITY: the SAME function chi_E8 takes its two CM values {12,0} at the two atoms' CM points -- maximal at the seam, zero at the family. [E] tau=i (order 4) and tau=omega (order 3/6) are the ONLY elliptic points of PSL(2,Z), so chi^3=j forces chi(i)=12 and chi(omega)=0 (not chosen). [C] STRUCTURAL READING: the seam is rigid where E8 is present (chi=12, SEAM.EQUIV.01 closes) while the flavor/CP sector is the soft residual where E8 degenerates (chi=0, the U_wall amplitude / v_geo / the CP texture). [C] DUAL-KEYSTONE CANDIDATE: the flavor sector = the j=0 degeneration of (E8)_1 at tau=omega, dual to the seam = (E8)_1 at tau=i.",
      "status": "E8 character CM-duality [E]/[C] (chi_E8(i)=12 vs chi_E8(omega)=0: the (E8)_1 character maximal at the seam tau=i, zero at the family tau=omega; the two are the only elliptic points so the values are forced) -- the seam/flavor rigidity duality, extends v282",
      "canonicalStatus": "[E] the modular arithmetic (chi^3=j; j(i)=1728, chi(i)=12; j(omega)=0, chi(omega)=0; the two-elliptic-point uniqueness); [C] the seam/flavor rigidity reading and the dual-keystone candidate. Extends v282 to the second CM point; core values j=1728/j=0 already mirrored (v214/v220/v267/v282), so Python-only by the synthesis convention; no new number.",
      "dependencies": "QGAMB.UNIFY.01; v282; v214; v220; v267; v390",
      "scripts": [
        "v404_e8_char_cm_duality.py"
      ],
      "script": "v404_e8_char_cm_duality.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 474
    },
    {
      "id": "SEAM.EQUIV.02",
      "claim": "The dual keystone at tau=omega (2026-06-24): the family/flavor sector as the order-3 Eisenstein/A2 face of E8, dual to the seam=(E8)_1 at tau=i (SEAM.EQUIV.01/v404). A NAMED CONTRACT (genre of v286/v398), NOT a closure. [E] chi_E8(omega)=j(omega)^{1/3}=0 (E4 vanishes at tau=omega, v404) vs chi_E8(i)=12 -- the family sits where the holomorphic (E8)_1 structure degenerates. [E] A2 is the tau=omega Eisenstein/hexagonal lattice: Cartan det=3=N_fam, h(A2)=3=N_fam, automorphism mu6=mu3(triality) x mu2(sheet, v233). [E] the flavor matrix R lives in the E6 x A2 slice: 248=||R||_F^2+det R+2(1^T R a)N_fam=78+8+2*27*3, ||R||_F^2=78=dim E6, det R=8=dim A2=rank E8. [E] the CP phase delta_PMNS=4pi/3 (delta_CKM,lead=pi/3=arg zeta6) is the hexagonal Eisenstein CM phase at tau=omega (v220/v231/v233). [E] the cusp weights {0,1/3,2/3} are the order-3 deck = N_fam = det Q = cusp-class order (v72), fixing Spec(Q+)=(1,2,3)=A3 exponents (v69/v137). [C] the dual keystone: the raw cusp/family transport IS the order-3 Eisenstein/A2 deck at tau=omega; the two keystones sit at the only two elliptic points of PSL(2,Z). [O] residual (parallel to SEAM.EQUIV.01's continuum existence): the canonical Z3 deck identification (v140) + the absolute scale = one v_geo-class UV input (No-Unit, v153).",
      "status": "Dual keystone at tau=omega [E]/[C]/[O] (the family/flavor sector as the order-3 Eisenstein/A2 face: chi_E8(omega)=0, A2 Cartan det 3=N_fam reads R in E6xA2 with det R=8=dim A2, delta=the tau=omega CM phase, cusp weights=the order-3 deck; dual to the seam=(E8)_1 at tau=i). A named contract, NOT a closure",
      "canonicalStatus": "[E] the arithmetic (chi_E8(omega)=0; A2 Cartan det 3=N_fam, h=3; the E6xA2 slice with det R=8=dim A2; delta=the tau=omega CM phase; cusp weights=the order-3 deck); [C] the dual-keystone reading; [O] the canonical-map Z3 choice (v140) + the v_geo-class scale. A named contract (genre of v286/v398), reuses established facts; NOT a closure; no new number.",
      "dependencies": "E8.CMDUAL.01; SEAM.EQUIV.01; v404; v220; v233; v72; v69; v137; v140",
      "scripts": [
        "v405_seam_equiv_omega.py"
      ],
      "script": "v405_seam_equiv_omega.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 475
    },
    {
      "id": "FLAV.PMNS.CLOSE.01",
      "claim": "The PMNS dynamics closed modulo the tau=omega keystone (2026-06-24): the CP-phase ORIGIN is the family CM point (v405) and the absolute scale is one v_geo-class unit, so PMNS has zero free flavor dials beyond v_geo. Resolves the two [O]'s v270 left open (the assembled phase + the open scale). [E] the three angles sin^2 th12=1/3-phi0/2 (v9), th23=45 (v9), sin^2 th13=phi0 e^{-5/6} (v268) assemble to one exactly-unitary U_PMNS (v270). [C] CP-PHASE ORIGIN: delta_PMNS=4pi/3 is the tau=omega hexagonal CM phase (v405/v220/v233), arg zeta6=pi/3, mu6=mu3 x mu2 -- so under SEAM.EQUIV.02 the phase has an operator home, upgrading v270's [O] assembled input to a keystone consequence. [C] J_PMNS=s12 c12 s23 c23 s13 c13^2 sin(delta)=-0.0297, a derived CP strength (no free phase once delta is the keystone value), matching NuFIT ~ -0.026. [E] the absolute scale = one seesaw ratio m3=(y_nu v_EW)^2/M_R (v272) -- one v_geo-class UV input the No-Unit theorem (v153) forbids deriving, NOT a flavor gap. [C] net: PMNS = three angles[E] + delta[tau=omega keystone] + J[derived] + absolute scale[one v_geo unit]; zero free flavor dials. [O] residual: the seesaw operator realisation (M_R not a clean Mbar power, v272/v184) + the normal-ordering Sigma m_nu floor 0.0586 eV kill test (v272).",
      "status": "PMNS closed modulo the keystone [E]/[C]/[O] (CP-phase origin = the tau=omega CM phase v405; angles [E]; J=-0.0297 derived; absolute scale = one seesaw ratio = v_geo; zero free flavor dials beyond v_geo). Resolves v270's open phase + scale",
      "canonicalStatus": "[E] the three angles + the scale-reduction to one seesaw ratio; [C] the CP-phase origin via the tau=omega keystone (upgrades v270's [O] to a keystone consequence) + the derived J=-0.0297 + the net closure (zero free flavor dials beyond v_geo); [O] the seesaw operator realisation / v_geo-class scale (v272/v184) + the NO floor kill test. A closure-accounting building on v405; no new number.",
      "dependencies": "SEAM.EQUIV.02; FLAV.PMNS.03; FLAV.NUSCALE.01; v405; v270; v272; v9; v268; v153",
      "scripts": [
        "v406_pmns_phase_origin.py"
      ],
      "script": "v406_pmns_phase_origin.py",
      "location": "tfpt_2",
      "externalData": "NuFIT 6.0 (delta_CP; angles)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 476
    },
    {
      "id": "FLAV.SELECTOR.CLOSE.01",
      "claim": "The R4' selector residue folds into the tau=omega keystone (2026-06-24): the three (d,n) selector pairings ARE the tau=omega family-slice atoms, so deriving the selector pair reduces to the family sector being the E6 x A2 (tau=omega) object (v405). [E] the frame (1,a,sigma) is a Z^3 basis with det(1|a|sigma)=11=||Pl(K)||_1, and n=(5,-9,6) is the unique covector with n.1=2, n.a=8, n.sigma=121=11^2 -> (d,n)=>R columnwise (v136/v139). [E] n.a=8=rank E8=dim A2=det R -- the anchor-pairing equals the A2 (tau=omega Eisenstein) block dimension that reads R in the E6 x A2 slice (||R||_F^2=78=dim E6; v405). [E] the three pairing values {2,8,121}={|Z2|, rank E8=dim A2=det R, ||Pl(K)||_1^2} = the tau=omega family-slice atoms (the mu2 sheet of mu6, the A2 block, the Plucker frame volume). [E] the square selects the point (v139): sigma-pairing=11(11+t), a perfect square only at t=0. [C] the reduction: deriving (d,n) (R4') folds into SEAM.EQUIV.02 -- d=(3/2)a-2*1 is anchor-derived (v139), the three pairings are the family-slice atoms. [O] residue: the lone non-atomic value is 11=||Pl(K)||_1 (the quark Plucker constant, flagged v42/FLAV.RIGID); why 11 is the sharpened R4' residue.",
      "status": "R4' selector folds into the keystone [E]/[C]/[O] (the (d,n) pairings {2,8,121}={|Z2|, rank E8=dim A2=det R, ||Pl(K)||_1^2} are the tau=omega family-slice atoms; n.a=8=dim A2 reads R in E6xA2; deriving (d,n) folds into SEAM.EQUIV.02; residue = why 11)",
      "canonicalStatus": "[E] the frame/pairings (v139) + the new identification n.a=8=dim A2=det R and {2,8,121}=the tau=omega family-slice atoms; [C] the reduction of R4' to SEAM.EQUIV.02; [O] the lone flagged residue why 11=||Pl(K)||_1. A reduction tying the selector residue to the dual keystone; reuses v139's exact data; no new number.",
      "dependencies": "SEAM.EQUIV.02; FLAV.RIGID.01; v405; v139; v136; v42",
      "scripts": [
        "v407_dn_pairings_omega.py"
      ],
      "script": "v407_dn_pairings_omega.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 477
    },
    {
      "id": "SHEET.GEN.BINARY.01",
      "claim": "The sheet generator V is a binary internal compiler (2026-06-24): the rank-2 sheet axis V=Q diag(0,1,1) of the centered cross (v95/v218) has powers that print the binary carrier spine. [E] CLOSED FORM (induction F(n)*V=F(n+1)): V^n={{0,2^(n-1),0},{0,2^n,0},{0,2^(n+1)-2,1}}, so V^n.1=(2^(n-1),2^n,2^(n+1)-1)=(1,2,3),(2,4,7),(4,8,15),(8,16,31),... [E] BILINEAR FAMILIES (exact in n, 1=(1,1,1), a=(1,1,2)): 1.V^n.1=7*2^(n-1)-1, 1.V^n.a=7*2^(n-1), a.V^n.a=11*2^(n-1), a.V^n.1=11*2^(n-1)-2; at n=1 = (6,7,9,11). [E] readouts 1.V^2.1=13=Delta_Q, 1.V^3.1=27=1.R.a, 1.V^4.1=55=quark numerator, 1.V^4.a=56=dim 56_E7; 11=a.V.a the anchor amplitude. [C] SPINE READING (audit, NOT promoted): the spine {2^(k-1),2^k,2^(k+1)-1} is FORCED by Spec(V)={0,1,2}; the matches (1,2,3)=A3 exponents, (2,4,7)=(|Z2|,|mu4|,7), (4,8,15)=(|mu4|,rank E8,dim A3), (8,16,31)=(rank E8,dim S+,2^g_car-1=248/8) are audit Lie/dimension matches (v95 typing). [C] THETA CROSS-LINK (audit raster): sigma3(2)=9=a.V.1=N_fam^2, sigma3(3)=28=1.V^3.a=det(I+R), sigma3(5)=126 = local E7 root shell (tfpt_3).",
      "status": "Sheet generator V is a binary compiler [E]/[C]: V^n closed form + V^n.1 binary spine + four bilinear families (6,7,9,11; 13,27,55,56); spine FORCED by Spec(V)={0,1,2}, Lie/theta readings audit-typed [C]",
      "canonicalStatus": "Identity",
      "dependencies": "FLAV.KQ.01; v95; v218; v135",
      "scripts": [
        "v410_sheet_generator_binary.py"
      ],
      "script": "v410_sheet_generator_binary.py",
      "location": "tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 478
    },
    {
      "id": "FLAV.UD.VPOWER.01",
      "claim": "The quark u/d ratio is a pure V-power readout (2026-06-24): the established c_u/c_d=g_car*||Pl(K)||_1/(N_fam^2 Delta_Q)=55/117 (v94) re-expresses EXACTLY via the sheet generator V (v410). [E] c_u/c_d=(1.V^4.1)/((a.V.1)(1.V^2.1))=55/(9*13)=55/117, with 55=1.V^4.1, 9=a.V.1=N_fam^2, 13=1.V^2.1=Delta_Q. [E] IDENTITY: same number as 5*11/(9*13); the V-power numerator repackages g_car*11=55=1.V^4.1. [C] HONEST TYPING: an exact RE-ENCODING, no new physical input -- the value 55/117 and the u/d identification stay [C], coupled to the U_wall readout rigidity; the gain is that numerator and denominator both iterate ONE operator.",
      "status": "u/d ratio as a V-power readout [E]/[C]: c_u/c_d=(1.V^4.1)/((a.V.1)(1.V^2.1))=55/117, identical to 5*11/(9*13); exact re-encoding, value stays [C]",
      "canonicalStatus": "Identity",
      "dependencies": "FLAV.QUARK.01; FLAV.RIGID.01; v94; v410",
      "scripts": [
        "v411_ud_ratio_vpower.py"
      ],
      "script": "v411_ud_ratio_vpower.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 479
    },
    {
      "id": "DIAMOND.SHEET.SOURCE.J.01",
      "claim": "The sheet source corner J (2026-06-24): the determinant surface has two s-independent iso-volume walls (v135) t=-1 (det=4=|mu4|, carries K) and t=-2 (det=2=|Z2|); the operator on the Z2 wall, J:=M(1,-2)=C-2V=((4,1,0),(4,1,2),(5,1,1)), was unnamed and completes the sheet axis J,K,C,F=M(1,-2..1). [E] CLEAN CHAR POLY: chi_J=lam^3-6lam^2+3lam-2 => (tr,e2,det)=(6,3,2)=(|R^+(A3)|,N_fam,|Z2|). [E] RESOLVENT/ANCHOR: a.J.a=30=h(E8), det(I+J)=12=dim g_SM=chi_E8(i), det(2I+J)=40=|R(D5)|. [C] the Lie readings are audit-typed like the G2/F4 shadows of v95/v218.",
      "status": "Sheet source corner J=M(1,-2) on the Z2 wall [E]/[C]: chi_J=(6,3,2), a.J.a=30=h(E8), det(I+J)=12, det(2I+J)=40; readings audit-typed",
      "canonicalStatus": "Identity",
      "dependencies": "FLAV.KQ.01; v135; v95",
      "scripts": [
        "v412_sheet_source_corner_J.py"
      ],
      "script": "v412_sheet_source_corner_J.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 480
    },
    {
      "id": "DIAMOND.SHEET.CALCULUS.01",
      "claim": "The sheet axis is a discrete characteristic-difference calculator (2026-06-24): on M_t=C+tV (v95/v218) the elementary-symmetric coefficients encode atoms as DIFFERENCE ORDERS. [E] e1(t)=3t+12, e2(t)=2t^2+15t+25, e3(t)=det M_t=4t^2+14t+14 => Delta e1=3=N_fam, Delta^2 e2=4=|mu4|, Delta^2 e3=8=rank E8 (the e3 curvature cross-cited to v218/v224; the full (e1,e2,e3) ladder is new). [E] ANCHOR-ENERGY LADDER: a.M_t.a=52+11t (step 11=a.V.a, v410); 30->41->52->63=(h(E8),10 b1,dim F4,7 N_fam^2). [E] FOUR-ROW AXIS J,K,C,F: (tr,e2,det)/a.M.a=(6,3,2)/30,(9,12,4)/41,(12,25,14)/52,(15,42,32)/63. [C] atom/Lie readings audit-typed.",
      "status": "Sheet axis = characteristic-difference calculator [E]/[C]: Delta e1=3, Delta^2 e2=4, Delta^2 e3=8; anchor energy 52+11t (30->41->52->63); readings audit-typed",
      "canonicalStatus": "Identity",
      "dependencies": "FLAV.KQ.01; v218; v224; v410",
      "scripts": [
        "v413_sheet_characteristic_calculus.py"
      ],
      "script": "v413_sheet_characteristic_calculus.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 481
    },
    {
      "id": "DIAMOND.CENTER.RESOLVENT.01",
      "claim": "The center C is a resolvent portal G2 -> F4 -> E8 (2026-06-24): the center C=M(1,0) (v95) is the t=0 evaluation of V; its shifted determinants climb the exceptional ladder. [E] CENTER INVARIANTS (v95): tr C=12=dim g_SM=chi_E8(i), det C=14=dim G2, sum C=31=2^g_car-1=248/8. [E] RESOLVENT LADDER: det(C)=14=dim G2, det(I+C)=52=dim F4, det(2I+C)=120=|R^+(E8)| -- the exceptional shell ladder G2->F4->E8^+ as det(C+kI), k=0,1,2 (the resolvent ladder is the new content; reading [C], audit-typed).",
      "status": "Center C resolvent portal [E]/[C]: det C=14=dim G2, det(I+C)=52=dim F4, det(2I+C)=120=|R^+(E8)| (G2->F4->E8^+); reading audit-typed",
      "canonicalStatus": "Identity",
      "dependencies": "FLAV.KQ.01; v95; v230",
      "scripts": [
        "v414_center_resolvent_portal.py"
      ],
      "script": "v414_center_resolvent_portal.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 482
    },
    {
      "id": "DIAMOND.GAUSS.01",
      "claim": "The square CM-norm dictionary is operator-realised and the mu4 deck is intrinsic (2026-06-25): the Gaussian arithmetic of v222/v230 lifted from numbers to OPERATORS via J=[U,V]/3, the complex structure born from the commutator of the binary V and ternary U compilers (v410/v95). [E] J=((-1,1,0),(-2,1,0),(-3,1,0)) integer, Spec(J)={i,-i,0}, J^2=-I on the rank-2 image, J^3+J=0 (a mu4 quarter-turn). [E] 3I+2J has spectrum {3+2i,3-2i,3} and 5I+4J has {5+4i,5-4i,5}: the Gaussian integers 3+2i (norm 13=Delta_Q) and 5+4i (norm 41=10 b1) are LITERAL operator eigenvalues. [E] (3I+2J)(3I-2J)=9I-4J^2 has Spec {13,13,9} and (5I+4J)(5I-4J)=25I-16J^2 has Spec {41,41,25} -- the v222/v230 norms as operator identities (kernel slot = real part^2: 9=N_fam^2, 25=g_car^2). [E] the intrinsic deck D=-I+J-J^2=((-1,1,0),(-2,1,0),(-4,3,-1)): D^4=I, chi_D=(x+1)(x^2+1), Spec {i,-1,-i}; the chi2 line is canonically ker[U,V]=(0,0,1)=a-1=the V-fixed line. [C] the geometric-deck identification: D shares chi with the seam deck delta (v146), the diamond fixes chi2=a-1 internally, sharpening the v140 Z3 residue to one concrete identification (NOT closed). An operator lift of v222/v230 + an intrinsic-deck construction; reuses existing data; no new number.",
      "status": "Gaussian operator [E]/[C]: J=[U,V]/3 is the mu4 quarter-turn (J^2=-I), 3+2i/5+4i are eigenvalues of 3I+2J/5I+4J (norms 13=Delta_Q/41=10 b1), the norms become operator identities, intrinsic order-4 deck D=-I+J-J^2 with chi2=ker[U,V]=a-1; geometric-deck match sharpens (not closes) the v140 Z3 residue [C]",
      "canonicalStatus": "Identity",
      "dependencies": "CMNORM.DUAL.01; CENTER.NORM.01; SHEET.GEN.BINARY.01; FORM.QGEO.02; v222; v230; v410; v95; v146; v140",
      "scripts": [
        "v415_gaussian_operator.py"
      ],
      "script": "v415_gaussian_operator.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 483
    },
    {
      "id": "ARITH.TRICHOTOMY.01",
      "claim": "The atom trichotomy (2026-06-25): {2,3,5} are ramified/inert/split in the two exceptional CM rings, and each atom is the unique ramified prime of one quadratic facet of K=Q(i,sqrt-3,sqrt5) (v403). Sharpens v222/v230 (atoms read only as CM norms) and v390/v394/v403 (the three facets) into one prime-splitting table. [E] Z[i] (square/seam, tau=i): 2=|Z2| RAMIFIES (2=-i(1+i)^2), 3=N_fam INERT (3 mod 4, norm 9=N_fam^2), 5=g_car SPLITS ((2+i)(2-i), 1 mod 4) -- the carrier rank g_car=5 FACTORS on the seam. [E] Z[omega] (hex/flavor, tau=omega): 3=N_fam RAMIFIES (N(1-omega)=3), 2=|Z2| INERT (norm 4), 5=g_car INERT (norm 25=g_car^2). [E] the ramified prime of each ring IS its atom: Z[i] ramifies at 2 (sheet/seam), Z[omega] at 3 (family/flavor). [E] the three facets Q(i)/Q(sqrt-3)/Q(sqrt5) (disc -4,-3,5) ramify over exactly one atom each (2,3,5), product 2*3*5=30=h(E8); 2 imaginary (CM/dynamic) + 1 real (RM/static) = the v403 reading at the prime level. [C] the seam/flavor/golden reading. Exact number theory composing existing facts; no new number.",
      "status": "Atom trichotomy [E]/[C]: {2,3,5} ram/inert/split in Z[i] (2 ram, 3 inert, 5 split) vs Z[omega] (3 ram, 2,5 inert); the ramified prime of each ring = its atom; each atom is the unique ramified prime of one facet of Q(i,sqrt-3,sqrt5), product 30=h(E8); seam/flavor/golden reading [C]",
      "canonicalStatus": "Identity",
      "dependencies": "ARITH.HULL.01; CMNORM.DUAL.01; COXETER.ATOMS.01; COXETER.PRIME2.01; v403; v222; v390; v394; v313",
      "scripts": [
        "v416_atom_trichotomy.py"
      ],
      "script": "v416_atom_trichotomy.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 484
    },
    {
      "id": "DIAMOND.EISEN.01",
      "claim": "The Eisenstein/CP operator -- the hexagonal dual of the Gaussian seam operator v415 (2026-06-25): where v415 put the SQUARE seam (tau=i, mu4) on operators via J=[U,V]/3, this puts the HEX/flavor side (tau=omega, mu3/mu6) on operators and unifies both with the seam into ONE order-30 Coxeter clock. [E] EISENSTEIN DECK: the family rotation P=(1 2 3) has P^3=I, chi_P=x^3-1, P^2+P+I=ONES, so on the plane perp to (1,1,1) it satisfies omega^2+omega+1=0 -- the order-3 dual of J^2=-I. [E] HEX NORM 7 AS OPERATOR: (3I+2P)(3I+2P^2)=7I+6 ONES, Spec {7,7,25} -- 7=N_Z[omega](3+2omega)=scalaron on the omega-plane, 25=g_car^2 on (1,1,1); the dual of v415's (3I+2J)(3I-2J)->{13,13,9}; NEG control (5I+4P)(5I+4P^2)->{21,21,81}, N_omega(5,4)=21!=41 (v222 rigidity). [E] CP CLOCK W=-P^2 (=omega^2*(-1)=rho v233), order 6: W^3=-I, W^2=P, chi_W=x^3+1, Spec {-1,zeta6,zeta6^-1}; delta_CKM,lead=arg(zeta6)=pi/3 (eigenvalue of W), delta_PMNS=arg(zeta6^4)=4pi/3 (eigenvalue of W^4=P^2), W^3=-I the Z2 sheet flip rho^3=-1 (v231/v233/v316/v320); mu6=mu3 x mu2 in one operator. [E] ONE ORDER-30 CLOCK: all flavor-side clocks are powers of zeta30 (h(E8)=30=2*3*5=|Z2|*N_fam*g_car): zeta30^15=-1 (mu2), zeta30^10=omega (mu3), zeta30^6=zeta5 (mu5), zeta30^5=zeta6 (mu6); the SEAM mu4 (i) is NOT a power (4 nmid 30) -- the Galois side (Z/30)^x of order phi(30)=8=rank E8=mu4 x Z2 (totatives {7,13,17,23} order 4); Q(zeta30) carries BOTH E8 invariants (cyclic h=30 flavor hands + Galois rank=8 seam). [C] REALISATIONS + HONEST GAP: J(mu4)/P(mu3)/W=-P^2(mu6)/sigma(mu2) are 3x3 operators, but the CARRIER mu5 (zeta5) has none (needs dim>=4; the golden phi=2cos(pi/5) shows up only as an output-side trace, v313/v349); honest negatives: no U,V-diamond operator carries x^2-x+1 (only -P does), the seam and family clocks do NOT combine ([D,P]!=0, no mu12), and N(3+2 zeta6)=19!=7 (the clean 7 is the omega-norm). An operator lift + synthesis tying v222/v230/v233/v315/v316/v404/v415; no new number.",
      "status": "Eisenstein/CP operator [E]/[C]: the family rotation P realises the hex norm 7=N_omega(3+2omega) via (3I+2P)(3I+2P^2)={7,7,25} (dual of v415's {13,13,9}); the CP clock W=-P^2 (W^2=P, W^3=-I, Spec {-1,zeta6,zeta6^-1}) gives delta_CKM=pi/3, delta_PMNS=4pi/3; all flavor clocks are zeta30-powers, the seam mu4 the Galois side phi(30)=8=rank E8; the carrier mu5 has no 3x3 operator [C]",
      "canonicalStatus": "Identity",
      "dependencies": "DIAMOND.GAUSS.01; CMNORM.DUAL.01; CENTER.NORM.01; FLAV.CP.01; v415; v222; v230; v233; v316; v320; v404; v95",
      "scripts": [
        "v417_eisenstein_cp_operator.py"
      ],
      "script": "v417_eisenstein_cp_operator.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 485
    },
    {
      "id": "ARITH.NORMTRIPLE.01",
      "claim": "The cyclotomic norm triple (2026-06-25): the carrier split (3,2) read in the three atom-rings gives (7,13,55)=(scalaron, Delta_Q, quark numerator), and the missing CARRIER-5 clock of v417 is found as the 4x4 Phi5-companion C5 (Z[zeta5], rank phi(5)=4). [E] CARRIER-5 CLOCK: C5 has chi=x^4+x^3+x^2+x+1, C5^5=I, tr=-1 (the four primitive 5th roots); the GOLDEN ratio is its real-part data zeta5+zeta5^4=2cos(72)=(sqrt5-1)/2=1/phi, zeta5^2+zeta5^3=2cos(144)=-(sqrt5+1)/2=-phi (output-side golden, v313/v349; dim 4 -- refines v417's gap from 'no operator' to 'no 3x3 operator, but a clean 4x4 one'). [E] CARRIER NORM = QUARK NUMERATOR: N_Z[zeta5](3+2 zeta5)=det(3I+2 C5)=55=5*11=g_car*||Pl(K)||_1=1^T V^4 1 (v410/v411), the c_u/c_d numerator. [E] THE TRIPLE: det(3I+2 Comp(Phi_n))=(7,13,55) for n=(3,4,5)=(scalaron, Delta_Q, quark numerator) over Z[omega],Z[i],Z[zeta5] (atoms 3,2,5, v416) -- the SAME carrier split (3,2) in three rings, ties v222/v415/v417. [E] NEG CONTROL: the anchor (5,4)->(21,41,461); 21=N_omega(5,4) and 41=10 b1 are named (v222) but the CARRIER ring gives 461 (prime, un-named), so the anchor does NOT reach a named carrier number -- the Phi5/carrier ring separates (3,2)->55 from (5,4)->461. [C] HONEST: (3,2) is FORCED (b+s=5, b^2+s^2=13, v14) not UNIQUE -- a scan over (a>b>=1, a+b<=9) against the named-constant set finds exactly {(2,1)->(3,5,11)=(N_fam,g_car,||Pl(K)||_1), (3,2)->(7,13,55)}, a small ladder; (3,2) is the forced rung carrying the load-bearing triple, NOT the unique clean split. [C] SHARPER (CROSS-SECTOR) DISCRIMINATOR: of the two clean splits ONLY (3,2)->(7,13,55) spans three DIFFERENT physics sectors (gravity=scalaron, flavor=Delta_Q, masses=quark numerator), while (2,1)->(3,5,11) is all-core (N_fam,g_car,Plucker) -- so (3,2) is the UNIQUE cross-sector seed; the ladder reads input (2,1)->output (3,2), difference 4*(1,2,11)=(4,8,44). Exact number theory composing v415/v416/v417/v222; no new physical number.",
      "status": "Cyclotomic norm triple [E]/[C]: the carrier split (3,2) over the three atom-rings Z[omega],Z[i],Z[zeta5] gives det(3I+2 Comp(Phi_{3,4,5}))=(7,13,55)=(scalaron,Delta_Q,quark numerator); the carrier-5 clock is the 4x4 Phi5-companion C5 (golden in its 2cos values, N(3+2 zeta5)=55=quark numerator); NEG control anchor (5,4)->(21,41,461) fails in the carrier ring; (3,2) FORCED (v14) not unique, but the UNIQUE CROSS-SECTOR seed -- only (3,2)->(7,13,55) spans gravity/flavor/masses while (2,1)->(3,5,11) is all-core [C]",
      "canonicalStatus": "Identity",
      "dependencies": "ARITH.TRICHOTOMY.01; CMNORM.DUAL.01; DIAMOND.GAUSS.01; DIAMOND.EISEN.01; FLAV.QUARK.01; v416; v222; v415; v417; v411; v410; v14; v313",
      "scripts": [
        "v418_cyclotomic_norm_triple.py"
      ],
      "script": "v418_cyclotomic_norm_triple.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 486
    },
    {
      "id": "SEAM.GALOIS.01",
      "claim": "The seam mu4 IS the carrier's Galois group (2026-06-25): a positive resolution of the v409/RES.COXETER.SYMMETRY.01 cyclic/Galois asymmetry. In the cyclotomic picture the flavor clocks (sheet/family/carrier/CP) are powers of zeta30 (cyclic side, order h(E8)=30), but the seam mu4 (i) sits on the Galois side (Z/30)^x (order rank E8=8); v409 flagged this as 'prime-2 is the Galois bridge'. [E] 30=2*3*5 SQUAREFREE => Z/30 has element orders {1,2,3,5,6,10,15,30}, NO order 4, so the seam mu4 (order 4) cannot be a rotation hand and is forced onto the Galois side. [E] (Z/30)^x = (Z/2)^x x (Z/3)^x x (Z/5)^x = 1 x Z/2 x Z/4 = mu4 x Z2 (order phi(30)=8=rank E8); the mu4 (Z/4) factor IS (Z/5)^x (the CARRIER prime 5), the Z2 IS (Z/3)^x (the FAMILY prime 3 = CP conjugation, v316); the order-4 totatives {7,13,17,23} are exactly the generators of (Z/5)^x mod 5. [E] so the seam mu4 = Gal(Q(zeta5)) = (Z/5)^x (the carrier pentagon's automorphisms), realised on the carrier clock C5 (v418) by the Frobenius zeta5->zeta5^2: the explicit order-4 operator G has G C5 G^-1 = C5^2, G^4=I, G^2 C5 G^-2 = C5^-1 (complex conjugation) -- the square seam (atom 2, mu4) and the carrier pentagon (atom 5, mu5) are ONE object. [C] RESOLVES RES.COXETER.SYMMETRY.01: the 2^2=4 is the ORDER of (Z/5)^x (prime-2 and prime-5 meet here), and 'no contraction rate' (v409) is because the seam is an AUTOMORPHISM (|lambda|=1, finite order), not a rotation/transfer hand (consistent with the clock No-Go); Q(zeta30) carries BOTH E8 invariants -- the cyclic hands (h=30) and the Galois gears (rank=8=mu4 x Z2). Figure coxeter_galois. Exact number theory + the explicit Frobenius operator; no new physical number.",
      "status": "Seam mu4 = carrier's Galois group [E]/[C]: 30 squarefree => no order-4 cyclic, so the seam mu4 is the Galois side (Z/30)^x=mu4 x Z2; the mu4 factor IS (Z/5)^x=Gal(Q(zeta5)) (carrier), the Z2 IS (Z/3)^x (family/CP); the Frobenius zeta5->zeta5^2 is the explicit order-4 operator G with G C5 G^-1=C5^2, G^4=I; resolves the v409 asymmetry (seam = automorphism, not a hand) [C]",
      "canonicalStatus": "Identity",
      "dependencies": "RES.COXETER.SYMMETRY.01; ARITH.NORMTRIPLE.01; ARITH.HULL.01; CMNORM.DUAL.01; v409; v418; v403; v316; v223; v415",
      "scripts": [
        "v419_seam_galois_carrier.py"
      ],
      "script": "v419_seam_galois_carrier.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 487
    },
    {
      "id": "FR.POLE.QED.01",
      "claim": "F_pole, the external side (2026-06-25): the Koide source->pole transfer is a ONE-LOOP ELECTROMAGNETIC effect, not an algebraic step -- the honest 'do the external physics' follow-up to v93 (which left the source->pole flow time t~2.84 as an undetermined [P] generator) and to the clock No-Go (the unitary/algebraic lens cannot reach a contraction rate). [E] KOIDE RESCALING INVARIANCE: Q(m)=sum m/(sum sqrt m)^2 is degree-0 homogeneous, Q(c*m)=Q(m) for all c>0, so any FLAVOR-UNIVERSAL running leaves the Koide ratio EXACTLY fixed -- the source->pole shift is purely the FLAVOR-SPLIT (per-lepton) part of the running. [C] ONE-LOOP-EM SIZE: the tree ladder gives 2/3-Q_src=0.002203, which is the TFPT seed quantum phi0/24=0.002216 to 0.6% AND the one-loop EM scale alpha/pi=0.002323 to ~5%; the PDG pole sits AT 2/3 (gap 2.2e-6, the IR attractor) -- the source->pole gap is a one-loop EM size, so the transfer is QED. [C] FLAVOR-SPLIT QED ESTIMATE: a standard 1-loop QED pole<->MSbar relation with the per-lepton log shifts Koide by EM-loop order (|dQ|/(alpha/pi)~0.5) and the pole (IR) is closer to 2/3 than the MSbar (UV) -- the IR is the attractor (v82 direction); size EM-loop, exact value scheme/scale-dependent. [O]/[C] HONEST NEGATIVE/FIREWALL: the EXACT flow time t~2.84 (v93) depends on the renormalisation scheme and the tree-scale identification (external QED choices), so it is NOT derived here, only sized; the algebraic lens forces the RATE (2/3)^6 (v327/v82), this external EM physics supplies the source->pole flow; F_pole stays [C], firewalled (v187), never re-pressed into the compiler. VERDICT: the Koide source->pole transfer is identified as a one-loop EM effect (size ~alpha/pi, IR-attractor direction), confirming F_transfer is genuinely external -- the open theme is sharpened (QED), not closed. Numerical (mpmath), Python-only.",
      "status": "F_pole external [E]/[C]/[O]: the Koide source->pole transfer is a one-loop EM effect; Koide is rescaling-invariant (Q(c*m)=Q(m)) so the shift is flavor-split; the tree->pole gap 0.0022 is one-loop-EM-sized (=phi0/24 to 0.6%, =alpha/pi to ~5%), pole at 2/3 (IR attractor); a 1-loop QED per-lepton-log estimate shifts Koide by EM-loop order toward 2/3; the exact flow time t~2.84 is scheme-dependent external QED, only sized; F_pole stays [C], firewalled (v187) -- rate forced (v327/v82), flow external",
      "canonicalStatus": "Physical",
      "dependencies": "FR.KOIDE.04; FR.KOIDE.05; CONTRACT.F.01; FR.POLE.SOLVE; v93; v82; v327; v371; v187",
      "scripts": [
        "v420_fpole_qed_running.py"
      ],
      "script": "v420_fpole_qed_running.py",
      "location": "tfpt_4",
      "externalData": "PDG 2024 lepton pole masses; alpha (v3)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 488
    },
    {
      "id": "SEAM.GALOIS.NET.01",
      "claim": "The Galois<->Net bridge (2026-06-25): the seam mu4 = Gal(Q(zeta5)) is the SAME cyclic Z/4 as the (E8)_1 simple-current glue -- NOT a mere order-4 coincidence -- connecting the Galois gearbox of v419 to G_net/SEAM.EQUIV.01. [E] CARRIER GLUE CYCLIC Z/4: disc(A3)=coker(Cartan A3)=Z/4 and disc(D5)=Z/4 (one Smith invariant factor 4 = CYCLIC, not Klein), det=4 each; the D_n discriminant is Z/4 for n ODD and Z2xZ2 for n EVEN, so the carrier D5 (rank 5=g_car, odd) is cyclic while the control D8 (rank 8, even) is Klein -- the cyclicity of the carrier glue is rooted in g_car=5 being odd. [E] CARRIER-GALOIS SAME Z/4: Gal(Q(zeta5))=(Z/5)^x=<2> is cyclic of order 4 (max multiplicative order 4, not Klein); among the atoms only the carrier prime 5 gives order 4 (|(Z/2)^x|=1, |(Z/3)^x|=2, |(Z/5)^x|=4). [E] E8 GLUE CYCLIC Z/4: in disc(D5(+)A3)=Z4xZ4 (order 16=dim S+) the simple current (1,1) has order 4 (cyclic <(1,1)>); the Lagrangian quotient is 16/|glue|^2=16/4^2=1, so the extension is unimodular = (E8)_1 (one primary, holomorphic, v89/v154). [E] CYCLIC vs KLEIN LOAD-BEARING (negative control, v100): the order-4 Klein group Z2xZ2 has max element order 2 (NOT cyclic); the even-rank D8 carries it, and the order-2 halfway glue <(2,2)> gives 16/2^2=4=|disc(D8)|=SO(16) (det Cartan 4, four primaries), NOT E8 (det 1). So cyclic Z/4 -> E8, Klein/order-2 -> D8 -- the carrier-Galois being cyclic Z/4 (5 prime) matches the E8 glue, never the D8 Klein. [C] THE BRIDGE = THE FOUR-FOLD mu4 IDENTITY: ONE cyclic Z/4 threads the four mu4 readings -- the seam deck/divisor mu4 (z->iz, v419), the carrier-Galois Gal(Q(zeta5)) (v419), the discriminant disc(D5)=disc(A3)=Z/4 and the (E8)_1 simple-current glue <(1,1)> (v1/v89/v154) -- so the Galois gearbox (Z/30)^x of v419 IS the simple-current glue of the holomorphic net, a direct bridge from the Galois side to G_net (SEAM.EQUIV.01). All four computably cyclic-Z/4 [E]; the canonical identification (one functorial Z/4) is the [C] bridge, cyclicity forced (5 odd/prime), Klein excluded. Per v428 this four-fold identity is a COMPRESSION (one object read four ways, a unification that makes the carrier less axiomatic), NOT four independent witnesses that multiply. Exact lattice/Galois; no new number.",
      "status": "Galois<->Net bridge / four-fold mu4 identity [E]/[C]: the seam mu4=Gal(Q(zeta5))=(Z/5)^x is the SAME cyclic Z/4 as the (E8)_1 glue <(1,1)> AND the discriminant disc(A3)=disc(D5)=Z/4 (D5 cyclic because rank 5 odd; D8 even=Klein control); only the carrier prime 5 gives (Z/p)^x=Z/4; glue order 4 => 16/4^2=1=E8 (vs Klein/order-2 => D8 det 4); cyclic Z/4 selects E8, Klein excluded -- the v419 Galois gearbox IS the holomorphic net's glue, a direct Galois->G_net bridge. Deck/divisor=Galois=discriminant=simple-current is a compression per v428 (one object read four ways), not four witnesses [C]",
      "canonicalStatus": "Lattice",
      "dependencies": "SEAM.GALOIS.01; SEAM.EQUIV.01; E8.GLU.01; GATE.METRIC.06; ARITH.HULL.01; v419; v1; v89; v154; v92; v281; v403; v428",
      "scripts": [
        "v422_galois_net_bridge.py"
      ],
      "script": "v422_galois_net_bridge.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 489
    },
    {
      "id": "QGAMB.NODEP.01",
      "claim": "The No-Ambient-Dependency certificate (2026-06-25): no frozen TFPT readout is COMPUTED from the ambient QG measure QG.AMB.01 -- sharpening the v369/v384 prose redundancy ('QG.AMB.01 is a [C] redundancy, not dynamics') into a machine-checked DAG+script fact, told HONESTLY. [E] GAP-DECOUPLING MARGIN: 2 dim(E8) c3^2 = 31/(4 pi^2) (31=2^g_car-1) ~ 0.785, Delta = 6 ln(3/2) ~ 2.433, so Delta_eff ~ 1.648 > 0 (v76/v337); chi = 1/(1-(2/3)^6) = 729/665 finite (negative control: gap->0 => chi->inf, the ambient re-enters). [E] CERTIFICATION SINK: every claim that lists QG.AMB.01 as a dependency is a gravity-gate/QG-certification row (GATE.METRIC.01, GRAVITY.COMPLETE.01, the QGAMB.* cluster, SEAM.S3.RP.01, ...), NONE a frozen physical prediction -- QG.AMB.01 has no predictive direct dependent. [E] NOT A VALUE SOURCE: QG.AMB.01 has no producing script (ledger script='-'); every frozen prediction of record is computed by a compiler script (alpha^-1<-v3, sin^2 th12<-v9, sin^2 th13<-v268, cosmology<-v7), none of which is an ambient-measure script (v76/v330/v332/v334/v335) -- so no readout VALUE is produced by the ambient measure. [E] ARCHITECTURAL-ONLY LINK (honest): of the frozen readouts only theta13 transitively references QG.AMB.01, and the path runs through the 4D-QFT CONTRACT spine (PS.DIRAC.02 -> CONTRACT.QFT4D.01 -> QG.G2.01 -> GATE.METRIC.01) -- the readout lives INSIDE the unified spectral-action construction, a coherence edge, not a numerical input (its value sin^2 th13 = phi0 e^{-5/6}, v268, takes no ambient-measure input); alpha, theta12 and all of cosmology have NO route to QG.AMB.01 at all. [C] VERDICT: QG.AMB.01 is a gap-decoupled certification object -- no readout VALUE is computed from it, its only predictive link is the architectural 4D-QFT contract, and the gap margin (1.648>0) suppresses its back-reaction -- v369 as a DAG/script fact. HONEST: this does NOT prove SEAM.EQUIV.01 (the readouts depend on the seam being the (E8)_1 net, the one open keystone); QG.AMB.01 specifically is dead.",
      "status": "No-Ambient-Dependency [E]/[C]: no frozen readout is computed from QG.AMB.01 -- it is a certification SINK (every dependent is a gravity-gate/QG row, no prediction), not a value source (script '-'; predictions computed by v3/v7/v9/v268, never an ambient-measure script), and its only predictive link is theta13's architectural 4D-QFT contract edge (coherence, not numerical); the gap margin Delta_eff~1.648>0 decouples its back-reaction (v337). v369 prose -> machine-checked DAG/script fact [E]; verdict [C], does NOT prove SEAM.EQUIV.01",
      "canonicalStatus": "Numerical",
      "dependencies": "QG.AMB.01; DECOUPLING.THEOREM.01; SEAM.EQUIV.01; v337; v369; v384",
      "scripts": [
        "v423_no_ambient_dependency.py"
      ],
      "script": "v423_no_ambient_dependency.py",
      "location": "tfpt_4",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 490
    },
    {
      "id": "SEAM.EQUIV.LTORP.01",
      "claim": "The (LTO-RP) reduction of the keystone (2026-06-25): an HONEST MMST-style reduction of SEAM.EQUIV.01's one open lemma -- the raw collar's intrinsic Bisognano-Wichmann property (the residual of v308/v309) -- to the recent commuting-projector theorem of Naaijkens-Penneys-Wallick [NPW26, arXiv:2605.10693], with the central mapping CORRECTED and the two open sub-steps made explicit. Does NOT close SEAM.EQUIV.01. [E] CORRECTED MAPPING: the mu4 clock rho is a modular SYMMETRY ([rho,K]=0, v309/v199), NOT the modular flow -- 'sigma^psi=rho' is type-wrong (a one-parameter group t->Delta^{it} vs a finite order-4 element); verified on an explicit character-block-diagonal K, with an off-character negative control. [E] CORRECT TARGET: NPW26's reflection-positivity axiom (LTO-RP) Theta(sigma^psi_{-i/2}(x^dagger))p_S=x p_S MEANS u_Theta=J (the reflection IS the Tomita-Takesaki modular conjugation) = the intrinsic BW condition theta K theta=-K, DISTINCT from omega o rho=omega; we refute the identification '(LTO-RP)<=>omega o rho=omega' by A-and-NOT-B: a positive char-block-diagonal K satisfies [rho,K]=0 (A) but, being positive definite, can never satisfy theta K theta=-K for any reflection (B), while B is non-vacuous (a +/- symmetric K' with a pairing reflection satisfies it). [E] NPW26 APPLICABILITY (make-or-break): NPW26 prove (LTO-RP) only for TOPOLOGICALLY ORDERED commuting-projector models (Toric Code |det K|=4, a TRACE state; Levin-Wen anyonic), whereas the seam is the opposite corner -- invertible (holomorphic (E8)_1, |det Cartan(E8)|=1, no anyons) AND a beta=1 KMS state (NOT a trace) -- so it sits in NEITHER proved bucket; neg control D8/SO(16) |det|=4. [E] established ingredients present: recovery gap 6 ln(3/2)>0 (v302), Z/2 reflection / OS (v240), mu4 clock from cross-ratio 2 (v214/v267), [rho,K]=0 (v309/v199). [C] VERDICT: SEAM.EQUIV.01's BW residual REDUCES to NPW26 Theorem A modulo TWO open sub-steps -- (i) realise the raw seam as a Z/2-reflection commuting-projector LTO; (ii) extend (LTO-RP) to the invertible beta=1-KMS case (outside NPW26's worked models). An MMST-style reduction (cf. v308/v336), NOT a closure; SEAM.EQUIV.01 stays [O].",
      "status": "(LTO-RP) keystone reduction [E]/[C]: SEAM.EQUIV.01's intrinsic-BW lemma reduced to the cited NPW26 commuting-projector theorem, mapping corrected -- (LTO-RP)=u_Theta=J=intrinsic BW (theta K theta=-K), NOT omega o rho=omega; the mu4 clock is a modular symmetry [rho,K]=0 (sigma^psi=rho type-wrong); A-and-NOT-B refutes clock-invariance => (LTO-RP). Make-or-break: NPW26 prove (LTO-RP) for topologically-ordered models (trace/anyonic), the seam is invertible (|det E8|=1) + beta=1 KMS (not a trace), in neither bucket. REDUCTION not closure; two open sub-steps (Z/2-reflection commuting-projector LTO realisation; invertible-KMS (LTO-RP)). SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.CHAIN.01; QGEO.MODHAM.01; v308; v309; v302; v240; v214; v267; v199",
      "scripts": [
        "v424_seam_lto_rp_reduction.py"
      ],
      "script": "v424_seam_lto_rp_reduction.py",
      "location": "tfpt_research_contracts",
      "externalData": "NPW26 (arXiv:2605.10693)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 491
    },
    {
      "id": "DYN.TRANSFER.UNIVERSAL.01",
      "claim": "The single-flow reduction of F_transfer (2026-06-25): the four frontier transfers (F_pole, F_Boltzmann, F_relic, F_QCD) do NOT each need a separate external solver -- their DYNAMICS is the ONE native flow of the theory (the seam modular/recovery semigroup, v238/v240, gap 6 ln(3/2)) restricted to each sector. Native flow + EXTERNAL ANCHORS, firewall-clean (the observable stays [C] until the anchor is supplied). [E] THE ONE NATIVE FLOW: recovery spec {1,(2/3)^6,(1/3)^6}, H_OS=-log T gap 6 ln(3/2)>0, e^{-Delta}=(2/3)^6 exact; an explicit 2-state column-stochastic T gives L=log T a Markov generator (off-diag>=0, columns sum 0). [E] F_pole IS THIS FLOW (lepton sector): the v99 Koide generator dq/dt=(Delta/N_fam)(q-2)(q-5) linearised at the attractor q=2 has rate g'(2)=-Delta, so the time-1 contraction is e^{-Delta}=(2/3)^6 (the recovery eigenvalue); q=5 unstable (g'(5)=+Delta). [E] F_Boltzmann washout = the native contraction (2/3)^6, CP phase delta_PMNS=4pi/3 native (v320). [E] F_QCD rate native: b3=-7 (v159), b0(n_f=6)=11-2*6/3=7=-b3, so the dimensional-transmutation exponent -2pi/(b0 alpha_s) is native (QCD.LAMBDA.01). [C] VERDICT: the transfer DYNAMICS is one native object (v238 recovery semigroup, v383 universal) -- native gap + native rates; only the ANCHORS (v_geo, lattice C_p, seesaw M_R) stay external. F_relic's free input is a cosmological initial condition theta_i (not theory-derivable; best case mu4-symmetry-reduced) -- the one genuine wall. A single-flow REDUCTION, NOT internalisation: no new premise, the firewall holds. Extends v383 (static universal gap) to a dynamic universal flow.",
      "status": "Single-flow reduction [E]/[C]: the four F_transfer flows are the ONE v238 recovery semigroup restricted to sectors (native gap 6 ln(3/2), native rates e^{-Delta}=(2/3)^6, b3=-7); F_pole=v99 generator at q=2, F_Boltzmann washout=that contraction (CP 4pi/3 native), F_QCD rate native -- only the ANCHORS (v_geo/C_p/M_R) external; F_relic's cosmological IC theta_i the one wall. Reduction not internalisation, firewall-clean; extends v383 static->dynamic",
      "canonicalStatus": "Numerical",
      "dependencies": "CONTRACT.F.01; DYN.SEMIGROUP.01; DYNAMICS.UNIVERSAL.01; QCD.LAMBDA.01; v238; v240; v99; v320; v159; v383",
      "scripts": [
        "v425_dyn_transfer_universal.py"
      ],
      "script": "v425_dyn_transfer_universal.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 492
    },
    {
      "id": "SEAM.EQUIV.LTORP.KMS.01",
      "claim": "The invertible beta=1-KMS variant of (LTO-RP) (2026-06-25): the one corner Naaijkens-Penneys-Wallick [NPW26, arXiv:2605.10693] leave open -- they prove (LTO-RP) only for TOPOLOGICALLY ORDERED commuting-projector models (Toric Code via a TRACE state; Levin-Wen anyonic), while the TFPT seam is the opposite corner (invertible, holomorphic (E8)_1, no anyons, AND a beta=1 KMS state, not a trace). A constructive finite-dim Tomita-Takesaki witness that on this corner the (LTO-RP)/Bisognano-Wichmann condition u_Theta=J is satisfiable and natural, advancing sub-step (ii) of v424; does NOT close SEAM.EQUIV.01. One-particle dictionary: modular operator Delta=e^{-K}, (LTO-RP)/BW <=> u_Theta=J <=> Theta K Theta^{-1}=-K (J Delta J=Delta^{-1}); the mu4 clock rho a modular SYMMETRY <=> [rho,K]=0. [E] INVERTIBLE-KMS COLLAR: K reflection-odd/mu4-even/gapped (gap 2k=6 ln(3/2)); C=1/(1+e^K) the beta=1 KMS covariance satisfies C/(1-C)=e^{-K} (a genuine KMS, not trace, state); rho order 4. [E] (LTO-RP)/BW HOLDS: Theta K Theta^{-1}=-K (reflection = modular conjugation, u_Theta=J) AND [rho,K]=0 -- both hold, so (LTO-RP) is satisfied by direct Tomita-Takesaki (no skein/UFC machinery). [E] INVERTIBLE, OUTSIDE NPW26: |det Cartan(E8)|=1 (one primary, invertible, no anyons) vs the anyonic |det|=4 of NPW26's worked models, and the state is NOT a trace (Delta=e^{-K}!=I, nontrivial modular flow). [E] NEG CONTROLS: a side-even reflection gives Theta K Theta=+K!=-K (BW fails); an off-mu4-block K mixing two marks gives [rho,K]!=0 (mu4 symmetry fails). [C] VERDICT: on the gapped invertible beta=1-KMS reflection-symmetric collar, (LTO-RP) reduces to (Theta K Theta=-K) and ([rho,K]=0), both structural -- the invertible-KMS variant NPW26 leave open is constructively satisfiable. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O] (the continuum/abstract realisation, sub-step (i), remains); a finite-dim witness, not the continuum theorem.",
      "status": "Invertible beta=1-KMS (LTO-RP) [E]/[C]: the corner NPW26 leave open (they do topologically-ordered; the seam is invertible + KMS, not a trace) is constructively satisfiable -- a gapped reflection-symmetric mu4-even collar has Theta K Theta=-K (u_Theta=J) and [rho,K]=0, so (LTO-RP) holds by direct Tomita-Takesaki; |det E8|=1 (no anyons), state not a trace; neg controls have teeth. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O] (continuum sub-step (i) remains), finite-dim witness not continuum theorem",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.LTORP.01; SEAM.EQUIV.01; SEAM.EQUIV.CHAIN.01; QGEO.MODHAM.01; v424; v309; v302; v240; v214",
      "scripts": [
        "v426_seam_lto_rp_kms.py"
      ],
      "script": "v426_seam_lto_rp_kms.py",
      "location": "tfpt_research_contracts",
      "externalData": "NPW26 (arXiv:2605.10693)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 493
    },
    {
      "id": "OVERDET.WITNESS.MAP.01",
      "claim": "The multiply-vs-compress FRAMEWORK for over-determination (2026-06-25; classification refined by v428/OVERDET.WITNESS.RECLASS.01): a reviewer's methodological correction made machine-explicit -- 'every independent restriction multiplies the implausibility of coincidence' is true ONLY for witnesses from genuinely DISJOINT grammars; witnesses that are projections of the SAME generator count as COMPRESSION, not Bayesian multiplication. [E] SEVEN GRAMMAR READOUTS, each computed from its own syntax and landing on the carrier skeleton: Gauss Z[i] N(3+2i)=13; Eisenstein Z[w] N(3+2w)=7; cyclotomy Q(z5) N(3+2 z5)=55; Galois |Gal Q(z5)|=4; root lattice |det Cartan(E8)|=1; Pascal/exterior C(4,<=2)=11 with 2^4=16=dim S+; Coxeter h(E8)=30=2*3*5 with phi(30)=8=rank E8. [E] SYNTACTIC DISTINCTNESS != INDEPENDENCE: by v236 (TOPO.BRIESKORN.01) these seven are facets of ONE (2,3,5)/E8 object, so they COMPRESS (settled in v428), like the anchor a=(1,1,2) emitting (e1,e2,e3)=(4,5,2)+power sums+the E8 data from one generator. [C] VERDICT: the framework stands but the classification of the seven is refined by v428 -- the genuine multiplicative strength is the multiply-forced inputs (the '8' four independent ways) + foreign witnesses (alpha^-1~137), not the seven readings; coincidence is still an expensive explanation of the discrete core, but this does NOT by itself establish physics (the seam/anchor/transfer bridge problem).",
      "status": "Witness map [E]/[C]: disjoint grammars multiply, same-generator projections compress; the seven arithmetic witnesses are refined by v428 to COMPRESSION (facets of one (2,3,5)/E8 object, v236). The genuine multiplication is the multiply-forced inputs + foreign witnesses (alpha^-1~137); coincidence is expensive for the discrete core; does not by itself establish physics",
      "canonicalStatus": "Numerical",
      "dependencies": "E8.REVERSE.AUDIT.01; ARITH.HULL.01; SEAM.GALOIS.01; CAR.PASCAL.01; TOPO.BRIESKORN.01; ARCH.CORE.01; v415; v417; v418; v419; v422; v45; v53",
      "scripts": [
        "v427_overdet_witness_map.py"
      ],
      "script": "v427_overdet_witness_map.py",
      "location": "tfpt_safeguards",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 494
    },
    {
      "id": "OVERDET.WITNESS.RECLASS.01",
      "claim": "The honest correction of v427 (2026-06-25): applying v427's own multiply-vs-compress test to v427 itself. [E] THE ONE GENERATOR (v236/TOPO.BRIESKORN.01): the (2,3,5) Brieskorn singularity generates E8/Coxeter-30 -- Milnor mu=(2-1)(3-1)(5-1)=8=rank E8, 30=2*3*5=h(E8), phi(30)=8. [E] WITNESS RE-CLASSIFICATION: each of v427's seven witnesses maps to a facet of that SAME object (primes 2,3,5 of 30, the clock 30, or E8), so all seven collapse to ONE generator (distinct objects=1) => COMPRESSION, not seven independent multiplications. [E] THE GENUINE MULTIPLICATION: the '8' in c3=1/(8 pi) forced four independent ways -- rank(E8), the carrier Coxeter h(D5)=2(5-1), the totient phi(30), the Milnor number -- plus the FOREIGN witness alpha^-1~137 (prime, not in the (2,3,5)-skeleton; v305). [C] VERDICT: v427's framework is right, its classification was wrong -- the arithmetic witnesses compress one (2,3,5)/E8 object; the honest over-determination is the multiply-forced inputs + foreign 137. Refines and partly walks back v427; consistent with v236/v305/v313.",
      "status": "Witness re-classification [E]/[C]: v427's seven arithmetic witnesses are facets of ONE (2,3,5)/E8 object (v236) => compression, not seven multiplications. The genuine multiplication is the '8' forced four independent ways (rank E8, h(D5), phi(30), Milnor) + the foreign alpha^-1~137. Honest correction of v427 (consistent with v305/v236/v313)",
      "canonicalStatus": "Identity",
      "dependencies": "OVERDET.WITNESS.MAP.01; TOPO.BRIESKORN.01; ARCH.CORE.01; v305; v313",
      "scripts": [
        "v428_overdet_reclass.py"
      ],
      "script": "v428_overdet_reclass.py",
      "location": "tfpt_safeguards",
      "externalData": "-",
      "supersedes": "OVERDET.WITNESS.MAP.01",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 495
    },
    {
      "id": "SEAM.EQUIV.BW.HSMI.01",
      "claim": "The half-sided modular inclusion (Borchers/Wiesbrock) as an INTRINSIC Bisognano-Wichmann route (2026-06-30) for face (ii) of SEAM.EQUIV.01 -- geometric modular flow WITHOUT presupposing covariance. Complementary to v323/v426 (which assume a rotation-covariant covariance) and to the Adamo-Giorgetti-Tanimoto covariance theorem. The standard-pair criterion (Borchers/Wiesbrock/Longo): a pair (K,P) with K self-adjoint (Delta=e^{-K}), P>=0 (positive-energy lightlike translation), [K,P]=P, and a reflection J with JKJ=-K and JPJ=-P is a +half-sided modular inclusion, by Borchers' theorem the modular flow is then GEOMETRIC. [E] STANDARD-PAIR ALGEBRA (ladder, exact): K=diag(n), P=raising shift give [K,P]=P and the Borchers scaling Delta^{it}P Delta^{-it}=e^{-t}P (the ax+b/Moebius relation). [E] BW REFLECTION: J(n->-n) gives J^2=I, JKJ=-K (u_Theta=J) and JPJ=P_- (opposite translation), the standard-pair reflection J U(s)J=U(-s). [E] POSITIVE ENERGY: in the unitary lowest-weight sl(2,R) rep the line translation P_line=L0-(L+ +L-)/2 is POSITIVE (min eig>0), the positive-energy generator the c3 one-sided Gauss-Bonnet origin demands; the boost direction is the indefinite negative control. [C] SYNTHESIS: P>=0 + [K,P]=P + J is a Longo STANDARD PAIR / +half-sided modular inclusion, so by Borchers the seam modular flow is geometric INTRINSICALLY, without presupposing the rotation-covariance of the vacuum; face (ii) becomes a STRUCTURE consequence, not a state condition. [C]/[O] VERDICT: an intrinsic-BW REDUCTION via Borchers/Wiesbrock; SEAM.EQUIV.01 stays [O] -- assembling the finite algebra and the positive-energy rep into ONE continuum standard pair is the infinite-dim continuum existence (the same wall as MMST/NPW26), a finite witness + cited theorem, not a closure.",
      "status": "Half-sided modular inclusion [E]/[C]: a Longo standard pair (P>=0, [K,P]=P, J:JKJ=-K, JPJ=P_-) is a +half-sided modular inclusion, so by Borchers the seam modular flow is geometric INTRINSICALLY (no covariance presupposed); face (ii) becomes a structure consequence. Finite witnesses: ladder algebra exact, sl(2,R) positivity, BW reflection; neg control indefinite boost. SEAM.EQUIV.01 stays [O] (continuum standard pair = continuum existence, same wall as MMST/NPW26)",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.LTORP.01; SEAM.EQUIV.LTORP.KMS.01; QGEO.MODHAM.01; v323; v426; v309",
      "scripts": [
        "v438_seam_hsmi_borchers.py"
      ],
      "script": "v438_seam_hsmi_borchers.py",
      "location": "tfpt_research_contracts",
      "externalData": "Borchers/Wiesbrock half-sided modular inclusion; Longo standard pairs; Adamo-Giorgetti-Tanimoto (arXiv:2508.07109)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 496
    },
    {
      "id": "SEAM.EQUIV.CONTINUUM.LR.01",
      "claim": "The Lieb-Robinson/Nachtergaele-Sims continuum-existence MOTOR for the gapped seam collar (2026-06-30) -- grounds the BULK half of the cited MMST existence (SEAM.MMST.INCLASS.01/v336) in the explicit v367 p+ip lattice model, narrowing the open analytic step to the chiral EDGE scaling limit. [E] UNIFORM GAP: the collar gap is independent of N (Delta(N)=const>0 across N=20..160), the lattice image of the OS transfer gap 6 ln(3/2)>0 (v302/v329) -- the Nachtergaele-Sims hypothesis. [E] EXPONENTIAL CLUSTERING: |<c_i^dag c_j>|~e^{-r/xi} with xi finite and STABLE across N (gapped => Hastings-Koma decay), the N-independent locality. [E] LIEB-ROBINSON LIGHT CONE: the single-particle propagator |(e^{-iHt})_ij| is confined to |i-j|<~v_LR t with finite v_LR (~2t), exponentially small beyond the cone -- finite-speed information flow. [C] NACHTERGAELE-SIMS LIFT: uniform gap + finite LR velocity => (Nachtergaele-Sims-Young) exponential clustering with N-independent xi AND existence+uniqueness of the thermodynamic-limit ground state with a finite light cone -- the bulk continuum-existence ingredient derived for OUR collar, not imported. [C]/[O] VERDICT: the bulk thermodynamic limit of the seam collar EXISTS by Lieb-Robinson; SEAM.EQUIV.01 stays [O], the open analytic step now narrowed to the CHIRAL massless EDGE scaling limit (the c=8 (E8)_1 CFT, MMST), distinct from the gapped bulk limit.",
      "status": "Lieb-Robinson motor [E]/[C]: the gapped p+ip collar (v367) has a uniform-in-N gap, exponential clustering with N-independent xi, and a finite light cone, so by Nachtergaele-Sims its bulk thermodynamic limit EXISTS -- grounding the bulk half of the cited MMST existence in OUR model. SEAM.EQUIV.01 stays [O], the open step narrowed to the chiral massless edge scaling limit",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.MMST.INCLASS.01; SEAM.S3.RP.01; SEAM.EQUIV.CONTINUUM.03; v367; v336; v302; v329",
      "scripts": [
        "v439_seam_lieb_robinson.py"
      ],
      "script": "v439_seam_lieb_robinson.py",
      "location": "tfpt_research_contracts",
      "externalData": "Lieb-Robinson/Nachtergaele-Sims bounds; MMST (scaling limit)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 497
    },
    {
      "id": "SEAM.EQUIV.LTORP.BETA.01",
      "claim": "The beta-interpolation connecting the NPW26-PROVEN trace case (beta=0) to the seam's beta=1 KMS case along an obstruction-free homotopy (2026-06-30) -- upgrading the single-point v426 witness for v424 sub-step (ii) to a PATH argument. NPW26 prove (LTO-RP) for the TRACE state (beta=0, C=1/2); the TFPT seam is the beta=1 KMS state; v426 showed (LTO-RP) holds AT beta=1; this module shows it holds ALONG the whole interpolation C_beta=1/(1+e^{beta K}), beta in [0,inf). [E] beta-INDEPENDENT (LTO-RP) DATA: Theta K Theta=-K and [rho,K]=0 are conditions on K alone, independent of beta. [E] THE KMS PATH: beta=0 -> C=1/2 I (the NPW26 TRACE state), beta=1 -> the seam KMS state, beta->inf -> the ground-state projection; at every beta Theta C_beta Theta=1-C (reflection), [rho,C_beta]=0 (mu4), 0<=C_beta<=1 (valid covariance/RP). [E] NO OBSTRUCTION: beta->C_beta is norm-continuous and stays in the (LTO-RP)-valid set -- a continuous homotopy from the proven trace endpoint to the seam KMS point, no topological wall. [E] NEG CONTROLS: a reflection-EVEN K' breaks Theta C_beta Theta=1-C for beta>0 (the trace endpoint is degenerate), an off-mu4-block K breaks [rho,C_beta]. [C]/[O] VERDICT: (LTO-RP)/BW connects the NPW26 trace case to the seam beta=1 KMS case along an obstruction-free homotopy -- a PATH argument for v424 sub-step (ii), stronger than the single point of v426; SEAM.EQUIV.01 stays [O], the residual being closedness of (LTO-RP) along the path in the continuum (uniform in the limit).",
      "status": "beta-interpolation [E]/[C]: the (LTO-RP)/BW data (Theta K Theta=-K, [rho,K]=0) are beta-independent; along the whole path C_beta=1/(1+e^{beta K}) from the NPW26 trace (beta=0) to the seam KMS (beta=1) every covariance is reflection-related, mu4-invariant and valid, norm-continuous, no obstruction -- a PATH argument for v424 (ii), stronger than v426. SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.LTORP.01; SEAM.EQUIV.LTORP.KMS.01; SEAM.EQUIV.01; v424; v426",
      "scripts": [
        "v440_seam_lto_rp_beta.py"
      ],
      "script": "v440_seam_lto_rp_beta.py",
      "location": "tfpt_research_contracts",
      "externalData": "NPW26 (arXiv:2605.10693)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 498
    },
    {
      "id": "SEAM.EQUIV.APPLICABILITY.01",
      "claim": "The applicability ledger (2026-06-30) -- a hypothesis-by-hypothesis audit of the THREE external theorems the seam keystone cites (MMST scaling limit, NPW26 (LTO-RP), Adamo-Giorgetti-Tanimoto covariance), isolating the SINGLE genuinely-imported analytic fact. Answers the reviewer worry that the cited theorems may carry hidden hypotheses, by classifying every hypothesis as either a TFPT-internal computed fact (with its establishing vN) or a genuinely external analytic assumption. [E] MMST HYPOTHESES: D=16 Majoranas (=2^{g_car-1}), c=8 (=g_car+N_fam), rank E8=8, range rank<=c<=D reads 8<=8<=16 (in range, saturating rank=c=8), gapped (6 ln(3/2)>0), quasi-free CAR -- all internal computed facts. [E] NPW26 HYPOTHESES: (LTO-RP)=u_Theta=J, gap, Z2 reflection, mu4 clock, [rho,K]=0 internal (v424/v426/v440); the seam is invertible |det Cartan E8|=1 + beta=1 KMS, OUTSIDE NPW26's topological/trace bucket (worked |det|=4) -- NPW26 used as a TEMPLATE, its theorem not directly imported. [E] ADAMO-GIORGETTI-TANIMOTO (arXiv:2508.07109): DHR reps of nets extending loop-group/Virasoro are AUTOMATICALLY positive-energy + diff-covariant; the seam net extends (D5)_1 x (A3)_1 (v154), so geometric covariance is a CONSEQUENCE -- the 'BW presupposes covariance' circularity (v215) is defused; covariance downgraded ASSUMPTION->[E]. [E] THE SINGLE IMPORTED FACT: of 12 audited hypotheses, 11 are TFPT-internal computed facts and exactly 1 is external -- the continuum EXISTENCE of the chiral massless scaling limit (MMST/v336, Lean FORM.SEAM.MMST.01). [C]/[O] VERDICT: the keystone's external dependence is pinned to ONE analytic fact; covariance is an Adamo consequence, not an input; SEAM.EQUIV.01 stays [O] -- that one fact is the honest closure ceiling.",
      "status": "Applicability ledger [E]/[C]: a hypothesis-by-hypothesis audit of MMST/NPW26/Adamo finds 11 of 12 hypotheses are TFPT-internal computed facts and exactly 1 is external -- the continuum existence of the chiral massless scaling limit; covariance is an Adamo consequence (defuses v215), not an input. SEAM.EQUIV.01's external dependence is pinned to ONE analytic fact; stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.MMST.INCLASS.01; SEAM.EQUIV.LTORP.01; SEAM.EQUIV.LTORP.KMS.01; v424; v426; v440; v336; v154; v215",
      "scripts": [
        "v441_seam_applicability_ledger.py"
      ],
      "script": "v441_seam_applicability_ledger.py",
      "location": "tfpt_research_contracts",
      "externalData": "MMST; NPW26 (arXiv:2605.10693); Adamo-Giorgetti-Tanimoto (arXiv:2508.07109)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 499
    },
    {
      "id": "SEAM.RIGIDITY.UNIFORM.01",
      "claim": "The uniform-in-N hardening of the Seam State Rigidity Theorem (2026-06-30; hardens SEAM.RIGIDITY.01/v398) -- the band-limited mu4-character rigidity is NOT a small-N artifact. v398 verified [rho,K]=0 by a finite sweep N=4..64; this module makes it uniform in N and adds an N-independent rigidity bound. [E] EXACT UNIFORM 4-SECTOR COMMUTATION: rho=diag(i^n) has EXACTLY 4 distinct eigenvalues {1,i,-1,-i} for ALL N (an order-4 clock, N-independent), and a genuinely character-block-diagonal K (random Hermitian within each n mod 4 class, not merely diagonal) satisfies [rho,K]=0 EXACTLY for N=4..256 (extends v398's N<=64). [E] UNIFORM GAP: the gapped collar proxy has Delta(N)=const>0 across N=20..160 (the Nachtergaele-Sims uniform gap, shared with v439). [E] N-INDEPENDENT RIGIDITY BOUND: a FIXED local clock-breaking perturbation (single off-mu4-block Hermitian entry) gives ||[rho,V]||=2 EXACTLY for all N=4..256 -- a uniform lower bound; any fixed-size clock-breaking is detected with N-independent strength. [E] NEG CONTROL: a clock-PRESERVING (same-sector) perturbation gives ||[rho,V']||=0; an order-2 (Z2) clock has only 2 distinct eigenvalues, not 4 -- the ORDER of the clock (the four Gauss-Bonnet marks) is the discriminator, uniformly in N. [C]/[O] VERDICT: the band-limited rigidity of v398 is uniform in N (exact 4-sector commutation to N=256, uniform gap, N-independent detection) -- not a small-N artifact; SEAM.RIGIDITY.01 / SEAM.EQUIV.01 stays [O]: that RP+gap+holomorphy FORCE Lambda_Sigma block-diagonal on the FULL L^2 (rather than merely permit it) is the cited rigidity/MMST step.",
      "status": "Uniform-in-N rigidity [E]/[C]: the band-limited mu4-character rigidity of v398 is uniform in N -- exact 4-sector commutation to N=256, a uniform gap, and an N-independent clock-breaking lower bound ||[rho,V]||=2; not a small-N artifact. SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O] (the full-L^2 forcing is the cited rigidity/MMST step)",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.RIGIDITY.01; SEAM.EQUIV.01; QGEO.MODHAM.01; v398; v439; v309; v199",
      "scripts": [
        "v442_seam_rigidity_uniform.py"
      ],
      "script": "v442_seam_rigidity_uniform.py",
      "location": "tfpt_research_contracts",
      "externalData": "MMST (rigidity/scaling limit)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 500
    },
    {
      "id": "SEAM.E8.DISCRIMINATOR.01",
      "claim": "The empirical/structural DISCRIMINATOR bundle (2026-06-30; mode D of the v347 closure-mode classification) -- concrete FALSIFIABLE kill tests separating the claimed seam=(E8)_1 from the nearest c=8 rival SO(16)_1=(D8)_1. v347 lists four closure modes; mode D is 'test the consequences', realised here as sharp discriminators. [E]/[X] PRIMARY/GSD DISCRIMINATOR: |det Cartan(E8)|=1 (ONE primary, holomorphic) vs |det Cartan(D8)=SO(16)|=4 (FOUR primaries 1,v,s,c); torus GSD = #primaries = 1 (E8) vs 4 (SO(16)) -- a seam with GSD>1 / >1 superselection sector falsifies (E8)_1. [E]/[X] CURRENT/THETA DISCRIMINATOR: level-1 current count dim E8=248 vs dim SO(16)=120; the E8 lattice theta has 240 norm-2 roots (=112 integer + 128 spinor, computed) vs D8's 112 -- the affine current spectrum and lattice theta DIFFER; the edge current multiplicity (248 vs 120) is observable. [E] c=8 IS NOT A DISCRIMINATOR: both nets have c=8 (the v277 ambiguity), so the central charge alone cannot decide; only the order-4 mu4 condensation + det K=1 (the genus-1 statement, SEAM.S3.MODULAR.01/v90) selects E8. [E] NON-VACUITY: the discriminators all fire on the rival (det 4!=1, currents 120!=248, roots 112!=240). [X]/[C] VERDICT: the (E8)_1 seam carries sharp falsifiable signatures distinct from every nearby c=8 net (mode D of v347 realised as concrete kill tests); SEAM.EQUIV.01 stays [O] -- the discriminators distinguish the TARGET but do not by themselves PROVE the seam realises it (continuum existence, residual [O]); they make the claim FALSIFIABLE.",
      "status": "E8 vs SO(16) kill tests [E]/[X]: mode D of v347 realised as discriminators -- torus GSD 1 vs 4, level-1 currents 248 vs 120, lattice theta 240 vs 112 norm-2 roots; c=8 alone does NOT decide (v277), only det K=1 (genus-1, v90) selects E8. Falsifiable: a seam with GSD>1 falsifies (E8)_1. SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Lattice",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.CLOSURE.01; SEAM.S3.MODULAR.01; v347; v90; v89; v281; v422",
      "scripts": [
        "v443_seam_e8_discriminator.py"
      ],
      "script": "v443_seam_e8_discriminator.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 501
    },
    {
      "id": "SEAM.EQUIV.CONTINUUM.EDGE.01",
      "claim": "The CHIRAL EDGE scaling-limit MOTOR (2026-06-30) -- the edge half of the one SEAM.EQUIV.01 continuum residual, attacking exactly the step v439 left open. v439 grounded the BULK thermodynamic limit (uniform gap, exponential clustering, finite light cone) and explicitly narrowed the open analytic step to the chiral massless EDGE scaling limit; this module attacks THAT residual on the SAME explicit v367/v368 p+ip collar, showing the lattice edge already carries the kinematic AND conformal data of a chiral c=1/2 Majorana CFT and that those data CONVERGE under refinement (the numerical hallmark of an existing scaling limit). [E] CHIRAL LINEAR EDGE DISPERSION: the in-gap branch is linear near its zero-crossing (top v=0.983, R^2=0.9999 over 21 pts) and the two boundaries carry OPPOSITE velocities (top +0.983, bot -0.983) -- one chiral Majorana per edge (relativistic, chiral kinematics). [E] ALGEBRAIC EDGE TWO-POINT: the equal-time edge correlator decays as r^{-eta} with eta=1.001 (the chiral free-fermion CFT exponent) -- a CRITICAL edge. [E] EXPONENTIAL BULK (neg control): the bulk-row correlator decays exponentially (xi~1.1 sites) -- only the edge is critical, so the power law is the edge CFT, not a geometric artefact. [E] TRIVIAL-PHASE NEG CONTROL: M=3 (no edge, v368) has an exponential edge-row correlator -- the algebraic decay is a CONSEQUENCE of the topological edge, not the boundary geometry. [E] SCALING-COLLAPSE / CONVERGENCE: eta is N-stable (1.001+-0.007) and the edge correlator profile is Cauchy-convergent under refinement (successive spread shrinks 0.028->0.013 across Nx=36..60) -- the lattice edge correlators converge to a fixed conformal profile (the edge analogue of v439's bulk thermodynamic limit). [C] EDGE CENTRAL CHARGE: eta=1 <=> one chiral Majorana (c=1/2) per edge, so the N_Maj=2^{g_car-1}=16-copy carrier has a chiral c_-=8 edge CFT (=(E8)_1, c_-=g_car+N_fam=8, consistent with v367/v368/v376) -- linear chiral dispersion + algebraic 1/r correlator + c_-=8 are the conformal data of the edge. [C]/[O] VERDICT: the EDGE half of the cited MMST scaling limit (v336) is now numerically GROUNDED on our explicit collar; with v439 (bulk) + v444 (edge) BOTH halves of the one continuum residual are grounded and pinned to a single cited theorem; SEAM.EQUIV.01 stays [O] -- the rigorous uniform convergence of all edge n-point functions (the MMST existence theorem) is not supplied here.",
      "status": "Chiral edge scaling motor [E]/[C]: the v367/v368 lattice edge carries the conformal data of a chiral c=1/2 Majorana CFT (linear chiral dispersion v~0.98 with opposite per-edge velocities; algebraic edge correlator eta~1 vs exponential bulk; trivial-phase neg control; N-stable Cauchy scaling-collapse), so the 16-copy edge is a chiral c_-=8 (E8)_1 CFT -- grounding the EDGE half of the cited MMST scaling limit the way v439 grounds the bulk. SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.MMST.INCLASS.01; SEAM.EQUIV.CONTINUUM.LR.01; SEAM.S3.SCALINGLIMIT.01; SEAM.EQUIV.CONTINUUM.02; v367; v368; v376; v392; v336; v439",
      "scripts": [
        "v444_seam_edge_scaling.py"
      ],
      "script": "v444_seam_edge_scaling.py",
      "location": "tfpt_research_contracts",
      "externalData": "MMST (scaling limit); bulk-edge correspondence / anomaly inflow",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 502
    },
    {
      "id": "FORM.SEAM.BW.HSMI.01",
      "claim": "LEAN 4 FORMALISATION of the Borchers/Wiesbrock standard-pair algebra (2026-06-30; the kernel-hardening of v438/SEAM.EQUIV.BW.HSMI.01) -- new module TfptCarrier/SeamStandardPair.lean, lake build OK, #print axioms CLEAN. It kernel-checks the four standard-pair RELATIONS on the centred 5-mode ladder over Z: [F] boost-translation commutator [K,P]=P (boost_translation_comm), [F] reflection involution J^2=1 (reflection_involution), [F] reflection reverses the boost JKJ=-K (reflection_reverses_boost), [F] reflection flips the translation JPJ=P_+ (reflection_flips_translation), bundled as standard_pair_relations -- ALL by kernel decide (axioms propext/Classical.choice/Quot.sound only, NO domain axiom); plus [F] boost_traceless tr K=0 (the structural reason a reflection CAN reverse the boost). The cited Borchers/Wiesbrock theorem (standard pair => +half-sided modular inclusion => geometric modular flow) and the one continuum input (assembling the finite ladder + positive-energy rep into a CONTINUUM standard pair, the same wall as MMST/NPW26) are declared as NAMED axioms (borchers_wiesbrock, seam_continuum_standard_pair); the composition geometricModularFlowFromStandardPair pins face (ii) of SEAM.EQUIV.01 to EXACTLY those two -- intrinsic Bisognano-Wichmann, no rotation-covariance presupposed. So the standard-pair RELATIONS are machine-checked; only the cited Borchers step + the continuum existence are assumed -- no sorry, no hidden assumption. Mirrors v438 (which verifies the same relations + sl(2,R) positivity numerically).",
      "status": "Formal [F] (Lean kernel; the standard-pair relations kernel-proved axiom-free, the Borchers step + continuum input named axioms)",
      "canonicalStatus": "Formal",
      "dependencies": "SEAM.EQUIV.BW.HSMI.01; SEAM.EQUIV.01; FORM.SEAM.MMST.01; v438",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/SeamStandardPair.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/SeamStandardPair.lean",
      "location": "tfpt_research_contracts",
      "externalData": "Borchers/Wiesbrock half-sided modular inclusion; Longo standard pairs",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 503
    },
    {
      "id": "SEAM.RIGIDITY.FORCING.01",
      "claim": "The rigidity FORCING theorem (2026-06-30; upgrades v442/SEAM.RIGIDITY.UNIFORM.01 from 'permit' to 'force') -- v442 (and v398) established that a mu4-block-diagonal K commutes with the order-4 clock rho=diag(i^n); this module proves the CONVERSE and exact structure, isolating the single remaining physical premise. [E] FORCING (commute <=> block-diagonal, EXACT): for rho=diag(i^n), [rho,K]=0 iff K_ij=0 whenever i!=j mod 4 -- verified ENTRYWISE for every (i,j), swept N=4..128; commuting FORCES mu4-block-diagonality, not merely permits it. [E] EXACT COMMUTANT DIMENSION: dim{K:[rho,K]=0}=sum_s n_s^2 (the four sector sizes), verified as the ad_rho nullspace for N=4..16 (e.g. N=16: 64=4*4^2 < 256=N^2); a PROPER subspace of the full N^2 algebra -- the forcing is a genuine, quantitative restriction. [E] OFF-BLOCK <=> NON-COMMUTING (sharpness): every off-block matrix unit E_ij (i!=j mod 4) has ||[rho,E_ij]||=|i^i - i^j|>=sqrt(2)>0 -- no off-block direction survives; the forcing leaves zero slack. [E] THE ORDER IS THE DISCRIMINATOR: the order-4 clock gives 4 sectors and commutant dim sum n_s^2(4); the order-2 clock diag((-1)^n) gives 2 sectors and a STRICTLY LARGER commutant (e.g. N=16: 64 vs 128) -- the FOUR Gauss-Bonnet marks (order 4=|mu4|=h(A3)) force strictly more block structure than two; only the order-4/index-4 extension is (E8)_1 (det 1; v281/v154/v351). [E] NONDEGENERACY: rho has EXACTLY 4 distinct eigenvalues {1,i,-1,-i} for all N>=4, so the four sectors never merge -- the block decomposition is unique. [C] SYNTHESIS (force, not permit): commuting with the order-4 mu4 clock is EQUIVALENT to mu4-block-diagonality (an iff, exact, uniform in N), so RP-definability (transfer fixed by RP+reflection, v54/v398) PLUS clock-invariance of the state FORCE the block-diagonal Lambda_Sigma -- v442's 'permit' upgraded to 'force' for the linear-algebra core. [C]/[O] VERDICT: the forcing is a proved iff; the SINGLE remaining premise is that the physical transfer Lambda_Sigma commutes with the clock on the full L^2 (clock-invariance = QGEO.SYM.01, downstream of SEAM.EQUIV.01 + BW; v323/v335). SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O] -- but the gap is now precisely 'transfer in commutant', not 'does block-diagonal permit'.",
      "status": "Rigidity FORCING [E]/[C]: commuting with the order-4 mu4 clock is EXACTLY mu4-block-diagonality (entrywise iff swept N=4..128; exact commutant dim sum n_s^2 < N^2; off-block slack zero; the order-2 commutant is strictly larger; 4 distinct eigenvalues), upgrading v442's 'permit' to 'force'. The single residual is the physical transfer lying in the commutant (clock-invariance=QGEO.SYM.01); SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.RIGIDITY.01; SEAM.RIGIDITY.UNIFORM.01; SEAM.EQUIV.01; QGEO.SYM.01; v442; v398; v54; v281; v154; v351; v323; v335",
      "scripts": [
        "v445_seam_rigidity_forcing.py"
      ],
      "script": "v445_seam_rigidity_forcing.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 504
    },
    {
      "id": "FORM.SEAM.APPLICABILITY.01",
      "claim": "LEAN 4 FORMALISATION of the applicability ledger (2026-06-30; the kernel-hardening of v441/SEAM.EQUIV.APPLICABILITY.01) -- new module TfptCarrier/SeamApplicabilityLedger.lean, lake build OK, #print axioms CLEAN. It turns the audit COUNT into a kernel-checked combinatorial fact: of the 12 cited-theorem hypotheses (MMST/NPW26/Adamo, mirrored from the v441 LEDGER in order), exactly 11 are TFPT-internal and exactly 1 external -- [F] total_count=12, [F] internal_count=11, [F] external_count=1, bundled as audit, ALL with NO axioms (pure kernel decide on a 12-element List of an inductive Src with derived DecidableEq); plus [F] mmst_range (rank<=c<=D reads 8<=8<=16) and [F] npw_outside_bucket (detE8=1 != detAnyon=4), also axiom-free. The single external fact -- the continuum existence of the chiral massless scaling limit -- is the named axiom ContinuumChiralScalingLimit; keystoneReducesToOneFact pins SEAM.EQUIV.01's external dependence to EXACTLY that one fact (+ the cited keystone_from_existence, already pinned in SeamScalingLimit/FORM.SEAM.MMST.01). So the very bookkeeping the reviewer worry is about (11 internal, 1 external) is itself machine-verified; no sorry. Mirrors v441.",
      "status": "Formal [F] (Lean kernel; the 11-internal/1-external audit count + MMST arithmetic kernel-proved axiom-free, the one external fact named)",
      "canonicalStatus": "Formal",
      "dependencies": "SEAM.EQUIV.APPLICABILITY.01; SEAM.EQUIV.01; FORM.SEAM.MMST.01; v441",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/SeamApplicabilityLedger.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/SeamApplicabilityLedger.lean",
      "location": "tfpt_research_contracts",
      "externalData": "MMST; NPW26 (arXiv:2605.10693); Adamo-Giorgetti-Tanimoto (arXiv:2508.07109)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 505
    },
    {
      "id": "SEAM.RIGIDITY.CLOCKINV.01",
      "claim": "DERIVING clock-invariance ('transfer in the commutant') from RP-symmetry (2026-06-30; closes the single residual of v445/SEAM.RIGIDITY.FORCING.01 one notch further) -- v445 reduced rigidity to the OPERATOR statement [rho,Lambda_Sigma]=0 (clock-invariance); this module shows that commutation is itself a CONSEQUENCE of a STRUCTURAL symmetry (mu4-invariance of the seam RP data, the four Gauss-Bonnet marks) via the OS canonical reconstruction (v54) and the free modular formula (v258), NOT an independent assumption. [E] OS RECONSTRUCTION INHERITS THE SYMMETRY: for a mu4-invariant RP covariance ([rho,C(t)]=0), the OS canonical one-step transfer Lambda=C(1)C(0)^-1=e^{-h} (the transfer fixed by RP+reflection, v54) satisfies [rho,Lambda]=0 -- swept N=4..32 over random block-diagonal h. [E] SHARPNESS/NEG CONTROL: a mu4-BREAKING covariance gives [rho,Lambda]!=0 -- iff-sharp, so the inheritance is non-vacuous; mu4-symmetry of the RP data is NECESSARY. [E] CONTRACTION/OS-VALIDITY: the reconstructed Lambda has spectrum in (0,1] (a positive contraction) -- the physical canonical transfer, not an arbitrary commuting matrix. [E] MODULAR IMPLEMENTATION (Tomita-Takesaki): the free modular Hamiltonian K=log((1-C)C^-1) (the v258 covariance->Dirac formula) of the mu4-invariant state satisfies [rho,K]=0 (the clock commutes with modular flow), while the symmetry-breaking control gives [rho,K]!=0. [C] SYNTHESIS: [rho,Lambda_Sigma]=0 is a CONSEQUENCE of mu4-invariance of the seam RP data (the four marks, v216/v323) via OS reconstruction (v54) + the modular fact (v258), not an extra assumption; with v445 (commute <=> block-diagonal), RP-definability PLUS the marks FORCE the block-diagonal Lambda_Sigma. [C]/[O] VERDICT: v445's residual is reduced from an OPERATOR commutation to the STRUCTURAL statement 'the seam RP data is mu4-symmetric' (QGEO.SYM.01); the continuum Tomita-Takesaki implementation on the full L^2 is the cited backbone. SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O].",
      "status": "Clock-invariance derived [E]/[C]: [rho,Lambda]=0 is a CONSEQUENCE of mu4-symmetry of the RP data (the four marks) via OS reconstruction (v54) + the modular formula (v258) -- the OS canonical transfer of mu4-symmetric data inherits the clock (swept N, iff-sharp neg control), is a contraction, and its modular Hamiltonian commutes with the clock. With v445 this reduces the rigidity residual from an operator commutation to 'the RP data is mu4-symmetric' (QGEO.SYM.01); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.RIGIDITY.FORCING.01; SEAM.RIGIDITY.01; SEAM.EQUIV.01; QGEO.SYM.01; v445; v442; v54; v258; v216; v323",
      "scripts": [
        "v446_seam_clock_invariance.py"
      ],
      "script": "v446_seam_clock_invariance.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 506
    },
    {
      "id": "SEAM.EQUIV.EDGE.CHERN.01",
      "claim": "An INDEPENDENT (topological) reading of the edge central charge c_-=8 (2026-06-30) -- corroborating v444/SEAM.EQUIV.CONTINUUM.EDGE.01 from a different observable. v444 read c_- from the CORRELATOR exponent (eta=1); this module reads the SAME c_-=8 from the bulk TOPOLOGY (Chern number) + the bulk-edge correspondence, a robust integer that does not depend on any fit, on the same v367/v368 p+ip collar. [E] INTEGER CHERN NUMBER: the M=1 collar lower band has Chern C=+1 by the Fukui-Hatsugai-Suzuki lattice link-variable method (an EXACT integer on the discretised BZ, no fit) -- one chiral channel per copy. [E] TRIVIAL-PHASE NEG CONTROL: M=3 (v368) has C=0 (no chiral edge); M=-1 gives C=-1 (opposite chirality, consistent with v444's opposite per-edge velocities). [E] BULK-EDGE CORRESPONDENCE: the open strip has EXACTLY |C|=1 in-gap edge branch crossing zero per boundary in the M=1 phase, 0 in the trivial phase -- the chiral edge channel count = |C|. [E] LI-HALDANE ENTANGLEMENT SIGNATURE: the half-cut bulk ground state's single-particle entanglement spectrum has in-gap modes at 1/2 in the topological phase and NONE in the trivial -- the entanglement edge tracks the physical edge. [C] EDGE CENTRAL CHARGE: c_- = N_Maj*(1/2)*|C| = 16*(1/2)*1 = 8 = g_car+N_fam, from the Chern integer + bulk-edge correspondence -- the SAME c_-=8 as v444's correlator route, from a logically independent topological observable. [C]/[O] VERDICT: v444's edge c_-=8=(E8)_1 reading is corroborated by an independent topological invariant; SEAM.EQUIV.01 stays [O] -- the topology fixes the edge central charge and channel count but does not supply the cited continuum existence theorem.",
      "status": "Independent topological c_-=8 [E]/[C]: the M=1 collar has integer Chern C=+1 (FHS, no fit; C=0 trivial, C=-1 opposite chirality), the open strip carries |C|=1 chiral edge branch per boundary, and the Li-Haldane entanglement spectrum tracks the edge; bulk-edge gives c_-=16*(1/2)*1=8=g_car+N_fam, corroborating v444 from a different observable. SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; v444; v367; v368; v376",
      "scripts": [
        "v447_seam_edge_chern.py"
      ],
      "script": "v447_seam_edge_chern.py",
      "location": "tfpt_research_contracts",
      "externalData": "bulk-edge correspondence / anomaly inflow",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 507
    },
    {
      "id": "SEAM.EQUIV.EDGE.FOURPOINT.01",
      "claim": "Pinning the ONE external MMST fact empirically (2026-06-30) -- the edge FOUR-point function is quasi-free and conformally convergent, extending v444's two-point scaling limit to the next order. v441 isolated the keystone's single EXTERNAL hypothesis (continuum existence of the chiral scaling limit, MMST); v444 grounded it at the TWO-point level; this module takes the FOUR-point step on the same v367/v368 collar. [E] QUASI-FREE/WICK (CAR hypothesis): the ground-state correlation matrix is an exact projector G^2=G (~1e-13), so the state is quasi-free and all n-point functions are Wick/Pfaffian sums of the two-point -- the MMST quasi-free CAR hypothesis (ledger entry 5) literally holds on the collar (structural: the BdG state is free). [E] FOUR-POINT CAUCHY CONVERGENCE: the edge four-point cross-ratio Q=[G01 G23]/[G02 G13] at commensurate proportional positions (1,2,4,7)*(Nx/12) is Cauchy-convergent under refinement (successive spread shrinks, Nx=36..60) -- the four-point converges, extending v444's two-point scaling-collapse. [E] CONFORMAL FOUR-POINT FORM: Q converges to the free-fermion conformal cross-ratio sin(5pi/12)/sin(pi/12)=2+sqrt(3)~3.732 (chord distance d(r)=(Nx/pi)sin(pi r/Nx)) -- the edge four-point has the chiral-CFT cross-ratio structure (eta=1), not just the two-point. [E] BULK NEG CONTROL: the bulk-row four-point is orders of magnitude off the conformal value (exponential decay, gapped) -- only the EDGE four-point is conformal. [C] VERDICT: the one external MMST fact (continuum existence) is now empirically supported at BOTH two-point (v444) and four-point order; every measured n-point converges to its conformal limit. SEAM.EQUIV.01 stays [O] -- the rigorous MMST uniform-convergence theorem (v336) is the cited backbone.",
      "status": "External MMST fact pinned at 4-point [E]/[C]: the edge four-point is quasi-free (G^2=G) and Cauchy-convergent to the free-fermion conformal cross-ratio 2+sqrt(3) (vs an off-scale exponential bulk control), extending v444's two-point scaling limit to 4-point order. The one external MMST existence fact is empirically supported at both orders; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.APPLICABILITY.01; v444; v441; v336",
      "scripts": [
        "v448_seam_edge_fourpoint.py"
      ],
      "script": "v448_seam_edge_fourpoint.py",
      "location": "tfpt_research_contracts",
      "externalData": "MMST (scaling limit)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 508
    },
    {
      "id": "FORM.SEAM.RIGIDITY.FORCING.01",
      "claim": "LEAN 4 FORMALISATION of the rigidity FORCING converse (2026-06-30; the kernel-hardening of v445/SEAM.RIGIDITY.FORCING.01, also hardening v442/SEAM.RIGIDITY.UNIFORM.01) -- new module TfptCarrier/SeamRigidityForcing.lean, lake build OK, #print axioms CLEAN. It kernel-checks the load-bearing forcing algebra over Fin 4 / Z: [F] commutator_zero_iff (the ENTRYWISE forcing clock4 a = clock4 b <=> a=b, i.e. i^a=i^b <=> a==b mod 4 -- residue-only, hence UNIFORM in N, which is exactly v442's uniformity), [F] clock_injective + four_distinct (the four eigenphases are pairwise distinct = nondegeneracy), [F] commutant_dim_16 + proper_16 (dim = sum n_s^2 = 4*4^2 = 64 < 256 = N^2, a PROPER subspace), [F] order2_dim_16 + order_discriminator_16 (the order-2 commutant 128 is strictly larger than the order-4 64 -- the four marks force strictly more structure), [F] marks_eq_four (|mu4|=h(A3)=4), bundled as forcing_kernel_facts -- ALL by kernel decide (axioms propext/Classical.choice/Quot.sound only, NO domain axiom). The lift of the finite-N forcing to the full operator L^2 (the cited rigidity/MMST step) is the single named premise ClockInvariantOnFullL2; forcingClosesRigidity pins the rigidity closure to EXACTLY that one premise -- which v446 (SEAM.RIGIDITY.CLOCKINV.01) further reduces to mu4-symmetry of the seam RP data. So the linear-algebra core is machine-checked; only the operator-level clock-invariance is assumed -- no sorry. Mirrors v445 (and Wolfram-mirrored in tfpt_readouts_extension.wl).",
      "status": "Formal [F] (Lean kernel; the entrywise forcing iff (uniform in N), the exact commutant dimension and the order discriminator kernel-proved axiom-free, the operator-level clock-invariance named)",
      "canonicalStatus": "Formal",
      "dependencies": "SEAM.RIGIDITY.FORCING.01; SEAM.RIGIDITY.UNIFORM.01; SEAM.EQUIV.01; v445; v442",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/SeamRigidityForcing.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/SeamRigidityForcing.lean",
      "location": "tfpt_research_contracts",
      "externalData": "Lean 4 + Mathlib (v4.29.1; #print axioms = propext/Classical.choice/Quot.sound for the kernel facts + the named operator premise)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 509
    },
    {
      "id": "SEAM.EQUIV.MMST.UNIFORM.01",
      "claim": "UNIFORM-IN-N control of the edge scaling limit (2026-06-30) -- the empirical content of the cited MMST uniform-convergence theorem (the single residual of SEAM.EQUIV.01 after v444/v447/v448), strengthened on the same v367/v368 p+ip collar. v444 grounded the two-point, v448 the four-point, each at a SINGLE conformal cross-ratio; this module adds the UNIFORMITY the abstract theorem actually asserts. [E] CROSS-RATIO A CONVERGES: the edge four-point Q(1,2,4,7)/12 -> sin(5pi/12)/sin(pi/12)=2+sqrt(3), deviation shrinks monotonically under refinement (Nx=36,48,60). [E] CROSS-RATIO B (independent positions): Q(1,3,5,9)/12 -> its OWN conformal value 2 -- a second observable with a DIFFERENT limit, so the convergence is not a single tuned point. [E] UNIFORM 1/N RATE: |Q(N)-Q_inf|*N <= 4 for BOTH cross-ratios and every Nx in {36,48,60} -- one N-independent constant bounds the deviation across two independent observables. [E] BULK NEG CONTROL: the bulk cross-ratio is off-scale (gapped). [C]/[O] the MMST uniform-convergence HYPOTHESIS is empirically satisfied (two-point v444, four-point v448, now multi-observable uniform rate); SEAM.EQUIV.01 stays [O] -- the abstract existence theorem (v336) is strengthened in evidence, not replaced by a proof.",
      "status": "Uniform-in-N edge convergence [E]/[C]: two independent edge four-point cross-ratios converge to distinct conformal values (2+sqrt(3) and 2) at one 1/N rate (|Q-Q_inf|*N bounded by a single constant), vs an off-scale bulk control -- the empirical content of the cited MMST uniform-convergence theorem; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.EDGE.FOURPOINT.01; v444; v448; v336; v367; v368",
      "scripts": [
        "v449_seam_mmst_uniform.py"
      ],
      "script": "v449_seam_mmst_uniform.py",
      "location": "tfpt_research_contracts",
      "externalData": "MMST (uniform convergence)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 510
    },
    {
      "id": "SEAM.EQUIV.EDGE.ENTANGLEMENT.01",
      "claim": "A THIRD independent reading of the edge central charge c_-=8 (2026-06-30) -- the Calabrese-Cardy entanglement scaling, after v444 (correlator) and v447 (topology). For a critical 1D free-fermion chain the Calabrese-Cardy law S(ell)=(c/3)ln[(L/pi)sin(pi ell/L)]+const fixes the central charge. [E] CALABRESE-CARDY LAW: the block entanglement entropy of the critical XX (Dirac) chain has a flat log slope -> c_Dirac=1.000 (fit ell=20..200, L=400) -- the universal scaling-limit signature. [E] c PER MAJORANA = 1/2: c_Dirac=1=2*c_Majorana (Dirac = two Majoranas), so c_Majorana=1/2 -- the chiral CFT of one p+ip edge mode (v444). [E] GAPPED NEG CONTROL: a dimerised (gapped) chain has saturating, ell-independent entanglement (c=0) -- the log growth is the CRITICAL edge CFT, not a lattice artefact. [C] EDGE CENTRAL CHARGE: c_- = N_Maj*c_Majorana = 16*(1/2) = 8 = g_car+N_fam, from entanglement -- the SAME c_-=8 as v444 (correlator) and v447 (topology), from a third independent observable. [C]/[O] three observables agree; SEAM.EQUIV.01 stays [O] -- entanglement fixes the central charge but not the cited continuum existence theorem.",
      "status": "Entanglement c_-=8 [E]/[C]: the Calabrese-Cardy law gives c_Dirac=1=2*(1/2) for the critical free-fermion chain (vs a saturating gapped control), so c_Majorana=1/2 and c_-=16*(1/2)=8=g_car+N_fam -- a third independent reading of the edge central charge after v444/v447; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.EDGE.CHERN.01; v444; v447",
      "scripts": [
        "v450_seam_edge_entanglement.py"
      ],
      "script": "v450_seam_edge_entanglement.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 511
    },
    {
      "id": "SEAM.EQUIV.EDGE.CARDY.01",
      "claim": "The Cardy finite-size conformal TOWER (2026-06-30) -- the edge spectrum reproduces the EXACT Ising primary dimensions {1/16,1/2}, naming the edge CFT. v450 fixed the central charge c=1/2; this module reads the OPERATOR CONTENT from the finite-size spectrum (Cardy: E_n - E_0 = (2 pi v / L) x_n). On the critical uniform Majorana ring (the v367/v368 edge universality class). [E] SPIN DIMENSION h_sigma=1/16: the periodic(R)-antiperiodic(NS) ground-energy split gives (E0_R-E0_NS)*L/(2 pi v)=0.06250 (v=2 from the ring dispersion), stable across L=200,400,800 -- the Ising spin field. [E] ENERGY DIMENSION h_epsilon=1/2: the lowest NS pair excitation gives 2*eps_min*L/(2 pi v)=0.50000 -- the Ising energy field. [E] CONFORMAL SCALING: both dimensions are L-INDEPENDENT (the gaps scale exactly as 1/L) -- a conformal (gapless) spectrum, not a gapped lattice. [C] ISING IDENTIFICATION: {c,h_sigma,h_epsilon}={1/2,1/16,1/2} is the chiral Ising (free Majorana) CFT; with N_Maj=16 copies and bulk-edge (v447), c_-=16*(1/2)=8=g_car+N_fam. [C]/[O] the edge CFT is named by both its central charge (v450) AND its operator content (this); SEAM.EQUIV.01 stays [O] -- the spectrum names the CFT but does not supply the cited continuum existence theorem.",
      "status": "Cardy tower {1/16,1/2} [E]/[C]: the edge finite-size spectrum reproduces the exact Ising chiral primary dimensions h_sigma=1/16 (R-NS split) and h_epsilon=1/2 (NS pair gap), L-independent (conformal 1/L scaling); with c=1/2 (v450) the edge is the Ising/Majorana CFT, 16 copies => c_-=8=g_car+N_fam; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.EDGE.ENTANGLEMENT.01; SEAM.EQUIV.EDGE.CHERN.01; v444; v450; v447",
      "scripts": [
        "v451_seam_edge_cardy_tower.py"
      ],
      "script": "v451_seam_edge_cardy_tower.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 512
    },
    {
      "id": "SEAM.EQUIV.E8.MODULAR.01",
      "claim": "The (E8)_1 torus MODULAR DATA (2026-06-30) -- one primary, S-invariant, T-phase e^{-2pi i/3}, the holomorphic c=8 signature, extending v156's character q-expansion to the modular transformation data. A holomorphic (single-primary) RCFT must have its torus character transform as a weight-0 modular form: S-invariant and T-multiplied by e^{-2 pi i c/24}. [E] S-INVARIANCE: chi(-1/tau)=chi(tau) for chi=E4/eta^8 (weight 0), to ~1e-30 at generic tau -- ONE primary, the S-matrix is [1]. [E] T-PHASE: chi(tau+1)/chi(tau)=e^{-2 pi i/3}=e^{-2 pi i c/24} (c=8), to ~1e-30 -- the central charge in the modular T eigenvalue. [E] LEADING POWER => c=8: chi*q^{1/3}->1, so the leading power is q^{-c/24}=q^{-1/3} => c=8=g_car+N_fam. [E] HOLOMORPHIC CONSTRAINT: c=8=0 mod 8 -- the minimal holomorphic chiral central charge, the even unimodular rank-8 lattice E8. [C]/[O] the (E8)_1 modular data (one primary, S-invariant, T-phase e^{-2pi i/3}, c=8) is the torus signature of the seam boundary net; with the edge readings (v444/v447/v450/v451) the (E8)_1 identity is pinned on the torus AND the edge; SEAM.EQUIV.01 stays [O] -- the cited continuum existence theorem (v336) is the residual.",
      "status": "(E8)_1 modular data [E]/[C]: the character chi=E4/eta^8 is S-invariant (one primary), T-multiplies by e^{-2pi i/3} (=e^{-2pi i c/24}, c=8) and has leading power q^{-1/3} (c=8=0 mod 8, holomorphic) -- the full torus modular signature of the seam (E8)_1 net; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Identity",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.EDGE.CHERN.01; v156; v444; v281; v336",
      "scripts": [
        "v452_seam_e8_modular.py"
      ],
      "script": "v452_seam_e8_modular.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 513
    },
    {
      "id": "SEAM.RIGIDITY.MU4FROMMARKS.01",
      "claim": "DERIVING the mu4-symmetry of the seam RP data (QGEO.SYM.01) from the four marks (2026-06-30) -- the LAST structural premise of the rigidity chain folded into the existing realisation premise. After v446 the rigidity residual was 'the seam RP data is mu4-symmetric' (QGEO.SYM.01); this module shows it is NOT an independent premise. [E] THE MARKS ARE mu4: the four Gauss-Bonnet marks are exactly {1,i,-1,-i} = roots of z^4-1 (v168/v216/v323). [E] THE CLOCK PERMUTES THE MARKS: z->iz is a 4-cycle of order 4 (1->i->-1->-i->1), fixing the configuration setwise (the order-4 clock of v445). [E] MOEBIUS STABILISER: z->iz is Moebius and preserves the marks' cross-ratio lambda=2 (in the D4 stabiliser, v168) -- the symmetry is geometric, not imposed. [E] THE FORM BASIS IS mu4-GRADED: omega_k=z^{k-1}dz/(z^4-1) -> i^k omega_k under z->iz (k=1,2,3 -> {i,-1,-i}, the three nontrivial mu4 characters; weights (1,2,3)=A3 exponents=Spec(Q_+)) -- the natural basis is mu4-graded, so any geometric datum is mu4-covariant. [C]/[O] SYNTHESIS: QGEO.SYM.01 ('the RP data is mu4-symmetric') is a CONSEQUENCE of marks=mu4 (this) + QGEO.REALIZE.01 (seam = the marked geometry, the EXISTING premise), NO new premise; the rigidity chain (this -> v446 -> v445) closes modulo QGEO.REALIZE.01, SEAM.EQUIV.01 stays [O].",
      "status": "mu4 from the marks [E]/[C]: the four marks ARE mu4 (roots of z^4-1), the order-4 clock z->iz is a 4-cycle in their D4 Moebius stabiliser (cross-ratio 2 fixed), and the form basis omega_k is mu4-graded (eigenvalue i^k) -- so QGEO.SYM.01 follows from marks=mu4 + the existing QGEO.REALIZE.01, no new premise; the rigidity chain closes modulo realisation, SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Identity",
      "dependencies": "SEAM.EQUIV.01; SEAM.RIGIDITY.CLOCKINV.01; SEAM.RIGIDITY.FORCING.01; QGEO.SYM.01; QGEO.REALIZE.01; v446; v445; v168; v216; v323",
      "scripts": [
        "v453_seam_mu4_from_marks.py"
      ],
      "script": "v453_seam_mu4_from_marks.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 514
    },
    {
      "id": "FORM.SEAM.EDGE.CHERN.01",
      "claim": "LEAN 4 FORMALISATION of the edge central-charge arithmetic (2026-06-30; the kernel-hardening of the edge readings v447/v450/v451/v452) -- new module TfptCarrier/SeamEdgeChern.lean, lake build OK, #print axioms CLEAN (propext only). It kernel-checks the integer backbone all four edge observables and the torus character converge on, over N/Z: [F] chern_topological/chern_trivial/chern_opposite (the FHS Chern integers +1/0/-1 for the M=1/3/-1 phases), [F] bulk_edge (the chiral edge-channel count = |C| = 1), [F] nMaj_eq_16 (N_Maj=2^(g_car-1)=16), [F] two_cMinus_eq (the doubled assembly 2*c_-=N_Maj*|C|, i.e. c_-=16*(1/2)*1=8), [F] cMinus_eq_eight/cMinus_eq_rank (c_-=8=g_car+N_fam=rank E8), [F] holomorphic (c=8=0 mod 8), [F] tphase_order_three (24/gcd(8,24)=3, the order-3 modular T-phase e^{-2pi i/3}), bundled as edge_kernel_facts -- ALL by kernel decide (axioms propext only, NO domain axiom). The continuum existence theorem (MMST, v336) that lifts these readouts to a rigorous net is NOT supplied and stays the single open residual of SEAM.EQUIV.01. Mirrors v447/v450/v451/v452 (and Wolfram-mirrored in tfpt_readouts_extension.wl).",
      "status": "Formal [F] (Lean kernel; the Chern integers, the bulk-edge count, the c_-=8=g_car+N_fam assembly, the holomorphic c mod 8 and the order-3 modular T-phase all kernel-proved axiom-free; the cited MMST continuum theorem stays the named residual)",
      "canonicalStatus": "Formal",
      "dependencies": "SEAM.EQUIV.EDGE.CHERN.01; SEAM.EQUIV.EDGE.ENTANGLEMENT.01; SEAM.EQUIV.EDGE.CARDY.01; SEAM.EQUIV.E8.MODULAR.01; SEAM.EQUIV.01; v447; v450; v451; v452",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/SeamEdgeChern.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/SeamEdgeChern.lean",
      "location": "tfpt_research_contracts",
      "externalData": "Lean 4 + Mathlib (v4.29.1; #print axioms = propext for the kernel facts)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 515
    },
    {
      "id": "SEAM.EQUIV.EDGE.VIRASORO.01",
      "claim": "The lattice level-1 current algebra with Sugawara c=8 (2026-06-30, G2 of the post-F next steps) -- testing the cited MMST 'Koo-Saleur lattice generators -> continuum Virasoro with the correct c' fact on our own free-fermion edge. v450/v451 read c=1/2 and {1/16,1/2}; this tests the ALGEBRA carries the correct central extension. [E] CASIMIR: the critical complex free-fermion ring E_0(L)=e_inf L - pi v c/(6L) (Affleck-Cardy) fits c_Dirac=1.000 => c per Majorana 1/2 (L=64..1024, parameter-free). [E] KAC-MOODY LEVEL: the lattice U(1) current J_n has <0|J_n J_-n|0>=n EXACTLY (n=1..8) -- the level-1 affine central extension closes; a gapped (dimerised) control gives a vanishing non-linear term. [E] SUGAWARA EXACT: c=dim G/(1+h^v)=8 for BOTH SO(16)_1 (120/15) and (E8)_1 (248/31) -- Wolfram-mirrored. [C] ASSEMBLY: 16 Majoranas x 1/2 = 8 = the level-1 Sugawara c; the edge IS a c_-=8 level-1 current-algebra CFT. SEAM.EQUIV.01 stays [O] (the continuum existence theorem v336 is the cited backbone).",
      "status": "Lattice Virasoro/current algebra c=8 [E]/[C]: Casimir c_Dirac=1.000 (1/2 per Majorana), lattice U(1) current Kac-Moody level 1 (<J_n J_-n>=n exact), Sugawara c=dim G/(1+h^v)=8 for BOTH SO(16)_1 120/15 and (E8)_1 248/31; 16*(1/2)=8=g_car+N_fam; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.EDGE.CARDY.01; SEAM.EQUIV.EDGE.ENTANGLEMENT.01; SEAM.MMST.INCLASS.01; v450; v451; v336",
      "scripts": [
        "v454_seam_edge_virasoro.py"
      ],
      "script": "v454_seam_edge_virasoro.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 516
    },
    {
      "id": "SEAM.EQUIV.BW.UNIFORM.01",
      "claim": "A UNIFORM-IN-N Tomita-Takesaki tower for the intrinsic Bisognano-Wichmann condition u_Theta=J (2026-06-30, G1 of the post-F next steps) -- advancing v424 sub-step (i) (the continuum lift), the inductive-limit companion to the single finite witness v426. One-particle: Delta=e^{-K}, conjugation J, reflection Theta; BW <=> u_Theta=J <=> Theta K Theta=-K. [E] UNIFORM BW: the reflection-odd, mu4-even, gapped collar of size 4m (m=2..32, N=8..128) has ||Theta K Theta + K||=0 to machine precision at EVERY N -- u_Theta=J uniform. [E] UNIFORM GAP: the gap of K is N-independent (=6 ln(3/2)=2 N_fam ln(3/2)) so Delta=e^{-K} stays uniformly bounded away from 1 (a genuine KMS, not trace, state). [E] UNIFORM CLOCK SYMMETRY: [rho,K]=0 exactly at every N. [E] N-INDEPENDENT NEG CONTROL: a side-even reflection gives ||Theta K Theta + K||=2||K|| (a fixed per-mode margin) and an off-mu4 K breaks [rho,K] at every N -- the success is non-vacuous. [C]/[O] the inductive (continuum) limit inherits u_Theta=J; SEAM.EQUIV.01 stays [O] (the rigorous continuum theorem NPW26/MMST v336 is the backbone).",
      "status": "Uniform-in-N BW tower [E]/[C]: ||Theta K Theta+K||=0 at every N (m=2..32), N-independent gap 6 ln(3/2), [rho,K]=0 exact, N-independent neg-control margin 2||K|| -- the inductive limit inherits u_Theta=J, advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.RIGIDITY.UNIFORM.01; SEAM.RIGIDITY.FORCING.01; v424; v426; v442; v336",
      "scripts": [
        "v455_seam_bw_uniform.py"
      ],
      "script": "v455_seam_bw_uniform.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 517
    },
    {
      "id": "SEAM.S3.FROM-P1.01",
      "claim": "The edge chirality (c_- != 0, the S3 topological phase) is FORCED by the one-sidedness that defines c3=1/(8pi) (P1) (2026-06-30, G6 of the post-F next steps) -- so S3 is a CONSEQUENCE of axiom P1, not an independent input. [E] c3's '8' IS THE ONE-SIDED COUNT: 8pi=|Z2| intK=2(2pi chi(S^2))=2*4pi (chi=2, topological), so 8=|Z2|*(intK/pi)=2*4 (Wolfram-mirrored, v73). [E] REFLECTION REVERSES C: the explicit 2-band p+ip Chern model has C=+1 and the orientation-reversed (ky->-ky) model C=-1. [E] TWO-SIDED => NON-CHIRAL: any GAPPED boundary carrying an orientation-reversing (reflection) symmetry must have C=-C=0 (c_-=0, gappable/non-chiral). [C] ONE-SIDED => CHIRAL: the one-sided seam is precisely the |Z2| quotient with NO reflection (the same |Z2| that gives c3 its 8pi), so C!=0 is forced -- S3 follows from P1. [C]/[O] magnitude c_-=8=rank E8=g_car+N_fam is the SAME integer 8 as c3's one-sided count; SEAM.EQUIV.01's continuum EXISTENCE (v336) stays [O].",
      "status": "S3 chirality from P1 [E]/[C]: c3's 8=|Z2|*(intK/pi)=2*4 is the one-sided count; a reflection sends Chern C->-C so a two-sided boundary forces C=0, the one-sided seam (no reflection, the |Z2| of c3's 8pi) forces C!=0 -- chirality is a CONSEQUENCE of P1, magnitude c_-=8; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.EDGE.CHERN.01; SEAM.S3.LATTICE.01; SEAM.EQUIV.CONTINUUM.03; v73; v367; v447; v336",
      "scripts": [
        "v456_seam_chirality_from_c3.py"
      ],
      "script": "v456_seam_chirality_from_c3.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 518
    },
    {
      "id": "SEAM.EQUIV.KILLTEST.01",
      "claim": "An explicit (E8)_1-vs-SO(16)_1 character/sector kill-test for the seam edge (2026-06-30, G7 of the post-F next steps) -- a FALSIFIABLE consistency test of the (E8)_1 identification of SEAM.EQUIV.01. [E] (E8)_1 CHARACTER TOWER: chi=E4/eta^8=q^{-1/3}(1+248q+4124q^2+34752q^3+...) the exact conformal-tower degeneracies (E4*prod(1-q^n)^{-8}, Wolfram-mirrored). [E] CURRENT-COUNT DISCRIMINATOR: the weight-1 multiplicity is 248=dim E8=120(SO16)+128(spinor), NOT the 120 of the free SO(16)_1. [E] SECTOR-COUNT DISCRIMINATOR: (E8)_1 is INVERTIBLE |det Cartan E8|=1 (one primary) vs SO(16)_1=D8 |det Cartan D8|=4 (four sectors). [X] KILL CRITERION: a measured weight-1 count of 120, OR a sector count of 4, OR a tower coefficient != {1,248,4124,...} would FALSIFY (E8)_1; current data give 248/1/{1,248,4124}, so PASSED. [C]/[O] the (E8)_1 identification survives the kill-test; SEAM.EQUIV.01's continuum EXISTENCE (v336) stays [O].",
      "status": "(E8)_1 character/sector kill-test [E]/[X]: chi=E4/eta^8={1,248,4124,34752}; the discriminators 248=120+128 (weight-1) and |det Cartan E8|=1 vs |det D8|=4 (sectors) -- a measured 120 or 4 sectors would FALSIFY (E8)_1, the data give 248/1 (PASSED); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Identity",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.E8.MODULAR.01; SEAM.S3.E8CHARACTER.01; SEAM.S3.MODULAR.01; v156; v281; v452; v336",
      "scripts": [
        "v457_seam_character_killtest.py"
      ],
      "script": "v457_seam_character_killtest.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 519
    },
    {
      "id": "SEAM.EQUIV.MMST.AUDIT.01",
      "claim": "The EXACT hypothesis-by-hypothesis citation audit of the MMST scaling-limit theorem against the collar (2026-06-30, G3 of the post-F next steps, the 'exakter Zitat-Abgleich') -- 'does the cited theorem close the c=8 limit, and if not, exactly which sub-statement remains?'. Cited: Osborne-Stottmeister, 'CFT from Lattice Fermions' (CMP 398 (2023) 219; arXiv:2107.13834; the legacy internal token 'MMST'/claim-id denotes this OS theorem, built on the MMST OAR framework arXiv:2010.11121). [C] H1 ALGEBRA CLASS: a gapped quasi-free CAR 16-Majorana system = MMST's lattice-fermion class (v155/v160/v302). [C] H2 MASSLESS LIMIT: the gapless chiral edge IS that massless scaling limit (v439/v444). [E] H3 PER-COPY (c=1/2): each of 16 chiral Majoranas is EXACTLY the proven Thm 4.11 c=1/2 case. [C] H4 DECOUPLED ADDITIVITY: MMST's limit decouples into Majoranas => L_k=sum_i L_k^(i) on the common core => the free c=8 SO(16)_1 Virasoro follows (16*1/2=8). [C] H5 WZW RANGE rank<=c<=D: 8<=8<=16 IN range, the 120 so(16) currents are bilinears -- covered. [O] H6 RESIDUAL = THE 128-SPINOR EXTENSION: the index-4 SO(16)_1->(E8)_1 adds 128 SPINOR currents (248=120+128), NOT bilinears -- OUTSIDE MMST's method (= the det K 4->1 holomorphy discriminator), handed to v459. [C]/[O] the audit CLOSES-by-citation the c=8 EXISTENCE and pins the open residual EXACTLY to the 128-spinor extension; SEAM.EQUIV.01 stays [O], its open part now this one holomorphic-extension statement.",
      "status": "MMST citation audit [C]/[O]: hypothesis-by-hypothesis (arXiv:2107.13834) -- H1 CAR class, H2 massless limit, H3 per-copy c=1/2 (Thm 4.11), H4 decoupled additivity 16*1/2=8, H5 WZW range 8<=8<=16 + 120 bilinear currents all MATCH; the SINGLE residual H6 is the 128-spinor extension SO(16)_1->(E8)_1 (det K 4->1), outside MMST's bilinear method -> v459; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Physical",
      "dependencies": "SEAM.EQUIV.01; SEAM.MMST.INCLASS.01; SEAM.EQUIV.CONTINUUM.03; SEAM.EQUIV.LATTICEVOA.01; v336; v356; v155; v160; v302",
      "scripts": [
        "v458_seam_mmst_citation_audit.py"
      ],
      "script": "v458_seam_mmst_citation_audit.py",
      "location": "tfpt_research_contracts",
      "externalData": "MMST arXiv:2107.13834 (CMP 397 (2023), doi:10.1007/s00220-022-04521-8)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 520
    },
    {
      "id": "SEAM.EQUIV.LATTICEVOA.01",
      "claim": "The SECOND, independent route to (E8)_1 -- the lattice-VOA / conformal-net A_Q(E8) construction supplies EXACTLY the 128-spinor extension MMST (v458) leaves open (2026-06-30, G5 of the post-F next steps). Cited: Adamo-Giorgetti-Tanimoto, 'Rational and non-rational 2d CFTs arising from lattices' (arXiv:2506.01008, 2025); OS for unitary lattice VOAs (Adamo-Moriwaki-Tanimoto, arXiv:2407.18222). [E] E8 EVEN UNIMODULAR: det Cartan(E8)=1 and even => A_{E8} HOLOMORPHIC (one sector, the det K=1 the residual demanded). [E] LATTICE CURRENTS 248=8+240: 8 Cartan + 240 roots = the full (E8)_1 weight-1 space, constructed from the lattice (not bilinears). [E] THE 240 ROOTS SPLIT 112+128: 112 INTEGER (D8) + 128 HALF-INTEGER; 8+112=120=so(16) and the 128 half-integer roots ARE the spinor (248=120+128=8+240) -- the lattice route builds the 128-spinor explicitly. [C] AGT THEOREM: A_Q of an even lattice IS a 2d conformal net; the holomorphic SO(16)_1->(E8)_1 extension is CONSTRUCTED, not assumed. [C]/[O] TWO complementary routes now cover (E8)_1 (MMST fermions v458 + lattice net A_{E8}); the residual reduces to the single realization input 'the collar's scaling limit = A_{E8}' (tied to P1 by v456). SEAM.EQUIV.01 stays [O].",
      "status": "Lattice-VOA route to (E8)_1 [E]/[C]: E8 even unimodular (det 1, holomorphic); 248=8+240 currents, the 240 roots split 112 integer + 128 spinor (248=120+128) -- A_{E8} (Adamo-Giorgetti-Tanimoto) constructs the 128-spinor extension MMST (v458) misses; two complementary routes cover (E8)_1, residual = the single realization input (v456); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Identity",
      "dependencies": "SEAM.EQUIV.01; SEAM.S3.MODULAR.01; SEAM.S3.FROM-P1.01; v458; v456; v281; v336",
      "scripts": [
        "v459_seam_lattice_voa_route.py"
      ],
      "script": "v459_seam_lattice_voa_route.py",
      "location": "tfpt_research_contracts",
      "externalData": "Adamo-Giorgetti-Tanimoto arXiv:2506.01008; Adamo-Moriwaki-Tanimoto arXiv:2407.18222",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 521
    },
    {
      "id": "SEAM.S3.LTO.01",
      "claim": "The EXPLICIT flat-band commuting-projector (local-topological-order) realisation of the S3 input (2026-06-30) -- advancing v424 sub-step (i) ('realise the raw seam as a Z/2-reflection commuting-projector LTO') from abstract existence to an explicit gap-protected quasi-local projector net carrying the Z/2 reflection and the mu4 clock. NPW26 (arXiv:2605.10693) prove the intrinsic Bisognano-Wichmann lemma for COMMUTING-PROJECTOR (LTO) models with a Z/2 reflection; this attacks the commuting-projector side of sub-step (i) on the same v367 collar. [E] FLAT-BAND PARENT: the flattened Bloch Q=h/|h| has Q^2=I (flat +-1 bands) and P=(I-Q)/2 is an exact projector EQUAL to v367's occupied ground-state projector -- a frustration-free commuting-projector parent with the same gapped ground state (machine precision over a k-grid). [E] GAP-PROTECTED QUASI-LOCALITY: the gapped projector kernel |P(n,0)| decays rapidly (xi~1.14 at M=1, exp R^2=0.995, |P(20,0)|<1e-6) and xi GROWS as the gap closes (2.7 at gap 0.02); at the GAPLESS point the kernel decays as a clean power law |R|^{-2.1} (R^2=0.999, tail ~2e-4) -- the projector net is quasi-local BECAUSE the bulk is gapped. [E] Z/2 REFLECTION + mu4 INHERITED: the flattened modular Hamiltonian K_flat=sign(K) inherits Theta K_flat Theta=-K_flat (BW, u_Theta=J) and [rho,K_flat]=0 (mu4) EXACTLY (sign is odd; rho commutes with any f(K)); side-even and off-mu4 neg controls fire. [C]/[O] CHIRAL OBSTRUCTION: the chiral c_-=8!=0 (FHS Chern |C|=1, trivial control 0) admits NO strictly finite-range commuting-projector Hamiltonian (Kapustin-Fidkowski arXiv:1810.07756; C!=0 forbids exp-localised Wannier, Thouless), so the realisation is the QUASI-LOCAL LTO net NPW26 use. Advances v424 sub-step (i); does NOT supply the continuum theorem; SEAM.EQUIV.01 stays [O].",
      "status": "S3 commuting-projector realisation [E]/[C]: v367's gapped collar has an exact flat-band parent (Q^2=I, P^2=P=P_gs) that is gap-protected quasi-local (xi~1.14 at M=1, power-law |R|^{-2.1} at the gapless point) and whose flattened modular Hamiltonian inherits Theta K_flat Theta=-K_flat and [rho,K_flat]=0 exactly; the chiral c_-=8 obstructs STRICT commuting-projector locality (Kapustin-Fidkowski), so the realisation is the quasi-local LTO net of NPW26 -- advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.S3.LATTICE.01; SEAM.EQUIV.BW.UNIFORM.01; v424; v426; v367; v455; v336",
      "scripts": [
        "v460_seam_s3_lto.py"
      ],
      "script": "v460_seam_s3_lto.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kapustin-Fidkowski arXiv:1810.07756; NPW26 arXiv:2605.10693",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 522
    },
    {
      "id": "FORM.SEAM.RESIDUAL.01",
      "claim": "LEAN 4 FORMALISATION reducing the SEAM.EQUIV.01 residual to a SINGLE named realization axiom (2026-06-30, G4 of the post-F next steps) -- new module TfptCarrier/SeamResidualAxiom.lean, lake build OK. It kernel-checks the arithmetic backbone of the whole G-block over Nat/Int: [F] c3_eight_one_sided (the one-sided count |Z2|*(intK/pi)=2*4=8, v456), [F] two_sided_nonchiral (a reflection sends C->-C so a two-sided boundary forces C=0), [F] cMinus_eq_eight (c_-=8=g_car+N_fam=rank E8), [F] mmst_range (rank<=c<=D, 8<=8<=16), [F] decoupled_additivity (16*1/2=8), [F] currents_fermionic (120 so(16) bilinears) + currents_lattice (248=8+240), [F] roots_split (240=112+128, 248=120+128), [F] holomorphic_discriminator (det K_E8=1 vs det K_SO16=4) -- ALL by kernel decide. UPDATE (2026-06-30, v461/v462 round): the two cited existence theorems are MERGED into ONE combined external axiom seam_realisation_theorem (dropping the intermediate ChiralCFT_c8), so #print axioms seamResidualClosed now lists exactly FOUR axioms (CollarRealizedAsLatticePhase, SeamIsE8, collar_realizes, seam_realisation_theorem), down from six; and two more kernel facts are added -- [F] strict_locality_forbidden (v461: the Wannier winding=|C|=1!=0 and c_-=8!=0, so a strict finite-range commuting projector is topologically forbidden) and [F] fock_sectors (v462: C(16,2)=120 currents, 2^8=256 Ramond states, spinor 128, 120+128=248). residual_arithmetic (now incl. v461/v462) depends only on [propext]. So the residual is reduced to ONE TFPT-internal realization axiom + ONE combined cited theorem. UPDATE (2026-06-30, v463/v464 round): two more kernel facts -- [F] c8_three_candidates (A8/D8/E8 each dim=8*(1+h^v), so c=8 is not unique at level 1) and [F] holomorphy_selects_e8 (the forced dim V_1=248=dim E8 != 80(A8) != 120(D8), so holomorphy promotes 'c=8' to 'E8' uniquely, v463); v464 (the one-particle realization rigidity) is NUMERICAL (symbol idempotency + Cauchy convergence + entanglement c-fit), so it sharpens the meaning of collar_realizes ('the unique quasi-free realization, modulo Araki/Shale-Stinespring') but is not a kernel fact. seamResidualClosed stays exactly FOUR axioms; residual_arithmetic (now incl. v463) depends only on [propext]. UPDATE (2026-07-02, v469 round): a PARALLEL peer-reviewed derivation seamResidualClosed' is added -- the invariant-level axiom collar_invariants (R1': quasi-free [C] + gap [E] + class D + c_-=8 [E from P1], replacing the model-level fiat) plus ONE combined 1995-2001 package crossedproduct_route_theorem (Kitaev/Kapustin-Spodyneiko phase classification o MMST+Boeckenhauer scaling limit o Longo-Rehren/BE local Z2 crossed product + KLM o Dong-Mason uniqueness); its joints are NEW kernel facts with NO axioms -- [F] lr_locality_integer (h_s = 16/16 in Z), [F] glue_locality_integers (mu4 h(J^k) = {1,1,1} in Z), [F] klm_holomorphic (4/2^2 = 1 and 16/4^2 = 1), [F] sixteenfold_pin (nu = 16 = 0 mod 16, c_- = nu/2 = 8), bundled as crossedproduct_arithmetic [propext only]. #print axioms seamResidualClosed' = exactly FOUR (CollarHasInvariants, SeamIsE8, collar_invariants, crossedproduct_route_theorem); the original seamResidualClosed (AGT/AMT witness) is kept unchanged as the independent second route. lake build OK (3285 jobs).",
      "status": "Formal [F] (Lean kernel): the G-block arithmetic (one-sided 8=|Z2|*intK/pi, reflection C->-C, c_-=8, MMST range 8<=8<=16, decoupled additivity, 248=8+240=120+128, 240=112+128, det K 1 vs 4, + v461 winding=|C|=1!=0 obstruction + v462 Fock counts 120/128/256 + v463 holomorphy selector forced dimV1=248=E8) all kernel-proved by decide; the two cited theorems MERGED into ONE combined axiom seam_realisation_theorem so seamResidualClosed depends on FOUR axioms (down from six); v464 one-particle rigidity is numerical; residual reduced to ONE realization axiom + ONE combined cited theorem; PLUS the v469 parallel peer-reviewed route seamResidualClosed' (invariant-level collar_invariants + crossedproduct_route_theorem; new kernel facts lr_locality_integer, glue_locality_integers, klm_holomorphic, sixteenfold_pin, all [propext only]); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Formal",
      "dependencies": "SEAM.EQUIV.01; SEAM.S3.FROM-P1.01; SEAM.S3.LOCALITY.01; SEAM.EQUIV.KILLTEST.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.SPINOR.01; SEAM.EQUIV.UNIQ.01; SEAM.EQUIV.ONEPARTICLE.01; SEAM.EQUIV.CROSSEDPRODUCT.01; v456; v457; v458; v459; v461; v462; v463; v464; v469",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/SeamResidualAxiom.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/SeamResidualAxiom.lean",
      "location": "tfpt_research_contracts",
      "externalData": "Lean 4 + Mathlib (v4.29.1)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 523
    },
    {
      "id": "SEAM.S3.LOCALITY.01",
      "claim": "The Kapustin-Fidkowski/Thouless strict-locality OBSTRUCTION made EXPLICIT (2026-06-30) -- sharpening v460's verdict from a cited statement to an exhibited topological obstruction on the same v367 p+ip collar. v460 realised the S3 input as a flat-band commuting-projector net and noted BY CITATION that the chiral c_-!=0 forbids strict finite-range locality; this module EXHIBITS that obstruction. [E] CHERN INTEGER: the FHS Chern is C=+1 (chiral M=1), 0 (trivial M=3), -1 (opposite mass) -- the invertible-phase invariant. [E] WILSON-LOOP / WANNIER-CENTRE WINDING = CHERN: the hybrid Wannier centre theta(ky) winds by |C|=1 (chiral) and 0 (trivial) over the BZ; exp-localised Wannier exist IFF the winding is 0 (Brouder-Panati et al. PRL 98 046402; Thouless), so the chiral collar has NONE. [E] OBSTRUCTION INTEGERS TIED: winding=|C|=1!=0 and c_-=16/2=8=g_car+N_fam=rank E8!=0; winding!=0 <=> C!=0 <=> c_-!=0 (the same 8 as c3's one-sided count, v456). [E] NEG CONTROL: BOTH phases are gapped, yet only the trivial (winding 0) admits exp-localised Wannier and a strict finite-range commuting projector -- it is the CHIRALITY, not the gap. [C]/[O] Kapustin-Fidkowski: a strictly finite-range commuting projector would force winding 0, C=0; since C=1!=0 NONE exists, so v424 sub-step (i)'s realisation is PROVABLY the quasi-local NPW26 LTO net (strict locality topologically forbidden, not a missing premise). SEAM.EQUIV.01 stays [O] (continuum theorem MMST v336 the backbone).",
      "status": "Strict-locality obstruction exhibited [E]/[C]: the chiral collar's Wilson-loop/Wannier-centre winding = Chern = 1!=0 (vs 0 for the gapped trivial control), the exact obstruction to exp-localised Wannier (Brouder et al./Thouless); so no strictly finite-range commuting projector exists for c_-=8!=0 (Kapustin-Fidkowski) -- v424 sub-step (i)'s realisation is PROVABLY the quasi-local NPW26 LTO net, sharpening v460 from citation to exhibited obstruction; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.S3.LTO.01; SEAM.S3.FROM-P1.01; SEAM.EQUIV.EDGE.CHERN.01; v460; v447; v456; v367",
      "scripts": [
        "v461_seam_strict_locality.py"
      ],
      "script": "v461_seam_strict_locality.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kapustin-Fidkowski arXiv:1810.07756; Thouless J. Phys. C 17 (1984) L325; Brouder et al. PRL 98 046402",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 524
    },
    {
      "id": "SEAM.EQUIV.SPINOR.01",
      "claim": "The 128-spinor extension SO(16)_1 -> (E8)_1 exhibited at CHARACTER level and as a finite-L -> continuum convergence (2026-06-30) -- the constructive companion of the v458 (MMST audit) / v459 (lattice-VOA) residual. v458 isolated the residual to the 128 spinor currents; v459 named the lattice-VOA route; this module EXHIBITS the extension three ways. [E] (E8)_1 TOWER FROM SO(16)_1 SECTORS: the exact Jacobi/E8 identity theta2^8+theta3^8+theta4^8=2E4 (~1e-30) gives chi_{(E8)_1}=E4/eta^8 = (1/2)[(t3/eta)^8+(t4/eta)^8] + (1/2)(t2/eta)^8 = chi_o^{SO16}+chi_s^{SO16} -- the holomorphic (E8)_1 character IS the SO(16)_1 vacuum+spinor sector sum (the conformal embedding SO(16)_1 in (E8)_1). [E] 128 SPINOR WEIGHT-1 STATES FILL 120 -> 248: chi_o gives weight-1 multiplicity 120 (=dim so(16)), chi_s gives 128 at weight 1, summing to the exact (E8)_1 tower {1,248,4124,34752} -- the 128 spinor currents ARE the (E8)_1 - SO(16)_1 deficit (248=120+128). [E] FINITE 16-MAJORANA FOCK: C(16,2)=120 NS bilinear (so(16)) currents and 2^{16/2}=256 Ramond ground states = 128 spinor + 128 cospinor (split by fermion parity) -- the spinor present already at finite N. [E] FINITE-L -> CONTINUUM CONVERGENCE: the critical free-fermion ring Casimir E_0(L)=e_inf L - pi v c/(6L) fits c_Dirac -> 1 (c_-=16*(1/2)=8) and the lowest scaled NS gap x_1 -> the weight-1 dimension 1 with |x_1-1| ~ 1/L^2 (L=64..1024) -- the lattice flows to the conformal tower. [C]/[O] the rigorous holomorphic extension (the simple-current chi_o+chi_s projection) is supplied by AGT/AMT (v459); SEAM.EQUIV.01's continuum existence (v336) stays [O] -- this constructs and converges, it does not prove the uniform scaling-limit theorem.",
      "status": "128-spinor extension exhibited [E]/[C]: the Jacobi/E8 identity theta2^8+theta3^8+theta4^8=2E4 makes chi_{(E8)_1}=E4/eta^8=chi_o+chi_s (vacuum 120 + spinor 128 = 248); the finite 16-Majorana Fock space carries C(16,2)=120 currents and 2^8=256=128+128 Ramond states; the critical ring converges c_- -> 8 and x_1 -> 1 as L grows; AGT/AMT (v459) supply the rigorous extension; SEAM.EQUIV.01 existence stays [O]",
      "canonicalStatus": "Identity",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.KILLTEST.01; v458; v459; v457; v336",
      "scripts": [
        "v462_seam_spinor_continuum.py"
      ],
      "script": "v462_seam_spinor_continuum.py",
      "location": "tfpt_research_contracts",
      "externalData": "Adamo-Giorgetti-Tanimoto arXiv:2506.01008; Adamo-Moriwaki-Tanimoto arXiv:2407.18222",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 525
    },
    {
      "id": "SEAM.EQUIV.UNIQ.01",
      "claim": "The c=8 HOLOMORPHIC-UNIQUENESS pin (2026-06-30, B of the A+B 'actually close it' round) -- the positive lever making the IDENTIFICATION half of SEAM.EQUIV.01 classification-forced rather than assumed. v458/v459 close the continuum EXISTENCE; this pins WHY the limit must be (E8)_1 and not a same-c rival. [E] c=8 NOT UNIQUE AT LEVEL 1: c=dim g/(1+h^v)=8 has THREE simple solutions A8 (dim 80), D8=so(16) (dim 120), E8 (dim 248) -- c=8 is necessary but NOT sufficient (the v457 det K=4 SO(16) rival is a genuine gleich-c competitor). [E] HOLOMORPHY FORCES dim V_1=248: a single-sector c=8 chiral CFT has an SL(2,Z)-covariant character; the only weight-0 form with leading q^{-1/3} and integer coeffs is j^{1/3}=E4/eta^8, q^1 coeff 248 (1-dim candidate space, vacuum a_0=1) -- forced. [E] dim 248 + level-1 c=8 => E8 uniquely (only E8 has dim 248; 248/(1+30)=8); A8/D8 excluded. [E] tower {1,248,4124,34752,213126}=E4/eta^8; E4/eta^8=j^{1/3} (numeric <1e-25). [C] Dong-Mason/Schellekens: the holomorphic VOA at c=8 is UNIQUE = V_{E8}; the even unimodular rank-8 lattice is unique = E8 (Mordell-Witt). So 'c=8 holomorphic limit = (E8)_1' is classification-forced; with v458/v459 the continuum/identification HALF of SEAM.EQUIV.01 is [C], leaving the realisation R1 (v464). SEAM.EQUIV.01 stays [O].",
      "status": "c=8 holomorphic-uniqueness pin [E]/[C]: c=8 has three level-1 candidates (A8/80, D8/120, E8/248) so c=8 is not sufficient; holomorphy forces the character E4/eta^8 => dim V_1=248 => E8 uniquely; Dong-Mason/Schellekens give holomorphic c=8 VOA unique = V_{E8}, so the identification half of SEAM.EQUIV.01 is [C] (classification-forced); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Identity",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.KILLTEST.01; v457; v458; v459",
      "scripts": [
        "v463_seam_c8_holomorphic_uniqueness.py"
      ],
      "script": "v463_seam_c8_holomorphic_uniqueness.py",
      "location": "tfpt_research_contracts",
      "externalData": "Dong-Mason (holomorphic VOA c=8 unique); Schellekens classification; Mordell-Witt (even unimodular rank-8 lattice unique = E8)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 526
    },
    {
      "id": "SEAM.EQUIV.ONEPARTICLE.01",
      "claim": "The one-particle RIGIDITY + scaling limit attacking R1 (2026-06-30, A of the A+B round) -- the REALISATION input of SEAM.EQUIV.01. The seam is quasi-free (v113/v160/v161), so by Araki self-dual CAR the whole net is the second quantisation of its one-particle symbol P, and R1 collapses to a finite-per-mode statement about P. [C] ARAKI SELF-DUAL CAR: a gauge-invariant quasi-free state <-> its symbol P (0<=P<=1) bijectively (Araki 1971; Shale-Stinespring 1965), so a realisation is fully determined by P. [E] RIGIDITY: the gapped ground state is the UNIQUE quasi-free state with P=projection onto K<0 (P^2=P, hermitian to <1e-12), K fixed by gap+chirality+reflection (v426) -- no second realisation. [E] ONE-PARTICLE SCALING LIMIT EXISTS: the real-space kernel is Cauchy on a fixed window (max|P_2N-P_N|~1/N -> 0), so P_inf exists (Gaussianity is why a computation reaches it). [E] LIMIT IS c=1 DIRAC (c_-=8): entanglement S(L)=(c/3)ln L gives c -> 1 (per Majorana 1/2; Peschel/Calabrese-Cardy, ties v450), 16 copies -> c_-=8. [C] Shale-Stinespring: gauge-invariant => no pairing block => the Bogoliubov map is implementable, GNS reps quasi-equivalent. R1 upgraded from a bald axiom to 'the unique quasi-free realisation, one-particle limit exhibited, modulo the named Araki/Shale-Stinespring functor'; with v463 the SEAM.EQUIV.01 residual is now entirely certification, zero open internal mechanism -- closed modulo cited theorems. SEAM.EQUIV.01 stays [O].",
      "status": "one-particle rigidity + scaling limit [E]/[C]: via Araki self-dual CAR (state<->symbol P, cited) the gapped ground state is the UNIQUE quasi-free realisation (P^2=P), its kernel is Cauchy (~1/N, limit exists), and the limit is the c=1 Dirac CFT (EE c-fit ~1 => c_-=16*1/2=8); Shale-Stinespring makes the Bogoliubov map implementable, upgrading R1 from a bald axiom to an audited [C]; with v463 the residual is entirely certification, zero open internal mechanism; SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "Numerical",
      "dependencies": "SEAM.EQUIV.01; SEAM.S3.LOCALITY.01; v113; v160; v161; v426; v450; v463",
      "scripts": [
        "v464_seam_oneparticle_rigidity.py"
      ],
      "script": "v464_seam_oneparticle_rigidity.py",
      "location": "tfpt_research_contracts",
      "externalData": "Araki self-dual CAR (1971); Shale-Stinespring (1965); Peschel (entanglement from correlations); Calabrese-Cardy",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 527
    },
    {
      "id": "FLAV.THIRDGEN.PATTERN.01",
      "claim": "THIRD-GENERATION LADDER-BASE PATTERN (2026-07-02, v467): the suite's THREE ~2 sigma mixing tensions (sin^2 theta13 +2.0 sigma, sin^2 theta23 +1.8 sigma, V_cb +1.8 sigma; v307) are ONE common ~ -phi0 suppression -- quantified and pre-registered as a CANDIDATE, the frozen record (v84/REG.FREEZE.01) is NOT changed. In existing vocabulary the pattern is parameter-free: the seed enters third-generation mixings through the LADDER BASE lambda_C^2 = phi0(1-phi0) (the Yukawa mass-ladder base, tfpt_2) instead of bare phi0, and the 2-3 texture carries the same -phi0/2 shift as theta12: sin^2 theta13 = lambda_C^2 e^{-5/6} = 0.021880 (+2.00 -> -0.12 sigma), sin^2 theta23 = (1-phi0)/2 <=> cos 2theta23 = phi0 = 0.473414 (+1.76 -> +0.20), V_cb = lambda_C^2/(1+lambda_C) = 0.041119 (+1.79 -> +0.01). Scope checks that make it a rule not a fit: V_ub = lambda_C^3/3 has no bare phi0, stays untouched and consistent (-0.23); the 1-2 channels must stay undressed and do (theta12 -0.02, lambda_C -0.25; dressing lambda_C would give -24 sigma). Weighted common deviation of the three affected channels -5.36% +- 1.67% (3.2 sigma non-zero, -0.03 sigma from -phi0); joint chi^2 over the four third-gen observables 10.4 -> 0.11 with ZERO new parameters; Daya Bay alone already prefers the pattern (Delta chi^2 = 4.3). HONEST: post-hoc (5/16 equally-simple dressings fix the two PMNS tensions; (1-phi0)/(1-c3)/resolvent degenerate <1.4%); suppression size phi0 = 5.32% sits INSIDE the seam-gapped correction band (2/3)^6 = 8.78% (v393) but NO seam-transport derivation exists (Bernoulli-variance reading lambda_Y^2 = Var[Bernoulli(phi0)] is an exact identity but only an interpretation). Pre-registered kills: theta13 stable at 0.0231 kills the pattern; at ~0.0219 (sigma_rel<1%) kills the record's bare-phi0 reading; exclusive |Vcb| ~0.0395 kills both; JUNO-era reactors + Belle II decide.",
      "status": "Numerical [N]/[C]/[O] (candidate pattern; record of v84 untouched)",
      "canonicalStatus": "Open [O] candidate pattern (post-hoc, quantified, pre-registered): three ~2 sigma third-gen mixing tensions = one common -phi0 suppression (3.2 sigma non-zero; ladder-base forms drop joint chi^2 10.4 -> 0.11, zero parameters); frozen record unchanged; needs a seam-transport derivation to become more than a candidate",
      "dependencies": "FLAV.TH13.PRESSURE.01; REG.FREEZE.01; CORRECTIONS.NUMERIC.01; FLAV.TH13.01; SEED.CROSSVAL.01",
      "scripts": [
        "v467_thirdgen_mixing_pattern.py"
      ],
      "script": "v467_thirdgen_mixing_pattern.py",
      "location": "tfpt_2",
      "externalData": "NuFIT 6.0 (theta13, theta23, theta12); PDG 2024 (|Vcb|, |Vub|, lambda_C); Daya Bay final",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 528
    },
    {
      "id": "FLAV.DM2RATIO.01",
      "claim": "SPLITTING RATIO = LEPTONIC JARLSKOG (2026-07-02, v468): the neutrino mass-squared splitting ratio equals the Jarlskog invariant of the FROZEN PMNS matrix -- dm21^2/dm31^2 = |J_PMNS| = 0.029653 vs NuFIT 6.0 NO measured 0.029805 +- 0.000792 -> -0.19 sigma; equivalently m2/m3 = sqrt|J| = 0.17220 vs 0.17264(229). A parameter-free CANDIDATE relation for a ratio the compiler does not otherwise predict: |J| is the canonical derived CP invariant of the already-frozen matrix (angles v9/v268 + delta = 4pi/3, assembled v270; zero dials). It SUPERSEDES the informal tfpt_2 table heuristic m2/m3 ~ pi phi0 = 0.16704 (a '~' entry, never frozen), which is now at -2.44 sigma vs NuFIT 6.0. Spectrum consistency: with r = |J| the NO floor Sigma m_nu = m3(1+sqrt|J|) = 0.0588 eV reproduces the documented 0.0586 eV (v272); the absolute scale stays the ONE seesaw ratio (no absolute number added). HONEST: post-hoc; as a free formula the window density is ~1.2e-2 (unremarkable alone) -- the content is structural (canonical invariant, zero dials); NO mechanism (the mu-tau texture gives theta13 = 0 at leading order and cannot carry this relation) -- [O] until an M_nu-level derivation exists. Pre-registered kill: JUNO shrinks the r-window ~4x; a stable offset from 0.029653 at >3 sigma kills the relation; the v467 pattern would shift |J| to 0.028849 (-1.21 sigma today), so the two candidates cross-discriminate at JUNO precision.",
      "status": "Numerical [N]/[C]/[O] (candidate relation; informal pi phi0 table heuristic superseded as comparator)",
      "canonicalStatus": "Open [O] candidate relation (post-hoc, parameter-free): dm21^2/dm31^2 = |J_PMNS(frozen)| at -0.19 sigma (the informal pi phi0 heuristic, -2.44 sigma, is retyped to this comparator); NO floor Sigma m_nu consistent; JUNO decides; needs an M_nu-level derivation",
      "dependencies": "FLAV.PMNS.03; FLAV.TH13.01; FLAV.NUSCALE.01; FLAV.THIRDGEN.PATTERN.01",
      "scripts": [
        "v468_dm2_ratio_jarlskog.py"
      ],
      "script": "v468_dm2_ratio_jarlskog.py",
      "location": "tfpt_2",
      "externalData": "NuFIT 6.0 (dm21^2 = 7.49(19)e-5, dm31^2 = +2.513(20)e-3, NO with SK-atm)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 529
    },
    {
      "id": "SEAM.EQUIV.CROSSEDPRODUCT.01",
      "claim": "The NET-LEVEL CROSSED-PRODUCT certification of the 128-spinor extension + the R1 -> R1' invariant-level reduction (2026-07-02, closure-route round) -- re-founds BOTH halves of the SEAM.EQUIV.01 residual on harder-to-reject ground; does NOT close it. [E] LR LOCALITY INTEGER (index-2): the SO(16)_1 spinor has h_s = N/16 = 16/16 = 1 in Z => statistics phase e^{2 pi i h_s} = +1 -- EXACTLY the Longo-Rehren locality criterion, so the Z2 simple-current crossed product SO(16)_1 x| {1,s} is a LOCAL index-2 extension (Longo-Rehren RMP 7 (1995); the level-1 so(N) net WITH its DHR sectors from even-CAR: Boeckenhauer RMP 8 (1996); simple-current extensions of nets: Boeckenhauer-Evans CMP 197 (1998)); KLM gives mu(B) = mu(SO16)/[B:A]^2 = 4/4 = 1 => B HOLOMORPHIC => B = (E8)_1 by the v463 classification pin (E4/eta^8 q^1 = 248 re-verified). THE 2024/2025-PREPRINT LEG (AGT arXiv:2506.01008 + AMT arXiv:2407.18222, v459) IS DEMOTED TO AN INDEPENDENT SECOND WITNESS -- the extension certification now rests on 1995-2001 peer-reviewed journals. [E] INDEX-4 PARALLEL: the mu4 glue J = (s, Lambda1) of (D5)_1 x (A3)_1 has h(J^k) = {1,1,1} for k=1,2,3 -- ALL integer (5/8+3/8=1; 1/2+1/2=1), the v125 isotropy q(k(1,1))=k^2 restated at net level; mu = 16/4^2 = 1, the SAME holomorphic endpoint. [E] R1' INVARIANTS COMPUTED: per-copy FHS Chern C(M=1)=1 vs C(M=3)=0 (control), nu = 16 x |C| = 16, c_- = nu/2 = 8 = g_car + N_fam = rank E8; nu mod 16 = 0 is EXACTLY the class-D phase whose edge admits a purely bosonic description = the (E8)_1 level-1 state (Kitaev Ann. Phys. 321 (2006), sixteen-fold way) -- the identification endpoint pinned from published condensed-matter theory. [E] THE CHAIN COMPOSES (v297 typing): T1 invariants -> nu=16 phase (Kitaev 2006/2009; c_- a PHASE invariant: Kapustin-Spodyneiko PRB 101 (2020), Kim et al. PRL 129 (2022)) -> T2 SO(16)_1 net (MMST CMP 397 (2023) + Boeckenhauer 1996) -> T3 local holomorphic extension (LR/BE + KLM CMP 219 (2001)) -> T4 (E8)_1 (Dong-Mason + v463). [C] R1 -> R1': the Lean axiom collar_realizes (model-level fiat 'the seam IS this lattice model') is replaceable by collar_invariants (invariant-level: quasi-free [C v155/v160] + gap [E v302] + class D + c_- = 8 [E v456, from P1]) -- any realisation with these invariants is phase-equivalent to the v367 stack, hence has the SAME edge scaling limit; quasi-freeness stays the ONE [C] hypothesis. [O] NOT CLOSED: the cited theorems stay cited (certification, not re-proof); the abstract continuum existence remains; SEAM.EQUIV.01 stays [O].",
      "status": "Net-level crossed-product certification + R1 invariant reduction [E]/[C]/[O]: h_s = 16/16 = 1 in Z => the LR locality criterion holds, the Z2 crossed product SO(16)_1 -> (E8)_1 is a LOCAL index-2 extension by 1995-2001 PEER-REVIEWED subfactor theory (LR/Boeckenhauer/BE/KLM), AGT/AMT demoted to second witness; mu4 glue h(J^k)={1,1,1} parallel; R1 reduced to computed invariants (FHS |C|=1, nu=16, c_-=8, the Kitaev-E8 16-fold-way class); SEAM.EQUIV.01 stays [O]",
      "canonicalStatus": "-",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.UNIQ.01; SEAM.EQUIV.ONEPARTICLE.01; v125; v154; v297; v336; v367; v456; v458; v459; v463; v464",
      "scripts": [
        "v469_seam_crossedproduct_route.py"
      ],
      "script": "v469_seam_crossedproduct_route.py",
      "location": "tfpt_research_contracts",
      "externalData": "Longo-Rehren RMP 7 (1995); Boeckenhauer RMP 8 (1996); Boeckenhauer-Evans CMP 197 (1998); KLM CMP 219 (2001); Kitaev Ann. Phys. 321 (2006); Kapustin-Spodyneiko PRB 101 (2020); Kim et al. PRL 129 (2022)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 530
    },
    {
      "id": "ALPHA.QUILLEN.INFLOW.01",
      "claim": "THE ALPHA^3 LEVEL = THE COMPUTED BULK CHERN INVARIANT + THE SEAM F-NORMALISATION = THE AFFINE EMBEDDING INDEX (2026-07-02, closure-route round) -- upgrades BOTH residuals left after v433/v434/v435; does NOT close ALPHA.QUILLEN.EXACT.01. [E] TARGET RE-VERIFIED: alpha^-1 = 137.0359992168 unique root; Quillen split a^3 - 2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at the root < 1e-35 (v341/v382). [E] THE INVARIANT COMPUTED: FHS Chern C(M=1) = 1, C(M=3) = 0 -- the same integers as v367/v461, here read as the quantised U(1) response of the collar phase. [C] k0 = |C| = 1 UNDER THE EM1 READING: the a^3 coefficient EQUALS the computed bulk invariant of the SAME p+ip model that realises S3 -- 'why exactly 1' is answered by COMPUTATION plus the cited inflow identification (TKNN PRL 49 (1982) / Avron-Seiler-Simon PRL 51 (1983): integer quantisation of the U(1) response; Callan-Harvey NPB 250 (1985): boundary level = bulk response; APS 1975 / Alvarez-Gaume-Della Pietra-Moore Ann. Phys. 163 (1985) / Witten RMP 88 (2016): the metric-independent part of delta log det is the eta/CS level; Quillen 1985 / Bismut-Freed CMP 106 (1986): determinant-line curvature integrality) -- REPLACING v435's minimality assumption ('1 is the unit level') by a measurement on the model. [E] k_Y = 5/3 EXACT: tr_5bar(Y^2)/tr(T3^2) = (5/6)/(1/2) = 5/3 (Ginsparg PLB 197 (1987), the string-GUT hypercharge level); (3/5) x (41/6) = 41/10 = b1 -- the 'GUT 3/5' in the closure IS the inverse embedding index, not an imported convention; carrier decomposition 41/6 = 20/3 (3 generations x 16-content) + 1/6 (Higgs doublet). [C] F-NORMALISATION RETYPED: once the seam net is (E8)_1 at level 1 (SEAM.EQUIV.01), the U(1)_Y current sits inside a level-1 current algebra and <J(z)J(w)> = k/(z-w)^2 is FIXED by the embedding index -- level-1 current-algebra rigidity, no continuous dial; the v434 [C] ('how does the gauge curvature couple to the boundary measure') has ZERO independent content: it is a face of SEAM.EQUIV.01 (consistent with the v382 typing), its value the embedding index. [E] THE STRUCTURAL UNIFICATION: the two named targets share ONE object -- the collar phase carries TWO quantised responses, gravitational c_- = 16 x 1/2 x |C| = 8 = g_car + N_fam (feeds SEAM.EQUIV.01/S3, v456) and U(1) C = 1 (feeds the a^3 level, EM1); 'the one theorem and the one number' are two faces of the SAME invertible phase, and closing R1' (v469) feeds both residuals simultaneously. [O] NOT CLOSED: the bridge lemma 'delta log det_zeta(seam) = the inflow response' stays the named cited step; the EM1 reading stays [C]; ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 = 137.0359992 stays [E] regardless.  UPDATED 2026-08-27 (v974): the inflow step 'boundary level = bulk response' is upgraded from cited to COMPUTED on the same model (probe laughlin_pump_em1_probe 20/20, PUMP_QUANTIZED_MATCHES_CHERN): Laughlin pump +1 per flux quantum (teleport bookkeeping with exact period closure), edge spectral flow -1/+1 antisymmetric as inflow demands, Streda dN/dPhi = C/(2 pi) exact integers over n_phi = 0..3 -- all equal to the FHS Chern, with the M = 3 controls all zero. The identification of the complex-copy U(1) with the physical Majorana collar stays part of the cited EM1 reading [C]; ALPHA.QUILLEN.EXACT.01 stays [O].",
      "status": "Inflow level + embedding rigidity [E]/[C]/[O]: the a^3 coefficient = the COMPUTED bulk Chern |C| = 1 of the v367 collar (TKNN/ASS quantisation + Callan-Harvey inflow + APS/Witten eta=CS), replacing v435's minimality; the seam F-normalisation = the affine embedding index k_Y = 5/3 (Ginsparg 1987, (3/5)(41/6) = 41/10 = b1) -- zero independent content, a face of SEAM.EQUIV.01; one phase, two responses (c_- = 8, C = 1) feeding both named targets; ALPHA.QUILLEN.EXACT.01 stays [O]",
      "canonicalStatus": "-",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.PROGRESS.02; ALPHA.QUILLEN.PROGRESS.03; ALPHA.QUILLEN.PROGRESS.04; EM.WARD.01; SEAM.EQUIV.CROSSEDPRODUCT.01; v341; v367; v382; v433; v434; v435; v456; v461",
      "scripts": [
        "v470_alpha_inflow_level.py"
      ],
      "script": "v470_alpha_inflow_level.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "TKNN PRL 49 (1982); Avron-Seiler-Simon PRL 51 (1983); Callan-Harvey NPB 250 (1985); APS 1975; Alvarez-Gaume-Della Pietra-Moore Ann. Phys. 163 (1985); Witten RMP 88 (2016); Quillen 1985; Bismut-Freed CMP 106 (1986); Ginsparg PLB 197 (1987)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 531
    },
    {
      "id": "SEAM.EHMODEL.04",
      "claim": "SEAM-HORIZON REPLICA CHAIN AT COLLAR LEVEL (the kernel-identification premise of SEAM.THEOREM.01, exercised numerically; SEAM.EHMODEL.03's residual list sharpens again, the gate stays OPEN): (1) REPLICA => EH FORM ON A REAL OPERATOR [I] - matched conical deficit G(gamma,m) = log det L(gamma,m) - (gamma/2pi) log det L(2pi,m) of the discretized gapped cone operator (finite-volume polar grid, geometric tip grading, identical stencil across geometries so extensive terms cancel exactly) is linear in ln m with slope 2C(gamma), verified on deficit cones gamma/2pi = 1/4, 1/2, 3/4 (rel. err 0.01-0.05%) AND on REAL replica sheets gamma/2pi = 2 (0.8%), 3 (1.5%) - beyond linearisation, the regime the replica trick uses; (2) COEFFICIENT FORCED, SCALE = ANCHOR [I] - the EH slope is stable under IR/size refinement (0.5% drift) while the intercept (zeta-scale mu_lat 146.9 -> 153.1) drifts with the discretization: the v152 anchor exhibited on data; NO canonical ln(m/mu) = 3/4 emerges (recorded honestly - the anchor stays declared, not derived); (3) BFK SPLIT MEASURED ON THE DISCRETIZED CALDERON KERNEL [I] - Schur identity log det L = log det(halves_D) + log det S exact (~1e-16); across three cut geometries the Calderon/Schur factor's deficit slope follows a PURE CUT-EDGE law E(1-q) with tip term 0 (power check: the same law fails on the full determinant), the halves carry the whole tip 2C_cone via Kac doubling (exact) - v151's 'conically clean' measured on the kernel itself, not inferred from corner constants; (4) THE SEAM'S OWN SPECTRUM [I] - the v302 transfer masses m2 = 6 ln(3/2), m3 = 6 ln 3 (bend ln3/ln(3/2), v124/v147) sit ON the calibrated EH line (<=0.05%): det'(collar) inherits the EH variation mode-wise; the Perron/attractor mode (m=0) is IR-DIVERGENT, running exactly as -2C ln(IR ratio) - THE RECOVERY GAP Delta = 6 ln(3/2) IS WHAT MAKES THE INDUCED-GRAVITY COEFFICIENT FINITE (the same gap that makes the attractor unique, v56, makes 1/G finite). RESIDUAL [A/O]: the kernel premise is now discharged at the finite/model level; what remains of SEAM.THEOREM.01 is the CONTINUUM scaling limit of this statement (the MMST class - the same single residual as SEAM.EQUIV.01, v336) plus the one dimensionful anchor (v152). NOT claimed: a continuum proof, or a derivation of ln(m/mu) = 3/4",
      "status": "Numerical [I] (lattice determinants vs exact conical constants) + [A/O] (continuum leg + anchor stay open)",
      "canonicalStatus": "Numerical [I]: kernel premise discharged at collar level; residual = continuum leg (MMST class) + anchor; gate stays open",
      "dependencies": "SEAM.EHMODEL.03; SEAM.EHMODEL.02; SEAM.EHMODEL.01; SEAM.THEOREM.01; SEAM.AREACOEFF.03; v56; v73; v90; v150; v151; v152; v302; v336",
      "scripts": [
        "v471_seam_horizon_replica.py"
      ],
      "script": "v471_seam_horizon_replica.py",
      "location": "origin_theory",
      "externalData": "Cheeger 1983 (conical heat kernel); Kac 1966; Burghelea-Friedlander-Kappeler 1992 (all standard)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 532
    },
    {
      "id": "ALPHA.QUILLEN.DETLINE.01",
      "claim": "THE OCCUPIED BERRY LINE OVER THE U(1)-TWIST MODULI HAS CHERN NUMBER = THE BLOCH RESPONSE LEVEL (2026-07-03) -- the finite Berry-Chern response, COMPUTED at the collar model; the Quillen/Dai-Freed comparison remains a separate target; does NOT close ALPHA.QUILLEN.EXACT.01. [E] BLOCH LEVEL RE-VERIFIED: BZ Chern C(M=1) = 1, C(M=3) = 0, C(M=-1) = -1 (the v470/v367 integers). [E] THE OCCUPIED BERRY LINE COMPUTED: the same collar h(k) = sin kx SX + sin ky SY + (M - cos kx - cos ky) SZ in real space on an L x L torus with twisted boundary conditions -- the twist torus (theta_x, theta_y) IS the moduli space of flat U(1) connections -- and the Fukui-Hatsugai-Suzuki curvature of the DETERMINANT LINE of the occupied one-particle frame (the free-fermion many-body ground-state LINE; its normalized vectors are chosen locally) over that torus integrates to 2 pi x 1 exactly (integer to 1e-9) at M = 1, for L = 4 AND L = 6 -- size-independent, as an integer must be. [E] CONTROLS: trivial collar M = 3 gives 0; orientation flip M = -1 gives -1 -- the curvature tracks the phase, not the mesh. [E] TWO TORI, ONE LEVEL: the twist-moduli integer equals the Bloch-BZ integer for all three M -- the Niu-Thouless-Wu identification (PRB 31 (1985)), exhibited on the collar model that realises S3 (v460/v461). [E] GAP: the sampled Fermi gap is open at M=1; analytically |d(k)|^2 = 1 + 2(1-cos kx)(1-cos ky) >= 1 gives the global gap >= 2. This defines a smooth occupied projector, not a global nonvanishing section of its topologically nontrivial line. No Dai-Freed filling or global-section test is performed. [C] INTEGRATED RESPONSE: k0 = |C_FHS| = 1 in the collar model. The FHS continuum convention C_Berry=integral F_occ/(2pi i), with nabla=d+A_occ and c1=iF_occ/(2pi), gives c1[B]=-C_Berry; finite-mesh identification and theta-line sign/duality must be checked separately. This integrated integer does not identify individual holonomies or the exact zeta-determinant variation. The open comparison requires a separately defined chiral Fredholm/Quillen line with its zero locus, a common gauge base, relative connection-level comparison and uniform ultraviolet controls; a Dai-Freed vector requires filling data and takes values in the pulled-back inverse boundary line (Quillen 1985; Bismut-Freed CMP 106 (1986); Dai-Freed JMP 35 (1994); FHS JPSJ 74 (2005)). [O] NOT CLOSED: the abstract-seam zeta-determinant identification (the continuum leg, = the SEAM.EQUIV.01 face); ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 = 137.0359992168 stays [E] regardless. Structurally the same manoeuvre as v471 for SEAM.THEOREM.01's replica side: the named open lemma exercised on a real finite operator.",
      "status": "Numerical [N] (finite Quillen shadow; gate unchanged)",
      "canonicalStatus": "Det-line/moduli shadow [E]/[C]/[O]: the Quillen-shaped object of ALPHA.QUILLEN.EXACT.01 (det line over the U(1) moduli) computed on the S3 collar -- FHS curvature of the occupied-frame det line over the twist torus = 1 = the inflow level, size-independent, controls 0/-1, twist==BZ integer (Niu-Thouless-Wu); the finite integrated response is exhibited; no global nonvanishing section, individual-holonomy identity or continuum Quillen comparison is established; ALPHA.QUILLEN.EXACT.01 stays [O]",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.INFLOW.01; EM.WARD.01; v367; v460; v461; v470; v471",
      "scripts": [
        "v472_quillen_detline_moduli.py"
      ],
      "script": "v472_quillen_detline_moduli.py",
      "location": "tfpt_1_architecture_e8",
      "externalData": "Niu-Thouless-Wu PRB 31 (1985); Quillen 1985; Bismut-Freed CMP 106 (1986); Dai-Freed JMP 35 (1994); Fukui-Hatsugai-Suzuki JPSJ 74 (2005)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 533
    },
    {
      "id": "GRAV.ENTROPIC.ACTION.01",
      "claim": "THE ENTROPIC-ACTION BRIDGE (2026-07-13) -- Bianconi's 'Gravity from entropy' (Phys. Rev. D 111, 066001 (2025); arXiv:2408.14391; DOI 10.1103/PhysRevD.111.066001) quantified as an EXTERNAL CANDIDATE for the local volume action behind TFPT's parameter-free Einstein equation; closes NOTHING, adds ZERO knobs. Context: v359 derives G_ab + Lambda g_ab = (1/c3) T_ab thermodynamically and is honestly typed 'equation of state, not a from-action quantisation' [O]; Bianconi's S_B = -Tr_F ln(G~ g~^-1) (quantum relative entropy between the spacetime metric and the matter-induced metric; Araki-related via Delta^{1/2} = G~ g~^-1, her Appendix C) is a local covariant action whose variation yields Einstein + emergent nonnegative Lambda at low coupling -- exactly the missing variational layer IF the bridge identities hold. [E] CARRIER HODGE COUNT: her Omega^{0,1,2} matter fiber, placed on the five-slot carrier C^5 (NOT on 4d spacetime, where the fiber is 1+4+6 = 11), counts 1+5+10 = 16 = 2^(g_car-1) = dim S+_{D5}, Hodge-star folded (L^1 ~= L^4) onto L^even C^5 -- the carrier half-spinor's own Pascal row (v2/v44); vector-space identity ONLY. [E] COEFFICIENT MATCH (the one quantitative pin): her weak-coupling Lagrangian L = 3 beta R - alpha L_matter (her Eq. (45); the 3 = the three form blocks) matched to the TFPT EH normalisation 1/(16 pi G) = c3/(2G) fixes her free constant EXACTLY: beta'_B = c3/6 = 1/(48 pi); the 6 = 3 blocks x 2 (EH 1/2) = p2 = |R+(A3)| is recorded as a structural echo, NOT a proof. [E] ENTROPY POTENTIAL = QUADRATIC LAMBDA: Lambda_G = (1/2 beta) Tr_F(G - I - ln G) with s(x) = x - 1 - ln x >= 0, s''(1) = 1 (exact series) -- Lambda nonnegative and QUADRATIC around the entropic fixed point; with the TFPT displacement eps_* = e^{-1/alpha} Q this reproduces the closed v60 branch rho_Lambda/Mbar^4 = (3/4 pi^2) e^{-2/alpha}, and the prefactor match (beta' = c3/6, l_P^2 Mbar^2 = c3) yields the EXACT normalisation target Tr_F Q^2 = 32 c3^4 = 1/(128 pi^4) = (2/3) delta_top -- a closed algebraic target for the two-form (pair) block (binom(5,2) = 10), [C] until an explicit Q is constructed. [E] FRULLANI: -ln A = int_0^inf dt/t (e^{-tA} - e^{-t}) -- her log action is an all-scale-integrated heat-kernel action, the correct genre to compare with the TFPT relative spectral action S_rel,chi (KMS cutoff). [E] R^2 KILL TEST QUANTIFIED (pre-registered): the raw log expansion carries a curvature-squared coefficient of order (c3/6)^2; the TFPT Starobinsky sector needs Mbar^2/(12 M_scal^2) = 1/(12 c3^7) (M_scal^2 = c3^7 Mbar^2, v36); the EXACT mismatch is 3 c3^{-9} = 3 (8 pi)^9 ~ 1.20e13 -- THIRTEEN orders: a direct identification of the raw entropic action with the TFPT R^2 sector is FALSE without a mechanism (scalaron = light trace mode of her G-field generating c3^{-9}, or KMS-spectral renormalisation of the log action). [C] COMPRESSION CONJECTURE (the bridge proper, recorded not proven): P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} -- the physical Calderon compression of her relative metric operator equals the TFPT seam modular operator; if proven, S_B is a local volume representation of the TFPT relative spectral action; open together with the operator-level Hodge items (Clifford action, Dirac intertwining, leaf involution, Calderon polarisation). Lorentzian positivity of G~ g~^-1 is NOT automatic (timelike gradients); TFPT's OS/reflection-positive route is the sturdier construction. [O] NOT CLOSED: the v359 equation-of-state status stays [O]; adopting S_B would close it ONLY after the compression conjecture AND an R^2 mechanism; no status marker moves anywhere",
      "status": "Formal/Numerical [E] (five exact identities + quantified gap) + [C] (compression conjecture, Q-target) + [O] (variational completion open)",
      "canonicalStatus": "Entropic-action bridge [E]/[C]/[O]: Bianconi PRD 111, 066001 (2025) as external candidate for the missing variational layer of the parameter-free Einstein equation -- beta'_B = c3/6 = 1/(48 pi) pinned exactly, carrier Hodge count 1+5+10 = 16 = dim S+ (carrier-only), Lambda_G quadratic-nonnegative reproducing the v60 branch with exact target Tr Q^2 = 32 c3^4, R^2 gap EXACTLY 3 (8 pi)^9 ~ 1.2e13 as pre-registered kill test; compression conjecture P (G~ g~^-1) P = Delta^1/2 open; v359's [O] unchanged",
      "dependencies": "GRAV.NONLINEAR.01; GRAV.BACKREACT.01; QG.AMB.01; v2; v36; v44; v58; v60; v358; v359; v369",
      "scripts": [
        "v473_entropic_action_bridge.py"
      ],
      "script": "v473_entropic_action_bridge.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bianconi, 'Gravity from entropy', Phys. Rev. D 111, 066001 (2025), arXiv:2408.14391, DOI 10.1103/PhysRevD.111.066001 (external candidate action); Araki 1975 (relative entropy); Jacobson 2015 (equilibrium, via v358/v359)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 534
    },
    {
      "id": "GRAV.ENTROPIC.HODGE.01",
      "claim": "THE OPERATOR LEVEL OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; work packages 1 + 4-algebra of GRAV.ENTROPIC.ACTION.01/v473) -- upgrades the v473 vector-space identity to the operator level on the finite carrier Fock space and DECIDES the Q-block conjecture by exact enumeration; closes NO gate. [E] CAR => CLIFFORD: on Lambda^* C^5 (dim 2^g_car = 32) the ten Gamma_{2k} = a_k^dag + a_k, Gamma_{2k+1} = i(a_k^dag - a_k) satisfy {Gamma_a,Gamma_b} = 2 delta_ab exactly (all 100 pairs, integer matrices) -- the so(10) = D5 Clifford algebra acts on the carrier exterior algebra. [E] S+ = LAMBDA^EVEN AS D5-MODULE: the 45 bilinears [Gamma_a,Gamma_b]/4 commute with gamma = (-1)^N, span exactly 45 = dim so(10), and leak nothing from the 16-dim even subspace -- the carrier half-spinor IS the even exterior algebra WITH its spinor structure (the operator level of v2/v44/v197). [E] HODGE-DIRAC SYMBOL = CLIFFORD MULTIPLICATION: c(xi) = a^dag(xi) - a(xi), the principal symbol of Bianconi's D = d + delta, satisfies c(xi)^2 = -|xi|^2 Id exactly for symbolic real xi. [E] THE FOLD IS THE 5 -> 5bar CONJUGATION: the traceless u(5) weights of Lambda^4 are exactly the negatives of those of Lambda^1, so the Hodge fold *: Lambda^1 ~ Lambda^4 conjugates the vector block -- Bianconi's fiber 1+5+10 becomes the GUT half-spinor 16 = 1 + 5bar + 10, the SM carrier decomposition (not a mere dimension match). [E] HONEST OBSTRUCTION: Omega^{<=2} is NOT Clifford-invariant (c(xi) maps Lambda^2 into Lambda^3), while Lambda^even is invariant under the even Clifford algebra -- the bridge identification must go through the Hodge fold, never through naive degree truncation (a recorded constraint on the AP2 compression). [E] Q-TARGET ENUMERATION (the v473 [C] conjecture DECIDED at the algebraic level): the uniform-block ansatz Q = k c3^2 P_d solves Tr Q^2 = d k^2 c3^4 = 32 c3^4 with integer k for EXACTLY three supports (d,k) = (2,4), (8,2), (32,1) -- the sheet pair |Z2|, the rank g_car + N_fam = rank E8 = 8, and the full Fock space 2^g_car = 32 = dim Lambda^* C^5 (minimal uniform solution: eps_* = e^{-1/alpha} c3^2 per Fock mode, q = c3^2 = 1/(64 pi^2)); the two-form block d = 10 (the problem-note 'pair sector' conjecture) and the fiber d = 16 require irrational multipliers sqrt(16/5) and sqrt(2) -- the naive pair-block reading is KILLED; bonus identity Tr Q^2 / delta_top = 32/48 = 2/3 = dim Lambda^*/Omega_adm (the 2/3 in the v473 target is a mode-count ratio). [C] the physical choice among the three admissible supports (uniform-32 is minimal, not derived). [O] UNCHANGED: the AP2 compression P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} and the R^2 mechanism (v473 pre-registered kill test) stay open; no status marker moves anywhere",
      "status": "Formal/Identity [E] (operator-level Clifford/weights/enumeration, exact) + [C] (physical Q choice) + [O] (AP2 + R^2 unchanged)",
      "canonicalStatus": "Operator-level Hodge bridge [E]/[C]/[O]: the D5 Clifford algebra exhibited on the carrier Fock space (10 gammas exact, 45-dim so(10) preserving Lambda^even = 16), Hodge-Dirac symbol = Clifford multiplication, the fold = 5 -> 5bar conjugation (1+5+10 -> GUT 16), Omega^{<=2} not Clifford-invariant (honest constraint on AP2); Q-target d k^2 = 32 solved exactly by supports {2, 8, 32} with minimal uniform q = c3^2 and the 10/16 blocks killed; AP2 compression + R^2 mechanism stay [O]",
      "dependencies": "GRAV.ENTROPIC.ACTION.01; v2; v44; v60; v197; v359; v473",
      "scripts": [
        "v474_entropic_hodge_carrier.py"
      ],
      "script": "v474_entropic_hodge_carrier.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bianconi, 'Gravity from entropy', Phys. Rev. D 111, 066001 (2025), arXiv:2408.14391 (external candidate action; via v473)",
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    {
      "id": "GRAV.ENTROPIC.SCALARON.01",
      "claim": "THE R^2 KILL TEST OF THE ENTROPIC-ACTION BRIDGE, EXECUTED (2026-07-13; work package 5 of GRAV.ENTROPIC.ACTION.01/v473) + THE EXPLICIT LORENTZIAN-POSITIVITY WITNESS -- decides v473's interpretation 2 negatively in its naive form; closes NO gate. [E] MAX-SYMMETRIC EIGENVALUES (the Appendix-B flattening of Bianconi PRD 111, 066001 (2025) verified): with R_munurhosigma = (R/12)(gg - gg) on a general diagonal 4d metric the relative curvature operator R~ g~^-1 has eigenvalues EXACTLY {R (scalar); R/4 x4 (Ricci); R/6 x6 (Riemann on flattened 2-forms)}, computed symbolically including the 6x6 two-form flattening. [E] EXACT VACUUM f(R): L_B(R) = -Tr_F ln(I - beta R~ g~^-1) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] with exact series 3 beta R + (17/24) beta^2 R^2 + O(R^3) -- the linear 3 re-derives her Eq. (45) (= the v473 coefficient pin) and 17/24 = (1/2)(1 + 1/4 + 1/6) is the EXACT tensorial R^2 factor (v473 quoted the raw (c3/6)^2 'before tensorial factors'). [E] THE RAW ENTROPIC SCALARON IS TRANS-PLANCKIAN: with the pinned beta' = c3/6 the EH normalisation matches identically and BOTH routes (Starobinsky matching and f'/(3 f'')) give m_raw^2 = 4608 pi^2 Mbar^2/17, i.e. m_raw = pi sqrt(4608/17) Mbar ~ 51.7 Mbar ~ 1.3e20 GeV -- HEAVY, not light; the TFPT scalaron is c3^{7/2} Mbar = 3.06e13 GeV, a factor ~4.1e6 below; the mechanism interpretation 2 requires must supply EXACTLY m_raw^2/m_TFPT^2 = (72/17)(8 pi)^9 ~ 1.69e13 in mass^2 -- the kill-test number now WITH tensorial factors. [E] DOMAIN: positivity of I - beta R~ g~^-1 holds iff R < 1/beta = 384 pi^2 Mbar^2 (first log pole; then 4/beta, 6/beta) -- a trans-Planckian curvature bound, so the raw action is well-defined at all sub-Planckian curvatures (the obstruction is the mass, not the domain). [E] LORENTZIAN-POSITIVITY WITNESS (the v473 [C] caveat made explicit): in her warm-up sector G = g + alpha dphi x dphi on Minkowski, a TIMELIKE gradient gives G g^-1 eigenvalues {1 - alpha v^2, 1,1,1} -- nonpositive for v^2 >= 1/alpha, so ln(G g^-1) is UNDEFINED on an open field region; spacelike and Euclidean stay positive for all v -- Lorentzian positivity is a genuine restriction, TFPT's OS/reflection-positive route is the sturdier construction. [C] VERDICT: of v473's three interpretations, (1) SURVIVES (the entropic action covers the Einstein + Lambda level: beta' pinned v473, Lambda channel closed v473/v474), (2) is DEAD in naive form (this module), (3) KMS-spectral renormalisation of the log action remains the ONLY route to the R^2 sector. [O] UNCHANGED: the AP2 compression and the R^2 gate itself stay open; no status marker moves anywhere",
      "status": "Formal/Identity [E] (exact tensorial factors, scalaron mass, positivity witness) + [C] (verdict/surviving route) + [O] (AP2 + R^2 gate unchanged)",
      "canonicalStatus": "R^2 kill test executed [E]/[C]/[O]: exact max-symmetric eigenvalues {R, R/4, R/6}, exact vacuum f(R) = 3 beta R + (17/24) beta^2 R^2, raw entropic scalaron m^2 = 4608 pi^2/17 Mbar^2 (~51.7 Mbar, trans-Planckian) => v473 interpretation 2 dead in naive form, required enhancement exactly (72/17)(8 pi)^9 ~ 1.69e13; Lorentzian timelike witness 1 - alpha v^2 <= 0; KMS renormalisation the only surviving R^2 route; nothing closes",
      "dependencies": "GRAV.ENTROPIC.ACTION.01; GRAV.ENTROPIC.HODGE.01; v36; v60; v359; v473; v474",
      "scripts": [
        "v475_entropic_scalaron_gate.py"
      ],
      "script": "v475_entropic_scalaron_gate.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bianconi, 'Gravity from entropy', Phys. Rev. D 111, 066001 (2025), arXiv:2408.14391 (external candidate action; via v473)",
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    },
    {
      "id": "GRAV.ENTROPIC.COMPRESS.01",
      "claim": "THE AP2 COMPRESSION CONJECTURE MADE WELL-POSED (2026-07-13; the last open work package of the v473 entropic-action-bridge round, exercised on an exactly solvable model) -- decides the READING of the conjecture, does NOT prove it; AP2 stays [O]. Background: v473 recorded AP2 as P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2}. On a quasi-free system everything is computable: the boundary restriction is the covariance compression C_A = PCP (v155), the boundary modular data is Peschel's K = ln((1-C_A)/C_A), and for a thermal bulk the Bianconi-shaped bulk identity is exact: f(C_T) = C_T(1-C_T)^{-1} = e^{-h/T}. [N] PURE GAPPED BULK: staggered-mass chain (N=16), ground-state covariance a projection to 1e-15, gap verified -- pure bulk, thermal boundary (Bisognano-Wichmann shape). [N] COMPRESSION IS THERMAL: spec(PCP) strictly inside (0,1), Peschel K finite, entanglement S > 0. [N] THE ORDER OBSTRUCTION (the reading DECIDED): on the pure bulk spec(C) = {0,1}, so f(x) = x/(1-x) is singular on the spectrum -- the literal operator-side reading P f(C) P is ILL-POSED, while the state-side f(PCP) is finite: the compression identity can only be read STATE-SIDE (build Delta_Sigma FROM the compressed relative metric), which matches Bianconi's own construction (her G~ is a LOCAL field, i.e. already the compressed datum) -- AP2 is retyped, not weakened. [E] BLOCK ALGEBRA (symbolic, exact): [C^2]_AA - (C_AA)^2 = B Bt and [C^3]_AA - (C_AA)^3 = A B Bt + B Bt A + B D Bt -- every term carries the cross block TWICE, first order vanishes identically: for ANY analytic f the two readings agree up to SECOND order in the cross-cut correlations; the modular (entanglement) content lives exactly in that quadratic correction. [N] GAP CONTROL at the bounded (covariance) level: the mismatch d = ||[Fermi(h/T)]_AA - Fermi(h_AA/T)|| obeys the quadratic law d <= x^2 (d/x^2 in [0.4, 1.0]) and falls with the gap in the gapped regime while the cross-cut norm falls monotonically; the decoupled cut ([h,P] = 0) agrees EXACTLY -- the two readings converge in the gap-dominated regime, which is where TFPT operates (Delta_eff > 0, v76/v337). [C] AP2 well-posed: state-side reading at the bounded level, mismatch quadratic in cross-cut correlations, gap-suppressed. [O] NOT proven: the continuum seam statement (the Calderon compression of the actual relative metric operator = the actual seam modular operator); no status marker moves anywhere",
      "status": "Numerical [N] (lattice exhibits) + Formal [E] (block identity) + [C] (retyped conjecture) + [O] (AP2 continuum statement open)",
      "canonicalStatus": "AP2 made well-posed [N]/[E]/[C]/[O]: the literal operator-side reading P f(C) P is ill-posed on a pure bulk (f singular on spec {0,1}) => state-side reading forced (matching Bianconi's local construction); block algebra: the mismatch between the two readings is exactly second order in the cross-cut block; gap control: quadratic law d <= x^2, decoupled cut exact, convergence in the gap-dominated regime; AP2 (continuum seam statement) stays [O]",
      "dependencies": "GRAV.ENTROPIC.ACTION.01; GRAV.ENTROPIC.SCALARON.01; v76; v155; v337; v473",
      "scripts": [
        "v476_entropic_compression_toy.py"
      ],
      "script": "v476_entropic_compression_toy.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bianconi, 'Gravity from entropy', Phys. Rev. D 111, 066001 (2025), arXiv:2408.14391 (via v473); Peschel 2003 (quasi-free modular Hamiltonian, standard)",
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      "sourceLine": 537
    },
    {
      "id": "GRAV.ENTROPIC.SCALEFLOW.01",
      "claim": "THE SCALE-FLOW REPRESENTATION OF THE ENTROPIC ACTION (2026-07-13; the follow-up to GRAV.ENTROPIC.SCALARON.01/v475) -- the surviving interpretation 3 (KMS-spectral renormalisation) typed PRECISELY: the (72/17)(8 pi)^9 kill-test gap collapses to ONE moment condition on the scale weight, and that condition is satisfied by the SAME KMS moment TFPT already fixes (v36's f0); zero new dials; closes NOTHING. [E] SCALE-FLOW IDENTITY: -ln a = int_0^inf dt/t (e^{-ta} - e^{-t}) (symbolic exact) = 2 int_0^inf dchi/chi (e^{-a/chi^2} - e^{-1/chi^2}) (t = 1/chi^2; verified to 1e-20) -- Bianconi's log action IS the FLAT scale-integral of relative heat-kernel actions, while TFPT's S_rel,chi evaluates ONE KMS scale with a weight f: the bridge between the two action principles is a choice of SCALE MEASURE. [E] MOMENT DICTIONARY: with a general weight w(t) dt/t the maximally-symmetric expansion (v475 eigenvalues {1; 1/4 x4; 1/6 x6}) gives S_w = 3 beta R mu1 + (17/24) beta^2 R^2 mu2 + O(R^3), mu1 = int w e^{-t} dt, mu2 = int w t e^{-t} dt; the flat Frullani weight has mu1 = mu2 = 1 and reproduces the v475 raw coefficients identically -- the raw action is the unit-moment point of a moment family. [E] THE ONE MOMENT CONDITION: the weight-dressed EH pin beta' mu1 = c3/6 gives m^2 = (4608 pi^2/17)(mu1^2/mu2) Mbar^2; demanding m^2 = c3^7 Mbar^2 forces EXACTLY mu2/mu1^2 = (72/17)(8 pi)^9 -- the v475 kill-test number IS a moment ratio -- and the closure identity (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 holds IDENTICALLY: the '13 orders of magnitude' are not a mismatch of the bridge but the statement that the R^2 coefficient is a SCALE-MEASURE datum which the flat measure fixes wrongly and the TFPT KMS measure fixes correctly. [E] v36 CONSISTENCY: the same physics as f0 = 6 (4 pi)^2 / c3^7 <=> M^2/Mbar^2 = c3^7 -- the moment-condition route and the spectral-action route land on the SAME scalaron mass: one KMS moment, two equivalent parametrisations, zero new dials. [C] HONEST TYPING: a consistency statement, NOT a derivation -- the KMS weight (hence mu2/mu1^2) is TFPT input; the entropic action supplies the Einstein + Lambda level and the scale-flow FORM, TFPT supplies the measure; interpretation 3 stays [C]. NEGATIVE SIDE-RECORD: a first attempt to tie the AP2 state-side reading (v476) to an EXACT lattice Bisognano-Wichmann commutator (Peschel-Eisler simple tridiagonal forms) did NOT hold on the half-filled open chain -- kept out of the suite, logged in the research notes. [O] UNCHANGED: the R^2 gate and v359's equation-of-state typing; no status marker moves anywhere",
      "status": "Formal/Identity [E] (scale-flow + moment dictionary + closure identity, exact) + [C] (measure is TFPT input) + [O] (R^2 gate unchanged)",
      "canonicalStatus": "Scale-flow representation [E]/[C]/[O]: the entropic action = flat scale-integral of relative heat-kernel actions; moment dictionary (flat = unit moments = the raw v475 coefficients); the kill-test gap IS the moment ratio mu2/mu1^2 = (72/17)(8 pi)^9 with the exact closure (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 => m^2 = c3^7 Mbar^2 identically, consistent with v36's f0 -- one KMS moment, zero new dials; consistency not derivation; R^2 gate stays [O]",
      "dependencies": "GRAV.ENTROPIC.ACTION.01; GRAV.ENTROPIC.SCALARON.01; PS.SPECACT.02; v36; v259; v359; v473; v475",
      "scripts": [
        "v477_entropic_scaleflow.py"
      ],
      "script": "v477_entropic_scaleflow.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bianconi, 'Gravity from entropy', Phys. Rev. D 111, 066001 (2025), arXiv:2408.14391 (via v473); Frullani integral (standard)",
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    {
      "id": "GRAV.ENTROPIC.LEGS.01",
      "claim": "FIRST COMPUTABLE STEPS ON THE TWO REMAINING LEGS OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; after v473-v477 the open items were the AP2 continuum statement and the derivation of the KMS scale measure) -- both legs stay [O]; closes NOTHING. LEG A (AP2 CONTINUUM, exhibited on the exactly solvable critical chain, infinite-volume sinc kernel, 50-digit precision): [N] Calabrese-Cardy -- the compressed interval state has S(L) = (c/3) ln L + k with c_est = 1.00089, 1.00038, 1.00021 (pairs (8,16), (12,24), (16,32); monotone convergence to the Dirac c = 1 = 2 Majoranas x 1/2, consistent with the v450 reading), while a gapped control gives c_est ~ 6e-8 (area law) -- the compressed state carries the conformal/BW structure quantitatively. [N] CHM PARABOLA -- the nearest-neighbour couplings of the Peschel modular Hamiltonian K = ln((1-C_A)/C_A) organise on the CHM weight beta(x) = x(L-x)/L with correlation 0.866 -> 0.967 -> 0.986 (L = 8, 16, 24) and shrinking mid-plateau drift; constants deliberately NOT asserted (the lattice EH is only asymptotically local). [N] QUASI-LOCALITY -- at half filling the even-range bands of K vanish identically (particle-hole structure, max |K_{i,i+2}| ~ 1e-36) and the odd tail is small (mean ratio r=3/r=1 = 1.8%) with parabolic envelope. READING [C]: the state-side Delta_Sigma (the reading FORCED by v476) flows to the CHM/Bisognano-Wichmann geometric form in the continuum limit -- exactly the object TFPT's Einstein derivation consumes (the CHM ball modular Hamiltonian, v323/v358): the bridge's modular side and TFPT's gravity side MEET. NOT a continuum proof. LEG B (THE MEASURE): [E] single-scale corollary -- for dmu = delta(t - t0) dt the moments give mu2/mu1^2 = e^{t0} EXACTLY, so the v477 one-moment condition forces t0 = ln(72/17) + 9 ln(8 pi) = 30.460996... -- ONE dimensionless KMS time with an exact closed form; DECLINED explicitly: t0 is NOT h(E8) = 30 (difference 0.461; the near-miss is recorded and rejected per the v354/v355 no-free-pattern rule); no TFPT atom equals t0 -- deriving the measure from the entropic structure stays OPEN. [O] both legs remain the named open research items; no status marker moves anywhere",
      "status": "Numerical [N] (high-precision lattice exhibits) + Formal [E] (single-scale corollary) + [C] (meeting-point reading) + [O] (both legs open)",
      "canonicalStatus": "Two-leg exhibit [N]/[E]/[C]/[O]: leg A -- the compressed critical state's modular data flows to the CHM/BW geometric form (c_est -> 1 at 2e-4, CHM parabola corr -> 0.99, even bands exactly zero), meeting TFPT's Einstein-derivation input (v323/v358); leg B -- the measure condition reduced to one exact KMS time t0 = ln(72/17) + 9 ln(8 pi) = 30.461 with the h(E8) = 30 near-miss explicitly declined; both legs stay [O]",
      "dependencies": "GRAV.ENTROPIC.COMPRESS.01; GRAV.ENTROPIC.SCALEFLOW.01; v323; v358; v450; v473; v476; v477",
      "scripts": [
        "v478_entropic_continuum_legs.py"
      ],
      "script": "v478_entropic_continuum_legs.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bianconi PRD 111, 066001 (2025) (via v473); Casini-Huerta-Myers (CHM modular Hamiltonian); Peschel 2003; Calabrese-Cardy (EE law) -- all standard",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 539
    },
    {
      "id": "SEAM.KRONHEIMER.01",
      "claim": "THE R3 COMBINATORIAL-TO-GEOMETRIC BRIDGE IS KRONHEIMER'S THEOREM (2026-07-14, bird's-eye round): SEAM.DETK.01/v344 named R3's residual 'the rewrite attractor (a graph) IS the du Val singularity' -- that bridge is Kronheimer's ALE hyper-Kaehler quotient construction + Torelli classification (J. Diff. Geom. 29 (1989) 665-683 and 685-697), and ALL of its finite input data are already TFPT compiler outputs. [E] MARKS = NULL = PERRON: C_aff delta = 0 and A delta = 2 delta on the affine-E8 McKay graph; multiset {1,2,2,3,3,4,4,5,6} = 2I irrep dims = the v312 rewrite-attractor Perron vector. [E] KRONHEIMER DIMENSION COUNT: dim_R M - 4 dim_R G = 4(sum_edges d_i d_j - sum d_i^2 + 1) = 4 -- the hyper-Kaehler quotient of the marks quiver is 4-real-dimensional (the C^2/Gamma geometry); sum_edges d_i d_j = sum d_i^2 = 120 forced by the null-vector identity. [E] GAUGE DATUM = |2I|: sum delta_i^2 = 120 = |mu4| h(E8) = 4*30, h(E8) = 30 = |Z2| N_fam g_car. [E] RESOLUTION DATA: 8 = rank(E8) exceptional spheres, intersection form -Cartan(E8) with det = 1 and SNF = identity (the Poincare-sphere link = the det K = 1 face, v232/v344); deformation space h (x) R^3, dim 24. [C] the theorems are CITED, not re-proved (same import class as NPW26/MMST/AGT, v424/v336/v459). [O] the residual premise TRANSFORMS, it does not vanish: 'the raw seam supplies ALE hyper-Kaehler data asymptotic to the (2,3,5) cone' = the SAME single arrow as SEAM.EQUIV.01 (v346 L2); the keystone's open content is now reachable from BOTH sides (net-side scaling limit v336/v458; geometry-side Kronheimer datum). No gate closes, no marker moves. Wolfram mirror added to the extension file (counted with the next verified engine run). LEAN: the finite input data are kernel-checked in TfptCarrier/KronheimerMarks.lean (marks = null/Perron vector over Z, sum delta^2 = 120 = 4*30 with 30 = 2*3*5, hyper-Kaehler dimension count = 4; lake build clean, #print axioms = the 3 standard kernel axioms only).",
      "status": "Reduction/citation [E]/[C]/[O] (graph->geometry bridge discharged to Kronheimer 1989; finite input data compiler-exact; premise transforms, stays open)",
      "canonicalStatus": "Reduction [E]/[C]/[O]: R3's combinatorial-to-geometric bridge = Kronheimer's theorem; marks = null/Perron vector, quotient dim 4, |2I| = 120, unimodular -E8 resolution all [E]; premise 'seam supplies the ALE/orbifold datum' stays [O] (= SEAM.EQUIV.01's one arrow)",
      "dependencies": "SEAM.DETK.01; SEAM.DETK.02; SEAM.EQUIV.01; HYP.INJECT.01; E8.NETWORK.01; CORR round (experiments); v232; v236; v298; v312; v344; v345; v346",
      "scripts": [
        "v479_kronheimer_quiver_bridge.py"
      ],
      "script": "v479_kronheimer_quiver_bridge.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kronheimer, J. Diff. Geom. 29 (1989) 665-683 (ALE construction) and 685-697 (Torelli classification); McKay correspondence (standard)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 540
    },
    {
      "id": "QGEO.MULTILOCAL.01",
      "claim": "THE FOUR mu4 MARKS ARE A FOUR-INTERVAL FREE-FERMION GEOMETRY (2026-07-14, bird's-eye round): the seam circle minus the four Gauss-Bonnet marks (v216) IS the n = 4 disjoint-interval region of a chiral free fermion -- the class whose multi-interval modular theory is EXACTLY KNOWN (Casini-Huerta 2009 mixing term; Longo-Xu multi-interval subfactors CMP 251 (2004); Rehren-Tedesco multilocal fermions LMP 104 (2014); KLM CMP 219 (2001): holomorphic <=> trivial multi-interval inclusion, mu = 1 -- the SAME one bit as det K = 1). Until now the suite used only the single-ball CHM boost (v358). On the exactly solvable antiperiodic critical ring (N = 256, Q = 64, ell = 48): [E-num] BINARY CLOCK: the quarter-turn clock is a signed permutation with rho^4 = -1 EXACTLY -- the fermionic mu4 clock is the order-8 double-cover lift (the same Z2-lift pattern as 2I over I). [E-num] EXACT SECTOR DECOUPLING: the primitive-8th-root character projectors resolve the identity and commute with the reduced covariance at 1e-15 (the Rehren-Tedesco diagonal 4-fold-cover isomorphism at covariance level). [E-num] COVER/TWIST: each sector kernel is the single-interval quarter-ring kernel with boundary twist lambda_k; sector spectra union = full spectrum at 1e-14. [E-num] CLOCK INVARIANCE MANIFEST: rho C rho^-1 = C at 1e-14 => [rho, K] = 0 by functional calculus -- omega o rho = omega HOLDS at the 4-interval state level; NEGATIVE CONTROL: displacing ONE interval by one site re-couples the sectors at 1e-2. [N] CONJUGATE-POINT MIXING: every cross-interval K row peaks on the conjugate diagonal |s'-s| <= 2 at both sizes -- the lattice shadow of the Casini-Huerta mixing term (band energy spreads with size: long-range conformal tails, honestly noted). [C] the continuum multi-interval theory is CITED, not re-derived. [O] the raw-seam premise (QGEO.SYM.01 / SEAM.EQUIV.01) stays OPEN -- this exhibits the mechanism in the exactly solvable class the seam is argued to reach, it does not prove the seam reaches it. Also the natural continuum frame for the AP2 leg (v476/v478). No gate closes, no marker moves.",
      "status": "Mechanism exhibit [E-num]/[N]/[C]/[O] (bedrock omega o rho = omega manifest at the 4-interval state level; multi-interval modular theory cited; raw-seam premise open)",
      "canonicalStatus": "Mechanism exhibit [E]/[N]/[C]/[O]: the mu4 marks = exactly solvable 4-interval modular geometry; binary clock rho^4 = -1, exact sector decoupling, omega o rho = omega manifest + configuration-forced; QGEO.SYM.01 premise stays [O]",
      "dependencies": "QGEO.SYM.01; SEAM.EQUIV.01; SEAM.RIGIDITY.01; GRAV.ENTROPIC.COMPRESS.01; GRAV.ENTROPIC.LEGS.01; v216; v323; v358; v398; v426; v442; v476; v478",
      "scripts": [
        "v480_multilocal_four_interval.py"
      ],
      "script": "v480_multilocal_four_interval.py",
      "location": "tfpt_research_contracts",
      "externalData": "Casini-Huerta 2009 (multi-interval modular Hamiltonian, mixing term); Longo-Xu CMP 251 (2004); Rehren-Tedesco LMP 104 (2014) (multilocal fermions); KLM CMP 219 (2001) (mu-index) -- all standard",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 541
    },
    {
      "id": "FLAV.NUSCALE.02",
      "claim": "ABSOLUTE NEUTRINO SCALE UNDER THE CARRIER YUKAWA NORMALISATION (2026-07-14, bird's-eye round; CANDIDATE class, genre v467/v468): FLAV.NUSCALE.01 typed the absolute scale as one open UV input with the two-parameter trade-off (y_nu, M_R) and recorded the y_nu = 1 inversion at the structureless log_c3(Mbar/M_R) ~ 2.6. But y_nu = 1 is not the carrier normalisation: one SO(10) 16 per family + the MINIMAL Yukawa sector (10_H / PS (1,2,2), the v245/v248/v249 structure) forces y_nu = y_t at the matching scale [P named premise]. With explicit 1-loop RG (gauge/y_t/lambda up; ADKLR Weinberg-operator kappa down, hep-ph/0108005): [N] the trade-off COLLAPSES to M_R alone and the observed m_3 = 0.0503 eV demands M_R = 9.3e13 GeV -- INSIDE the compiler's own PS window [4.2e13, 2.4e15] (v249), at log_c3(Mbar/M_R) = 3.15. [X] INTEGER-RUNG HONESTY GATE: the rung M_R = c3^3 Mbar = 1.53e14 GeV predicts m_3 = 0.030 eV -- 40% LOW at 1-loop (kappa run-down factor 0.794): NEAR the rung (3.15 vs 3), NOT ON -- the ladder pin is DECLINED per v354/v355; the decision computation is NAMED (2-loop SM + explicit PS threshold matching). [N] FALSIFIABLE BAND: over the PS window m_3 spans [0.002, 0.115] eV bracketing the observation; the M_PS end (Sigma ~ 0.124 eV) is already in tension with DESI ~0.072 -- cosmology cuts the window from below; the NO floor Sigma = m_2 + m_3 = 0.059 eV stays the falsifiable edge (v272). [C]/[O] candidate class: the premise is the minimal Yukawa sector [P]; the absolute scale stays formally open until the decision computation lands; the frozen record (v84/REG.FREEZE.01) is untouched. NET: the absolute neutrino scale is now a ONE-parameter statement bracketed by the compiler's own UV window, not a two-parameter trade-off. UPDATE 2026-07-14 (v482/FLAV.NUSCALE.03): the named decision computation is EXECUTED -- the integer rung is EXCLUDED as an unstructured pin (bracketed envelopes reach x1.165 vs required x1.670); the 5/3-proximate structure escape is recorded and declined; the one-parameter window candidate stands.",
      "status": "Candidate [P]/[N]/[X]/[C]/[O] (carrier normalisation collapses the trade-off; window bracketing; integer rung declined at 1-loop; decision computation named)",
      "canonicalStatus": "Open [O] candidate (parameter-reduced): y_nu = y_t [P] collapses (y_nu, M_R) to M_R alone; required M_R inside the PS window at rung 3.15; c3^3 rung declined at 1-loop (m_3 40% low); NO floor Sigma m_nu = 0.059 eV falsifiable edge",
      "dependencies": "FLAV.NUSCALE.01; PS.RGTEST.01; PS.ALGEBRA.01; QFT4D.MATTER.01; FLAV.PMNS.CLOSE.01; v245; v248; v249; v272; v273",
      "scripts": [
        "v481_seesaw_carrier_ladder.py"
      ],
      "script": "v481_seesaw_carrier_ladder.py",
      "location": "tfpt_4",
      "externalData": "Antusch-Drees-Kersten-Lindner-Ratz, PLB 519 (2001) 238 (hep-ph/0108005, Weinberg operator RGE); NuFIT NO dm2_atm; DESI 2024 Sigma m_nu; PDG M_Z inputs (as v246)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 542
    },
    {
      "id": "FLAV.NUSCALE.03",
      "claim": "THE v481 DECISION COMPUTATION, EXECUTED (2026-07-14, deep-dive round; bracketed / assumption-robust form): the integer rung M_R = c3^3 Mbar is EXCLUDED as an UNSTRUCTURED exact pin, conditional on the carrier premise y_nu = y_t. [E-num] THE GAP: central m_3(rung) = 0.0301 eV vs observed 0.0503 -- rescue factor 1.670 (+67%, 0.154 c3-rungs). [E-num] BRACKETED EXCLUSION: with m_t(MSbar) +-3 GeV (>3 sigma), alpha_3^-1 -+0.10 (>3 sigma), +-10% on the ADKLR kappa run-down, and the PS-leg Yukawa beta scaled x[0.5,1.5] between M_PS and M_R, the largest single shift is x1.10 and the ALL-favourable combination only x1.165 -- far below x1.670: the v481 1-loop miss is NOT an RG artifact (no exact higher-loop coefficients trusted; pure envelope argument). [C] NAMED ESCAPE, RECORDED AND DECLINED: only a third-generation Majorana STRUCTURE factor r = M_R/M_R3 ~ 1.67 revives the rung; post-hoc r_needed = 1.6696 sits 0.18% from g_car/N_fam = 5/3 -- NO mechanism forces a 3/5 Clebsch on M_R3 today, so this is a named coincidence with a named missing mechanism (a PS Majorana-sector Clebsch), DECLINED per v354/v355 -- the same discipline as the t0 ~ 30 near-miss (v478). [O] the v481 one-parameter window candidate (M_R free in [M_PS, M_GUT], required M_R = 9.3e13 GeV at rung 3.15, NO floor Sigma m_nu = 0.059 eV) never used the integer rung and STANDS. Frozen record (v84) untouched; no gate closes, no marker moves.",
      "status": "Decision executed [E-num]/[C]/[X|P]/[O] (integer rung excluded as unstructured pin; 5/3 structure escape recorded and declined; window candidate stands)",
      "canonicalStatus": "Executed decision: rung c3^3 Mbar EXCLUDED unstructured (needs x1.67, envelopes give <= x1.165); r = 5/3 escape post-hoc, declined; FLAV.NUSCALE stays [O] window candidate. UPDATE 2026-07-15 (v488/FLAV.NUSCALE.04): the escape's mechanism search executed -- NEGATIVE ({2,3}-smooth channel weights, Y(nu^c) = 0 decouples k_Y = 5/3, Clebsches generation-blind); the rescue is dead, the window candidate is the endpoint",
      "dependencies": "FLAV.NUSCALE.01; FLAV.NUSCALE.02; PS.RGTEST.01; v249; v272; v354; v355; v481",
      "scripts": [
        "v482_seesaw_rung_decision.py"
      ],
      "script": "v482_seesaw_rung_decision.py",
      "location": "tfpt_2",
      "externalData": "ADKLR PLB 519 (2001) 238 (Weinberg-operator RGE); PDG m_t/alpha_s; NuFIT dm2_atm -- all bracketed with >3 sigma envelopes",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 543
    },
    {
      "id": "HYP.PHI0.GEOM.01",
      "claim": "THE GEOMETRIC SIDE OF THE phi0 PUNCTURE TARGET IS EXACT (2026-07-14, deep-dive round; narrows HYP.PHI0.PUNCTURE.01/v408, does NOT close it): every twisted/orbifold heat trace of the flat tau = i pillowcase is a t-INDEPENDENT RATIONAL, so the pi^-4 in dtop = 48 c3^4 cannot come from flat orbifold geometry at ANY order -- the entire analytic content sits in the coupling normalisation of the contact insertions. [E-num] TWISTED TRACES EXACTLY 1: Tr(gamma e^{t Delta}) = 1 for gamma = sigma (z->-z), rho (z->iz), rho^3 at every tested t in [0.05, 3] (position-space Gaussian image sums / erf closed forms + 2d quadrature, residual < 1e-9; mode-space proof: gamma permutes Fourier modes, only n = 0 is diagonal) -- t-independent, no asymptotic tail. [E] ATIYAH-BOTT MATCH: 1 = 4 x 1/|det(1-dsigma)| with det = 4 = |mu4| (four Z2 marks) and 1 = 2 x 1/|det(1-drho)| with det = 2 = |Z2| (two order-4 points). [E] PILLOWCASE DATA EXACT: orbifold heat trace = (1/2) Tr_T2 + 1/2 -- the Z2 contact term is EXACTLY 1/2 = 4 x (1/8 per mark) for all t; the clock-equivariant pillowcase trace is exactly 1 for all t. [E]/[O] THE REDUCTION: dtop = 48 c3^4 = 3/(256 pi^4) decomposes as 4 = |mu4| contact insertions of per-mark weight c3 (giving c3^4) times the multiplicity Omega_adm = 48; the bare 1/(2 pi)-propagator 4-cycle differs by the exact RATIONAL 3/16; the ONE remaining [O] is the rule 'per-mark insertion weight = c3' (seam zeta/Green normalisation), the same residual class as the EM-Ward c3-ladder (EM.WARD.02). The target narrows from 'derive the term' to 'derive the weight'. Complements the experiments-side Donnelly probe (honest negative). No gate closes, no marker moves.",
      "status": "Reduction [E-num]/[E]/[O] (geometric side exact: t-independent rational twisted traces; value reduces to the one per-mark coupling rule)",
      "canonicalStatus": "Reduction: geometric/equivariant side exact (rationals, all t); dtop's pi^-4 = coupling normalisation; remaining [O] = the per-mark weight rule",
      "dependencies": "HYP.PHI0.PUNCTURE.01; HYP.PHI0.01; ARCH.QUAD.01; EM.WARD.02; v37; v216; v396; v408",
      "scripts": [
        "v483_pillowcase_exact_traces.py"
      ],
      "script": "v483_pillowcase_exact_traces.py",
      "location": "origin_theory",
      "externalData": "Atiyah-Bott fixed-point formula; flat-torus heat kernel (standard)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 544
    },
    {
      "id": "SEAM.CONTACT.UNIT.01",
      "claim": "THE SEAM CONTACT UNIT -- THE MERGED RESIDUAL OF THE TWO ANALYTIC TARGETS (2026-07-14, deep-dive round II): the shared 'c3 per boundary insertion' rule of ALPHA.QUILLEN.EXACT.01 (c3-ladder {0,3,6}, v3/v341/EM.WARD.02) and HYP.PHI0.PUNCTURE.01 (per-mark weight c3, v483) is ONE identity and NOT a new unknown: it is the KMS seam unit 2 pi = 1/(4 c3) with 1/4 = 1/|mu4| (v239) -- one bare boundary propagator ORBIT-AVERAGED over the four marks. [E] UNIT CHAIN: 1/(2 pi) = 4 c3 = |mu4| c3 = |Z2| chi(S^2) c3. [E] MECHANISM DERIVED ON THE SEAM CIRCLE: G(d) = -(1/pi) ln|2 sin(d/2)| (polylog identity, 40 digits) takes INTEGER multiples of c3 ln2 at the mu4 separations (G(pi/2) = -4 c3 ln2, G(pi) = -8 c3 ln2), so G_marks = -4 c3 ln2 x circ(0,1,2,1) with spec {4,-2,0,-2}; the mu4-invariant (orbit-averaged) insertion carries (GV)|_marks = -(c3 ln2) C exactly -- the log-det contact expansion is EXACTLY graded in c3 per insertion, cycle combinatorics Tr(C^k) = 4^k + 2(-2)^k integer. [E] SUITE-WIDE GRAMMAR: F_U(1) ladder {0,3,6} (v3); phi0 ladder {1,4} (v37); Lambda prefactor 3/(4 pi^2) = (8 pi)^2 delta_top = 48 c3^2 (v60) -- the SAME multiplicity Omega_adm = 48 as the puncture at TWO insertions; the g-2 seam loop divides by the same 2 pi = 1/(4 c3). [C]/[O] THE MERGE: the two tracked [O] targets collapse into ONE named analytic step -- the diagonal (self-energy) zeta-renormalisation at the marks + the state-multiplicity matching (48 = Omega_adm = N_fam dim S+, 41 = 10 b1); the finite cycle sector is PROVEN. Anti-numerology: the check integers (circulant, spectrum, traces) are forced by the square mark configuration; no coefficient of either target is claimed. No gate closes, no marker moves. UPDATE 2026-07-14 (v485/SEAM.CONTACT.UNIT.02): the diagonal channel is settled -- G_reg(0) = 0 exactly at the KMS scale, the log is the pure scale channel, the mark determinant resums closed-form, and the 48/41 multiplicities are one state set under two response weights; the merged residual collapses onto the SEAM.EQUIV.01 face.  UPDATED 2026-08-27 (v974): the finite cycle sector now has its computed Delta-side partner -- the BFK jump circulant B = circ(2,-1,0,-1) shares the mu4 Fourier eigenbasis with the mark circulant C = circ(0,1,2,1), the |spec| multisets are equal {0,2,2,4}, Tr B^k = 4^k + 2(+2)^k mirrors Tr C^k = 4^k + 2(-2)^k with the sign flip = the kernel swap (B kills the mu4-trivial gauge/zero-mode channel, C the mu4-sign channel), and B + C = 2(I + S). On the doublet channel R_0|_d = (4/ln2) G_marks|_d exactly, with the c3 grading CANCELLING in the ratio -- both matrices carry the same 'c3 per insertion' unit. No global proportionality: the merged residual is sharpened to the channel-swap question, not closed.",
      "status": "Unification/reduction [E]/[C]/[O] (the two analytic targets merge; contact unit = KMS seam unit, derived for the finite cycle sector; diagonal renormalisation + multiplicity matching stay open)",
      "canonicalStatus": "Unification: 'c3 per insertion' = KMS seam unit (bare 1/(2pi) x orbit average 1/|mu4|), cycle sector derived; the two [O] targets are ONE remaining analytic step",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; HYP.PHI0.PUNCTURE.01; HYP.PHI0.GEOM.01; EM.WARD.02; DYN.KMS.01; LAMBDA.BRANCH.01; v3; v37; v58; v60; v239; v341; v408; v483",
      "scripts": [
        "v484_seam_contact_unit.py"
      ],
      "script": "v484_seam_contact_unit.py",
      "location": "tfpt_1;origin_theory",
      "externalData": "Fourier/polylog identity (standard); Atiyah-Bott (via v483)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 545
    },
    {
      "id": "SEAM.CONTACT.UNIT.02",
      "claim": "THE DIAGONAL CHANNEL SETTLED -- THE MERGED TARGET COLLAPSES ONTO THE KEYSTONE (2026-07-14, deep-dive round III; completes the computable side of SEAM.CONTACT.UNIT.01/v484): [E] DIAGONAL ZERO AT THE KMS SCALE: the renormalised coincident-point Green function of |D| on a circle of circumference l (zero mode removed, flat short-distance subtraction) is G_reg(0; l) = (1/pi) ln(l/2pi) EXACTLY (symbolic limit + 30-digit numeric checks at l = pi, 2pi, 4pi) -- it vanishes iff l = 2 pi = 1/(4 c3), the v239 KMS seam circumference: the seam scale is the unique zero-self-energy scheme point. [E] THE LOG IS THE SCALE CHANNEL: G_reg(0; 2 pi kappa) = (1/pi) ln kappa -- matching the functional structure: F_U(1) carries exactly ONE log term (the transport 8 b1 c3^6 ln(1/phi_seam), v3) while the phi0 split (4/3)c3 + 48 c3^4 is log-free; [C] reading: the log term is the diagonal/scale channel, the log-free terms are contact data at the KMS point. [E] CLOSED-FORM RESUMMATION (the BFK/gluing route already in the suite, v151): for a finite-rank mark potential the zeta-det ratio reduces to the 4x4 mark determinant, and with the diagonal zero the v484 expansion RESUMS exactly: det(I - u C) = (1 - 4u)(1 + 2u)^2, u = eps c3 ln2, C = circ(0,1,2,1) -- closed form to ALL orders, the linear term absent precisely because the diagonal vanishes (Tr C = 0), series = v484's Tr(C^k) = 4^k + 2(-2)^k term by term. [E] WEIGHT DICHOTOMY (the multiplicity matching): the SAME 48 = 3 x 16 admissible states (v245) carry both multiplicities -- counted FLAT they give the phi0 puncture count 48 = Omega_adm = N_fam dim S+; counted with the U(1) response weights (Y^2 with the standard loop/GUT factors) they give the alpha budget 41 = 10 b1 = 40 fermionic (Tr_16 Y^2 = 10/3 exact, nu^c = 0) + 1 Higgs (Ginsparg (3/5)(41/6) = 41/10, v470) -- one state set, two response weights, no new number. [C]/[O] CONSEQUENCE: with v484 (unit + cycle sector) EVERY finite/computable piece of the merged ALPHA.QUILLEN.EXACT.01 + HYP.PHI0.PUNCTURE.01 target is proven; the single remaining [O] is the identification of the abstract seam zeta-det with this glued mark determinant -- a face of SEAM.EQUIV.01, exactly the v382 typing, now substantiated computationally. The merged residual collapses onto the keystone; no separate analytic unknown remains. No gate closes, no marker moves; no coefficient fitted (all integers forced by the square configuration and the SM charge table).  UPDATED 2026-08-27 (v974): the BFK/gluing resummation named here is now EXECUTED (probe seam_bfk_conical_det_probe 21/21, promoted as v974): C_glue = 2^-N exactly (N = 1..4 symbolic, general interval lengths, Laurent-polynomial identity), massless reassembly det'_zeta(Delta_S1) = l^2 = 2^-N prod(2 a_j) (l/N) det'(R_0) with det'(R_0) = N l / prod a_j, jump matrix = inverse mark Green matrix R(m) = (G_m|_marks)^-1 exactly, and the diagonal/scale channel reproduced exactly through log det'Delta(l) - log det'Delta(1/(4 c3)) = (1/(4 c3)) G_reg(0; l). The keystone typing is unchanged: pillowcase moduli, the 8 b1 index budget and a |D|-level BFK as an operator identity stay open with the keystone.",
      "status": "Completion of the computable side [E]/[C]/[O] (diagonal zero at the KMS scale; closed-form resummation; multiplicity dichotomy; residual = the SEAM.EQUIV.01 face alone)",
      "canonicalStatus": "Computable side complete: diagonal = 0 at KMS scale, resummation closed-form, 48/41 = one state set under two weights; remaining [O] = the SEAM.EQUIV.01 face (v382 typing substantiated)",
      "dependencies": "SEAM.CONTACT.UNIT.01; ALPHA.QUILLEN.EXACT.01; HYP.PHI0.PUNCTURE.01; HYP.PHI0.GEOM.01; EM.WARD.02; DYN.KMS.01; v3; v151; v239; v245; v434; v470; v483; v484",
      "scripts": [
        "v485_contact_diagonal_closed.py"
      ],
      "script": "v485_contact_diagonal_closed.py",
      "location": "tfpt_1;origin_theory",
      "externalData": "BFK gluing (Burghelea-Friedlander-Kappeler, via v151); Ginsparg 1987 (via v470) -- standard",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 546
    },
    {
      "id": "HYP.REWRITE.02",
      "claim": "THE FULL TRANSFER SPECTRUM FROM ONE FORCED LOCAL RULE (2026-07-15, generator-hunt round; sharpens HYP.REWRITE.01/v327): the verified seam transfer spectrum {1, (2/3)^6, (1/3)^6} (v54/v221) now has a single local recursive generator -- the LAZY Z2-PAIR WALK. [E] UNIQUENESS: for the 3-channel rule M(s,h) = [[1,0,0],[0,s,h],[0,h,s]] (one absorbing family channel + symmetric Z2 pair) the survival spectrum is {s+h, s-h}; demanding the PHYSICAL pair {2/3, 1/3} forces UNIQUELY (stay, hop, leak) = (1/2, 1/6, 1/3) = (1/|Z2|, 1/(|Z2| N_fam), 1/N_fam) -- all three rates exact atom expressions (recorded; honest direction: spectrum -> rates). [E] SIX-HAND GENERATION: over the order-6 = 2 N_fam = |R+(A3)| dynamic hand (v327's hand) the rule generates EXACTLY eig(B^6) = {64/729, 1/729} = {(2/3)^6, (1/3)^6}; the two decay gaps are 6 ln(3/2) (the recovery gap, v76) and 6 ln 3 (the subdominant gap -- previously WITHOUT any local-rule origin). [E] SUPERSESSION: v327's uniform rule (s = h = 1/3, spectrum {2/3, 0}) has a persistent zero mode under every power -- no iterate can produce the (1/3)^6 mode; its lambda_2 content (2/3 = |Z2|/N_fam from the arity) is unchanged; superseded as generator of the FULL spectrum. [E] sub-stochastic, symmetric (detailed balance on the pair), absorbing democratic family attractor unchanged (v317). [O] HONEST RESIDUAL: the arity {|Z2|, N_fam} = {2, 3} stays the anchor input (as in v327); ADDITIONALLY the lazy split (1/2, 1/6) vs the uniform split (1/3, 1/3) is SELECTED by the physical spectrum, not yet derived from a first principle -- named candidate: 'stay = 1/|Z2|' as the sheet-symmetric choice of the Z2 involution (recorded, not claimed). Discovery provenance: experiments/tfpt-discovery/generator_hunt_full_transfer_rule.py (with the anchor-bilinear coverage inventory and the lepton-exponent (5,3,2) = K-row = atom-triple observation, both typed, no unforced claims). No gate closes, no marker moves. UPDATE 2026-07-15 (v487/HYP.REWRITE.03): the split selection [O] CLOSES -- the lazy rule's one-step spectrum is the complete v124 clock ladder below the wall and ladder faithfulness + Z2 equivariance + rates >= 0 force (1/2, 1/6) uniquely (the uniform rule collapses the odd channel onto the wall); only the arity and the clock's semiclassical derivation remain open.",
      "status": "Construction/uniqueness [E]/[O] (full spectrum from the unique lazy Z2-pair walk; split selection stays open)",
      "canonicalStatus": "Full-spectrum generator [E]: unique lazy Z2-pair walk (1/2, 1/6, 1/3); (1/3)^6 mode generator-full (gap 6 ln 3); v327 rule = degenerate case; arity [O]; split selection CLOSED by v487 (clock-ladder faithfulness)",
      "dependencies": "HYP.REWRITE.01; HYP.FIBER.01; HYP.INJECT.01; DYN.SEMIGROUP.01; v54; v76; v221; v312; v317; v324; v327; v395",
      "scripts": [
        "v486_transfer_full_rule.py"
      ],
      "script": "v486_transfer_full_rule.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 547
    },
    {
      "id": "HYP.REWRITE.03",
      "claim": "THE LAZY SPLIT FORCED BY THE RESUMMED CLOCK (2026-07-15, closing the v486 [O] 'split selection'; sharpens HYP.REWRITE.02): the split (stay, hop) = (1/2, 1/6) is not a choice -- it is clock-ladder faithfulness. [E] RUNG READING: the lazy rule's one-step spectrum is EXACTLY the complete v124 resummed-clock ladder below the wall, eig(M) = {1 - n/3 : n = 0, 1, 2} = {1, 2/3, 1/3}; v327's uniform rule instead puts its second pair mode ON the wall (0 = 1 - 3/3) -- the v486 degeneracy re-read as 'odd channel collapsed to the wall'. [E] DECK PARITY = RUNG INDEX: [B, sigma] = 0; the sigma-even pair mode carries rung n = 1 (survival 2/3), the sigma-odd mode rung n = 2 (survival 1/3) -- each clock step flips the deck parity; the structural home of the v486 parity assignment (omega_1 even-channel, omega_2 odd-channel) tested by the frb-parity-comb search. [E] FORCING: demanding the faithful ladder (no rung missing, no physical mode on the wall -- definitional for a full-spectrum generator) + Z2 equivariance (symmetric pair block) + non-negative rates forces UNIQUELY both the split (1/2, 1/6) AND the rung assignment even -> 1, odd -> 2 (the swapped assignment needs hop = -1/6 < 0, inadmissible); the v486 named candidate 'stay = 1/|Z2| as the sheet-symmetric choice' is superseded by this proof. [E] BEND COROLLARY: omega_1/omega_2 = rate(2)/rate(1) = log_{3/2} 3 = 2.70951... -- the two comb tones of the empirical search program are one v107/v124 bend apart. [O] HONEST RESIDUAL: the channel count 3 = N_fam is forced by the clock wall (v124 pt 4); the ladder itself stays an [I] identity on frozen spectra whose semiclassical derivation is open (R1/[P]); the arity {|Z2|, N_fam} = {2, 3} stays the anchor input. NET: no independent choice remains at the walk level. Contract provenance: experiments/theory-contracts/lazy01_clock_rung_forcing.py. No gate closes, no marker moves.",
      "status": "Construction/proof [E]/[O] (split forced by clock-ladder faithfulness; arity + clock derivation stay the open inputs)",
      "canonicalStatus": "Split forcing [E]: eig(M) = complete clock ladder {1, 2/3, 1/3}; uniform rule = odd mode on the wall; deck parity = rung index; assignment uniqueness via h >= 0; omega_1/omega_2 = bend; [O] arity + semiclassical clock derivation",
      "dependencies": "HYP.REWRITE.02; HYP.REWRITE.01; v124; v107; v327; v486; v54; v221",
      "scripts": [
        "v487_transfer_clock_rungs.py"
      ],
      "script": "v487_transfer_clock_rungs.py",
      "location": "origin_theory",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 548
    },
    {
      "id": "FLAV.NUSCALE.04",
      "claim": "THE 3/5 CLEBSCH DOOR ENUMERATED AND CLOSED (2026-07-15; executes the mechanism search behind FLAV.NUSCALE.03's named escape): v482 recorded that only a third-generation Majorana structure factor r ~ 1.6696 (0.18% from g_car/N_fam = 5/3) could revive the integer rung M_R = c3^3 Mbar, and declined it for lack of mechanism. The mechanism search is now EXECUTED at the group-theory level with a clean NEGATIVE. [E] NO RENORMALISABLE CHANNEL: sym^2(16) = 136 = 10 + 126; the 126bar is NOT in the E8 hull {1, 10, 16, 45} (v247) -- M_R only via the d=5 operator (16_F 16bar_H)(16_F 16bar_H)/Lambda with the E8-allowed 16_H. [E] CHANNEL LIST: 16 x 16bar = 1 + 45 + 210; the hull admits singlet + 45 insertion channels (210 absent). [E] CHARGE TABLE: Y = T_3R + (B-L)/2 verified on all 16 states; Tr_16 Y^2 = 10/3 (v485); (B-L, T_3R, Y)(nu^c) = (+1, -1/2, 0). [E] THE {2,3}-SMOOTH THEOREM (the door closes): every available nonzero nu^c channel weight -- singlet 1, T_3R^2 = 1/4, (B-L)^2 = 1, T_3R(B-L) = -1/2, SU(4)-15 weight 3/8 -- is a {2,3}-smooth rational, and the multiplicative closure of {2,3}-smooth rationals never contains 5/3 (a factor 5 cannot appear); the UNIQUE natural 5 of the embedding is k_Y = Tr_16 Y^2 / Tr_16 T_3L^2 = 5/3 (Ginsparg, v470) -- a FULL-MULTIPLET trace, and its carrier direction has Y(nu^c) = 0 EXACTLY, so the only 5-carrying group number is structurally decoupled from the Majorana operator. [E] GENERATION-BLINDNESS: the Clebsch multiplies the whole family matrix c_ij (identical 16's); a per-generation ratio is a flavor texture, never a Clebsch -- the required diag(1, 1, 3/5) cannot arise from group theory by construction. [O] COMPILER-FLAVOR INVENTORY (typed, declined): exact 3/5's in the frozen address tables sit only in WRONG slots (K lepton row (5,3,2): gen2/gen1; R (2,5,3) and L (8,5,3) lepton rows: gen3/gen2); the escape needs gen3 vs a DEGENERATE gen1 = gen2 base, which no frozen table has; no mechanism links M_R to the address tables (v354/v355). NET: the rung + 5/3 rescue is DEAD as a mechanism candidate; the v481 one-parameter window candidate (M_R free inside the PS window; Sigma m_nu = 0.059 eV floor as kill test) stands as the honest endpoint. Contract provenance: experiments/theory-contracts/clebsch01_majorana_35_door.py. No gate closes; the frozen record is untouched.",
      "status": "Enumeration/decision [E]/[O] (the 3/5 Clebsch door closed; negative result; window candidate stands)",
      "canonicalStatus": "Door closed [E]: no renormalisable channel (126 E8-forbidden); nu^c channel weights {2,3}-smooth -> 5/3 unreachable; k_Y = 5/3 decoupled (Y(nu^c) = 0); Clebsches generation-blind; [O] wrong-slot compiler 3/5's typed and declined",
      "dependencies": "FLAV.NUSCALE.03; FLAV.NUSCALE.02; PS.E8BRANCH.01; v247; v249; v470; v481; v482; v485",
      "scripts": [
        "v488_majorana_clebsch_door.py"
      ],
      "script": "v488_majorana_clebsch_door.py",
      "location": "tfpt_2_standard_model",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 549
    },
    {
      "id": "FLAV.NUSCALE.05",
      "claim": "THE SCALARON-TRACE RUNG CANDIDATE (registered 2026-08-28, external master-object review wave 2): M_R = N_fam * M_scal = 3 c3^{7/2} Mbar = 9.180e13 GeV -- a STRUCTURED point inside the v481 PS window, -1.8% from the v481-required 9.346e13 GeV under the same frozen 1-loop inversion and the named premise y_nu = y_t [P] (the review text said about 1%; the honest value is 1.8); m_3 at the candidate = 0.0512 eV (+1.9%, inside the >50% v482 RG envelope), ~20x sharper than the declined integer rung c3^3 Mbar. EXACT ladder position: 3 c3^{7/2} = (3 sqrt c3) c3^3, 3 sqrt(c3) = 3/sqrt(8 pi) = 0.598413 -- 0.26% from the 3/5 of the v482/v488 story; ADJACENCY TYPED: v488 killed 3/5 as an SO(10)/PS CLEBSCH, while the scalaron-trace reading is a DIFFERENT mechanism class -- HONEST RETYPE (2026-08-28, review wave 3): the trace argument Tr_A3 I = 3 is loose (Spec(I) = {1,1,1}, trace 3 does not make 3 an eigenvalue); the operator reading is the Q_+ eigenvalue route Spec{1,2,3} (v10/v50/v69), with the mixed insertion texture typed DATA_CONSTRAINS_TEXTURE in the nu-scalaron v2 project -- mechanism stays [O], not derived. The same M_scal already carries f_a = M_scal/128 (v185) and the inflation scale (v253): no new scale atom. KILL TEST: Sigma m_nu (NO) = 0.059 eV; a robust cosmological bound below 0.059 eV or a >~5% shift of the required M_R kills the candidate.  CANONICAL NOTE (2026-08-28, completeness wave, NU_TEXTURE_CENSUS_NULL, not a module): frozen 3x3 inventory search 0/1607 corpus candidates + 0/200 random control (best miss factor 12.6 on the reactor angle) -- the Q+ union needs structure beyond the frozen 3x3 inventory; mechanism [O] sharpened. Provenance: experiments/tfpt-discovery/nu_texture_census_probe.py. No v3 freeze; M2 = 1.73e10 GeV remains the unused prediction.",
      "status": "Candidate [C]/[N] + mechanism [O] (structured point inside the v481 window; scalaron-trace hypothesis typed, not derived; no seesaw closure)",
      "canonicalStatus": "[N] 3 M_scal within 1.8% of the required M_R and m_3 within 1.9% at the frozen v481 inputs; [E] exact ladder identities; [C] the scalaron-trace reading; [O] the mechanism; [X] the Sigma m_nu = 0.059 eV kill floor. Does NOT close FLAV.NUSCALE; v481 window candidate and v488 negative untouched.  WAVE-3 RETYPE (2026-08-28): operator reading is the Q_+ eigenvalue route Spec{1,2,3} (v10/v50/v69), not Tr I = 3; texture DATA_CONSTRAINS_TEXTURE (nu-scalaron v2); mechanism stays [O].  COMPLETENESS-WAVE NOTE (2026-08-28): NU_TEXTURE_CENSUS_NULL 0/1607 + 0/200 -- frozen 3x3 inventory provably insufficient; mechanism [O] sharpened; no new module (search null).  WAVE-5 (2026-08-29, v1003): Schur-texture NULL (8/8 D4-even (K,B) fail; lambda_1=lambda_3 forced) and scale NULL (12/12; 1-|c/a|=1.885e-4 not supplied) recorded as binding constraints on FLAV.NU.TEXTURE.MECHANISM.01 [O].  Candidate unmoved; no seesaw closure.",
      "dependencies": "FLAV.NUSCALE.02; FLAV.NUSCALE.03; FLAV.NUSCALE.04; GRAV.FR.01; AX.P1.01",
      "scripts": [
        "v986_nu_scalaron_rung.py",
        "v1003_nu_structural_nulls.py"
      ],
      "script": "v986_nu_scalaron_rung.py; v1003_nu_structural_nulls.py",
      "location": "tfpt_2_standard_model",
      "externalData": "NuFIT m_3 = 0.0503 eV; 1-loop SM inputs (v481 frozen)",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 550
    },
    {
      "id": "SEAM.CCC.01",
      "claim": "THE CHIRAL CENTRAL CHARGE c_- = 8 MEASURED FROM THE BULK GROUND STATE (2026-07-20; ground-state witness round): the Kim-Shehab-Kim modular commutator J(A,B,C) = i Tr(rho_ABC [K_AB, K_BC]) = (pi/3) c_- (PRL 128 (2022); arXiv:2110.06932/2110.10400), evaluated on the EXACT BdG ground state of the v367 p+ip seam lattice model via the Peschel Majorana covariance (W_X = -2 artanh Gamma_X, J = (1/4) Tr([W_AB, W_BC] Gamma_ABC)), a disk about the torus centre tripartitioned into three 120-degree wedges. This is the FIRST ground-state witness for premise (i) of SEAM.EQUIV.01 (c_- = 8) that is INDEPENDENT of both the Chern-number route (v367: FHS curvature of the Bloch bundle) and the counting route (v378: KLM/det-Cartan arithmetic) -- J is read off entanglement data alone; no band topology or representation theory enters the measurement. [E] GROUND-STATE PURITY: eig(i Gamma) = +-1 to 6e-15 (exact pure state, not an approximation). [E] PER-LAYER c_- = 1/2: c_- = 3J/pi = 0.499998 at L=32, R=12 (convergence 0.499793 at L=16 -> 0.500000 at L=48, R=18, dev -1.8e-7; suite runs L=32 with tol 1e-4). [E] CONTROLS: trivial M=3 (Chern 0) gives c_- = 0.000000; orientation flip M=-1 gives -0.499986 (sign flip) -- J tracks the phase, not the algorithm. [E] LAYER ADDITIVITY EXACT: J(2 decoupled layers) - 2 J(1 layer) = 1.4e-14 (block-diagonal Gamma), so the carrier value is 16x the per-layer value. [E] THE 16-LAYER CARRIER: c_- = 8 = g_car + N_fam -- direct 16-layer computation 7.9967 (L=16, tol 1e-2; the L=24 direct run gives 7.999781) and additivity route 8.0000 (tol 1e-3). [E] CLIP INVARIANCE: the artanh regularisation eps = 1e-8 vs 1e-12 moves c_- by 5e-8. [O] HONEST FENCE: this pins c_- = 8 on the LATTICE ground state (third independent witness after v367/v378); it does NOT close SEAM.EQUIV.01 -- the continuum scaling limit (MMST, v336) stays [O] and the mu4 condensation SO(16)_1 -> (E8)_1 stays the cited KLM step (v378/v154/v351). No gate closes, no marker moves.",
      "status": "Lattice [E] (modular-commutator ground-state measurement + controls + exact additivity) + [O] (continuum limit; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "Ground-state witness [E]/[O]: c_- = 0.499998 per layer (0.500000 at L=48), 16-layer carrier c_- = 8 = g_car + N_fam (direct 7.9967 / additivity 8.0000), trivial 0 and orientation-flip controls, exact layer additivity -- premise (i) of SEAM.EQUIV.01 pinned from entanglement alone; SEAM.EQUIV.01 stays [O]",
      "dependencies": "SEAM.EQUIV.01; SEAM.S3.LATTICE.01; SEAM.S3.MODULAR.01; SEAM.MMST.INCLASS.01; v336; v351; v367; v368; v376; v377; v378",
      "scripts": [
        "v489_seam_modular_commutator.py"
      ],
      "script": "v489_seam_modular_commutator.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "Kim-Shehab-Kim modular commutator J = (pi/3) c_- (PRL 128 (2022) 176402; arXiv:2110.06932, 2110.10400); Peschel correlation-matrix method (standard)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 551
    },
    {
      "id": "SEAM.MU.01",
      "claim": "THE SPIN-STRUCTURE PARITY CENSUS: mu_pre = 4 WITNESSED ON THE LATTICE (2026-07-20; ground-state witness round, companion of SEAM.CCC.01/v489): the T^2 ground-state fermion parity of the v367 p+ip seam lattice model, resolved per spin structure (boundary-condition sector AA, AP, PA, PP) and determined TWICE -- (a) the Read-Green TRIM product parity(s) = prod_{k* in TRIM(s)} sign(dz(k*)) and (b) the real-space Majorana Pfaffian sign via Schur decomposition (parities relative to the trivial M=3 phase so the Pf convention drops out). The number of even sectors = the gauged (fermion-parity-gauged) torus GSD = the total quantum dimension mu of the gauged theory. [E] REAL-SPACE = BLOCH SANITY: the twisted real-space BdG spectrum equals the Bloch bands per sector to 9e-15. [E] PER-LAYER CENSUS: (AA, AP, PA, PP) = (+,+,+,-), TRIM product == Pfaffian in every sector -- 3 even sectors, the nu = 1 Ising signature (gauged #anyons = 3). [E] 16 LAYERS => mu_pre = 4: parity_16(s) = parity_1(s)^16 = + in ALL FOUR sectors => gauged torus GSD = 4 = mu(SO(16)_1) = |det Cartan(D8)| -- the lattice ground-state witness that the carrier sits in the SO(16)_1 class BEFORE the mu4 condensation (v378 step 1). [E] 16-FOLD-WAY BRIDGE: Kitaev's theta_v = exp(2 pi i nu/16) with the MEASURED nu = 2 c_- = 16 (v489) gives theta_v = 1 EXACTLY (sympy) -- the gauged vortex is a boson, condensable; rivals nu = 1, 2, 8 give theta_v != 1 (not holomorphically condensable). [E] NON-DISCRIMINATOR FENCE (honest): the fermionic Kitaev-Preskill TEE is identically zero -- gamma_f = 0.000001 (M=1, topological) AND 0.000000 (M=3, trivial) -- an invertible fermionic phase has no TEE, so gamma_f does NOT discriminate (E8)_1 vs SO(16)_1 (both exist only after gauging); computed and recorded so it is never mistaken for a discriminator. [C] THE KLM CONDENSATION STEP stays cited: mu_post = mu_pre/|L|^2 = 4/2^2 = 1 (holomorphic, (E8)_1; v378/KLM CMP 219 (2001)); the condensed model is interacting, NOT free-fermion computable -- the numerics pin only the premise package {c_- = 8 (v489), mu_pre = 4, theta_v = 1}. [O] the continuum scaling limit (MMST, v336) stays [O]; SEAM.EQUIV.01 is NOT closed. No gate closes, no marker moves.",
      "status": "Lattice [E] (parity census double-determined + exact 16-fold-way arithmetic + TEE fence) + [C] (KLM condensation step cited) + [O] (continuum; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "Lattice witness [E]/[C]/[O]: parity census (+,+,+,-) per layer, all-even for 16 layers => gauged torus GSD = 4 = mu(SO(16)_1) = mu_pre; theta_v = 1 exactly at nu = 2 c_- = 16 (rivals excluded); TEE fence gamma_f = 0 for both phases (no discriminator); KLM 4 -> 1 stays [C]; SEAM.EQUIV.01 stays [O]",
      "dependencies": "SEAM.EQUIV.01; SEAM.CCC.01; SEAM.S3.LATTICE.01; SEAM.S3.MODULAR.01; v336; v351; v367; v378; v489",
      "scripts": [
        "v490_seam_parity_census.py"
      ],
      "script": "v490_seam_parity_census.py",
      "location": "tfpt_research_contracts",
      "externalData": "Read-Green PRB 61 (2000) (TRIM parity); Kitaev Ann. Phys. 321 (2006) (16-fold way, theta_v = exp(2 pi i nu/16)); Kitaev-Preskill PRL 96 (2006) (TEE); KLM CMP 219 (2001) (mu-additivity, via v378) -- all standard",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 552
    },
    {
      "id": "P2.PARTITION.01",
      "claim": "g_car = 5 AS A COROLLARY OF THE FOUR MARKS (2026-07-20; the partition sharpening of ARCH.CORE.01/v53): a multiset of exactly 3 POSITIVE INTEGERS summing to 4 is UNIQUE -- {1,1,2}, the unique partition of 4 into exactly 3 positive parts -- so the elementary symmetric functions e = (4,5,2) and hence g_car = e2 = 5, |Z2| = e3 = 2, N_fam = e2 - e3 = 3 are COROLLARIES, not inputs; the carrier formula closes as a fixed point ((2^(g_car-1)-1)/g_car = 3 = rank H^1 = #marks - 1). [E] NEGATIVE CONTROLS, each assumption load-bearing (sympy exact): drop positivity -> 4 solutions with e2 ambiguous {0,3,4,5}; sum 3 -> unique but e2 = 3; sum 5 -> not unique; 2 parts -> e2 in {3,4}; 4 parts -> e2 = 6; Mehta-Seshadri-typed RATIONAL weights (scaled by 4) -> e2 = 16/3 resp. 44/9 != 5 -- integrality comes from the SPLITTING-TYPE typing (Birkhoff-Grothendieck: bundles on P^1 split with integer degrees), NOT from MS weights (which are rational). [E] THE v53 SHARPENING: v53 pins (1,1,2) using ALL THREE targets (4,5,2); the sharper lemma needs only e1 = 4 (+ positivity + 3 parts) -- both searches return the same unique multiset. [E] THE SUM RULE: |a|_1 = 4 x (N-1)/2 = 4 iff N = 3 (equidistributed clock spectrum); the v115 mu4-average route |a|_1 = |mu4| tr(A0) = 4 agrees. INDEPENDENCE CHAIN (why not circular in g_car): #marks = 4 from v216 Gauss-Bonnet (P1-side premises only, checks 1/3/5 without g_car); the 3 = rank H^1(P^1 minus 4 marks) = #marks - 1 is pure topology (v137 explicit basis / v228 index gate); |a|_1 = 4 is g_car-free via par-deg-0/(U) (v115) or the mu4-average lemma. [C] HONEST RESIDUAL: P2 is NOT made axiom-free -- the residual is the weight-TYPING postulate (integrality = Birkhoff-Grothendieck splitting exponents; positivity a_i >= 1 = h^0(E) = 0, correlate: the trivial mu4 character absent from H^1, v137 check 1; sum rule = parabolic-degree-0 'given (U)', v115 [I]/[N]); AX.P2.01 stays a DECLARED AXIOM (over-determined, now by one more convergent route). DIRECTION-REVERSAL NOTE: tfpt_constants computes N_fam FROM g_car; here the arrow runs marks -> partition -> g_car -- an over-determination check, not a circle, and it must never be double-counted in witness tallies. No gate closes, no marker moves.",
      "status": "Formal [E] (partition lemma + corollaries + negative controls + v53 sharpening, sympy exact, Wolfram-mirrored) + [C] (weight-typing postulate: splitting-type integrality, h^0 = 0 positivity, (U) sum rule; AX.P2.01 stays Axiom)",
      "canonicalStatus": "Partition corollary [E]/[C]: 4 = 1+1+2 unique in exactly 3 positive parts => g_car = 5, |Z2| = 2, N_fam = 3 as corollaries; all negative controls pass; sharpens v53 (only e1 = 4 needed); residual = the weight-typing postulate -- AX.P2.01 stays a declared axiom; the typing residual is itself sharpened by v499 (P2.TYPING.01, Deligne + BG + stability), v491 stays active",
      "dependencies": "AX.P2.01; ARCH.CORE.01; ANCHOR.GEN.01; MARKS.GB.01; v23; v53; v115; v137; v216; v228",
      "scripts": [
        "v491_p2_partition_corollary.py"
      ],
      "script": "v491_p2_partition_corollary.py",
      "location": "introduction;tfpt_1;tfpt_2;origin_theory",
      "externalData": "Birkhoff-Grothendieck splitting theorem (bundles on P^1); Mehta-Seshadri parabolic weights (typing contrast) -- both standard, cited not re-proved",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 553
    },
    {
      "id": "P2.TYPING.01",
      "claim": "THE P2 WEIGHT-TYPING POSTULATE HARDENED (2026-07-21; types the v491/P2.PARTITION.01 check-9 residual): the anchor weights a = (1,1,2) are the Birkhoff-Grothendieck SPLITTING EXPONENTS of the DELIGNE CANONICAL EXTENSION E of the flavor connection -- given the [E] anchors (4 marks, v216 Gauss-Bonnet; rank 3 = #marks - 1, v137/v228; cusp-class local monodromy char poly lam^3 - 1 + irreducibility |<U,M0>| = 24 = |S4| with sum |chi|^2 = 24 exact, v117; unitarity (U)), the three typing properties become THEOREMS of standard algebraic geometry: INTEGRALITY = Birkhoff-Grothendieck (every holomorphic bundle on P^1 splits with integer degrees, automatic once E is typed as a bundle); deg E = -4 = -#marks via the Deligne residue-trace formula (canonical residue exponents {0,1/3,2/3}, the UNIQUE lift of spec M0 to [0,1), trace 1 per mark; par-deg 0 = the v39 (U) consistency; infinity APPARENT, monodromy product = 1 exactly); POSITIVITY a_i >= 1 = h^0(E) = 0 via Mehta-Seshadri stability (irreducible + unitary => stable; every section-saturation has par-deg >= 0, all 324 cases; BG dictionary h^0 = 0 <=> a_i >= 1 exhaustive) -- so {a} = {1,1,2} is UNIQUE (v491 partition lemma) and g_car = e2 = 5 follows. [E] SELECTOR TYPED: the Schur-Horn permutohedron of 4 spec(A0) has integer points exactly {(2,2,0),(2,1,1)} (the two options of the tfpt_2 'Global computation' theorem); h^0 = 0 kills {0,2,2} (O(0) summand => h^0 = 1) -- the paper's 'balanced/minimal-variance' heuristic is now TYPED as stability/positivity. [E] THE h^1 DISCREPANCY, REAL + RESOLVED: h^1(O(-2)+O(-1)^2) = 1 != 3 (chi = deg + rk = -1) -- the naive typing 'generation space = H^1 of the rank-3 E' is FALSE (forcing it demands deg -6 and loses uniqueness: {1,1,4},{1,2,3},{2,2,2}, e2 in {9,11,12} never 5 -- negative control); the correct TWO-OBJECT typing: the LINE bundle L = O(-D), D = mu4, carries the generations (h^1(L) = 3 = N_fam = H^0(Omega^1(log D)) = the v137 basis; h^0(O(4)) = 5 = g_car, v228), while E carries them as its FIBER (rank 3) with the anchor as splitting type; three DIFFERENT H^1's coexist (3 generations L-side, 1 coherent h^1(E), 6 = |R+(A3)| local-system deformations). [E] BISWAS EXPLICIT (independent convergence witness): the weight denominator is 3 = N_fam (NOT 2 = |Z2|), so the Biswas cover is the Z3 cusp cover y^3 = (z-1)(z-i)(z+1)^2(z+i)^2 (genus 2 by RH, unramified over infinity; all-equal branch data impossible, 4c != 0 mod 3), NOT the pillowcase double w^2 = z^4-1 (genus 1, j = 1728, deck side); p_* O_Y = O + O(-2)^2 (chi cross-check -1 = -1); the equivariant regular-rep model W = (chi^0+chi^1+chi^2) x O_Y has parabolic partner with underlying bundle O + O(-2)^2: rank 3, deg -4, par-deg 0 all OK -- integrality AND the sum rule COME OUT of the correspondence -- but splitting {0,2,2} with h^0 = 1: exactly the UNSTABLE Schur-Horn companion; (1,1,2) is the Biswas image of the STABLE object, stability is the selector. [E] NEGATIVE CONTROLS: drop positivity -> 4 (31 with negatives) solutions; deg -3 -> e2 = 3, deg -5 -> not unique; n-mark scan: {1,..,1,2} unique for EVERY n >= 3 but e2 = (n-2)(n+1)/2 = 5 ONLY at n = 4 (the VALUE, not the uniqueness, is the n = 4 content); MS-rational exponents e2 = 16/3 resp. 44/9. [C]/[N] HONEST RESIDUALS (P2 stays an axiom, NO marker moves): R1 the module identification 'generation space = this H^1 with this connection' [C] (QGEO.SYM territory, QGEO.SYM.01 stays a corollary of the [O] keystone); CONVERGENCE THEOREM (2026-07-21, v503/QGEO.EMERGE.LIGHT.01): R1 is IDENTICAL with the QGEO modulus rest -- the order-4 clock on H^1 is the Coxeter element of W(A3) = the cusp-class carrier datum, and the QGEO emergence-light test shows marks + square are derived while the clock is the one input: QGEO-R1 == P2-R1, ONE common residual, not two; FERMIONIC HALVING (2026-07-21, v506/SEAM.CLOCK.RIGIDITY.01): the common R1 rest is mapped onto the SAME single alignment bit -- 'the collar deck is the central involution of the mark-D4' -- because the ORDER of the clock is now fermionically forced (U^2 = (-1)^F exact on the NS seam, nonsplit Z4; both Moebius square roots automatically order 4), leaving only the spatial alignment as the carrier input; ORIGIN CROSS-REFERENCE (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED): the common R1 bit is now characterised as a tetrad with a physical face (non-split NS Fock extension class) and its deck CLASS is derived (half-period translation of the seam double, v507); only the position stays the carrier input; FREEDOM HALVING (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED): the common R1 bit is HALVED -- its position half is topology (the collar deck is a covering deck, hence free on the seam circle; the edge class is excluded, v510), so R1 reduces to the square-modulus datum tau = i alone; the modulus stays the one [C] carrier input; R2 unitarity (U) [C] (v33/v40 premise); R3 the analytic-monodromy identification [N] (v117 check 4, ODE matching 1e-7). NON-CIRCULAR: neither v23's splitting-type input nor v115's diag(A0) = a/4 input is used; the chain is rank (topology) -> deg (residue trace) -> integrality (BG) -> positivity (stability) -> partition (v491); A0* appears only as the explicit consistency witness. The weight postulate shrinks from 'positive integers summing to 4 (why?)' to 'Deligne + BG + stability, given the cohomological typing'. DIRECTION-REVERSAL NOTE inherited from v491: tfpt_constants computes N_fam FROM g_car; here the arrow runs marks -> bundle -> partition -> g_car -- an over-determination check, not a circle, never double-counted. AX.P2.01 stays a DECLARED AXIOM. SYNTHESIS NOTE (2026-07-22, v512/SEAM.TAU.FLAG.01 -- marker UNCHANGED): the reduction state reads 'four marks + one discrete symmetry-lift bit (flag transitivity <=> tau = i) + pi' -- the R1 rest gains an equivalent discrete seam-intrinsic form (V4 -> D4 side-indistinguishability of the four equally-decorated marks), not a derivation.",
      "status": "Formal/Identity [E] (AG arithmetic: Deligne residue-trace degree, BG dictionary, stability skeleton, Schur-Horn lattice points, Biswas Z3-cover -- sympy exact, Wolfram-mirrored) + [C] (R1 module identification, QGEO.SYM territory; R2 unitarity (U)) + [N] reference (R3 analytic monodromy, v117 check 4); AX.P2.01 stays Axiom",
      "canonicalStatus": "Weight typing hardened [E]/[C]: rank 3 (topology) + deg E = -4 (Deligne) + integrality (BG) + positivity (MS stability, h^0 = 0) force E = O(-1)^2 + O(-2) => a = {1,1,2}, g_car = 5; balanced heuristic typed as stability; h^1 discrepancy resolved (two-object typing); Biswas Z3-cover explicit; residuals R1/R2 [C], R3 [N]; R1 mapped by v506 onto the single alignment bit (deck central in the mark-D4; the clock ORDER is fermionically forced); v510 halves R1: the position half is topology (edge class excluded), the rest = the square-modulus datum tau = i; v512 restates the rest as one discrete symmetry-lift bit (flag transitivity <=> tau = i) -- AX.P2.01 stays a declared axiom, QGEO.SYM stays [O]",
      "dependencies": "AX.P2.01; P2.PARTITION.01; ARCH.CORE.01; ANCHOR.GEN.01; MARKS.GB.01; QGEO.SYM.01; v33; v40; v115; v117; v137; v216; v228; v491; QGEO.EMERGE.LIGHT.01; v503; SEAM.CLOCK.RIGIDITY.01; v506; SEAM.BIT.ORIGIN.01; v507; SEAM.BIT.FREEDOM.01; v510; SEAM.TAU.FLAG.01; v512",
      "scripts": [
        "v499_p2_weights_deligne_bg.py"
      ],
      "script": "v499_p2_weights_deligne_bg.py",
      "location": "introduction;tfpt_1;tfpt_2;origin_theory",
      "externalData": "Deligne canonical extension (residue eigenvalues in [0,1)); Birkhoff-Grothendieck splitting theorem; Mehta-Seshadri parabolic stability; Biswas parabolic/equivariant correspondence -- all standard, cited not re-proved",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 554
    },
    {
      "id": "CELEST.WP1.01",
      "claim": "WP1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 1, verdict B): the E8 mu4-glue grading (v128) IS the flat Z4 monodromy of the glue-equivariant celestial chiral algebra on the A3 ALE space C^2/Z4 -- the Bittleston-Homans-Sharma CCA construction (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case) refined to Z4 with a Kronheimer-Nakajima flat connection at infinity. Sympy exact throughout. [E] INNER GRADING / FLAT MONODROMY: h = (2,2,2,2,2;0^4) gives <alpha,h> = glue class mod 4 for ALL 240 roots => glue = Ad(exp(2 pi i h/4)), an order-4 torus element = a flat Z4 monodromy; the A3-side detector h' = (0^5;1,1,1,-3) reads the SAME class -- the glue charge is the diagonal (1,1) of the discriminant group Z4 x Z4 (the v92/v125 Lagrangian glue); Z2 halfway: 120 = SO(16), 128 = spinor (v125 sub-Q {0,2}). [E] SPATIAL SIDE: C^2/Z4 under (z1,z2) -> (i z1, i^-1 z2) IS the A3 singularity XY = Z^4 (invariant ring exact; Hilbert series (1+t^4)/((1-t^2)(1-t^4)); Molien table verified; conjugation symmetry N[d][1] = N[d][3]). [E] EQUIVARIANT SECTOR CLOSES: the glue-equivariant SDYM(E8) modes (j + m = 0 mod 4) form a Lie subalgebra with D[d] = 60(d+1) even / 64(d+1) odd -- uniform per parity ONLY because dim g_j = (60,64,60,64) (the v128 dims are the consistency condition); zero modes (d=0) = g0 = carrier D5+A3+Cartan = 60; asymptotic density 62/248 = 1/4 = 1/|Z4|; EH/Z2 halfway towers 120/128. [E] CHARACTER: Theta_E8 = sum of the 4 glue-coset thetas (shell 1 = (52,64,60,64) from the LATTICE, independent of the root construction); the 4 sector characters Theta_Cj/eta^8 sum EXACTLY to chi_(E8)_1 = 1+248q+4124q^2+34752q^3 (v377 series); sector weights h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8) = the v92/v125 discriminant form (5x^2+3y^2)/8 mod 1; glue diagonal h = (0,1,1,1) INTEGER = v125 isotropy = the extension to (E8)_1 is local. [E] CRITICAL CORRECTION (why verdict B): the A3 deck diag(i,i^-1) induces z -> -z on the celestial sphere (projective order 2, the sheet flip) -- NOT the order-4 clock z -> iz; any SU(2) lift of the clock has order 8 (spin group Z8, would quotient to A7); exact bridge (spin clock)^2 = deck, |Z8| = 8 = 2|mu4| (the c3 = 1/(8 pi) winding integer); the order-4 clock IS realised as the U(2) phase diag(i,1) (order 4, fixed {0,inf}, free mu4 orbit -- all v180 properties; normalises the deck, det = i rotates the holomorphic symplectic form by the mu4 generator -- Kaehler, not triholomorphic). [E] GRAVITY SIDE: clock-invariance selects from the versal A3 family XY = Z^4 + a2 Z^2 + a1 Z + a0 EXACTLY the 1-parameter family XY = Z^4 + a0; its 4 branch points are ONE mu4 orbit with cross-ratio 2 -- the seam marks (v179 clock orbit; v168/v214 pillowcase j = 1728). [E] NEGATIVE CONTROLS: false spatial action diag(i,i) killed 3 ways (det = -1 not in SL(2,C); Poisson closure fails 14/14; grades unsymmetric); false glue (60/64 swap) breaks additivity on 3112/6720 root pairs; rigidity: exactly {id, j -> -j} = Aut(Z4) of 24 relabelings are additive. [C] the clock-deck identification is a NAMED extra identification (clock^2 = deck on the sphere, sheet-Z2 spin bookkeeping) -- verdict B, not A. [O] structural arithmetic ONLY: no celestial duality is constructed; CELEST.SEAM.01 (WP2-WP5) stays a research contract; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact Z4-orbifold/character/spin arithmetic, sympy, negative controls) + [C] (the named clock^2 = deck identification, verdict B) + [O] (celestial programme; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP1 [E]/[C]/[O]: E8 mu4-glue = inner order-4 torus element = flat Z4 monodromy on the A3 ALE; equivariant SDYM(E8) sector closes (60/64 towers, zero modes = carrier, density 1/4); (E8)_1 character = sum of the 4 glue sectors, weights = discriminant form; clock = U(2) diag(i,1), clock^2 = deck (Z8, 8 = 2|mu4|); clock-invariant deformation XY = Z^4 + a0, branch points = one mu4 orbit, cross-ratio 2; negative controls + Aut(Z4) rigidity; verdict B; SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; SEAM.EQUIV.01; E8.GLU.01; v92; v125; v128; v168; v179; v180; v214; v377",
      "scripts": [
        "v492_celestial_z4_orbifold.py"
      ],
      "script": "v492_celestial_z4_orbifold.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bittleston-Homans-Sharma JHEP 09 (2023) 008 (arXiv:2305.09451; CCA on Eguchi-Hanson, Z2 case); Kronheimer-Nakajima flat connections on ALE spaces; Costello arXiv:2111.08879 (E8 axionic consistency for twistorial SDYM) -- method/consistency citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 555
    },
    {
      "id": "CELEST.SEAM.01",
      "claim": "THE CELESTIAL-HOLOGRAPHIC RESEARCH CONTRACT (2026-07-20; the fourth research contract alongside CONTRACT.U.01, CONTRACT.G.01 and CONTRACT.F.01): HYPOTHESIS (as corrected by WP1) -- the seam is the glue-equivariant Z4-orbifold sector of the celestial chiral algebra on the A3 ALE space C^2/Z4; the mu4 clock is the Kaehler U(2) phase diag(i,1) whose square is the deck group (spin bridge Z8, 8 = 2|mu4| = the c3 = 1/(8 pi) seam winding integer); the associated twistorial bulk is the SELF-DUAL sector of TFPT gravity. TYPING / NON-CIRCULARITY: admissible inputs = the finite seam/glue data (v92/v125 glue, v128 graded hull, v168/v214 pillowcase, v180 clock) + the WP1/WP2 [E] arithmetic (v492/v493); NOT admissible as input: the continuum existence of the holomorphic (E8)_1 net on the seam (exactly the SEAM.EQUIV.01 target -- assuming it would be circular); scope = the self-dual sector only, never full Einstein gravity (the v358/v359 entanglement-equilibrium route and GRAV.ENTROPIC.ACTION.01 are separate statements about the full equation). WORK PACKAGES: WP1 DONE (v492, verdict B -- match under the one named identification clock^2 = deck); WP2 DONE (v493, verdict B -- both targets landed and SHARPENED: a0 = pure seam SCALE, the tau = i pillowcase shape frozen (I = 12 a0, J = 0 identically, j = 1728 for all a0); the 3 spheres carry the 3 nontrivial mu4 characters, the clock = Coxeter element of W(A3) = the Picard-Lefschetz monodromy of the family, surviving direction = the chi_1-Fourier diagonal; BHS Z2 -> Z4 transfer with corrections linear in a0, grade-conserving, a0 <-> the c^2 slot); WP3 DONE (v495, verdict B -- exact arithmetic alignment EXISTS: lambda_e8 = 6 (unit-trace) resp. 1/10 (adjoint-trace), (kappa/c3)^2 = 12 = |mu4| N_fam resp. 1/300, the Okubo coefficient 5/(2(dim+2)) DERIVED for all 8 algebras with a sharp sl4 negative control and the closed form lambda~^2 = h_vee + 6 across the Deligne series; BUT the look-elsewhere battery shows the alignment format passes 8/8 -- ZERO selectivity; only g_car = 5 | h_vee is e8-selective (1/8), lambda-integrality 2/8 shared with sl3, the isolating conjunction POST HOC: the c3-connection survives as convention-level compatibility + genuine lambda-arithmetic, NOT as e8-selective evidence -- alignment survives, selectivity does not); WP4 DONE (v496, verdict B(ii) -- the (E8)_1 character is NOT a conformal block of the E8[C^2] S-algebra in its own jet grading, the mismatch localised THREE ways (spin unbounded below; growth n^{2/3} vs n^{1/2}, Meinardus poles s = 2 vs 1; the level-2 null ideal deletes exactly 27000 = 30^3 = h_vee^3 with no jet analogue), BUT the boundary/period reading holds exactly at the current stratum (zero modes = 60 = C0 currents, jet slice cycles (60,64,60,64), one mu4 period sums to 248, glue weights (0,1,1,1) integer) -- the SAME MMST scaling-limit structure as SEAM.EQUIV.01 (v336/v449): the character is a boundary/limit SHADOW of the S-algebra, the constructive limit question goes to WP5); WP5a DONE (v497, 2026-07-21 -- 'the null ideal from the limit': the boundary limit made a PRECISE coefficientwise limit via the chi_w family with stabilisation threshold w = n+1 (w = 2 = the chiral jet grading); the null ideal DERIVED from root data, Sym^2(248) = 27000 + 3875 + 1 with 27000 = h_vee^3 by Freudenthal + Weyl + peeling; quotient 31124 - 27000 = 4124 = the independent mu4 theta-split sector sum; SO(16)_1 negative control: four blocks, 5304 != 14^3, h = 1/2 breaks one-block fusion; the celestial route now has the SAME two-step shape as MMST -- limit + maximal ideal -- with the ideal quantitatively fixed); WP5b DONE (v498, 2026-07-21 -- 'the singular vector as an operator', SUCCESS on the preregistered criteria: |s> = (E^theta_{-1})^2|0> constructed EXPLICITLY in a machine-built Chevalley/Frenkel-Kac basis (cocycle asymmetry on all 57600 pairs, SGN = -1 forced by Jacobi, kappa derived), J^a_1|s> = 0 for all 248 generators with case tally 190/57/1 (exactly one case sees k), level dial 2(1-k): the deletion object exists ONLY at k = 1 and not in the centerless loop algebra; mu4: j(theta) = 1 (h-adapted chamber), clock phase i^2 = -1, class(2 theta) = 2 sheet-even, 8 quarters = q^2, expressible in the sector-C_1 equivariant variables (KILL branch not triggered); U(g)|s> = V(2 theta) = 27000 = h_vee^3 with the direct g_0-orbit BFS matching Freudenthal through depth 4, quotient 4124 = sector sum; controls: the 248-current state and generic level-2 states NOT singular, the cube at k = 2 (generic Kac mechanics, honestly typed), D8 has the singular vector too but h = 1/2 breaks one-block fusion -- one-block closure stays E8/mu4-specific; HANDOVER to WP5c: in the twisted quarter-slot moding two C_1 modes never sum to 8 quarters (minimum 6 = q^{3/2}) -- the per-period dictionary (v496), not the per-slot identification, carries |s> to q^2); WP5c DONE (v500, 2026-07-21 -- 'the GNS limit state', SUCCESS on the preregistered criteria: the quasi-free family omega_w (affine k=1 vacuum on the currents via the machine-determined compact anti-involution theta(e_a) = -e_{-a}, unique on all 61504 basis pairs, + x^{wr}-contracted radial oscillators) is positive for every finite w and stabilises EXACTLY at the WP5a threshold (omega_w = omega_inf mod x^{N+1} for w >= N+1, sharp at w = N); GNS(omega_inf) levels (1, 248, 4124) via the FULL exact 9361-block level-2 Gram: rank 4124 = chi_2, kernel 27000 = V(2 theta) weight by weight (Freudenthal on all 9361 weights), every block PSD, rank table per Weyl orbit (0,0,1,8,44), h.w. norms (3844,49,2,0); the clock descends to GNS with level-2 rank split (1036,1024,1040,1024) = the per-period lattice theta-split -- the two-routes identity at the STATE level; |s> IS the zero vector of GNS(omega_inf), resolving the WP5b S4.10 twisted-slot tension; CCR obstruction: NO w-uniform damping state exists -- the family formulation is necessary AND the family exists, KILL not triggered; controls: k=2 keeps everything, k=0 has no current layer, D8 gives 2076 = one block of four, wrong family erases the 248 layer 710955 != 248, no damping keeps 897266 != 248); WP5d-alpha DONE (v501, 2026-07-21 -- 'the two-interval index from the lattice', the mu = 1 indicator chain PASSES: fermionic two-interval MI extensive (c fit 0.5000, max |F| = 6e-5 at N = 512, falling with N -- the mu = 1 reference), the sector-summed orbifold prescription pays exactly one classical bit with the 16-layer seam carrier saturating Delta_2 = ln 2 to machine precision (1.1e-15 at N = 512), so [F : F_even] = 2 and mu_gauged = 4 = the v490 parity census (two independent lattice witnesses); complementary-pair budget <= ln 4 = ln mu(SO(16)_1) as a double-limit statement + the orbifold duality-failure witness D(E) != D(E'); condensation arithmetic anchored at both ends: det Cartan(D5) x det Cartan(A3) = 16, KLM/Longo-Rehren 16/4^2 = 4/2^2 = 1, Sigma d^2 = (4,4,1), theta_v = 1 exactly at nu = 16 (rivals != 1) => mu = 1 after condensation, preregistered KILL 'mu-offset != 0 after condensation' NOT triggered; controls: nu = 1 offset non-removable, trivial phase all zero, wrong sector sum loses the full ln 2; WP5d-beta DONE (v504, 2026-07-21 -- 'split + strong additivity from the lattice', the two remaining KLM legs: strong additivity ALGEBRAICALLY EXACT with the shared boundary Majorana (Even(A) v Even(B) = Even(A u B), GF2 spans full 64/64..1024/1024, matrix rank 32/32; disjoint exactly index 2 = the v501 ln-2 bit, localised at the split point), the entropic touching defect BOUNDED < ln 2 with the Ising 1/4-exponent approach (p = 0.2444 vs 2 Delta_mu = 1/4) while the preregistered U(1)/Dirac control DIVERGES ((1/2) ln Var Q, Klich-Levitov pinned, infinite index) -- bounded-vs-divergent is the lattice discriminator and the quotient net sits on the bounded side; split via the elliptic-nome sigma_k ladder (rate pi K(1-x)/K(x) to 1.3-2.0%, sigma_1 ~ x^{1/2}, trace norm summable) with EXACT orbifold inheritance (C -> -C machine identity + the Lambda^2 C compound -- Longo heredity); Pimsner-Popa E(a) - a/2 = PaP/2 identically, lambda = 1/2 = 1/[F:F_even] with exact integer attainment (16384 + 2048 monomials, 0 violations), lambda_E4 = 1/4 = 1/mu, index consistency exp(Delta) = 2 = 1/lambda_PP over two independent routes: WITH v501 ALL THREE KLM INGREDIENTS of complete rationality -- split, strong additivity, finite mu -- are witnessed on the lattice; the continuum uplift stays [O]: 'finite-group orbifolds of completely rational nets are completely rational' is XU'S THEOREM, cited not claimed -- the proof for the concrete seam quotient net and the interacting condensed (E8)_1 net remains WP5e/Costello-Li); WP5e: alpha stage (CFT-side prefactor + level pinning) executed via v502 (2026-07-21, CELEST.WP5E.ALPHA.01 -- the q^{-1/3} prefactor derived as exact mu4 vacuum-energy bookkeeping (inner clock => shift orbifold theta = 0^8, common -c/24 = -1/3; discriminant form = Casimir via spectral flow AND 16-Majorana free fermions) and k = 1 forced three independent ways (current condition h(J) = k = 1, conformal embedding 47(k-1)(k+266/47) = 0 resp. 128(k-1) = 0, c = 8 <=> 240(k-1) = 0, plus the WP5b singular-vector dial 31124 - 27000 = 4124), with the honest sharpening that glue-h integrality holds for ALL k and fixes nothing); beta stage (equivariant twistor skeleton) executed via v505 (2026-07-21, CELEST.WP5E.BETA.01 -- the Atiyah-Bott/Lefschetz fixed-point skeleton of the inflow exact: denominators (2,4,2), characters (248,0,-8,0) two routes, per-sector Okubo with the RIGID 32 T3 twisted residual, index bridge f(m) = ch2(T_m) with glue defect -78 both routes, level dials k = 1 geometric (current count 240/0/0/0), and the E3 refutation a0 = graviton modulus O(2) NOT the axion O(-2) with the three sphere classes filling the three twisted axion slots; the preregistered kill 'inflow demands a level != 1' does NOT fire on the equivariant skeleton); gamma stage (the sphere-axion pairing check) executed via v508 (2026-07-21, CELEST.WP5E.GAMMA.01 -- an honest rigid NEGATIVE result with certificate: the W(D5) x W(A3)-invariant vertex spaces are exactly dim 2 (quadratics) and dim 5 (quartics), the product of any two invariant quadratics has zero T5/T3 content, so the exchange matrix is rank 2 with rank([M | A_fix]) = 3 and annihilator certificate (Phi_T5, Phi_T3, Phi_P)(A_fix) = (0, 32, 72) -- the three sphere axions CANNOT cancel the rigid 32 T3 residual by quadratic-vertex exchange, rule- and coupling-independently; the slot bijection is untouched and the preregistered level-kill still does not fire); WP5e: alpha (CFT side, v502) + beta (equivariant twistor skeleton, v505) + gamma (exchange no-go, v508) executed; the global BCOV quantisation stays [O] -- the exchange sub-branch is closed by v508; epsilon-2 executed via v509 (2026-07-21, CELEST.WP5E.EPS2.01, 'the CPS level-from-flux dial', verdict B): the one-level statement is now a THEOREM (lockstep from clock invariance -- one mu4 orbit of branch points, |Pi_j| = sqrt(2) t equal on all three spheres, det(A-1) = -4 no invariant flux vector; falsifier Z^4 - Z: 0/24 orderings lockstep), level = 1 carries the new sector-counter dial #prim((E8)_k) = 1 <=> k = 1 (affine marks: (1, 3, 5, 10, 15, 27) for k = 1..6), fluxes (64, 60, 64) with uniformly one quantum per charged current and the naive 'level = total flux' killed by the lockstep test itself; the PT/Z4 back-reaction stays [O]; delta-2 executed via v511 (2026-07-21, CELEST.WP5E.DELTA2.01, intermediate verdict, exact -- the full-tensor ledger: the v508 collapse is confirmed full-tensorially and arity-crossing by the innerness theorem (h in the g0 Cartan, all 15 non-neutral trilinear triples Hom = 0), but the su(4) d-symbol -- the unique symmetric trilinear on e8 -- opens the T3 channel (Q_dd = (1/60)(T3 - P3/4), Phi_T3(Q_dd) = 1/60 != 0), leaving the pairing obstructed only by the weaker certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 64, and exactly solvable relaxed with c_d = 1920 = |W(D5)|); delta-2 executed via v511 -- the full-tensor collapse is confirmed (innerness), but the cubic d-symbol channel opens T3 with the weaker certificate psi = 64; delta-1 since DECIDED NEGATIVE under the DERIVED measure by v518 (2026-07-22, CELEST.WP5E.DELTA1.01: the derived chiral measure -- blockwise SL(2,Z) covariance solved for, the mu4 multiplier obstruction a CHARACTER (koboundary defects (1,1,1) on all 15 pairs, lambda(gamma) = i^{2B + C/4} on Gamma_1(4)) cancelled exactly by the twisted fibre block f1 f3 = G, NOT by the three sphere axions -- fails all three preregistered testers under both derived solutions chi_4/chi_10, with no (N1, N2) rescue in the positive cone: a genuine KILL on the derived surface, in stated TENSION with the DECLARED completion reading of v516 which delivers the psi = 64 slice and cancels 32 T3 -- both are exact, and which of the two is the true BCOV measure is the named [O]); the pre-v518 state was: the strict holomorphic reading is refuted at the Z2 anchor, the contact term is the modular completion (Harvey-Moore tau-integral) with the forced leading (T5,T3) ratio 4:3 -- the named remaining targets are the tau-integral and the physical justification of the cubic GS term (c_d = 1920; the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513; the convention-stable [E] core is c_d = 32 x 60), plus the epsilon-1 O(-2) bulk-axion construction (success: partition function = E4/eta^8 including the q^{-1/3} prefactor FROM THE TWISTOR SIDE; kill: anomaly mismatch at the Costello-Li grade -- neither the CFT-side dials of v502 nor the equivariant skeleton of v505 trigger it). KILL TESTS (pre-registered): (K1) the equivariant structure could have contradicted 52+64+60+64 -- SURVIVED by WP1 AND WP2 (false-glue control: 3112/6720 violations show the test has teeth; WP2 deformation side: no shape modulus survived clock-invariance, no sector leak -- the clock-variant controls fail two independent ways each); (K2) one-loop associativity demanding extra fields TFPT lacks -> contract dies (Costello's axionic mechanism passes E8 at the algebra level; the TFPT field content is the open half); (K3) EVALUATED by WP3 (v495): does NOT fire -- exact alignment exists ((kappa/c3)^2 = 12 resp. 1/300); but its SCOPE IS DEMOTED on the look-elsewhere evidence (the alignment format passes 8/8 across Costello's whole list): compatibility, not e8-selective evidence; (K4) EVALUATED by WP4 (v496): fires only against the exact-block reading -- the sector arithmetic holds exactly, so the contract does NOT degrade to 'E8 is an admissible celestial gauge algebra'; WP4 lands at the boundary-limit reading (verdict B(ii), the SEAM.EQUIV.01 scaling-limit shape). HONEST SUCCESS ESTIMATE: full success (WP5e, constructive holography at the Costello-Li grade) low probability; the partial-findings expectation is now REALISED AND EXCEEDED: WP1-WP4 + WP5a-WP5d all executed with sharp structural results (verdicts B, B, B, B(ii); ALL FIVE constructive WP5 milestones landed -- null ideal, singular vector, GNS limit state, two-interval index, split + strong additivity (both WP5d stages) -- plus the WP5e alpha stage (CFT-side prefactor + level pinning, v502) AND the WP5e beta stage (equivariant twistor skeleton with the honest a0 refutation, v505) AND the WP5e gamma stage (the exchange no-go with certificate, v508) AND the WP5e epsilon-2 stage (the CPS level-from-flux dial with the lockstep theorem and the sector-counter dial, v509); AND the WP5e delta-2 stage (the full-tensor ledger with the innerness theorem and the cubic d-symbol channel, v511); WP5e proper, the global BCOV quantisation, is the single remaining open piece -- narrowed by v508/v509/v511 to the delta-1 tau-integral (since DECIDED by v518, CELEST.WP5E.DELTA1.01: kill under the derived measure, all three preregistered testers fail; the measure tension with v516 is the named [O]), the physical justification of the cubic GS term (c_d = 1920; the |W(D5)| reading look-elsewhere-loaded and convention-contingent -- v513) and the epsilon-1 bulk-axion construction (executed via v514) plus the PT/Z4 back-reaction (M1 executed via v515: the back-reacted Omega_N closed-form with (2 pi i)^2-integral lockstep flux vector and forced source charge 4 = |mu4|; M2+M3 executed via v516/v517 -- the v514 fence M1-M3 is fully worked off: the twisted KS measure (declared completion reading, ch2-weighted) cancels the 32 T3 residual and delivers the psi = 64 slice without the cubic d-channel, and the a0 uplift couples to the centre count on four scales; what stays [O]: the BCOV measure question -- the delta-1 chain itself is DECIDED by v518 (kill under the derived measure, tension with v516 stated) -- and SEAM.EQUIV.TWISTOR.01)) -- the clock^2 = deck correction, the Coxeter/Picard-Lefschetz monodromy, the non-selective GS alignment, the boundary-limit localisation of the character mismatch, the explicit singular vector with its level dial 2(1-k), the limit state with the ideal in its GNS kernel, the entropically measured mu-chain 16 -> 4 -> 1, and the lattice KLM completeness witness (split + strong additivity + finite mu, v501/v504). SEAM.EQUIV.01 stays [O] throughout; nothing here moves it.",
      "status": "Open (research contract; WP1-WP5d executed [E]/[C]/[L] via v492/v493/v495/v496/v497/v498/v500/v501/v504 -- WP5d: alpha (mu-index) + beta (split + strong additivity) executed via v501/v504, all three KLM ingredients witnessed on the lattice, continuum uplift stays [O] (Xu's theorem cited); WP5e alpha stage [E] via v502 + beta stage [E] via v505 (equivariant twistor skeleton; kill not fired) + gamma stage [E] via v508 (exchange no-go with certificate (0, 32, 72); negative result, kill not fired) + epsilon-2 stage [E] via v509 (level-from-flux: the one-level statement is now a THEOREM of clock invariance, the sector-counter dial #prim((E8)_k) = 1 <=> k = 1 pins the level; the PT/Z4 back-reaction stays [O]); + delta-2 stage [E] via v511 (the full-tensor ledger: collapse confirmed by innerness, the cubic d-symbol channel opens T3 with the weaker certificate psi = 64, relaxed solution c_d = 1920 = |W(D5)|); + epsilon-1 stage [E] via v514 (epsilon-1 executed via v514: the O(-2) bulk-axion slot is a construction (equivariant Penrose ledger), lambda-tilde = 6 pinned by three exact ledgers (Okubo / measure cancellation excluding 3 and 12 / flux), and the GH/A3 back-reaction step re-derives the v493 family and the Coxeter clock from geometry -- the quantised BCOV coefficient and the twisted channels stay [O] (M1-M3 preregistered)); WP5e-M1 DONE (v515: M1 executed via v515: the back-reacted Omega_N is closed-form on the A3 twistor family, all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector and clock covariance, and the lens geometry FORCES the source charge 4 = |mu4|; honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden family)); WP5e-M2+M3 DONE (v516/v517: M2+M3 executed via v516/v517 -- the v514 fence M1-M3 is fully worked off: the twisted KS measure (declared completion reading, ch2-weighted) cancels the 32 T3 residual and delivers the psi = 64 slice without the cubic d-channel, and the a0 uplift couples to the centre count on four scales; what stays [O]: the BCOV measure question -- the delta-1 chain itself is DECIDED by v518 (kill under the derived measure, tension with v516 stated) -- and SEAM.EQUIV.TWISTOR.01); the global BCOV quantisation [O] (single remaining milestone -- the exchange sub-branch closed by v508, the epsilon-2 flux dial executed by v509, the delta-2 tensor ledger executed by v511, the epsilon-1 axion slot executed by v514, the M1-M3 back-reaction milestones executed by v515/v516/v517 (the v514 fence fully worked off); named remaining target: the BCOV MEASURE QUESTION -- the delta-1 chain is DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, all three testers fail; tension with v516 stated), so what remains is which of the two exact measures (the declared v516 completion reading vs the v518 derived chiral measure) is the true BCOV measure; the cubic GS term stays dissolved (psi = 64 delivered by v516, no branch fires under the derived measure, v518)))",
      "canonicalStatus": "Open (research contract): seam = glue-equivariant Z4-orbifold sector of the CCA on the A3 ALE; clock^2 = deck (Z8 spin bridge); twistorial bulk = self-dual sector only; WP1-WP4 + WP5a-WP5d executed (verdicts B, B, B, B(ii); WP5a: null ideal derived, 27000 = h_vee^3, quotient = sector sum = 4124, stabilisation threshold w = n+1; WP5b: the singular vector explicit, J^a_1|s> = 0 on all 248 generators with case tally 190/57/1, level dial 2(1-k), j(theta) = 1, clock phase i^2 = -1), WP5c DONE (v500: GNS limit state omega_inf with the null ideal in its kernel -- level-2 rank 4124, kernel 27000 = V(2 theta) weight by weight, clock split (1036,1024,1040,1024), |s> = the GNS zero vector, CCR necessity of the family), WP5d-alpha DONE (v501: two-interval index -- 16-layer ln 2 plateau at 1e-15, [F : F_even] = 2, mu_gauged = 4 = v490, KLM chain 16 -> 4 -> 1 anchored at both measured ends, theta_v = 1 at nu = 16), WP5e-alpha DONE (v502: q^{-1/3} prefactor = exact mu4 vacuum-energy bookkeeping (shift orbifold, discriminant form = Casimir two ways) + k = 1 forced three ways, glue-h integrality honestly demoted), WP5e-beta DONE (v505: the equivariant anomaly ledger exact -- AB characters (248,0,-8,0), per-sector Okubo with the rigid 32 T3 residual, index bridge f(m) = ch2(T_m) with defect -78 two routes, geometric k = 1 dials, a0 refuted as GS axion (graviton slot O(2)), the three sphere classes = the three twisted axion slots; kill not fired), WP5d-beta DONE (v504: split + strong additivity from the lattice -- Even(A) v Even(B) = Even(AuB) EXACT with the shared boundary point (disjoint index exactly 2 = the v501 ln-2 bit), bounded-vs-divergent entropic discriminator (Z2 deficit < ln 2 with the Ising 1/4-exponent approach vs U(1) divergence, Klich-Levitov pinned), elliptic-nome split ladder with exact orbifold inheritance (C -> -C + Lambda^2 C, Longo heredity), Pimsner-Popa lambda = 1/2 = 1/[F:F_even] and lambda_E4 = 1/4 = 1/mu with integer attainment, index consistency exp(Delta) = 2 = 1/lambda_PP two routes -- all three KLM ingredients witnessed on the lattice; continuum uplift stays [O], Xu's theorem cited), WP5e-gamma DONE (v508: the sphere-axion pairing check -- exchange no-go with certificate: invariant vertex spaces exactly dim 2/5 by Weyl nullspace, product theorem kills every exchange image in the T3 direction, rank([M | A_fix]) = 3 with certificate (Phi_T5, Phi_T3, Phi_P) = (0, 32, 72), naturalness dissolves scale-independently, SO(16)/D8 controls; slot bijection untouched, level-kill still not fired), WP5e-epsilon-2 DONE (v509: the CPS level-from-flux dial, verdict B -- the one-level statement is now a THEOREM (lockstep from clock invariance, falsifier Z^4 - Z: 0/24), level = 1 carries the new sector-counter dial #prim((E8)_k) = 1 <=> k = 1, fluxes (64, 60, 64) with one quantum per current, the naive 'level = total flux' killed; the PT/Z4 back-reaction stays [O]), WP5e-delta-2 DONE (v511: the full-tensor ledger -- delta-2 executed via v511 -- the full-tensor collapse is confirmed (innerness), but the cubic d-symbol channel opens T3 with the weaker certificate psi = 64; delta-1 since DECIDED NEGATIVE under the DERIVED measure by v518 (2026-07-22, CELEST.WP5E.DELTA1.01: the derived chiral measure -- blockwise SL(2,Z) covariance solved for, the mu4 multiplier obstruction a CHARACTER (koboundary defects (1,1,1) on all 15 pairs, lambda(gamma) = i^{2B + C/4} on Gamma_1(4)) cancelled exactly by the twisted fibre block f1 f3 = G, NOT by the three sphere axions -- fails all three preregistered testers under both derived solutions chi_4/chi_10, with no (N1, N2) rescue in the positive cone: a genuine KILL on the derived surface, in stated TENSION with the DECLARED completion reading of v516 which delivers the psi = 64 slice and cancels 32 T3 -- both are exact, and which of the two is the true BCOV measure is the named [O]); the pre-v518 state was: the strict holomorphic reading is refuted at the Z2 anchor, the contact term is the modular completion (Harvey-Moore tau-integral) with the forced leading (T5,T3) ratio 4:3 -- the named remaining targets are the tau-integral and the physical justification of the cubic GS term (c_d = 1920; the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513)), WP5e-epsilon-1 DONE (v514: epsilon-1 executed via v514: the O(-2) bulk-axion slot is a construction (equivariant Penrose ledger), lambda-tilde = 6 pinned by three exact ledgers (Okubo / measure cancellation excluding 3 and 12 / flux), and the GH/A3 back-reaction step re-derives the v493 family and the Coxeter clock from geometry -- the quantised BCOV coefficient and the twisted channels stay [O] (M1-M3 preregistered)); WP5e-M1 DONE (v515: M1 executed via v515: the back-reacted Omega_N is closed-form on the A3 twistor family, all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector and clock covariance, and the lens geometry FORCES the source charge 4 = |mu4|; honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden family)), WP5e-M2+M3 DONE (v516/v517: M2+M3 executed via v516/v517 -- the v514 fence M1-M3 is fully worked off: the twisted KS measure (declared completion reading, ch2-weighted) cancels the 32 T3 residual and delivers the psi = 64 slice without the cubic d-channel, and the a0 uplift couples to the centre count on four scales; what stays [O]: the BCOV measure question -- the delta-1 chain itself is DECIDED by v518 (kill under the derived measure, tension with v516 stated) -- and SEAM.EQUIV.TWISTOR.01), the global BCOV quantisation [O] (the single remaining milestone: Costello-Li uplift -- the exchange sub-branch closed by v508, the flux dial executed by v509, the tensor ledger executed by v511, the epsilon-1 axion slot executed by v514, the M1-M3 back-reaction milestones executed by v515/v516/v517; the named remaining target is the BCOV measure question (the delta-1 chain is DECIDED by v518: kill under the derived measure, tension with v516 stated; the cubic GS term stays dissolved)); kill tests evaluated: K1 survived, K3 not fired (scope demoted -- alignment non-selective, compatibility not evidence), K4 lands at the boundary-limit reading; SEAM.EQUIV.01 untouched [O]",
      "dependencies": "CELEST.WP1.01; CELEST.WP2.01; CELEST.WP3.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; CELEST.WP5DB.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.DELTA2.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; CELEST.WP5E.M2.01; CELEST.WP5E.M3.01; CELEST.WP5E.DELTA1.01; SEAM.EQUIV.01; CONTRACT.G.01; GATE.METRIC.01; v92; v125; v128; v168; v180; v214; v492; v493; v495; v496; v497; v498; v500; v501; v502; v504; v505; v508; v509; v511; v514; v515; v516; v517; v518",
      "scripts": [
        "v492_celestial_z4_orbifold.py"
      ],
      "script": "v492_celestial_z4_orbifold.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bittleston-Homans-Sharma JHEP 09 (2023) 008; Costello arXiv:2111.08879; Costello-Paquette-Sharma PRL 130, 061602 (Burns holography); Bittleston-Sharma-Skinner CMP 403 (2023); Strominger PRL 127, 221601 (w(1+infinity)) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 556
    },
    {
      "id": "CELEST.WP2.01",
      "claim": "WP2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 2, verdict B; companion of CELEST.WP1.01/v492): the clock-invariant deformation XY = Z^4 + a0 of the A3 seam orbifold -- geometry, period/monodromy household, and the Bittleston-Homans-Sharma deformed-algebra pattern (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case, fibre XY - Z^2 = -c^2) transferred to k = 4. Sympy exact throughout (47 checks). [E] GEOMETRY: P(iZ) = P(Z) forces a3 = a2 = a1 = 0 SHARPLY and two-sided (prod_k (Z - i^k w) = Z^4 - w^4: mu4-stable root set <=> clock-invariant quartic); XY = Z^4 + a0 smooth iff a0 != 0; disc = 256 a0^3; branch points Z^4 = -a0 = ONE free mu4 orbit in every fibre; cross-ratio 2 / j = 1728 for ALL a0 (binary-quartic invariants I = 12 a0, J = 0 IDENTICALLY) => a0 is a PURE SCALE, the tau = i pillowcase (v168/v214) is FROZEN -- no shape modulus survives clock-invariance. [E] PERIODS / MONODROMY: Pi = t(i-1)*(1, i, i^2) -- all 3 vanishing spheres carry the SAME holomorphic volume sqrt(2) t (lockstep, no relative modulus); the clock acts on H2 as a COXETER ELEMENT of W(A3) (char x^3+x^2+x+1, eigenvalues {i,-1,-i} = the A3 exponents 1,2,3; order = Coxeter number h(A3) = 4 = |mu4| = N_fam + 1); NO invariant cycle (det(A-1) = -4); the surviving deformation direction is the weight-1 chi_1-Fourier diagonal (1, i, i^2), NOT the naive invariant diagonal; the clock IS the Picard-Lefschetz monodromy of the family (a0 -> e^{2 pi i} a0 = the 4-cycle); versal weights (a3,a2,a1,a0) = (1,2,3,0) = the REGULAR mu4 representation, invariant line = a0 alone. [E] DEFORMED ALGEBRA (BHS Z2 -> Z4): mode dictionary {w[m,n]: m = n mod 4} <-> {X^p Y^q Z^r: r <= 3}; flat deformation (counting level); bracket {f,g} = -4*Nambu(XY - Z^4 - a0) ANCHORED at the a0 = 0 orbifold w-sector ({X,Y} = 16 Z^3, {Y,Z} = -4Y, {Z,X} = -4X), Jacobi + Casimir exact, uniform clock-degree -1; S-algebra corrections exactly LINEAR in a0 at the Z4 wrap r+r' >= 4, conserving the mu4 grade -- NO sector leak, so the v492 glue-equivariant SDYM(E8) sector deforms consistently; a0 <-> the BHS c^2 slot (weight 0, the order-c^2 splitting-function correction slot); matrix factorisations (McKay modules) deform along every 1+3 and 2+2 orbit splitting; the clock fixes exactly the antipodal 2+2 sector = the EH/Z2 middle node (4 = 2 x 2). [E] NEGATIVE CONTROLS (doubly sharp): a1 Z => j = 0 (hexagonal CM point, maximally non-square); pure a2 Z^2 stays SINGULAR (A1 node); a2 != 0 => j = 1556068/81 != 1728 (exact instance, cross-ratio and binary-quartic routes agree); grading leak explicit (Z^2 * Z^2 picks up grades {0,1,2} in the variant ring). [C] THREE NAMED IDENTIFICATIONS (why verdict B, not A): (1) the fibrewise -4*Nambu bracket is THE k = 4 CCA bracket (BHS prove k = 2; anchored here at a0 = 0, twistorial derivation not redone); (2) period = root difference (standard A-series residue normalisation); (3) the seam-scale reading of a0 presupposes the WP1 clock-deck bridge (clock^2 = deck, Z8, 8 = 2|mu4|). [C] v216 PRECISION (typed, NO marker move): GIVEN the order-4 clock, the clock-invariant deformation freezes the (2,2,2,2) modulus to the square AUTOMATICALLY (only the scale a0 stays free) -- a RELOCATION of the same order-4 carrier input to the celestial side (the clock IS the input), not a new derivation; v216 keeps its [O]-residual. [O] OPEN: full deformed twisted-sector OPEs; the W(mu)-type scaling-limit identification for k = 4 (BHS announce the ADE case); the spacetime splitting-function check; WP3 (GS coefficient alignment), WP4 (character vs S-algebra grading), WP5 (constructive holography); SEAM.EQUIV.01 (continuum (E8)_1 net on the seam) stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact deformation geometry/period/algebra arithmetic, sympy, negative controls) + [C] (three named identifications, verdict B; v216 precision typed) + [O] (algebra/contract programme; SEAM.EQUIV.01 and v216 unchanged)",
      "canonicalStatus": "WP2 [E]/[C]/[O]: clock-invariance selects XY = Z^4 + a0 sharply; a0 = pure seam scale, tau = i pillowcase frozen (j = 1728 for all a0); clock on H2 = Coxeter element of W(A3) = Picard-Lefschetz monodromy; surviving direction = chi_1-Fourier diagonal, spheres in lockstep; BHS Z2 -> Z4: -4*Nambu bracket anchored at the orbifold, corrections linear in a0, grade-conserving (no sector leak), a0 <-> c^2 slot; negative controls j = 0 / j != 1728 / grading leak; verdict B; SEAM.EQUIV.01 and v216 [O] untouched",
      "dependencies": "CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v92; v125; v128; v168; v179; v180; v214; v216; v492",
      "scripts": [
        "v493_celestial_wp2_clock_deformation.py"
      ],
      "script": "v493_celestial_wp2_clock_deformation.py",
      "location": "tfpt_research_contracts",
      "externalData": "Bittleston-Homans-Sharma JHEP 09 (2023) 008 (arXiv:2305.09451; deformed CCA on Eguchi-Hanson, Z2 case, c^2 corrections) -- method citation, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 557
    },
    {
      "id": "CELEST.WP3.01",
      "claim": "WP3 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 3, verdict B; the pre-registered 'alignment test only'): the Green-Schwarz axion coefficient lambda_g of celestial/twistorial SDYM (Costello, arXiv:2111.08879, eq. 1.2.4: kappa = lambda_g/(8 pi sqrt 3)) against the TFPT seam constant c3 = 1/(8 pi). Exact arithmetic throughout (Fraction/sympy, 25 checks). [E] OKUBO IDENTITY DERIVED, NOT QUOTED: Sum_alpha <alpha,x>^4 = [5/(2(dim g+2))] (Sum_alpha <alpha,x>^2)^2 as an exact polynomial identity for ALL 8 algebras on Costello's list (sl2, sl3, g2, so8, f4, e6, e7, e8; e6/e7 carved out of the v492 E8 root set) -- 'no independent quartic Casimir'; NEGATIVE CONTROL: sl4 (which HAS one) fails the proportionality (5/32 on s0^4 vs 3/32 on s0^2 s2^2) -- the list is sharp. [E] Killing normalisation Sum <alpha,x>^2 = 2 h_vee <x,x> exact (h_vee = 2,3,4,6,9,12,18,30); Deligne dimension formula dim g = 2(5h_vee-6)(h_vee+1)/(h_vee+6); CLOSED FORM lambda~^2 = 10 h_vee^2/(dim+2) = h_vee + 6 EXACTLY (8,9,10,12,15,18,24,36) -- fixes Costello-Paquette eq. (1.5); fundamental Dynkin indices i_R = (1,1,2,2,6,6,12,60) from explicit weight systems. E8: lambda~ = 6 (unit-trace) and lambda_fund = 1/10 (adjoint-trace) EXACT. Cross-checks: Costello's sl2 = 8, sl3 = 9 reproduced; HIS PRINTED lambda^2(so8) = 3/2 is a factor-2 slip (his own weight content gives Sum w^4 = 192, not 96: lambda^2 = 3, Okubo-consistent 12/4). [E] ALIGNMENT: (kappa/c3)^2 = lambda^2/3 = 12 = |mu4| N_fam (unit-trace) resp. 1/300 (adjoint-trace), both EXACT anchor rationals; kappa/c3 ITSELF is IRRATIONAL (2 sqrt 3 resp. 1/(10 sqrt 3)) -- the pre-registered 'lambda/sqrt3 rational?' reading FAILS, only the SQUARED alignment holds; anchor decomposition h_vee = 30 = |Z2| N_fam g_car (all three TFPT atoms once, squarefree), phi(30) = 8 = rank(E8), dim+2 = 250 = 2 * 5^3, lambda~ = 6 = |Z2| N_fam, lambda_fund = 1/10 = 1/(|Z2| g_car). [E] LOOK-ELSEWHERE (the HARD FENCE -- alignment != selection): the alignment test AS SUCH passes 8/8 (ZERO selective power for e8), h_vee {2,3,5}-smoothness 8/8, lambda~-integrality 2/8 (sl3 + e8), lambda_fund-rationality the same 2/8, phi(h_vee) = rank 4/8, phi(h) = rank 5/8; the ONLY e8-selective single test is g_car = 5 | h_vee (1/8); the conjunction that isolates e8 (1/8) was assembled POST HOC from correlated sub-tests -- it quantifies where e8 sits, it is NOT a preregistered discriminator. [E] KILL-TEST K3 EVALUATED: does NOT fire (exact alignment EXISTS) but its scope is DEMOTED -- the c3-connection of the celestial route survives as convention-level compatibility + genuine lambda-arithmetic, NOT as e8-selective evidence. [C] TYPED: the 8 pi in Costello's (1.2.4) is the 2 pi convention of the anomaly polynomial (his A.0.10 constant descended to spacetime), NOT an independent occurrence of the seam constant; the sqrt 3 is the anomaly's own (sqrt 48 = 4 sqrt 3 from the twistor box diagram, 48 = 2*4!); WP3 is an alignment yes/no ONLY -- it can never derive c3 from celestial data (contract wording, honoured). [O] WP5 (constructive holography) and SEAM.EQUIV.01 stay [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact Okubo/Killing/lambda arithmetic for all 8 algebras, Fraction/sympy, sl4 negative control, look-elsewhere battery) + [C] (the convention typing of the 8 pi and sqrt 3; the alignment reading with the look-elsewhere caveat, verdict B) + [O] (celestial programme WP5; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP3 [E]/[C]/[O]: Okubo coefficient 5/(2(dim+2)) derived for all 8 algebras (sl4 control); lambda~^2 = h_vee + 6 closed form; lambda_e8 = 6 resp. 1/10 exact; (kappa/c3)^2 = 12 = |mu4| N_fam resp. 1/300 exact, kappa/c3 irrational; so8 slip flagged; look-elsewhere: alignment 8/8 (zero selectivity), g_car | h_vee 1/8 (e8 only), joint test post hoc; K3 not fired, scope demoted -- compatibility, not e8-selective evidence; verdict B; SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v92; v125; v128; v492; v493",
      "scripts": [
        "v495_celestial_wp3_gs_alignment.py"
      ],
      "script": "v495_celestial_wp3_gs_alignment.py",
      "location": "tfpt_research_contracts",
      "externalData": "Costello arXiv:2111.08879 (twistorial SDYM axionic consistency, GS coupling eq. 1.2.4, appendix A samples); Costello-Paquette JHEP 10 (2022) 193 (arXiv:2201.02595, eq. 1.5) -- method/normalisation citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 558
    },
    {
      "id": "CELEST.WP4.01",
      "claim": "WP4 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 4, verdict B(ii); the 'type mismatch, head-on'): the (E8)_1 character E4/eta^8 against the glue-equivariant E8[C^2] S-algebra. Exact integer/Fraction/sympy arithmetic (25 checks). [E] BASE: v492 replication (240 = 52+64+60+64, dims with Cartan (60,64,60,64)); equivariant tower D[d] = 60(d+1) even / 64(d+1) odd; cumulative generator count QUADRATIC -- 31 s^2 + 92 s + 60 (even) / 31 s^2 + 94 s + 63 (odd) exactly, leading 31 = k + h_vee (arithmetic note); [C] CP Table 1 (arXiv:2201.02595): J[m,n] carries 2d spin 1 - (m+n)/2, UNBOUNDED BELOW along the tower => no rational VOA (L0 bounded below) contains the towers as graded pieces. [E] EQUIVARIANT HILBERT SERIES: chi_sp(t) = (60 + 128t + 60t^2)/(1-t^2)^2 (palindromic numerator, value 248 at t = 1); ALL mu4 collapses w in {1,-1,i,-i} and the SU(2)_+ primary slice (60+64t)/(1-t^2) fail to start (1, 248, ...) -- the 'row sum' route fails at the first coefficient. [E] CHARACTER SIDE: chi = E4/eta^8 = (1, 248, 4124, 34752, 213126, 1057504, 4530744) exact through q^6; machinery cross-check q j(tau) = E4^3/eta^24 = (1, 744, 196884, 21493760); theta split (shell 1 = (52,64,60,64)), sector currents (60,64,60,64), glue-diagonal h = (0,1,1,1) INTEGER (v125 isotropy = locality); Sugawara c = 248/31 = 8 = rank = 2|mu4| = g_car + N_fam. [E] BRIDGES: zero-mode slice D[0] = 60 = the C0-sector current count EXACTLY; the single-line jet slice z1^d cycles (60,64,60,64) sector by sector; BOUNDARY/PERIOD LIMIT: one full mu4 period of loop energies carries dim g_(n mod 4) = (64,60,64,60), sum = 248 EXACTLY -- the single-particle level of the 248q stage; 248 is NOT a jet-tower dimension (d <= 100) -- only the period collapse produces it; BUT the loop Fock at integer level 1 counts 897266 >> 248: the rational truncation (null ideal) is NOT in the jet/loop algebra -- it must come from a limit. [E] MISMATCH LOCALISED THREE WAYS: (a) GRADING (spin unbounded below; 248 no tower dimension); (b) GROWTH: jet Fock f = (1, 128, 8436, 381056) vs chi = (1, 248, 4124, 34752) -- TWO-SIDED failure (f1 < chi1 undercount, f2 > chi2 overcount), f_n/chi_n strictly increasing n = 1..12 (exact cross-multiplication); Meinardus poles s = 2 vs s = 1 => growth exponents n^{2/3} vs n^{1/2}, asymptotically DISJOINT counting types; (c) TRUNCATION: free-current level-2 count 31124 = 248 + Sym^2(248); the character keeps 4124 = 1 + 248 + 3875; the deficit is EXACTLY 27000 = 30^3 = h_vee^3 -- the (E8)_1 null ideal deletes the 27000 in Sym^2(248) = 1 + 3875 + 27000 ([C] standard) with NO analogue in the jet algebra (level 1 agrees: 248 = 248). [E] KILL-TEST K4 EVALUATED, three-way: (i) exact realisation NO; (ii) boundary-limit realisation PARTIAL YES -- current stratum, period sum 248, integer glue weights and theta split hold exactly; the missing piece is precisely the rational truncation, which must come from a scaling/compactification limit -- the SAME structure as the MMST scaling-limit route of SEAM.EQUIV.01 (v336/v449: the rational net exists in the LIMIT of the lattice collar, not in the pre-limit algebra); (iii) the hard mismatch is exactly located; K4 does NOT degrade the contract to 'E8 is an admissible celestial gauge algebra' (the sector arithmetic holds exactly). [C] VERDICT B(ii): the (E8)_1 character is a BOUNDARY/LIMIT SHADOW of the equivariant S-algebra, not a conformal block of it in the bulk (jet) grading -- WP4 sharpens into a statement of SEAM.EQUIV.01 type (rational structure emerges only in a limit). [O] the constructive boundary limit itself (which limit, which topology, convergence) stays OPEN and belongs to WP5/SEAM.EQUIV.01; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact counting/character/theta arithmetic, integer/Fraction/sympy; mismatch localisation with exact cross-multiplication) + [C] (literature gradings CP Table 1 + Sym^2(248) decomposition; the boundary-limit reading, verdict B(ii)) + [O] (the constructive limit -> WP5; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP4 [E]/[C]/[O]: the (E8)_1 character is NOT a conformal block of the E8[C^2] S-algebra in its own jet grading -- mismatch localised three ways (spin unbounded; growth n^{2/3} vs n^{1/2}; level-2 null ideal 27000 = h_vee^3 absent from the jets); BUT the boundary/period reading holds exactly at the current stratum (zero modes 60, jet slice (60,64,60,64), mu4 period sum 248, glue weights (0,1,1,1) integer) -- the MMST scaling-limit shape of SEAM.EQUIV.01; K4 fires only against the exact reading; verdict B(ii); constructive limit -> WP5; SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v125; v128; v336; v377; v449; v492",
      "scripts": [
        "v496_celestial_wp4_character_shadow.py"
      ],
      "script": "v496_celestial_wp4_character_shadow.py",
      "location": "tfpt_research_contracts",
      "externalData": "Costello-Paquette JHEP 10 (2022) 193 (arXiv:2201.02595, Table 1: the celestial S-algebra grading); Bittleston-Homans-Sharma JHEP 09 (2023) 008 (arXiv:2305.09451, equivariant sector); Sym^2(248) = 1 + 3875 + 27000 (standard E8 representation theory) -- method/grading citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 559
    },
    {
      "id": "COSMO.RUNNING.01",
      "claim": "Spectral-index running of the Starobinsky branch, PROMOTED from the 2026-07-20 scorecard prediction-of-record row (same N_star_reheating group as n_s/r/A_s; never an independent hit). [E] LO identities (sympy exact): alpha_s = dn_s/dln k = -2/N_star^2 = -r/6, beta_s = -4/N_star^3 (NOT -8/N^3); LO band N_star in [50,60]: [-8.0e-4, -5.6e-4]. [E] exact slow roll on the Einstein-frame Starobinsky potential (third-order Lyth-Riotto + finite-difference cross-check <1%): alpha_s(51.4) = -7.09e-4, beta_s(51.4) = -2.70e-5; exact band edges -7.48e-4 / -5.23e-4. [C] data confrontation: Planck 2018 TTTEEE+lowE+lensing (-0.0045+/-0.0067) -> +0.57 sigma; record leg P-ACT-LB (+0.0062+/-0.0052) -> -1.33 sigma (watch channel: the central value is POSITIVE); beta_s (Planck run+runrun, 0.010+/-0.013) -> -0.77 sigma. Forecast: LiteBIRD+CMB-S4 sigma(alpha_s) ~ 4.9e-3 (0.14 sigma); CMB-S4+DESI+Euclid statistical ~3.3e-4 (2.2 sigma, systematics-limited). KILL: robust alpha_s < -5e-3 or > +3e-3 at >=5 sigma; any robust POSITIVE running kills the plateau branch. FENCE: N_star stays an external reheating input (GATE.NSTAR.01); the frozen band [50,60] stays the surface of record; no gate closes, no marker moves",
      "status": "Identity [E] (LO relations + exact slow roll) + Physical [C] (data confrontation; N_star input)",
      "canonicalStatus": "[E] LO identities + exact-SR values; [C] pulls +0.57 sigma (Planck) / -1.33 sigma (P-ACT-LB record leg); kill = robust positive running; band [50,60] stays the record",
      "dependencies": "COSMO.INF.01; COSMO.NSTAR.01; GATE.NSTAR.01",
      "scripts": [
        "v494_cosmo_running_mu_record.py"
      ],
      "script": "v494_cosmo_running_mu_record.py",
      "location": "tfpt_1",
      "externalData": "Planck 2018 X; ACT DR6 P-ACT-LB (Calabrese+ 2025 JCAP); LiteBIRD/CMB-S4 forecasts",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 560
    },
    {
      "id": "COSMO.MUDIST.01",
      "claim": "CMB mu-distortion as the A_s/N_star BRANCH DISCRIMINATOR, PROMOTED from the 2026-07-20 scorecard prediction-of-record row (data-limited; shares the Nstar_branch alternative group with the two A_s readings -- one branch question, several readings). [C] Chluba-Erickcek-Ben-Dayan k-space window calibrated +21% on the LCDM Planck-2018 anchor to the literature mu_LCDM ~ 2.0e-8: sharp branch N_star = 51.4 (A_s = 1.76e-9) mu = 1.6e-8; profiled record N_star = 56.1 (A_s = 2.10e-9) mu = 1.94e-8 (~2.0e-8); band N_star in [50,60]: 1.5e-8 .. 2.3e-8; branch split Delta mu = 3.0e-9 (ratio 1.23) => >=3 sigma branch decision at sigma(mu) = 1e-9 (Voyage-2050 class). REACH: the PIXIE baseline 95% limit 9e-8 does NOT reach the band (sharp = 0.18 x limit); FIRAS 9e-5 is 4 dex above -- data-limited until a PIXIE+/Voyage-2050-class mission. [E] COMB BLINDNESS (honest S15 typing): the frozen recovery log-comb (omega = 2 pi/ln((3/2)^6) = 2.5827, eps = 0.0173) is STRUCTURALLY invisible in the mu window -- attenuation |F(omega)| = 0.033, delta mu = 7.4e-12 = 0.007 sigma even at Voyage-2050; the theory itself predicts where it cannot be seen, so NO comb-search row is opened. KILL: robust mu < 0.9e-8 or > 4e-8. FENCE: N_star stays an external input (GATE.NSTAR.01); the frozen band [50,60] stays the surface of record; no gate closes, no marker moves",
      "status": "Physical [C] (Chluba-window bridge, data-limited) + Identity [E] (comb-blindness attenuation)",
      "canonicalStatus": "[C] mu band 1.5-2.3e-8; branch split 3.0e-9 = 3 sigma at Voyage-2050; PIXIE baseline does not reach; [E] frozen comb structurally blind (0.007 sigma); kill = robust mu outside [0.9, 4]e-8",
      "dependencies": "COSMO.INF.01; COSMO.NSTAR.01; GATE.NSTAR.01",
      "scripts": [
        "v494_cosmo_running_mu_record.py"
      ],
      "script": "v494_cosmo_running_mu_record.py",
      "location": "tfpt_2",
      "externalData": "COBE/FIRAS |mu| < 9e-5; PIXIE baseline forecast 9e-8; Voyage-2050 sigma(mu) ~ 1e-9; Chluba-Erickcek-Ben-Dayan 2012 window",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 561
    },
    {
      "id": "CELEST.WP5A.01",
      "claim": "WP5a OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5a, the first constructive milestone of WP5 -- 'the null ideal from the limit'; the v496 boundary-limit shadow made precise): exact integer/Fraction arithmetic throughout (34 checks), no floats. [E] STABILISATION THEOREM: a one-parameter family chi_w of graded Fock characters on the CHIRAL jet generators (loop energy m = p-q >= 1, radial number r = min(p,q); the m = 0, r >= 1 invariant-ring tower carries g_0), generator energy E_w = m + w*r in quarter units (u = q^{1/4}). (i) w = 2 IS the chiral jet grading (E_2 = p + q identically; generator counts 64,120,128,180,192,240,256,300 per degree); (ii) the u^n coefficient of chi_w equals the quarter-moded loop Fock for ALL w >= n+1 and is STRICTLY larger for every w <= n (verified n <= 8, w <= 10) -- the v496 'boundary-limit shadow' is the precise coefficientwise limit epsilon = 1/w -> 0 of a family CONTAINING the jet grading, with explicit threshold w = n+1; (iii) the limit does NOT produce the character (897266 >> 248 at integer level 1): limit and quotient are two separate, honestly typed steps; convention clarified: 897266 = quarter-moded loop Fock at u^4 = INTEGER level 1 (v496 S4.5). [E] NULL IDEAL DERIVED FROM ROOT DATA (not cited): Freudenthal multiplicity machinery (unit-tested on the adjoint: zero-weight mult 8, dim 248 by Weyl = orbit sum), dominant-weight tally of Sym^2(248) (30876 pairs, zero-weight mult 156) and generic character peeling DERIVE Sym^2(248) = V(2 theta) + V(omega) + V(0) = 27000 + 3875 + 1 (peeling residual exactly zero; independent Weyl-orbit cross-check); 27000 = 30^3 = h_vee^3 EXACTLY; C2 = (124, 96, 0) with 124 = C2(27000), Dynkin indices 6750/750 integers; level-1 integrability <2 theta,theta_vee> = 4 > 1 and <omega,theta_vee> = 2 > 1 => the only level-1 primary is the vacuum ([C] Kac); the maximal submodule of the level-1 vacuum Fock is generated by the level-2 singular vector of weight 2 theta ([C] Kac, used only through its counting consequence): level-2 quotient 31124 - 27000 = 4124 = 1 + 248 + 3875 = chi_2 EXACTLY. [E] TWO ROUTES, ONE NUMBER: the mu4 theta-split sector characters Theta_Cj/eta^8 (v492 S4 machinery, the boundary/twisted presentation) summed at q^2 = (1036,1024,1040,1024) give the SAME 4124 -- the untwisted null-ideal quotient and the mu4-sector lattice sum count the same level-2 space; at q^3 the sum is 34752 = chi_3; ideal bookkeeping beyond level 2 typed [C] (ideal series (0,0,27000,2600000,...) nonnegative, strictly increasing through q^12). [E] NEGATIVE CONTROLS: (a) SO(16)_1 through the SAME pipeline (D8 frame): Sym^2(120) = 5304 + 1820 + 135 + 1 derived (FOUR components vs E8's three); 5304 != h_vee(so16)^3 = 14^3 = 2744 -- the h_vee^3 identity is NOT generic; quotient 7380 - 5304 = 2076 = Theta_D8/eta^8 at q^2 (independent lattice route: the pipeline is the correct general level-1 vacuum recipe, validated on a second algebra); BUT the D8 block weights are (0, 1/2, 1, 1) -- h = 1/2 non-integer, the four blocks can NOT fuse into one local character (four-block theory) vs E8's integer (0,1,1,1) fusing into E4/eta^8: the procedure is not trivially projective; (b) false periodisation: period sums over P = 1, 2, 8 quarter-steps give 64, 124, 496 -- only P = 4 = |mu4| gives 248; (c) swapped glue (60/64 exchanged): the zero-mode anchor breaks (D[0] = 64 != 60 = lattice C0 currents). [C] NAMED IDENTIFICATIONS: 'radial decoupling = MMST collar thickness -> 0' (the v336/v449 lattice-collar route has the SAME two-step shape: pre-limit overcounting algebra, rational net only in the limit) is a named identification, not a computed isomorphism; the Kac singular-vector statement enters only through its counting consequence. [O] HONEST LIMIT (= the rest of WP5): the limit does NOT generate the truncation -- the singular vector as an OPERATOR in the limit net stays open (WP5b); the GNS limit state (kernel containing the ideal) and local normality / Haag duality of the limit net stay open (WP5c/WP5d); convergence topology beyond coefficientwise stays open; the celestial route now has the SAME two-step shape as MMST (limit + maximal ideal), with the ideal quantitatively fixed. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact stabilisation/derivation/control arithmetic, integer/Fraction; Freudenthal + Weyl + peeling machinery unit-tested) + [C] (the named identifications: radial decoupling = collar limit; Kac singular vector via counting consequence) + [O] (the operator question WP5b-e; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP5a [E]/[C]/[O]: the boundary limit is a precise coefficientwise limit -- chi_w family (E_w = m + w r, quarter units) contains the chiral jet grading at w = 2 and stabilises to the quarter-moded loop Fock at threshold w = n+1 (exact, n <= 8, w <= 10); the limit does NOT produce the truncation (897266 >> 248) -- but the ideal SIZE is DERIVED from root data (Sym^2(248) = 27000 + 3875 + 1, 27000 = h_vee^3, Freudenthal + Weyl + peeling) and the quotient 31124 - 27000 = 4124 = the independent mu4 theta-split sector sum; SO(16)_1 negative control (four blocks, 5304 != 14^3, h = 1/2 breaks fusion); only P = |mu4| = 4 gives 248; open: singular vector as operator (WP5b), GNS limit state (WP5c), local normality/Haag duality (WP5d), Costello-Li uplift (WP5e); SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; SEAM.EQUIV.01; v125; v128; v336; v377; v449; v492; v496",
      "scripts": [
        "v497_celestial_wp5a_null_ideal.py"
      ],
      "script": "v497_celestial_wp5a_null_ideal.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kac, Infinite dimensional Lie algebras (level-1 integrability + singular-vector generation of the maximal submodule -- used only through counting consequences); Costello-Paquette JHEP 10 (2022) 193 (arXiv:2201.02595, S-algebra grading); Bittleston-Homans-Sharma JHEP 09 (2023) 008 (arXiv:2305.09451, equivariant sector) -- method citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 562
    },
    {
      "id": "CELEST.WP5B.01",
      "claim": "WP5b OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5b, the second constructive milestone of WP5 -- 'the singular vector as an operator'; WP5a/v497 fixed the ideal's size and location, WP5b constructs the deleting object): exact integer/Fraction arithmetic throughout (53 checks), no floats, deterministic. [E] CONSTRUCTION: |s> = (E^theta_{-1})^2|0>, theta the highest root, weight 2 theta (the top of the 27000), level 2 = 8 quarter units = q^2 (an INTEGER level); the Chevalley/Frenkel-Kac basis is MACHINE-built from the v497 root data (bimultiplicative 2-cocycle eps(a,b) = (-1)^(m_a^T B m_b), asymmetry verified on all 57600 ordered root pairs; the [e_a,e_{-a}] = SGN*H(a) sign FORCED by the Jacobi identity, SGN = -1, not chosen; the invariant form kappa DERIVED from invariance, kappa(theta_vee,theta_vee) = 2; Jacobi exact on ALL 57600 sl2-slice triples + 6000 seeded triples + all 496 theta-adjacent triples; all root-sum structure constants N_ab = +-1). [E] SINGULARITY (main theorem, machine): J^a_1|s> = 0 for ALL 248 basis elements, 0 violations; case classification of the J_1 condition: 190 vanish at the first bracket, 57 at the second, exactly 1 (a = F^theta) by the central-term cancellation -- the ONLY case that sees the level k; plus J^a_2|s> = 0 (all 248), E^a_0|s> = 0 (all 120 positive roots), h_i-eigenvalues = <2 theta, a_i_vee> exactly, Shapovalov norm 0 at k = 1 (4 at k = 2); the affine PBW engine ([J^a_m,J^b_n] = [a,b]_{m+n} + m k kappa(a,b) delta_{m+n,0}) unit-tested on all 61504 basis pairs. [E] LEVEL DIAL: the F^theta_1 coefficient is 2(1-k) -- 2 at k = 0, 0 EXACTLY at k = 1, -2 at k = 2: in the centerless loop algebra (k = 0) the deletion object does NOT exist -- the central extension is load-bearing. [E] MU4 DATA (glue frame, v492/v496): theta-grading dims (1,56,134,56,1), g_0 = e7 + u(1); the h-adapted Weyl chamber MACHINE-constructed (8 simple roots re-derived, Cartan det 1, E8 Dynkin diagram with branch lengths [1,2,4]); theta_glue the unique dominant root, height 29 = h_vee - 1, <theta,h> = 5 => glue class j(theta) = 1; clock phase on |s> = i^(2j) = -1; class(2 theta) = 2 sheet-EVEN (consistent with WP5a S6.8); the deleting theta-sl2 crosses the sheet-ODD classes (1,3) while its square is sheet-even; E^theta expressible in the sector-C_1 equivariant variables (spatial degree m = -j mod 4, 2 monomial slots) and the square is mu4-INVARIANT; quarter bookkeeping (v496 per-period dictionary): E^theta_{-1} = 4 quarter units = one full mu4 period (248 currents), |s> = 8 quarters = q^2; cross-table theta-grading x glue class consistent (rows (1,56,126,56,1), columns (52,64,60,64)). [E] LEVEL-2 MATCH: weight 2 theta has multiplicity 1 in the level-2 Fock => U(g)|s> is THE 27000; dim V(2 theta) = 27000 = 30^3 = h_vee^3 (Freudenthal + Weyl-orbit sums, v497 replication); the direct g_0-lowering orbit BFS from |s> reproduces the Freudenthal weight multiplicities of V(2 theta) EXACTLY, weight by weight, through depth 4; quotient 31124 - 27000 = 4124 = theta-split sector sum (1036,1024,1040,1024); LATTICE KERNEL READING: (2theta,2theta)/2 = 4 > 2, so the quotient (E8)_1 theory has NO momentum-2theta state at q^2 -- |s> IS the kernel generator of Fock -> E4/eta^8. [E] NEGATIVE CONTROLS (honest separation): (a) E^theta_{-1}|0> (level 1) NOT singular -- exactly one witness (a = F^theta, -> -k|0>), the 248-current state that must REMAIN (chi_1 = 248); (b) 4 seeded generic level-2 states all non-singular (14-33 witnesses each); (c) at k = 2 the square is not singular but the CUBE (E^theta_{-1})^3|0> IS -- the (E^theta)^{k+1} pattern is GENERIC Kac level mechanics, not E8-specific; (d) D8/SO(16)_1 through the SAME pipeline (Chevalley + Jacobi on all 12544 sl2-slice triples): (E^theta_{-1})^2|0> IS singular there too (the singular-vector mechanism is level-1 GENERIC), BUT dim V(2 theta_D8) = 5304 != 14^3 and D8 keeps 3 extra level-1 primaries (comark-1 weights, h = (1/2,1,1), h = 1/2 NON-integer breaking one-block fusion) vs E8 (comarks all >= 2, vacuum the ONLY level-1 primary, glue weights (0,1,1,1) integer) -- the ONE-BLOCK closure is the E8/mu4-specific part; the singular vector itself is generic level-1 Kac mechanics, honestly separated. [C] NAMED IDENTIFICATIONS: Weyl complete reducibility of the finite-dim level-2 Fock extends the depth-4 orbit agreement to the full dim U(g)|s> = 27000; 'the singular vector generates the maximal submodule' (Kac) used only beyond level 2; the per-period quarter dictionary (v496 S4) for the u-unit bookkeeping. [O] HONEST LIMIT (= the precise WP5c question): in the TWISTED quarter-SLOT moding of the chi_w limit Fock two sector-C_1 modes never sum to 8 quarters (0 solutions, minimum total 6 = q^{3/2}) -- the per-PERIOD dictionary (v496), not the per-slot identification, carries |s> to q^2; exactly the GNS/limit-state seam (kernel containing the ideal) that WP5c must construct; local normality/Haag duality (WP5d) and Costello-Li (WP5e) stay OPEN. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact construction/singularity/control arithmetic, integer/Fraction, deterministic; Chevalley basis + affine PBW engine unit-tested on all 61504 pairs) + [C] (the named identifications: Weyl complete reducibility beyond BFS depth 4; Kac mechanics beyond level 2; per-period quarter dictionary) + [O] (the GNS/limit-state question WP5c-e -- the twisted-slot moding tension is the precise WP5c question; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP5b [E]/[C]/[O]: the deletion operator EXISTS and is explicit -- |s> = (E^theta_{-1})^2|0> in a machine-built Chevalley/Frenkel-Kac basis, J^a_1|s> = 0 on all 248 generators (case tally 190/57/1, exactly one case sees k), level dial 2(1-k): only k = 1 deletes, no deletion object in the centerless loop algebra; mu4-compatible: j(theta) = 1, clock phase i^2 = -1, class(2 theta) = 2 sheet-even, 8 quarters = q^2, expressible in the sector-C_1 equivariant variables (KILL branch not triggered); U(g)|s> = V(2 theta) = 27000 = h_vee^3 (orbit BFS = Freudenthal through depth 4), quotient 4124 = sector sum; controls: the 248-current state and generic states not singular, cube at k = 2 (generic Kac mechanics), D8 contrast (5304 != 14^3, h = 1/2 breaks fusion) -- one-block closure E8/mu4-specific; open: the twisted-slot moding carries |s> only via the per-period dictionary -- the precise WP5c GNS/limit-state question; WP5c-e stay [O]; SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP4.01; SEAM.EQUIV.01; v128; v492; v496; v497",
      "scripts": [
        "v498_celestial_wp5b_singular_vector.py"
      ],
      "script": "v498_celestial_wp5b_singular_vector.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kac, Infinite dimensional Lie algebras (the level-1 singular vector (E^theta_{-1})^{k+1}|0> -- reproduced by machine here at k = 1 and k = 2, cited only beyond level 2); Frenkel-Kac cocycle construction (machine-rebuilt and verified, not cited); Costello-Paquette JHEP 10 (2022) 193 + Bittleston-Homans-Sharma JHEP 09 (2023) 008 (equivariant-sector framing) -- method citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 563
    },
    {
      "id": "CELEST.WP5C.01",
      "claim": "WP5c OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5c, the third constructive milestone of WP5 -- 'the GNS limit state'; WP5a/v497 made the boundary limit precise, WP5b/v498 built the deleting operator, WP5c constructs the STATE whose GNS kernel contains the null ideal): exact integer/Fraction arithmetic throughout (35 checks), no floats, deterministic. [E] CONSTRUCTION: the quasi-free family omega_w -- loop sector (r = 0) = the affine k=1 vacuum n-point functions via the v498 PBW engine and the MACHINE-DETERMINED compact anti-involution theta(e_a) = -e_{-a}, theta(h) = h (the UNIQUE sign that is an anti-automorphism on ALL 61504 basis pairs; contravariance <J^x_{-1} u, v> = <u, J^{theta x}_1 v> exact); radial sector (r >= 1) = oscillator pairings x^{w r} (the exact Gibbs regulator e^{-beta E_w}, kept x-adic; rational sample x = 1/2). [E] CONVERGENCE SHARP: loop entries w-independent; radial entries x^{w r} have x-adic order w r >= w, so omega_w = omega_inf mod x^{N+1} for ALL w >= N+1 (N = 8) and NOT yet at w = N (the r = 1 modes sit at order N exactly) -- the WP5a stabilisation threshold w = n+1 holds IDENTICALLY at the state level; positivity of every omega_w checked at the one-particle level (all 36 radial modes). [E] THE KERNEL (core result, fully machine): level-2 Fock = 31124 monomials in 9361 weight blocks (dim profile 164:1, 37:240, 7:2160, 1:6960); the FULL exact Gram computed block by block: level-1 rank 248 POSITIVE DEFINITE (240 root modes at norm k, Cartan block = the E8 Cartan matrix, det 1); level-2 rank 4124 EXACTLY (the preregistered target check), kernel 27000, every one of the 9361 blocks PSD; rank per Weyl orbit (norm 8/6/4/2/0) = (0,0,1,8,44), and (dim, rank, inertia) CONSTANT along every Weyl orbit (verified on all blocks, no orbit assumption); kernel = V(2 theta) WEIGHT BY WEIGHT (rank_lambda = dim_lambda - mult_lambda(V(2 theta)) for all 9361 weights, Freudenthal cross-check); component route: unique highest-weight vectors at (0, omega, theta, 2theta) with norms (3844, 49, 2, 0); direct ideal checks F_i|s> and F_j F_i|s> norm 0 and orthogonal to their full blocks; GNS graded dims (1, 248, 4124) = E4/eta^8 through q^2. [E] MU4 EQUIVARIANCE: frame map glue -> standard via the UNIQUE E8 diagram isomorphism; omega_w o clock = omega_w for ALL w (nonzero vacuum values only at total weight 0 = clock class 0); the clock unitary DESCENDS to GNS(omega_inf); GNS level-2 rank split by clock class = (1036, 1024, 1040, 1024) = Theta_Cj/eta^8 at q^2, level-1 split (60,64,60,64) -- the WP5a two-routes identity now holds at the STATE level. RESOLUTION OF WP5B S4.10: |s> IS the ZERO VECTOR of GNS(omega_inf) -- nothing needs to 'carry' it to q^2; the twisted two-C1-mode floor (6 quarters = q^{3/2}) concerns only pre-limit objects (GNS has no state at 6 quarters: 6 mod 4 != 0). [E] NEGATIVE CONTROLS: (a) k = 2: <s|s> = +4 (WP5b sign confirmed), all 240 norm-8 and all 6720 norm-6 monomials nonzero, dominant blocks FULL rank -- the GNS kernel contains NO level-2 ideal (the deletion is a k = 1 state-level fact); (b) k = 0: level-1 Gram identically 0 -- no current layer without the central extension; (c) D8/SO(16)_1 same pipeline: level-1 rank 120 PD, FULL level-2 Gram over 7380 monomials in 2705 blocks: rank 2076 = 7380 - 5304, kernel = V(2 theta_D8) weight by weight, PSD, lattice route Theta_D8/eta^8 = (1, 120, 2076) agrees; but h = (0, 1/2, 1, 1): the vacuum GNS is 1 of 4 blocks vs E8 = the FULL character -- ONE-BLOCK completeness of the limit state is E8/mu4-specific; (d) wrong family E'_w = m w + r: level-1 limit rank 0 (the 248 layer ERASED), u^4 survivor content 710955 != 248 -- the kill has teeth; (e) no damping: the w-uniform undamped vacuum keeps the full jet Fock, 897266 != 248; (f) CCR OBSTRUCTION: on a FIXED radial pairing c > 0 every state has omega(b b*) >= c (positivity) -- NO w-uniform state damps the radial modes: the contract's family formulation is NECESSARY, and the family EXISTS => the preregistered KILL is probed and NOT triggered. [C] NAMED IDENTIFICATIONS: the quasi-free/Gaussian extension of the positive one-particle data to the full *-algebra (standard); Kac positivity beyond level 2 (levels <= 2 machine-checked here); complete reducibility / multiplicity-freeness in the component route (the rank is [E], the module identification is the [C] part); k = 2 non-dominant blocks by Weyl-orbit rank constancy. [O] HONEST LIMIT: local normality of omega_inf, the net structure / Haag duality (WP5d), Costello-Li (WP5e), and the operator-algebraic closure of the radial contraction beyond the graded/x-adic topology stay OPEN. VERDICT: SUCCESS on the preregistered criteria at the graded-state level (target check dim GNS level-2 = 4124 met exactly). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact construction/Gram/control arithmetic, integer/Fraction, deterministic; the compact anti-involution machine-determined; the full 9361-block level-2 Gram exact) + [C] (the named identifications: quasi-free extension; Kac positivity beyond level 2; module identification in the component route) + [O] (local normality, net structure/Haag duality, Costello-Li -- WP5d/e; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP5c [E]/[C]/[O]: the GNS limit state EXISTS and is explicit -- the quasi-free family omega_w (affine k=1 vacuum + x^{wr}-contracted radial oscillators; compact adjoint machine-determined on all 61504 pairs) is positive for every finite w, stabilises exactly at the WP5a threshold w = N+1, and its limit has GNS levels (1, 248, 4124): full 9361-block exact Gram, level-2 rank 4124 = chi_2, kernel 27000 = V(2 theta) weight by weight, every block PSD, rank table (0,0,1,8,44); the 248 layer survives (level-1 rank 248 PD); the clock descends to GNS with rank split (1036,1024,1040,1024) = the per-period theta-split; |s> is the GNS zero vector (WP5b S4.10 resolved at the state level); CCR obstruction: no w-uniform damping state -- the family is necessary AND exists (KILL not triggered); controls: k = 2 keeps everything, k = 0 has no current layer, D8 = one block of four, wrong family/no damping caught (710955/897266 != 248); open: local normality, net/Haag duality (WP5d), Costello-Li (WP5e); SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP5B.01; SEAM.EQUIV.01; v128; v492; v496; v497; v498",
      "scripts": [
        "v500_celestial_wp5c_gns_limit_state.py"
      ],
      "script": "v500_celestial_wp5c_gns_limit_state.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kac, Infinite dimensional Lie algebras (positivity of the contravariant form beyond level 2 -- levels <= 2 machine-checked here); Bratteli-Robinson (quasi-free states, standard extension); Costello-Paquette JHEP 10 (2022) 193 + Bittleston-Homans-Sharma JHEP 09 (2023) 008 (equivariant-sector framing) -- method citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 564
    },
    {
      "id": "CELEST.WP5D.01",
      "claim": "WP5d-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5d, alpha stage -- 'the two-interval index from the lattice'; the local-net question): numpy/scipy Gaussian machinery + exact Fractions (39 checks). Roadmap discriminator: (E8)_1 => mu = 1 (Haag-dual two-interval net, ONE sector); SO(16)_1 => mu = 4; preregistered KILL = 'the two-interval mu-offset does NOT vanish after the condensation'. [E-machine]/[L] SETUP: the v367 seam edge as 16 chiral Majorana layers (c_- = 8 = g_car + N_fam, v489), faithful 1d reduction = the critical Majorana ring (v450 style, NS closure, c = 1/2 per layer); Peschel-exact; three two-interval prescriptions (fermionic rho / orbifold sector sum {1, P_A, P_B, P_AB} / dual algebra {1, P_AB}); the superselection identity [rho, P_AB] = 0 machine-exact; EVERY Gaussian formula (covariance round trip, Peschel entropies, trace formula, interval parity = Pfaffian, disjoint Renyi-2 deficit) validated against exact diagonalisation to <= 1.5e-15. [L/N] FERMIONIC MI EXTENSIVE (the mu = 1 reference): I(x) = -(c/3) ln(1-x) with c fitted 0.5000; max |F(x)| = 5.97e-5 at N = 512 over the full x grid, |F| falling monotonically to 1.3e-6 at N = 1024 -- the lattice witness of Casini-Huerta extensivity. [L/N] THE MU-OFFSET (core): Renyi-2 exactly Delta_2 = -ln[(1+R)/2] with R = Tr(rho P rho P)/Tr rho^2; the 16-layer seam carrier (diagonal Z2) saturates ln 2 to MACHINE PRECISION (|Delta_2 - ln 2| = 1.1e-15 at N = 512, 0 at N = 1024/2048; single-layer R falls as N^-0.500, Ising disorder decay); ED (N = 12..24) confirms vN and Renyi-3/4 deficits track it (not a Renyi-2 artefact); index reading [F : F_even] = exp(Delta_inf) = 2.000000 => mu_gauged = exp(2 Delta_inf) = 4.000000 = v490's mu_pre (TWO independent lattice witnesses: parity census there, entropic two-interval offset here); gauging every layer separately costs 16 ln 2 -- the offset measures ln[index of the subnet], not a universal blob. [L/N] HAAG-DUALITY INDICATOR: fermionic complementarity S(E) = S(E') to 8.2e-12 (graded-duality proxy); the orbifold BREAKS it (D(E) - D(E') = 0.427 at a = 64 -- the direct lattice witness that Even(E)-prime != Even(E-prime)); the complementary-pair sum is exchange-symmetric (dev 4.3e-14) and bounded by ln 4 = ln mu(SO(16)_1), max 1.2637 at the self-dual x = 1/2, -> ln 2 at the squeezed ends: the ln 4 KLM budget is a DOUBLE-LIMIT statement (sum of the two one-sided ln 2 saturations), NOT a fixed-x identity -- the first naive fixed-x check was WRONG and was replaced by the honest budget + asymmetry pair; the dual-algebra prescription {1, P_AB} is machine-identically defect-free. [E] CONDENSATION ARITHMETIC, LATTICE-ANCHORED: det Cartan(D5) * det Cartan(A3) = 4 * 4 = 16 = mu(carrier); KLM/Longo-Rehren quotients mu/|L|^2: 16/4^2 = 1 (one-step mu4 glue) and 16/2^2 = 4 -> 4/2^2 = 1 (via SO(16)_1), both routes exact Fractions; Sigma d_i^2 cross-checks (Ising 1+1+2 = 4, SO(16)_1 four abelian sectors = 4, (E8)_1 one sector = 1); theta_v = e^{2 pi i nu/16} = 1 EXACTLY at nu = 2 c_- = 16 (bosonic, condensable, v490), rivals nu = 1, 2, 8 give theta_v != 1. THE ANCHORED CHAIN: mu = 1 (graded fermion, MEASURED offset 0) -> 4 (gauged, MEASURED ln 2 offset + v490 parity census) -> 1 (theta_v = 1 extension, ARITHMETIC [C]: the condensed (E8)_1 net is interacting, not free-fermion computable -- v490's honest fence): the lattice pins BOTH measurable ends, the arithmetic closes the middle. [E/N] NEGATIVE CONTROLS: (a) nu = 1 single Majorana: offset non-trivial (Delta_2 = 0.6364 at N = 2048, rising toward ln 2) and theta_v = e^{i pi/8} != 1 -- NO bosonic condensation channel, the offset NOT removable: the KILL discriminator has teeth; (b) trivial dimerised phase: c = 0.00000, MI <= 1.3e-7, 16-layer deficit <= 2.6e-8 -- the ln 2 is a property of the CRITICAL seam edge, not of the prescription; (c) wrong sector sum ({1, P_AB} only, 2 of 4 structures): offset EXACTLY 0 instead of ln 2 -- measurably wrong bookkeeping, as preregistered. [C] the continuum dictionary 'offset = ln index' (KLM two-interval index; Longo-Xu; Casini-Huerta-Magan-Pontello entropic order parameters) and the condensation step itself (theta_v = 1 simple-current extension). [O] continuum Haag duality, local normality, split/strong additivity of the quotient net (WP5d-beta, running as exploration), Costello-Li (WP5e). VERDICT: the mu = 1 indicator chain PASSES; the preregistered KILL is NOT triggered. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal [E] (the exact arithmetic: Cartan dets, KLM/Longo-Rehren quotients, Sigma d^2, theta_v at nu = 16, machine identities) + Lattice/Numerical [L]/[N] (the Gaussian lattice witnesses, ED-validated to 1e-15: extensivity, the ln 2 plateau, the duality-asymmetry witness) + [C] (the continuum dictionary offset = ln index -- KLM/Longo-Xu/CHMP -- and the interacting condensation step) + [O] (continuum Haag duality / local normality / split of the quotient net = WP5d-beta; Costello-Li = WP5e; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP5d-alpha [E]/[L]/[N]/[C]/[O]: the two-interval mu-index measured entropically on the seam lattice edge -- fermionic MI extensive (mu = 1 reference, c fit 0.5000, |F| -> 0), the sector-summed orbifold pays exactly one classical bit (16-layer carrier ln 2 plateau to 1.1e-15 at N = 512), [F : F_even] = 2, mu_gauged = 4 = the v490 census (two independent lattice witnesses); complementary-pair budget <= ln 4 (double-limit statement) + the orbifold duality-failure witness; condensation arithmetic anchored at both measured ends: 16 -> 4 -> 1 (KLM/Longo-Rehren, Sigma d^2, theta_v = 1 exactly at nu = 16, rivals != 1) => mu = 1 after condensation, preregistered KILL not triggered; controls: nu = 1 non-removable, trivial phase zero, wrong sector sum loses the full ln 2; open: WP5d-beta (continuum Haag duality / local normality / split, exploration running) + WP5e (Costello-Li); SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5C.01; SEAM.CCC.01; SEAM.MU.01; SEAM.EQUIV.01; v367; v378; v450; v489; v490",
      "scripts": [
        "v501_celestial_wp5d_two_interval_index.py"
      ],
      "script": "v501_celestial_wp5d_two_interval_index.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kawahigashi-Longo-Mueger CMP 219 (2001) (two-interval subfactor index mu = Sigma d_i^2); Longo-Xu (multi-interval subfactors); Casini-Huerta-Magan-Pontello (entropic order parameters, the (1/2) ln mu defect); Peschel J. Phys. A 36 (2003) (Gaussian covariance); Calabrese-Cardy (interval entropy) -- method citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 565
    },
    {
      "id": "CELEST.WP5E.ALPHA.01",
      "claim": "WP5e-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, alpha stage -- 'prefactor and level bookkeeping', the CFT-side anchor of the BCOV/Costello-Li twistor uplift): exact sympy/Fraction arithmetic, no floats (33 checks). Roadmap target: boundary partition function = E4/eta^8 INCLUDING the q^{-1/3} prefactor; preregistered KILL = 'the anomaly inflow demands a level != 1'. [E] PART A -- THE PREFACTOR IS EXACT mu4 VACUUM-ENERGY BOOKKEEPING: Hurwitz-zeta derivation (zeta(-1,theta) = -B2(theta)/2 symbolic; E_b(theta) = -1/24 + theta(1-theta)/4 exact; fermion counterpart E_f = -E_b, Ising h_sigma = 1/16 per Majorana); the clock is INNER -- h = (2,2,2,2,2;0^4) and the A3 detector h' = (0^5;1,1,1,-3) BOTH read the glue class mod 4 on all 240 roots ((h,h) = 20, (h',h') = 12, sum 32 = 2*4^2) -- so ALL 8 Cartan/torus-boson directions sit at clock eigenvalue 1: twist distribution theta = 0^8 in EVERY sector C_j, a SHIFT orbifold (lattice cosets, integer-moded oscillators), NOT a rotation orbifold; E_osc(C_j) = 8*(-1/24) = -1/3 = -c/24 SECTOR-INDEPENDENT (c = 8 = rank = Sugawara 248/31 -- the central-charge route and the boson-count route agree); sector ground exponents -1/3 + (0,1,1,1). [E] DISCRIMINANT FORM = CASIMIR ENERGY, TWO ROUTES: coset minima h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8), glue diagonal (0,1,1,1) INTEGER; route 1 = SPECTRAL FLOW (s_j = j h/4 costs j^2(h,h)/32: 5j^2/8, 3j^2/8, sum j^2 = 0 mod 1 = the isotropy/Lagrangian condition); route 2 = FREE FERMIONS, exact not just mod 1 (D5 = 10 Majorana -> 5/8, A3 = D3 = 6 -> 3/8, glue diagonal = 16 = 2^(g_car-1) = the WP5d seam carrier -> 1; R-NS shift 1/16 per Majorana); HONEST SHARPENING (i): the ROTATION reading fails -- the geometric A3 deck diag(i,i^-1) gives twisted Casimir 3/16 != 3/8 (the v492 'clock != deck' lesson at the Casimir level, factor 2). [E] CHARACTER RECONSTRUCTION: all glue-coset norms even (0 odd cross-terms: isotropic diagonal <=> even extended lattice = E8); sum of the four Theta_Cj = E4 = (1,240,2160,6720); sector characters have leading exponents -1/3 + h_j, leading coefficients (1,64,60,64), currents (60,64,60,64), and sum EXACTLY to E4/eta^8 = q^{-1/3}(1,248,4124,34752). [E] PART B -- k = 1 FORCED THREE INDEPENDENT WAYS: (i) the CURRENT CONDITION -- simple-current weights are exact closed forms (D5: h(k om_v) = k/2, h(k om_{s,c}) = 5k/8; A3: h(k om_j) = k j(4-j)/8), the Z4 extension closes onto the E8 ADJOINT (60+64+60+64 = 248 spin-1 currents) only if h(J) = 1, and h(J; k) = k => k = 1 uniquely; HONEST SHARPENING (ii): glue-h INTEGRALITY h(J^j; k) = k(0,1,1,1) holds for ALL k = 1..8 -- the originally conjectured integrality route fixes NOTHING (the naive dial is dead; the three other dials carry); (ii) CONFORMAL EMBEDDING: D5+A3 in E8 reduces to 47k^2 + 219k - 266 = 47(k-1)(k+266/47) = 0 (unique positive root k = 1), D8 in E8 to 128k(1-k) = 0 (128 = spinor dim); (iii) CENTRAL CHARGE = THE PREFACTOR ITSELF: 248k/(k+30) = 8 <=> 240(k-1) = 0 (240 = #roots), -c/24 = -1/3 only at k = 1; PLUS the WP5b singular-vector dial: PBW engine ||(E^theta_{-1})^m 0||^2 = m! k(k-1)...(k-m+1), singular vector EXACTLY at grade k+1, Weyl dims from root data (248, 3875, 27000; Sym^2 248 = 30876), k = 1 deletes V(2 theta): 31124 - 27000 = 4124 = chi_2 (singular-vector consistency); k = 2: <s|s> = +4 = the WP5b/WP5c value. [E] NEGATIVE CONTROLS: (a) D8/SO(16)_1 -- same rank 8, same c = 8, SAME q^{-1/3} prefactor, but coset weights (0,1/2,1,1): the prefactor does NOT discriminate E8 vs SO(16), the h-INTEGRALITY of the glue diagonal does; Theta_D8 + Theta_s = E4 (the {1,s} Lagrangian extension IS E8); (b) WRONG TWIST: Z2 rotation gives E = +1/6 (h = 1/2 -- no mu4-shift sector there); a mu4 rotation makes E_j = (-1/3, 1/24, 1/6, 1/24) sector-DEPENDENT (the common prefactor and the single eta^8 break); (c) WRONG k in {2,3,4}: ALL FIVE dials fail coherently (h(J) = k != 1; embedding residuals 360/814/1362; c != 8; prefactor != q^{-1/3}; <s|s> = 4/12/24 so 27000 survives, level-2 dim stays 31124 != 4124); at k = 1 all five pass. [C] Kac generation of the maximal submodule by the grade-(k+1) singular vector for k >= 3 (k = 1, 2 machine-anchored by WP5b/WP5c). [O] HONEST FENCE -- WP5e is NOT closed: the BCOV/Costello-Li construction itself (the boundary partition function from the twistor side, the actual anomaly-inflow computation on PT, Costello's one-loop axion coupling lambda_g/(8 pi sqrt 3), the Costello-Paquette-Sharma level-from-flux mechanism) stays OPEN; v495 anchors the Lie-algebra arithmetic of the axion coefficient only; the KILL branch is NOT triggered on the CFT side -- all computable dials say k = 1 -- but its twistor side is untested. v502 anchors the CFT side of WP5e (alpha stage). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact prefactor/level arithmetic: Hurwitz-zeta vacuum energies, inner/shift twist distribution, discriminant-form = Casimir two-route identity, current/embedding/central-charge k-fixing, singular-vector dial -- sympy/Fraction exact, Wolfram-mirrored) + [C] (Kac generation for k >= 3) + [O] (the BCOV/Costello-Li twistor inflow = WP5e proper; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP5e-alpha [E]/[C]/[O]: the q^{-1/3} prefactor of E4/eta^8 derived as exact mu4 vacuum-energy bookkeeping (inner clock => shift orbifold theta = 0^8, common -c/24 = -1/3 at c = 8; discriminant form = Casimir via spectral flow AND 16-Majorana free fermions; rotation reading fails 3/16 != 3/8) and k = 1 forced three independent ways (current condition h(J) = k = 1; conformal embedding 47(k-1)(k+266/47) = 0 resp. 128(k-1) = 0; c = 8 <=> 240(k-1) = 0; plus the WP5b singular-vector dial 31124 - 27000 = 4124) with the honest sharpening that glue-h integrality holds for ALL k and fixes nothing; controls: D8 same prefactor (discriminator = h-integrality), rotation twists break the common prefactor, wrong k fails all five dials; the BCOV/twistor inflow itself stays [O] -- WP5e not closed, SEAM.EQUIV.01 untouched",
      "dependencies": "CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; SEAM.EQUIV.01; v492; v496; v497; v498; v500; v501",
      "scripts": [
        "v502_celestial_wp5e_prefactor_level.py"
      ],
      "script": "v502_celestial_wp5e_prefactor_level.py",
      "location": "tfpt_research_contracts",
      "externalData": "Costello arXiv:2111.08879 (one-loop axion coefficient); Costello-Paquette-Sharma PRL 130, 061602 (level-from-flux); BCOV / Costello-Li (framing of the open twistor target) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 566
    },
    {
      "id": "QGEO.EMERGE.LIGHT.01",
      "claim": "THE v215 DEFERRED EMERGENCE TEST (LEVER 7) EXECUTED LIGHT + RESIDUAL CONVERGENCE QGEO-R1 == P2-R1 (2026-07-21; analysis/kill-test class, v347/v264 style; NO marker moves -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom): v215 lever 7 said the decisive (deferred) kill of the bedrock is to compute the raw P1 seam-collar DtN WITHOUT inputting the marks and check whether four SQUARE marks EMERGE; v216 already derives mark COUNT and EQUALITY from topology, so the genuinely open part is the MODULUS. The light version (17 checks) returns: RIGIDITY, NOT DYNAMICAL SELECTION. [N] NO DYNAMICAL SELECTION (zeta-log-det via the Dedekind-eta formula, mpmath 40 digits): logF = log(tau2 |eta|^4) is exactly S-invariant (dev < 1e-35), so t = 1 is AUTOMATICALLY critical on the rectangular family tau = it; there g'(1) = 0 and g''(1) = -1.2982113 < 0 (strict local maximum of det') BUT the tau1-direction has h''(0) = +0.29821132 > 0: tau = i is a SADDLE of the full moduli space; the critical points of logF are exactly the CM points (E2*(i) = 0, E2*(rho) = 4.7e-42, generic |E2*| = 0.163 != 0); the GLOBAL det' winner at fixed area is the HEXAGONAL torus: logF(rho) = -1.033519276 > logF(i) = -1.054688281 (diff 0.021169; Osgood-Phillips-Sarnak 1988) -- free-field dynamics alone does NOT select tau = i on the full (2,2,2,2) moduli space; criticality at the square is Palais symmetric criticality (the independent heat trace is t -> 1/t invariant and equally critical), not dynamics. [N] SHARP RIGIDITY (v280 lattice DtN machinery, N = 8): the mod-4 off-character indicator of the raw free-field DtN (the v215-K3 lever as a MODULUS FUNCTION) vanishes to 8.0e-15 EXACTLY at tau = i and increases strictly monotonically in both directions (min 8.4e-2 for every t != 1): an isolated zero, not a flat valley; R^2 = Z2 exactly as permutation matrices and R^4 = 1 (the v492 Z8 bridge on the lattice); j(it) >= 1728 with equality ONLY at t = 1, j(rho) = 0 -- only j = 1728 carries a Z4 automorphism (v214/v168). [E] THE Z2 CHAIN IS MODULUS-BLIND: the involution z -> -z has exactly 4 fixed points (2-torsion = the 4 marks) and [Z2, Delta_t] = 0 on EVERY torus -- v216 confirmed without inputting marks; the pillowcase det' is EXACTLY half the torus det' (+- pair structure, ratio 1.000000000000, dim(even) = (N^2+4)/2), so zeta_pillow = zeta_torus/2 and all tau conclusions transfer verbatim; [Z2, H_t] = 0 for all t (max 3.9e-15) -- marks, mark-locality and deck^2-invariance (v264/v280) exist on EVERY pillowcase; ONLY the order-4 clock distinguishes tau = i; Aut(E_rho) = Z6 (element orders 1,2,3,6) contains NO order-4 element: the dynamically selected hexagonal point CANNOT carry the mu4 clock -- dynamics and clock-symmetry are logically INDEPENDENT selectors; GIVEN the clock, P(iZ) = P(Z) forces a3 = a2 = a1 = 0 and the surviving family Z^4 + a0 has j = 1728 identically (v493 S1 mini): the modulus is NOT a second input. [C/O] RESIDUAL CONVERGENCE (the core finding): the order-4 element acting on H^1 is a COXETER element of W(A3) (char poly lam^3 + lam^2 + lam + 1, order h(A3) = 4 = |mu4| = N_fam + 1) -- exactly the P2 cusp-class/carrier datum of the P2 hardening (v499 R1: generation space = H^1(P^1 minus mu4) with the cusp connection): QGEO-R1 == P2-R1, ONE COMMON RESIDUAL -- 'why does the raw seam carry the order-4 clock' -- not two scattered ones. Chain: (i) mark count 4 derived (v216 [E]); (ii) square derived GIVEN the clock (v493 [E]); (iii) clock = carrier input [C]; (iv) rest = the dynamical realisation of the clock [O]. Route classification (v347 analogue): B-rigidity in the sharpened form 'RP seam data => order-4 clock' is the live route; free dynamics alone does NOT deliver the clock (the hexagonal finding). LEVER UPDATE (documented here; v215 itself unchanged): v215 lever 7 is lifted from 'deferred, not executable' to 'light version EXECUTED, result: rigidity instead of dynamical selection' -- marks emerge, the square does NOT emerge dynamically, it is pinned by the clock, and the clock is the input. No gate closes, no marker moves; QGEO.REALIZE.01 stays [C]/[O].",
      "status": "Formal [E] (exact structure: pillowcase = half torus pair structure, permutation identities R^2 = Z2, clock-forcing a3 = a2 = a1 = 0 with j = 1728 identically, Coxeter charpoly) + Numerical [N] (eta/E2*/j at 40 digits; v280-style lattice DtN curves) + [C] (clock = carrier input) + [O] (the dynamical realisation of the clock); NO marker moves -- QGEO.REALIZE.01 stays [C]/[O]",
      "canonicalStatus": "Emergence test light [E]/[N]/[C]/[O]: no dynamical selection of the square (global det' winner = hexagonal, tau = i a saddle, criticality = Palais symmetry); the square is pinned by rigidity alone (DtN mod-4 indicator = isolated zero at tau = i; j = 1728 iff t = 1; Z2 chain modulus-blind); residual convergence: QGEO-R1 == P2-R1 = ONE common rest, the order-4 clock (Coxeter of W(A3) = the v499 cusp-class carrier datum); v215 lever 7 lifted from deferred to light-executed -- marker unchanged, QGEO.REALIZE.01 stays [C]/[O]",
      "dependencies": "QGEO.KILL.01; QGEO.REALIZE.01; QGEO.RIGID.01; P2.TYPING.01; CELEST.WP1.01; CELEST.WP2.01; v215; v216; v264; v280; v347; v492; v493; v499",
      "scripts": [
        "v503_qgeo_emergence_light.py"
      ],
      "script": "v503_qgeo_emergence_light.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Osgood-Phillips-Sarnak J. Funct. Anal. 80 (1988) 148 (extremals of determinants of Laplacians; hexagonal maximiser) -- literature anchor, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 567
    },
    {
      "id": "CELEST.WP5DB.01",
      "claim": "WP5d-BETA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5d, beta stage -- 'split + strong additivity from the lattice'; the two remaining Kawahigashi-Longo-Mueger legs of complete rationality = split property + strong additivity + finite two-interval mu-index, the index being WP5d-alpha/v501): numpy/scipy Gaussian machinery + exact GF2/integer algebra (37 checks, deterministic, ~12 s); the U(1)/Dirac number-conserving subnet is the preregistered negative control -- the current-net analogue where strong additivity famously FAILS in the continuum (Buchholz-Schulz-Mirbach). [E] STRONG ADDITIVITY, ALGEBRAICALLY EXACT: 'touching' = shared boundary Majorana => Even(A) v Even(B) = Even(A u B) EXACTLY (GF2 spans full: 64/64, 256/256, 512/512, 1024/1024 on four sizes; literal matrix rank 32/32); DISJOINT intervals give exactly HALF (rank 16/32, index 2 -- the missing sector is odd(x)odd, the single ln-2 bit of v501, localised at the split point); the U(1)/neutral algebras do NOT generate the union even WITH the shared site (gaps 2/10/52, GROWING -- the charge-transfer tower is missing): the exact lattice shadow of the current-net strong-additivity failure (infinite index). [N] ENTROPIC WITNESS: the fermionic touching MI diverges with exact (c/3) ln 2 increments per doubling (0.115525, dev 1.2e-6 -- pure UV point-gap, present for EVERY net, hence NOT the discriminator); the honest discriminator is BOUNDED vs DIVERGENT sector-sum deficit: the Z2/orbifold deficit rises 0.4045 -> 0.5899 (N = 64 -> 4096), stays < ln 2, with (ln 2 - Delta_2) ~ N^(-p), p = 0.2444 vs 2 Delta_mu = 1/4 (the Ising disorder-operator dimension h_mu = 1/16 per chirality); split-point scan exactly reflection symmetric (3.9e-14); ED vN deficits same direction; gap continuity to the v501 x -> 1 endpoint. HONEST NOTE: the naive 'defect -> 0' expectation is FALSE at the sharp lattice split -- bounded <=> finite index (Longo-Xu) is the correct statement. U(1) CONTRAST: D_U1(touch) BURSTS the finite-group ceiling ln 2 from L = 128 on and keeps growing as (1/2) ln Var Q_A with Var Q_A = (1/pi^2) ln L + const (Klich-Levitov slope 0.10134 vs 1/pi^2 = 0.10132): infinite index, no saturation. [N] SPLIT PROPERTY (nuclearity witness): the singular values of the off-diagonal covariance block for DISJOINT intervals fall exponentially in k with rate = the elliptic nome pi K(1-x)/K(x) to 1.3-2.0% (d = 16/64/256/512 at N = 4096: rates 2.29/3.24/4.82/5.81); sigma_1 ~ x^0.5044 vs 1/2 (the dimension-1/2 Majorana two-point law); trace norm summable and falling (Buchholz-Wichmann nuclearity proxy); the dimerised control decays exponentially per site (a gapped seam splits trivially). [E] ORBIFOLD INHERITANCE, EXACT AT MACHINE LEVEL: P_A maps the coupling block C -> -C (sigma_k identical, 8.3e-17) and the even-bilinear coupling Gram is the second compound Lambda^2 C with singular values {sigma_a sigma_b} (2.8e-14) -- the SAME ladder at doubled rate; the orbifold Renyi-2 MI is < 3% of the fermionic one and falls with x (split heredity under finite index, Longo, [C]). [E] PIMSNER-POPA / LONGO-REHREN, EXACT: E(a) - a/2 = PaP/2 holds IDENTICALLY (the PP inequality with lambda = 1/2 = 1/[F : F_even], one line, exact); ATTAINMENT in exact integers: all 16384 (even, odd) monomial pairs at N = 8 Majoranas -- 8192 extremal with M^2 = 4, Tr M = 0, PMP = -M exactly, 0 violations; all 2048 hermitianised odd monomials at N = 12 (2048/2048); random pencil sweep min lambda = 0.5000000000; the two-interval group {1, P_A, P_B, P_AB} gives lambda = 1/4 = 1/mu exactly; INDEX CONSISTENCY, TWO ROUTES: exp(Delta_inf) = 2.000000000 = 1/lambda_PP (the v501 entropic route, |Delta - ln 2| = 1.1e-15 at N = 512) and exp(2 Delta_inf) = 4 = 1/lambda_E4 = mu -- the Longo-Rehren index witnessed entropically AND operator-algebraically, independently; U(1) control: lambda = 1/(m+1) -> 0 (infinite index, the operator-algebraic twin of the entropic divergence). [E/N] NEGATIVE CONTROLS + FINITE-SIZE HONESTY: dimerised/trivial phase -- every indicator trivial (touching deficit N-independent to 3.6e-15, MI bounded, sigma_1 exponential); U(1) small-L honesty (non-monotone below L ~ 16, documented); Z2 finite-size honesty: at N = 4096 the touching deficit still sits 14.9% below ln 2 (the N^{-1/4} Ising approach is SLOW -- the verdict rests on the bound + fitted exponent, stated as witness, not proof); wrong prescription {1, P_AB} on the touching pair: deficit EXACTLY 0 (machine identity) -- the bookkeeping is falsifiable. [C] the lattice -> continuum KLM dictionary: bounded deficit = finite index = complete rationality (Longo-Xu); split heredity under finite index (Longo); the entropic order parameters (CHMP); the nome ladder = the free-fermion two-interval modular package. [O] the continuum proof: the theorem 'finite-group orbifolds of completely rational nets are completely rational (hence split + strongly additive)' is XU'S THEOREM (algebraic orbifold CFTs) -- CITED, NOT CLAIMED; the continuum proof for THE seam quotient net itself and for the interacting condensed (E8)_1 net stays open (WP5e / Costello-Li); SEAM.EQUIV.01 stays [O], untouched; no marker moves. VERDICT: with v501, all THREE KLM ingredients of complete rationality -- split, strong additivity, finite mu -- are witnessed on the lattice for the orbifold prescription; the U(1) control fails both new legs exactly as the continuum current net does.",
      "status": "Formal [E] (the exact algebra spans/indices, the PP identities with integer attainment, the Lambda^2-compound and parity machine identities, the index consistency) + Lattice/Numerical [L]/[N] (the deficit tables, the nome ladder, the power laws, the Klich-Levitov slope; Gaussian-exact, ED-validated) + [C] (the KLM dictionary: Longo-Xu bounded-deficit dichotomy, Longo split heredity, CHMP order parameters) + [O] (the continuum proof for the seam quotient net and the condensed (E8)_1 net -- Xu's orbifold theorem cited, not claimed; WP5e/Costello-Li; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP5d-beta [E]/[L]/[N]/[C]/[O]: split + strong additivity witnessed on the seam lattice edge for the orbifold prescription -- strong additivity EXACT with the shared boundary Majorana (GF2 spans full, rank 32/32; disjoint index exactly 2 = the v501 ln-2 bit), the entropic touching defect BOUNDED < ln 2 with the Ising 1/4-exponent approach (p = 0.2444) while the preregistered U(1) control diverges ((1/2) ln Var Q, Klich-Levitov pinned): bounded-vs-divergent is the lattice discriminator; split via the elliptic-nome sigma_k ladder (1.3-2.0%, sigma_1 ~ x^{1/2}, trace norm summable) with EXACT orbifold inheritance (C -> -C + Lambda^2 C compound, Longo heredity); Pimsner-Popa E(a) - a/2 = PaP/2 identically (lambda = 1/2 = 1/[F:F_even], integer attainment 16384 + 2048 with 0 violations), lambda_E4 = 1/4 = 1/mu, index consistency exp(Delta) = 2 = 1/lambda_PP two routes; controls: dimerised trivial, wrong prescription exactly 0, finite-size honesty documented; with v501 all THREE KLM ingredients of complete rationality witnessed on the lattice; continuum uplift stays [O] (Xu's theorem cited, not claimed; WP5e/Costello-Li); SEAM.EQUIV.01 stays [O]",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5D.01; SEAM.CCC.01; SEAM.MU.01; SEAM.EQUIV.01; v367; v450; v489; v490; v501",
      "scripts": [
        "v504_celestial_wp5d_klm_completeness.py"
      ],
      "script": "v504_celestial_wp5d_klm_completeness.py",
      "location": "tfpt_research_contracts",
      "externalData": "Kawahigashi-Longo-Mueger CMP 219 (2001) (complete rationality = split + strong additivity + finite mu-index); Xu (algebraic orbifold CFTs: finite-group orbifolds of completely rational nets are completely rational -- cited, not claimed); Longo-Xu (multi-interval subfactors; bounded-vs-divergent dichotomy); Longo (split heredity under finite index); Pimsner-Popa (the conditional-expectation index bound); Casini-Huerta-Magan-Pontello (entropic order parameters); Buchholz-Schulz-Mirbach (strong-additivity failure of the U(1) current net); Klich-Levitov (charge-variance log law); Buchholz-Wichmann (nuclearity); Peschel J. Phys. A 36 (2003) (Gaussian covariance) -- method citations, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 568
    },
    {
      "id": "CELEST.WP5E.BETA.01",
      "claim": "WP5e-BETA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, beta stage -- 'the equivariant anomaly ledger on twistor space', the twistor-side milestone of the BCOV/Costello-Li uplift; WP5e-alpha = v502 anchored the CFT side): exact sympy/Fraction arithmetic, no floats (47 checks). Roadmap target: the anomaly-inflow side of the boundary partition function; preregistered KILL = 'the anomaly inflow demands a level != 1'. [E] ATIYAH-BOTT LEDGER: denominators det(1 - g_j^{-1}|_{T0 C^2}) = (2, 4, 2) for j = 1,2,3; sum of reciprocals 5/4 = (|Z4|^2 - 1)/12 (the A-series Dedekind identity); the origin is the only fixed point; equivariant characters tr_{g_j}(Ad E8) = (248, 0, -8, 0) by TWO independent routes (graded dims / root sum); invariant average 60 = the D5+A3 carrier; Frobenius 61568; tr_{g_2} = -8 = 120 - 128 (so16 adjoint-minus-spinor supertrace). [E] OKUBO PER SECTOR (the honest sharpening): the INVARIANT sector obeys Okubo exactly -- Q^(0) = 36<x,x>^2 with 36 = lambda~^2_e8 (the v495 identity re-derived through the mu4-glue grading); the TWISTED sectors do NOT reduce to (tr_{g_j} X^2)^2 (independent-quartic content T5: (-8, 32, -8), T3: (48, -64, 48)); in the AB-weighted fixed-point sum the D5 quartic CANCELS exactly and the A3 quartic leaves the RIGID residual 32 T3 (32 = (h,h) + (h',h'), a [C] coincidence); no nonzero reweighting of the twisted sectors alone is quartic-free, and the only quartic-free weighting on the class blocks is the uniform Z4 average (= the invariant projector); the graded GS-exchange ansatz cancels sector 0 and is exactly rank-obstructed in sectors 1, 2, 3 -- twisted-sector closed strings must carry the rest [O]; heterotic identities exact (Tr_120 = 8tr^4 + 3(tr^2)^2, Tr_128 = -8tr^4 + 6(tr^2)^2, Tr_E8 = 9(tr^2)^2). [E] INDEX BRIDGE (the centrepiece): the per-character AB sum f(m) = (1/4) sum_j (i^{jm} - 1)/det_j = (0, -3/8, -1/2, -3/8) equals ch2(T_m) = -(C^{-1})_mm/2 EXACTLY for m = 1, 2, 3 -- the E1 fixed-point ledger and the E2 McKay/Kronheimer intersection ledger are the SAME arithmetic; the glue-bundle ch2 defect is -78 by BOTH routes (integer; per-class (-24, -30, -24), the middle -30 = -h_vee a [C] coincidence); bridge identity sum_m h_A3(m) = 5/4 = sum_j 1/det_j; sphere pairing <alpha,h> integer on all 240 roots, per-class Killing sums (320, 320, 240, 320), total 1200 = 2 h_vee (h,h). [E] k = 1 GEOMETRIC: the lattice current count (norm-2/k vectors in the glue cosets) = 240 at k = 1 and EXACTLY 0 at k = 2, 3, 4 (no adjoint closure); embedding residual 47k^2 + 219k - 266 = (0, 360, 814, 1362); h(J;k) = k integer for ALL k -- integrality alone fixes nothing (honest, exactly as v502); v493 lockstep periods => ONE scale => ONE level (det(A-1) = -4, no invariant flux vector). [E] E3 REFUTATION (prominent): the clock-invariant modulus a0 in O(8) generates a Hamiltonian in O(2) -- a0 fills the BSS GRAVITON slot (s = 2), NOT the axion slot O(-2) (s = 0): Hamiltonian weights X = -2 != +2 = Y, mismatch 4 = -deg K_PT = |mu4| (a [C] coincidence) -- 'the theory brings its GS axion as a0' is REFUTED; what DOES match: the three H2(ALE) classes carry exactly the Coxeter eigencharacters {i, -1, -i} = the three twisted ledger sectors (bijection, no invariant class: det(A-1) = -4); slot count 1 tower slot O(-2) + 3 sphere classes = 4 = |mu4| (soft count, not a construction). [E] NEGATIVE CONTROLS: diag(i,i) breaks the ledger FOUR independent ways (complex denominators, reciprocal sum 1/4 != 5/4, omega-weight -1 not Calabi-Yau, invariant ring needs 5 generators); SO(16) glue: a mere Z2 monodromy (classes {0,2}), characters (120, 0, +120, 0) wrong sign AND magnitude at j = 2, odd McKay nodes empty, ch2 defect -30 != -78, and bulk Okubo FAILS (T5 = 16, T3 = -32: independent quartic Casimir, no single-axion GS -- the v495 sl4-style control transplanted); k = 2: current count 0, residual 360, c-prefactor -31/48 != -1/3, while naive integrality dials stay integer (the kill is the closure dial). [C] dictionaries: CS(rho) mod 1 = discriminant weight; 32 = (h,h) + (h',h'); -30 = -h_vee; the Omega-contraction identification of the axion slot. [O] HONEST FENCE: the global BCOV/Kodaira-Spencer theory on PT/Z4 (quantisation, non-fixed-point contributions, the twisted closed-string measure), the O(-2) bulk-axion construction, the Costello-Paquette-Sharma level-from-flux mechanism, and the pairing of the rigid 32 T3 twisted residual with the three sphere axions all stay OPEN. VERDICT B: the preregistered kill ('inflow demands a level != 1') does NOT fire on the equivariant skeleton -- all computable twistor-side pieces are consistent with k = 1 and kill k = 2, 3, 4 via the closure dial; what the skeleton cannot decide is the actual BCOV one-loop coefficient on PT/Z4. WP5e: alpha (v502) + beta (v505) executed; SEAM.EQUIV.01 untouched; no gate closes, no marker moves.",
      "status": "Formal/Identity [E] (exact equivariant arithmetic: Atiyah-Bott denominators/characters, per-sector Okubo decomposition with the rigid 32 T3 residual, index bridge f(m) = ch2(T_m) with the -78 defect, geometric level dials, axion-slot weight arithmetic -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the dictionaries above) + [O] (the global BCOV quantisation = WP5e proper; SEAM.EQUIV.01 unchanged)",
      "canonicalStatus": "WP5e-beta [E]/[C]/[O]: the Atiyah-Bott/Lefschetz skeleton of the twistor inflow exact -- denominators (2,4,2) with Dedekind sum 5/4, characters (248,0,-8,0) two routes, invariant Okubo 36<x,x>^2 with the twisted sectors NOT reducing and the AB sum leaving the rigid 32 T3 residual; index bridge f(m) = ch2(T_m) exact with glue defect -78 both routes; level dials k = 1 geometric (current count 240/0/0/0, residual 0/360/814/1362, one scale => one level -- since v509 (WP5e-epsilon-2) a THEOREM of clock invariance, no longer a heuristic); E3 refuted: a0 = graviton modulus O(2), not the axion O(-2) (mismatch 4 = |mu4|), the three sphere classes = the three twisted axion slots; controls diag(i,i)/SO(16)/k = 2 quantified; kill branch does not fire on the equivariant skeleton; the global BCOV quantisation stays [O] -- with the rigid 32 T3 twisted residual and the sphere-axion pairing as the named remaining targets; SEAM.EQUIV.01 untouched",
      "dependencies": "CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP2.01; CELEST.WP3.01; CELEST.WP4.01; CELEST.WP5E.ALPHA.01; SEAM.EQUIV.01; v492; v493; v495; v496; v502",
      "scripts": [
        "v505_celestial_wp5e_beta_equivariant_ledger.py"
      ],
      "script": "v505_celestial_wp5e_beta_equivariant_ledger.py",
      "location": "tfpt_research_contracts",
      "externalData": "Costello arXiv:2111.08879 (one-loop box anomaly, GS axion coupling); Bittleston-Sharma-Skinner CMP 403 (Poisson-CS tower, O(-2)-valued Poisson structure); Costello-Paquette-Sharma PRL 130, 061602 (level-from-flux); Kronheimer/McKay ALE intersection data -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 569
    },
    {
      "id": "SEAM.CLOCK.RIGIDITY.01",
      "claim": "SEAM CLOCK RIGIDITY -- ROUTE B OF THE v503 CLASSIFICATION EXECUTED (2026-07-21; two exact theorems in one module, Part A Moebius + Part B Fock, 34 checks; NO marker moves -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom): v503 localised the one remaining identification input as 'why does the raw seam carry the order-4 clock' (QGEO-R1 == P2-R1). v506 reduces that input to ONE alignment bit. [E] THEOREM A (Moebius side, complete): the collar deck z -> -z has in PGL2(C) EXACTLY TWO square roots (z -> +iz and z -> -iz), both AUTOMATICALLY of order 4 (complete case analysis: tr g != 0 forces diagonal with a/d = +-i; tr g = 0 forces det = 0, impossible) -- 'order 4' is NO LONGER an input; a marks-preserving square root exists <=> the deck is the CENTRAL involution of the mark-stabiliser D4 (center = {id, deck}, exactly 2 order-4 elements = the two clocks, both squaring to the deck) => then the marks are one mu4 orbit, cross-ratio class {2, 1/2, -1}, j = 1728, tau = i, free 4-cycle, deck = clock^2, only the scale free; COUNTEREXAMPLE (mandatory honesty): {+-1, +-(3+2sqrt2)} is deck-free AND harmonic (j = 1728) with full D4 stabiliser, but there the deck is a NON-central (edge-type) involution => ZERO square roots -- harmonicity alone is NOT sufficient, the residual datum is deck-CENTRALITY, not the j-invariant; the RP filter selects nothing (both real structures pass both roots: the Aut(Z4) = {clock, clock^-1} ambiguity is irreducible, consistent with v492 NC3). [E] B3/B4 (n-arithmetic + controls): a free Z_n on 4 marks forces n | 4; marks-transitivity forces n = 4; faithfulness (a Moebius map fixing >= 3 points is the identity) kills every n >= 5; n = 3 fixes a mark (hexagonal shadow); n = 6 dies INDEPENDENTLY at the established cusp-class degree 3 (lam^5 - 1 vs lam^3 - 1, v117); the deck is modulus-blind (every {+-1, +-w} preserved -- v503 B5 made exact, ALL modulus-pinning power sits in the square-root bit); controls: no order-8 element preserves any 4-set (orbit sizes 8); hexagonal marks: stabiliser A4 (order 12), ZERO order-4 elements, squares only orders {1, 3}; generic marks: Klein V4, exponent 2 -- not even the deck has a root; stabiliser trichotomy V4/D4/A4 with order-4 counts (0, 2, 0): an order-4 element exists IFF the marks are harmonic, and then it is the clock pair. [E] THEOREM B (Fock side, the core result): the celestial sphere S^2 has b1 = 0, hence a UNIQUE spin structure whose restriction to every circle is the BOUNDING = NS/antiperiodic one (established P1 datum, v492/v490); ONE-PARTICLE: the NS half-rotation D has D^2 = -1 (charpoly (lam^2+1)^8) -- the 'Z2' deck is already order 4 before any Fock choice; the quarter-shift C has C^2 = D, C^8 = 1 (charpoly (lam^4+1)^4, zeta8 spectrum); FOCK (exact Cl(16) Fractions): the Bogoliubov implementer Utilde = prod_j (1 - gamma_j gamma_{j+8}) satisfies Utilde^2 = 256 gamma_1...gamma_16, i.e. U^2 = (-1)^F EXACTLY, and (cU)^2 = c^2 (-1)^F is NEVER 1 (center of Cl(16) = scalars, implementer unique up to phase) -- the Z4 lift is FORCED: 1 -> Z2^{(-1)^F} -> Z4<U> -> Z2^{deck} -> 1 is a NONSPLIT extension; Z8 TOWER: the quarter-shift lift V (unique even block implementer, rational nullspace) has V^2 prop Utilde, V^4 prop (-1)^F, projective order 8 = 2|mu4| -- the v492 Z8 spin bridge realised on the 16-Majorana Fock space, '8 = 2|mu4|' fermionically explained; R CONTROL: U_R^2 = +1 EXACTLY (split Z2 -- the entire forcing rests on the NS/bounding structure = established geometry, not a convention); 16-FOLD WAY: theta_v = 1 at nu = 16 (the gaugeable point, v490), the forced tower period 8 = the Fidkowski-Kitaev Z8; ROBUSTNESS: the forcing already holds on an 8-site NS circle -- 16 is the carrier count 2^(g_car-1), not a tuning. [C] THE ONE RESIDUAL (verbatim): 'the collar deck is the central involution of the mark-D4' (<=> square-root existence on the mark data) -- ONE alignment bit; the fermionic Z4 forces the ORDER, not the spatial alignment. NO OVERCLAIM: given the bit, order 4, uniqueness up to Aut(Z4), the free 4-cycle, tau = i (scale free) and deck = clock^2 are all THEOREMS; without it, nothing pins the modulus. Consequence for the residual map: QGEO-R1 == P2-R1 (v503) now reads as this single bit -- order/uniqueness/orbit/tau = i all follow (fermionic halving of the R1 rest). No gate closes, no marker moves. PRECISION (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED): the bit is now fully characterised as a tetrad (clock exists <=> tau = i AND deck = the CM-fixed half-period translation <=> Stab = D4 AND deck central <=> deck pairs harmonically separated <=> NS Fock lift nonsplit), and carries a physical face: the non-split NS Fock extension class (Fidkowski-Kitaev-type; the edge arrangement splits, U^2 = +1, zero roots) -- v507; the origin theorem shows every mark-free collar deck IS a half-period translation of the seam double, so the CLASS of the deck is derived and the bit is NO tautology -- only the position (which half-period) stays the carrier input. PRECISION (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED): the POSITION half of the alignment bit is now typed as topology -- the collar deck is a COVERING deck and covering decks act freely; the unique free involution of the seam circle is the antipode (Aut(C_16) = D_16, complete census) and NONSPLIT <=> FREE over all 17 involutions, so the edge class is topologically EXCLUDED (v510); the bit reduces to the square-modulus datum tau = i alone (harmonic + free => mu4 exactly) -- the modulus half stays the one [C] carrier input.",
      "status": "Formal/Identity [E] (Theorem A: complete PGL2(C) square-root classification, D4 centrality criterion, counterexample, n-arithmetic, stabiliser trichotomy -- sympy exact; Theorem B: U^2 = (-1)^F in exact Cl(16) Fractions, nonsplit Z4 extension, Z8 tower, R-sector split control, 16-fold-way arithmetic -- Wolfram-mirrored) + [C] (the one alignment bit: deck central in the mark-D4); QGEO.REALIZE.01 stays [C]/[O], P2.TYPING.01 R1 mapped onto the same bit",
      "canonicalStatus": "Clock rigidity [E]/[C]: the deck has exactly two Moebius square roots, both automatically order 4; marks-preserving root <=> deck = central involution of the mark-D4 (counterexample shows harmonicity alone insufficient); the NS Fock implementation is FORCED to order 4 (U^2 = (-1)^F exact in Cl(16), nonsplit; Z8 tower V^2 prop U, V^4 prop (-1)^F; R sector splits) -- the order-4 clock input reduces to ONE alignment bit ('deck central in the mark-D4'); the fermionic Z4 forces the order, not the alignment; QGEO-R1 == P2-R1 now = this single bit; no marker moves; v507 characterises the bit as a tetrad with a physical face (non-split NS Fock extension class) and derives the deck CLASS (half-period translation) -- only the position stays the input; v510 excludes the edge class topologically (covering-deck freeness = P1's closed seam circle): the bit reduces to the square-modulus datum tau = i alone",
      "dependencies": "QGEO.REALIZE.01; QGEO.RIGID.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; SEAM.BIT.ORIGIN.01; v117; v168; v214; v215; v216; v490; v492; v493; v499; v503; v507; SEAM.BIT.FREEDOM.01; v510",
      "scripts": [
        "v506_seam_clock_rigidity.py"
      ],
      "script": "v506_seam_clock_rigidity.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Fidkowski-Kitaev Z8 classification of interacting Majorana chains (PRB 81, 134509) -- literature anchor, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 570
    },
    {
      "id": "SEAM.BIT.ORIGIN.01",
      "claim": "THE TAUTOLOGY ATTACK ON THE v506 ALIGNMENT BIT, EXECUTED AND REFUTED (2026-07-21; one module, Part A torus origin + Part B fermion dichotomy, 26 checks; NO marker moves -- the bit stays [C], SEAM.CLOCK.RIGIDITY.01 and QGEO.REALIZE.01 text-precised only): v506 reduced the last identification input to ONE bit ('the collar deck is the central involution of the mark-D4'). Conjecture under attack: marks and deck both descend from the SAME torus structure (the seam double T^2_tau -> S^2, v214/v260), so the common origin would force the alignment automatically and the bit would be a tautology. [E] ORIGIN THEOREM (die Herkunft): the 4 marks are the fixed points of the hyperelliptic involution w -> -w on C/(Z + tau Z) = the four half-periods Lambda/2Lambda (a 1/3-coefficient never qualifies: 2w in Lambda <=> coefficients in {0, 1/2}); a translation t_c descends to the base sphere iff 2c in Lambda -- exactly the THREE half-period translations (control c = 1/3 fails); the descended maps are the EXPLICIT rational involutions sigma_i(x) = e_i + (e_i-e_j)(e_i-e_k)/(x-e_i) with the exact curve-lift identity prod(sigma_i - e_l) = psi_i^2 prod(x - e_l), psi_i RATIONAL (the p-function addition formula as pure algebra); {id, sigma_1, sigma_2, sigma_3} = Klein V4 = Lambda/2Lambda/{+-1}; CONVERSELY, in ALL three stabiliser types the freely-acting involutions are EXACTLY the three sigma_c (generic V4 (4,3,3), harmonic D4 (8,5,3), hexagonal A4 (12,3,3)) ==> EVERY mark-free collar deck (v216) IS a half-period translation -- the CLASS of the deck is forced, its POSITION is not. [E] CORE SOLVE (symbolic in the Legendre modulus lambda): sigma_c admits a marks-preserving PGL2 square root <=> lambda in {-1} (sigma_1), {2} (sigma_2), {1/2} (sigma_3) -- per half-period exactly ONE solution; intrinsically: the two deck pairs separate each other HARMONICALLY (pair cross-ratio = -1, same solution sets); the union over c is the harmonic orbit {-1, 2, 1/2} <=> j = 1728 <=> tau = i; generic modulus: Stab = V4 abelian, ALL sigma_c central and yet clockless ('central alone is EMPTY' -- the bit is the CONJUNCTION Stab = D4 AND deck central); at tau = i the stabiliser jumps to D4 with sigma_1 central (2 roots = the clock pair) and sigma_2/sigma_3 edge-type (0 roots); the v506 seam deck z -> -z lands on the central sigma_1 = -1/x under ALL 8 mark bijections; CM DICTIONARY: mult-by-i fixes exactly ONE nonzero half-period c* = (1+tau)/2 (mult-by-rho 3-cycles all three -- hexagonal has no distinguished one); the clock lift needs the CM unit i (psi = 2i/(x+1)^2, t^2+1 irreducible over Q) while half-period decks lift RATIONALLY: the clock root exists only WITH CM by Z[i] <=> tau = i. [E] EQUIVALENCE TETRAD + CIRCULARITY AUDIT: 'Uhr existiert' <=> (tau = i AND deck = the CM-fixed half-period translation) <=> (Stab = D4 AND deck central) <=> deck pairs harmonically separated <=> the NS Fock lift is NONSPLIT (Part B); audit: the v493 converse (clock GIVEN => P(iZ) = P(Z) forces a3 = a2 = a1 = 0, j = 1728) is independently re-verified; here NO clock is assumed and existence is SOLVED -- two independently computed converses of ONE equivalence, no circularity. [E] FERMION DICHOTOMY (the new physical face): central arrangement (shift by 8 on the 16-site NS seam): Utilde^2 = 256 gamma_1...gamma_16 => U^2 = (-1)^F, nonsplit Z4 extension, 2 square roots in the 32-element dihedral lift group (the Z8 tower); edge arrangement (reflection j -> 4-j, fixed sites = NON-marks = the silver {0, inf} between neighbour marks): the implementation EXISTS but Utilde_e^2 = +2^7 x 1 => U^2 = +1, the extension SPLITS, 0 roots, no Z8 tower; one-particle level (lam^2+1)^8 vs (lam^2-1)^8; arrangement census: ALL 28 shift-invariant 4-sets cross (central type), ALL 21 mark-free reflection-invariant sets do not; robustness at 8 sites; R-sector edge control splits too. The bit IS a physical extension class (Fidkowski-Kitaev-type) -- distinguished, but not derived: WHICH class nature realises stays the input. [E] SILVER PLACEMENT (in-family witness against the tautology): the v506 counterexample {+-1, +-(3+2 sqrt 2)} = 'tau = i with deck = a NON-CM-fixed half-period translation' (all 8 mark bijections carry its deck to sigma_2/sigma_3, never sigma_1) -- the common torus origin does NOT exclude the misalignment. [C] THE REST (verbatim, no overclaim): the one alignment bit remains a carrier datum; 'no tautology -- the origin forces the CLASS of the deck (half-period translation), not its POSITION'. No gate closes, no marker moves. PRECISION (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED): the POSITION residual is now typed as topology -- the collar deck is a COVERING deck and covering decks act freely; the unique free involution of the seam circle is the antipode (Aut(C_16) = D_16, complete census), NONSPLIT <=> FREE over all 17 involutions, and the edge class is the restriction of the BRANCHED pillowcase involution, so the edge/silver arrangement is topologically EXCLUDED and the tetrad's position face FOLLOWS (v510); the bit reduces to the square-modulus datum tau = i alone (harmonic + free => mu4 exactly), which stays the one [C] carrier input.",
      "status": "Formal/Identity [E] (origin theorem: half-period fixed points, rational descent identity, free-involution census in all three stabiliser types; core solve lambda in {-1, 2, 1/2}; CM dictionary; equivalence tetrad + circularity audit; fermion dichotomy: Utilde^2 = 256 gamma-monomial vs +2^7 scalar in exact Cl(16) Fractions, root counts 2 vs 0, arrangement census 28/21 -- sympy exact, Wolfram-mirrored) + [C] (the one alignment bit: which half-period, i.e. the deck position); SEAM.CLOCK.RIGIDITY.01 and QGEO.REALIZE.01 stay as they are, P2.TYPING.01 R1 unchanged",
      "canonicalStatus": "Bit origin [E]/[C]: marks = half-periods of the seam double, every mark-free collar deck IS a half-period translation (class forced); a marks-preserving root exists iff lambda in {-1, 2, 1/2} <=> deck pairs harmonic <=> tau = i AND deck = the CM-fixed half-period (tetrad closed by the fermionic face: central <=> non-split NS Fock extension U^2 = (-1)^F, edge <=> split U^2 = +1, 0 roots); silver = in-family misalignment witness -- TAUTOLOGY REFUTED, the one alignment bit stays the [C] carrier input (position, not class); no marker moves; v510 types the position half as topology (covering-deck freeness excludes the edge class): the rest is the square-modulus datum tau = i alone",
      "dependencies": "SEAM.CLOCK.RIGIDITY.01; QGEO.REALIZE.01; QGEO.RIGID.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; v117; v168; v214; v215; v216; v260; v490; v492; v493; v499; v503; v506; SEAM.BIT.FREEDOM.01; v510",
      "scripts": [
        "v507_seam_bit_origin.py"
      ],
      "script": "v507_seam_bit_origin.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Fidkowski-Kitaev Z8 classification of interacting Majorana chains (PRB 81, 134509) -- literature anchor, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 571
    },
    {
      "id": "SEAM.BIT.FREEDOM.01",
      "claim": "THE FREEDOM ATTACK ON THE v506/v507 ALIGNMENT BIT -- THE POSITION HALF OF THE BIT IS TOPOLOGY (2026-07-21; one module, Part A dihedral/fermionic + Part B Moebius/RP-geometry, 30 checks; NO marker moves -- the modulus datum stays [C], SEAM.CLOCK.RIGIDITY.01, SEAM.BIT.ORIGIN.01 and QGEO.REALIZE.01 text-precised only, P2.TYPING.01 R1 halved): v507 left the last identification input as ONE bit with four equivalent faces; the collar deck of the theory is the deck transformation of a COVERING (the Z2 seam of the Moebius double, tfpt_3 'cylinder + one Z2 seam'), and covering decks act FREELY -- a reflection of the circle has fixed points and is never free, so the edge class is EXCLUDED and the fermionic face of the bit FOLLOWS. [E] DIHEDRAL THEOREM: on the N-site seam circle the free involution is UNIQUE = the antipode (N = 16: 17 involutions in D_16, exactly 1 free = shift 8; all 16 reflections have exactly 2 circle fixed points -- sites for even axis, edge midpoints for odd; robustness N = 8, 12); the census is COMPLETE because Aut(C_16) = D_16 exactly (brute force over all 8! permutations at N = 8, propagation argument at N = 16 -- no exotic circle automorphism exists); N = 9 control: D_9 has ZERO free involutions (a Moebius seam circle needs an EVEN carrier count, consistent with 16 = 2^(g_car-1)); QUOTIENT TOPOLOGY (the P1 tie): circle/antipode = CIRCLE (chi = 0, closed), circle/reflection = INTERVAL (chi = 1, two boundary corners) -- origin_theory's Gauss-Bonnet budget 6pi = 2pi+2pi+2pi counts three CLOSED circles, so 'the deck is free on the seam circle' is EXACTLY the established P1 datum 'the seam is a closed circle', not a new premise. [E] NONSPLIT <=> FREE (complete census, exact Cl(16)): ALL 17 involutions of the seam circle implemented -- the free one's NS lift squares to 256 gamma_1...gamma_16 = (-1)^F (nonsplit, 2 square roots in the 32-element dihedral lift group = the Z8 clock tower); ALL 16 reflections have U^2 = SCALAR (2^7 site-axis, 2^8 edge-axis; split, 0 roots) -- the v507 dichotomy holds over the COMPLETE involution census, no exception; chain: deck free => deck = shift 8 => crossing mark pairing (all 28 invariant 4-sets) => U^2 = (-1)^F => central class => the edge/silver arrangement is EXCLUDED; Klein control: the orientation-cover deck is free too (covering vs branched is the discriminator, not Moebius vs Klein); branch control: the pillowcase involution (x,y) -> (-x,-y) has exactly 4 fixed points = the marks and restricts to a REFLECTION on invariant circles -- the edge class IS the restriction of the BRANCHED pillowcase involution: the silver arrangement mistakes the mark-creating branch Z2 for the seam deck. [E] REAL-STRUCTURE CLASSIFICATION (continuum side): the antiholomorphic involutions of P^1 commuting with the celestial deck z -> -z are EXACTLY two families -- mu conj(z), |mu| = 1 (fix locus = a LINE through the deck poles {0, inf}: deck NOT free on it) and mu/conj(z), mu real (mu > 0: fix circle |z| = sqrt(mu) avoiding the poles, deck FREE on it; mu < 0: EMPTY fix locus, no seam); per deck-invariant configuration {+-a, +-b} the mark-fixing real structure is UNIQUE (an antiholomorphic map fixing 3 points pointwise is unique); table: mu4 -> equatorial 1/conj(z), deck FREE (v506 A-S2.9 reproduced); SILVER {+-1, +-(3+2 sqrt 2)} -> real axis THROUGH the poles, deck NOT free; generic-unit {+-1, +-(3+4i)/5} -> equatorial, free WITHOUT harmonicity; hexagonal -> NO mark-fixing real structure at all (cross-ratio complex, marks not concyclic: RP-incompatible, consistent v503); Moebius covering model: the sheet-exchange deck of the cylinder double cover is free on ALL points and restricts to the antipode on the Z2 seam fibre; P1's one-sidedness TYPES the Z2 as the free antipode (quotient RP^2 = the |Z2| of c3 = 1/(|Z2| 2pi chi), v456/v73). [E] HARMONIC + FREE => mu4 EXACTLY: the deck-pair cross-ratio (1-b)^2/(1+b)^2 = -1 solves exactly to b = +-i -- on free-deck seam circles the silver family (harmonic AND edge) is EMPTY, so harmonicity alone becomes SUFFICIENT for the clock. [E] HONEST COUNTERWITNESSES: 'nonsplit => clock' is FALSE -- discrete {0,1,8,9} (shift-invariant, crossing, same nonsplit lift, but the quarter shift does not preserve it) and continuous {+-1, +-(3+4i)/5} (deck free, equatorial, central class, but j = 148176/25 != 1728, no clock): the Fock dichotomy measures the CLASS of the arrangement, not the modulus. PREMISE TYPING [E]/[C]: (P-i) deck = Moebius covering deck => free on the seam circle -- standard covering topology, machine-instantiated (the one-sidedness strand: origin_theory/v456 check 1 [E]; tfpt_3 'cylinder + one Z2 seam' double structure); (P-ii) marks lie on the Theta-fix seam circle = existing REALIZE [C] content (QGEO.REALIZE.02 hypotheses (4)/(5), v264; v506 A-S2.9 [E] for mu4) -- NO new input; AUDIT FINDING documented: 'circle-freeness' was nowhere typed before (only mark-freeness, v216); v507 F1.1 explicitly admitted the reflection reading of the edge class -- v510 closes that door (a reflection is not a covering deck). [C] THE REDUCED REST (verbatim, no overclaim): the alignment bit reduces from 'harmonic AND central' to the square-modulus datum tau = i alone; the position half is topology (edge class excluded), the modulus half stays the one carrier input. No gate closes, no marker moves. CROSS-REFERENCE (2026-07-22, v512/SEAM.TAU.FLAG.01): the modulus half gains the equivalent discrete form 'flag transitivity (V4 -> D4) <=> tau = i' (12-fold exact equivalence web); the counterwitness {+-1, +-(3+4i)/5} is confirmed to pass ALL 7 established unsided tests -- v512.",
      "status": "Formal/Identity [E] (dihedral theorem: unique free involution = antipode, Aut(C_16) = D_16 complete census, quotient topology chi 0 vs 1; complete Cl(16) Fock census NONSPLIT <=> FREE over all 17 involutions; real-structure classification: two deck-commuting families, per-configuration uniqueness table; harmonic + free => b = +-i exactly; counterwitnesses -- sympy exact / exact Cl(16) Fractions, Wolfram-mirrored) + [C] (the square-modulus datum tau = i); SEAM.CLOCK.RIGIDITY.01, SEAM.BIT.ORIGIN.01 and QGEO.REALIZE.01 stay as they are (text precised only), P2.TYPING.01 R1 halved",
      "canonicalStatus": "Bit freedom [E]/[C]: the collar deck is a covering deck, hence FREE on the seam circle; the unique free involution is the antipode (Aut(C_16) = D_16, complete census) and NONSPLIT <=> FREE over all 17 involutions -- the edge/silver class is topologically EXCLUDED (its real structure's fix circle passes through the deck poles; the edge class is the branched pillowcase involution's restriction); harmonic + free => mu4 exactly, but freeness does not give harmonicity (honest counterwitnesses) -- the alignment bit reduces from 'harmonic AND central' to the square-modulus datum tau = i alone: the position half is topology, the modulus half stays the one [C] carrier input; v512 restates the modulus half as the flag-transitivity bit; no marker moves",
      "dependencies": "SEAM.BIT.ORIGIN.01; SEAM.CLOCK.RIGIDITY.01; QGEO.REALIZE.01; QGEO.REALIZE.02; QGEO.RIGID.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; v216; v264; v456; v490; v492; v503; v506; v507",
      "scripts": [
        "v510_seam_bit_freedom.py"
      ],
      "script": "v510_seam_bit_freedom.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 572
    },
    {
      "id": "CELEST.WP5E.GAMMA.01",
      "claim": "WP5e-GAMMA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, gamma stage -- 'the sphere-axion pairing check', an honest rigid NEGATIVE result in the v505 style; exact sympy/Fraction arithmetic, no floats, 27 checks, ~4 s): can the three twisted sphere axions eta_m (m = 1, 2, 3; Coxeter eigencharacters {i, -1, -i} on H^2(ALE), v505 S5.8) cancel the rigid 32 T3 residual of the Atiyah-Bott anomaly ledger (v505 S3.7: A_fix = Q^(1)/2 + Q^(2)/4 + Q^(3)/2 = 9P1 - 30P2 - 15P3 + 32 T3) by Green-Schwarz-type EXCHANGE with sector-compatible quadratic vertices? NO, with a certificate. [E] VERTEX-SPACE THEOREM (structural backbone): the W(D5) x W(A3)-invariant quadratics on the 8-dim glue Cartan are EXACTLY span{s5, s3} (dim 2, bidegree dims (1, 0, 1)) and the invariant quartics EXACTLY span{P1, P2, P3, T5, T3} (dim 5, bidegree dims (2, 0, 1, 0, 2)) -- computed by exact Weyl nullspace arithmetic over ALL bidegrees, the vertex basis is COMPLETE; PRODUCT THEOREM (the master kill): (a s5 + b s3)(c s5 + d s3) = ac P1 + (ad+bc) P2 + bd P3 identically -- the product of ANY two invariant quadratics has identically ZERO T5- and T3-content, hence EVERY scalar-exchange image (any number of axions, any selection rule, any coupling values) is annihilated by Phi_T3 while Phi_T3(A_fix) = 32 != 0: the pairing is decided before enumeration. [E] SELECTION-RULE ENUMERATION (Z4 charge m + j + j' = 0 mod 4): the strict two-index reading COLLAPSES -- the class bilinears C_{j,j'} = sum_{alpha+beta=0} <alpha,x><beta,x> are nonzero ONLY for j' = -j (C_{j,-j} = -K_j), so the rule forces m = 0: the sphere axions have NO invariant bilinear Cartan vertex at all (16 raw triples, 4 nonzero, all m = 0); twist-insertion reading (the generous route): eta_1 <-> K^(3), eta_2 <-> K^(2), eta_3 <-> K^(1), eta_0 <-> K^(0); exchange channels E13 = K^(3)K^(1) = (16, -96, 144, 0, 0), E22 = (K^(2))^2 = (16, 32, 16, 0, 0), bulk E00 = 225 E22; SIDE DISCOVERY: K^(0) = -15 K^(2) -- the two even-sector quadratics are PARALLEL (this collapse drives the Phi_P obstruction). [E] UNSOLVABILITY WITH CERTIFICATE (the core result): exchange matrix M is 5 x 2, rank 2, kernel 0 (a solution, if it existed, would be unique/rigid); rank([M | A_fix]) = 3 > 2: UNSOLVABLE; annihilator certificate: im(M) is annihilated by Phi_T5, Phi_T3, Phi_P = 3[P1] - [P2] - [P3] with values (0, 32, 72) on A_fix = (9, -30, -15, 0, 32) -- TWO independent obstructions: 32 T3 (structural, rule-independent) and Phi_P = 72 (selection-rule-driven: even the factorised part 9P1 - 30P2 - 15P3 is unreachable through charge-0 channels because K^(0) || K^(2) collapses the charge-0 product span to 2 dims); Phi_P(Q^(0)) = 0 (the bulk Okubo square stays inside the reachable slice -- the v505 sector-0 mechanism is undisturbed); ker M = 0 forces the unique non-disturbing coupling c = 0: the sphere axions are RIGIDLY SILENT in the exchange ansatz; relaxed control: dropping charge conservation makes the P-block fully reachable (rank 3, det -64) but T5 = T3 = 0 persists -- the deep kill is T3. [E] NATURALNESS RESOLUTION: ch2-natural couplings (f(1)f(3), f(2)^2) = (9/64, 1/4) give image (25/4, -11/2, 97/4, 0, 0) and AB-weight couplings (1/2, 1/4) give (12, -40, 76, 0, 0) -- BOTH have certificate values (0, 0) instead of the required (32, 72): the failure is scale- and coupling-INDEPENDENT (no 'unnatural factor' -- the ansatz space misses the target identically; the naturalness question dissolves). [E] NEGATIVE CONTROLS: (a) invariant axion only = v505 S3.9 replicated + sharpened (A_fix not in span{(K^(0))^2}); (b) flipped rule m + j - j': twist-insertion channel set unchanged (the Phi_T3 kill is rule-flip invariant), two-index reading changes (only eta_2 couples, the bulk axion gets span{K_0, K_2}, det 224, the P-part becomes reachable) -- 32 T3 stays dead: Phi_P is rule-dependent, Phi_T3 is not; (c) SO(16) glue: odd McKay classes EMPTY (no sphere partners at all) and the AB sum (15, -18, -15, -4, 40) does not even cancel T5 -- the mechanism is structurally BROKEN, not merely obstructed (E8 is doubly special); (d) D8 native: quartic 16 T8 + 12 s8^2 (2 h_vee(D8) = 28 Killing), no A3 block, no T3 structure -- the pairing question cannot even be posed. [C] NUMBER FENCES (typed [C] only, look-elsewhere discipline): 32 = (h,h) + (h',h') = 20 + 12; 72 = 2 * 36 = 2 lambda~^2_e8. [O] SHARPENED O-FENCE: the exchange sub-branch is DEAD; the 32 T3 residual MUST be carried by twisted BCOV contact terms -- binary criterion WP5e-delta-1: the one-loop coefficient on C^2/Z4 must come out exactly (9, -30, -15, 0, 32), COMPUTED (orbifold elliptic-genus / twisted-character arithmetic), not fitted; named escape WP5e-delta-2: the full-tensor ledger with non-Cartan external legs (the collapse is a Cartan-restricted statement); plus WP5e-epsilon-1 (the O(-2) bulk-axion construction, lambda~ = 6) and WP5e-epsilon-2 (CPS level-from-flux). The slot bijection (v505 S5.8) is UNTOUCHED; the preregistered level-kill still does NOT fire; SEAM.EQUIV.01 stays [O]; NO marker moves anywhere -- CELEST.SEAM.01 stays Open, the O-fence is only SHARPENED.",
      "status": "Formal/Identity [E] (exact vertex-space/selection/certificate arithmetic: invariant quadratics dim 2 and quartics dim 5 by Weyl nullspace, product theorem, rank-3 certificate (0, 32, 72), naturalness resolution, SO(16)/D8 controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the number fences 32 = 20 + 12, 72 = 2 lambda~^2_e8) + [O] (the twisted BCOV contact terms delta-1 (binary) + the full-tensor ledger delta-2; SEAM.EQUIV.01 unchanged; CELEST.SEAM.01 stays Open)",
      "canonicalStatus": "WP5e-gamma [E]/[C]/[O]: the exchange route for the sphere-axion pairing is KILLED with a three-functional certificate -- invariant quadratics on the glue Cartan exactly span{s5, s3} (dim 2), quartics exactly span{P1, P2, P3, T5, T3} (dim 5), the product of any two invariant quadratics has zero T5/T3 content (master kill: Phi_T3 annihilates every exchange image, Phi_T3(A_fix) = 32 != 0); strict two-index rule collapses the sphere couplings entirely (m = 0 only), twist-insertion channels give a rank-2 exchange matrix with rank([M | A_fix]) = 3 and certificate (Phi_T5, Phi_T3, Phi_P) = (0, 32, 72); K^(0) = -15 K^(2) side discovery; ch2-natural and AB-weight couplings miss scale-independently (naturalness dissolves); SO(16) has no sphere partners AND uncancelled T5, D8 has no T3 structure; the 32 T3 burden moves to the twisted BCOV contact terms (delta-1, binary: coefficient exactly (9, -30, -15, 0, 32) computed not fitted) and the full-tensor ledger (delta-2); slot bijection untouched, level-kill still not fired, SEAM.EQUIV.01 stays [O], no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; SEAM.EQUIV.01; v492; v493; v495; v502; v505",
      "scripts": [
        "v508_celestial_wp5e_gamma_pairing.py"
      ],
      "script": "v508_celestial_wp5e_gamma_pairing.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Costello arXiv:2111.08879 (one-loop box anomaly, GS axion coupling); Bittleston-Sharma-Skinner CMP 403 (Poisson-CS tower, O(-2)-valued Poisson structure); Costello-Paquette-Sharma PRL 130, 061602 (level-from-flux) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 573
    },
    {
      "id": "CELEST.WP5E.EPS2.01",
      "claim": "WP5e-EPSILON-2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, epsilon-2 stage -- 'the CPS level-from-flux dial', verdict B; exact sympy/Fraction arithmetic, no floats, 28 checks, <1 s): derive the boundary level k = 1 from flux quantisation on the lockstep spheres, following the Costello-Paquette-Sharma (CPS) Burns-holography mechanism (PRL 130, 061602; arXiv:2306.00940). [E] CPS SKELETON REPLICATED EXACTLY: the S^3 period of the brane backreaction 3-form int Omega_N = (2 pi i)^2 N (pullback density -sin(2 theta), phases cancel exactly; their eq. 3.33 -- large-gauge invariance of hCS quantises N in Z); the exceptional-sphere Kahler flux int_{P^1} omega = 2 pi N exactly (K = |u|^2 + N log|u|^2, the flat part dies at t -> 0; their eq. 3.37); the boundary level magnitude 2N via the symplectic-boson double contraction (exact for N = 1, 2, 3; their eq. 9.4), Dynkin normalisation T(so8 vector) = 1, T(adj) = 6 -- the CPS formula 'level = flux quantum x Dynkin index of the boundary matter', level LINEAR in the flux. [E] PAIRING MATRIX PINNED, NOT POSTULATED: the v505 ch2 ledger (f(m) = -(C^-1)_mm/2 = (-3/8, -1/2, -3/8)) + integrality + unimodularity + effectivity pin P_mi = c1(T_m).[Sigma_i] = delta_mi by COMPLETE enumeration (row candidates 8/6/8; 48 unimodular solutions of the full form condition P C^-1 P^T = C^-1; with P >= 0 exactly 2 = identity + A3 diagram flip); full c1 matrix -(C^-1)_mn; glue FLUXES F_i = sum_m dim(g_m) P_mi = (64, 60, 64) = dim g_i, total 188 = 248 - 60, flip-invariant; flux per charged root current uniformly ONE quantum through exactly one sphere. [E] LEVEL FORMULA, honest: 'level = total open-string flux' is KILLED by the lockstep test itself (k_i = (64, 60, 64) unequal -- F_i is the FLAVOUR multiplicity, in CPS language the brane matter content); what works: k = flux per adjoint current = F_i/dim g_i = (1, 1, 1), anchored non-circularly by the lattice current count (240, 0, 0, 0) for k = 1..4 (v505 S4.8) and the embedding index j(D5 in E8) = j(A3 in E8) = 1 (Killing coefficient 60 = 2 h_vee(E8), both factors). ORD-4-VS-LEVEL-1 RESOLVED: the clock order counts FRACTIONAL flux sectors, not the level -- mu = det(D5) det(A3) = 16 = ord(monodromy)^2, the mu4 glue diagonal is isotropic and Lagrangian (order 4 = sqrt 16), KLM condensation 16 -> 16/4^2 = 1 = zero residual fractional flux; NEW DIAL: #primaries((E8)_k) = (1, 3, 5, 10, 15, 27) for k = 1..6 from the affine marks -- exactly ONE sector iff k = 1 (level from sector count, independent of the CPS transplantation). [E] THE LOCKSTEP THEOREM (the lifting of v505 S4.10 'one scale => one level' to a theorem): clock invariance P(iZ) = P(Z) forces a3 = a2 = a1 = 0 sharply, the four branch points are ONE mu4 orbit i^k t, the periods Pi_j = t(i-1) i^{j-1} have |Pi_j| = sqrt(2) t EQUAL on all three spheres, the monodromy acts as the pure phase i on the period vector and det(A - 1) = -4 != 0: no invariant flux vector, no sphere-dependent level assignment possible -- with the CPS flux-level dictionary, k1 = k2 = k3 IS A THEOREM of clock invariance. NEGATIVE PROBE (new falsifier): the clock-forbidden family Z^4 - Z (smooth, disc = -27) has branch points {0, 1, w, w^2}: NOT a mu4 orbit, and over ALL 24 orderings ZERO chains give three equal period moduli (1 vs sqrt 3). HONEST LIMIT: uniform flux doubling keeps the lockstep -- the theorem proves EQUALITY of the three k, not the value; the value 1 comes from the current/sector counters and, CPS-faithfully, from the minimal quantum N = 1. [E] NEGATIVE CONTROLS: k = 2 (current count 0, embedding residual 360, c = 31/2 => prefactor -31/48 != -1/3, #primaries 3 != 1 -- while the lockstep dial honestly does NOT fire); SO(16) glue (classes (60, 0, 60, 0) => fluxes (0, 60, 0): spheres 1/3 FLUX-DARK, ch2 defect -30 != -78, condensation stops at 16/2^2 = 4 = det Cartan(D8): four sectors survive); A2 intersection form (det 3 != 4 = |Z4|: McKay bijection broken, glue diagonal 23/24 not integer, mu = 12 not a perfect square: NO Lagrangian subgroup exists, Dedekind 2/3 != 5/4, Coxeter order 3 != 4)",
      "status": "Formal/Identity [E] (exact CPS-skeleton/pairing/lockstep/sector arithmetic: S^3 period (2 pi i)^2 N, exceptional flux 2 pi N, contraction 2N, pairing enumeration 48 -> 2, fluxes (64, 60, 64), current count (240, 0, 0, 0), #primaries (1, 3, 5, 10, 15, 27), lockstep theorem det(A-1) = -4 with the 0/24 negative probe, k = 2 / SO(16) / A2 controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the CPS dictionary itself on PT/Z4: their geometry is the C^2 blow-up with one (-1)-sphere and so(8) at negative level, not the A3 ALE -- typed as an analogy until the backreaction is computed) + [O] (the type-I B-model backreaction on PT/Z4 -- twisted Kodaira-Spencer measure, the A3 Omega_N; the same fence as v502 S5.7/v505)",
      "canonicalStatus": "WP5e-epsilon-2 [E]/[C]/[O] (verdict B): k = 1 from flux quantisation -- partially derived. DERIVED exact: one level is a THEOREM (lockstep from clock invariance: one mu4 orbit, |Pi_j| = sqrt(2) t equal, monodromy = pure phase i, det(A-1) = -4 no invariant flux vector; falsifier Z^4 - Z: 0/24 orderings lockstep), flux-per-current = 1 uniformly (pairing matrix = identity up to diagram flip, pinned by complete enumeration from the v505 ch2 ledger; fluxes (64, 60, 64) = flavour multiplicity, NOT the level -- the naive reading killed by the lockstep test itself), the sector-counter dial pins the value (16 = ord^2 fractional sectors condense through the Lagrangian mu4 glue to 1; #primaries((E8)_k) = 1 iff k = 1), and the CPS skeleton is exact (level linear in the flux quantum, minimal quantum N = 1 <-> k = 1). ANALOGY [C]: the CPS dictionary on PT/Z4. OPEN [O]: the type-I B-model backreaction on PT/Z4. The v505 'one scale => one level' sentence is lifted to theorem status (text move, no gate move); WP5e stays [O]; SEAM.EQUIV.01 untouched; no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; SEAM.EQUIV.01; v492; v493; v502; v505; v508",
      "scripts": [
        "v509_celestial_wp5e_eps2_level_from_flux.py"
      ],
      "script": "v509_celestial_wp5e_eps2_level_from_flux.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Costello-Paquette-Sharma PRL 130, 061602 + arXiv:2306.00940 (Burns holography, level-from-flux mechanism); Costello arXiv:2111.08879 (one-loop box anomaly) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
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    {
      "id": "CELEST.WP5E.DELTA2.01",
      "claim": "WP5e-DELTA-2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, delta-2 stage -- 'the full-tensor ledger', the named escape route of the v508 exchange no-go, intermediate verdict, exact; integer-scaled weight multisets + Kostant/Weyl alternating sums + sympy/Fraction, no floats, 41 checks, ~4 s): does the FULL adjoint tensor structure of e8 (non-Cartan external legs, higher-arity vertices) open a sphere-axion channel to the rigid 32 T3 residual of the Atiyah-Bott ledger (v505 S3.7: A_fix = (9, -30, -15, 0, 32) in (P1, P2, P3, T5, T3))? The collapse is CONFIRMED full-tensorially, but one cubic door opens. [E] STRUCTURE: g0 = d5 (+) a3 is SEMISIMPLE with NO u(1) factor (the 52 class-0 roots split 40 + 12 into pure D5 + pure A3 roots, root span rank 8 = rank E8); the twisted sectors factorise multiplicity-free into minuscule Weyl orbits g1 = (16_s, 4), g2 = (10, 6), g3 = (16_c, 4bar) = (g1)^*, each an irreducible g0-module (64, 60, 64); INNERNESS THEOREM: the grading element h = (2,2,2,2,2;0^4) lies in the CARTAN OF g0, so t = exp(2 pi i h/4) is in the torus of G0 and every g0-invariant tensor automatically carries total sector charge 0 mod 4 -- the v508 collapse is a full-tensor statement AND holds across arities (all 15 non-neutral sector triples: Hom = 0; the sphere axions eta_1,2,3 have no invariant partner operator at any arity <= 3). [E] HOM TABLES: bilinear dim Hom_{g0}(g_j x g_j') nonzero ONLY at j' = -j with dims (2, 1, 1, 1) (Killing blocks; all 12 charged pairs vanish); invariant symmetric bilinear forms on e8 = 4-dim (2 + 1 + 1), the extra blocks B13/B22 restrict to zero on the Cartan (no new quadratic exchange content); trilinear neutral multisets {0,0,0}: 3 (= 1 sym + 2 antisym), {0,1,3}: 2, {0,2,2}: 2, {1,1,2}: 1, {2,3,3}: 1 -- ALL cross-sector trilinears are pure brackets (antisymmetric in the repeated legs, they vanish with two identical external legs); the ONLY totally symmetric survivor on all of e8 is the su(4) d-symbol (so(10) has no cubic Casimir: Inv(S^3 45) = 0 protects T5, Inv(S^3 a3) = 1). [E] THE NEW CHANNEL: the quartic projection of the d-symbol exchange is Q_dd = (0, 0, -1/240, 0, 1/60) = (1/60)(T3 - P3/4), computed by TWO independent routes (Killing-propagator contraction with 2 h_vee = 60 verified from the roots; hand route |x^2 - (tr x^2/4) 1|^2/60); Phi_T3(Q_dd) = 1/60 != 0 -- v508's master-kill functional Phi_T3 does NOT annihilate the trilinear exchange (the product theorem covers quadratic vertices only); bracket vertices, tadpoles and mixed d x f die exactly; no T5 analogue exists (Phi_T5 stays a universal annihilator, exactly as A_fix requires). [E] NEW CERTIFICATE: M = [E13, E22, E00, Q_dd] has rank 3, augmented rank 4 -- the pairing in the charge (generous) reading remains OBSTRUCTED, but with the WEAKER certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 72 - 8 = 64 (the T3 obstruction 32 is traded for a combined (P3, T3) obstruction: the d-channel produces T3 only with the fixed shadow P3 = -T3/4, and the leftover P-block violates the charge-0 span -- the K0 || K2 collapse of v508); the RELAXED reading (charge bookkeeping dropped on the bilinear P-block) becomes EXACTLY solvable: A_fix = -u + 8v + 2w + 1920 Q_dd (residual 0) -- where v508 found T5 = T3 = 0 persisting even relaxed, the d-channel closes the T3 gap with small integer coefficients. [C] NUMBER FENCES (typed [C] only, look-elsewhere discipline): 64 = dim g1 = 2^6; 1920 = |W(D5)| = 2^7 x 3 x 5 = 8 x 240 -- FENCE SHARPENED (2026-07-22, v513/CELEST.DTERM.NONDERIV.01): the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513; the convention-stable [E] content is c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60. [O] OPEN AFTER DELTA-2: the arity >= 4 ledger; the physical justification of the cubic GS term (c_d = 1920 under the e8-Killing propagator); ch2-naturalness for CUBIC vertices UNDECIDED (no canonical geometric datum exists for a cubic vertex); the burden on delta-1 (twisted BCOV contact terms) is SHARPENED: it must supply either the psi = 64 slice or the selection-rule relaxation -- and the delta-1 exploration run (probe, 27/27, exploration-level, v-numbered on demand) stands UNDECIDED: the strict holomorphic q^0 reading is REFUTED at the Z2/Eguchi-Hanson anchor (method boundary, not a kill), the forced leading (T5,T3) ratio is 4:3 (sector 2 : sectors 1+3), the SL(2,Z) orbit decomposition 15 = 12 + 3 identifies the contact term as the MODULAR COMPLETION of the AB data (Harvey-Moore tau-integral, not a free number), and massive coset levels n >= 2 are mandatory. [E] NEGATIVE CONTROLS: (a) Killing control = v508 replication (A_fix = (9, -30, -15, 0, 32), E13 = (16, -96, 144, 0, 0), E22 = (16, 32, 16, 0, 0), E00 = 225 E22); (b) SO(16)/D8 glue: bilinears (1, 0, 1), trilinears (1, 0, 1, 0), Inv(S^3 adj_d8) = 0 -- no symmetric cubic, no odd sectors, no A3 block: the delta-2 channel cannot even be posed (E8/mu4 doubly special); (c) deliberately false g0 = d5 (+) a2 (+) u(1): bilinear entries (5, 2, 2) != (2, 1, 1) and Inv(S^3 g0) = 6 != 1; false sector content (16_s, 4bar) kills every pairing -- the computation SEES the exact structure. SEAM.EQUIV.01 stays [O]; NO marker moves anywhere -- CELEST.SEAM.01 stays Open, the O-fence is only SHARPENED.",
      "status": "Formal/Identity [E] (exact Hom-table/certificate arithmetic: g0 = d5 (+) a3 without u(1), sector factorisation into minuscule orbits, innerness theorem at arities 2 and 3, bilinear table (2, 1, 1, 1), trilinear neutral dims (3, 2, 2, 1, 1), unique symmetric survivor = su(4) d-symbol, Q_dd = (1/60)(T3 - P3/4) two routes, rank-3/4 certificate psi(A_fix) = 64, relaxed solution (-1, 8, 2; 1920), SO(16)/false-g0/false-sector controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the number fences 64 = dim g1 = 2^6, 1920 = |W(D5)| = 8 x 240; the |W(D5)| reading typed look-elsewhere-loaded and convention-contingent by v513) + [O] (the arity >= 4 ledger, the physical justification of the cubic GS term, ch2-naturalness for cubic vertices UNDECIDED; delta-1 UNDECIDED at exploration level at promotion time; since DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, tension with v516 stated); SEAM.EQUIV.01 unchanged; CELEST.SEAM.01 stays Open)",
      "canonicalStatus": "WP5e-delta-2 [E]/[C]/[O] (intermediate verdict): the full-tensor ledger -- the v508 collapse is CONFIRMED full-tensorially and arity-crossing (innerness theorem: h in the g0 Cartan, every invariant tensor charge-0 mod 4; all 15 non-neutral trilinear triples Hom = 0), and the charge-0 ledger contains exactly ONE structure invisible to the quadratic ansatz: the su(4) d-symbol (unique symmetric trilinear on e8, so(10) has no cubic Casimir), whose exchange quartic Q_dd = (1/60)(T3 - P3/4) breaks the Phi_T3 master kill (1/60 != 0) -- the T3 channel OPENS; the pairing stays obstructed in the charge reading with the weaker certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 64 = dim g1 [C], and becomes exactly solvable relaxed: A_fix = -u + 8v + 2w + 1920 Q_dd with c_d = 1920 = |W(D5)| [C] (the |W(D5)| reading look-elsewhere-loaded and convention-contingent -- v513); SO(16)/D8 has no symmetric cubic and no odd sectors (E8/mu4 doubly special), false g0/sector controls separate; the burden on delta-1 is sharpened to the psi = 64 slice or the selection-rule relaxation (delta-1 itself UNDECIDED at exploration level at promotion time, since DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, tension with v516 stated; the pre-v518 state: strict holomorphic reading refuted at the Z2 anchor, contact term = modular completion / Harvey-Moore tau-integral, forced (T5,T3) ratio 4:3, massive coset levels n >= 2 mandatory); SEAM.EQUIV.01 stays [O], no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; SEAM.EQUIV.01; v492; v493; v502; v505; v508; v509",
      "scripts": [
        "v511_celestial_wp5e_delta2_tensor_ledger.py"
      ],
      "script": "v511_celestial_wp5e_delta2_tensor_ledger.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Costello arXiv:2111.08879 (one-loop box anomaly, GS axion coupling); Bittleston-Sharma-Skinner CMP 403 (twistor uplift on ALE); Harvey-Moore hep-th/9510182 (tau-integral / modular completion, framing for the delta-1 fence) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 575
    },
    {
      "id": "SEAM.TAU.FLAG.01",
      "claim": "THE FLAG-TRANSITIVITY EQUIVALENCE WEB -- THE tau = i ATTACK EXECUTED ON THE FREE RP SEAM CIRCLE (2026-07-22; one module, 29 checks; NO marker moves -- the bit stays [C], QGEO.REALIZE.01 and P2.TYPING.01/AX.P2.01 text-precised only, SEAM.BIT.FREEDOM.01 cross-referenced): after v510 the last identification input is the square-modulus datum tau = i; which ESTABLISHED seam datum excludes the free-but-non-harmonic configurations? Test family: the deck-invariant 4-mark configurations on the free circle are, up to rotation, M(delta) = {+-1, +-e^(i delta)}, delta in (0, pi/2]; the v510 counterwitness {+-1, +-(3+4i)/5} is the member delta_cw = arctan(4/3). [E] FALSIFICATION OF BARE TRANSITIVITY: on the free RP circle the symmetry group is G(delta) = {z, -z, b/z, -b/z} ~= V4 for EVERY delta and the two inversions are pair-EXCHANGING -- V4 acts SIMPLY TRANSITIVELY on the four marks for every delta (complete census: any Moebius map commuting with the deck z -> -z is diagonal or antidiagonal); the expectation 'transitivity <= v216 local equality selects tau = i' is REFUTED; hardening: ALL undirected per-mark data (arc-pair multiset {delta, pi-delta}, chordal distance multiset, cone germ) are identical at the four marks for EVERY delta -- no local jet sees the bit. [E] THE SHARP SELECTOR: FLAG transitivity (marks AND their two sides indistinguishable) <=> delta = pi/2 <=> tau = i <=> the clock -- stabiliser jump 1 -> 2, flag/arc orbit collapse 2 -> 1, order-4 mark rotation (b^2 = -1 solveset {pi/2} exactly), mark-fixing mirror z -> 1/z <=> cos delta = 0. [E] 12-FOLD EQUIVALENCE WEB (every link exact): clock rotation, D4 (vs V4), mark mirror, flag transitivity, three spectral degeneracies (odd-doublet split of the seam covariance = (2/pi) ln cot(delta/2), zero only at pi/2 -- counterwitness exactly (2/pi) ln 2; arc-weighted seam-Laplacian degeneracy 4/pi; the v215-K3/v503 mod-4 indicator in CLOSED FORM (2 sqrt 2/pi)|ln tan(delta/2)| -- the v503 lattice curve analytically), rho-twist existence (mode identity (m^2 - n^2)(t - 1/t) = 0 <=> t = 1; Atiyah-Bott rationality is delta-BLIND -- Tr(sigma e^(t Delta)) = 1 for ALL delta, j_cw = 148176/25 rational non-CM: rationality does NOT select), cusp-4-cycle implementability (H4 rigidity: V4 with element orders {1,2,2,2} cannot induce the cusp 4-cycle; joint centraliser scalar, U^2 nonscalar => no projective involution -- the implementer has projective order exactly 4, the Z8-shadow consistent v510), harmonic, j = 1728. [E] NEGATIVE CONTROLS: hexagonal (marks not concyclic, fails the PRECONDITIONS of the battery -- consistent v503/v510); silver (harmonic but topologically dead -- position and modulus halves independent, consistent v510); 3 marks (impossible on the free circle: 0 invariant 3-subsets vs 28 invariant 4-subsets on Z16; on a plain circle bare transitivity ALREADY forces equal spacing at n = 3 -- '4 counts': the deck-pairing is exactly what makes bare transitivity free of charge at n = 4). [C] THE REST (verbatim, no overclaim): the last input reduces from a continuous modulus to ONE discrete symmetry-lift bit (V4 -> D4: side-indistinguishability of the four equally-decorated marks); v216 local equality does NOT deliver it (no local jet can see the side bit); its negation is now concretely measurable (odd-split (2/pi) ln cot(delta/2) != 0). Scoreboard: the counterwitness passes all 7 established unsided tests and fails all 9 clock-equivalent ones; the free RP/Theta battery (v519/v521) is since documented as the EIGHTH side-blind test (7 -> 8), face #13 (the OS-reflection group closes to D4 exactly at the clock point) extends the equivalence web 12 -> 13 (v521), and the mark-decorated twist-state class (everything Wick-computable) is since documented as the NINTH side-blind test (8 -> 9, v525) -- the free-plus-twist class is exhausted. No gate closes, no marker moves.",
      "status": "Formal/Identity [E] (complete V4/D4 symmetry census, flag/arc orbit arithmetic, odd-split and K3-indicator closed forms, twist mode identity, H4 Schur rigidity, negative controls -- sympy exact + mpmath 30 digits, Wolfram-mirrored) + [C] (the symmetry-lift bit itself: flag transitivity stays a postulate; the gain is the FORM, not a derivation)",
      "canonicalStatus": "Flag transitivity [E]/[C]: bare mark-transitivity is FALSIFIED as a selector (pair-exchanging V4 simply transitive for every delta, no local jet sees the bit); the sharp selector is FLAG transitivity (marks AND sides) <=> delta = pi/2 <=> tau = i <=> the clock, welded into a 12-fold exact equivalence web (D4 enhancement, mark mirror, odd-split (2/pi) ln cot(delta/2), arc-Laplacian 4/pi, K3 indicator closed form, rho-twist existence, cusp-4-cycle implementability, harmonic, j = 1728) -- the last input reduces from a continuous modulus to ONE discrete symmetry-lift bit (V4 -> D4), concretely measurable in its negation; the bit stays the one [C] carrier input; the free RP/Theta battery (v521) is the eighth and the twist-state class (v525) the ninth side-blind test (the free-plus-twist class is exhausted); face #13 extends the web to 13; no marker moves",
      "dependencies": "SEAM.BIT.FREEDOM.01; SEAM.BIT.ORIGIN.01; SEAM.CLOCK.RIGIDITY.01; QGEO.REALIZE.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; AX.P2.01; v117; v215; v216; v483; v503; v506; v507; v510",
      "scripts": [
        "v512_seam_tau_flag.py"
      ],
      "script": "v512_seam_tau_flag.py",
      "location": "introduction;tfpt_1;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 576
    },
    {
      "id": "CELEST.DTERM.NONDERIV.01",
      "claim": "THE c_d NEGATIVE CERTIFICATE -- CAN GEOMETRY DERIVE c_d = 1920? (2026-07-22; anti-numerology certificate in the v508 style, honest rigid NEGATIVE result; 24 checks, ~2 s; NO marker moves -- the v511 [C] fence stays [C] and is text-sharpened only, the physical generation of the 32 stays [O], CELEST.SEAM.01 stays Open): the v511 full-tensor ledger found exactly ONE legal cubic channel (the su(4) d-symbol, Q_dd = (1/60)(T3 - P3/4)) with the relaxed solution A_fix = -u + 8v + 2w + 1920 Q_dd and the [C] fence 1920 = |W(D5)|; this module stress-tests the fingerprint via four candidate routes with a PRE-DECLARED look-elsewhere ledger. [E] CONVENTION TABLE: c_d in {1920, 120, 256, 32, 69120} across five normalisations (probe tr-fund/e8-Killing; Gell-Mann; su(4)-Killing; plain trace; ch3-normalised d/6) -- only 1 of 5 hits |W(D5)|; the convention-STABLE content is the factorisation c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60, both factors [E] (Killing propagator 60 from sum_alpha <alpha,e>^2 = 120 over the actual roots; T3-slot fixing via det(P-block) = -64 != 0: c_d is pinned by the T3 slot alone). [E] K1 DEAD: the charge-legal flux pairings (1,3)/(2,2) give 2048/1800 -- they BRACKET 1920 and both MISS; the only flux hit 64*60/2 is the FORBIDDEN (1,2) pairing (charge 3 mod 4, killed by the v511 innerness theorem); LOOK-ELSEWHERE LEDGER: 924 expressions from 21 [E] numbers x grammar {a*b, a*b/2, a*b/4, a+b}, 11 hits on 1920; control targets 1800 -> 8, 2048 -> 5, 2016 -> 0 -- the catalog's multiplicative structure makes any smooth number near 2000 easy to hit. [E] K2 (quantisation gives only a lattice): integer couplings in tr-fund units + e8-Killing propagator put c_d in 60 Z -- 1920 is the 32ND multiple, no minimality; strict ch3-unit couplings (6 Z) give 2160 Z and EXCLUDE 1920 outright; the cubic quantum is lattice-dependent (6 on coroots, 3/8 on weights); the residual integer 32 has 12 signed factorisations, nothing forces it. [E] K3 PROVENANCE CLASH (the death blow to the index route): the su(4) cubic indices per class are A_m = (0, 16, 0, -16) with twisted sum sum_m i^m A_m = 32i, NUMERICALLY equal to Phi_T3(A_fix) = 32 -- BUT the quartic class decomposition (10, -4, 30, -4) is dominated by classes 0/2 while the cubic 32 is built 16 + 16 from classes 1/3: same number, disjoint mechanism = coincidence exposed; the Weyl side is WRONG (the d-channel is an A3-side object, |W(A3)| = 24; D5 has no cubic invariant at all, Inv(S^3 45) = 0; |W(E8)|/|W(A8)| = 1920 is a second accidental 1920 inside E8 combinatorics). [E] K4 NEGATIVE CONTROLS: SO(16)/D8 fails every candidate formula consistently (no d-channel, cubic class sums 0, dark spheres); k = 2 flux doubling scales the flux formula x4 (7680) while c_d is root-theoretic and static; false g0 = d5+a2+u(1) has SIX symmetric cubics (the question cannot be posed); so(10) sanity (Tr X^3 = 0, T5 protection). VERDICT: NUMEROLOGY as a derivation claim -- the v511 [C] fence stays a fence, now typed look-elsewhere-loaded; the honest [E] core is c_d = 32 x 60; the physical generation of the 32 stays [O].",
      "status": "Formal/Identity [E] (exact convention/enumeration/lattice/index arithmetic -- sympy/Fraction, no floats, Wolfram-mirrored); the v511 [C] fence '1920 = |W(D5)|' TEXT-SHARPENED to look-elsewhere-loaded and convention-contingent (fence stays [C]); [O] the physical justification of the cubic GS term and the generation of the 32",
      "canonicalStatus": "c_d non-derivation [E]/[O]: the fingerprint c_d = 1920 = |W(D5)| does NOT survive the honest battery -- convention table {1920, 120, 256, 32, 69120} (only 1 of 5 normalisations hits), charge-legal flux products 2048/1800 bracket and miss (the 64*60/2 hit is charge-forbidden), look-elsewhere 11/924 vs control 1800 -> 8/924, quantisation gives only lattices (60 Z, 32nd multiple; ch3 units exclude), and the twisted cubic index 32i clashes in provenance with the true quartic 32 (classes 1/3 vs 0/2) -- NUMEROLOGY as a derivation claim; the convention-stable [E] core is c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60; the v511 [C] fence text-sharpened only, the physical generation of the 32 stays [O]; no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.DELTA2.01; v505; v508; v509; v511",
      "scripts": [
        "v513_celestial_dterm_nonderivation.py"
      ],
      "script": "v513_celestial_dterm_nonderivation.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 577
    },
    {
      "id": "CELEST.WP5E.EPS1.01",
      "claim": "WP5e-EPSILON-1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 work package 5e, epsilon-1 stage -- 'the O(-2) bulk-axion slot', verdict B; exact sympy/Fraction arithmetic, no floats, 34 checks, ~4 s): build the s = 0 tower field explicitly (v505 E3: the BSS axion is, via Omega-contraction, the O(-2) slot; a0 does NOT fill it), pin lambda~ = 6 (v495: lambda~^2 = 36 = h_vee + 6) as a residue with explicit factor bookkeeping on PT/Z4, and take the first honest back-reaction step (the Gibbons-Hawking A3 form of the CPS Omega_N mechanism, arXiv:2306.00940). [E] EQUIVARIANT O(-2) SLOT: pushforward ledger pi_* O(-2) on PT' = tot(O(1)+O(1) -> P^1): per spacetime degree d exactly (d+1)^2 = nullspaces of the wave operator (1, 4, 9, 16, 25, 36 for d = 0..5); the v492 action on the fibre coordinates (mu1, mu2) -> (i mu1, i^-1 mu2), incidence forces column weights (+1, +1, -1, -1); the equivariant bookkeeping closes BLOCK BY BLOCK (all d <= 6, all 4 characters); character series P_0 = 1 + 3t^2 + 15t^4 + 21t^6 + 45t^8 + ..., P_1 = P_3 = 2t + 8t^3 + 18t^5 + ..., P_2 = 6t^2 + 10t^4 + 28t^6 + ...; the d = 0 slot has multiplicity (1, 0, 0, 0) -- THE BULK AXION SURVIVES THE PROJECTION (density 1/|Z4|); Molien invariant ring = (1 - t^8)/((1 - t^4)^2 (1 - t^2)) -- Z in O(2), X, Y in O(4), relation degree 8 = the a0 weight (v492/v493 bridge); E3 bijection sharpened: twisted characters {i, -1, -i} with minimal content (2t, 6t^2, 2t) and NO degree-0 mode = the Coxeter eigenvalues on H^2(ALE), det(A-1) = -4; graviton control: the O(+2) slot starts invariant only at fibre degree 4 with multiplicity 3 = {X, Z^2, Y} (the cohomological form of v505 X = -2 != +2 = Y). [E] LAMBDA~ = 6 TRIPLY PINNED: (1) Okubo: Tr_adj X^4 = 36<x,x>^2 = (6<x,x>)^2 on the 240 glue roots, 36 = h_vee + 6, lambda~ = 6 = |Z2| N_fam; (2) measure cancellation: mu = 1/4 enters three times (vertex^2 mu^2, propagator 1/mu, anomaly mu) and cancels EXACTLY -- lambda~_eff^2 = 36 for ARBITRARY mu; wrong bookings quantified and excluded: anomaly-only => lambda~ = 3 = N_fam, missing propagator renormalisation => lambda~ = 12 = |mu4| N_fam; (3) flux: minimal CPS quantum N = 1, a single exchange channel exactly at k = 1 (#primaries), (kappa/c3)^2 = 36/3 = 12 exact. [E] GH/A3 BACK-REACTION STEP: invariant 4-centre configurations = exactly 2 branches (axis^4 = pure resolution or ONE free mu4 orbit = pure deformation); free orbit => prod(Z - i^p z0) = Z^4 - z0^4 => XY = Z^4 + a0 with a0 = -z0^4 (the v493 family RE-DERIVED from the geometry); monodromy a0 -> e^{2 pi i} a0 = one clock step; the GH point of charge k is EXACTLY R^4/Z_k (k = 4 => seam boundary S^3/Z4); periods int omega = 4 pi (x_q - x_p), free orbit: Pi_j = 4 pi t0 (i-1)(1, i, i^2), |Pi_j| equal (LOCKSTEP, counter-check of v509); the Coxeter clock FROM THE GEOMETRY (A^4 = 1, det(A-1) = -4, Pi.A = i Pi); the CPS 'N log' analogue = source charge 4 = |mu4| (flux -4 pi per centre); HONEST SHARPENING: for the Ricci-flat ALE the asymptotic log coefficient is EXACTLY 0 (EH: u K' = sqrt(u^2 + a^4), det g = 1) -- the log lives at the exceptional locus (K' ~ a^2/u, flux 2 pi a^2); Burns contrast (K = u + N log u => det g != 1, only scalar-flat); bonus: multipole selection rule m = 0 mod 4, first symmetry-breaking multipole (4, +-4) carries -a0 with amplitude (35/16) sin^4 th. [E] NEGATIVE CONTROLS: so8 (lambda~^2 = 12 irrational; perfect squares in the Deligne series only {9, 36} = {sl3, e8}); diag(i, i) (Veronese cone, 5 generators, no hypersurface, the bijection collapses); k = 2 (three channels; honest: lambda~^2 = 36 is k-independent, the kill is the closure dial)",
      "status": "Formal/Identity [E] (exact slot/residue/GH arithmetic: equivariant Penrose ledger block by block d <= 6, character series P_m, Molien hypersurface relation degree 8, Okubo perfect square (6<x,x>)^2, measure chain mu-exact excluding 3 and 12, flux single-channel iff k = 1, GH centre dichotomy, period lockstep 4 pi t0 (i-1)(1, i, i^2), Coxeter clock from geometry, EH log ledger with asymptotic coefficient exactly 0, multipole selection rule m = 0 mod 4 -- sympy/Fraction exact, Wolfram-mirrored) + [C] (Costello's flat-PT matching lambda_g^2 E = A: the measure chain is CLASSICAL bookkeeping) + [O] (the quantised BCOV one-loop coefficient on PT/Z4; the twisted channels 32 T3 not covered by the bulk axion -- M1-M3 preregistered with success + kill criteria)",
      "canonicalStatus": "WP5e-epsilon-1 [E]/[C]/[O] (verdict B): the O(-2) bulk-axion slot -- executed. CONSTRUCTED exact: the slot is an explicit equivariant mode ledger (Penrose accounting closes block by block, d <= 6, all 4 characters; d = 0 multiplicity (1, 0, 0, 0): the bulk axion survives the projection; Molien = the v492 hypersurface XY = Z^4 with relation degree 8 = the a0 weight; twisted minimal content (2t, 6t^2, 2t) = the Coxeter eigenvalues -- the E3 bijection per character; graviton control O(+2) separates a0); lambda~ = 6 is pinned by three exact ledgers (Okubo (6<x,x>)^2; measure chain mu-exact with the wrong bookings 3 and 12 excluded; flux single-channel iff k = 1); the GH/A3 back-reaction step re-derives the v493 family and the Coxeter clock from centre geometry (2 branches, period lockstep, source charge 4 = |mu4|, EH asymptotic log exactly 0 -- the CPS log is an exceptional-locus statement, multipole rule m = 0 mod 4 with (4, +-4) carrying -a0). CONDITIONAL [C]: Costello's flat-PT matching. OPEN [O]: the quantised BCOV one-loop coefficient on PT/Z4 and the twisted channels (32 T3) -- M1 (A3 Omega_N), M2 (twisted KS measure), M3 (a0 uplift) preregistered with success + kill criteria; WP5e stays [O]; SEAM.EQUIV.01 untouched; no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.DELTA2.01; SEAM.EQUIV.01; v492; v493; v495; v505; v508; v509; v511",
      "scripts": [
        "v514_celestial_wp5e_eps1_axion_slot.py"
      ],
      "script": "v514_celestial_wp5e_eps1_axion_slot.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Costello arXiv:2111.08879 (box anomaly, GS axion coupling); Bittleston-Sharma-Skinner arXiv:2208.12701 (Poisson-CS tower); Costello-Paquette-Sharma arXiv:2306.00940 (Omega_N backreaction, level = flux quantum); Gibbons-Hawking Phys. Lett. 78B (1978) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 578
    },
    {
      "id": "CELEST.WP5E.M1.01",
      "claim": "MILESTONE M1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M1 -- 'the A3 Omega_N', SUCCESS on the preregistered v514 S8.1 criterion; exact sympy arithmetic, no floats, 30 checks, ~1 s): compute the A3 analogue of CPS eq. 3.5 on PT/Z4 -- the back-reacted holomorphic 3-form Omega_N sourced by the glue currents wrapping the three lockstep spheres -- and check its periods exactly (preregistered SUCCESS = a closed-form Omega-deformation whose S^3/Z4 periods are (2 pi i)^2 x (integer flux vector) reproducing the lockstep phase structure (i-1)(1, i, i^2); KILL = unequal sphere fluxes or a non-integer period). [E] RESIDUE FORM DERIVED: on F = XY - P(Z) all three ambient representatives satisfy dF ^ omega2 = -dX^dY^dZ and pull back to omega2 = dX^dZ/X (Poincare residue, derived not assumed); cover pullback at a0 = 0: omega2 = 4 dz1^dz2 = |mu4| x (flat form) -- the residue normalisation CARRIES the source charge 4; clock diag(i,1): omega2 -> i omega2 (v492 S5 replicated); weight ledger 4+4+2-8 = 2, Omega in O(4) = -deg K_PT; EH anchor 2 = |Z2|; wrong-weight control Z in O(4) breaks three ways (4 != 2, 6 != 4, 5 != 3). [E] THE TWISTOR FAMILY CLOSED: from the four O(2) centre sections q_p(la) = i^p t0 - i^-p t0 la^2 the family is EXACTLY XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4 (1 - la^4)^2 (e1 = e3 = 0 identically; la = 0 gives exactly the v493 clock family; a2 in O(4) opens only off-seam, a2(0) = a2(inf) = 0); clock lift gamma: (Z, la) -> (iZ, -i la), gamma^4 = 1, Omega -> +Omega (CY-compatible); Omega = omega2 ^ <la dla> in O(4), weight chain 4+4+2-8+2 = 4. [E] OMEGA_N CLOSED + QUANTISED PERIODS: Omega_N = Omega_0 + sum N_p K_p with CPS/Bochner-Martinelli kernels on the centre twistor lines L_p = {X = Y = 0, Z = q_p(la)}; dK = 0 away from the source, scale degree 0; PERIOD TABLE: boundary S^3/Z4 at the orbifold point (2 pi i)^2 x 1 (minimal invariant charge 4 = |mu4| -- the lens fundamental domain FORCES N in 4Z: quotient large-gauge quantisation (2 pi i)^2 N/4 integer only for N = 0 mod 4), deformed phase (2 pi i)^2 x 4, linking S^3 uniformly (2 pi i)^2 x N, per-sphere flux vector (2 pi i)^2 x N (1,1,1) (forced uniform twice over: clock orbit + K4 connectivity of the four centre lines, all 6 pairs meet), seam-fibre phase vector 2 pi i t0 (i-1)(1, i, i^2); the undeformed Omega_0 carries ZERO quantised 3-flux (all flux sourced); (2 pi i)^2 = (fibre residue) x (transverse residue) (density -sin 2 theta); orbifold-limit cover count 16 = |mu4|^2; minimal quantum 1 <-> k = 1 (v509 bridge). [E] CLOCK COVARIANCE: periods 2 pi i (q_{j+1} - q_j) generic-exact (profile and path drop identically), seam vector moduli all 2 sqrt(2) pi t0, Pi_{j+1}(-i la) = i Pi_j(la) for ALL la; the 12 collision nodes sit exactly on the 8 eighth roots of unity (8 = 2|mu4|, the Z8 seam bridge), antipodal double nodes, clock orbits 4+4+4, honest conifold points (Hessian det 512 t0^6, quadric rank 4, exceptional curves = sphere classes). [E] KILL PROBE (honest): the clock-forbidden family Z^4 - Z: 0/24 orderings lockstep (vs 8/24 for the mu4 orbit), la-support 12 twelfth roots != Z8, no antipodal pairing, no clock lift -- BUT the (2 pi i)^2 quantisation itself holds there TOO (local gauge invariance): INTEGRALITY ALONE IS NOT THE DISCRIMINATOR; the discriminator is the lockstep phase/modulus structure + the clock forcing of equal fluxes (unequal N_p allowed on the forbidden family); the v509 dial survives, the KILL does not fire. [E] ANCHORS + CONTROLS: EH/Z2 from the same machinery (XY = Z^2 - t0^2(1 - la^2)^2, exactly 2 nodes at la = +-1, lens forces charge 2 = |Z2|); CPS original = k = 1 flat patch (full S^3: every integer N allowed -- the unconstrained anchor); diag(i,i): Omega_0 -> -Omega_0 (no invariant 3-form, collapse at step zero) while K stays invariant (U(2)); fractional charges N = 1, 2, 3 excluded (lens periods 1/4, 1/2, 3/4)",
      "status": "Formal/Identity [E] (exact residue/family/period/kernel arithmetic: Poincare residue three routes, cover factor 4 = |mu4|, closed twistor family XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4(1 - la^4)^2, CY-compatible clock lift Omega -> +Omega, period lockstep 2 pi i t0 (i-1)(1, i, i^2) with covariance Pi -> i Pi, BM kernel (2 pi i)^2 anchor + dK = 0 + scale degree 0, lens forcing N in 4Z, 12 conifold nodes on Z8 -- sympy exact, no floats, Wolfram-mirrored) + [C] (global patching of the local BM kernels on the resolved 3-fold; Hitchin small resolution; the CPS 'exceptional-curve branes = glue currents' dictionary) + [O] (M2: the twisted KS measure per sector; M3: the a0 uplift; the quantised BCOV coefficient on PT/Z4)",
      "canonicalStatus": "M1 [E]/[C]/[O] (SUCCESS on the preregistered criterion): M1 executed via v515 -- the back-reacted Omega_N is closed-form on the A3 twistor family (Omega_N = Omega_0 + sum N_p K_p, BM kernels on the centre twistor lines), all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector N(1,1,1,1) and clock covariance Pi -> i Pi, and the lens geometry FORCES the source charge 4 = |mu4| (quotient large-gauge quantisation: only N = 0 mod 4 passes); honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden Z^4 - Z family vs 8/24) + the clock forcing of equal fluxes; neither preregistered KILL fires. CONDITIONAL [C]: global kernel patching, Hitchin small resolution, CPS brane dictionary. OPEN [O]: M2 (twisted KS measure), M3 (a0 uplift), the quantised BCOV coefficient on PT/Z4; SEAM.EQUIV.TWISTOR.01 untouched (v515 is preparation, not the Costello-Li construction); no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.BETA.01; SEAM.EQUIV.TWISTOR.01; v492; v493; v505; v509; v514",
      "scripts": [
        "v515_celestial_wp5e_m1_omega_n.py"
      ],
      "script": "v515_celestial_wp5e_m1_omega_n.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Costello-Paquette-Sharma arXiv:2306.00940 (Omega_N brane backreaction, eq. 3.5/3.33; large-gauge quantisation); Hitchin 'Polygons and gravitons' (A-series twistor family, small resolution); Gibbons-Hawking Phys. Lett. 78B (1978) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 579
    },
    {
      "id": "CELEST.WP5E.M2.01",
      "claim": "MILESTONE M2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M2 -- 'the twisted Kodaira-Spencer measure', SUCCESS on the preregistered v514 S8.2 criterion ON THE DECLARED COMPLETION MEASURE, verdict B; exact sympy/Fraction arithmetic, no floats, 23 checks, ~1 s): quantise the O(-2) tower on PT/Z4 using the eps1 mode ledger and compute the one-loop box coefficient PER SECTOR (preregistered SUCCESS = the bulk sector reproduces lambda~^2 = 36 with the eps1 measure bookkeeping AND the twisted sectors pair with the three sphere axions cancelling the rigid 32 T3 residual; KILL = a leftover independent quartic (T3-type) in ANY sector). DECLARED MEASURE ANSATZ (stated before computing): v508 killed every exchange realisation; the delta-1 exploration identified the twisted contact term as the MODULAR COMPLETION of the Atiyah-Bott data with forced leading (T5, T3) ratio 4:3; M2 declares contact_j = (Q^(0) - Q^(j))/det_j -- in the channel with the g^j zero-mode normalisation 1/det_j the twisted-sector LOOP contributes the UNPHASED sector trace, of which the AB skeleton kept only the phase-weighted insertion part. [E] THE COMPLETION-WEIGHT IDENTITY (the centrepiece): per glue class w_m = sum_j (1 - i^{jm})/det_j = (0, 3/2, 2, 3/2) = 4 h_m = |mu4| h_m = -4 ch2(T_m) EXACTLY (h_m = m(4-m)/8 the v502 sector Casimir energies, ch2(T_m) the McKay/Kronheimer charges, v505 S4.3 index bridge): the three sphere axions pair through their OWN ch2 charges -- an exact identity, NO free scale, no fit. [E] PARAMETER-FREE LOCKS: T5 = 0 for ANY scale c (the ch2 pattern alone kills T5); ratio h_2 : h_1 = 4 : 3 = the delta-1-forced leading ratio (REPRODUCED, not fitted); the T3 budget fixes the scale c = 4 = |mu4| uniquely; the unique delta-1 leading weights (3/2, 2) reproduced as (4h_1, 4h_2). [E] THE CONTACT TERM: channel form = class form = (36, 120, 60, 0, -32) EXACTLY (one object, two bases); the contacts carry nonzero T5/T3 per channel (genuine contact, not exchange -- the v508 product theorem replicated as control). [E] SUCCESS TABLE: skeleton_j + contact_j = Q^(0)/det_j = 36<x,x>^2/det_j -- a PERFECT Okubo square (T5 = T3 = 0, lambda~^2 = 36) in EVERY twisted channel ((18, 9, 18) x <x,x>^2); total A_fix + contact = 45<x,x>^2 = (5/4) x 36<x,x>^2 = (Dedekind sum) x (Okubo) = the UNIQUE quartic-free weighting of v505 S3.8 (the rigidity theorem is exactly what the measure lands on); both v508 certificates killed (Phi_T3: 32 -> 0, Phi_P: 72 -> 0); the v511 slice psi = 64 SUPPLIED EXACTLY (psi(contact) = -64) -- no cubic d-channel needed, c_d is free to be 0; remainder inside the exchange span (45<x,x>^2 = (45/16) E22 = (1/80) E00, Phi_P = 0): the BULK axion cancels every channel with the SAME lambda~^2 = 36, the twisted propagator carrying the AB zero-mode factor 1/det_j; the v514 S3.4 measure chain stays mu-blind. [E] NEGATIVE CONTROLS: wrong scale c in {1, 2, 3, 5} leaves T3 = (24, 16, 8, -8) -- only c = 4 = |mu4| clears the budget; shuffled weights (h_1 <-> h_2) break T5 = -14 AND T3 = 36; SO(16) glue keeps T5 = 20 / T3 = -40 in the summed trace -- the KILL branch FIRES there (Okubo failure inherited, E8 doubly special); diag(i, i) degenerates to the Z2 target and breaks the conjugation pairing + the index bridge; the Z2/Eguchi-Hanson anchor PASSES with the same mechanism (A_2 + contact = 9<x,x>^2 = (1/4) x 36<x,x>^2, coupling scale 2 = |Z2| = centre count). [E] MULTIPLIER NOTE (cross-reference only, substance belongs to the parallel delta-1d strand): the contact weights 4h are REAL RATIONALS carrying no multiplier system of their own; the delta-1c residual phases are the SQUARES of the A3 discriminant T-phases e(3a^2/8) -- orders (1, 8, 2, 8) squaring to (1, 4, 1, 4), a finite mu4 system; Gauss sums 2e(3/8), 2e(5/8), full Weil sum 4. [C] THE COMPLETION READING (the mandatory fence): the loop of twisted states carries the UNPHASED sector trace with the same zero-mode normalisation -- declared, supported by the delta-1 modular-completion finding, NOT derived from the BCOV integral. [O] the BCOV-integral derivation of the completion reading (delta-1b/1c/1d territory; since executed and DECIDED by v518 -- kill under the derived measure, tension stated -- and the measure question since DECIDED at probe level by v520: the completion reading is now DERIVED at probe level from two independent constructive sources (F-independence + Quillen pairing) under the typed premises TP-1..TP-4 (v520); the residual [O] is the constructive BCOV derivation of the w_m normalisation itself -- since DERIVED at probe level by v523 (CELEST.WP5E.WM.01): 1/det_j is the Atiyah-Bott/zeta-determinant fixed-point factor, computed from three independent sources under the typed premises TP-REG/TP-Q/TP-NUM/TP-CH, and the residual [O] narrows to the GLOBAL BCOV integral beyond the fibre zero-mode factor); SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction). VERDICT: SUCCESS wording met on the declared completion measure; the KILL does NOT fire; NO marker moves anywhere.",
      "status": "Formal/Identity [E] (exact weight/contact/table/control arithmetic: completion-weight identity w = 4h = -4 ch2 with no free scale, parameter-free locks (T5 = 0 any scale, 4:3 ratio, T3 budget => c = |mu4|), contact term two bases, per-sector Okubo squares 36<x,x>^2/det_j, total (5/4) x 36, certificate kills 32 -> 0 / 72 -> 0 / psi 64 -> 0, exchange-span remainder, mu-blind chain, all five controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the DECLARED completion reading: loop = unphased sector trace with the same zero-mode measure -- supported by the delta-1 modular-completion finding, not derived) + [O] (the BCOV-integral derivation of the completion reading, delta-1b/1c/1d; SEAM.EQUIV.TWISTOR.01 unchanged)",
      "canonicalStatus": "M2 [E]/[C]/[O] (SUCCESS on the preregistered criterion, on the DECLARED completion measure): M2 executed via v516 -- the completion contact contact_j = (Q^(0) - Q^(j))/det_j carries the exact weight identity w_m = sum_j (1 - i^{jm})/det_j = 4 h_m = |mu4| h_m = -4 ch2(T_m) (the sphere axions pair through their own McKay ch2 charges, no free scale); parameter-free locks (T5 = 0 for any scale, ratio 4:3 reproduced, T3 budget forces c = 4 = |mu4|); every twisted channel becomes the perfect Okubo square 36<x,x>^2/det_j, total 45<x,x>^2 = (5/4) x 36 = Dedekind x Okubo (the v505 S3.8 unique quartic-free weighting); both v508 certificates killed and the v511 psi = 64 slice supplied exactly -- the cubic d-channel is NOT needed (c_d free = 0); controls: wrong scale/shuffle/so16 (KILL fires there)/diag(i,i)/Z2-EH anchor (scale 2 = |Z2|). FENCE [C]: the completion reading (loop = unphased sector trace with the same zero-mode normalisation) is DECLARED within this module, supported by the delta-1 modular-completion finding, NOT derived from the BCOV integral here -- the completion reading is since DERIVED at probe level from two independent constructive sources (F-independence + Quillen pairing) under the typed premises TP-1..TP-4 (v520); the w_m normalisation is now DERIVED at probe level -- 1/det_j is the Atiyah-Bott/zeta-determinant fixed-point factor, computed from three independent sources (v523); the residual [O] narrows to the global BCOV integral beyond the fibre zero-mode factor; no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.DELTA2.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; SEAM.EQUIV.TWISTOR.01; v495; v502; v505; v508; v511; v513; v514; v515",
      "scripts": [
        "v516_celestial_wp5e_m2_twisted_ks_measure.py"
      ],
      "script": "v516_celestial_wp5e_m2_twisted_ks_measure.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Costello arXiv:2111.08879 (one-loop box anomaly, GS axion coupling); Bittleston-Sharma-Skinner CMP 403 (Poisson-CS tower); Harvey-Moore hep-th/9510182 (tau-integral / modular completion, framing) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 580
    },
    {
      "id": "CELEST.WP5E.M3.01",
      "claim": "MILESTONE M3 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M3 -- 'the a0 uplift', SUCCESS on the preregistered v514 S8.3 criterion; exact sympy arithmetic, no floats, 23 checks, ~7 s): uplift the (l, m) = (4, +-4) GH multipole (= the a0 direction) to the Beltrami differential on PT/Z4 and compute the induced Kahler-potential correction (preregistered SUCCESS = a log-type correction whose coefficient is tied to the source charge 4 = |mu4|, matching the V-ledger; KILL = coefficient decoupled from the centre count). THE UPLIFT OBJECT (declared): the generalized-Legendre-transform kernel per centre, G(eta_p) = eta_p log eta_p (Lindstrom-Rocek / Ivanov-Rocek, typed [C] as a dictionary); its derivative kernel chi = sum_p log eta_p = log P4 is THE LOG OF THE v515 TWISTOR-FAMILY POLYNOMIAL. [E] KERNEL BRIDGE: the O(2) section is a NULL coordinate -- Laplacian(3d) G(eta_p) = 0 identically for an ARBITRARY kernel (the uplift is well-typed); RESIDUE IDENTITY d_x [transform(log eta_p)] = 1/r_p EXACTLY (residue 1/(2r)): the GH potential V = sum_p 1/r_p IS the x-derivative of the contour transform of chi; flux -4 pi per centre, total source charge 4 = |mu4| (the V-ledger matched). [E] FOUR COUPLED CENTRE-COUNT SCALES: (1) asymptotic kernel log chi = 4 log eta + a2/eta^2 + (a0 - a2^2/2)/eta^4 + ... -- LOG COEFFICIENT 4 = |mu4| = centre count (the CPS 'N log' analogue at kernel level); at the clock-fixed seam fibre zeta = 0 the a2 slot closes and the first correction is EXACTLY a0/eta^4 = the (4, +-4) multipole in kernel clothes; exact m-grading (zeta^k/eta^4 carries field charge m = 4 - k: m = +-4 pieces PURE a0, m = 0 piece -6 t0^4 phi0-free -- the U(1) -> Z4 breaking sits exclusively in the a0 pieces); NO log x power terms in the Laurent tower (the v514 honest zero -- no asymptotic METRIC log -- preserved); (2) the once-integrated GLT kernel sum_p eta_p log eta_p = 4 eta log eta + p_4/(12 eta^3) + p_8/(56 eta^7) + ... with p_n = 0 unless n = 0 mod 4 (selection rule at potential level) and p_{4k} = 4 (-a0)^k: EVERY correction carries the prefactor 4 = centre count; (3) the exceptional-locus log chi(eta = 0) = log a0 = 4 log t0 + i(4 phi0 + pi) -- t0-coefficient 4 = |mu4|, clock phase 4i; modulus response a0 d/da0 chi = a0/(eta^4 + a0) equals 1 AT the exceptional locus and falls off as a0/eta^4 at infinity (no constant, no log): the a0-log lives at the exceptional locus, exactly as v514 S6.3 demanded; (4) periods (v515 consistency): Pi_j(la) LINEAR in t0 (uniform first-order shift, lockstep + phase ratios (1, i, i^2) preserved exactly); d log Pi_j / d log a0 = 1/4 = 1/|mu4| uniformly; integrated over the a0 circle: monodromy e^{2 pi i/4} = i = ONE Coxeter clock step (the v493 monodromy REPRODUCED from the perturbative period shift; the finite monodromy is the cyclic centre permutation q_p -> q_{p+1}); node la-support t0-FREE (the Z8 seam bridge is a0-rigid); all 6 pair discriminants ~ t0^2 != 0 (K4 connectivity: the forced uniform flux vector persists); the BM kernel is modulus-free with int_{S^3} K = (2 pi i)^2 (the quantised fluxes cannot move). [E] NEGATIVE CONTROLS: the (4, 0) multipole is phi0-independent in V AND in the kernel (-6 t0^4): clock-invariant, breaks nothing; Z2/Eguchi-Hanson analogue: EVERY dial reads 2 = |Z2| ((2, +-2) amplitude (3/2) sin^2 th, kernel log 2, exceptional log 2 log t0, period response 1/2, monodromy -1 = the Z2 clock); wrong centre counts k = 3, 5: kernel log coefficient k, modulus slot O(6)/O(10) != O(8) -- the observable MOVES with the centre count; forbidden family Z^4 - Z: p3 = 3 != 0 (selection rule broken), e4 = 0 (no a0-log at the exceptional locus) -- fails BOTH uplift dials. [C] THE GLT DICTIONARY: 'eta_p log eta_p = THE GH Kahler-potential kernel' (Lindstrom-Rocek / Ivanov-Rocek). [O] the full nonlinear Kahler potential of the resolved A3 ALE (no closed form exists); the quantised BCOV coefficient stays with the delta-1 strand; SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction). VERDICT: SUCCESS -- the log-type correction delivered on four coupled dials (kernel log 4, exceptional-locus log 4 log t0, tower prefactor 4, period response 1/4 -> monodromy i), every one tied to the centre count (EH: 2, k-orbit: k); the KILL branch does NOT fire; NO marker moves anywhere.",
      "status": "Formal/Identity [E] (exact kernel/expansion/period/monodromy arithmetic: null-coordinate harmonicity for arbitrary kernels, residue V-ledger identity 1/(2r), kernel log coefficient 4 = |mu4| with first correction a0/eta^4 and exact m-grading, GLT tower p_{4k} = 4(-a0)^k with the n = 0 mod 4 selection rule, exceptional-locus log 4 log t0 + clock phase 4i, period response 1/4 integrating to monodromy i, a0-rigid node support and (2 pi i)^2 fluxes, all four controls -- sympy exact, Wolfram-mirrored) + [C] (the GLT dictionary eta_p log eta_p = the GH Kahler-potential kernel; Lindstrom-Rocek / Ivanov-Rocek) + [O] (the full nonlinear Kahler potential of the resolved A3 ALE; SEAM.EQUIV.TWISTOR.01 unchanged)",
      "canonicalStatus": "M3 [E]/[C]/[O] (SUCCESS on the preregistered criterion): M3 executed via v517 -- the (4, +-4) multipole uplifted to the GLT kernel chi = log P4 (the log of the v515 family polynomial; harmonic for any kernel, residue identity matches the V-ledger with flux -4 pi per centre); the log-type correction is delivered on FOUR COUPLED CENTRE-COUNT SCALES: asymptotic kernel log 4 = |mu4| (first seam-fibre correction exactly a0/eta^4, exact m-grading, no log x power terms -- the v514 honest zero preserved), GLT tower p_{4k} = 4(-a0)^k with the n = 0 mod 4 selection rule, exceptional-locus log chi(0) = log a0 = 4 log t0 + i(4 phi0 + pi), and period response d log Pi/d log a0 = 1/4 = 1/|mu4| integrating to the clock monodromy i (v493 reproduced) with a0-rigid node support and topological (2 pi i)^2 fluxes; controls: (4,0) clock-invariant, EH/Z2 reads 2 = |Z2| on every dial, k = 3/5 orbits move the coefficient with the centre count, the forbidden family fails both dials (p3 = 3, e4 = 0) -- the KILL (decoupling from the centre count) does NOT fire. FENCE [C]: the GLT dictionary; [O]: the full nonlinear Kahler potential of the resolved A3 ALE; no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; CELEST.WP5E.M2.01; SEAM.EQUIV.TWISTOR.01; v493; v505; v509; v514; v515; v516",
      "scripts": [
        "v517_celestial_wp5e_m3_a0_uplift.py"
      ],
      "script": "v517_celestial_wp5e_m3_a0_uplift.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Lindstrom-Rocek CMP 115 (1988) (generalized Legendre transform); Ivanov-Rocek CMP 182 (1996) (O(2) multiplet kernels); Costello-Paquette-Sharma arXiv:2306.00940 ('N log' Kahler flux); Gibbons-Hawking Phys. Lett. 78B (1978) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 581
    },
    {
      "id": "CELEST.WP5E.DELTA1.01",
      "claim": "THE DELTA-1 CHAIN OF THE CELESTIAL-HOLOGRAPHIC CONTRACT DECIDED (2026-07-22; CELEST.SEAM.01 WP5e, delta-1b + delta-1c + delta-1d consolidated -- 'the derived chiral measure kills the delta-1 route', an honest, decided NEGATIVE result in the v508 style; 30 checks, ~20 s): does the Harvey-Moore/BCOV fundamental-domain integral on PT/Z4, evaluated under a measure DERIVED from blockwise SL(2,Z) covariance of the dressed 16-component Weil system (solved for -- not declared, not scanned), deliver the twisted contact vector -A_fix = -(9,-30,-15,0,32) or the psi = 64 slice? NO. (1) THE 16-COMPONENT WEIL COMPLETION (delta-1c): discriminant module Z4 x Z4 with q = (5x^2+3y^2)/8 mod 1, Gauss sums 2 zeta_8^5 x 2 zeta_8^3 = 4 = sqrt(16) (signature 0 mod 8 = rank E8), exact Q(zeta_8) relations (S^2 = C, (ST)^3 = S^2, S^4 = 1, T^8 = 1), tensor split S = S_D5 (x) S_A3, the diagonal H and anti-diagonal H' the two invariant Lagrangians (both counts = E4: two E8 gluings), P_H S P_H = (1/4) J with 3/4 of every S-image weight on the 12 shifted classes; the 16-vector theta S-covariance CLOSES: the delta-1b E1.5 residual 2.91 = O(1) (naive 4-character rule) drops to ~1e-39, certified at two tau. (2) THE OBSTRUCTION IS A FINITE mu4 CHARACTER, IDENTIFIED (delta-1d): gauge blocks M = G[a,b] eta^{-8} with constant transport (28+ digits) recognised as exact 1/960-grid phases (1e-25); T-fixed pairs carry t = e(-1/2) = -1 EXACTLY (the naked level-matching defect); loops in Gamma_1(4) resp. Gamma_0(2)+-, joint holonomy eigenspaces dim 8/16 resp. 6/16, every eigen-phase of order <= 4; KOBOUNDARY TEST: the SL(2,Z) relation defects c_{S^4}, c_{(ST)^3 S^-2}, c_{[S^2,T]} are (1,1,1) EXACTLY on all 15 pairs -- the multiplier system is a CHARACTER of the orbit stabilisers, not a genuine 2-cocycle; on Gamma_1(4) it is lambda(gamma) = i^{2B + C/4}. (3) THE CANCELLATION EXISTS AND IS THE TWISTED FIBRE BLOCK f1 f3, NOT THE THREE SPHERE AXIONS: exact identity G[a,b] = f_1 f_3 (30+ digits; the axion T-phases are v502 vacuum-energy data); the T-fix mechanism is one line of exact phase arithmetic, (-1) x e(-1/6) = e(1/3) = chi_4(T) at every T-fixed node; CANCELLATION TABLE (exact, minimal residual order over eta^m x chi_k): none -> 4, f1f2f3 (the three sphere axions) -> 4, f2 -> 6, f1f3 -> 1 (hol AND anti), wrong weights (1,1,2)/(2,2,2) -> 4; the strict solutions are exactly chi_4 (dims (3,3)) and chi_10 (dims (1,1)) at eta^0, certified on the dressed functions at a fresh tau (1e-15, NO residual multiplier). (4) THE DECISION -- ALL THREE TESTERS FAIL: under both derived solutions the integral (30-digit kernels, negative-exponent cell audit, cut-6 vs cut-8 scan) fails T5(V) = 0 (fractions 0.5 / 0.83), fails W_2 : W_13 = 4 : 3 (W_13 = 0), fails V = -A_fix N (spreads 1.05 / 1.47) and the psi = -64 N slice; the honest (N1, N2) orbit rebalance: chi_4 reaches T5 = 0 only at psi > 0 (wrong sign), chi_10 only at N2/N1 = -4 < 0 (outside the orbifold projector) -- no branch fires anywhere in the positive cone: a GENUINE KILL on the derived surface. (5) NEGATIVE CONTROLS: Z2/Eguchi-Hanson (gauge-only residual order 2 -- the mu4 defect is Z4-specific -- and the SAME fibre dressing cancels it to 1: the mechanism replicates at the anchor); SO(16)/D8 (only even sectors survive, all axion b-characters in mu_2: the order-4 supply structurally absent); wrong discriminant form (x^2+y^2)/4 (|Gauss|^2 = 64 != 16, its S-rule fails at O(1) on the true thetas); wrong signature factor (S^2 = C breaks exactly). [C] FENCES: the f1f3 = KS-weight identification (deck weights (1,3) = the twistor fibre weights (+1,-1 mod 4) forced by the v514 incidence ledger -- exact match, NOT a complete BCOV derivation); the kernel-family convention (J(Delta,0/1/2) from weight bookkeeping); the discrete chi_k choice (both members evaluated, both fail -- the kill does not depend on it). TENSION, STATED HONESTLY (the mandatory fence): the DECLARED completion reading (v516/M2) delivers the psi = 64 slice and cancels 32 T3; the DERIVED chiral measure (v518) fails all three testers -- both are exact; the sharp open question is which of the two is the true BCOV measure (the declared reading is supported by the delta-1 modular-completion finding but not derived; the derived measure rests on the f1f3 identification). Neither result is hidden behind the other. [O] the BCOV-integral derivation on PT/Z4 that decides between the two measures -- the single open item of the delta-1 strand; the twisted-contact route of v508 S6.2 is DECIDED NEGATIVE under the derived measure. SINCE DECIDED at probe level by v520 in favour of the declared reading: single-valuedness is derived from two independent constructive sources (F-independence + Quillen pairing), the chi_4 route was never a well-defined nonvanishing moduli integral -- the kill of this module is SHARPENED (the column-canonicalised chi_4 member is boundary-consistent yet tester-dead); the residual [O] is the constructive w_m derivation. Discovery provenance: delta-1b (30/30), delta-1c (31/31), delta-1d (36/36) probes in experiments/tfpt-discovery/ (2026-07-22). NO marker moves anywhere.",
      "status": "DECIDED: kill under the derived measure, tension with v516 stated ([E] exact Weil/Q(zeta_8) algebra, loop words + koboundary defects + 1/960 phase arithmetic, ledgers, G = f1 f3, the decision table; [C] the f1f3/KS-weight identification + kernel-family convention + chi_k choice; [O] the BCOV-integral derivation that decides between the declared (v516) and derived (v518) measures -- since decided at probe level by v520 for the declared reading, residual [O] = the constructive w_m derivation; NO marker moves)",
      "canonicalStatus": "DECIDED (negative under the derived measure): the delta-1 chain is closed -- the derived chiral measure kills the route, the v516 declared-measure tension is the named [O]; since decided at probe level by v520 in favour of the declared reading (kill sharpened; residual [O]: the constructive w_m derivation)",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.M2.01; CELEST.WP5E.M3.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.DELTA2.01; CELEST.DTERM.NONDERIV.01; SEAM.EQUIV.TWISTOR.01; v502; v505; v508; v511; v513; v514; v515; v516; v517",
      "scripts": [
        "v518_celestial_wp5e_delta1_derived_kill.py"
      ],
      "script": "v518_celestial_wp5e_delta1_derived_kill.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "Harvey-Moore Nucl.Phys. B463 (1996) 315 (fundamental-domain unfolding); Costello-Li BCOV arXiv:1201.4501; Bittleston-Homans-Sharma JHEP 09 (2023) 008 -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 582
    },
    {
      "id": "SEAM.EQUIV.MMST.01",
      "claim": "THE MMST ROUTE OF THE SEAM EQUIVALENCE TARGET (2026-07-22 route split, carved out of SEAM.EQUIV.01 -- an ID restructuring, NO status change in substance; this row carries exactly the statement previously attached to 'SEAM.EQUIV.01 closed modulo a cited theorem'): the conditional scaling-limit route. The explicit lattice model (v367/v368) and the S3 closure stack -- central charge c = 8 (v376), the (E8)_1 character with 248 currents and one primary (v377), torus GSD = 1 (v378), reflection positivity (v379), the ground-state witnesses c_- = 8 (v489) and mu_pre = 4 (v490) -- pin the target at every computable level; the continuum leg is checked hypothesis-by-hypothesis (v356/v366: D = 16 Majoranas, rank <= c <= D reads 8 <= 8 <= 16, gap 6 ln(3/2) > 0 derived, v302); the whole route is Lean-formalised as FORM.SEAM.MMST.01 (collar hypotheses kernel-proved; the scaling-limit theorem (Osborne-Stottmeister, arXiv:2107.13834, CMP 398 (2023) 219, on the MMST OAR framework arXiv:2010.11121) and the AMT OS/holomorphy selector as NAMED cited axioms; #print axioms clean); the 128-spinor extension leg is crossed-product-certified on peer-reviewed theorems with the realisation input reduced to invariant level (v469). STATUS AS DOCUMENTED (typing taken verbatim from the pre-split rows/prose): CLOSED MODULO CITED THEOREMS -- [C] conditional; the ONLY residual that stays [O] is the abstract continuum existence of the scaling limit (the cited MMST/OS theorems, v336). The parent SEAM.EQUIV.01 stays [O] as an unconditional claim.  UPDATED 2026-08-27 (v973): the cited-theorem coverage is CARVED EXACTLY -- OS23 COVERS the bilinear half (the so(16)_1 fermion-bilinear current algebra with STRONG convergence on the finite-particle core, Thm 5.1 + Eq. 229 incl. Majorana algebras; Virasoro/Koo-Saleur Thms A/4.11/4.16; correlators Thms B/C; polynomial error estimates Error^2 <= C 2^(-2 delta N), delta in [0,2]) and does NOT cover four NAMED items: N1 the mu4/GSO extension SO(16)_1 -> (E8)_1 as a NET statement (the 128 spinor currents are not bilinears), N2 the twist/spin-field sector (deferred by the paper), N3 the one-sided chiral edge net of a 2+1D bulk, N4 the OS/holomorphy selector det K = 1 vs 4; NEW MEASUREMENT: the collar edge-correlator convergence rate p = 3.48 (Richardson 3.1-4.2) lies inside the preregistered band [0.5, 4.0] consistent with the cited polynomial form (HONEST: the ratio observable superconverges past delta_max = 2 -- the band's upper edge is calibration-informed); the M = 3 control does not converge (min rel dev 0.836). The [O] residual is now the NAMED list N1-N4 instead of an unstructured 'cited continuum existence'; status UNCHANGED [C]/[O] residual.  EXECUTED UPDATE (2026-08-28, completeness wave, v994): 5 of 7 MMST identification criteria executed [E-measured] -- measured c = 8.005 (16 Majorana copies), conformal edge spectrum [0.506, 1.516, 2.523, 3.524] with degeneracies [1,0,1,1,2], current normalisation k = 1.000000000000, sector collapse det D8 = 4 -> det E8 = 1, and mu4/GSO-projected lattice character sums reproduce the (E8)_1 vacuum character coefficients [1, 0, 248, 0, 4124, 0, 34752] three orders deep. C6 = the quasi-free convergence theorem is now WRITTEN OUT in articles/2026-08-28/psi_lambda_convergence_theorem_en.tex (analytic log coefficient 2/pi^2 = 0.20264 vs measured 0.203, 0.18%); C7 modular invariance stays cited/open. Display UNCHANGED [C]. Python-only / Wolfram mirror deferred.  EXECUTED UPDATE (2026-08-29, review wave 4, v998): C7 ARITHMETIC SHADOW now exact -- two-edge HS remainder bounded by 2.11 (plateau Nx in (16,24,32,48,64,96)); lattice character theta_E8/eta^8 EXACT four orders [1,0,248,0,4124,0,34752]; Theta_E8(tau)=E4(tau) q-expansion match to >= 8 orders exact (Eisenstein; Jacobi identity); j^{1/3} numeric transform at 60 digits with S-weight bookkeeping.  Net-level C7 modular covariance of the completed extension stays cited/open.  Display stays [C]/[O] as booked; no marker move.",
      "status": "Conditional [C] (closed modulo cited theorems: the MMST/OS continuum existence, v336; every computable level pinned [E]; Lean FORM.SEAM.MMST.01)",
      "canonicalStatus": "[C] closed modulo cited theorems (the MMST route of SEAM.EQUIV.01; route split 2026-07-22, no status change in substance); residual [O] = the cited continuum existence (v336)  EXECUTED UPDATE (2026-08-28, v994): 5/7 identification criteria [E-measured]; C6 written-out [C] (psi_lambda_convergence_theorem_en.tex; 2/pi^2=0.20264 vs 0.203); C7 cited [O]; status UNCHANGED [C].  | 2026-08-29 v998: C7 arithmetic shadow exact (HS bound 2.11, Theta_E8=E4 through q^8, character four orders, 60-digit S); net-level C7 stays cited.  Display unmoved.",
      "dependencies": "SEAM.EQUIV.01; SEAM.MMST.INCLASS.01; SEAM.EQUIV.CONTINUUM.03; SEAM.S3.LATTICE.01; SEAM.S3.INFLOW.01; SEAM.S3.CENTRALCHARGE.01; SEAM.S3.E8CHARACTER.01; SEAM.S3.MODULAR.01; SEAM.S3.RP.01; SEAM.CCC.01; SEAM.MU.01; FORM.SEAM.MMST.01",
      "scripts": [
        "v366_mmst_seam_collar.py",
        "v994_mmst_criteria_battery.py",
        "v998_seam_modular_closure.py"
      ],
      "script": "v366_mmst_seam_collar.py; v994_mmst_criteria_battery.py; v998_seam_modular_closure.py",
      "location": "tfpt_research_contracts;introduction;tfpt_1;tfpt_3;tfpt_4",
      "externalData": "Osborne-Stottmeister arXiv:2107.13834, CMP 398 (2023) 219; MMST OAR arXiv:2010.11121; AGT arXiv:2506.01008 -- cited theorems, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 583
    },
    {
      "id": "SEAM.EQUIV.TWISTOR.01",
      "claim": "THE TWISTOR ROUTE OF THE SEAM EQUIVALENCE TARGET (2026-07-22 route split, the second continuum route named in its own right -- an ID restructuring, NO status change in substance): the open Costello-Li-grade construction -- the interacting holomorphic open+closed twistorial algebra on PT/Gamma (the global BCOV+SDYM quantisation, WP5e proper of CELEST.SEAM.01) whose boundary partition function is E4/eta^8 INCLUDING the q^{-1/3} prefactor, OS-reconstructed to the ACTUAL (E8)_1 seam net (not only its character). PREPARED, NOT CONSTRUCTED: the CELEST.SEAM.01 work packages (v492-v518) fix the null ideal, the deleting operator, the GNS limit state, the two-interval index, the KLM triple, both faces of the inflow, the level (a theorem of clock invariance, value pinned by the sector counter), the bulk-axion slot, the back-reacted Omega_N (M1, v515: closed form, (2 pi i)^2-integral lockstep periods, forced source charge 4 = |mu4|), the twisted KS measure on the declared completion reading (M2, v516: per-sector Okubo squares, the 32 T3 cancelled, psi = 64 supplied) and the a0 uplift (M3, v517: log correction coupled to the centre count on four scales); the delta-1 chain is DECIDED by v518 (CELEST.WP5E.DELTA1.01: the derived chiral measure kills the delta-1 route; the stated v516 tension -- declared vs derived measure -- is the sharp named [O] the global quantisation must resolve); the global quantisation and the OS/real-structure bridge (WOIT.OS.TWISTOR.01) stay open. [O] throughout; nothing executed so far moves it; the parent SEAM.EQUIV.01 stays [O] as an unconditional claim.  UPDATED 2026-08-27 (v973): the declared measure is RIGID on the computable sector -- an 18-parameter deformation of the declared measure data (contact normalisation, 4 sector weights, residue normalisation, 4 linking charges, 4 centre positions) with 4 gauge directions has executed-constraint Jacobian ranks C1 clock 9 / C2 lockstep+integrality 8 / C3 charge 2 / C4 null-ideal 2, joint rank 13 => the moduli kernel is EXACTLY 1 = the uniform sector-weight shift dv = (1,1,1,1), mapping to the untwisted Okubo square Q0 = 36(1,2,1,0,0) and annihilated by every executed certificate functional BY CONSTRUCTION (exactly flat to all orders); ablation ledger (full,-C1,-C2,-C3,-C4,-C1&C2) = (1,3,1,2,3,10); byproduct Phi_P = 3 Phi_T5 + (9/4) Phi_T3 on the deformation image. HONEST: this is sector-level rigidity, NOT a BCOV derivation -- the computable sector is the finite v508/v515 shadow, C4 pins level sets of EXECUTED functionals only, and the W2:W13 = 4:3 Harvey-Moore tester is typed NOT-TRANSCRIBABLE; the global quantisation stays the open closure condition; status UNCHANGED [O].",
      "status": "Open (research route [O]: the Costello-Li construction; preparation [E]/[C] via the CELEST.SEAM.01 work packages v492-v518)",
      "canonicalStatus": "Open [O]: the twistor/Costello-Li route of SEAM.EQUIV.01; the CELEST.SEAM.01 work packages are preparation; closes only with the global BCOV quantisation + the WOIT.OS.TWISTOR.01 bridge",
      "dependencies": "SEAM.EQUIV.01; CELEST.SEAM.01; WOIT.OS.TWISTOR.01; CELEST.WP1.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; CELEST.WP5DB.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.DELTA2.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; CELEST.WP5E.M2.01; CELEST.WP5E.M3.01; CELEST.WP5E.DELTA1.01",
      "scripts": [
        "v492_celestial_z4_orbifold.py"
      ],
      "script": "v492_celestial_z4_orbifold.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "Costello-Li BCOV arXiv:1201.4501; Bittleston-Homans-Sharma JHEP 09 (2023) 008; Costello arXiv:2111.08879 -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 584
    },
    {
      "id": "WOIT.OS.TWISTOR.01",
      "claim": "THE OSTERWALDER-SCHRADER TWISTOR BRIDGE (2026-07-22; the new central research contract after the external review of the celestial/twistor branch -- THIS IS THE ACTUAL BRIDGE FROM COMPILER TO PHYSICS; everything in CELEST.SEAM.01 is preparation for it). INPUT: X_+ = (PT/Gamma)_+ (Gamma = Gamma_ALE = <Deck> ~ Z4 in SU(2), normalised by the clock rho = diag(i,1) with rho^2 = Deck); A_hol = the INTERACTING open+closed twistorial algebra (SDYM(E8) + BCOV); rho = the order-4 clock; Theta = the anti-linear real structure induced by the seam reflection (the free covering deck, v510); mu_BCOV+SDYM = the interacting functional (global BCOV+SDYM quantisation on PT/Gamma). TARGET THEOREM: Theta^2 = 1, Theta rho Theta = rho^-1; the gauge-invariant algebra is reflection-positive NAMED TOY THREAT (v529 SEAM.INT.FKTOY.01, NO status change): on the first genuinely interacting 16-Majorana Fidkowski-Kitaev seam toy, Kill-Test 1 does NOT fire (Theta exists with Theta^2=+1 and clock-tower inversion) but Kill-Test 2 FIRES AT TOY LEVEL -- RP breaks for every g>0 by an interference mechanism, with the STRADDLE LAW (RP fails exactly on quartet-straddled cuts) as a concrete hurdle for beta3/gamma; fence: one toy / one interaction class / [C] flat-band parent -- narrows the route (every A_hol must protect RP against exactly this mechanism), does not close it.; the OS quotient produces (H, Omega, U(P^up_+), A_Mink) with: positive Hilbert metric, positive energy, local causality, ONE chiral fermion generation WITHOUT mirror doubling, the TFPT charge lattice, an internal SU(3) x SU(2) x U(1) action, and the (E8)_1 seam net as an ACTUAL boundary net (not only as a character). KILL TESTS (all seven, preregistered): (1) Theta incompatible with the clock; (2) RP fails after gauge fixing; (3) the reconstruction produces a vector-like mirror generation; (4) the Penrose transform reaches only free/self-dual states; (5) the reconstructed net has the right character but the wrong OPE / fails net equivalence; (6) the internal SU(2) remains a spacetime factor; (7) the four mu4 marks are not incidence-compatibly extendable over spacetime. SCOPE FENCE (explicit non-claims of the celestial/twistor branch until this closes): both helicities, generic amplitudes, local matter, full Einstein dynamics, EWSB, confinement. EXTERNAL FRAME: Woit's Euclidean Twistor Unification (arXiv:2104.05099 -- chiral Wick rotation problematic for purely chiral theories; Euclidean-holomorphic theory on PT, Lorentzian theory via conjugation structure/boundary values) is a NAMED reference frame, never a confirmation: until Theta + interacting RP exist, the Woit connection is a strong analogy, not a mathematical integration. ALPHA STAGE EXECUTED (2026-07-22) via v519: Theta exists with Theta^2 = +1 and Theta rho Theta = rho^-1 on all four levels (sphere, C^4, Z8 spin, Cl(16) Fock), free RP holds on the bond cut and selects the SAME family as the clock-inversion relation; kill test 1 does NOT fire at the free level and stays live on the interacting algebra; beta/gamma milestones named (beta-1 Theta on the gauge-invariant subalgebra + gauge-fixed RP on the equivariant SDYM(E8) sector; beta-2 the OS quotient of the free system explicit; beta-3 the PT <-> PT* duality vs sigma_std typed; gamma the chirality theorem + the mark incidence). TWO CONTRACT PRECISIONS (prose, no marker): (i) 'Theta induced by the seam reflection' means the seam-circle REFLECTION, not the deck -- the deck furnishes the (-1)^F Kramers class (the v510 bridge); (ii) the marks sit RP-side on the BOND MIDPOINTS of the 16-Majorana circle. BETA-1 EXECUTED (2026-07-22) via v522: verdict UNDECIDED per the frozen preregistration -- the clock-tower reading of 'gauge' fails Hermiticity structurally (the mu4 clock is euclidean-time-like: all 16 mirror axes invert it); the gaugeable part of the Z8 tower is exactly the GSO/fermion-parity Z2, under which gauge-fixed RP HOLDS ((29,0,0) PD at N = 16, (8,0,0) PD full N = 8 algebra); kill test (2)'s free shadow does NOT fire, kill test (1) stays discharged also gauge-invariantly; the clock-equivariant statement is re-routed to beta-2 (OS quotient first, then the clock as reconstructed transfer operator). THIRD CONTRACT PRECISION (prose, no marker): (iii) 'gauge-invariant subalgebra' at the free level means the GSO/fermion-parity Z2 -- the only theta-compatible part of the mu4 tower; the clock is the euclidean rotation itself (timelike), its equivariant statement moves to beta2. BETA-2 EXECUTED (2026-07-23) via v524: verdict SUCCESS, [C]-typed per precision (iii) -- the OS quotient of the free system is EXPLICIT: H_phys nondegenerate and PD at both levels (dim 37 = 29+8 at N = 16 deg <= 2; dim 16 = 4^2 at N = 8 complete half algebra -- compact euclidean time reconstructs a thermal/KMS representation), the Klein-Landau local transfer semigroup is exactly Hermitian with the site/bond dichotomy as its positivity pattern (even steps PSD via T(2j) = A*A, the one-step transfer NOT positive -- the chirality datum), the mu4 clock = T^{N/4} is a positive self-adjoint transfer step with spectral calculus (N = 8 spectrum exactly {1, sqrt(2)-1} = {1, 1/delta_Silver}) and a reconstructed unitary rotation group U(s) = exp(isH); the v522 non-Hermiticity is resolved as EXACTLY the domain/wrap artifact; the pre-declared KMS deviation (no contraction on the compact circle) carries exactly the silver witnesses; Theta_phys^2 = +1 on every sector (Kramers-free); kill tests (1)/(2) strengthened, (3) shadow sharpened, none fires -- all seven stay live on A_hol; beta-3 is EXECUTED (v565, WOIT.BETA3.DUALITY.01, 2026-07-30, verdict SUCCESS): the PT <-> PT* duality typed against sigma_std -- the compatible class is the two-block Hermitian family (clock-forced), the rho_tw pairing splits it into an Euclidean branch (sigma_std's home, no OS cut possible) and a Minkowski branch whose null locus meets the seam line exactly in the seam circle (the OS cut forced) with total signature (2,2) by the mirror theorem; the contract kill branch (family A) is EMPTY BY ALGEBRA (rho4^2 not scalar), and every seam-preserving Minkowski member induces the unique conjugation z -> -conj(z) = theta_perp = the v524 Theta_phys carrier (regressions re-verified; role separation: sigma_std ~ the OS cut/metric, the Phi-twisted duality ~ the descending mirror); gamma (the chirality theorem + the mark incidence) is the next milestone. [O] throughout; no marker moves anywhere. [2026-08-01 XXVI: the gamma stage gains its FREE SLICE -- v608: mirror modes are strictly OS-negative sector-exactly (kill test 3 cannot fire at the free level), the chiral generation is multiplicity-free ({1, sqrt2-1}), and the mark transport exists (tau_2 = mark-A -> mark-B, PD; kill test 7 does not fire free); gamma itself stays open on A_hol; see WOIT.GAMMA.FREE.01.]  UPDATED 2026-08-27 (v972): the kill-test-2 counterweight NARROWED -- on the same v529 toy the quartet-AVOIDING cut family of the clock-invariant m = 4 member generates the full even algebra (rank 15, dim 32768) and carries a POSITIVE OS/GNS structure at g = 1 (min +4.4338e-5, GNS 37/37) while the straddled control genuinely fails (SEAM.INT.OSAVOID.01); the straddle law itself refines to the odd-split law (SEAM.INT.ODDSPLIT.01).  The threat paragraph STANDS: every A_hol must still protect RP against the odd-split mechanism -- but the toy now EXHIBITS an OS-viable cut geometry selected by its own mu4 clock.  Contract stays Open; no marker move.",
      "status": "Open (research contract)",
      "canonicalStatus": "Open (research contract): the OS twistor bridge -- Theta real structure + interacting RP + OS reconstruction to (H, Omega, U(P^up_+), A_Mink) with one chiral generation, the TFPT charge lattice and the (E8)_1 net as an actual boundary net; seven preregistered kill tests; the actual bridge from compiler to physics; alpha stage executed via v519 (Theta + free RP, kill test 1 discharged on the free level only -- stays live on the interacting algebra); beta-1 executed via v522 (UNDECIDED: the clock reading was the wrong operationalisation, the clock is time-like, GSO-fixed RP holds, kill test 2 does not fire; contract precision (iii) added); beta-2 executed via v524 (SUCCESS, [C]-typed per precision (iii): H_phys explicit and PD at both levels, the clock a positive transfer step with spectral calculus -- N = 8 spectrum exactly {1, sqrt(2)-1} -- and a reconstructed unitary rotation group; the pre-declared KMS deviation carries exactly the silver witnesses; no kill fires); beta-3/gamma milestones next",
      "dependencies": "CELEST.SEAM.01; SEAM.EQUIV.TWISTOR.01; SEAM.EQUIV.01; SEAM.BIT.FREEDOM.01; SEAM.CLOCK.RIGIDITY.01",
      "scripts": [
        "v519_woit_theta_rp_free.py"
      ],
      "script": "v519_woit_theta_rp_free.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "Woit arXiv:2104.05099 (Euclidean Twistor Unification) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 585
    },
    {
      "id": "WOIT.THETA.FREE.01",
      "claim": "THE WOIT-ALPHA MILESTONE (2026-07-22): the real structure exists, and free reflection positivity picks the SAME family -- the alpha stage of WOIT.OS.TWISTOR.01 executed as one module (Part W = Theta, Part R = free RP). [E] THETA CLASSIFICATION COMPLETE: exactly TWO families of anti-linear structures on C^2 normalise the clock rho = diag(i,1) -- family D (z -> mu conj(z), |mu| = 1): Theta rho Theta = rho^-1 EXACTLY, Theta^2 = +1 (-1 IMPOSSIBLE: M conj(M) = diag(|mu|^2,1), Kramers-free), inverts the deck, fixed set = the great circle through the deck poles; family A (z -> mu/conj(z)): centralises the clock projectively (NEVER inverts), Theta^2 = +-1 per mu, fixed set |z| = 1 resp. empty; ROLE SEPARATION: family A defines the euclidean section (Woit's rho_tw side: rho_tw^2 = -1, no real points -- replicated exactly on C^4), family D reflects it (the OS conjugation, sigma_std = conjugation with real points RP^3). [E] MARK PINNING: mu in mu4 iff the marks are preserved -- a 4-element mu4-torsor (2 mark axes + 2 silver-midpoint axes), clock conjugation rho Theta_mu rho^-1 = Theta_{-mu} => 2 clock orbits. [E] SPIN/FOCK: Z8 plane without phase leaks (conj(zeta8^k) = zeta8^-k); Cl(16): Theta_Fock = U_r o K with Theta_Fock^2 = 2^7 > 0 (normalised +1), inverts the clock tower (V -> 4096 V^-1) and normalises the deck; KRAMERS DICHOTOMY: the deck-induced candidate Theta_t = Utilde o K has Theta_t^2 = (-1)^F -- the v510 split/nonsplit dichotomy IS the Theta^2 = +1 vs (-1)^F dichotomy; the deck does NOT furnish Theta, the seam-circle REFLECTION does. [E] KILL TEST 1 (free level): does NOT fire -- Theta exists with Theta^2 = +1 AND Theta rho Theta = rho^-1 and satisfies marks + deck normalisation + spin/Fock simultaneously. [E] FREE RP: NS vacuum C(d) = (2/N)/sin(pi d/N), C^2 = -1 exact; preconditions exact (RCR^T = -C, SCS^T = C, RSR^-1 = S^-1; the deck fails anti-invariance); BOND CUT: one-particle Gram (8,0,0) PD (min EV 1.888e-3, 40 digits), even deg <= 2 (29,0,0) PD, N = 8 full half-sided algebra PD (even + odd each (8,0,0)) -- RP without degree truncation; twist eta = +i forced (eta = 1 non-Hermitian); CONTROLS: site cut fails exactly (det = 0, inertia (3,3,1); continuum Cauchy-Stieltjes control strictly PD -- a lattice-placement artifact, RP selects the site placement relative to the marks); the clock-centralising family-A structure fails RP structurally ((4,4,0)) -- RP and Theta rho Theta = rho^-1 select the SAME family; bonus: anti-chiral state -> odd sector (0,8,0) ND (the free shadow of kill test 3). [C]/[O] NOT shown: the interacting algebra A_hol (kill test 1 stays live there), gauge-fixed RP (kill test 2), OS reconstruction, chirality without mirrors (kill tests 3/6), mu4 incidence (kill test 7) -- the WOIT-beta/gamma roadmap. NO marker moves; WOIT.OS.TWISTOR.01 stays Open.",
      "status": "Identity/Lattice (exact sympy + exact Cl(16); 40-digit gap-certified RP inertia certificates)",
      "canonicalStatus": "[E] the free/equivariant alpha stage (Theta classification, exact relations, kill test 1 discharged at the free level, free RP on the bond cut selecting the same family); [C]/[O] everything interacting -- kill test 1 stays live on A_hol; no marker moves",
      "dependencies": "WOIT.OS.TWISTOR.01; SEAM.BIT.FREEDOM.01; SEAM.CLOCK.RIGIDITY.01; SEAM.BIT.ORIGIN.01; CELEST.SEAM.01",
      "scripts": [
        "v519_woit_theta_rp_free.py"
      ],
      "script": "v519_woit_theta_rp_free.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "Woit arXiv:2104.05099 (Euclidean Twistor Unification) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 586
    },
    {
      "id": "CELEST.WP5E.MEASURE.01",
      "claim": "THE WP5E BCOV MEASURE QUESTION DECIDED AT PROBE LEVEL (2026-07-22; delta-1e + delta-1f consolidated in the v518 style, one module, 39 checks, ~20 s; ERFOLG-A on the preregistered decision layer): which of the two exact, mutually contradictory measures on PT/Z4 is the true BCOV measure -- (A) the DECLARED completion reading (v516: unphased sector trace, w_m = 4 h_m, supplies psi = 64) or (B) the DERIVED chiral measure (v518: multiplier character chi_4 on Gamma_1(4); all three testers fail)? ANSWER: single-valuedness of the physical one-loop lattice factor is DERIVED from two independent constructive sources, not postulated, and the completion reading (A) wins. (1) THE INVARIANT SUBSPACE (delta-1e D2): dim_Q {v : rho(T) v = v, rho(S) v = v} = 8 = 4 x 2 in the dressed 16-component Weil system (exact rational nullspace); EVERY kernel vector lies in the Q(zeta_8)-span of {e_H, e_H'} -- the two Lagrangian gluings, both theta = E4, level-1 quartic = Q^(0) = the unphased trace; strict trivial-character solutions at physical weight (m, k) = (0, 0): ZERO for every dressing (11 combinations, both orientations). (2) SOURCE 1 -- THE F-LEMMA (delta-1f A1): the only strict chiral families are chi_4 (dims (3,3)) and chi_10 (dims (1,1)), both with nontrivial T-character EXACTLY; the canonical column-matched chi_4 member's total block sum obeys G(gamma tau) = chi_4(gamma) G(tau) pointwise (certificate 3.9e-16 over two fresh tau, nonvacuous |G| >= 59.6); EXACT change of variables: fundamental-domain independence forces int = 0 for EVERY chiral integrand -- under TP-1 + TP-2 the chiral route can never source psi = 64: the v518 route was never a well-defined nonvanishing moduli integral (the kill is SHARPENED, its unfolded values are convention bookkeeping). (3) SOURCE 2 -- THE QUILLEN PAIRING (delta-1f A2): |t conj(t) - 1| < 8.3e-40 on all 15 pairs and both generators (measured unitarity); |chi_4|^2 = 1 exactly vs one-sided chi_4^2 = e(2/3) != 1 (T-fixed: 640 != 0 mod 960, 15/15 defective); the doubled 256-dim hol x antihol Weil system closes STRICTLY at trivial character and physical weight (nullspaces (28, 17) >= 11), contains the unphased diagonal (~2e-15) AND the physical columns w_b (x) conj(w_b) (sigma 4.9e-16/8.3e-16); negative control hol (x) hol: EMPTY nullspace (0, 0). (4) CONSISTENCY CLOSURE (delta-1f A3): the forced unphased numerator inside the v518 scaffold reproduces sum_m V prop (1, 2, 1, 0, 0) = <x,x>^2 EXACTLY on every level n <= 8 (levelwise Okubo/4-design); J-weighted spreads ~5e-41, T5/T3 fractions ~7e-42, zero negative-exponent cells; psi(skeleton) = +0.2302 = -psi(contact) -- the J-weighted mirror image of psi(A_fix) = +64 / psi(contact) = -64; linearity 4.6e-41: the circle to v516 is CLOSED. (5) D3 -- THE Z2/EH ANCHOR (delta-1e): only the declared reading hits the anchor (9<x,x>^2 = (1/4) x 36<x,x>^2, psi(A_2) = -8, scale tooth c = 1/2 = |Z2| h^(A1) unique); ALL THREE derived Z2 instantiations fail the anchor testers (spreads 1.12/0.895/0.516, psi sign wrong). (6) D1 -- COLUMN CANONICALISATION (delta-1e): the physical AB column sum_m i^(bm) e_(m,m) is contained EXACTLY in the chi_4 family (sigma 1.8e-16/1.7e-16, one scalar per orbit -- canonicalising the (N1, N2) freedom); the canonical member STILL fails every tester (sharpened kill). NEGATIVE CONTROLS: SO(16)/D8 (T5 = 20/T3 = -40 exact, mu_2-only supply), wrong form (|Gauss|^2 = 64 != 16, S^2 = C breaks), wrong weight (transport dev 0.78 at exponent 2), shuffled weights (T5 = -14/T3 = 36), planted family detected (1.5e-16), hol (x) hol empty. [C] TYPED PREMISES (the mandatory fence): TP-1 (the twisted one-loop contribution is an F-independent moduli-space integral), TP-2 (it is nonzero -- it must source psi = 64), TP-3 (the v518 kernel-family weight convention), TP-4 (BCOV F1 is Quillen/Ray-Singer: the one-loop factor pairs hol x conj(hol)). [O] the constructive BCOV derivation of the w_m normalisation itself (the residual open item) -- since executed and DERIVED at probe level by v523 (CELEST.WP5E.WM.01: Atiyah-Bott mode ledger + zeta/Quillen determinant + consistency chain, typed premises TP-REG/TP-Q/TP-NUM/TP-CH); the residual [O] narrows to the GLOBAL BCOV integral on PT/Z4 beyond the fibre zero-mode factor; SEAM.EQUIV.TWISTOR.01 stays Open. Discovery provenance: delta-1e (28/28) and delta-1f (23/23) probes in experiments/tfpt-discovery/ (2026-07-22). NO marker moves anywhere.",
      "status": "DECIDED at probe level for the declared reading ([E] exact Weil/Q(zeta_8) algebra, exact invariant subspace, characters + 1/960 grid, ledgers, levelwise 4-design, Z2 anchor arithmetic, change-of-variables corollary, controls; numeric documented: transports 28+ digits, SVD memberships 1e-8/1e-3, certificates fresh tau, HM kernels 30+ digits; [C] the typed premises TP-1..TP-4; [O] the constructive BCOV derivation of w_m -- since DERIVED at probe level by v523, narrowing the [O] to the global BCOV integral beyond the fibre zero-mode factor; NO marker moves)",
      "canonicalStatus": "DECIDED (ERFOLG-A at probe level): single-valuedness is derived from two independent constructive sources (F-independence + Quillen pairing) under the typed premises TP-1..TP-4; the v516 completion reading wins, the v518 kill is sharpened; residual [O]: the w_m normalisation is since DERIVED at probe level by v523 (1/det_j = the Atiyah-Bott/zeta-determinant fixed-point factor, three independent sources); what remains open is the global BCOV integral beyond the fibre zero-mode factor",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.M2.01; CELEST.WP5E.DELTA1.01; SEAM.EQUIV.TWISTOR.01; v502; v505; v508; v511; v514; v515; v516; v517; v518",
      "scripts": [
        "v520_celestial_wp5e_measure_decision.py"
      ],
      "script": "v520_celestial_wp5e_measure_decision.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "Harvey-Moore Nucl.Phys. B463 (1996) 315 (fundamental-domain unfolding); Costello-Li BCOV arXiv:1201.4501; Quillen Funct. Anal. Appl. 19 (1985) (determinant metric) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 587
    },
    {
      "id": "SEAM.BIT.RPBLIND.01",
      "claim": "THE RP/THETA ATTACK ON THE ALIGNMENT BIT DECIDED (2026-07-22; one module, 25 checks, ~25 s; an honest, decided NEGATIVE result in the v508 style): does free reflection positivity (v519 RP) or anti-linear Theta existence (with the v519 side conditions) BREAK for delta != pi/2 on the deformed mark family M(delta) = {+-1, +-e^(i delta)} -- which would DERIVE the v512 alignment bit from OS positivity? NO: the entire free RP/Theta battery is SIDE-BLIND, hereby documented as the EIGHTH side-blind test on the v512 scoreboard (7 -> 8); the bit REMAINS genuine discrete input. (1) CENSUS (exact solvesets, symbolic delta): for EVERY delta exactly the two mark-SWAPPING reflections Theta_(+-e^(i delta)) exist (axes delta/2, delta/2 + pi/2; all 20 cut/mark incidence solvesets EMPTY); mark-FIXING reflections exist exactly at delta = pi/2 (solveset cos delta = 0 = {pi/2}, lost immediately at pi/2 - eps); the v510/v512 counterwitness {+-1, +-(3+4i)/5} carries the swap pair exactly and is NOT excluded. (2) RP IS delta-BLIND: bond-cut Gram inertias (8, 0, 0) PD for all tested delta at N = 16, even deg <= 2 sector (29, 0, 0); the full sorted 40-digit spectra are IDENTICAL across all axes (deviation 0.0) -- the ENTIRE free-RP spectral datum carries ZERO delta information; the N = 16 odd-m failure (site axes, inertia (3,3,1), det = 0) is a lattice-PLACEMENT artifact: at N = 32 ALL m = 1..7 (delta = m pi/8) have mark-compatible BOND axes with PD inertia (16, 0, 0); continuum: the OS kernel in axis-adapted coordinates is EXACTLY 1/sin((s+t)/2) -- the axis position drops out identically (Cauchy-Stieltjes factorisation PD); the counterwitness PASSES free RP (6-point Gram (6, 0, 0)). (3) THETA EXISTENCE IS delta-BLIND: Theta^2 = +1, deck normalisation and Fock implementability (U^2 = +2^7/2^8 > 0, exact Cl(16)) hold for every delta; the ONLY delta-sensitive clause of the v519 battery (Theta rho Theta = rho^-1) is for delta != pi/2 not violated but NOT FORMULABLE (a mark-compatible order-4 rotation exists iff delta = pi/2) -- it PRESUPPOSES the bit instead of deriving it; same for the clock-tower clause. (4) THE BATTERY HAS TEETH AT EVERY delta: RP separates the two real-structure FAMILIES (family A/antipode: zero diagonal, indefinite (4, 4, 0); chirality-eta pinning: anti-chiral state (0, 8, 0) ND) -- but never the SIDES within family D. (5) PRECONDITION CONTROLS: hexagonal marks not concyclic (|b_hex| = 2 - sqrt(3) != 1, no seam circle -- rejected before positivity is asked); silver marks all real (deck not free -- excluded by topology); lifting audit: B (pullback) = index relabelling (delta-blind by construction), B' (stretched sampling) fails the pi/2 negative control (not the chiral vacuum -- discarded). [C] FACE #13 of the v512 equivalence web: the group generated by the admissible OS reflections is V4 for every delta < pi/2 and closes to D4 exactly at delta = pi/2 (Theta_i o Theta_1 = the clock) -- a typed REFORMULATION of the bit (12 -> 13 faces (since 13 -> 15 by v528 facets #14/#15)), not a derivation from positivity. [O] HONEST GAP: marks decorating the STATE (defect/twist insertions, interacting A_hol) are outside the free class = exactly v519's [O] scope -- there, and only there, could positivity still see the bit; since DECIDED by v525 (SEAM.BIT.TWISTBLIND.01): the mark-decorated free-plus-twist state class (sigma gauge, Kadanoff-Ceva 4-twist insertions, mu4/Bogoliubov defects -- everything Wick-computable) is side-blind too, the NINTH side-blind test (8 -> 9); the free-plus-twist class is EXHAUSTED, the remaining door is a genuinely interacting A_hol with non-Pfaffian-reducible OS data. VERDICT: KILL exactly as preregistered. Discovery provenance: seam_alignment_bit_rp_selector_probe.py (25/25) in experiments/tfpt-discovery/ (2026-07-22). NO marker moves.",
      "status": "Formal/Identity [E] (exact solveset censuses, exact Gram entries, exact Cl(16) implementers, 40-digit gap-certified inertia spectra, continuum axis elimination, precondition controls) + [C] (face #13 = a reformulation of the bit) + [O] (mark-decorated states / interacting A_hol); the bit stays the one [C] carrier input; NO marker moves",
      "canonicalStatus": "KILL (honest negative, preregistered): free RP and Theta existence are side-blind on the whole M(delta) family -- the eighth side-blind test on the v512 scoreboard (7 -> 8), face #13 added [C] (12 -> 13; since 13 -> 15 by v528); the alignment bit is NOT derivable from OS positivity and remains genuine discrete input; [O] gap: mark-decorated/interacting states -- the mark-decorated (free-plus-twist) half since decided side-blind by v525 (the ninth side-blind test, 8 -> 9); what remains is the genuinely interacting A_hol",
      "dependencies": "SEAM.TAU.FLAG.01; SEAM.BIT.FREEDOM.01; SEAM.BIT.ORIGIN.01; SEAM.CLOCK.RIGIDITY.01; WOIT.THETA.FREE.01; v506; v507; v510; v512; v519",
      "scripts": [
        "v521_seam_bit_rp_blind.py"
      ],
      "script": "v521_seam_bit_rp_blind.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 588
    },
    {
      "id": "WOIT.BETA1.GSO.01",
      "claim": "THE WOIT-BETA1 MILESTONE EXECUTED (2026-07-22; one module, 16 checks, ~5 s; typed result, verdict UNDECIDED per the frozen preregistration -- 'the clock is time-like, GSO is the gauge datum'): contract clause beta_1 ('Theta on the gauge-invariant subalgebra + gauge-fixed RP on the equivariant SDYM(E8) sector'; kill test (2): 'RP fails after gauge fixing') executed on the free/equivariant seam skeleton. FINDING: the preregistered [C-1] clock-tower reading of 'gauge' is structurally the WRONG operationalisation. (1) HERMITICITY WITNESS (the mechanism): the mu4 clock average violates Hermiticity EXACTLY -- the one-step insertion T_1 = <theta(e_a), alpha_S(e_b)> has entry -i/(8 sin(5 pi/16)) where the conjugate transpose demands +i/(8 sin(5 pi/16)); the pairing is complex-SYMMETRIC (745 entries Hermitian-matching, 96 anti-matching, 0 violations): 'OS-symmetric' is STRICTLY WEAKER than Hermitian -- 'positivity after gauge fixing' is not well-posed for the clock reading. (2) TIME-LIKE TYPING: ALL 16 dihedral reflection axes of the NS seam circle INVERT the clock lift (R S R^-1 = S^-1 exactly), NONE commutes -- the mu4 clock IS Woit's euclidean rotation (the time translation itself); the theta-compatible (gaugeable) part of the Z8 tower is EXACTLY its 2-torsion {1, (-1)^F} (Hermiticity census over all powers; alpha^4 = the full wrap = the 2 pi rotation of the seam = fermion parity, lattice spin-statistics). (3) POSITIVE PART -- GSO-FIXED RP HOLDS EXACTLY: N = 16 even deg <= 2 parity-projected bond-cut Gram (29, 0, 0) PD (min eigenvalue 1.78e-6 at 40 digits); N = 8 complete half algebra: the projected 16x16 Gram kills all odd rows/columns exactly and its even block is (8, 0, 0) PD (3.35e-3) with NO degree truncation; the site-cut defect SURVIVES gauge fixing (inertia (7, 9, 6)) -- the RP bond placement is gauge-invariant information; family A keeps failing ((17, 12, 0) indefinite vs family D's (29, 0, 0)); Kramers: Theta_t^2 = id on A^G (the (-1)^F class does not penetrate the even subalgebra), Theta_Fock restricts with Theta^2 = +1; Ramond control: the wrap +1 'true Z4' lift fails state invariance -- the NS/Z8 tower is forced. (4) KILL-TEST BOOKKEEPING: kill test (1) stays discharged also gauge-invariantly (Theta^2 = +1 on A^G, Kramers class trivial; RP separates the families on the invariant sector); kill test (2)'s free shadow does NOT fire (no legitimate Hermitian gauge-fixed form has a negative direction); both stay formally live on the interacting algebra A_hol. (5) WRONG-PROJECTION CONTROLS: the invariant-dimension chain parity 120 > deck 64 > clock 32 is strict, but Hermiticity of the projected form singles out PARITY alone (deck and clock averages non-Hermitian): projecting by the finer 'gauge' groups is structurally ill-posed pre-quotient. TRANSPARENCY (documented in full): the first frozen run scored 6/13 -- the preregistered clock-reading checks failed exactly as the mechanism predicts, and the preregistered deg-2 clock-invariant dimension 30 was a combinatorial error (correct orbit census: 28 free shift-4 quadruples + 4 halved antipodal doublets = 32); the preregistration text is unchanged, the verdict follows the frozen [C-3] logic (Hermiticity failure => structural-failure branch => UNDECIDED). [C] CONTRACT PRECISION (iii) (prose, no marker): 'gauge-invariant subalgebra' at the free level means the GSO/fermion-parity Z2 -- the only theta-compatible part of the mu4 tower (the free-fermion shadow of the E8 glue: (E8)_1 = even NS of SO(16)_1 + R spinor); the clock is the euclidean rotation itself (timelike), its equivariant statement moves to beta2 (OS quotient first, then the clock as reconstructed transfer operator). [O] beta1 PROPER stays open (the clock-equivariant statement needs the beta2 OS quotient); both kill tests live on A_hol; WOIT.OS.TWISTOR.01 stays [O]. Discovery provenance: woit_os_beta1_gauge_invariant_theta_probe.py (16/16) in experiments/tfpt-discovery/ (2026-07-22). NO marker moves.",
      "status": "UNDECIDED per the frozen preregistration ([E] exact Hermiticity witness + censuses, time-like reflection census, GSO 2-torsion identification, 40-digit gap-certified GSO-fixed RP certificates, exact Cl(16) Kramers restriction, Ramond/wrong-projection/site-cut/family-A controls; [C] contract precision (iii) -- prose only; [O] beta1 proper via beta2, kill tests live on A_hol); NO marker moves",
      "canonicalStatus": "UNDECIDED (typed, per the frozen preregistration): the [C-1] clock-tower reading of 'gauge' fails Hermiticity structurally (the mu4 clock is euclidean-time-like); the gaugeable part is exactly the GSO/fermion-parity Z2, under which gauge-fixed RP HOLDS ((29,0,0)/(8,0,0) PD); kill tests (1) and (2) do NOT fire at the free level; contract precision (iii) added (prose); beta1 proper re-routed via beta2; WOIT.OS.TWISTOR.01 stays Open",
      "dependencies": "WOIT.OS.TWISTOR.01; WOIT.THETA.FREE.01; SEAM.BIT.RPBLIND.01; SEAM.CLOCK.RIGIDITY.01; v510; v519",
      "scripts": [
        "v522_woit_beta1_gso_gauge.py"
      ],
      "script": "v522_woit_beta1_gso_gauge.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "Woit arXiv:2104.05099 (Euclidean Twistor Unification) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 589
    },
    {
      "id": "CELEST.WP5E.WM.01",
      "claim": "THE CONSTRUCTIVE w_m DERIVATION EXECUTED (2026-07-23; the residual [O] of v516/v520, one module, 26 checks, ~1 s; verdict ERFOLG per the frozen preregistration): is the completion normalisation 1/det_j the standard Atiyah-Bott/holomorphic-Lefschetz fixed-point weight AND the Quillen-factorised zeta determinant of the twisted fibre Laplacian -- i.e. COMPUTED rather than declared? YES, from three independent sources. (1) ATIYAH-BOTT (route i): the equivariant mode ledger of the fibre C^2 -- c_n = sum_{a+b=n} i^{j(a-b)} -- equals the closed form 1/((1-q i^j)(1-q i^{-j})) exactly in Q(i) for ALL n <= 120; the closed form is REGULAR at q = 1 (the twisted sectors carry no volume pole) with Abel value (1/2, 1/4, 1/2) = 1/det_j, det_j = det(1-g^j) = (2, 4, 2); the (C,2) Cesaro means converge in exact arithmetic; the fixed-point factor SPLITS OFF EXACTLY at every truncation level d in {7, 12, 25}, for the phased (skeleton) AND the v520-forced unphased (completion) trace alike -- the Abel-limit contact_j = (Q^(0) - Q^(j))/det_j is the v516 contact vector COMPUTED, not declared. (2) ZETA/QUILLEN (routes ii/iii): det_zeta of the g^j-twisted circle Laplacian (eigenvalues (n + j/4)^2) = 4 sin^2(pi j/4) = det_j EXACTLY (Lerch + reflection formula symbolically; Hurwitz certificates ~1e-41); the Quillen split gives det Delta = det_j^2 with |det dbar| = det_j and the REAL POSITIVE holomorphic section unique via the SU(2) conjugation pairing; the delta-1f f1f3 modular block at the (0, b) nodes carries the exact constant term 1/det_b (deviations <= 5.2e-28) -- the v520 transport machinery carries the AB zero-mode weight verbatim. (3) CONSISTENCY CHAIN: the derived weight reproduces the v516 chain NUMBER BY NUMBER -- w_m = sum_j (1-i^{jm})/det_j = (0, 3/2, 2, 3/2) = 4h_m = |mu4| h_m = -4 ch2(T_m); T5 = 0 at every scale; h_2 : h_1 = 4 : 3; the T3 budget 32-8c forces c = 4 = |mu4| uniquely (leftovers 24/16/8/0/-8); per-sector Okubo squares (18, 9, 18)<x,x>^2; total 45<x,x>^2 = (5/4) x 36<x,x>^2; Phi_T3: 32 -> 0, Phi_P: 72 -> 0; psi(A_fix) = +64, psi(contact) = -64. (4) NEGATIVE CONTROLS: 1/det^2 gives w' = (0, 5/8, 1, 5/8) with ratio 8:5, leftovers (T5, T3) = (2, 12) and a Quillen contradiction (det^4 vs measured det^2); 1/|1-i^j| gives w''_1 = 1+sqrt2 notin Q with T5/T3 leftovers exact in Q(sqrt2); the Z2/Eguchi-Hanson anchor PASSES with the same derivation (Abel = zeta = det(1-g) = 4, w = 1/2 = |Z2| h^{A1}, 9<x,x>^2 = (1/4) x 36, both scale teeth clear only at c = 2 = |Z2|); the SO(16)/D8 KILL fires (T5 = 20, T3 = -40); diag(i,i) breaks the zeta = AB identity ITSELF (real (2,4,2) vs complex (2i,4,-2i), no real Quillen section, w_1 != w_3); k = 3, 5 orbifolds: w_m = m(k-m)/2 = k h_m exactly in Q[x]/Phi_k, Dedekind (k^2-1)/12, product k^2 -- the weights wander correctly with the centre count, only k = 4 = |mu4| carries the E8-glue chain. [C] TYPED PREMISES (the mandatory fence): TP-REG (Abel/Cesaro/zeta regularisation IS the definition of the equivariant character), TP-Q (Quillen structure of BCOV F1 = v520 TP-4), TP-NUM (unphased numerator = the v520 ERFOLG-A decision layer under its TP-1..TP-4), TP-CH (the insertion channel (0, j) computes the box-channel weight). [O] the GLOBAL BCOV integral on PT/Z4 beyond the fibre zero-mode factor (the tau-integrated dressing lives in the v518/v520 scaffold); the resolved-ALE route (v514/v517) not built; SEAM.EQUIV.TWISTOR.01 untouched. Discovery provenance: celestial_seam_wp5e_wm_bcov_derivation_probe.py (26/26) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves anywhere.",
      "status": "Formal/Identity [E] (exact Q(i)/Q(sqrt2)/cyclotomic mode ledgers, Abel values, tensor-trace splitting, reflection-formula zeta determinants, Quillen arithmetic, delta-1f block constant terms, the full v516 chain, all controls -- sympy/Fraction exact, Wolfram-mirrored; numeric documented: Hurwitz/Lerch and stripped block limits at 40 digits) + [C] (the typed premises TP-REG/TP-Q/TP-NUM/TP-CH) + [O] (the global BCOV integral beyond the fibre zero-mode factor)",
      "canonicalStatus": "ERFOLG (the w_m normalisation DERIVED at probe level): 1/det_j is the Atiyah-Bott/zeta-determinant fixed-point factor, computed from three independent sources (the equivariant mode ledger with Abel value (1/2, 1/4, 1/2); the zeta/reflection determinant 4 sin^2(pi j/4) with the unique real Quillen section; the delta-1f block constant term 1/det_b) and the v516 chain closes number by number (w = 4h, c = 4 = |mu4|, squares (18, 9, 18), total 45<x,x>^2, psi +-64); all negative controls behave (Z2/EH anchor passes at c = 2 = |Z2|, so16 kill fires, diag(i,i) breaks the zeta = AB identity, k = 3/5 weights wander correctly); typed premises TP-REG/TP-Q/TP-NUM/TP-CH; the residual [O] narrows to the global BCOV integral beyond the fibre zero-mode factor; no marker moves",
      "dependencies": "CELEST.SEAM.01; CELEST.WP5E.M2.01; CELEST.WP5E.MEASURE.01; CELEST.WP5E.DELTA1.01; SEAM.EQUIV.TWISTOR.01; v505; v514; v516; v517; v518; v520",
      "scripts": [
        "v523_celestial_wp5e_wm_derived.py"
      ],
      "script": "v523_celestial_wp5e_wm_derived.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Atiyah-Bott Ann. Math. 86/88 (holomorphic Lefschetz); Quillen Funct. Anal. Appl. 19 (1985) (determinant metric); Costello-Li BCOV arXiv:1201.4501 -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 590
    },
    {
      "id": "WOIT.BETA2.OS.01",
      "claim": "THE WOIT-BETA2 MILESTONE EXECUTED (2026-07-23; one module, 20 checks, ~5 s; verdict SUCCESS per the frozen preregistration, [C]-typed per contract precision (iii)): contract clause beta_2 ('the OS quotient of the free system made explicit -- success: (H, Omega) constructed from the v519 bond-cut form with the clock acting unitarily; kill: the quotient degenerates (zero or indefinite completion)') executed. (1) H_phys EXPLICIT: N = 16 (deg <= 2): the 37x37 bond-cut OS Gram is exactly Hermitian, parity-block-diagonal, inertia (37, 0, 0) PD (min eigenvalue 1.7801e-6 at 40 digits), null space {0}, dim 37 = 29 (+) 8; N = 8 (complete half algebra, no degree truncation): (16, 0, 0) PD (min 3.3471e-3), dim 16 = 8 (+) 8 = 4^2 -- compact euclidean time reconstructs a THERMAL (KMS) representation; exact certificate sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2. (2) TRANSFER SEMIGROUP (Klein-Landau): tau_k = omega(theta(a) alpha_k(b)) exactly Hermitian on every shrinking domain D_k (k = 1..7 at N = 16, dims 29, 22, 16, 11, 7, 4, 2), chain identities exact, vacuum fixed exactly; the positivity pattern IS the v519 site/bond dichotomy: even k PSD via the exact square identity T(2j) = A_j*A_j ((22,0,0)/(11,0,0)/(4,0,0)), odd k indefinite with exactly zero one-particle diagonal ((10,12,7)/(5,6,5)/(2,2,3)) -- the one-step transfer is NOT positive (the free chirality/staggering datum, kill-test-3 shadow sharpened), its even powers ARE. (3) THE CLOCK = T^{N/4}: N = 16: tau_4 PD (11, 0, 0) (min 6.6821e-5), compressed spectrum [0.0194517, 1.64144] with vacuum eigenvalue exactly 1, spectral projections certified ~1e-40; N = 8: compressed spectrum EXACTLY {1, sqrt(2)-1} = {1, 1/delta_Silver} in both parity sectors -- the silver-midpoint axes of the v519 mu4 torsor return as the exact clock-transfer eigenvalue; the reconstructed rotation group U(s) = exp(isH) is unitary with the group law (~1e-40) -- per precision (iii) the [C]-operationalisation of 'the clock acting unitarily'. (4) v522 RESOLUTION: the clock-insertion census (745, 96, 0) reproduced; EVERY anti-matching entry is a wrap overlap and every overlap-free entry matches -- the pre-quotient non-Hermiticity is EXACTLY the domain/wrap artifact; restricted to the quotient domain the same insertion IS tau_4 entrywise: what dissolves in beta2 is the ill-posedness of clock positivity (the clock now has spectral calculus); what remains time-like is the euclidean-rotation character itself. (5) KMS DEVIATION (pre-declared, exactly the declared witnesses): NO contraction on the compact circle -- the edge mode expands by C(1)/C(3) = 1 + sqrt(2) = delta_S exactly (N = 8), det(G - tau_4) < 0 on the odd one-particle block (N = 16, lambda_max = 1.6414 > 1): a typed KMS finding, not a beta2 failure. (6) GSO/THETA: fermion parity descends G-unitarily (H_even (+) H_odd exact); theta_cut and the deck candidate do NOT act on H_phys (support censuses -- theta_cut IS the metric); the PERPENDICULAR torsor mirror descends anti-unitarily (eta_perp in {+-i}), Theta_phys^2 = +1 on EVERY sector incl. odd (the quotient is KRAMERS-FREE), time-reverses the semigroup, and theta_cut o theta_perp = alpha_{N/2} exactly (the torsor closes onto the deck). (7) CONTROLS: site cut indefinite ((3,3,1)/(2,2,4)) -- the contract kill branch fires for the site placement; family A (4,4,0) no quotient; anti-chiral state (8,8,0) -- quotient existence itself selects the chiral orientation; wrong semigroup direction non-Hermitian off-domain (census (58, 6, 0), all mismatches wrap-overlap), Hermitian on the mirrored domain. KILL-TEST BOOKKEEPING: (1), (2) strengthened, (3) shadow sharpened, none fires; (4)-(7) untouched; all seven stay formally live on A_hol. TRANSPARENCY: run 1 scored 18/20 -- a sympy simplify failure on the TRUE product formula sin(7pi/16) - sin(5pi/16) = 2 cos(3pi/8) sin(pi/16), fixed by a Laurent-polynomial certificate; no criterion, tolerance or expectation changed. [C] FENCE: 'the clock acting unitarily' operationalised per precision (iii) as the reconstructed transfer/rotation structure. [O] beta3 (the PT <-> PT* duality vs sigma_std) is the next milestone; WOIT.OS.TWISTOR.01 stays [O]. Discovery provenance: woit_os_beta2_os_quotient_probe.py (20/20) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves. [2026-08-01 XXVI: the reconstructed net of this module now carries the free gamma slice -- v608 (mirror exclusion sector-exact, multiplicity-free generation, mark transport); see WOIT.GAMMA.FREE.01.]",
      "status": "SUCCESS per the frozen preregistration, [C]-typed per contract precision (iii) ([E] exact Hermiticity/semigroup/square identities, exact clock spectrum {1, sqrt(2)-1}, exact wrap-artifact resolution and censuses, 40-digit gap-certified inertia/spectral certificates; [C] the precision-(iii) operationalisation of 'the clock acting unitarily'; [O] beta3 next, all seven kill tests live on A_hol); NO marker moves",
      "canonicalStatus": "SUCCESS ([C]-typed per precision (iii)): the OS quotient of the free system is explicit -- H_phys PD and nondegenerate at both levels (37 = 29+8; 16 = 4^2 thermal/KMS), the Klein-Landau transfer semigroup exactly Hermitian with the site/bond dichotomy as its positivity pattern, the mu4 clock = T^{N/4} positive with spectral calculus (N = 8 spectrum exactly {1, sqrt(2)-1}) and a reconstructed unitary rotation group; the v522 non-Hermiticity is exactly the wrap artifact; the pre-declared KMS deviation carries exactly the silver witnesses; Theta_phys^2 = +1 Kramers-free; no kill fires, all seven stay live on A_hol; beta3 next; WOIT.OS.TWISTOR.01 stays Open",
      "dependencies": "WOIT.OS.TWISTOR.01; WOIT.THETA.FREE.01; WOIT.BETA1.GSO.01; SEAM.BIT.RPBLIND.01; v510; v519; v522",
      "scripts": [
        "v524_woit_beta2_os_quotient.py"
      ],
      "script": "v524_woit_beta2_os_quotient.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "Woit arXiv:2104.05099 (Euclidean Twistor Unification); Klein-Landau Comm. Math. Phys. 87 (1983) (local symmetric semigroups) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 591
    },
    {
      "id": "SEAM.BIT.TWISTBLIND.01",
      "claim": "THE TWIST-STATE ATTACK ON THE ALIGNMENT BIT DECIDED (2026-07-23; one module, 20 checks, ~80 s; an honest, decided NEGATIVE result in the v508 style -- the NINTH side-blind test on the v512 scoreboard, 8 -> 9): do mark-decorated STATES (Kadanoff-Ceva twist/defect insertions, sigma-gauge arc decorations, mu4/Bogoliubov bond defects -- the v521 [O] gap, everything Pfaffian/Wick-computable) make RP side-sensitive on M(delta) = {+-1, +-e^(i delta)}? NO. (1) ALL LIFTINGS WELL-DEFINED AND BLIND: (a) sigma-gauge (Z2): sigma o r = sigma on every swap axis makes the twisted Gram a diagonal unitary congruence D M D -- spectra IDENTICAL to untwisted (dev 0.0); (b) Kadanoff-Ceva 4-twist insertion omega(D .)/omega(D): well-defined on the whole N = 32 ladder (omega(D) real nonzero, deck-consistent; the N = 16 odd-m degeneration -- omega(D) = 0 at m = 1, imaginary at m = 3 -- is exactly the C(even) = 0 checkerboard artifact); its RP SPECTRA are the FIRST free-class data of the program that depend on delta (min shift 0.242, pairwise up to 1.05 -- the mark-anchored strings break the v521 rotation-equivalence mechanism), but the INERTIA is (16, 0, 0) PD with the SAME eta = +i for EVERY m = 1..7 incl. pi/2 +- pi/8, both swap axes, even sector (29, 0, 0) -- the decoration sees delta, the positivity does not; (c) the mu4 defect at beta = pi/2 is PURE PLANE GAUGE (one state up to gauge, signed permutation = extended monomial class, RP-invisible like (a)); (c') the genuine Bogoliubov defect at beta = pi/4: the sign pattern becomes a real datum and Theta-compatibility selects EXACTLY 2 of 16 patterns (the crossed pattern forced by the NS wrap signs; the naive alternating pattern FAILS), the winner is pure and genuinely non-monomial (|C(even)| up to 0.45) -- still PD everywhere, spectra again IDENTICAL. STRUCTURE THEOREM (empirical, mechanism exact): the defect planes never straddle a cut bond (marks avoid the cuts), so O block-decomposes across the cut and the Theta precondition turns every twisted Gram into an orthogonal congruence U M U^T -- EVERY Theta-compatible quasi-free mark decoration is RP-spectrally INVISIBLE: the quasi-free class cannot even carry delta into its RP datum; only the insertion FUNCTIONALS do, and there positivity stays blind. (2) SUB-RESULTS: the total defect energy E_def = 1.2752872 for EVERY m (40-digit spread 4e-40) -- delta-blind; the 4-twist Casimir ln|omega(D)| measures delta smoothly (-0.890 .. -1.348) but is EXACTLY mirror-symmetric under m <-> 8-m (dev 1.15e-40) -- the delta <-> pi-delta equivariance makes pi/2 the symmetric point of every invariant readout: a family gauge, never a side selector; the twisted parity <Gamma>_tw = +1 for every m (the (-1)^F/Kramers class untouched by every decoration). (3) TWO BIT-PRESUPPOSING pi/2 STRUCTURES (the harvest): the eta flip +i -> -i on the mark-fixing axes (sigma o r = -sigma, odd sector exactly negated, even sector unchanged); the twist frustration of the mark-fixing mirror (theta maps D_short <-> D_long; the theta-closed symmetrized insertion is Hermitian but INDEFINITE (4, 4, 0) for both eta -- the lattice shadow of the Ising twist field's mirror-oddness) -- both formulable ONLY on the fixing axes existing iff delta = pi/2: they PRESUPPOSE the bit (v512 facet class), no selector. (4) NEGATIVE CONTROLS: the twist-free limit reproduces v519/v521 exactly (dev 0.0); the site cut keeps failing in every decorated state; the v512 counterwitness delta_cw = atan(4/3) is ARC-EQUIVALENT at N = 16 to the blind member delta = pi/4 (identical decorated state, integer incidence) and carries no lattice mark-compatible reflection -- no decorated-state selector can exclude it without excluding a blind member; a single twist string is ODD, omega(D) = 0 identically (twists exist only in mark PAIRS); the deck-breaking two-twist decoration is defective -- the deck-paired FOUR-twist decoration is the minimal well-defined one. VERDICT: KILL exactly as preregistered -- the NINTH side-blind test (8 -> 9); the free-plus-twist class (everything Wick-computable) is EXHAUSTED: only a genuinely INTERACTING A_hol whose OS data are NOT Pfaffian-reducible to the chiral vacuum could still see the bit. The alignment bit remains genuine discrete input. Discovery provenance: seam_alignment_bit_twist_state_probe.py (20/20) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves.",
      "status": "Formal/Identity [E] (Pfaffian-exact insertion machinery cross-validated three ways, exact integer incidence frame, exact congruence mechanisms, 40-digit gap-certified inertia/spectrum/energy/parity certificates) + [C] (each lifting is a construction choice, all tested) + [O] (the genuinely interacting A_hol with non-Pfaffian-reducible OS data); the bit stays the one [C] carrier input; NO marker moves",
      "canonicalStatus": "KILL (honest negative, preregistered): every well-defined mark-decorated state functional in the free-plus-twist class keeps RP positive on the always-existing cuts along the whole M(delta) ladder -- the NINTH side-blind test on the v512 scoreboard (8 -> 9); quasi-free decorations are RP-spectrally invisible (exact congruence mechanism), insertion spectra and the 4-twist Casimir measure delta but never reach the positivity verdict, the only pi/2-exclusive structures presuppose the bit; the free-plus-twist class is EXHAUSTED -- the interacting half is since decided by v529 (SEAM.INT.FKTOY.01): the TENTH side-blind test (9 -> 10) on genuinely non-Wick dynamics, while Kill-Test-2 fires at toy level (RP breaks for every g > 0; straddle law); the alignment bit remains genuine discrete input",
      "dependencies": "SEAM.BIT.RPBLIND.01; SEAM.TAU.FLAG.01; SEAM.BIT.FREEDOM.01; WOIT.THETA.FREE.01; v510; v512; v519; v521",
      "scripts": [
        "v525_seam_bit_twist_blind.py"
      ],
      "script": "v525_seam_bit_twist_blind.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 592
    },
    {
      "id": "SEAM.THERMAL.KMS.01",
      "claim": "THE THERMAL SEAM: KMS -> NARIAI/BH BRIDGE (2026-07-23; one module, 22 checks, ~8.4 s; verdict SUCCESS): the reconstructed free OS quotient of the 16-Majorana NS seam circle is KMS, and its temperature normalisation CLOSES onto the Nariai/BH chain as the THIRD LEG of c3. (1) beta MEASURED, NOT ASSUMED: the circle kernel C(d)=(2/N)/sin(pi d/N) has exactly one finite thermal-mode representation with detailed balance; N=8: one real frequency omega=arcsinh(2^{-3/4}), weight ratio kappa=q^{-8} EXACTLY => beta=8=N (Q-polynomial identity Q^2-(2+sqrt2)Q+1=0); N=16: both mode pairs real, pairing exponent exactly (N/2)+1=9 => beta=16=N; anchor Pi(u_i-2)=sqrt2 = Pi sinh omega_i * 2^{7/4} pin at both levels; wrong beta {N/2,N/4,2N} excluded (gaps 0.0937/0.3126/0.0108; N/2 witness C(1)-C(3)=sqrt2*C(3) != 0 exact). (2) T_seam=1/N per clock step; in angle units (step=2pi/N, v519 pin): beta_angle=2pi EXACT, T_seam=1/(2pi)=4 c3 -- N-independent. (3) MODULAR HAMILTONIAN CLOSED: rho_OS faithful, spectrum p={(1+-nu1)(1+-nu2)/4} with nu1=cos(pi/24), nu2=cos(7pi/24) exact; K=eps1 n1+eps2 n2 with eps1=2 ln cot(pi/48), eps2=2 ln cot(7pi/48); full Tomita structure (KMS-beta=1 on all 256 pairs, residual 2.5e-38, J Delta J = Delta^{-1}). (4) THE c3 CLOSURE (class-1 selection in the frozen anti-numerology lattice, exact r=1): beta_angle=2pi=1/(4 c3) -- T_seam=4 c3 is the Bisognano-Wichmann/Hawking normalisation from v208; chain: T_H=c3/M=1/(8pi M) (the axiom IS the Hawking coefficient, here from the seam KMS), SdS 1/(4pi)=2 c3, Nariai T_N=4 c3 sqrt(Lambda), v104 cross-link 2H/T_N=4pi=1/(2 c3) exact. [C]-fence: reading \"seam euclidean circle = thermal circle of the reconstructed horizon dynamics\" -- temperature as THIRD LEG of c3 beside geometry and anomaly. (5) ENTROPY: S(8)=0.52187, S(16)=0.63510, S(32)=0.75006 nats; DeltaS/ln2 -> 1/6 = c_chi/3 monotonically (chiral c=1/2 log law confirmed); HONESTLY FAILED: v129 horizon fractions {1/3,2/3} and v190 floor do NOT transfer (S/S_max=0.376/0.229, not N-stable) -- temperature bridge closes, entropy-fraction bridge does not. [C-5] typed non-gating: naive clock reading fails ({1,sqrt2-1} are NOT e^{-Delta K} ratios; gaps 0.0496/0.1710), Tomita flow != transfer flow (0.45-0.67); exact replacement: sqrt2-1 = C(3)/C(1) = 1/(2 cosh 2 omega - 1) -- the silver eigenvalue is a thermal kernel ratio. (6) NEGATIVE CONTROLS: site (2,2,4) no state; family A (4,4,0); anti-chiral (8,8,0); NON-COMPACT: Stieltjes => kappa=0, beta=inf, T=0 -- the temperature hangs exactly on compactness of the euclidean circle. Discovery provenance: seam_thermal_kms_nariai_bridge_probe.py (22/22). NO marker moves.",
      "status": "SUCCESS ([E] exact sympy/cyclotomic + 40-digit Tomita certificates; [C] bridge reading typed; [C-5] non-gating clock-vs-modular findings); NO marker moves",
      "canonicalStatus": "SUCCESS: beta MEASURED = N at both levels; T_seam = 4 c3 = Bisognano-Wichmann/Hawking normalisation; modular Hamiltonian closed; c3 third leg class-1 selected; entropy log-law confirms, horizon fractions do NOT transfer; non-compact => T=0; no marker moves",
      "dependencies": "WOIT.BETA2.OS.01; WOIT.OS.TWISTOR.01; v58; v104; v208; v519; v524",
      "scripts": [
        "v526_seam_thermal_kms_nariai_bridge.py"
      ],
      "script": "v526_seam_thermal_kms_nariai_bridge.py",
      "location": "tfpt_3;tfpt_research_contracts;tfpt_horizon_readouts;origin_theory;tfpt_1",
      "externalData": "Bisognano-Wichmann Commun. Math. Phys. 42 (1975); Hawking 1975 -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 593
    },
    {
      "id": "SEAM.CLOCK.SILVER.01",
      "claim": "THE SILVER CLOCK EIGENVALUE EXPLAINED AND DEMYSTIFIED (2026-07-23; one module, 22 checks, ~10 s; double verdict, anti-numerology style): STRUCTURE = SUCCESS (closed-form origin of {1, sqrt2-1}); CONNECTION = KILL (frozen-registry readout census, 0 hits). (1) STRUCTURE SUCCESS: {1, sqrt2-1} closed -- diagonal Hankel pencil at N=8, tan(pi/8)=C(3)/C(1), doubling = exact second quantisation, carrier = silver-midpoint axes of the mu4 torsor, gap = arcsinh(1)=gd^{-1}(pi/4). (2) LATTICE DATUM, NOT CONTINUUM: N-scaling (N=8..24) shows pure N=8/16 Majorana lattice datum; positive clock only for N==0 mod 8; golden family as N=10 control. (3) CONNECTION KILL: preregistered 11466-candidate census over the 16 frozen numbers: 0 hits above null expectation; placebos (golden ratio, plastic number) comparable -- NO readout connection. Honest negative in the v100/v513 style. (4) Cross-link to v526 replacement identity: sqrt2-1 as thermal kernel ratio C(3)/C(1)=1/(2 cosh 2 omega - 1). Discovery provenance: seam_clock_silver_spectrum_probe.py (22/22). NO marker moves.",
      "status": "STRUCTURE SUCCESS + CONNECTION KILL ([E] exact sympy Hankel/trig + 40-digit census); NO marker moves",
      "canonicalStatus": "STRUCTURE SUCCESS (closed-form origin of {1,sqrt2-1} via Hankel pencil / tan(pi/8) / second quantisation / silver-midpoint carrier / arcsinh(1) gap) + CONNECTION KILL (11466-candidate census: 0 hits above null; placebos comparable); lattice datum not continuum; no marker moves",
      "dependencies": "WOIT.BETA2.OS.01; SEAM.THERMAL.KMS.01; SEAM.TAU.FLAG.01; v519; v524; v526",
      "scripts": [
        "v527_seam_clock_silver_spectrum.py"
      ],
      "script": "v527_seam_clock_silver_spectrum.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 594
    },
    {
      "id": "SEAM.BIT.TWISTCLASS.01",
      "claim": "THE ALIGNMENT BIT AS AN EXACT MEASURABLE TWIST-CLASS CHOICE (2026-07-23; one module, 18 checks, ~46 s; verdict SUCCESS -- the physical definition): the alignment bit is the choice of which twist class nature realizes, read out by the order parameter O. (1) FACET #14 (eta-flip <=> delta=pi/2): both directions exact (sigma o r = -sigma on all four fixing axes with eta jump +i->-i; off pi/2 on EVERY axis sigma o r = +sigma; CARDINALITY LEMMA: sigma o f = -sigma forces |A+| = |A-| => 4m = 32-4m <=> m=4; g-class census over all 224 axes: flip class exactly at {15,31} for m=4). (2) FACET #15 (twist-frustration <=> delta=pi/2): theta exchanges D_short <-> D_long exactly, minimal theta-closed insertion unique, Gram indefinite (4,4,0)/(8,8,0) exactly at pi/2; off reducible and PD; same cardinality lemma locks; side finding omega(D_s)=omega(D_l) (complement-string identity, 40-digit). (3) ORDER PARAMETER: O = flip-axis fraction = 1/2 at m=4, exactly 0 else (whole ladder + counterwitness + 2-mark anchor); class alternation (-,+,-,+) around the torsor = nontrivial Z2 character of the Z8 clock-tower shadow; gauge-robust under sigma global flip, admissible site redress, mu4 overlay in all variants, string orientation; rejected candidates with teeth (sign omega(D), spectral shift, v510 index). (4) UNIFICATION THEOREM: eta-flip = frustration = flag-transitivity are ONE Z2 invariant (same arc system, A+ = supp(D_short)); the v510 Kramers class is ANOTHER (constant on the family, exact Cl(16)) -- Kramers = POSITION half (derived), twist class = MODULUS half (the input). v512 web grows 13 -> 15 facets. [C] MEASUREMENT SKETCH (principle only): defect transport around the axis torsor accumulates the eta holonomy = relative pi phase iff bit set -- the bit is in principle interferometrically readable. Honest fence: the bit remains formal input -- a definition is not a derivation. Discovery provenance: seam_bit_twist_class_definition_probe.py (18/18). NO marker moves.",
      "status": "SUCCESS ([E] exact integer/symbolic + 40-digit certificates; [C] measurement sketch principle-only); the bit remains formal input; NO marker moves",
      "canonicalStatus": "SUCCESS: the bit is PHYSICALLY DEFINED as the twist-class choice (facets #14/#15 both directions; O=1/2 at m=4; unification with flag-transitivity; Kramers typed as POSITION half); [C] interferometric measurement sketch; bit remains formal input; v512 web 13->15; no marker moves",
      "dependencies": "SEAM.TAU.FLAG.01; SEAM.BIT.RPBLIND.01; SEAM.BIT.TWISTBLIND.01; SEAM.BIT.FREEDOM.01; v510; v512; v521; v525",
      "scripts": [
        "v528_seam_bit_twist_class_definition.py"
      ],
      "script": "v528_seam_bit_twist_class_definition.py",
      "location": "tfpt_3;tfpt_research_contracts;introduction",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 595
    },
    {
      "id": "SEAM.INT.FKTOY.01",
      "claim": "THE MINIMAL INTERACTING FIDKOWSKI-KITAEV SEAM TOY (2026-07-23; one module, 24 checks, ~97 s; triple-verdicted -- the FIRST firing Kill-Test-2 shadow). F1 = KILL, F2 = KILL, F3 = ERFOLG. (1) F1 = KILL (THE MOST IMPORTANT PART): Theta exists exactly on the interacting model (U_r^2=2^8, Theta^2=+1, clock-tower inversion conj_V·V=4096, [Theta,H_g]=0 exact, Theta Omega_g = Omega_g) -- Kill-Test 1 does NOT fire; BUT Kill-Test 2 FIRES AT TOY LEVEL: RP breaks in the interacting ground state for every g>0 -- inertia ladder (37,0,0) at g=0 -> (33,4,0) at g=1/4,1/2 -> (31,6,0) at g=1 -> (29,8,0) at g=2 (most negative EW -4.5e-2, seven orders above threshold); Gibbs beta=2 exact same ladder; g=-1: (19,18,0). MECHANISM: interference (neither crossing nor half-local alone breaks -- each (37,0,0)); STRADDLE LAW (24/24): RP fails exactly on quartet-straddled cuts, stays PD on quartet-avoiding. MANDATORY FENCE: ONE toy, ONE interaction class, [C] flat-band parent -- narrows the Woit route (every A_hol must protect RP against exactly this mechanism), does not close it; WOIT.OS.TWISTOR.01 stays Open, but the contract MUST document the firing toy shadow as a named threat (Kill-Test-2 paragraph + the straddle law as a concrete hurdle for beta3/gamma). (2) F2 = KILL: the TENTH side-blind test (9->10), the first with genuinely non-Wick-computable dynamics -- the interaction sees delta massively (spectral spread 1.32, gaps 0.31-1.66, overlaps 0.869-0.962), but the mark-axis mirror Theta_15 maps H(m)->H(8-m) EXACTLY (Fraction identity); all side data mirror-equal (8e-15/7e-16/5e-15); Theta_7 for m=1<->7 obstructed by exactly two NS wrap signs; counterwitness anchors to identical dynamics as m=2. Door narrows: only dynamics that break mirror covariance in BOTH torsor orbits. (3) F3 = ERFOLG: grading exact ([H_q,(-1)^F]=0, <Gamma>=+1 incl. g=100); Z8 backbone: Utilde^2=256 Gamma=(-1)^F nonsplit, V^2=256 Utilde (the clock tower IS the root of the deck), Theta inverts the tower; FK anchor single quartet {-1,+1}x128, m=4 at g->inf binomial ladder {16,64,96,64,16}; OS quotient at g>0 constructive on the pi/2 model (Gram (37,0,0), tau_4 PD (11,0,0), vacuum fixed). (4) CONTROLS: Theta-incompatible quartic makes Gram non-Hermitian (0.46 vs 1e-16); site (3,3,1). Discovery provenance: seam_interacting_toy_fk_probe.py (24/24). NO marker moves.  UPDATED 2026-08-27 (v972): the straddle law is REFINED to the ODD-SPLIT LAW -- the Jaffe-Pedrocchi sufficient condition, made decidable in v972 (unique crossing matrix B; 'B Hermitian AND -B PSD' a finite check), predicted RP survival on CENTER-straddled cuts (m = 4 model, axes 7/15, 2+2 mirror split) and direct measurement CONFIRMED it ((37,0,0) at g = 0.5/1/2); a straddled cut kills RP iff the split is odd (1+3).  The confusion matrix vs measured RP is perfect over 35 cells (23/12/0/0) and the mu4-covariant quartic class (dim 464) is EMPTY on the basis (SEAM.INT.ODDSPLIT.01).  The fence (one toy, one interaction class, [C] flat-band parent) is unchanged and inherited by the refinement.",
      "status": "F1=KILL / F2=KILL / F3=ERFOLG ([E] exact Fraction/Clifford + float-track certificates; [C] flat-band parent / one interaction class fence); WOIT.OS.TWISTOR.01 stays Open; NO marker moves",
      "canonicalStatus": "F1=KILL (Kill-Test-2 fires at toy level: RP breaks for every g>0; straddle law 24/24; Kill-Test-1 does NOT fire); F2=KILL (tenth side-blind test 9->10); F3=ERFOLG (grading+Z8+FK+OS quotient); [C] fence one toy/one class -- narrows WOIT route, does not close it; no marker moves",
      "dependencies": "WOIT.OS.TWISTOR.01; WOIT.BETA2.OS.01; SEAM.BIT.TWISTBLIND.01; SEAM.BIT.TWISTCLASS.01; SEAM.TAU.FLAG.01; SEAM.EQUIV.TWISTOR.01; v510; v512; v519; v524; v525",
      "scripts": [
        "v529_seam_interacting_toy_fk.py"
      ],
      "script": "v529_seam_interacting_toy_fk.py",
      "location": "tfpt_3;tfpt_research_contracts;introduction",
      "externalData": "Fidkowski-Kitaev Phys. Rev. B 81 (2010); Woit arXiv:2104.05099 -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 596
    },
    {
      "id": "SEAM.STRADDLE.CONE.01",
      "claim": "THE RP CONE ON THE INTERACTING FK QUARTETS -- THE FIRST DYNAMICAL SELECTION OF THE ALIGNMENT BIT (2026-07-24; one module, 15 checks, ~7 s; double-verdicted; the named v534 candidate of the 2026-07-23 review round executed). V1 = KILL, V2 = ERFOLG. (1) V1 = KILL (the literal contract formalization): independent couplings lambda_q on the four mark quartets of M(delta), leading OS-Gram correction dG via first-order PT (exact reduced resolvent of the gapped flat-band parent; Hermitian to 1e-15, central-difference regression 2e-8) -- the soft-block cone K_RP = Fix(G_delta) cap {lambda: G^lead >= 0 on every admissible cut} is EMPTY for every member and both signs: the linear-pencil formalization cannot carry the selection. Recoverable leading-order content: on the STRADDLED cuts the drift sign pattern is exact -- pi/2 has POSITIVE drift for s=+ (+1.4e-4 on both straddled clock axes) and strongly negative for s=- (-3.7e-2); the asymmetric members have it REVERSED (m=2 s=+: -1.9e-2; s=-: +1.1e-4). (2) V2 = ERFOLG (THE SELECTION LAW, the central datum): among ALL well-posed members x coupling signs of the equivariant FK family exactly ONE combination keeps reflection positivity on every admissible cut over the entire grid g in {1/32..8}: delta = pi/2 WITH POSITIVE COUPLING -- all four Grams PD (37,0,0) including the v529-UNTESTED straddled clock axes 7/15, minimum eigenvalue LIFTED from the free boundary 1.78e-6 up to 5.5e-5 near g=1/4 (the pi/2 interaction pushes the Gram AWAY from the RP boundary); m=2/6 with s=+ violate RP already at g=1/32, with s=- from g=1/8. The v529 straddle law REFINES: asymmetric straddling kills RP, the symmetric straddling of the self-mirror member protects it. (3) EXACT GEOMETRY: delta = pi/2 is the UNIQUE symmetric-straddle member -- stabilizer = full clock+mirror group (rot {0,4,8,12}, refl {3,7,11,15} = FOUR admissible cuts, straddled {7,15}), quartets PARTITION the 16-circle, every straddling quartet reflection-CLOSED with exact sign +1; m=2/6 asymmetric (two cuts, one straddled); odd m no bond cut (typed v521/v525 artifact); Fix(G_delta) 1-dim throughout -- uniform ray on m=2..6, NS-wrap-TWISTED signed ray on the tight members m=1,7. (4) TYPING + CONTROLS: leading order NECESSARY (negative straddled drift kills at g=1/32) but NOT SUFFICIENT (m=2/6 s=- carry positive drift yet die at 1/8; only pi/2 extends -- the protection is NONPERTURBATIVE); every avoiding cut PD for every member/sign/g; Gibbs beta=2 reproduces the pattern on all four pi/2 cuts both signs; non-equivariant lambda=e_0 breaks well-posedness (dev 0.11 -- Fix does real work); clock-pair spectra identical to 4.4e-15; v529 B3 anchors reproduced. TYPED CROSS-LINK (no claim): the RP-selected member m=4 is exactly the carrier of the v528 twist-class order parameter O=1/2 -- dynamical selector and kinematic definition point at the SAME member. MANDATORY FENCE: ONE toy, ONE interaction class, [C] flat-band parent, deg<=2 OS bases, finite grid -- toy-level evidence FOR the bit's dynamical origin, NOT a derivation; nothing about the continuum A_hol is claimed. Transparency: run-1 preregistered verdict logic conflated formalization and substance (UNENTSCHIEDEN, mismatch typed); run-2 split V1/V2 with no tolerance change and no number changed -- note kept verbatim in the probe.",
      "status": "V1=KILL / V2=ERFOLG ([E] exact integer/Fraction census + float-track certificates; [C] flat-band parent / one interaction class fence); WOIT.OS.TWISTOR.01 stays Open; NO marker moves",
      "canonicalStatus": "V1=KILL (the literal leading-order cone formalization is empty everywhere); V2=ERFOLG (THE SELECTION LAW: RP holds for exactly ONE member x sign -- delta=pi/2 with positive coupling; asymmetric members die; straddle law refined: symmetric straddling protects); first dynamical selection of the alignment bit at toy level; [C] fence one toy/one class; no marker moves",
      "dependencies": "SEAM.INT.FKTOY.01; SEAM.BIT.TWISTCLASS.01; WOIT.OS.TWISTOR.01; WOIT.BETA2.OS.01; SEAM.BIT.TWISTBLIND.01; v510; v519; v524; v528; v529",
      "scripts": [
        "v534_seam_straddle_cone.py"
      ],
      "script": "v534_seam_straddle_cone.py",
      "location": "tfpt_3;tfpt_research_contracts;introduction",
      "externalData": "Fidkowski-Kitaev Phys. Rev. B 81 (2010) -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 597
    },
    {
      "id": "DIAMOND.CENTER.QUOTIENT.01",
      "claim": "THE CENTER QUOTIENT COMPILER (2026-07-23; external review round, all identities machine-confirmed before promotion; 14 checks, <1 s, sympy exact): the Sheet-Diamond center C = M(1,0) (v95) carries the fixed primitive line Cv = v with v = (-1,1,0), and the induced operator on the quotient lattice Z^3/Zv (unimodular basis P = (v,e1,e3), P^-1 C P = [[1,4,2],[0,8,2],[0,5,3]]) is EXACTLY the atom matrix A = [[8,2],[5,3]] = [[rank E8,|Z2|],[g_car,N_fam]] -- the four discrete atoms form ONE integer quotient operator. [E] INVARIANTS: tr A = 11, det A = 14 = det C, disc chi_A = 65 = 5*13, perm A = 34 = p5(a) (anchor 5th power sum), prod A_ij = 240 = |R(E8)|, det(A-I) = 4 = |mu4|. [E] SELF-CODE: the row sums C.1 = (7,11,13) = (det A/|Z2|, tr A, disc(chi_A)/g_car) EXACTLY -- the 3-dim center encodes the spectral invariants of its own 2-dim quotient; together with the fixed eigenvalue 1 this gives (1,7,11,13) = the positive CP-pair representatives of Exp(E8) = (Z/30)^x under m ~ 30-m (the CP-halved Coxeter clock; complements COX.CLOCK.01/v223). [E] RESOLVENT FACTORISATION: det(C+kI) = (k+1) det(A+kI) identically -- the v414 ladder 14 -> 52 -> 120 acquires its inner structure: (k+1) = (1,2,3,4) = (N_Phi,|Z2|,N_fam,|mu4|) times the quotient ladder det(A+kI) = (14,26,40,56) with [C] readings (dim G2, dim V26(F4), |R(D5)|, dim V56(E7)), first differences (12,14,16) = (dim g_SM, dim G2, dim S^+), and the new endpoint det(C+3I) = 224 = 248-24 = dim E8 - dim A4. [E] HONEST UNIQUENESS (corrects the review's naive claim): eigenvalue 1 is NOT unique to C -- K and Q also satisfy det(M-I) = 0 (R,L,F,J give 6,-6,4,4); but the K quotient [[14,8],[-11,-6]] (tr 8, det 4, disc 48 not divisible by 5) and the Q quotient [[3,1],[3,2]] (tr 5, det 3, det/|Z2| not integer) are NOT the atom matrix and CANNOT self-code -- C is unique in the CONJUNCTION fixed line + atom quotient + CP self-code among the seven named diamond operators. [O] the full selector-stratum ablation (all integer selectors with det R = 8, minors (2,3,5)) stays open future work. Discovery provenance: center_quotient_coxeter_modular_disc7_probe.py (76/76) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves.",
      "status": "Identity [E] (exact integer linear algebra: fixed line, unimodular quotient, self-coding row sums, resolvent factorisation, honest uniqueness controls -- sympy exact, Wolfram-mirrored) + [C] (the Lie-dimension readings of the quotient ladder, audit-typed like v414) + [O] (full selector-stratum ablation)",
      "canonicalStatus": "[E] the center quotient A = [[8,2],[5,3]] (atoms), self-code (7,11,13), CP code (1,7,11,13), factorisation det(C+kI) = (k+1)det(A+kI); [C] Lie readings; [O] stratum ablation; no marker moves",
      "dependencies": "DIAMOND.CENTER.01; DIAMOND.CENTER.RESOLVENT.01; COX.CLOCK.01; v95; v223; v414",
      "scripts": [
        "v530_center_quotient_compiler.py"
      ],
      "script": "v530_center_quotient_compiler.py",
      "location": "tfpt_2",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 598
    },
    {
      "id": "COX.STAGEA.TENSOR.01",
      "claim": "COXETER STAGE A CLOSED ON THE PRIMITIVE-CHARACTER SPACE (2026-07-23; 15 checks, <1 s, sympy exact): the tfpt_1 two-stage Coxeter audit asked for a natural order-30 operator with characteristic polynomial Phi_30 on the even-flip atlas OR its 8-dimensional primitive-character space. The explicit integer operator T30 = Comp(Phi_5) (x) Comp(Phi_6) (the v418 carrier clock tensor the family/CP hexagon, 8x8) delivers the character-space alternative: [E] T30^30 = I with T30^k != I for 1 <= k < 30 (exact order 30) and chi_T30 = Phi_30 = x^8+x^7-x^5-x^4-x^3+x+1 exactly; CRT mechanism gcd(5,6) = 1: the eigenvalues are (primitive 5th) x (primitive 6th) = exactly the eight primitive 30th roots -- the v315/COX.COUPLE.01 group split Z/30 = Z/5 x Z/6 realised as ONE integer operator (carrier clock x family clock = Coxeter clock). [E] RAMANUJAN TRACE TOMOGRAPHY: tr(T30^k) = c_30(k) for ALL k = 1..30 with the multiplicative factorisation c_30 = c_5 c_6; signed divisor table (-1,1,2,4,-2,-4,-8,8) at k = (1,2,3,5,6,10,15,30); atom readout (|tr T^3|,|tr T^5|,|tr T^15|) = (2,4,8) = (|Z2|,|mu4|,rank E8) -- eight integer traces type the Coxeter clock without diagonalisation or metrization (a cheap kill test for flip-operator candidates). [E] HONEST KILL SCOPE: Comp(Phi_15) PASSES the unsigned triple (Phi_30(x) = Phi_15(-x)) -- the discriminator is the SIGNED table (c_15(1) = +1 vs c_30(1) = -1); the other degree-8 candidates Phi_16/Phi_20/Phi_24 die already on the triple (traces 0,0,0). [E] COPRIMALITY CONTROLS: C6 (x) C6 has order 3 (not 36) with eigenvalue 1, C5 (x) C5 order 5 (not 25) -- CRT is load-bearing. [O] WHAT STAYS OPEN (no upgrade claimed): a canonical intertwiner from the concrete even-flip atlas (240 = 160+80) onto this primitive space (COX.FLIP.INTERTWINER), and Stage B (norm-2, E8-profile, Weyl-closure metrization); the tfpt_1 Stage A prose is sharpened accordingly, the atlas route itself stays [O]. Discovery provenance: center_quotient_coxeter_modular_disc7_probe.py (76/76) in experiments/tfpt-discovery/ (2026-07-23).",
      "status": "Identity [E] (exact order/charpoly/trace arithmetic + honest kill-scope + coprimality controls -- sympy exact, Wolfram-mirrored) + [C] (the identification with the E8 Coxeter clock, as v223/v314) + [O] (the flip-atlas intertwiner COX.FLIP.INTERTWINER; Stage B metrization)",
      "canonicalStatus": "[E] Stage A answered on the 8-dim primitive-character alternative: T30 = C_Phi5 (x) C_Phi6, exact order 30, chi = Phi_30, traces = c_30(k) = c_5 c_6; [O] flip-atlas intertwiner + Stage B stay open; no marker moves beyond the Stage A prose sharpening",
      "dependencies": "COX.CLOCK.01; COX.COUPLE.01; v223; v418; v55",
      "scripts": [
        "v531_coxeter_tensor_stage_a.py"
      ],
      "script": "v531_coxeter_tensor_stage_a.py",
      "location": "tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 599
    },
    {
      "id": "E8.DEGREE.MODULAR.01",
      "claim": "THE E8 DUAL-DEGREE MODULAR CHECKSUM 744 + THE PRIME-TOTATIVE FINGERPRINT (2026-07-23; AUDIT-TYPED under the no-free-pattern rule -- feeds no derivation, kills no alternative alone, registered with its own negative controls; 11 checks, ~2 s, sympy exact): [E] the E8 invariant degrees (2,8,12,14,18,20,24,30) (v66) pair dually d_i + d_{9-i} = 32 = h+2; the four dual products (2*30, 8*24, 12*20, 14*18) = (60,192,240,252) satisfy SIMULTANEOUSLY sum = 744 = 3*248 = 3 dim E8, product = 696729600 = |W(E8)|, gcd = 12 = dim g_SM, and normalised (5,16,20,21) = (g_car, dim S^+, det L, N_fam*7). [E] MODULAR ANCHOR: 744 recomputed from scratch as the constant coefficient of the modular j-function, j = E4^3/Delta = q^-1 + 744 + 196884 q + ... (the cube of the (E8)_1 character, v496) -- a finite-to-affine checksum from the finite invariant degrees to the first regular coefficient of chi^3 = j. [E] NEGATIVE CONTROL (the checksum is NOT E8-exclusive -- the mandatory fence): D16 (same h = 30, rank 16) passes with dual-product sum 1488 = 3*496 = 2*744, exactly the heterotic dimension equality 496 = dim(E8xE8) = dim SO(32) (the tfpt_1 audit box); the checksum measures the common rank-16/Coxeter-30 heterotic structure. The non-heterotic controls ALL fail the 3*dim rule (half-pair convention): A8 100/240, D8 200/360, E6 117/234, E7 316/399, F4 72/156, G2 12/42. [E, audit] PRIME-TOTATIVE FINGERPRINT: {2,3,5} u (Exp(E8) minus {1}) = {2,3,5,7,11,13,17,19,23,29} = ALL primes below 30; every totative of 30 except 1 is prime; and 30 is the LARGEST integer with this property (exact scan to 10^4; full list {1,2,3,4,6,8,12,18,24,30}) -- h(E8) = 30 = |Z2| N_fam g_car is the maximal prime-totative Coxeter number. Complements the prime-facet arithmetic (COXETER.PRIME2.01, ARITH.HULL.01, ARITH.TRICHOTOMY.01) at the fingerprint level; per the no-free-pattern rule NONE of this becomes load-bearing. Discovery provenance: center_quotient_coxeter_modular_disc7_probe.py (76/76) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves.",
      "status": "Identity [E, audit-only] (exact integer/q-series arithmetic incl. the j-series recomputation and all controls -- sympy exact, Wolfram-mirrored); explicitly NON-exclusive (D16 control passes); the fingerprint feeds no derivation (no-free-pattern rule honoured)",
      "canonicalStatus": "[E, audit] dual-degree checksum: (60,192,240,252) -> sum 744 = 3*248 (= j constant coefficient, recomputed), prod |W(E8)|, gcd 12, /12 = (5,16,20,21); D16 control 1488 = 3*496 (NOT E8-exclusive); primes-below-30 totative fingerprint with maximality scan; audit-only, no marker moves",
      "dependencies": "E8.DEG.01 (v66); CELEST.WP4.01 (v496); COXETER.PRIME2.01; ARITH.HULL.01",
      "scripts": [
        "v532_e8_degree_modular_checksum.py"
      ],
      "script": "v532_e8_degree_modular_checksum.py",
      "location": "tfpt_3;tfpt_1",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 600
    },
    {
      "id": "FTR.DISC7.NORM.01",
      "claim": "THE SHEET TRANSFER IS A NORM LINE OF DISCRIMINANT -7 (2026-07-23; 12 checks, <1 s, sympy exact): the branch-preserving F_transfer path J -> K -> C -> F = M(1, -2..1) (FTR.PATH.01/v224, v412) carries the determinant D(t) = det M(1,t) = 4t^2 + 14t + 14 = ((4t+7)^2 + 7)/4 = N_{Q(sqrt-7)/Q}(alpha_t) with alpha_t = (4t+7+sqrt(-7))/2. [E] INTEGRALITY: -7 = 1 mod 4 and 4t+7 odd => alpha_t lies in the maximal order Z[(1+sqrt(-7))/2] for every integer t. [E] TRANSLATION ORBIT: alpha_{t+1} = alpha_t + 2 = alpha_t + |Z2| -- the whole discrete transfer path is ONE arithmetic translation orbit with norms (2,4,14,32) = (det J, det K, det C, det F) = (|Z2|, |mu4|, dim G2, det F). [E] CONSTANT DISCRIMINANT: the characteristic polynomial x^2 - (4t+7)x + D(t) of alpha_t has discriminant EXACTLY -7 for every t; vertex t = -7/4 with D_min = 7/4 > 0 (the branch-preserving path has no real zero); curvature D'' = 8 = rank E8 and sqrt(-disc) = sqrt(7) -- the sheet coefficients (8,7) reappear as (curvature, fixed quadratic discriminant), the scalaron 7 as arithmetic discriminant [C readings]. [E, audit] -7 = 1 mod 8 => 2 SPLITS in Q(sqrt-7) (class number 1, Heegner field) -- the J endpoint norm 2 = |Z2| is realised by the split prime above 2. [E] NEGATIVE CONTROLS: the winding axis det M(s,0) = 6s+8 is LINEAR (no norm form) and the anchor block det B(1,t) = 3t^2+21t+28 has POSITIVE discriminant 105 (real roots) -- only the sheet determinant is a complex norm line. [C]/[O] THE ARITHMETIC F_TRANSFER FRAME (the new solution perspective): the discrete data of the transfer path live in ONE quadratic order of discriminant -7; the preregistered next test is whether the four external transfer solvers (Koide, eta_B, axion, m_p/m_e) act on alpha_t by a COMMON fractional-linear action ('one functor, four interfaces' as an algebraic contract), with the kill criterion 'four incompatible arithmetic actions'; GATE.FTRANSFER unchanged, the external solver stays external. Discovery provenance: center_quotient_coxeter_modular_disc7_probe.py (76/76) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves. [2026-08-01 XXIV, cross-reference audit: the preregistered 'common fractional-linear action' test is EXECUTED on its discrete half -- v571: the four discrete kernels lie on the alpha_t translation orbit and the cross-ratio 4/3 is Moebius-forced (y_F = 53 fit-free); v575: the continuous export is uniflow-degenerate (closure onto the single v425 flow); v578: each external instance is projective IN ITS OWN CLOCK and the joint obstruction is exactly the clock anchor -- so the residual of this preregistration is the same external-anchor-jet export that CONTRACT.F.01 fences; no independent open remains here.] [2026-08-02 XLI: the preregistered PGL2-action test EXECUTED in its literal group form by v632 -- ONE common parabolic action on the discrete half (exact intertwining with the base translation; CR = 4/3; the F corner 53 Moebius-forced), uniflow degeneration on the native continuous half (g_Boltzmann = g_pole identically, g_QCD exactly conjugate to the base translation, relic trivial); the kill criterion 'four incompatible arithmetic actions' is NOT met; see FTR.PGL2.01.]",
      "status": "Identity [E] (exact norm/integrality/translation/discriminant arithmetic + negative controls -- sympy exact, Wolfram-mirrored) + [E, audit] (split prime 2, Heegner class number) + [C]/[O] (the arithmetic F_transfer frame: common Moebius action of the four solvers = preregistered next test; GATE.FTRANSFER unchanged)",
      "canonicalStatus": "[E] the sheet path J->K->C->F is one translation orbit alpha_{t+1} = alpha_t + 2 in Z[(1+sqrt(-7))/2] with norms (2,4,14,32) and constant discriminant -7; [C]/[O] common-solver Moebius action = preregistered next test; no marker moves",
      "dependencies": "FTR.PATH.01; DIAMOND.SHEET.SOURCE.J.01 (v412); DIAMOND.SHEET.CALCULUS.01 (v413); v224",
      "scripts": [
        "v533_ftransfer_disc7_norm.py"
      ],
      "script": "v533_ftransfer_disc7_norm.py",
      "location": "tfpt_2;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 601
    },
    {
      "id": "CP.CHANNEL.IDENT.01",
      "claim": "THE CP CHANNEL / INVARIANT IDENTITIES (2026-07-29; one module, 25 checks, ~1 s; promotion of the exact cores of discovery T177 (contract CP.INVARIANT, probe cp_invariant_probe.py, 49/49, verdict DEGENERATE) -- THE CP MIRROR AS ALGEBRA: the prime-front toolkit (gauge-degree bookkeeping under declared group actions) turned on the four CP ledger rows REDTEAM.D.01 / E8.CHAN.01 / CP.MOD.01 / DUAL.FRAME.01; every statement is a NARROWING of an existing reading, executed in the same change, with NO marker moves and NO CP derivation claimed. CONTENT: (1) THE HEXAGONAL CHANNEL IDENTITY -- on the CM unit rho = e^{i pi/3} (CP.MOD.01) the E8 split 248 = 120 + 128 (E8.CHAN.01/v227) has an EXACT fiber realisation: sum_k rho^{d_k} = 4 rho (arg = pi/3 exactly; the degrees mod 6 are four 0s and four 2s, 1 + rho^2 = rho) and sum_k rho^{m_k} = 4 (arg = 0 exactly), with the parity typing forced by integer parity -- all degrees EVEN (phase channel SHEET-EVEN under the sheet flip rho -> rho^4 = -rho), all exponents ODD (magnitude channel SHEET-ODD); CONSEQUENCE: the sheet-even phase-channel angle cannot distinguish quark from lepton (it carries pi/3, not the CP.MU6.01 sheet split); mutation controls: the identity breaks on the non-CM unit e^{i pi/5} and on a mutated degree list (30 -> 31), reality breaks on a mutated exponent list (29 -> 30); HONEST CAVEAT wired as a check: reality of the exponent sum alone is NOT hexagon-specific (the exponent residues are conjugation-closed mod 10 too) -- only the degree half (= 4 rho) is; (2) ZERO PHASE IN THE MAGNITUDE DATA + THE BRANCH-ORDER LAW (the REDTEAM.D.01 narrowing): arg(prod c) = 0 EXACTLY on both TFPT sectors (lepton triple (16/7, 4/3, 7/6), product 32/9; up-quark triple (1/2, 34/41, 4/13)) -- the CP residual is not hidden in the product -- and the mu3 branch group is EXACTLY the kernel of the magnitude data because its order divides N_fam = 3 (w^{3j} = 1; a mu4 branch already shifts the product phase by -pi/2, mutation): the residual is a FIBER datum, not a magnitude datum; (3) TIER-1 EMPTINESS BY THE BLINDNESS THEOREM: both complex kernels commute with conjugation (conj sum rho^{d_k} = 4 conj rho exactly; Im det(1, d, conj(rho) n) = -Im det(1, d, rho n) exactly), so EVERY pi/3-carrier is kappa-odd and invariant + odd forces Im F = 0: strict invariance under all four T177 groups is EMPTY on pi/3-carriers, per algebra and as preregistered; (4) THE FRAME REMOVAL (the DUAL.FRAME.01 narrowing): det(1, d, n) = 21 = N_fam * scalaron and Im det(1, d, rho n) = 21 sin(pi/3) exactly (v225 rebuilt); the orientation functional is homogeneous of DEGREE +1 in the frame normalisation (Im det(1, d, rho(t n)) = t 21 sin(pi/3) symbolically), so the NUMBER 21 sin(pi/3) is FRAME data; dividing by the oriented volume removes the frame IDENTICALLY (Im(rho v)/v = Im rho symbolically for every real v != 0): the invariant core is the fiber datum Im(rho) = sin(pi/3), the sign a declared orientation; sheet control Im det(1, d, rho^4 n) = -21 sin(pi/3); mutation: n' = (5, -8, 6) gives det = 39/2 != 21; (5) THE TWO TIER-2 CHARACTERS + THE JARLSKOG ANCHOR: P2 = arg(sum rho^{d_k}) carries (chi(kappa), chi(sigma)) = (-1, +1) and V5 = Im(rho det)/det carries (-1, -1), both verified as character HOMOMORPHISMS on the full four-element group (16 products) -- TWO INEQUIVALENT invariants carry the same value pi/3 (the DEGENERATE verdict as algebra) and NEITHER couples to the frame; the classical anchor: the TFPT CKM rebuild (s12 = lambda_C, s23 = phi0/(1+lambda_C), s13 = lambda_C^3/3, frozen delta = pi/3 + 3 lambda_C^2 per REG.FREEZE.01) is unitary to 1.1e-41 and gives J = 3.327021e-5 = the v88 lattice value; J passes rephasing invariance (incl. the mu3 branch group) and CP-oddness and fails ONLY the pi/3-carrier test (its phase content is delta, not pi/3) -- the benchmark VALIDATES the criterion instead of the value, and the full delta has NO character under the parity involution (a CP-even correction on a CP-odd angle: the CP.MOD.01 scope narrowing). NAMED LIMITS AS LOAD-BEARING CONTENT: (i) NARROWINGS ONLY -- no marker of any pre-existing contract moves, the CP residual is NOT closed, the word 'derived' appears nowhere, REG.FREEZE.01 and the 3-vs-2 lambda_C^2 coefficient question are untouched; (ii) R1 STAYS OPEN: whether a frame-coupled, scale-free, character-covariant pi/3-carrier can exist at all (the T177 obstruction: the (1, d, n) frame has only ONE oriented volume to normalise against); R2 STAYS OPEN: which of the two Tier-2 characters is physical -- the sheet-odd (orientation) reading reproduces CP.MU6.01's quark/lepton split, the sheet-even (channel) reading cannot: a structural preference, NOT a derivation; (iii) the D4 deck column of T177 is a DECLARED MODEL and is not promoted (only the model-free statements are). HONEST FENCES: no RH statement anywhere (finite exact statements about roots of unity, 3x3 determinants and one 3x3 unitary matrix); Jarlskog 1985, Wolfenstein 1983, Weyl 1946 named CLASSICAL as addresses. Discovery: T177 (experiments/tfpt-discovery/cp_invariant_probe.py, 49/49, 0.2 s, verdict DEGENERATE). UPDATE (2026-07-30, T178/v564, CP.FIBER.PIN.01): R1 is DECIDED -- the natural (metric-free) class is fiber-only BY THEOREM (one bracket generator), the metric class couples but is NON-CANONICAL (a continuum of normaliser-relative values) -- and R2 is SELECTED conditional on the classical anchor (the Jarlskog character tuple (-1,-1,-1) matches the sheet-odd orientation class only); the single-sine lemma types this row's DEGENERATE verdict as FORCED on crystallographic fibers. Markers unchanged.",
      "status": "Identity [E] (exact identities and certified numerics with mutation controls -- the hexagonal channel identity sum rho^{d_k} = 4 rho / sum rho^{m_k} = 4 sympy-exact with the sheet-even/odd parity typing, arg(prod c) = 0 exactly on both magnitude sectors with the branch-order law, Tier-1 emptiness by the conjugation-commutation theorem, the frame removal Im(rho det)/det = Im rho exact with the degree +1 account, the two inequivalent Tier-2 characters as verified homomorphisms, and the Jarlskog anchor J = 3.327021e-5 = v88 at unitarity 1.1e-41) -- NARROWINGS of four existing ledger readings only; no marker moves; R1/R2 typed OPEN",
      "canonicalStatus": "[E] sum_k rho^{d_k} = 4 rho (arg = pi/3 exactly, sheet-EVEN) and sum_k rho^{m_k} = 4 (arg = 0 exactly, sheet-ODD) on the hexagonal CM unit -- the E8 channel typing has an exact fiber realisation and the phase channel is sheet-blind; arg(prod c) = 0 exactly on both TFPT sectors (the CP residual is a FIBER datum, not a magnitude datum); Tier-1 invariance EMPTY per algebra (every pi/3-carrier is kappa-odd); Im(rho det)/det = Im rho = sin(pi/3) exactly (the orientation reading is degree +1 in the frame normalisation, 21 sin(pi/3) is frame data, the sign a declared orientation); two inequivalent Tier-2 characters (-1,+1) vs (-1,-1) carry the same pi/3 (DEGENERATE), neither frame-coupled; Jarlskog anchor J = 3.327021e-5 = v88 passes (a)-(c), fails only the pi/3-carrier test. Four ledger rows narrowed, no marker moves; R1/R2 OPEN.",
      "dependencies": "REDTEAM.D.01; E8.CHAN.01; CP.MOD.01; DUAL.FRAME.01; CP.MU6.01; FLAV.CP.01; REG.FREEZE.01; v88; v220; v225; v227; T177",
      "scripts": [
        "v561_cp_channel_identities.py"
      ],
      "script": "v561_cp_channel_identities.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 602
    },
    {
      "id": "CP.FIBER.PIN.01",
      "claim": "THE FRAME DOOR OF THE CP MIRROR, DECIDED (2026-07-30; T178, contract FRAME.CARRIER, probe cp_frame_carrier_probe.py 31/31, verdict FIBER-PINNED; one module, 22 checks, ~2 s, sympy-exact generic-frame algebra + one 3x3 unitary matrix at 40 dps). T177/v561 left R1 (can a frame-coupled, scale-free, character-covariant pi/3-carrier exist at all) and R2 (which of the two degenerate Tier-2 characters is physical) open; both are decided here, and NOTHING about CP violation is derived. (1) EULER FACTORISATION: the fiber acts on the frame by a scalar on the n-row, so P(rho n) = rho^m P(n) for every homogeneous block (fully generic degree 1..4 polynomials, symbolic coefficients): every scale-free functional decomposes as sum_k A_k Im(rho^k) with A_k real degree-0 frame ratios -- the fiber content is QUANTISED. (2) THE NATURAL CLASS IS FIBER-ONLY (R1, natural half, NEGATIVE BY THEOREM): metric-free invariants of vector tuples are generated by volume-form brackets (Weyl 1946 FFT, the one cited class boundary [C]); the exact enumeration on {1, d, n} gives ONE generator (the six permutations = +-Delta; the other 21 brackets vanish identically in a GENERIC n), so every scale-free ratio is constant: no frame-coupled scale-free pi/3-carrier exists without a metric -- T177's obstruction sentence ('one determinant only') is now a theorem. (3) THE METRIC DOOR IS NON-CANONICAL (R1, metric half): declared metrics admit frame-coupled Tier-2 carriers, non-unique at BOTH levels -- characters: Delta/<d,n> and Delta/<1,n> carry (kappa, sigma, deck-Ref) = (-1,-1,+1) (the odd Delta-power absorbs the reflection) while Delta^2/<n,n> carries (-1,-1,-1); values: the admissible normalisers contain the continuum q_t = t n1^2 + n2^2 + n3^2 (deck-invariant for EVERY t) with carrier values 441 sin(pi/3)/(25t+117), strictly monotone in t -- NO canonical frame factor exists; anti-numerology wired as a check (21/8, 21/2, 441/142 pairwise distinct, none claimed -- normaliser-relative frame data). (4) THE SINGLE-SINE LEMMA (the pattern): {|Im(rho_n^k)|} has exactly ONE nonzero element iff n is crystallographic (n in {1,2,3,4,6} = {n : phi(n) <= 2}; Barlow 1894), by the exact identity sin(4pi/n) - sin(2pi/n) = 2 sin(pi/n) cos(3pi/n) with its only zero at n = 6 on n >= 5, enumerated exactly to n = 24; the compiler's two clocks mu4 (seam deck, j = 1728) and mu6 (CP fiber, j = 0) are the two MAXIMAL single-sine fibers (the two CM points with extra automorphisms): value-based pinning of functionals is impossible BY THEOREM on both -- T177's DEGENERATE verdict was FORCED, not found (mutation: on mu5 values DO separate carriers). (5) THE ANCHOR SELECTION (R2, SELECTED): the Jarlskog invariant on the TFPT CKM rebuild (v88, frozen delta per REG.FREEZE.01) has full character tuple (-1,-1,-1) under (kappa = conjugation, sigma = delta -> delta+pi, deck reflection V -> V Ref), each certified at 40 dps below 1e-25: it matches the sheet-ODD orientation class V5 = (-1,-1,-1) and NOT the sheet-even channel class P2 = (-1,+1,-1) -- of T177's two degenerate classes exactly ONE matches; the (-1,-1,+1) metric carriers are ANCHOR-REJECTED and the matching (-1,-1,-1) family carries the continuum, so the UNIQUE anchor-matching CANONICAL carrier is the fiber-pure V5 = Im(rho) = sin(pi/3); consistency: Im(rho^4) = -Im(rho) flips exactly as sin(delta+pi) = -sin(delta) (the CP.MU6.01 wiring). NAMED LIMITS AS CONTENT: the selection is CONDITIONAL ON THE ANCHOR (Jarlskog 1985 is imported physics -- SELECTED, not derived; the word 'derived' appears nowhere); the deck-reflection column is the T177 DECLARED MODEL (kappa/sigma model-free); the CP residual is NOT closed, it is RELOCATED with a proof: the invariant CP core is the fiber datum Im(rho) = sin(pi/3) and provably cannot be made frame data without a non-canonical metric import; REG.FREEZE.01 and the 3-vs-2 lambda_C^2 question untouched; NO marker of any pre-existing contract moves; NO RH statement (finite exact statements about roots of unity, 3x3 determinants, one 3x3 unitary matrix and one trigonometric identity).",
      "status": "Identity [E] (exact theorems and certified character accounts with mutation controls -- Euler factorisation on fully generic homogeneous blocks, the one-generator bracket enumeration, the character split and the q_t continuum, the single-sine lemma with the crystallographic equivalence, the anchor tuple at 40 dps) + [C] (the Weyl 1946 FFT class boundary; the selection conditional on the Jarlskog anchor; the deck column a declared model); R1 decided, R2 selected, no marker moves",
      "canonicalStatus": "[E] natural class fiber-only (6 brackets = +-Delta, 21 vanish identically; generic degree 1..4 Euler factorisation exact); metric door non-canonical (character split (-1,-1,+1) vs (-1,-1,-1); continuum 441 sin(pi/3)/(25t+117)); single-sine <=> n in {1,2,3,4,6} = phi(n) <= 2 (identity + enumeration to 24); anchor tuple (-1,-1,-1) selects sheet-odd, J = 3.327021e-5 = v88; unique canonical anchor-matching carrier = fiber-pure sin(pi/3); 22/22 checks",
      "dependencies": "CP.CHANNEL.IDENT.01; REDTEAM.D.01; E8.CHAN.01; CP.MOD.01; DUAL.FRAME.01; CP.MU6.01; REG.FREEZE.01; v88; v225; v561; T177; T178",
      "scripts": [
        "v564_cp_fiber_pin.py"
      ],
      "script": "v564_cp_fiber_pin.py",
      "location": "tfpt_5_redteam",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 603
    },
    {
      "id": "WOIT.BETA3.DUALITY.01",
      "claim": "THE WOIT-BETA3 MILESTONE EXECUTED (2026-07-30; one module, 22 checks, ~2 s; probe woit_os_beta3_pt_dual_probe.py 25/25, verdict SUCCESS per the frozen preregistration): 'the PT <-> PT* duality typed against sigma_std' -- contract wording: success = Woit's Minkowski conjugation identified with the family-D conjugation on the quotient; kill = the duality forces a clock-centralising (family A) structure. (1) THE COMPATIBLE CLASS (block theorem): a duality is a nondegenerate Hermitian form Phi on C^4 (sigma(Z) = Z^dagger Phi); clock compatibility (rho4 = diag(i,1,i,1), the v519 W-S4 blockwise lift) has entrywise residual (conj(lam_j) lam_k - 1) Phi_jk and forces the four cross-eigenspace entries to ZERO: two 2x2 Hermitian blocks on {e1,e3} (+) {e2,e4}, 16 -> 8 real parameters; deck compatibility adds nothing; the wrong clock lift diag(i,i,1,1) forces a different split (the compiler clock is load-bearing). (2) THE DICHOTOMY + THE FORCED (2,2): the rho_tw pairing M_tw^dagger Phi M_tw = eps conj(Phi) (rho_tw = Woit's Euclidean structure, v519 W-S4.1) splits the class exactly in two -- EUCLIDEAN (eps = +1; sigma_std = Phi = 1 lives here): seam-line norm a(|z|^2 + 1), NO null point, no OS cut ever; MINKOWSKI (eps = -1): seam-line norm a(|z|^2 - 1) -- the null locus meets the seam line EXACTLY in the seam circle |z| = 1 for EVERY member (the OS cut is FORCED, not imposed) -- and Phi_1 = -conj(Phi_i) exactly (charpoly_1(x) = charpoly_i(-x)): the TOTAL SIGNATURE IS (2,2) for every nondegenerate member -- Woit's Minkowski signature is a THEOREM of clock + rho_tw pairing, not an input. (3) THE KILL IS EMPTY BY ALGEBRA: s_Phi = Phi^T o conj inverts clock AND deck identically in the block parameters (family D, both branches); family A would need rho4^-1 = lambda rho4, i.e. rho4^2 = scalar, but rho4^2 = diag(-1,1,-1,1): NO solution -- decided, not scanned; s_Phi^2 = diag(a^2,a^2,c^2,c^2) > 0: the Kramers-free class. (4) THE INDUCED MEMBER IS UNIQUE AND PERPENDICULAR TO sigma_std: seam preservation <=> b = 0; every seam-preserving Minkowski member induces the SAME projective conjugation z -> -conj(z) (mu = -1, fixed points +-i), perpendicular to sigma_std's z -> +conj(z) (mu = +1, fixed points +-1); the clock conjugation exchanges exactly these two (mu -> -mu) -- the projective shadow of theta_cut o theta_perp = alpha_{N/2}. (5) THE QUOTIENT IDENTIFICATION: exact placement arithmetic (cut axis k = 7 fixes bond midpoints {7pi/8, 15pi/8}, perp axis k = 3 fixes {3pi/8, 11pi/8}; offsets exactly -+pi/2 -- under the v519 precision-(ii) RP-side placement the perp axis carries the +-i mark pair) + coordinate transport (axis angle 11pi/8 = 3pi/8 mod pi): THE INDUCED MINKOWSKI MEMBER IS theta_perp, the mirror whose descent v524 certified (Theta_phys) -- sigma_std's member transports to the CUT axis (the OS metric itself): role separation complete; v524 regressions re-verified at N = 8 (Gram PD (16,0,0) min 3.3471e-3 at 40 digits, theta_perp anti-unitary wrt the Gram exactly with eta_perp = +i, Theta_phys^2 = +1 on BOTH parity sectors, time reversal exact, torsor closure exact at support level). (6) CONTROLS: family A has no quotient ((4,4,0) one-particle, v524 C4.2 verbatim); the Euclidean branch cannot cut; a non-mu4 reflection (e^{2 pi i/5}) is not mark-anchored. NAMED LIMITS AS LOAD-BEARING CONTENT: free/quotient level ONLY -- nothing constructs the interacting algebra A_hol, all seven contract kill tests stay formally live there, WOIT.OS.TWISTOR.01 stays [O], gamma (the chirality theorem + the mark incidence) is the next milestone; the declared [C] conventions (seam line span(e1,e2) with the v519 blockwise lift; 'the duality respects the seam' read as seam-line preservation; the RP-side bond-midpoint placement, v519 precision (ii); the clock relabeling of the torsor) are cited from v492/v519/v524, not invented; 'identified' means inside these declared compatibility classes, NOT a derivation of Minkowski space; the word 'derived' appears nowhere for any physics statement.",
      "status": "Identity [E] (exact sympy algebra on generic block parameters -- block theorem, branch dichotomy, signature-mirror theorem, kill emptiness, unique induced member, exact rational-pi placement arithmetic, exact lattice regressions) + two 40-digit inertia certificates + [C] (the declared conventions: seam line, seam preservation, RP-side placement, torsor relabeling; contract-level identification, not a derivation); verdict SUCCESS, kill branch empty by algebra, no marker moves",
      "canonicalStatus": "[E] block theorem (4 cross entries forced zero; 16 -> 8 params); dichotomy eps = +-1 with the OS cut forced on the Minkowski branch (seam norm a(|z|^2 - 1)) and signature (2,2) by the mirror theorem; kill empty (rho4^2 = diag(-1,1,-1,1) not scalar); induced member unique z -> -conj(z) = theta_perp = the v524 Theta_phys carrier (placement offsets exactly -+pi/2; axis 11pi/8 = 3pi/8 mod pi); N = 8 regressions: Gram PD (16,0,0) min 3.3471e-3, eta_perp = +i, Theta^2 = +1 both sectors, time reversal + torsor closure exact; family A (4,4,0); 22/22 checks",
      "dependencies": "WOIT.OS.TWISTOR.01; WOIT.THETA.FREE.01; WOIT.BETA1.GSO.01; WOIT.BETA2.OS.01; SEAM.BIT.RPBLIND.01; v492; v519; v522; v524",
      "scripts": [
        "v565_woit_beta3_pt_duality.py"
      ],
      "script": "v565_woit_beta3_pt_duality.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 604
    },
    {
      "id": "DIAMOND.PARA.SELFCODE.01",
      "claim": "THE PARABOLIC ANCHOR SELF-CODE (2026-07-31; external structural review audited by discovery probe parabolic_selfcode_audit_probe.py 23/23, verdict SELFCODE-VERIFIED, and promoted; one module, 14 checks, ~1 s, exact integer/symbolic sympy, no floats). The v218 sheet-diamond direction operators U = Q diag(1,0,0) (rank-1 winding) and V = Q diag(0,1,1) (rank-2 sheet, Spec V = {0,1,2}) carry a previously unstated exact operator structure. (1) PARABOLIC THEOREM: A = Q<I,U,V> is the FULL line-stabiliser (maximal parabolic) algebra of block type 2|1 -- seven words independent (coordinate det -81), multiplicatively closed (49/49), all seven matrix units contained, dim 7 = 9-2; mutation of Q's third column breaks it. (2) DOUBLE SELF-CODE: Pi2 = V(V-I)/2 idempotent trace-1; H = I + Pi2 has spectrum EXACTLY {1,1,2} = a (ANCHOR.GEN.01/v23); chi_H = x^3 - 4x^2 + 5x - 2 with (e1,e2,e3) = (4,5,2) = (|mu4|, g_car, |Z2|) AND = (dim_R(GL3/P), dim(A/J), dim J) -- the anchor reproduced spectrally (dominant projector of V) and categorically (flag/Levi/radical of A) from ONE structure. (3) UNIQUENESS THEOREM: x^3 - 2m x^2 + (m^2+1) x - m = (x-m)(x^2-mx+1) identically; positive-integer spectra force m^2 - 4 = k^2, (m-k)(m+k) = 4, m = 2 UNIQUELY -- a = (1,1,2) is the only anchor this parabolic family can code. (4) THE 81 AS LATTICE INDEX: the SATURATED integral order (closure stable at word length 4) has elementary divisors (1,1,1,3,3,3,3): [P_Z : Z<U,V>] = 3^4 = 81, cokernel (Z/3)^4 -- equal to the v218 determinant-staircase sum 3+4+8+14+20+32 = N_fam^4 (cross-link typed; the mark-local mechanism, one Z/3 per mu4 mark, stays [O]: if no canonical local decomposition exists the 81 stays a fingerprint). (5) A2 BIT CARRIER: the basic algebra e A e is the A2 path algebra (dim Ext^1 = 1): dim-vector-(1,1) modules fall into EXACTLY two iso classes (split / one nonsplit) -- the alignment bit gains a canonical algebra carrier; uniqueness [E], SELECTION stays with RP/geometry (v528/v534) [C]. (6) TRACE CODE (re-encoding): p_n(a) = 2 + 2^n = 1 + tr(V^n); recurrence, Hankel 2^{n+1}; (1+trV)(1+trV^2)(1+trV^3) = 240 = |R(E8)|; tr(V^4-V^3) = 8 = rank E8 -- AnchorLadder read through V, one object fewer, NO new content. (7) CR TEST DESIGN (CONTRACT.F.01): CR(alpha_{-2},alpha_{-1};alpha_0,alpha_1) = 4/3 exactly on the v533 line; honesty wired (universal arithmetic-progression cross-ratio -- content = preregistered solver-state export); negative control 28/25; y_F forced fit-free. (8) LORENTZ FORM (reformulation): det X = t^2 - x^2 - y^2 under Phi; D(P,Q) = 2<Phi,Phi>_{1,2}; delta = sech^2(atanh r); pencil identities det(B-S) = det B det(I-Z), D(B,S) = det B tr Z, det S = det B det Z -- eigenvalue locking as the named prime-front instrument (no measurement, no RH statement). NAMED LIMITS AS CONTENT: NO P2 REDUCTION -- 'g_car = dim(A/J) derived' is conditional on the OPEN 'No Carrier Reconstruction' audit (rebuild Q/V from mu4, D4, Q_+ = A3 exponent grading, H^1, Q_- = sheet monodromy, WITHOUT g_car, a, rank E8 or carrier-aware budgets; Q's historical provenance runs through such budgets); AX.P2.01 and ANCHOR.GEN.01 markers UNTOUCHED; the 2..9 dimension table is COMPRESSION of one operator system, not nine independent hits (no-free-pattern rule); the bit selection, the mark-local 81 mechanism and the CR-test execution stay open and typed open; the word 'derived' appears nowhere for any physics statement. UPDATE (2026-07-31, v567): the review's Priority-1 audit is EXECUTED at Stage A -- verdict RECONSTRUCTS-TO-FAMILY: the anchor self-code is carrier-free; Q reconstructs up to one binary sheet-coupling choice; see QGEO.QRECON.01. FURTHER UPDATE (2026-07-31, v568): Priority 3 executed -- the A2 Ext arrow is NONZERO on the whole reconstruction class (the split member collapses to dim 5 and is excluded); Priority 2 sharpened -- the mu4-grading candidate fails, the F3[V]-action on the cokernel exists (see DIAMOND.BIT.SELECT.01). [2026-07-31: the S8 pencil identities are now executed on the REAL Paper-II surface by v569 (machine-tight at 2e-14 on 70 windows, verdict ONE-MODE); see PRIME.PENCIL.ONEMODE.01.] [2026-08-01 X: the per-mark Z/3 mechanism appears geometrically -- v597: the cusp-class group of the mu3-cover is (Z/3)^4 of order 81 with four free local classes; see QGEO.COVER.01.] [2026-08-01 XVI: the self-code algebra has a geometric realization -- v600: (U, V) as Gamma-even degree-2 cusp-Gram operators on the mu3-cover, GL3-conjugate to the compiler pair, with the SAME saturated Z-order structure (index 81 = 3^4, divisors (1,1,1,3,3,3,3)); existence [E], canonicity open; see QGEO.EMBED.01.]",
      "status": "Identity [E] (exact integer/symbolic algebra with mutation controls -- parabolic closure, double self-code, uniqueness enumeration + divisor argument, saturated Smith form, A2 quiver, trace code, cross-ratio identities, Lorentz/pencil identities) + [C] (the TFPT readings of the invariants; the bit-carrier and CR-test interpretations) + [O] (the No-Carrier-Reconstruction audit, the mark-local 81 mechanism, the CR-test execution); NO marker moves",
      "canonicalStatus": "[E] A = full 2|1 parabolic, dim 7, word det -81, closed 49/49, mutation-fragile; a = Spec(I + V(V-I)/2) = (1,1,2) with chi_H = x^3 - flag x^2 + Levi x - radical (4,5,2 double-coded); m = 2 unique (enumeration to 2000 + divisor argument); saturated Smith (1,1,1,3,3,3,3), index 81 = 3^4 = staircase sum; A2 Ext^1 = 1 (two module classes); p_n = 1 + tr V^n (240, rank 8 as traces); CR = 4/3 design with 28/25 control; Lorentz/pencil identities exact; 14/14 checks",
      "dependencies": "DIAMOND.AXIS.01; ANCHOR.GEN.01; AX.P2.01; FLAV.QGEO.01; CONTRACT.F.01; SEAM.BIT.RPBLIND.01; FORM.LADDER.01; v23; v218; v224; v528; v533; v534",
      "scripts": [
        "v566_parabolic_anchor_selfcode.py"
      ],
      "script": "v566_parabolic_anchor_selfcode.py",
      "location": "tfpt_1;tfpt_2;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 605
    },
    {
      "id": "QGEO.QRECON.01",
      "claim": "THE NO-CARRIER-RECONSTRUCTION AUDIT, STAGE A (2026-07-31; Priority 1 of the self-code review; one module, 9 checks, ~1 s, exact enumeration with integer prefilters + sympy verification; discovery probe carrier_free_q_reconstruction_probe.py 10/10, verdict RECONSTRUCTS-TO-FAMILY). THE QUESTION: can Q -- hence V and the v566 parabolic anchor self-code -- be reconstructed WITHOUT carrier/anchor inputs? THE CIRCULARITY LOCATED: v11's uniqueness certificate for Q enumerates over row sums (4,5,6), column sums (9,5,1) and chi_Q = (t-1)(t^2-5t+3) -- the 5 = g_car sits in the budgets and in chi_Q; all of it FORBIDDEN here. ALLOWED (each carrier-free, with corpus source): (S) Sigma = diag(1,-1,-1) [v10 sheet]; (W) Q(I+Sigma) = 2 N_fam W [v10 winding]; (E) Spec(Q_+) = {1,2,3} [FLAV.QGEO.01, A3 exponents]; (M) Q_- != 0, Q_-^2 = N_fam P, P^2 = P [FLAV.QGEO.01]; (Z) nonneg integers, bound 6 (robust at 9); (L) the seam-line demand [geometric, [C]]; (X) the saturation index N_fam^4 = 81 [v566 S4]. RESULTS [E]: (1) exactly 52 admissible matrices under (S)(W)(E)(M)(Z), corpus Q among them, first column forced to N_fam(1,1,1) by (W); (2) the seam-line demand cuts 52 -> 28 (real teeth); (3) THE ANCHOR IS CARRIER-FREE: all 28 parabolic members code the SAME anchor -- H' = I + Pi_dom(V') with char poly x^3 - 4x^2 + 5x - 2, spectrum (1,1,2) = a, ONE code class, no exceptions: the anchor self-code sits UPSTREAM of the carrier budgets, conditional on the declared geometric inputs; (4) THE RESIDUAL BIT: the 81-pin selects EXACTLY TWO members -- the corpus Q (q32 = 2 = |Z2|) and Q_alt (q32 = 0), difference matrix 2 E32 (the sheet-block coupling); every classical carrier-free certificate coincides on the pair (anticommutator spectrum {6,-4,-2} = (|R+(A3)|, -|mu4|, -|Z2|); commutator charpoly t(t^2+12); det = 3; saturation 81), while the ONE separating certificate is the carrier column budget (9,5,1) vs (9,3,1) -- exactly the forbidden input: the residual freedom is ONE binary coupled-vs-uncoupled sheet-block choice; (5) ablations: dropping (W) widens the bound-3 set 28 -> 1376 over 36 first columns (the winding input carries real load); bound robustness 6 -> 9 adds zero new (L)+(X) members; the forbidden inputs were evaluated nowhere. NAMED LIMITS AS LOAD-BEARING CONTENT: STAGE A ONLY -- whether (S)(W)(E)(M)(L) themselves derive from mu4/D4/H^1 parabolic geometry without family/sheet inputs is EXACTLY the open gate GATE.QGEO.01: the audit is RELOCATED there, not closed (N_fam = 3 is used throughout; family data was never forbidden -- the forbidden list is the CARRIER list); the carrier-free selector of the residual bit is OPEN, and its structural resonance with the split/nonsplit alignment-bit dichotomy (v528/v534, v566 S5) is a TYPED OBSERVATION, not a claim; NO P2 reduction -- AX.P2.01 and ANCHOR.GEN.01 markers untouched; the word 'derived' appears nowhere for any physics statement. UPDATE (2026-07-31, v568, DIAMOND.BIT.SELECT.01): the residual bit is located and given a canonical witness -- the twins generate different integral orders, and the ladder self-consistency (dominant mode = its own eigenvalue ladder (1,2,4)) selects the compiler Q uniquely on all 28 members; the split geometry (q32 = 1) is excluded by the demands already imposed; the physical selection stays open. Markers unchanged. [2026-07-31: Stage B narrowed by v574 -- the involution type is forced by a rank theorem and N_fam = 3 by the double self-consistency (ladder + sat = N^4); (E)(M)(L) stay the remaining gate; see QGEO.WINDING.SELF.01.]",
      "status": "Identity/enumeration [E] (exact admissible-set census 52/28/2 with integer prefilters verified by sympy on survivors; one anchor code class on all 28; separator dissection exact; ablations quantified) + [C] (the seam-line demand as declared geometric input; the bit-resonance observation) + [O] (Stage B = GATE.QGEO.01 relocation; the binary selector); verdict RECONSTRUCTS-TO-FAMILY; no marker moves",
      "canonicalStatus": "[E] 52 admissible under (S)(W)(E)(M)(Z) at B = 6; 28 under the seam-line demand; ONE anchor code x^3-4x^2+5x-2 = spectrum (1,1,2) on all 28 (carrier-free anchor); 81-pin -> exactly 2 (corpus Q vs q32 = 0 twin; all carrier-free certificates equal, separator = the forbidden column budget (9,5,1)); winding ablation 28 -> 1376 at bound 3; bound-stable at 9; 9/9 checks; verdict RECONSTRUCTS-TO-FAMILY",
      "dependencies": "DIAMOND.PARA.SELFCODE.01; FLAV.QGEO.01; GATE.QGEO.01; FLAV.KQ.01; AX.P2.01; ANCHOR.GEN.01; v10; v11; v218; v566",
      "scripts": [
        "v567_carrier_free_q_reconstruction.py"
      ],
      "script": "v567_carrier_free_q_reconstruction.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 606
    },
    {
      "id": "DIAMOND.BIT.SELECT.01",
      "claim": "THE RESIDUAL RECONSTRUCTION BIT: LOCATED, SEPARATED, AND GIVEN A CANONICAL EXACT WITNESS (2026-07-31; Priorities 2+3 of the self-code review fused with the v567 residual bit; one module, 11 checks, ~1 s; discovery probe bit_selector_probe.py 10/10, verdict SELECTOR-FOUND). (1) THE BIT IS INTEGRALLY VISIBLE [E]: the two 81-twins of the v567 audit (q32 = 2 = |Z2| vs q32 = 0) generate DIFFERENT integral orders -- Hermite bases differ at EQUAL index 81 and equal Smith divisors (1,1,1,3,3,3,3): real integral structure where every classical carrier-free invariant coincides. (2) THE SELECTOR [E]: the dominant mode of the sheet direction satisfies V_c (1,2,2c)^T = 2 (1,2,2c)^T identically in the coupling c; the LADDER SELF-CONSISTENCY -- the dominant mode reproduces its own eigenvalue as the geometric ladder (2^0,2^1,2^2) = (1,2,4) -- forces c = 2: on ALL 28 members of the v567 parabolic carrier-free class EXACTLY ONE satisfies it, the compiler Q, with no 81-pin and no column budget. (3) THE WITNESS IS PRIOR STRUCTURE [C] (no-free-pattern): (1,2,4) is the binary AnchorLadder step sequence p_{n+1} - p_n = 2^n (Lean-formalised), 2 = |Z2| the sheet datum, and the v566 trace code p_n(a) = 1 + tr(V^n) = 2 + 2^n rides the same ladder (n = 1..6 re-verified exactly). (4) THE SPLIT GEOMETRY LOCATED AND EXCLUDED [E]: an invariant module complement of the seam line exists IFF q32 = 1 (exact solve: a = 0, b = 1, q32 = 1; the plane span((1,0,0),(0,1,1)) exactly invariant there), and that member's direction algebra COLLAPSES to dimension 5 (Smith (1,1,1,3,3), no full parabolic, no 81): the seam-line + saturation demands admit ONLY nonsplit geometry -- the v566 A2 Ext arrow is NONZERO on the whole reconstruction class; neither twin splits (the complement system is inconsistent for q32 in {0,2}): the bit does NOT toggle split/nonsplit -- it toggles WHICH nonsplit integral order is realised and whether the dominant mode carries the full binary ladder. (5) PRIORITY-2 CANDIDATE TESTS [E], honest both ways: the mu4-grading candidate deg(E_jk) = g_k - g_j mod 4 FAILS (the order is not a graded sublattice: 5 of 7 Hermite vectors homogeneous) -- the review's kill condition does NOT fire and the 81 stays a FINGERPRINT pending a mechanism; left multiplication by V PRESERVES the order lattice and induces a canonical F3-linear action on the cokernel (Z/3)^4 -- the F3[V]-character computation is EXECUTED: the induced action has characteristic polynomial x(x-1)^2(x+1) mod 3 -- eigenvalues {0,1,1,-1}, NOT the regular mu4 pattern x^4 - 1 that 'one Z/3 per mark' would require (not even invertible), U acts as ZERO, and both twins give the identical structure: after two candidates (grading, V-action) the 81 stays a FINGERPRINT; the kill condition (no canonical local decomposition at all) is closer but not proven. NAMED LIMITS AS LOAD-BEARING CONTENT: the PHYSICAL selection is NOT derived -- why the dominant mode should satisfy the ladder self-consistency is the relocated open question, the same structural slot as the RP-selection of the alignment bit (v528/v534); the resonance between the reconstruction bit and the alignment bit is a TYPED OBSERVATION, not an identification; Stage B of v567 (GATE.QGEO.01) untouched; NO P2 reduction; AX.P2.01 and ANCHOR.GEN.01 markers untouched; the word 'derived' appears nowhere for any physics statement. [2026-07-31: the RP form is spectrally blind but integrally ANCHOR-VISIBLE on the twins (v572, Smith slot 18 vs 2 = 9 = N_fam^2); the dynamical RP-slot stays open; see DIAMOND.BIT.RPFORM.01.] [2026-07-31 II: the review's Ext-sensitive marked resolvent chi_Ext(z) = tr(P_seam R N_Ext R) vanishes IDENTICALLY for both twins with the natural markings (P_seam = E33, N_Ext = the radical lattice generators -- which are IDENTICAL between the twins: span{3E31, 3E32}); the review's own kill criteria fire -- the Gibbs pencil (v572) remains the one working marked object; sandbox record, no module.]",
      "status": "Identity [E] (exact integer/symbolic algebra -- Hermite/Smith lattice comparison, the symbolic dominant-mode identity, the uniqueness census on the 28-member class, the split-complement solve, the candidate tests) + [C] (the anchored ladder selector as candidate canonical selection; its physical justification open) + [O] (the selection mechanism; the F3[V]-character question; Stage B); verdict SELECTOR-FOUND; no marker moves",
      "canonicalStatus": "[E] twins integrally distinct (H0 != H2 at equal index 81); v(c) = (1,2,2c) identically; ladder self-consistency <=> c = 2, unique on 28/28; witness = AnchorLadder 2^n / |Z2| / trace code; split member q32 = 1 (complement solve a=0,b=1) collapses to dim 5 -- excluded; mu4-grading candidate fails (5/7 homogeneous); F3[V]-action computed: charpoly x(x-1)^2(x+1) mod 3, eigenvalues {0,1,1,-1} != regular mu4, U-action zero, twins identical; 11/11 checks; verdict SELECTOR-FOUND",
      "dependencies": "QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; DIAMOND.AXIS.01; SEAM.BIT.RPBLIND.01; FORM.LADDER.01; v10; v218; v528; v534; v566; v567",
      "scripts": [
        "v568_bit_selector_ladder.py"
      ],
      "script": "v568_bit_selector_ladder.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 607
    },
    {
      "id": "FTR.CRDISC.01",
      "claim": "The preregistered cross-ratio check (self-code review Priority 4 / v566 S9), executed on its DISCRETE half (one module, 9 checks, <1 s; discovery probe cr_discrete_corners_probe.py 9/9, verdict CR-PASSES): the four solver corners are the in-repo Sheet-Diamond kernels J,K,C,F = M(1,t) = R + Q diag(1,t,t), t = -2..1 (FTR.PATH.01/v224, dets (2,4,14,32)); the corpus's one canonical declared solver reading, the Koide anchor response ell(M) = a^T M 1 (v183), reads (26, 35, 44, 53): CR = 4/3 EXACTLY, and the fit-free reconstruction forces y_F = 53 from (26, 35, 44) alone -- the Koide numerator is Moebius-determined by the three upstream corners (a NEW exact cross-corner identity); honesty theorem: ell is affine on the path (step 9 = N_fam^2) and EVERY linear functional passes -- the check separates linear from nonlinear exports; negative controls with teeth: the determinant path fails at 28/25 exactly as preregistered, the dominant-eigenvalue export fails at 1.322162.  THE HONEST BOUNDARY: the DISCRETE half only -- the external continuous solvers (pole transfer, Boltzmann, relic ODE, lattice) export no states here; the continuous half of the check stays open exactly as CONTRACT.F.01 fences it; contract markers unchanged. [2026-07-31: the CONTINUOUS half is since executed by v575 -- UNIFLOW-DEGENERATE, closing onto the v425 single-flow theorem; see FTR.CRCONT.01.]",
      "status": "Identity [E] + typed boundary [C] (exact symbolic identities on compiler kernels; the continuous-solver half open)",
      "canonicalStatus": "CR-PASSES (discrete half): declared export (26,35,44,53), CR = 4/3 exact, y_F = 53 Moebius-forced; controls 28/25 and 1.3222 fail as they must; continuous half open; 9/9 checks",
      "dependencies": "CONTRACT.F.01; FTR.PATH.01; DIAMOND.PARA.SELFCODE.01; v183; v224; v533; v566",
      "scripts": [
        "v571_cr_discrete_corners.py"
      ],
      "script": "v571_cr_discrete_corners.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 608
    },
    {
      "id": "DIAMOND.BIT.RPFORM.01",
      "claim": "The RP form on the 81-twins -- spectrally BLIND, integrally ANCHOR-VISIBLE (one module, 11 checks, ~1 s; discovery probe rp_bit_form_probe.py 11/11, verdict ANCHOR-VISIBLE): with the corpus reflection theta(X) = Sigma X^T Sigma (v10) and the declared states (trace; anchor (1,1,2); ladder (1,2,4); unit), the RP kernel K(x,y) = omega(theta(x) y) on the Hermite basis of each twin order has IDENTICAL inertia on both twins for every state (trace (4,0,3); vector states (3,1,3)) and is never PSD -- no kinematic v534-style positivity selection exists (honest negative, consistent with v567's certificate coincidence and v534's nonperturbative lesson); but the INTEGRAL Smith towers of the anchor-state Gram SEPARATE the twins -- q32 = 2: (2,2,18,18,18,18,0) vs q32 = 0: (2,2,2,18,18,18,0), one slot 18 vs 2 = a factor 9 = N_fam^2 (the compiler twin carries an EXTRA 3^2 of divisibility); the ladder state separates identically while trace/unit are blind: the bit is visible to the RP form ONLY through anchor-coupled states; trace discriminant -6561 = -81^2 on both twins; q32 = 1 collapses to a 5-dim order (v568 reproduced); and the blindness is ROBUST under the compiler's own flow -- the V-semigroup-deformed kernels K_n = omega(theta(x) V^n y), n = 1..3, keep identical inertia on both twins for every state (the deformation moves the inertia, never differentially): a dynamical selection needs genuine interaction, not a semigroup weighting.  THE GIBBS PENCIL SEPARATES (the first one-parameter dynamical distinction): along the anchor-Hamiltonian Gibbs family omega_g ~ tr(e^{-gH} .), H = I + Pi_dom (v566), det(K_0 + s K_Pi) factors (6561/4096)(s^2+16s+16)^2(15s^2-16s-16) on the compiler twin vs -(6561/4096)(s^2+16s+16)^3 on the alt (6561 = 81^2) -- the compiler's extra factor has its root s* = (8 - 4 sqrt 19)/15 INSIDE the physical Gibbs range: the compiler twin's kernel degenerates at a finite critical temperature, the alt twin's never does (a rank-drop marker, honest: not a positivity selection) -- and THE CRITICAL MODE IS IDENTIFIED: the null space at s* is one-dimensional, rank 1, supported entirely on the seam row (the radical / A2-Ext-carrier row of v566 S5), fixes the seam line (X e3 = e3), is a fixed point of the sheet flow (V X = X) and is annihilated by the dominant projector: the Gibbs criticality lives exactly in the Ext-carrier slot -- the typed resonance with the alignment bit deepens to a SHARED CARRIER; and the half-sided restrictions stay blind (Levi (3,2) / radical (1,1) inertia identical on both twins across the Gibbs range): the RP-positivity selection resists every kinematic and half-sided design tried -- what remains is the genuinely INTERACTING construction (the v534 FK-quartet analogue), typed open.  NAMED LIMITS: no physical selection claimed; the dynamical/interacting RP question stays the open RP-slot; the alignment-bit resonance stays typed; no marker moves. [2026-07-31 V: the interacting follow-up is executed -- v584: rank-one Ext-source coupling separates the twins directionally in the unit box (exact thresholds, pencil-factor numerators, positive-coupling healing of exactly the compiler twin); see DIAMOND.BIT.EXTSOURCE.01.]",
      "status": "Identity [E] (exact integral/spectral computations; honest negative on the kinematic selection; RP-slot open)",
      "canonicalStatus": "ANCHOR-VISIBLE: spectrally blind (same inertia, never PSD), integrally separated exactly through anchor/ladder states (Smith slot 18 vs 2, factor 9 = N_fam^2); trace discriminant -81^2; 11/11 checks (half-sided restrictions blind too; the interacting construction is the one remaining RP route) (V-flow robustness; Gibbs pencil: compiler Gibbs-critical at s* = (8-4sqrt19)/15, alt never; critical mode = seam-row flow-fixed direction, the Ext-carrier slot); dynamical RP-slot open",
      "dependencies": "DIAMOND.BIT.SELECT.01; QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; v10; v23; v534; v568",
      "scripts": [
        "v572_rp_bit_form.py"
      ],
      "script": "v572_rp_bit_form.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 609
    },
    {
      "id": "QGEO.WINDING.SELF.01",
      "claim": "Stage B of the No-Carrier-Reconstruction audit, first narrowing (one module, 11 checks, ~6 s; discovery probe stageb_winding_probe.py 12/12, verdict FORCED): the involution TYPE is killed by a rank theorem -- any involution with a two-dimensional positive eigenspace gives V' = Q P_- of rank <= 1 (char poly x^2 (x - t) identically in Q), so the binary trace code Spec V = {0,1,2} (v566) is unreachable: only the corpus type carries the self-code; (E) BOUNDS the winding normalisation (the winding demand forces the first column to (N,N,N), so q11 = N must itself be an A3 exponent -- N in {4,5,6} give EMPTY classes at the declared B = 6); the ladder self-consistency (v568) is ROBUST -- for every N in {1,2,3} it selects EXACTLY ONE parabolic member (28/16/14 -> 1/1/1); and the saturation self-consistency sat = N^4 (v567's (X) read as a self-consistency) holds ONLY at N = 3 (N=1: 32 != 1; N=2: 512 != 16; N=3: 81 = 3^4): the DOUBLE self-consistency pins (N, Q) = (3, compiler Q) uniquely across the whole N-ablated family -- N_fam = 3 relocates from declared input to consistency-forced within the reconstruction; equivalently the V-spectra {0,2,3}/{0,1,3}/{0,1,2}: only N = 3 carries the binary trace code p_n = 2 + 2^n.  THE HONEST DECOMPOSITION (spectrum ablation): freeing the exponent spectrum itself, the surviving double-selfconsistent family is EXACTLY spectrum = {1,2,N} with shape Q(N) = [[N,1,0],[N,2,0],[N,2,1]] (the full 35-spectrum scan in the probe finds no other) -- the self-consistency forces the SHEET spectrum {1,2} (binary trace code) and the compiler SHAPE, and the A3 exponent demand (E) contributes exactly ONE datum, N = 3.  REDUNDANCY -- (M) AND (L) FALL TOO: relative to the self-code demands (parabolic + ladder + sat = 81) the monodromy budget and the seam-line demand are REDUNDANT at the declared bound (dropping both widens the class to 1274 members and the self-code still picks exactly the compiler Q): the declared geometric input list shrinks to (S)-existence, (W)-form and (E) -- and THE MAXIMAL ABLATION drops (E) too: (W: column 3(1,1,1)) + integrality + the self-code package alone pin the compiler Q uniquely (29 ladder candidates, one hit): what stays declared is the involution EXISTENCE, the winding FORM and N_fam = 3 (independently anchored in the E8 glue, v1: 240 = 16 x 5 x 3); and the winding FORM's content is located exactly: dropping it entirely leaves a FOUR-element column family {(0,1,1),(0,3,3),(3,3,3),(5,1,5)} of which exactly ONE is uniform -- the compiler: the winding demand is precisely the UNIFORMITY of the column (1 of 4), a genuine declared input with sharply located content.  NAMED LIMITS: (E) stays DECLARED ((M)/(L) relocate to redundant-given-the-self-code at B = 6) geometric inputs -- deriving them from mu4/D4/H^1 geometry is exactly the remaining GATE.QGEO.01 (narrowed by two items, NOT closed); NO P2 reduction, AX.P2.01/ANCHOR.GEN.01 markers untouched. [2026-08-01: the involution-existence input is DERIVED -- v590: rigidity (Z/2)^3, narrowing to a unique explicit S0 via the v574 type theorem plus two named demands; GATE.QGEO.01 narrows further; see QGEO.INVOL.01.]",
      "status": "Identity [E] + typed gate [C] (rank theorem symbolic; enumerations at declared bound; the remaining gate named)",
      "canonicalStatus": "FORCED: type killed by rank theorem; N <= 3 by (E); double self-consistency (ladder + sat = N^4) pins (N, Q) = (3, compiler Q) uniquely; 11/11 checks; (E)/(M)/(L) ALL redundant, winding form = the uniformity choice (1 of 4) at B = 6 -- the reconstruction needs only involution existence, winding form with N_fam = 3 (E8-glue) and the self-code; gate narrowed by FIVE items; GATE.QGEO.01 narrowed by two items, not closed",
      "dependencies": "QGEO.QRECON.01; DIAMOND.BIT.SELECT.01; DIAMOND.PARA.SELFCODE.01; FLAV.QGEO.01; v567; v568",
      "scripts": [
        "v574_stageb_winding.py"
      ],
      "script": "v574_stageb_winding.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 610
    },
    {
      "id": "FTR.CRCONT.01",
      "claim": "The CONTINUOUS half of the preregistered cross-ratio check, executed against the in-repo solver dynamics (one module, 7 checks, <1 s; discovery probe cr_continuous_uniflow_probe.py 7/7, verdict UNIFLOW-DEGENERATE): the frozen protocol exports -- the fixed-point multipliers of the native contractions, recomputed from primitives (F_pole = e^{-Delta} = (2/3)^6 via the v99 generator; F_Boltzmann = the SAME contraction by the v425 single-flow theorem; F_QCD b3 = -7; F_relic adiabatic 1) -- form the quadruple {(2/3)^6, (2/3)^6, 1, -7} with a FORCED repetition, and a repeated quadruple gives CR in {0, 1, oo} in every corner assignment (theorem + all 24 permutations): the continuous half can never pass natively.  THE CLOSURE: it closes onto v425 -- four independent projective solver states do not exist natively; the only four independent external data are the dimensionful anchors (v_geo, M_R, C_p, theta_i), [O] walls, not canonical projective states; the meaningful CR content was and remains the discrete half (FTR.CRDISC.01, y_F = 53 forced).  A future continuous test would need anchor-level exports from genuinely external runs -- outside the native scope, stated honestly; contract markers unchanged. [2026-07-31 II: the jet-level successor is executed natively by v578 (MOEBIUS-IN-OWN-CLOCK: no shared wall-time Schwarzian, each instance projective in its own clock); see FTR.JETS.01.]",
      "status": "Identity [E] + typed closure [C] (exports recomputed; degeneracy theorem exact; the closure onto v425 typed)",
      "canonicalStatus": "UNIFLOW-DEGENERATE: exports {(2/3)^6,(2/3)^6,1,-7} forced degenerate by the single-flow theorem; CR in {0,1,oo} in every assignment; the continuous half closes onto v425; 7/7 checks; anchors stay [O]",
      "dependencies": "FTR.CRDISC.01; CONTRACT.F.01; DYN.TRANSFER.UNIVERSAL.01; v99; v159; v320; v425; v571",
      "scripts": [
        "v575_cr_continuous_uniflow.py"
      ],
      "script": "v575_cr_continuous_uniflow.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 611
    },
    {
      "id": "FTR.JETS.01",
      "claim": "The jet/Schwarzian contract, executed natively (one module, 6 checks, <1 s; discovery probe native_jets_probe.py 6/6, verdict MOEBIUS-IN-OWN-CLOCK; the second review's Priority 6): the exact v99 Koide trajectory solves the generator identically and has wall-time Schwarzian {q,t} = -Delta^2/2 = -18 log(3/2)^2 exactly (constant, trajectory-independent; F_Boltzmann carries the SAME value by the v425 single-flow theorem); in its own exponential clock u = e^{Delta t} the flow is Moebius (Schwarzian 0); the 1-loop QCD flow is Moebius in its OWN RG time already (Schwarzian 0; b0 = 7 = -b3); the relic instance is adiabatic-constant (jets degenerate).  The native wall-time quadruple {-Delta^2/2, -Delta^2/2, 0, degenerate} shares NO Schwarzian, and the discrepancy is exactly one exponential clock change: each instance is PROJECTIVE IN ITS OWN CLOCK -- the shared structure is Moebius-in-own-clock, the obstruction is the clock, and the clocks are anchored by the external data (v_geo, M_R, alpha_s(M_Z), theta_i).  A future joint jet test needs external anchor-level jets, exactly as v575 fenced; contract markers unchanged.",
      "status": "Identity [E] + typed closure [C] (exact trajectories and Schwarzians; the joint test external)",
      "canonicalStatus": "MOEBIUS-IN-OWN-CLOCK: seam {q,t} = -Delta^2/2 (own clock 0), QCD Schwarzian 0 in own time, relic degenerate; no shared wall-time jet invariant; clock = the obstruction; 6/6 checks",
      "dependencies": "FTR.CRCONT.01; FTR.CRDISC.01; DYN.TRANSFER.UNIVERSAL.01; v99; v159; v425; v575",
      "scripts": [
        "v578_native_jets.py"
      ],
      "script": "v578_native_jets.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 612
    },
    {
      "id": "DIAMOND.BIT.EXTSOURCE.01",
      "claim": "The interacting follow-up of v572, executed -- RANK-ONE EXT-SOURCE COUPLING ON THE GIBBS PENCIL SEPARATES THE 81-TWINS DIRECTIONALLY INSIDE THE UNIT COUPLING BOX (one module, 8 checks, ~1 s; discovery probe ext_source_response_probe.py 8/8, verdict UNIT-BOX-SEPARATED): the interacting kernel K(s,J) = K0 + s K_Pi + J v v^T (v from the shared Ext arrows E31/E32 of the radical row) has EXACT rational flip thresholds J_c(s) = -1/(v^T K(s)^{-1} v) whose numerators are exactly the v572 pencil factors (compiler: 15s^2-16s-16; alt: s^2+16s+16); numerically verified (16 flip events, one eigenvalue negative -> positive each).  THE SEPARATION: in the crossed phase s in (-1, s*] the compiler twin is healed by positive coupling J_c <= 15/16 < 1 through either arrow; the alt twin cannot be flipped by any positive coupling in the open unit box (E31 threshold >= 1 with equality exactly at s = 0, E32 threshold negative throughout): positive Ext coupling inside |J| < 1 selects exactly the compiler twin.  Exact anchors: J_c(-1) = 15/16; alt J_c(0) = 1 and -1; both compiler thresholds vanish exactly at s*.  The direction is strictly positive (negative coupling never heals) -- consistent with the v534 toy RP selection of positive seam coupling.  Observed structure: the compiler E31 response pole sits on the golden polynomial s^2 - s - 1 (s = -1/phi = -0.618, 0.011 above s* = -0.629); typed as observed, no claim built on it.  Mechanism: the v572 critical mode is Ext-visible (pairings 1/4 and 1/2 + sqrt(19)/4), hence J_c(s*) = 0 exactly.  HONEST TYPING: a RESPONSE-level selection (inertia restoration in a declared coupling box), NOT an OS/RP positivity theorem -- no kernel is ever PSD; the alignment-bit contract stays open.",
      "status": "Exact [E] + typed consistency [C] (thresholds closed rational; verified numerically; response level only)",
      "canonicalStatus": "UNIT-BOX-SEPARATED: positive Ext coupling inside |J| < 1 heals exactly the compiler twin (J_c <= 15/16, thresholds exact with v572 pencil-factor numerators); alt twin rigid; golden-polynomial response pole observed; response-level, contract stays open; 8/8 checks",
      "dependencies": "DIAMOND.BIT.RPFORM.01; DIAMOND.BIT.SELECT.01; QGEO.QRECON.01; v534; v568; v572",
      "scripts": [
        "v584_ext_source_response.py"
      ],
      "script": "v584_ext_source_response.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 613
    },
    {
      "id": "QGEO.INVOL.01",
      "claim": "The sheet-involution existence question of GATE.QGEO.01, narrowed to a two-demand selection with UNCONDITIONAL existence (one module, 7 checks, <1 s; discovery probe involution_existence_probe.py 7/7, verdict EXISTENCE-UNCONDITIONAL): RIGIDITY -- the integral V-commuting involutions are exactly the spectral sign group (Z/2)^3: all eight sign choices on V's eigenspaces {0,1,2} are integer matrices, genuine involutions, commute with V and stabilize the integral order Z<U,V>, on BOTH twins (nontrivial: generic integral operators do not have integral spectral projectors).  THE NARROWING CHAIN: (i) the v574 rank theorem (two-dim-positive involutions cannot carry the binary trace code) kills the trace +1 half, 6 -> 3, all remaining of exact Sigma-signature (1,-1,-1); (ii) the seam-reversal demand S e3 = -e3 narrows 3 -> 2; (iii) the zero-mode-fixing demand (+1 on ker V, flipping exactly the dynamical sheet pair {1,2}) singles out S0 = [[1,-1,0],[0,-1,0],[0,0,-1]] uniquely -- explicit, integral, order-stabilizing: EXISTENCE IS UNCONDITIONAL.  HONEST SCOPE: the corpus RP reflection Sigma = diag(1,-1,-1) (an anti-automorphism) does NOT commute with V -- a different structure with a different role; the alt twin admits the same classification (the involution does not separate the twins, consistent with v567/v572).  RELOCATION: GATE.QGEO.01's involution-existence input is DERIVED; the residual freedom is the finite two-demand selection, machine-checked; deriving the two demands from the continuum geometry (mu4/D4/H1) is the remaining open [C] of the gate -- named, not claimed. [2026-08-01 V: the grading shortcut is falsified -- NO V-commuting involution implements the corpus mu4 grading S Q S = Sigma Q Sigma (all six fail); Sigma and S0 are genuinely different structures whose product is the infinite-order shear [[1,-1,0],[0,1,0],[0,0,1]]: the two v590 demands stay independent; the continuum derivation remains open.] [2026-08-01 X: a live geometric candidate -- v597: the mu3-cover y^3 = x^4-1 hits four compiler fingerprints (rank-3 Eisenstein, projective D4, (Z/3)^4 = 81 cusp classes, Lorentz (1,2)); see QGEO.COVER.01.] [2026-08-01 XIX: the seam-reversal demand is now GEOMETRIC -- v603: on the mu3-cover the seam line is real (R-fixed) and the two-demand involution reverses it exactly; the zero-mode demand is built into the curve-side classification (v601); the continuum derivation of the demands has its first geometric anchor; see QGEO.SEAMMARKS.01.] [2026-08-01 XXV: BOTH demands now have canonical cover carriers -- v603: the seam-reversal demand is 'reverse the R-real line'; v607: the zero-mode demand is 'fix the only unlabeled mode' (seam = R-real, top = deck-eigen, zero = unlabeled); typed [C] readings, the continuum derivation of the demands from the raw seam stays the gate residual; see QGEO.SELRULE.01.]",
      "status": "Exact [E] + typed relocation [C] (exact sympy; demands declared; continuum derivation open)",
      "canonicalStatus": "EXISTENCE-UNCONDITIONAL: rigidity (Z/2)^3 on both twins; narrowing 6 -> 3 -> 2 -> 1 via v574 type theorem + seam reversal + zero-mode fixing; S0 explicit with Sigma-signature; the two demands' continuum derivation stays open; 7/7 checks",
      "dependencies": "QGEO.WINDING.SELF.01; QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; v566; v567; v574",
      "scripts": [
        "v590_involution_existence.py"
      ],
      "script": "v590_involution_existence.py",
      "location": "tfpt_1;tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 614
    },
    {
      "id": "QGEO.COVER.01",
      "claim": "Strategy I/II, first executed step -- the mu3-cover of P^1 branched at mu4 (the curve y^3 = x^4 - 1) carries FOUR independent compiler fingerprints (one module, 6 checks, <1 s; discovery probe mu4_cover_model_probe.py 6/6, verdict FINGERPRINTS-HIT): (1) genus 3 with five cusps (mu4 + infinity; monodromies (w,w,w,w,w^2) multiplying to 1) and Chevalley-Weil H^{1,0} = (1,2) per character -- H_1 is a RANK-3 module over the Eisenstein integers, the compiler's 3-dimensional integral frame; (2) the D4 rotation closes PROJECTIVELY VIA THE DECK: in the reduced Burau model at t = omega (braid relations machine-verified) the rotation braid satisfies r^4 = omega x identity and r^12 = 1; (3) THE TORSION: the cusp-class group is (Z/3)^5 modulo one global relation = (Z/3)^4 of order 81 = 3^4 -- the v566 saturation index with exactly the per-mark Z/3 mechanism the self-code program asked for (four free local classes); det_Z(1 - deck) = 27 complementarily; (4) THE SIGNATURE: the invariant hermitian form of the Burau module is unique up to scale with eigenvalues {-2, 2-2sqrt2, 2+2sqrt2} -- LORENTZ SIGNATURE (1,2), matching the Chevalley-Weil Hodge numbers AND the rank-3 polarization signature of Paper II: Strategy II's first prediction (RP positivity as Hodge/Riemann positivity) confirmed at the form level.  First operator pass: the single-cusp Gram has all-integer spectrum {0,0,2}; the compiler V-spectrum {0,1,2} is not yet located -- the operator dictionary (which canonical homology operator is Q/V) and the real-structure/RP map to the v572 Gram data are the NAMED NEXT STEPS.  Kill criteria live (Smith/spectral mismatch for I; signature mismatch for II -- not triggered).  GATE.QGEO does NOT move; no marker changes. [2026-08-01 XI, the dictionary's first entry: U IS LOCATED -- the cusp Gram (1-T2)(1-T2)^dagger has characteristic polynomial x^2(x-3) = char(U) exactly (the winding operator as a cusp Gram; fifth fingerprint); additional exact structure: opposite-twist products (1-T1)(1-T3) are NILPOTENT (char x^3), and G1+G3 has spectrum {0,1,3}; V/H/Q not located in the scanned canonical ranges (products/sums/omega-twists of length <= 2) -- the V-dictionary likely requires the parabolic flag (the seam-cusp filtration), named as the next step.] [2026-08-01 XIII: dictionary round 2 -- v598: THREE entries now exact (U ~ G2; Q_- ~ G1-G3, the sheet coupling as the diamond-diagonal Gram difference; the binary ladder ~ 1+G1+G3); the grading half absent from 7296 canonical combinations; see QGEO.DICT.01.] [2026-08-01 XV: the real structure is constructed and the invariant form is pinned -- v599: R unique with R^2 = +1, det J = 8 (a sheet-diamond determinant list member), Lorentz (1,2) confirmed exactly; see QGEO.REAL.01.] [2026-08-01 XVI: the operator dictionary lands at ALGEBRA level -- v600: the compiler pair (U, V) realized on the cover with the exact v566 saturation structure (index 81, divisors (1,1,1,3,3,3,3)); the [P]-model of GATE.QGEO has an explicit construction; canonicity and the two-demand derivation stay open; see QGEO.EMBED.01.] [2026-08-01 XXI: the torsion picture completes -- v604: 81 = 3^4 localizes at the finite marks (divisor classes; per-frame word order) and 27 = 3^3 at infinity (monodromy-defect cokernel = equivariant-order cokernel = det(1-deck)); see QGEO.EQORDER.01.] [2026-08-01 XXVIII: the curve's wider landscape is mapped -- v610: the Jacobian splits into a Q(zeta_12)-CM surface (the field of deck AND marks) times the Q(omega)-CM elliptic quotient, point-count-confirmed; the integer-spectrum census honestly relativizes single-char hits (the anchor class is the most frequent nontrivial family; the joint structure is the distinction); see QGEO.LANDSCAPE.01.] [2026-08-01 XXXV: the bedrock direction reverses -- v617: the conformal seam axioms FORCE the mu3-cover (Z4-Mobius rigidity, uniform weights j in {1,2}, genus 3, seam reflections = the v599 pattern); the QGEO.SYM.01 residue narrows to the cyclic-3 choice and the physical<->conformal seam identification; see QGEO.SEAMFORCE.01.]",
      "status": "Exact [E] fingerprints + typed candidate [C] (Burau convention validated; Q-identification open)",
      "canonicalStatus": "FINGERPRINTS-HIT: rank-3 Eisenstein module; projective D4 via deck (r^4 = omega); cusp-class group (Z/3)^4 = 81 with the per-mark mechanism; Lorentz (1,2) polarization (Chevalley-Weil + Paper II simultaneously); single-cusp Gram {0,0,2}; Q/V dictionary and RP map named open; 6/6 checks",
      "dependencies": "QGEO.INVOL.01; QGEO.WINDING.SELF.01; DIAMOND.PARA.SELFCODE.01; GATE.QGEO.01; v566; v574; v590",
      "scripts": [
        "v597_mu4_cover_model.py"
      ],
      "script": "v597_mu4_cover_model.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 615
    },
    {
      "id": "QGEO.DICT.01",
      "claim": "The cover dictionary, round 2 -- THREE compiler objects located as canonical cusp-Gram combinations on the mu3-cover (one module, 4 checks, <1 s; discovery probe cover_dictionary_probe.py 4/4, verdict DICT-THREE-ENTRIES): with T_1..T_4 the D4-conjugate cusp twists and G_k their Grams -- U ~ G_2 (char x^2(x-3), the winding operator as a single-cusp Gram; v597 entry); Q_- ~ G_1 - G_3 (char x(x^2-3)): THE MU4 SHEET COUPLING IS THE DIFFERENCE OF THE OPPOSITE CUSP GRAMS -- the diamond diagonal; ladder ~ 1 + G_1 + G_3 (char (x-1)(x-2)(x-4)): the binary AnchorLadder (1,2,4) = (2^0,2^1,2^2) of v568 is identity plus the diagonal sum.  Supporting exact structure: opposite-twist products (1-T_1)(1-T_3) are nilpotent (cube zero), Q_-^2 has char x(x-3)^2.  THE NEGATIVES QUANTIFIED: Q_+ ({1,2,3}), V ({0,1,2}), H ({1,1,2}) and Q itself are NOT contained in 7296 scanned canonical combinations (the integer Gram lattice a + bG_1 + cG_2 + dG_3 + eG_4 and its non-hermitian (1-T_i)(1-T_j) extension; numeric prefilter, exact confirmation; 0 matches) -- the grading half needs a generator outside the finite-cusp twist algebra, with the anti-holomorphic D4 reflections (the real structure) as the named suspect.  The Strategy-I kill criterion is NOT triggered (the canonical space is not exhausted).  GATE.QGEO does not move; no marker changes. [2026-08-01 XV: the named suspect is CONSTRUCTED -- v599: the real structure exists and is unique up to scale (R^2 = +1); its grading functor closes all three gaps at char level (V, H with the anchor spectrum, Q_+); the joint algebra embedding stays open (quantified negative); see QGEO.REAL.01.] [2026-08-01 XVI: the dictionary completes at algebra level -- v600: a Gamma-even degree-2 pair realizes (U, V) jointly (conjugation exact, index 81); canonicity open; see QGEO.EMBED.01.] [2026-08-01 XXVIII: retroactive typing confirmed -- v610's census shows single integer spectra are common on the cover; this module's char-level entries were correctly typed candidates, upgraded to the joint embedding only by v600; see QGEO.LANDSCAPE.01.]",
      "status": "Exact [E] entries + quantified negatives (numeric prefilter, symbolic confirmation)",
      "canonicalStatus": "DICT-THREE-ENTRIES: U ~ G2, Q_- ~ G1-G3 (diamond diagonal), ladder ~ 1+G1+G3 -- all exact; grading half absent from 7296 canonical combinations (kill not triggered; real structure the named suspect); 4/4 checks",
      "dependencies": "QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v566; v568; v597",
      "scripts": [
        "v598_cover_dictionary.py"
      ],
      "script": "v598_cover_dictionary.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 616
    },
    {
      "id": "QGEO.REAL.01",
      "claim": "The real structure on the mu3-cover -- the v598 suspect constructed (one module, 18 checks, ~3 s; discovery probe real_structure_probe.py 18/18, verdict REAL-STRUCTURE-LANDED).  EXISTENCE AND UNIQUENESS [E]: complex conjugation on the base fixes the punctures {1,-1} and swaps {i,-i} (an orientation-reversing D4 reflection); it must act semi-linearly, R(v) = M conj(v), sending each cusp twist to the INVERSE of its reflected partner, R T_k R^-1 = T_(5-k)^-1.  The intertwining system has a 1-DIMENSIONAL solution space: R exists and is unique up to scale; the canonical normalization satisfies R^2 = +1 exactly (a genuine real structure, not quaternionic); all four twist intertwinings hold exactly.  THE GRADING FUNCTOR [E]: Gamma(A) = M conj(A) M^-1 (the curve analogue of the compiler grading A -> Sigma A Sigma) closes the three v598 dictionary gaps at char level: V ~ G2 + G3 - Gamma(G3) with char x(x-1)(x-2); H ~ 1 + G2 - Gamma(G3) with char (x-1)^2(x-2) -- THE ANCHOR SPECTRUM (1,1,2); Q_+ ~ 1 + G2 + G3 - Gamma(G3) with char (x-1)(x-2)(x-3).  THE CONSTANTS [E]: the Gamma-even/odd cusp elements E_k = G_k + Gamma(G_k), O_k = G_k - Gamma(G_k) carry the compiler constants: E2 spectrum {0,1,5} (the carrier g_car = 5), E4 {0,3,9} = {0, N_fam, N_fam^2}, O3 nilpotent (char x^3), O2 char x(x^2-5), O4 char x(x^2-27).  THE JOINT NEGATIVE [X, quantified]: char-matching is NOT yet an algebra embedding -- in the +-1 Gamma-extended integer lattice (8 generators) there are exactly 4 U-candidates and 6 V-candidates, and ZERO pairs satisfy the joint compiler relations tr(UV) = 3 and char(U+V) = (x-1)(x^2-5x+3); the natural pair (G2, G2+G3-Gamma(G3)) gives char x(x^2-6x+4), an exact documented mismatch.  The flag-adapted simultaneous-conjugation search remains the named next step; the Strategy-I kill criterion is NOT triggered.  THE LORENTZ FORM [E, Strategy II]: the invariant hermitian form J (T_k^dagger J T_k = J for all four twists) is unique up to real scale, integral over Z[omega], with det J = 8 -- A MEMBER OF THE SHEET-DIAMOND DETERMINANT LIST (3,4,8,14,20,32) -- and char (x+2)(x^2-4x-4): LORENTZ SIGNATURE (1,2), matching the Paper-II rank-3 polarization signature AND the Chevalley-Weil Hodge prediction (h^{1,0}, h^{0,1}) of Strategy II simultaneously.  GATE.QGEO does not move; no marker changes. [2026-08-01 XVI: the degree-2 Gamma-even sector closes the joint gap -- v600: the compiler pair (U, V) is realized (GL3-conjugation solved exactly, saturation index 81 on the R-fixed lattice); the v599 joint negative is thereby located precisely at degree 1; see QGEO.EMBED.01.]",
      "status": "Exact [E] existence/uniqueness/char identities + [X] quantified joint negative + [C] the TFPT readings (exact sympy; numeric prefilter with symbolic confirmation for the scan)",
      "canonicalStatus": "REAL-STRUCTURE-LANDED: R unique (R^2 = +1), four intertwinings exact; Gamma closes V/H/Q_+ at char level (H carries the anchor spectrum); constants 5/9/27 in the Gamma-decomposition; joint embedding open (4x6 candidates, 0 joint pairs); det J = 8, Lorentz (1,2); 18/18 checks",
      "dependencies": "QGEO.DICT.01; QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v566; v597; v598",
      "scripts": [
        "v599_real_structure.py"
      ],
      "script": "v599_real_structure.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 617
    },
    {
      "id": "QGEO.EMBED.01",
      "claim": "The joint embedding -- the compiler pair (U, V) is realized on the mu3-cover (one module, 15 checks, ~4 s; discovery probe joint_embedding_probe.py 15/15, verdict JOINT-EMBEDDING-LANDED).  THE PAIR [E]: in the Gamma-even DEGREE-2 sector (the v599 grading functor applied to cusp-Gram products), U* = -2E1 + (G1 Gamma(G1) + Gamma(G1) G1) + 3 and V* = -E3 + (1/2)(G2 Gamma(G2) + Gamma(G2) G2) + 1 are both integral over Z[omega], Gamma-even, with char U* = x^2(x-3), char V* = x(x-1)(x-2), char(U*+V*) = (x-1)(x^2-5x+3) = char Q, and the compiler joint traces tr(U*V*) = 3, tr(U*V*^2) = 6.  THE ALGEBRA EMBEDDING [E]: the seven words {1, U*, V*, U*V*, V*U*, V*^2, V*U*V*} are independent (rank 7 = the full 2|1 parabolic) and the simultaneous-conjugation system is EXACTLY solved: an explicit invertible g with g U* g^-1 = U_c and g V* g^-1 = V_c -- the v566 parabolic self-code algebra lives on the cover.  THE INTEGRAL FORM [E]: the real structure R as an integer matrix on Z^6 has R^2 = 1; its fixed lattice has the explicit saturated basis (1,2,2,0,0,0), (0,1,1,-1,0,0), (0,0,1,1,-1,1) (SNF certificate (1,1,1)); U*, V* restrict to INTEGER 3x3 matrices with the same chars.  THE 81 [E, the hard test]: the Z-order of the seven restricted words has Smith elementary divisors (1,1,1,3,3,3,3) -- saturation index 81 = 3^4, EXACTLY the v566 compiler self-code index with the same divisor pattern.  HONEST LIMITS: must-break control passes (the v599 degree-1 pair fails the joint criterion); CANONICITY IS OPEN and typed open -- the degree-2 scan admits many joint pairs, so this is an EXISTENCE claim, not uniqueness; GATE.QGEO does not move until the canonicity round and the two-demand continuum derivation land; no marker changes. [2026-08-01 XVII: the canonicity question is answered -- v601: the J-adapted projector presentation is D4-equivariant by construction (mark choice = winding choice) and realizes the DUAL parabolic exactly; the Klein group lives integrally on the curve; see QGEO.CANON.01.] [2026-08-01 XVIII: the integer core is Lean-certified -- FORM.QGEO.04 in TfptCarrier/CoverEmbedding.lean: W = T*S, Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P, S-primitivity via unimodular minor (determinant-free, kernel decide; lake build green).]",
      "status": "Exact [E] existence realization (integrality, conjugation, saturation index all exact sympy) + typed open canonicity [O]; NO marker moves",
      "canonicalStatus": "JOINT-EMBEDDING-LANDED: (U*, V*) Gamma-even degree-2, Z[omega]-integral, chars + joint traces compiler-exact, word rank 7, conjugation solved, integer on saturated Fix(R), Smith (1,1,1,3,3,3,3) index 81 = v566; canonicity open; 15/15 checks",
      "dependencies": "QGEO.REAL.01; QGEO.DICT.01; QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; GATE.QGEO.01; v566; v597; v598; v599",
      "scripts": [
        "v600_joint_embedding.py"
      ],
      "script": "v600_joint_embedding.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 618
    },
    {
      "id": "QGEO.CANON.01",
      "claim": "The canonical equivariant realization (one module, 14 checks, ~12 s; discovery probe equivariant_dual_probe.py 14/14, verdict EQUIVARIANT-DUAL-LANDED).  THE REPAIR: the naive cusp-Gram presentation of v600 is not D4-equivariant because the standard hermitian dagger is basis-dependent -- the Burau image preserves J; the canonical building blocks are the J-ADAPTED cusp projectors GJ_k = (1-T_k) J^-1 (1-T_k)^dagger J.  THE PROJECTORS [E]: all four are idempotent rank-1 J-self-adjoint projectors (char x^2(x-1)) rotating exactly under D4 (r GJ_k r^-1 = GJ_(k+1)); opposite cusps J-orthogonal (tr(GJ1 GJ3) = 0), neighbors overlap with trace 1.  THE EQUIVARIANT PAIR [E]: U_k = 3 GJ_k, V_k = 1 + GJ_(k-1) - GJ_k GJ_(k+1) -- Z[omega]-integral, compiler spectra, compiler joint data (char(U+V) = charQ, tr(UV) = 3, tr(UV^2) = 6), seven-word rank 7, and D4-EQUIVARIANT BY CONSTRUCTION (r^k-conjugation = mark shift, verified for all marks): the mark choice IS the compiler's winding choice.  THE DUALITY [E]: the equivariant pair is NOT conjugate to (U_c, V_c) (the intertwiner space is 1-dimensional with identically singular solutions -- must-fail documented) but EXACTLY conjugate to the transposed pair (U_c^T, V_c^T): the equivariant presentation realizes the DUAL (plane-stabilizer) parabolic, the v600 Gamma-even pair the line-stabilizer -- the cover carries BOTH parabolic types, exchanged by duality.  THE KLEIN GROUP ON THE CURVE [E]: all eight spectral-sign involutions of V_1 are Z[omega]-integral (v590 rigidity reproduced); the two v590 demands select the explicit integral S_0 (trace -1, commuting with V), and A B = S_0 closes the Klein four-group on the cover.  HONEST OPENS [C]: the naive J-word Gram has inertia (3,2,2), not the corpus v572 RP pattern (4,0,3) -- the Hodge/RP comparison needs the Weil-twisted polarization (named); the seam-line geometric identification stays open.  GATE.QGEO does not move; no marker changes. [2026-08-01 XVIII: the duality mechanism is identified -- v602: the J-adjoint exchanges the two parabolic types exactly ((U1, V1^J) ~ compiler pair); the dictionary completes in the J-projector language; Weil positivity holds exactly; see QGEO.DUALFORMS.01.]",
      "status": "Exact [E] equivariance/duality/Klein identities + [C] typed opens (exact sympy; one numeric inertia count of an exact hermitian matrix)",
      "canonicalStatus": "EQUIVARIANT-DUAL-LANDED: GJ_k idempotent equivariant projectors; pair family compiler-exact and equivariant by construction; conjugate to the TRANSPOSED compiler pair (dual parabolic); Klein group integral on the curve (8/8, S_0 explicit, A B = S_0); J-word-Gram inertia (3,2,2) typed open vs v572; 14/14 checks",
      "dependencies": "QGEO.EMBED.01; QGEO.REAL.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v590; v597; v599; v600",
      "scripts": [
        "v601_equivariant_dual.py"
      ],
      "script": "v601_equivariant_dual.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 619
    },
    {
      "id": "QGEO.DUALFORMS.01",
      "claim": "The duality-and-forms round (one module, 17 checks, ~8 s; discovery probe duality_forms_probe.py 17/17, verdict DUALITY-FORMS-LANDED).  THE DUALITY MECHANISM [E]: U_1 = 3 GJ_1 is J-self-adjoint; the J-adjoint of V_1 = 1 + GJ_4 - GJ_1 GJ_2 is the exact product reversal 1 + GJ_4 - GJ_2 GJ_1; and (U_1, V_1^J) is EXACTLY conjugate to the compiler line-stabilizer pair while (U_1, V_1) is conjugate to the transposed pair (v601) -- the J-adjoint exchanges the two parabolic types inside ONE equivariant structure: the cover's realization is J-self-dual as a pair of pairs.  DICTIONARY COMPLETION [E]: H = 1 + V(V-1)/2 has char (x-1)^2(x-2) -- the corpus anchor formula holds VERBATIM on the curve; Q = U + V has char (x-1)(x^2-5x+3); Q_+ ~ 2 - GJ_1 + GJ_2 GJ_3 has char (x-1)(x-2)(x-3); the binary AnchorLadder ~ 2 - GJ_1 + 2 GJ_2 GJ_3 has char (x-1)(x-2)(x-4); the S_0-grading halves of Q are computed and documented.  WEIL POSITIVITY [E, Strategy II at module level]: the Weil operator C from J's spectral split (Hodge numbers (1,2)) gives a POSITIVE DEFINITE twisted form JC with exact eigenvalues {2sqrt2-2, 2, 2+2sqrt2} = |Spec J| -- the Riemann bilinear positivity statement holds on the omega-sheet.  THE CANONICAL BILINEAR FORM [E]: B(v,w) = (Rv)^dagger J w has matrix B = M^dagger J, SYMMETRIC and nondegenerate with det B = 8 -- the curve's complex orthogonal structure combining the real structure with the polarization.  HONEST NEGATIVES [X/O, quantified]: (i) g^T g is NOT proportional to B -- the corpus transpose structure is not the (R,J)-bilinear form (corpus RP word-Gram inertia (4,3,0) documented); (ii) the naive mark-local cokernel map fails (U_2, U_3 lie outside the seven-word algebra, U_4 inside with integer coefficients) -- the carrier of the mark-local Z/3 mechanism stays open; (iii) the seam-plane normal is explicit but NOT a deck eigenvector -- the seam identification stays open.  COMPANION LEAN CERTIFICATE (FORM.QGEO.04): the v600 integer core is certified in TfptCarrier/CoverEmbedding.lean -- compiler relations of (C_U, C_V), the word factorization W = T*S, the Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P (P*Pinv = 1), and S-primitivity via a unimodular minor: determinant-free, kernel decide, lake build green (3375 jobs).  GATE.QGEO does not move; no marker changes. [2026-08-01 XIX: the seam question advances -- v603: the seam line is a REAL line of the cover (R-fixed), S_0 reverses it geometrically; the infinity cusp separates in the unreduced module; the first per-mark Z/3 witness lands (E_1, order 3); see QGEO.SEAMMARKS.01.] [2026-08-01 XXII: the corpus-transpose translator question is RESOLVED -- v605: theta_corpus = Jadj o inner(c) exactly, c unique up to scale, 14c integral; c's independent geometric identity is the remaining open; see QGEO.TRANSLATOR.01.]",
      "status": "Exact [E] duality/dictionary/positivity identities + [X/O] quantified negatives and typed opens (exact sympy; two numeric inertia counts of exact matrices) + Lean certificate FORM.QGEO.04",
      "canonicalStatus": "DUALITY-FORMS-LANDED: J-adjoint exchanges the parabolic types (exact conjugations both ways); dictionary complete in J-projector language (anchor formula verbatim); Weil-twisted form positive definite (exact); B = M^dag J symmetric, det 8; corpus-transpose / mark-local / seam questions typed open; Lean FORM.QGEO.04 green; 17/17 checks",
      "dependencies": "QGEO.CANON.01; QGEO.EMBED.01; QGEO.REAL.01; DIAMOND.PARA.SELFCODE.01; v572; v590; v600; v601",
      "scripts": [
        "v602_duality_forms.py"
      ],
      "script": "v602_duality_forms.py",
      "location": "tfpt_1;tfpt_2;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 620
    },
    {
      "id": "QGEO.SEAMMARKS.01",
      "claim": "The seam-and-marks round (one module, 15 checks, ~7 s; discovery probe seam_marks_probe.py 15/15, verdict SEAM-REAL-LANDED).  THE SEAM LINE IS REAL [E]: the line-stabilizer pair (U_1, V_1^J) has a UNIQUE common eigenline v with U v = 0 and V v = v -- the compiler seam span(e3), transported; explicitly v = (0, (sqrt3-i)/(sqrt3+i), 1).  The real structure FIXES this line (R v ~ v): the seam is a real line of the cover.  The two-demand involution REVERSES it (S_0 v = -v): the v590 seam-reversal demand holds GEOMETRICALLY on the curve.  Honest scope: S_0 does not commute with R (the sheet involution is not Gamma-even), and the seam's identification with the infinity cusp stays open.  THE INFINITY CUSP SEPARATED [E]: the unreduced Burau of B_4 at t = omega (braid relations verified) has invariant column (1,1,1,1) and invariant row (1, omega, conj omega, 1); the full twist r^4 acts with eigenvalues {1 (x1), omega (x3)} -- the infinity direction canonically separated from the omega-sheet.  THE FIRST PER-MARK Z/3 WITNESS [E]: E_1 = G_1 + Gamma(G_1) restricts to an INTEGER matrix on the saturated fixed lattice and lies in the seven-word algebra with coefficient-denominator lcm EXACTLY 3 -- an ORDER-THREE class in the word-order cokernel P/O: the v566/v597 per-mark mechanism gets its first exact witness.  Honest structure: E_4's class is trivial (integer coefficients), E_2/E_3 lie OUTSIDE the mark-1-anchored algebra -- the naive global map fails; the equivariant per-mark framing (each mark's class in its own rotated frame) is the typed reading.  TRANSLATOR NEGATIVES [X, quantified]: the pullback form F_2 = g^T Sigma g is symmetric but not proportional to any of 7 canonical menu forms, and neither B^-1 F_2 nor F_2 B^-1 lies in the word algebra -- the corpus-transpose translator needs a new canonical object, named open.  GATE.QGEO does not move; no marker changes. [2026-08-01 XXI: the integral infinity bridge lands -- v604: the 3-torsion localizes at infinity (coker(1-T_inf) = Z^2 + (Z/3)^3, finite cusps torsion-free), the four frames generate a FULL order of index 27 = 3^3, and R is an anti-isometry (J_fix integer, det 72); see QGEO.EQORDER.01.] [2026-08-01 XXII: the seam-infinity identification lands INTEGRALLY -- v605: the seam lift is a NONZERO pure 3-torsion class in coker(1-T_inf) (free coords zero, torsion (2,2,0)); see QGEO.TRANSLATOR.01.] [2026-08-01 XXIV: the zero-mode selection gains its carrier -- v606: the infinity torsion separates all three modes (pure torsion, distinct classes); the canonical '+1' rule is the remaining step; see QGEO.MODESEP.01.]",
      "status": "Exact [E] seam/infinity/witness identities + [X] quantified translator negatives + [C] typed per-mark framing (exact sympy)",
      "canonicalStatus": "SEAM-REAL-LANDED: seam line unique, real (R-fixed), S_0-reversed (v590 demand geometric); infinity cusp separated (unreduced Burau, full twist {1, omega x3}); E_1 order-3 cokernel class (first per-mark witness); E_4 trivial, E_2/E_3 outside; translator negatives quantified; 15/15 checks",
      "dependencies": "QGEO.DUALFORMS.01; QGEO.CANON.01; QGEO.EMBED.01; QGEO.INVOL.01; v590; v600; v601; v602",
      "scripts": [
        "v603_seam_marks.py"
      ],
      "script": "v603_seam_marks.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 621
    },
    {
      "id": "QGEO.EQORDER.01",
      "claim": "The equivariant order and the infinity bridge (one module, 12 checks, ~4 s; discovery probe equivariant_order_probe.py 12/12, verdict EQUIVARIANT-ORDER-LANDED).  CORRECTION + THE INTEGRAL FORM [E]: R IS an anti-isometry of the polarization -- the correct compatibility equation M^dagger J M = J^T holds EXACTLY (a v599-diary side remark had tested the wrong equation M conj(J) M^dagger vs J); consequently J restricts to a REAL SYMMETRIC INTEGER form on the saturated fixed lattice: J_fix = [[16,2,4],[2,-2,2],[4,2,-2]], det 72 = 8 x 9 (the polarization determinant times the (1-omega)-covolume factor).  THE EQUIVARIANT ORDER IS FULL [E]: the Z-lattice generated by the seven words of ALL FOUR mark frames (the r^k-conjugates of the v601/v602 equivariant pair; all 28 words Z[omega]-integral) has FULL rank 18 in M_3(Z[omega]) with Smith divisors (1^15, 3, 3, 3) -- saturation index 27 = 3^3: one frame gives the 7-dim parabolic of index 81 = 3^4 (v600), all four give the full matrix algebra of index 27.  THE 3-TORSION LOCALIZES AT INFINITY [E]: on the unreduced integral module Z[omega]^4, coker(1 - T_inf) = Z^2 + (Z/3)^3 (Smith divisors (1,1,1,3,3,3,0,0), torsion 27) while each FINITE cusp twist is torsion-free (divisors (1,1,0^6)) -- three objects share (Z/3)^3 = 27: the infinity-monodromy cokernel, the equivariant-order cokernel, and det_Z(1-deck) (v597).  THE READING [C]: the two compiler constants localize on the cover -- 81 = 3^4 at the FINITE marks (cusp divisor classes, v597; per-frame word order, v600) and 27 = 3^3 at INFINITY (monodromy defect; full equivariant order): the integral infinity <-> order-torsion bridge.  TRANSLATOR NEGATIVES EXTENDED [X, quantified]: the corpus RP metric pulled back to the integral fixed lattice is NOT proportional to J_fix or its Klein twists (6 tested), and neither J_fix^-1 F2 nor F2 J_fix^-1 lies in the word algebra -- after the (B, Gamma, Klein, J_fix) sweep the corpus transpose structure resists all single-sheet constructions; the conjugate character sheet (omega-bar) is the named next candidate.  GATE.QGEO does not move; no marker changes. [2026-08-01 XXII: the translator is FOUND -- v605: the corpus RP anti-automorphism is the J-adjoint twisted by an explicit inner element c (14c integral; the omega-bar detour was unnecessary); the seam is 3-torsion at infinity; the 81 survives frame-joining; see QGEO.TRANSLATOR.01.] [2026-08-02 XXXIX: the fixed-lattice form J_fix gains an arithmetic identity -- v624: P^T J_det P = J_fix EXACTLY with det P = -6 (J_det the prime-front determinant form): the two Lorentz lattices are the same rational quadratic form, the cover an index-6 sublattice (72/2 = 6^2, 6 = p_2(a)); P-canonicity and the Hodge-chamber question preregistered; see EXTREV.LATTICE.01.]",
      "status": "Exact [E] anti-isometry/order/localization identities + [C] typed localization reading + [X] quantified translator negatives (exact sympy; Smith normal forms over Z)",
      "canonicalStatus": "EQUIVARIANT-ORDER-LANDED: M^dag J M = J^T exact (R anti-isometry); J_fix integer, det 72; four-frame order FULL rank 18, divisors (1^15,3,3,3), index 27 = 3^3; coker(1-T_inf) = Z^2+(Z/3)^3, finite cusps torsion-free; 81 at marks / 27 at infinity; translator negatives extended (omega-bar named); 12/12 checks",
      "dependencies": "QGEO.SEAMMARKS.01; QGEO.DUALFORMS.01; QGEO.EMBED.01; QGEO.COVER.01; v597; v600; v601; v602; v603",
      "scripts": [
        "v604_equivariant_order.py"
      ],
      "script": "v604_equivariant_order.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 622
    },
    {
      "id": "QGEO.TRANSLATOR.01",
      "claim": "The translator round (one module, 11 checks, ~11 s; discovery probe translator_probe.py 11/11, verdict TRANSLATOR-LANDED).  THE TRANSLATOR EXISTS [E, centerpiece]: pulling the corpus RP anti-automorphism theta(X) = Sigma X^T Sigma back along the compiler conjugation and composing with the J-adjoint gives an algebra automorphism phi = Jadj o theta_pull -- and phi is INNER: the intertwiner space is 1-DIMENSIONAL, its generator c is invertible, and phi(A) = c A c^-1 holds EXACTLY on the pair.  Hence THE CORPUS RP STRUCTURE IS THE J-ADJOINT COMPOSED WITH THE INNER TWIST BY c -- the RP anti-automorphism is fully expressed in curve language (polarization + one explicit element); the normalization 14c is Z[omega]-INTEGRAL (denominator 14 = 2 x 7, the parabolic dimension).  Honest scope: c is unique up to scale by construction but not Gamma-even and not in the word algebra -- its independent geometric identity is the named open (the earlier single-sheet negatives v603/v604 are thereby RESOLVED: the missing object was inner, not a new form; the omega-bar detour was unnecessary).  THE SEAM-INFINITY BRIDGE [E]: in the Smith coordinates of coker(1 - T_inf) = Z^2 + (Z/3)^3, the seam lift (-conj omega, -conj omega, 0, -omega) has ZERO free coordinates and torsion coordinates (2,2,0) mod 3: the seam line is a NONZERO pure 3-torsion class of the infinity monodromy cokernel -- the integral seam <-> infinity identification.  THE 81 SURVIVES [E]: inside the frame-1 span the order chain O_1 subset (O_joint cap span_1) subset P_1 has indices 3^5 = 243 and 3^4 = 81: joining all four frames refines each frame's order by exactly 3^5 and the residual saturation defect is EXACTLY the v566 index 81.  THE J_FIX DUALITY CERTIFIED [E]: C_V^T J_fix = J_fix C_Vdual with the explicit integer C_Vdual = [[1,0,0],[-2,0,0],[2,1,2]] (char x(x-1)(x-2), dual joint trace 3, char(C_U + C_Vdual) = x(x-2)(x-4)); companion Lean addendum in TfptCarrier/CoverEmbedding.lean (Jfix_symmetric, Jfix_det = 72, CV_dual_certificate, CVdual_cubic, dual_joint_trace; lake build green, 3375 jobs).  GATE.QGEO does not move; no marker changes. [2026-08-01 XXIV: the mode separation lands -- v606: all three V-eigenlines are pure 3-torsion at infinity with pairwise distinct classes (the selection carrier R could not provide); c's arithmetic-identity route is closed (cubic field S_3, not abelian); see QGEO.MODESEP.01.] [2026-08-01 XXVII: the geometric identity of c is CLOSED -- v609: c = lambda K_1 d with K_1 = J^-1 conj(F_2)^T M^-1 and d the UNIQUE unimodular Z[omega]-integral Gamma-involutive mirror intertwiner of the realized pair; see QGEO.CFACTOR.01.]",
      "status": "Exact [E] translator/bridge/chain/duality identities + [C] typed opens (exact sympy; Smith transforms over Z; Lean addendum green)",
      "canonicalStatus": "TRANSLATOR-LANDED: theta_corpus = Jadj o inner(c) exact (c unique, 14c integral); seam = nonzero 3-torsion at infinity ((2,2,0) pattern); order chain indices 3^5/3^4 (the 81 survives); J_fix duality certified + Lean; c's geometric identity open; 11/11 checks",
      "dependencies": "QGEO.EQORDER.01; QGEO.SEAMMARKS.01; QGEO.DUALFORMS.01; QGEO.EMBED.01; v572; v600; v601; v602; v603; v604",
      "scripts": [
        "v605_translator.py"
      ],
      "script": "v605_translator.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 623
    },
    {
      "id": "QGEO.MODESEP.01",
      "claim": "The mode-separation round (one module, 7 checks, ~2 s; discovery probe mode_separation_probe.py 7/7, verdict MODES-SEPARATED).  ALL THREE MODES ARE PURE TORSION AT INFINITY [E]: the three eigenlines of the sheet operator V (eigenvalues 0, 1, 2), lifted to the unreduced integral module along the unique intertwiner (1-parameter, rank 3), all have ZERO free part in coker(1 - T_inf) = Z^2 + (Z/3)^3 -- like the seam (v605), the zero mode and the top mode are pure 3-torsion classes at infinity.  THE SEPARATION [E]: the three classes are PAIRWISE DISTINCT elements of (Z/3)^3 (fixed Smith frame: (2,0,2), (2,2,0), (1,2,0) for V = 0, 1, 2; distinctness is frame-invariant) and none is trivial -- the infinity-cusp torsion is a SELECTION CARRIER distinguishing the zero mode from the top mode: exactly the separation the real structure could NOT provide (only the seam is R-fixed; the 0/2 eigenlines are neither R-fixed nor R-swapped, v605-followup).  HONEST SCOPE [C]: the specific coordinate triples are Smith-frame-dependent; the canonical rule singling out the zero mode's class (the '+1' of the v590 demand) from the structure alone is the named remaining step.  THE ARITHMETIC ROUTE FOR c IS CLOSED [X, honest negative]: the canonical cubic invariant char(7 Gamma(c) c) = y^3 + 3y^2 + 6y - 1 (integer monic, irreducible) has discriminant -783 = -3^3 x 29, NOT a square -- Galois group S_3, not abelian, so no cyclotomic-subfield identity for c exists (Kronecker-Weber); c's geometric identity stays the named open.  GATE.QGEO does not move; no marker changes. [2026-08-01 XXV: the canonical rule LANDS -- v607: the deck rotation has exactly ONE projective eigenline on the torsion (char (x+1)(x^2+1) mod 3) and it is the top mode; the zero mode is the only unlabeled mode; see QGEO.SELRULE.01.] [2026-08-01 XXVII: the c-identity question is resolved geometrically -- v609 (factorization through the unique mirror intertwiner); the arithmetic negative of this module stands as the reason the cyclotomic route had to fail; see QGEO.CFACTOR.01.]",
      "status": "Exact [E] torsion/separation identities + [C] typed frame caveat + [X] honest arithmetic negative (exact sympy; Smith transforms over Z)",
      "canonicalStatus": "MODES-SEPARATED: all three V-eigenlines pure 3-torsion at infinity, classes pairwise distinct ((2,0,2)/(2,2,0)/(1,2,0)); canonical '+1' rule open; cubic field of c disc -783, S_3, not abelian (arithmetic route closed); 7/7 checks",
      "dependencies": "QGEO.TRANSLATOR.01; QGEO.SEAMMARKS.01; QGEO.EQORDER.01; v590; v603; v604; v605",
      "scripts": [
        "v606_mode_separation.py"
      ],
      "script": "v606_mode_separation.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 624
    },
    {
      "id": "QGEO.SELRULE.01",
      "claim": "The selection rule round (one module, 10 checks, ~2 s; discovery probe selection_rule_probe.py 10/10, verdict SELECTION-RULE-LANDED).  THE F_3 STRUCTURE [E]: the torsion of coker(1 - T_inf) is a genuine F_3 vector space -- omega-multiplication acts as the IDENTITY ((1-omega) acts as zero) -- and the deck rotation r (commuting with T_inf = r^4) acts F_3-linearly on it.  THE RIGIDITY [E]: the induced action is the explicit matrix [[0,1,0],[1,0,2],[2,2,2]] mod 3; its only fixed class is 0; char mod 3 = (x+1)(x^2+1) with x^2+1 IRREDUCIBLE over F_3: the action has EXACTLY ONE projective eigenline (eigenvalue -1) -- structural uniqueness, census-confirmed.  THE LABELING [E]: the unique deck eigenline IS the top-mode class (V = 2); the zero mode and the seam are NOT projectively fixed (neither under r nor r^2).  With v603/v605 the three modes carry MUTUALLY EXCLUSIVE canonical labels: seam (V = 1) = the R-REAL line; top (V = 2) = the unique DECK-EIGEN torsion class; zero (V = 0) = the only UNLABELED mode.  THE RULE [C, typed reading]: the two-demand involution S_0 acts as -1 exactly on the two labeled modes and +1 on the unlabeled one -- the v590 zero-mode demand ('+1 on ker V') is the statement 'fix the unlabeled mode', the seam-reversal demand is 'reverse the R-real line' (v603): BOTH demands are now expressed through canonical cover structure.  Honest scope: a typed reading of the selection, not a continuum derivation; eigen-ness, uniqueness and distinctness are Smith-frame-invariant, the coordinate triples are not.  GATE.QGEO does not move; no marker changes.",
      "status": "Exact [E] F_3/rigidity/labeling identities + [C] typed selection reading (exact sympy / F_3 arithmetic; Smith transforms over Z)",
      "canonicalStatus": "SELECTION-RULE-LANDED: torsion = F_3 space (omega trivial); r-action explicit, char (x+1)(x^2+1) mod 3, unique projective eigenline = top mode; seam/zero not fixed; labels mutually exclusive (R-real / deck-eigen / unlabeled); S_0 flips the labeled pair; 10/10 checks",
      "dependencies": "QGEO.MODESEP.01; QGEO.TRANSLATOR.01; QGEO.INVOL.01; v590; v603; v605; v606",
      "scripts": [
        "v607_selection_rule.py"
      ],
      "script": "v607_selection_rule.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 625
    },
    {
      "id": "WOIT.GAMMA.FREE.01",
      "claim": "The free gamma slice of the WOIT roadmap (one module, 9 checks, ~1 s; discovery probe gamma_free_slice_probe.py 9/9, verdict GAMMA-FREE-SLICE-LANDED; the v524 machinery imported read-only).  THE BASELINE [E]: the chiral N = 8 OS Gram is PD (16,0,0) -- H_phys exists (v524 Q1.3 reproduced).  THE FLIP LOCALIZES [E]: the anti-chiral (mirror) state's Gram has inertia (8,8,0) and the split is SECTOR-EXACT: even (8,0,0) PD, odd (0,8,0) STRICTLY NEGATIVE DEFINITE -- the v519 odd-sector definiteness flip stated sector-exactly on the full N = 8 half algebra.  MIRROR EXCLUSION [E, the gamma-A statement at the free level]: every nonzero mirror fermion vector has strictly negative OS norm; hence in the doubled (vector-like) system any OS-positive subspace meets the mirror odd sector in {0}: NO mirror fermion mode survives OS reconstruction -- kill test (3) ('the reconstruction produces a vector-like mirror generation') CANNOT fire at the free level, upgraded from the v519 state-level shadow to the SECTOR level.  ONE GENERATION, MULTIPLICITY-FREE [E]: the compressed clock spectrum on the odd (fermion) transfer domain is EXACTLY {1, sqrt(2)-1} -- two distinct eigenvalues, no degeneracy.  MARK INCIDENCE AT THE FREE LEVEL [E]: the two positive-half mark algebras (monomials on {4,5} and {6,7}) have PD OS Grams (4,0,0) each (dim 4 = |mu4|); their monomial products generate the FULL 16-dim H_phys basis (the marks jointly generate the net, meeting in the vacuum line); and the quarter-turn transfer tau_2 IS the mark-A -> mark-B transport (alpha_2: {4,5} -> {6,7}), Hermitian PD on the mark algebra -- the incidence-compatible extension exists at the free level, kill test (7) does not fire there.  FIREWALL: the FREE slice only -- the gamma milestone itself (A_hol, the interacting net, the spacetime extension) stays OPEN; all seven contract kill tests stay formally live on A_hol; WOIT.OS.TWISTOR.01 does not move; no marker changes. [2026-08-01 XXXIV: the interacting gamma slice lands at toy level -- v615: on the RP-surviving interaction class (the alignment bit, v534) the mirror odd sector is strictly ND on every cut and every coupling (kill test (3) cannot fire on the surviving class), the mark transport is Hermitian PD at every g (kill test (7) does not fire), and the odd transfer spectrum stays multiplicity-free; the dead member has no RP home -- bit selection and chirality data co-locate; see WOIT.GAMMA.TOY.01.]",
      "status": "Exact [E] sector/spectrum/incidence statements on the explicit v524 net + [C] the gamma reading (free slice; A_hol untouched)",
      "canonicalStatus": "GAMMA-FREE-SLICE-LANDED: chiral (16,0,0); anti-chiral (8,8,0) sector-exact (odd strictly ND -- mirror exclusion); clock spectrum {1, sqrt2-1} multiplicity-free; mark algebras PD dim 4, products generate, tau_2 = mark transport PD; gamma open on A_hol; 9/9 checks",
      "dependencies": "WOIT.OS.TWISTOR.01; WOIT.BETA2.OS.01; WOIT.THETA.FREE.01; v519; v524; v565",
      "scripts": [
        "v608_gamma_free_slice.py"
      ],
      "script": "v608_gamma_free_slice.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 626
    },
    {
      "id": "QGEO.CFACTOR.01",
      "claim": "The translator factorization (one module, 8 checks, ~8 s; discovery probe translator_factorization_probe.py 8/8, verdict C-FACTORIZATION-LANDED).  THE MIRROR INTERTWINER [E]: the intertwining system d A = Gamma(A) d for the realized pair (A in {U_1, V_1^J}; Gamma = the v599 real-structure conjugation) has a 1-DIMENSIONAL solution space -- d exists, is unique up to scale, and intertwines the pair with its mirror image exactly.  UNIMODULAR AND INTEGRAL [E]: in the canonical normalization det d = 1 exactly and d is integral over Z[omega].  GAMMA-INVOLUTIVE [E]: Gamma(d) d = 1 exactly -- the mirror twist squares to the identity through d.  THE FACTORIZATION [E]: with K_1 = J^-1 conj(F_2)^T M^-1 (the polarization J, the transported RP metric F_2 = g^T Sigma g, the real structure M), the v605 translator element factorizes EXACTLY: c = lambda K_1 d (lambda the free scale), and inner(K_1 d) reproduces phi on the pair -- structurally theta_corpus = Jadj o inner(K_1) o inner(d).  THE READING [C]: the geometric identity of c -- the last of the three v605 residuals -- is CLOSED: c is not an unexplained element but the product of the canonical form comparison (polarization x RP metric x real structure) with the unique unimodular integral mirror intertwiner of the realized pair.  Honest scope: F_2 carries the corpus datum Sigma by definition of the translator (nothing else could); the v606 arithmetic negative (cubic field S_3) stands as the reason the cyclotomic route had to fail.  GATE.QGEO does not move; no marker changes.",
      "status": "Exact [E] uniqueness/unimodularity/integrality/involutivity/factorization identities + [C] the closed-identity reading (exact sympy)",
      "canonicalStatus": "C-FACTORIZATION-LANDED: d unique (1-dim), det 1, Z[omega]-integral, Gamma(d)d = 1; c = lambda K_1 d exact with K_1 = J^-1 conj(F_2)^T M^-1; theta_corpus = Jadj o inner(K_1) o inner(d); c geometrically identified; 8/8 checks",
      "dependencies": "QGEO.TRANSLATOR.01; QGEO.MODESEP.01; QGEO.REAL.01; v599; v605; v606",
      "scripts": [
        "v609_translator_factorization.py"
      ],
      "script": "v609_translator_factorization.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 627
    },
    {
      "id": "QGEO.LANDSCAPE.01",
      "claim": "The curve landscape round -- what else the mu3-cover carries (one module, 9 checks, ~1 s; discovery probe curve_landscape_probe.py 9/9, verdict LANDSCAPE-MAPPED).  THE JACOBIAN SPLITS [E chars + C reading]: the three holomorphic differentials of C: y^3 = x^4 - 1 carry (deck, rotation) characters of combined orders (12, 12, 6) (dx/y ~ (w^-1, i), dx/y^2 ~ (w, i), x dx/y^2 ~ (w, -1)); by standard CM theory the Jacobian splits up to isogeny as Jac(C) ~ A_2(CM Q(zeta_12)) x E(CM Q(omega)) -- with Q(zeta_12) = Q(i, sqrt3) THE FIELD GENERATED BY THE DECK AND THE MARK ROTATION TOGETHER, and E: y^3 = u^2 - 1 the x -> -x quotient curve.  THE SPLITTING CONFIRMED BY POINT COUNTS [E]: a_p(C) = a_p(E) EXACTLY for all ten tested primes p != 1 mod 12 (5,7,11,17,19,23,29,31,41,43) -- the zeta_12 surface is supersingular there, as CM theory demands; the surface contributes nonzero traces exactly at p = 1 mod 12 ((13, 37, 61, 73) -> (-8, 4, -20, -32)); and a_p(C) = 0 for every tested p = 2 mod 3 (inert in Q(omega)).  THE INTEGER-SPECTRUM CENSUS [E + honest relativization]: in the declared degree <= 2 J-projector pool, integer characteristic polynomials are COMMON (7894 entries) -- single spectra are CHEAP on the curve; the ANCHOR CLASS ((x+1)^2(x+2), i.e. {1,1,2} up to negation/shift) is the MOST FREQUENT nontrivial family (286 = the maximum), and non-compiler spectra exist ({1,1,3}: 254 entries): the compiler's distinction is the JOINT algebra structure (v600: rank 7, joint traces, saturation 81), not individual characteristic polynomials -- the v598/v599 char-level hits were correctly typed candidates.  NEW NAMED RESOURCES [C]: the zeta_12 CM surface and the omega CM elliptic curve are two canonical objects of the cover not yet used by the program; their periods are classical Beta values -- the named resource for future RP/Hodge rounds.  GATE.QGEO does not move; no marker changes. [2026-08-01 XXIX: the period resource is made concrete -- v611: all five basic periods are exact Beta values on the zeta_12 grid, the rotation acts with the exact characters, Omega_E is the classical omega-CM period, and the two CM factors are period-independent (PSLQ null), as the splitting demands; see QGEO.PERIODS.01.]",
      "status": "Exact [E] characters/point-counts/census + [C] CM-theory reading and named resources (deterministic integer computations; no floats beyond the census prefilter)",
      "canonicalStatus": "LANDSCAPE-MAPPED: differential characters (12,12,6); Jac ~ A_2(zeta_12) x E(omega) point-count-confirmed (10 exact zero-differences; surface traces only at p = 1 mod 12); anchor class most frequent nontrivial integer family (286); single spectra cheap, joint structure the distinction; periods named; 9/9 checks",
      "dependencies": "QGEO.COVER.01; QGEO.DICT.01; QGEO.EMBED.01; v597; v598; v600",
      "scripts": [
        "v610_curve_landscape.py"
      ],
      "script": "v610_curve_landscape.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 628
    },
    {
      "id": "QGEO.PERIODS.01",
      "claim": "The periods round -- the analytic layer of the cover is classical (one module, 6 checks, ~1 s; discovery probe periods_probe.py 6/6, verdict PERIODS-CLASSICAL).  EXACT BETA IDENTITIES [E, symbolic]: the five basic period integrals I_{k,j} = int_0^1 x^k (1-x^4)^{-j/3} dx equal (1/4) B((k+1)/4, 1-j/3) EXACTLY; the Gamma denominators (Gamma(11/12), Gamma(7/12), Gamma(5/12), Gamma(1/12)) all live on the ZETA_12 GRID -- the analytic layer carries the same 12 = |mu4| x |mu3| structure as the algebra.  THE ROTATION ACTS EXACTLY [E, 40-digit certificates]: substituting x -> ix maps the [0,1]-period to the [0,i]-period and multiplies by EXACTLY i^(k+1) -- the v610 differential characters act on actual periods (three cases).  THE OMEGA-CM PERIOD [E, symbolic]: the quotient curve E: y^3 = u^2 - 1 has real period Omega_E = B(1/3,1/6)/2 = Gamma(1/6)Gamma(1/3)/(2 sqrt pi) exactly -- the classical CM period of the omega curve.  CROSS-FACTOR INDEPENDENCE [E, must-fail consistency]: the ratio I_{1,2}/I_{0,2} -- periods of differentials on DIFFERENT CM factors (v610) -- admits NO integer polynomial relation of degree <= 4 with coefficients up to 10^6 at 40 digits (PSLQ null): the zeta_12 surface and the omega curve are period-independent, exactly as the splitting demands.  THE PERIOD DICTIONARY [C]: every cover period is a Beta/Gamma monomial on the 12-grid -- the named analytic resource for the future Hodge/RP period comparisons.  GATE.QGEO does not move; no marker changes. [2026-08-01 XXX: the form-level bridge lands -- v612: J diagonalizes exactly in the rotation eigenbasis and the unique positive (Hodge) direction is the primitive zeta_12 eigenspace; the canonical period normalization (Beta weights) is the named remaining step; see QGEO.POLFRAME.01.] [2026-08-01 XXXII: the canonical period normalization lands -- v613: r = deck o rotation exactly (char(r) = char(omega*M)), the period rows realize the rotation eigenframe, the Hodge weights are explicit Gamma monomials, and J = -h per character (Riemann-Hodge positivity realized by the compiler polarization); v612's named step B5 executed; see QGEO.PERNORM.01.]",
      "status": "Exact [E] symbolic Beta identities + 40-digit certificates + PSLQ-null consistency + [C] the period-dictionary reading",
      "canonicalStatus": "PERIODS-CLASSICAL: five Beta identities exact (zeta_12-grid Gamma denominators); rotation characters act on periods exactly; Omega_E = B(1/3,1/6)/2; cross-factor PSLQ null (splitting-consistent); period dictionary named; 6/6 checks",
      "dependencies": "QGEO.LANDSCAPE.01; QGEO.COVER.01; v597; v610",
      "scripts": [
        "v611_periods.py"
      ],
      "script": "v611_periods.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 629
    },
    {
      "id": "QGEO.POLFRAME.01",
      "claim": "The polarization frame round (one module, 9 checks, ~2 s; discovery probe polarization_frame_probe.py 9/9, verdict POLARIZATION-FRAMED).  THE ROTATION SPECTRUM [E]: the braid rotation r has three SIMPLE eigenvalues = zeta_12 x {1, -1, -i} (e^{i pi/6}, e^{-i 5pi/6}, e^{-i pi/3}), det r = -1 exactly -- the twelfth roots organize the spectrum.  J DIAGONALIZES IN THE ROTATION FRAME [E]: in the left-eigenvector basis W of r the polarization becomes EXACTLY diagonal, (W^dag)^-1 J W^-1 = diag((1+sqrt3)/2, (1-sqrt3)/2, -1) (canonical normalization).  Honest typing: the diagonality is the standard consequence of r being J-unitary with simple unimodular eigenvalues (eigenspaces to distinct eigenvalues are J-orthogonal); the CONTENT is the exact values and the signature distribution.  THE HODGE DIRECTION IS THE ZETA_12 EIGENSPACE [E]: the unique POSITIVE direction of J (signature (1,2), v597/v599) is EXACTLY the eigenspace of the primitive twelfth root e^{i pi/6} -- the h^{1,0} = 1 Hodge line of the omega-sheet sits on the zeta_12 eigenvector of the rotation: the analytic Hodge structure and the finite rotation agree on where positivity lives.  CONSISTENCY [E]: r^dag J r = J re-verified; the product of the diagonal values equals det J / |det W|^2 = 1/2 exactly.  THE READING [C]: with v611 (periods = Beta values on the zeta_12 grid) the analytic bridge is one step from closing -- the canonical period normalization of the eigenvectors (fixing the diagonal weights as explicit Beta monomials) is the named next step.  GATE.QGEO does not move; no marker changes. [2026-08-01 XXXII: the canonical period normalization lands -- v613: r = deck o rotation exactly (char(r) = char(omega*M)), the period rows realize the rotation eigenframe, the Hodge weights are explicit Gamma monomials, and J = -h per character (Riemann-Hodge positivity realized by the compiler polarization); v612's named step B5 executed; see QGEO.PERNORM.01.]",
      "status": "Exact [E] spectrum/diagonalization/signature identities + [C] the bridge reading (exact sympy)",
      "canonicalStatus": "POLARIZATION-FRAMED: r eigenvalues zeta_12 x {1,-1,-i} simple; J exactly diagonal in the rotation frame (values (1+sqrt3)/2, (1-sqrt3)/2, -1); the positive Hodge direction = the primitive zeta_12 eigenspace; product = det J/|det W|^2 = 1/2; period normalization named; 9/9 checks",
      "dependencies": "QGEO.PERIODS.01; QGEO.LANDSCAPE.01; QGEO.REAL.01; v597; v599; v610; v611",
      "scripts": [
        "v612_polarization_frame.py"
      ],
      "script": "v612_polarization_frame.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 630
    },
    {
      "id": "QGEO.PERNORM.01",
      "claim": "The canonical period normalization -- the analytic bridge closes (one module, 19 checks, ~6 s; discovery probe canonical_periods_probe.py 19/19, verdict PERIOD-NORMALIZED; executes v612's named step B5).  DECK-FREE ROTATION [E, 40-digit certificates]: the quarter rotation x -> ix maps the four branch-point segments of y^3 = x^4 - 1 cyclically with NO deck correction (all 12 wrap factors exactly 1); the single omega-sheet homology relation is seg1+seg2+seg3+seg4 = 0; a MUST-FAIL control documents that the raw period rows are NOT r-eigenvectors (naive ansatz falsified, residual 0.42).  THE EXACT DICTIONARY r = deck o rotation [E, exact]: char(r) = char(omega*M) EXACTLY for the induced companion rotation M (M^4 = 1) -- the reduced Burau rotation at t = omega is GL3-conjugate (explicit conjugator G) to the geometric quarter rotation composed with EXACTLY ONE deck step; explains v597's r^4 = omega structurally; the deck power is UNIQUE among the four scalar twists c*M with (cM)^4 = omega.  THE EIGENFRAME REALIZED [E]: the period rows (hol dx/y^2, hol x dx/y^2, ANTIHOL conj(dx/y)) are left eigenvectors of M with characters (i, -1, -i); transported through G they are r-eigenvectors with eigenvalues omega*(i,-1,-i) = spec(r); character match: antihol <-> zeta_12.  THE HODGE WEIGHTS ARE GAMMA MONOMIALS [E]: h0 = (3/4) pi G(1/4)G(1/3)G(5/12)/(G(3/4)G(2/3)G(7/12)), h1 = (3 sqrt3/16 pi) G(1/3)^2 G(1/6)^2, h2 = (3/4) pi G(1/4)G(2/3)G(1/12)/(G(3/4)G(1/3)G(11/12)) (complex Beta/Gelfand-KLT formula, verified to <= 1e-25); omega-sheet signature (2,1), unique NEGATIVE = the antiholomorphic line conj(dx/y).  THE SIGN BRIDGE [E]: all three character-matched sign products are negative -- J = -h PER CHARACTER: the compiler polarization is the NEGATIVE of the analytic Hodge form up to per-line positive scales; J's unique positive line (zeta_12, v612) = h's unique negative line (antihol): Riemann-Hodge positivity is realized by J.  THE RESIDUE [C]: the canonical fork-basis identification (basepoint bookkeeping for G), named, not claimed.  GATE.QGEO does not move; no marker changes. [2026-08-01 XXXIII: the residue F1 is closed at the transport level -- v614: the deck step is DERIVED as the boundary monodromy t^4 = omega (Gauss-Manin transport along the rotation loop; transported periods factorize exactly, the canonical transport matrix is omega*M): the dictionary r ~ omega*M is canonical; see QGEO.TRANSPORT.01.]",
      "status": "Exact [E] dictionary/uniqueness identities + 40-digit certificates + Gamma-monomial closed forms + [C] the fork-basis residue (sympy + mpmath)",
      "canonicalStatus": "PERIOD-NORMALIZED: rotation deck-free on segments (12 wrap factors = 1); r = deck o rotation exactly (char(r) = char(omega*M), deck power unique); period rows realize the eigenframe (antihol <-> zeta_12); Hodge weights = explicit Gamma monomials, signature (2,1); J = -h per character (Riemann-Hodge positivity realized); 19/19 checks",
      "dependencies": "QGEO.POLFRAME.01; QGEO.PERIODS.01; QGEO.LANDSCAPE.01; QGEO.REAL.01; v597; v599; v610; v611; v612",
      "scripts": [
        "v613_canonical_periods.py"
      ],
      "script": "v613_canonical_periods.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 631
    },
    {
      "id": "QGEO.TRANSPORT.01",
      "claim": "The Gauss-Manin transport round -- the deck step IS the boundary monodromy (one module, 12 checks, ~7 s; discovery probe transport_deck_probe.py 12/12, verdict TRANSPORT-CANONICAL; closes v613's named residue F1 at the transport level).  THE FACTORIZATION [E, symbolic]: transporting the segment cycles along the rigid quarter-rotation loop of the four branch points (p_k(tau) = e^{i pi tau/2} p_k, the rotation braid), the discriminant lambda(tau) = e^{2 pi i tau} winds ONCE around 0 and the transported periods factorize EXACTLY: I_m(tau) = e^{i pi tau (m+1)/2} e^{-2 pi i tau j/3} I_m(0) (rotation substitution identity).  THE DECK STEP = BOUNDARY MONODROMY [E]: at tau = 1 the deck factor is omega = t^4 on the t = omega sheet, UNIFORM across all three rows (j = 2 forms AND the conjugated antiholomorphic row), and omega-bar = (omega-bar)^4 on the conjugate sheet -- the deck step is t^4 on EVERY sheet; must-fail controls: t^3 = 1 != omega and t^5 != omega (the exponent 4 = |mu_4| is forced by the puncture count).  THE CANONICAL MATRIX [E]: the transport matrix in the segment basis is EXACTLY omega*M -- no conjugator freedom left: the v613 dictionary r ~ omega*M is CANONICAL (Gauss-Manin transport); (omega*M)^4 = omega*1 re-derives v597's r^4 = omega; char(transport) = char(r) exactly.  CERTIFICATES [E, 25-digit]: direct branch-tracked integration at tau = 1/3, 1/2, 2/3, 1 (dps 80) matches the closed factorization (max rel diff 5e-28); transported seg_1 at tau = 1 equals omega * (static seg_2) to 2.7e-81.  THE READING [C]: the braid fixes the boundary, the rigid rotation does not -- their homological difference is one boundary loop of local-system monodromy (t^4 = omega); the rotation-braid convention (rigid quarter rotation realizes delta = s1 s2 s3) is the typed literature identification, carried by the exact char identity.  GATE.QGEO does not move; no marker changes.",
      "status": "Exact [E] factorization/deck/uniqueness identities + 25-digit transport certificates + [C] the boundary-monodromy reading (sympy + mpmath)",
      "canonicalStatus": "TRANSPORT-CANONICAL: transported periods factorize exactly (rotation substitution identity); deck step = boundary monodromy t^4 = omega on every sheet (exponent 4 = |mu_4| forced); canonical transport matrix = omega*M (v613 dictionary canonical, r^4 = omega re-derived); 25-digit certificates; 12/12 checks",
      "dependencies": "QGEO.PERNORM.01; QGEO.POLFRAME.01; v597; v612; v613",
      "scripts": [
        "v614_transport_deck.py"
      ],
      "script": "v614_transport_deck.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 632
    },
    {
      "id": "WOIT.GAMMA.TOY.01",
      "claim": "The interacting gamma slice at toy level (one module, 11 checks, ~2 s; discovery probe gamma_toy_interacting_probe.py 11/11, verdict GAMMA-TOY-LANDED).  The gamma battery executed ON the RP-surviving interaction class (the alignment bit: delta = pi/2, positive coupling, v534) of the 16-Majorana FK seam toy (v529).  TWO PARENTS [float]: chiral (2-point I + iC, the v519 kernel) and MIRROR (I - iC), both gap 1.  SURVIVOR ANCHOR [float, v534 regression]: chiral interacting Grams PD (37,0,0) at the forced twist eta = +i on all four admissible cuts for g in {1/32, 1/4, 1, 8}.  THE INTERACTING MIRROR FLIP [float, central]: at the SAME eta the mirror state splits sector-exactly on EVERY cut and EVERY g -- even (29,0,0) PD, odd (0,8,0) STRICTLY ND with max odd eigenvalue <= -1.5e-3 (bounded away from zero over the whole grid): every mirror fermion vector has strictly negative OS norm AT THE INTERACTING LEVEL -- in the doubled vector-like system any OS-positive subspace meets the mirror odd sector in {0}: KILL TEST (3) CANNOT FIRE on the surviving interaction class (upgrade of v608 G3 from g = 0 to the full coupling grid).  CONVENTION CONTROL [float]: eta = -i swaps the roles exactly -- the datum is the RELATIVE chirality.  MARK TRANSPORT INTERACTING [exact + float]: alpha_4 maps the mark-A quartet algebra exactly into the mark-B sites (16/16), and both the mark Gram G_A and the transport form T_4 are Hermitian PD (16,0,0) at every g -- KILL TEST (7) DOES NOT FIRE at the interacting toy level.  MULTIPLICITY-FREE [float]: the generalized odd-block transfer spectrum has 8 SIMPLE positive eigenvalues at every g (min pairwise gap 2.3e-5 > 1e-5; honest finding: strong-coupling near-doubling at g = 8 without closing); g = 0 baseline locked (top 1.641438 -- numerically the v524 compressed-clock top: the quarter-turn transfer datum recurs across N).  DEAD-MEMBER CONTROL [float]: (m = 2, s = +, g = 1/2) is indefinite (21,16,0) on its straddled cut while the avoiding cut stays PD -- off the survivor the gamma battery has no RP home: the dynamical bit selection (v534) and the chirality data CO-LOCATE on the same member.  FIREWALL [C]: one toy, one interaction class (FK quartets, flat-band parent), deg <= 2 cut bases + the full 16-dim mark algebra, finite grid -- NO continuum claim, NO statement about A_hol beyond this model class; kill tests (3)/(6)/(7) stay formally live on A_hol; kill test (6) untouched; WOIT.OS.TWISTOR.01 does not move. [2026-08-01 XXXV: the last untouched kill test gets its first slice -- v616: the internal SU(2) separates from the Euclidean rotation group at the kinematic twistor level (rho_tw = right-j, clock = left-u o right-u, deck = left-i, internal SU(2) broken to U(1) by the complex choice with the mark torsor as its mu4); kill test (6)'s kinematic branch does not fire, the dynamical half stays live on A_hol; see WOIT.SU2.KIN.01.] [2026-08-01 XXXVI: the net-dynamics slice lands -- v621: the Klein-Landau semigroup contracts at g > 0 to the mu4-commensurate steps {4, 6, 7} (the survivor interaction is alpha_4-invariant exactly, breaking translation invariance to the quartet stabilizer), and the clock step k = 4 stays a positive transfer ((11,0,0)) over the whole grid: the reconstructed rotation group of the interacting toy is the clock tower; see WOIT.GAMMA.SEMI.01.]",
      "status": "Float-track certificates (tolerances declared) + exact combinatorics + [C] toy fence (numpy/sympy)",
      "canonicalStatus": "GAMMA-TOY-LANDED: interacting mirror flip sector-exact on every cut/coupling (odd strictly ND, bounded away from zero -- kill test (3) cannot fire on the survivor); mark transport Hermitian PD at every g (kill test (7) does not fire); odd transfer spectrum multiplicity-free; dead member has no RP home (bit selection and chirality co-locate); 11/11 checks",
      "dependencies": "WOIT.GAMMA.FREE.01; SEAM.STRADDLE.CONE.01; SEAM.INT.FKTOY.01; WOIT.THETA.FREE.01; WOIT.BETA2.OS.01; v519; v524; v529; v534; v608",
      "scripts": [
        "v615_gamma_toy_interacting.py"
      ],
      "script": "v615_gamma_toy_interacting.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 633
    },
    {
      "id": "WOIT.SU2.KIN.01",
      "claim": "The internal SU(2), kinematic separation at twistor level (one module, 17 checks, <1 s; discovery probe su2_internal_kinematic_probe.py 17/17, verdict SU2-KINEMATIC-SEPARATED; the first executed slice of gamma kill test (6)).  THE QUATERNION SPLIT [E]: on C^4 = H^2 the right multiplications close the (opposite) quaternion algebra exactly, and Woit's Euclidean structure IS an internal direction: rho_tw = right-j VERBATIM (v519/v565 conventions).  THE INTERNAL ALGEBRA [E]: right-sp(1) is a 3-dim real Lie algebra with exact su(2) brackets preserving EVERY fiber of PT -> S^4 (trivial on spacetime; control: left-j moves a generic fiber).  THE SEPARATION [E, the kill-test datum]: the internal su(2) commutes EXACTLY with the full spacetime quaternion action and the two algebras intersect in {0} -- kill test (6)'s KINEMATIC branch does not fire.  THE CLOCK FACTORIZATION [E]: RHO4 = left-u o right-u with u = e^{i pi/4} EXACTLY (must-fail: neither factor alone); the ALE deck is PURELY spacetime-side (DECK4 = left-i) and RHO4^2 = DECK4 o right-i.  THE BREAKING [E]: the complex-linear part of the internal algebra is exactly span{right-i} -- the Euclidean/complex choice breaks the internal SU(2) to U(1), and the v519 mark torsor is exactly {right-mu o sigma_std : mu in mu4}.  OS COMPATIBILITY [E]: sigma_std fixes right-j, inverts the internal charge, and inverts both clock factors simultaneously.  THE READING [C]: kinematic twistor level only -- the dynamical half of kill test (6) stays open and formally live on A_hol; consistent with (not derived) the electroweak reading.  WOIT.OS.TWISTOR.01 does not move.",
      "status": "Exact [E] operator identities (sympy) + [C] the kinematic fence",
      "canonicalStatus": "SU2-KINEMATIC-SEPARATED: rho_tw = right-j; internal su(2) fiber-preserving, commutes with spacetime, intersection {0} (kill test (6) kinematic branch does not fire); clock = left-u o right-u exact; deck = left-i; broken to U(1) with mark torsor = internal mu4; 17/17 checks",
      "dependencies": "WOIT.GAMMA.TOY.01; WOIT.BETA3.DUALITY.01; WOIT.THETA.FREE.01; v519; v565; v608; v615",
      "scripts": [
        "v616_su2_internal_kinematic.py"
      ],
      "script": "v616_su2_internal_kinematic.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 634
    },
    {
      "id": "QGEO.SEAMFORCE.01",
      "claim": "The seam forcing round -- the conformal seam axioms force the mu3-cover y^3 = x^4 - 1 up to the conjugate sheet (one module, 14 checks, <1 s; discovery probe seam_cover_forcing_probe.py 14/14, verdict SEAM-FORCES-COVER; a constructive slice of the bedrock premise QGEO.SYM.01).  Z4-MOBIUS RIGIDITY [E]: any 4-point configuration on P^1 cyclically permuted by an order-4 Mobius map is Mobius-equivalent to mu4 (multiplier forced primitive); the cross-ratio census is the HARMONIC orbit {2, 1/2, -1} (must-fail: 4/3 not in the orbit).  THE WEIGHT CENSUS [E]: among 81 mu3-weight vectors exactly FOUR are clock-equivariant; the mark-EQUIVALENCE demand kills the alternating pair -- weights FORCED uniform, j in {1,2} = the conjugate sheet pair (v613).  THE CURVE [E]: j = 1 gives y^3 = x^4 - 1 EXACTLY; full ramification at infinity (v603); Riemann-Hurwitz genus 3 exactly.  THE SEAM GEOMETRY [E]: marks ON the unit circle, doubling reflection fixes seam and marks; the OS-cut (bond) reflection z -> -i conj(z) permutes the marks by EXACTLY the (k,5-k) pattern of the v599 real structure and meets the circle at the two v519 cut points; the mark-fixing reflection is the site-type axis v519 excludes.  THE READING [C]: the seam axioms (disk + Z4 clock + four equivalent marks + cyclic-3 grading) force the cover; the bedrock residue narrows to (i) the cyclic-3 choice (N_fam = 3, anchored in the E8 glue 240 = 16 x 5 x 3) and (ii) the physical-seam <-> conformal-seam identification.  GATE.QGEO does not move. [2026-08-01 XXXVI: residue (i) is made CONDITIONAL -- v620: the cyclic-N census shows the seam admits exactly N in {3, 5} as primitive-Lorentz orders (N = 6 reduces to N = 3), and the compiler's machine-checked integral constants (27 = 3^3 deck determinant, 81 = 3^4 saturation) pin N = 3 uniquely; the residue moves to N_fam = 3 (E8-anchored); see QGEO.NCENSUS.01.] [2026-08-01 XXXVII: residue (ii) closes at the kernel + dictionary level -- v622: the v519 seam kernel is EXACTLY the discrete NS chiral mode sum (flat band = NS spectrum, Ramond must-fail), the mark-bond midpoints sit exactly at mu4, the group dictionary is exact, and the straddle law is literal geometry (straddled = axis through marks); the remaining continuum residue is the covering-level identification; see QGEO.SEAMID.01.]",
      "status": "Exact [E] rigidity/census/genus identities (sympy) + [C] the residue typing",
      "canonicalStatus": "SEAM-FORCES-COVER: Z4-Mobius rigidity (mu4 forced, harmonic CR orbit); weights forced uniform j in {1,2}; y^3 = x^4-1 exact, genus 3; seam reflections = the v599 (k,5-k) pattern with the v519 cut points; residue = cyclic-3 choice + physical<->conformal identification; 14/14 checks",
      "dependencies": "QGEO.COVER.01; QGEO.REAL.01; QGEO.SEAMMARKS.01; QGEO.PERNORM.01; v597; v599; v603; v613",
      "scripts": [
        "v617_seam_cover_forcing.py"
      ],
      "script": "v617_seam_cover_forcing.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 635
    },
    {
      "id": "QGEO.NCENSUS.01",
      "claim": "The cyclic-N census -- the seam admits N in {3, 5} as primitive-Lorentz orders and the compiler's integral constants pin N = 3 (one module, 9 checks, <1 s; discovery probe cyclic_n_census_probe.py 9/9, verdict N3-PINNED; makes the first v617 bedrock residue CONDITIONAL).  RAMIFICATION [E]: points over infinity = gcd(N, 4): full ramification everywhere iff N in {3, 5} among N = 2..6.  GENUS [E]: Riemann-Hurwitz table g = 1, 3, 3, 6, 7 with complete character decomposition.  SHEET DIMENSIONS [E]: primitive sheets have the compiler's rank 3 exactly for N in {3, 5, 6}, rank 2 for N in {2, 4}.  CHEVALLEY-WEIL [E]: N = 3 has BOTH primitive sheets Lorentz ((1,2)/(2,1), the v599/v613 signature); N = 5 has a Lorentz pair; N = 6 has NO primitive Lorentz sheet (its Lorentz content = the order-3 characters: N = 6 REDUCES to N = 3) -- the seam-admissible orders with a primitive Lorentz sheet are EXACTLY {3, 5}.  THE PINNING [E]: det_Z(1 - deck) = Phi_N(1)^3 = N^3 for prime N: 27 for N = 3 (the machine-checked v597 constant) vs 125 for N = 5; saturation 81 = 3^4 vs 625 = 5^4: N = 3 UNIQUE.  THE READING [C]: the residue moves one level down -- WHY the compiler is 3-adic is N_fam = 3, anchored in the E8 glue (240 = 16 x 5 x 3), named, not derived.  GATE.QGEO does not move.",
      "status": "Exact [E] ramification/genus/Chevalley-Weil/pinning identities (sympy) + [C] the conditional typing",
      "canonicalStatus": "N3-PINNED: seam-admissible primitive-Lorentz orders = {3, 5} exactly (N = 6 reduces to N = 3); compiler constants 27/81 pin N = 3 uniquely (vs 125/625 for N = 5); residue = N_fam = 3 (E8-anchored); 9/9 checks",
      "dependencies": "QGEO.SEAMFORCE.01; QGEO.COVER.01; QGEO.PERNORM.01; v597; v600; v613; v617",
      "scripts": [
        "v620_cyclic_n_census.py"
      ],
      "script": "v620_cyclic_n_census.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 636
    },
    {
      "id": "WOIT.GAMMA.SEMI.01",
      "claim": "The interacting Klein-Landau semigroup -- the clock survives the interaction as a positive transfer exactly on the mu4-commensurate steps (one module, 6 checks, ~2 s; discovery probe interacting_semigroup_probe.py 6/6, verdict CLOCK-SURVIVES-INTERACTING; the gamma net-dynamics slice after v615/v616).  FREE BASELINE [E-float]: at g = 0 all seven steps tau_k (dims 29, 22, 16, 11, 7, 4, 2) are Hermitian with normalized vacuum; even steps PD; odd steps k = 1, 3, 5 indefinite with EXACTLY ZERO one-particle diagonal; k = 7 parity-trivial -- the v524 pattern at N = 16.  THE INTERACTING LAW [MEASURED, central]: at g > 0 exact Hermiticity survives PRECISELY on {4, 6, 7} and is lost on {1, 2, 3, 5} (dev > 0.1): the semigroup CONTRACTS to the mu4-commensurate window.  THE CLOCK ACTS [MEASURED]: k = 4 stays Hermitian PD ((11,0,0)) over g in {1/32..8} -- the mu4 clock is a positive transfer on the interacting net.  THE MECHANISM [E, exact]: the survivor interaction is alpha_4-invariant EXACTLY and not alpha_1/alpha_2 invariant -- translation invariance breaks to the quartet stabilizer {0, 4, 8, 12}.  SUBSTEP TYPING [E/measured]: k = 6 even quartet-contained (protected); k = 5 quartet-contained but odd (NOT protected -- both conditions needed); k = 7 parity-trivial.  THE READING [C]: the reconstructed rotation group of the interacting toy is the CLOCK tower, not the full circle; one toy, one interaction class; gamma proper stays open on A_hol.  WOIT.OS.TWISTOR.01 does not move.",
      "status": "Float-track certificates (declared tolerances) + exact Clifford combinatorics + [C] toy fence",
      "canonicalStatus": "CLOCK-SURVIVES-INTERACTING: free v524 pattern reproduced at N = 16; at g > 0 Hermiticity survives exactly on {4, 6, 7} (mu4-commensurate); clock step PD over the whole grid; mechanism = alpha_4-invariance exact (quartet stabilizer); 6/6 checks",
      "dependencies": "WOIT.GAMMA.TOY.01; WOIT.BETA2.OS.01; SEAM.STRADDLE.CONE.01; v524; v534; v615; v616",
      "scripts": [
        "v621_interacting_semigroup.py"
      ],
      "script": "v621_interacting_semigroup.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 637
    },
    {
      "id": "QGEO.SEAMID.01",
      "claim": "The seam identification round -- the physical seam IS the conformal seam (one module, 10 checks, ~1 s; discovery probe seam_identification_probe.py 10/10, verdict SEAM-IDENTIFIED; closes the second v617 bedrock residue at the kernel + dictionary level).  THE KERNEL IDENTITY [E, exact]: the v519 chiral seam kernel c(d) = (2/N)/sin(pi d/N) for odd d and 0 for even d is EXACTLY the antiperiodic (NS) chiral mode sum on the 16-site circle, for ALL d = 1..15 INCLUDING the even-distance zeros -- the flat-band structure IS the NS spectrum (closed form sum_{j<8} sin((2j+1)x) = sin^2(8x)/sin(x) exact).  MUST-FAIL [E]: the Ramond (periodic) mode sum does NOT reproduce the kernel -- NS is forced, not conventional.  THE SITE MAP [E]: theta_a = (2a+1) pi/16 puts the four mark-bond midpoints EXACTLY at mu4 and no site coincides with a mark (v519 precision (ii), literal geometry).  THE GROUP DICTIONARY [E]: alpha_1 = rotation by 2 pi/16, clock alpha_4 = z -> iz, refl_map(k) = the anti-holomorphic reflection about the axis (k+1) pi/16; cuts {3, 7, 11, 15} map to axes {pi/4, pi/2, 3pi/4, pi}.  THE STRADDLE LAW IS GEOMETRY [E]: the through-mark axes are exactly the v534 STRADDLED cuts {7, 15}, the between-mark axes the avoiding cuts {3, 11}; v617's bond reflection is the cut k = 11 with the v599 (k, 5-k) mark permutation.  THE READING [C]: the remaining continuum residue is the COVERING-level identification (the mu3-cover of the seam double carrying the full interacting theory), named, not claimed.  GATE.QGEO does not move. [2026-08-01 XXXVIII: the covering residue gets its lattice skeleton -- v623: the lifted seam is ONE 48-site NS circle (uniform weights only; the alternating pair disconnects it -- a new independent kill), the lifted clock has exact order 12 with L^4 = deck (r^4 = omega as lattice combinatorics), and the deck-character mode grid is the 12-grid with the untwisted sector = the base seam verbatim; the residue narrows to the interacting/CFT orbifold statement; see QGEO.COVERSEAM.01.]",
      "status": "Exact [E] mode-sum/dictionary identities (sympy) + must-fail control + [C] the covering residue typing",
      "canonicalStatus": "SEAM-IDENTIFIED: v519 kernel = discrete NS mode sum exactly (flat band = NS spectrum; Ramond must-fail); mark-bond midpoints = mu4 exactly; group dictionary exact; straddle law = axis-through-marks geometry; residue = covering-level identification; 10/10 checks",
      "dependencies": "QGEO.SEAMFORCE.01; QGEO.NCENSUS.01; WOIT.THETA.FREE.01; SEAM.STRADDLE.CONE.01; v519; v534; v617; v620",
      "scripts": [
        "v622_seam_identification.py"
      ],
      "script": "v622_seam_identification.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 638
    },
    {
      "id": "QGEO.COVERSEAM.01",
      "claim": "The covered seam round -- the lattice skeleton of the covering-level identification (one module, 9 checks, ~4 s; discovery probe covered_seam_probe.py 9/9, verdict COVERED-SEAM-SKELETON; the named residue of v622, first executed slice).  CONNECTIVITY CENSUS [E]: lifting the 16-site seam to the mu3-cover, the covered seam is ONE 48-site circle for the uniform weights j = 1, 2 and DISCONNECTS into 3 circles for the trivial AND the alternating weights (1,2,1,2) -- a NEW independent kill of the alternating pair (seam connectivity; total monodromy 6 = 0 mod 3).  THE WALK [E]: closes after exactly 48 steps, covers the base three times, crosses marks 12 times.  THE LIFTED CLOCK [E]: L^4 = deck and L^12 = id on ALL 48 points with proper order 12 -- v597's r^4 = omega and v614's boundary-monodromy transport are LATTICE COMBINATORICS on the covered seam.  THE KERNEL [E]: the v622 NS mode-sum identity generalizes verbatim to N = 48 (all d = 1..47).  THE 12-GRID [E]: the NS modes split under the deck into THREE classes of 16 modes with offsets {1/6, 1/2, 5/6}; the offset-1/2 class equals the BASE NS frequencies EXACTLY (the untwisted sector IS the base seam); the fermionic deck lift satisfies deck^3 = -1 (the spin double, resonating with the v510/v519 (-1)^F dichotomy) and the bilinear deck eigenvalue of the twisted pair is exactly omega -- the t = omega/omega-bar sheets live on the H^1/bilinear level.  THE LIFTED MARKS [E]: the 12 crossings are equally spaced and form a SINGLE 12-cycle under the lifted clock.  THE READING [C]: the remaining residue is the interacting/CFT-level orbifold statement (twist-field OPE, the full Z3 orbifold of the seam CFT), named, not claimed.  GATE.QGEO does not move.",
      "status": "Exact [E] combinatorics + mode-sum identities (sympy) + [C] the orbifold residue typing",
      "canonicalStatus": "COVERED-SEAM-SKELETON: one 48-site covered seam (uniform weights; alternating pair disconnects -- independent kill); lifted clock exact order 12 with L^4 = deck (r^4 = omega as lattice combinatorics); N = 48 kernel identity; 12-grid mode split with untwisted = base verbatim, twisted pair = omega on bilinears; 12 marks in a single 12-cycle; 9/9 checks",
      "dependencies": "QGEO.SEAMID.01; QGEO.SEAMFORCE.01; QGEO.NCENSUS.01; QGEO.PERNORM.01; v597; v613; v614; v617; v620; v622",
      "scripts": [
        "v623_covered_seam.py"
      ],
      "script": "v623_covered_seam.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 639
    },
    {
      "id": "EXTREV.LATTICE.01",
      "claim": "The external-review lattice audit (one module, 14 checks, <1 s; discovery probe external_lattice_audit_probe.py 14/14, verdict LATTICE-AUDIT-PASSED; external review 2026-08-02, three claimed derivations machine-checked, one follow-up retyped honestly).  THE FOUR-MARK CLOSURE [E, conditional]: generalizing the glue to D_{d+1} + A_{d-1}, THREE independent selectors each force d = 4 -- the disc-group census (Z_d = Z_4 with cyclic D-side; negative controls d = 3, 5, 8), the even-glue norm equation (d+1)/4 + (d-1)/d = 2 (unique positive solution; no other even target admits an integer d), and the Pythagorean selector (d-1)^2 + d^2 = (d+1)^2 (the (3,4,5) = (cycles, marks, carrier) triple); (1,1,2) is the unique 3-part partition of 4 and c_3 = 1/(2 e_1 pi) = 1/(8 pi): under the named architecture the two-input presentation reduces to ONE boundary degree d = 4 plus pi; honest: conditional on the glue architecture, the residue moves to why the boundary realizes the minimal cyclically-glueable genus-0 divisor.  THE ANCHOR BYTECODE [E]: chi_a = (t-1)^2(t-2) carries (4, 5, 2); p_n = 2 + 2^n with p_{n+2} = 3p_{n+1} - 2p_n; Newton recovers the e_k; p_1 p_2 p_3 = 240, p_4 - p_3 = 8, 240 + 8 = 248 (H-realization v602).  THE LORENTZ CONGRUENCE [E + X]: P^T J_det P = J_fix EXACTLY with det P = -6 -- the prime-front determinant form and the v604 cover polarization lattice are the SAME rational quadratic form, the cover an index-6 sublattice (36 = 6^2, 6 = p_2(a)); the review's follow-up (integer R to B = M^dagger J) is ILL-POSED as stated (B COMPLEX symmetric, det 8 verified); partial positive: the third symmetric-Gauss ratio is an exact Q(omega) square; P-canonicity and the Hodge-chamber question are the named opens.  CITATIONS VERIFIED: Suzuki arXiv 2606.09096 (screw-function Weil framework, Yoshida odd-criterion) and arXiv 2607.24830 (numerical realization) -- the basis of the PRIME.WEIL.OPERATOR contract.  No marker moves; AX.P2.01 keeps its typing; no RH statement. [2026-08-02 XL: the prime-shadow audit lands -- v625: the 'primes appear AFTER E8' reading is exact at the theta level (Theta_E8 = E_4, shell counts 240 sigma_3(n) multiplicative, commuting Hecke channels with the theta as simultaneous eigenvector, L(E_4,s) = zeta(s) zeta(s-3)); the direction of explanation (geometry first, primes as readout) is fixed within the narrative; see PRIME.SHADOW.01.] [2026-08-02 XLI: the Lorentz congruence P^T J_det P = J_fix is additionally machine-proved in Lean (experiments/lean4-carrier-rigidity/TfptCarrier/LorentzCongruence.lean, kernel decide on the explicit integer matrices, lake build green, no sorry); P-canonicity and the operator construction of P executed by v635/v636 (see PRIME.PCANON.01 / PRIME.PCONSTRUCT.01).] [2026-08-02 XLV: look-elsewhere quantification -- v664: the observation 6 = Lorentz-congruence index = p_2(a) carries p_max = 0.169 under the most conservative null model: retyped as an observation WITHOUT statistical weight (the exact congruence P^T J_det P = J_fix itself is [E] and untouched).]  [2026-08-28 CXL: v993 lifts the architecture conditionality at the census level -- full rank-8 ADE with cyclic Z4 both sides selects (D5, A3) uniquely; the D+A split is an OUTPUT.  EXTREV.LATTICE.01's own v624 checks and typing UNCHANGED (they remain the architecture-conditional selectors).]",
      "status": "Exact [E] closure/bytecode/congruence identities + [X] the ill-posed retype + [C] conditional typing (sympy; citations web-verified)",
      "canonicalStatus": "LATTICE-AUDIT-PASSED: four-mark closure forces d = 4 three ways (conditional); bytecode exact (240 + 8 = 248); Lorentz congruence P^T J_det P = J_fix exact (index 6); B-test retyped ill-posed; Suzuki citations verified; 14/14 checks | 2026-08-28 v993: architecture conditionality lifted at census level (this row's v624 checks unchanged)",
      "dependencies": "AX.P2.01; QGEO.EQORDER.01; QGEO.DUALFORMS.01; PRIME.UNIFC.01; v47; v602; v604; v618",
      "scripts": [
        "v624_external_lattice_audit.py"
      ],
      "script": "v624_external_lattice_audit.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 640
    },
    {
      "id": "E8.CODE.01",
      "claim": "The E8 code round (one module, 7 checks, ~2 s; discovery probe e8_code_probe.py 7/7, verdict E8-IS-A-CODE). Construction A on the self-dual extended Hamming code [8,4,4] yields the even unimodular rank-8 lattice = E8 (shells 240/2160 = the v625 theta); every single-bit error is exhaustively correctable (16 x 8, unique nearest codewords); the compiler ties (8 = rank, 4 = d = |mu4|, 16 = half-spinor, 14 = 2 x 7 parabolic) are typed observations. [2026-08-02 XLI: the code reading sharpens to an exact dictionary -- v638: the naive v626 placement itself is DEAD for the family 3-cycle (only 2/30 placements carry both compiler symmetries), and the unique equivariant placement C* is Reed-Muller RM(1,3) on AG(3,2) with information bits one per mu4 pair (3 families + 1 anchor), a syndrome flag carrying the sigma 3-cycle, and decode = projection verbatim; the v626 theorems are placement-independent, the SEMANTICS is not; see E8.CODESEM.01.] [2026-08-02 XLV: v664 types 14 = A_4 = 2 x 7 parabolics and 16 = |C| as FORCED (mathematically determined by the unique self-dual [8,4,4] code -- no free coincidence mass; their value is the theorem, not the number); LEAN ROUND 3: the code layer is machine-certified (HammingCode.lean: self-duality, weight distribution {0:1, 4:14, 8:1}, coset-leader weight distribution {0:1, 1:8, 2:7} with covering radius 2, all by kernel decide on explicit witnesses); lake build green.]",
      "status": "Exact [E] code/lattice identities + [C] typed compiler ties (integer combinatorics; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "E8-IS-A-CODE: Construction A(Hamming [8,4,4]) = E8 exactly; single-bit errors correctable; the error-correction reading is a theorem; no physics claim, no marker moves.",
      "dependencies": "PRIME.SHADOW.01; EXTREV.LATTICE.01; v1; v625",
      "scripts": [
        "v626_e8_code.py"
      ],
      "script": "v626_e8_code.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 641
    },
    {
      "id": "QGEO.ORBCAS.01",
      "claim": "The orbifold Casimir round (one module, 5 checks, ~1 s; discovery probe orbifold_casimir_probe.py 5/5, verdict ORBIFOLD-CASIMIR-EXACT). The covered seam's three deck-mode classes {1/6, 1/2, 5/6} carry EXACTLY the Z3-twist Casimir data: exact closed form E0(nu, N) = -cos(pi(2nu-1)/(2N))/(2 sin(pi/(2N))); finite-size coefficient exactly pi(6nu^2 - 6nu + 1)/12; sector table (1/72, -1/24, 1/72); degenerate twisted pair; ratio -1/3; twist gap 1/18. [2026-08-02 XLI: the named interacting residue executed -- v639: twist-field weights from three exact routes (h_sigma = 1/36), lattice two-point 2 Delta = 2/9 at < 0.11%, crossing exactly symmetry-protected, closed four-point form, OPE data (c1 = 2/9 model-bound); RP validated in the real Z2 class, the complex Z3 pairing (parafermionic Klein twist) typed open; GATE.QGEO does not move; see QGEO.TWISTOPE.01.]",
      "status": "Exact [E] closed forms and expansions (sympy exact; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ORBIFOLD-CASIMIR-EXACT: the kinematic half of the Z3 orbifold statement is exact; the interacting statement (twist-field OPE, crossing, RP) stays the named residue; GATE.QGEO does not move.",
      "dependencies": "QGEO.COVERSEAM.01; QGEO.SEAMID.01; v622; v623",
      "scripts": [
        "v628_orbifold_casimir.py"
      ],
      "script": "v628_orbifold_casimir.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 642
    },
    {
      "id": "E8.INCIDENCE.01",
      "claim": "The root incidence census (one module, 6 checks, ~1 s; discovery probe root_incidence_probe.py 6/6, verdict INCIDENCE-CENSUS-DECIDED). The naive 48 x 5 incidence to the 240 E8 roots is KILLED for the compiler-canonical order-12 element (orbit census 19 x 12 + 3 x 4, not 20 free zeta_12 orbits); positive residue: the mu4 clock alone acts FREELY with EXACTLY 60 orbits (60 = D_start) and the family 3-cycle fixes exactly 12 roots (the lifted marks). [2026-08-02 XLI: the positive residue sharpened two levels -- v633: the 60 free clock orbits ARE the hermitian-unimodular Z[i]-E8 with the ST31 hyperplane system (and the 60 -> 6 quotient-cascade route is killed); v634: G31 = the full unitary stabilizer with sigma = c^4, J = c^9 exact, and the honest sharpening that the compiler clock c is NOT the Springer-regular 12-element (the 19 x 12 + 3 x 4 census reproduced at the group level; the free 20-comb belongs to a different conjugacy class); see E8.ZIQUOTIENT.01 / E8.ST31.01.]",
      "status": "Exact [E] census (Fraction combinatorics) + honest kill + [C] typed observations (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "INCIDENCE-CENSUS-DECIDED: the naive 48 x 5 construction fails at the first hurdle; 48 x 5 = 240 stays an audit fingerprint; 60 = D_start is a free clock-orbit count on the roots.",
      "dependencies": "EXTREV.LATTICE.01; QGEO.COVERSEAM.01; v623; v624; v626",
      "scripts": [
        "v629_root_incidence.py"
      ],
      "script": "v629_root_incidence.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 643
    },
    {
      "id": "FTR.PGL2.01",
      "claim": "The F_transfer PGL2 round (one module, 25 checks, ~1 s; discovery probe ftransfer_pgl2_probe.py 25/25, verdict COMMON-ACTION-DISCRETE / UNIFLOW-CONTINUOUS -- NOT KILLED). The v533 preregistration executed in its LITERAL group form: the discrete transfer path alpha_t (disc -7 norm line, norms 2/4/14/32) carries the parabolic base translation T: alpha -> alpha + 2; the declared v183 Koide reading induces the parabolic step +9 = N_fam^2 and is EXACTLY intertwined with T (the reading map Phi is itself Moebius, Phi.T = g.Phi); CR = 4/3 exact; fit-free Moebius transport forces the F corner 53. Continuous half: the native time-1 maps constructed as explicit PGL2 elements -- g_pole hyperbolic with multiplier (2/3)^6, g_Boltzmann = g_pole IDENTICALLY (v425 single-flow theorem), g_QCD parabolic and EXPLICITLY conjugate to T (translation +7/(2pi) in the 1/alpha coordinate), g_relic = identity (degenerate); the v575 uniflow degeneration and the v578 own-clock equivalence reproduced. Five must-fail controls (raw determinant reading killed: transport predicts -16 != 32, CR 28/25; perturbed reading, elliptic clock, deck, dominant-eigenvalue export all killed).",
      "status": "Exact [E] PGL2/intertwining identities + [C] typed reading (sympy exact; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "COMMON-ACTION-DISCRETE / UNIFLOW-CONTINUOUS -- NOT KILLED: the kill criterion 'four incompatible arithmetic actions' is not met; one parabolic class on the discrete half (exact conjugations), uniflow degeneration on the native continuous half; residual = the external anchor clocks/jets, fenced by CONTRACT.F.01; no marker moves.",
      "dependencies": "CONTRACT.F.01; FTR.DISC7.NORM.01; FTR.CRDISC.01; FTR.CRCONT.01; v533; v571; v575; v578",
      "scripts": [
        "v632_ftransfer_pgl2.py"
      ],
      "script": "v632_ftransfer_pgl2.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 644
    },
    {
      "id": "E8.ZIQUOTIENT.01",
      "claim": "The mu4-quotient round (one module, 17 checks, ~3 s; discovery probe orbit60_quotient_probe.py 17/17, verdicts ZI-E8-QUOTIENT-CONFIRMED + CASCADE-KILLED). The quotient of the 240 E8 roots by the free mu4 clock IS the hermitian-unimodular Z[i]-E8 with the 60-hyperplane system of ST31: H(x,y) = <x,y> + i<x,Jy> has H(v,v) = 2 on all 60 line reps; the Z[i]-span of the reps is the FULL E8 lattice (HNF covolume 256 in doubled coordinates) and a hermitian-UNIMODULAR Z[i]-basis exists among the reps (det Gram = 1); sigma acts on the 60 lines with census 19 x 3 + 3 fixed, the 3 fixed lines being exactly the J-orbits of the 12 sigma-fixed roots (v629 R4 closure); all 60 order-2 unitary reflections preserve the root set (order-4: none), and the line-permutation group has order 11520 = 46080/4 (the ST31 fingerprint: |G31| = 8*12*20*24 = 46080, hyperplanes 60 = 7+11+19+23; ST32 killed by its 40-hyperplane count). THE CASCADE HARD TEST: the 60 -> 6 route on the quotient is KILLED -- the hermitian Gram has rank 4 (no rank reduction), both |H|^2 graphs are REGULAR (any degree rule ill-defined), pair elimination has no canonical maximum (tie counts 14..360), no spectral run on the cascade grid.",
      "status": "Exact [E] quotient/lattice identities + [E, KILL] typed cascade negative (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ZI-E8-QUOTIENT-CONFIRMED + CASCADE-KILLED: the 'complex E8' reading of the 60 free clock orbits is exact (hermitian-unimodular Z[i]-E8, ST31 hyperplane system); the quotient-cascade route 60 -> 6 carries no canonical structure -- a typed negative; no marker moves.",
      "dependencies": "E8.INCIDENCE.01; EXTREV.LATTICE.01; v626; v629",
      "scripts": [
        "v633_orbit60_quotient.py"
      ],
      "script": "v633_orbit60_quotient.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 645
    },
    {
      "id": "E8.ST31.01",
      "claim": "The ST31 structure round (one module, 56 checks, ~10 s; discovery probe st31_structure_probe.py 56/56). G31 = <the 60 unitary reflections> has order 46080 (BFS + independent Schreier-Sims) and IS the FULL unitary stabilizer of the mu4-quotient structure (exact backtracking count over Gram-matching images of a Z[i]-unimodular basis); degrees (8,12,20,24) UNIQUE via Molien among all divisor quadruples with product 46080 (exactly 60 reflections, all order 2); Springer regular elements of orders 8/12/20/24 with centralizers 192/288/20/24; the compiler data embeds canonically: J, sigma, c = J o sigma in G31 with the EXACT clock identities sigma = c^4 and J = c^9 (<c> = <J,sigma> = Z12, the mu4 center is a POWER of the clock); root stabilizer = G(4,2,3) of order 192 (Steinberg, generated by its 15 fixing reflections), line stabilizer = Z4 x G(4,2,3); the computed structure is G31 = (Z4 o 2^{1+4}).Sp4(2). THE NUMEROLOGY KILL: |G31| = |W(D5)| x |W(A3)| = 46080 carries NO structure -- G31 is NOT isomorphic to W(D5) x W(A3) (centers 4 vs 1, abelianizations 2 vs 4, element-order spectra differ), NOT a direct or semidirect product in either order (the join of all small normal closures has order 64: Lagrange kill), and W(D5) does not even EMBED in G31 (its smallest faithful complex rep has dimension 5 > 4); W(A3) = S4 does embed and sigma, c do NOT lie in the glue product. HONEST: the v629 compiler clock c is NOT the zeta12-regular element (census 19 x 12 + 3 x 4, |C(c)| = 72 != 288) -- the free 20 = 240/12 comb belongs to the REGULAR 12-clock of G31, a different conjugacy class. LEAN: the order identities 8*12*20*24 = 46080 = 1920*24 = |W(D5)|*|W(A3)| and 60 = 7+11+19+23 (+ 240 = 4*60) are machine-proved in experiments/lean4-carrier-rigidity/TfptCarrier/G31Orders.lean (lake build green, no sorry). [2026-08-02 XLIII: the degree-24/-20 round lands -- v647: the mu4 center is a POWER of both regular clocks (w^6 = +-J with chi_w = x^4 + i x^2 - 1 exactly primitive-zeta24; u^5 = +-J central), the clock question c = w^2 is KILLED by a parity theorem (census(c) = 19 x 12 + 3 x 4 has odd counts -- impossible even in S_240), and u^4 acts freely with census {5:48}, 48 = 240/5 (typed observation); see E8.ST31DEG.01.] [2026-08-02 XLV: the degree-8 round completes the clock census -- v654: the unified d/4-theorem (every Springer-regular d-clock has x^{d/4} = +-i Id, a generator of mu4; converse exact at d in {8, 20, 24}; the unique exception at d = 12 is the compiler-clock class, chi_c = (x-i)^2(x^2+ix-1) -- the only non-regular clocks whose d/4-th power generates the center); see E8.ST31D8.01.]  [2026-08-03 L: G31 is the holomorphic mu4 shadow at the invariant-ring level -- v690: the restrictions F8|, F12|, F20|, F24| of the W(E8) root power sums to V^{1,0} are a system of BASIC G31 invariants (algebraically independent, degrees Molien-unique, product 46080 = |G31|, Chevalley cited) -- the holomorphic restriction generates the FULL G31 invariant ring; the vanishing half {2,14,18,30} is true but trivially mu4 (honest typing); the Jacobian carries the 60 mirror lines; see E8.QUARTICHALF.01.]",
      "status": "Exact [E] group/Molien/Springer census + [E, KILL] the numerology kill + honest clock typing (Python-only per GATE.WOLFRAM.02; Lean order identities).",
      "canonicalStatus": "ST31-STRUCTURE-DECIDED: G31 = full unitary stabilizer, degrees (8,12,20,24) Molien-unique, sigma = c^4 / J = c^9 exact, real structure (Z4 o 2^{1+4}).Sp4(2) computed; |G31| = |W(D5)|*|W(A3)| stays NUMEROLOGY (killed three ways); the compiler clock is NOT the regular 12-element; no marker moves.",
      "dependencies": "E8.ZIQUOTIENT.01; E8.INCIDENCE.01; v629; v633",
      "scripts": [
        "v634_st31_structure.py"
      ],
      "script": "v634_st31_structure.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 646
    },
    {
      "id": "E8.CODESEM.01",
      "claim": "The code-semantics rounds, kill + stage 2 (one module, 27 checks, ~3 s; discovery probes code_semantics_kill_probe.py 13/13 CODE-SEMANTICS-RELABELLED and code_semantics_stage2_probe.py 27/27 DICTIONARY-SHARP). Stage 1 (kill): the naive v626 placement is DEAD for the family 3-cycle pisig; exactly 2/30 S8-placements of the extended Hamming code carry BOTH compiler symmetries mod 2 (piJ = mu4 in-pair swap, pisig = family pair 3-cycle), exchanged by the anchor transposition (67) -- ONE placement C* up to anchor orientation. Stage 2 (the dictionary, DICTIONARY-SHARP): under binary position labels C* IS Reed-Muller RM(1,3) on AG(3,2) (piJ = translation by 001, pisig affine, the 14 weight-4 supports = the 14 planes; 30 = |S8|/|AGL(3,2)|); NO equivariant 4+4 block split exists (all 6 pair-unions are codeword supports), the UNIQUE minimal Z6-orbit of information sets is the transversal pair 'one bit per mu4 pair, anchor-flipped' (3 family bits + 1 anchor bit); the 12 sigma-fixed roots reduce mod 2 to exactly {0, t_even, t_odd} (4 each); the clock-orbit fibration reproduces 60 = 7 x 4 + 4 x 8 and the sigma census 3 + 19 (the v629 tower 240 -> 60 -> 3+19); SYNDROME SEMANTICS: all 3840 single-coordinate errors sit at squared distance EXACTLY 1 with the tie pair collapsing mod 2; the syndrome space factors as the flag 0 < <q> < F2^4 (q = the universal in-pair bit) with sigma acting on the pair directions as the 3-CYCLE + fixed anchor (P3 = P0+P1+P2) and piJ trivial on the quotient; DECODE = PROJECTION verbatim on the single-error class (3840/3840, exact Fraction arithmetic; boundary regime honest: mismatches need >= 2 rounding errors). KILL CONTROL: {universal in-pair syndrome} = {zero fragmented Z6 orbits} = {both-invariant} = the two C* relabellings -- the equivariance is load-bearing; the whole dictionary dies for every non-equivariant placement. [2026-08-02 XLIII: the reverse reading executed -- v646: the sigma-fixed syndromes carry the anchor decomposition exactly (leader weights (0,1,1,2), norm 6, anchor = family sum; the SELECTION is equivariance-specific, the raw (1,1,2)/norm-6 weights are generic for any [8,4,4] placement), while the bytecode p-sequence p_n = 2 + 2^n has NO code reading (0/11 families; 18/27/81 without any hit -- the bingo is buried); the kill control fires selectively; see E8.RM13REV.01.] [2026-08-02 XLV: look-elsewhere quantification -- v664: the CODESEM number battery (anchor norm 6 et al.) carries p_max = 0.593 under the conservative null model -- the v646 finding quantified: the numeric coincidences of this row are observations without statistical weight; the [E] structure theorems (RM(1,3) dictionary, decode = projection) are untouched.]  [2026-08-03 L: the message space is IDENTIFIED as a lattice congruence -- v689: E8(Z[i])/(1+i) = F2^4 carries this module's RM(1,3) info-bit semantics exactly (240 = 15 x 16 census, zero class provably empty; sigma = c^4 = the family 3-cycle + fixed anchor on the quotient; residual gauge order 18; coordinate block = F1+F2+F3), presentation-independent and with all must-fail controls firing; see E8.GAUSSCODE.01.]",
      "status": "Exact [E] code/dictionary identities + [E, kill] the 30-census control (exact arithmetic; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "DICTIONARY-SHARP: the equivariant placement is RM(1,3) on AG(3,2) with an exact dictionary (information bits one per mu4 pair, syndrome flag with the sigma 3-cycle, decode = projection); only 2/30 placements carry it; residual: the anchor-pair orientation; no physics claim, no marker moves.",
      "dependencies": "E8.CODE.01; E8.INCIDENCE.01; v626; v629",
      "scripts": [
        "v638_code_semantics.py"
      ],
      "script": "v638_code_semantics.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 647
    },
    {
      "id": "QGEO.TWISTOPE.01",
      "claim": "The twist-field/OPE round (one module, 25 checks, ~1 s; discovery probe orbifold_twist_ope_probe.py 25/25, verdict TWIST-OPE-COHERENT-RP-OPEN). The interacting/CFT-level slice of the Z3 orbifold residue (v623/v628): h_sigma = 1/36 from THREE independent exact routes (v628 Casimir coefficient = 2(h - c/24); zeta-function mode sum -zeta(-1,nu)/2; bosonization pair h(1/2-eta)+h(1/2+eta) = eta^2/2), with the convention traps resolved exactly (eta(1-eta)/2 = the Ramond-deficit identity); lattice dictionary Delta = eta^2 = 1/9 = 4 h_sigma; the lattice twist two-point (U(1) string determinants on the half-filled NS seam double; equal-time covariance = half the v519 seam kernel, dev 8e-16) hits 2 Delta = 2/9 at 0.06% (N -> 1536) and 0.11% (within-ring); Ising control (eta = 1/2) at 0.02%; the four-point: exact lattice crossing (gap rotation = w -> 1-w, |G4| invariant at 1.4e-14: symmetry-protected), conformal collapse onto w (< 1%), the exact closed form A^2 [w(1-w)]^{-2 Delta} at < 0.1% after N^{-2/3} extrapolation (deviation identified as the Fisher-Hartwig artifact, ratio 0.637 vs 2^{-2/3}); OPE data: cross-channel block (1-w)^{-2/9} at 0.19%, c1 = 2 Delta = 2/9 model-bound at 0.19% (the exact vertex value 2 eta^2), identity normalization C -> 1 at 0.07%; HONEST OPEN [O]: the model-free c1 extractors bracket 2/9 only within +-15%; the Majorana-level block split not fixed by G4 alone. REFLECTION POSITIVITY DIFFERENTIATED: all mirror Grams Hermitian at machine precision; POSITIVE CONTROL: the real Z2 string class exactly PSD at every N; HONEST NEGATIVE [X-slice]: the naive antiunitary Z3 pairing fails positivity by an N-STABLE amount (-4.5e-3 charged; indefinite already in the continuum) -- the positivity-carrying pairing (parafermionic Klein twist, Jaffe-Pedrocchi-type) is a named OPEN item. [2026-08-02 XLIII: the named open item is RESOLVED at the lattice OS level -- v645: the positivity-carrying pairing IS the parafermionic Klein twist (mirror composed with charge conjugation, omega^{k q} zero-mode phases, eta = (1,1) unique among the 36 scanned pairs), all charged blocks real symmetric and PSD, N-stable 48..384, both axes, full mixed identity+twist+dipole OS Gram PSD; the naive pairing keeps its -4.5e-3 violation as the must-fail control; continuum OS reconstruction/modularity stay typed open; GATE.QGEO does not move; see QGEO.KLEINRP.01.] [2026-08-02 XLV: the orbifold front is COMPLETE at the lattice level -- modular data (v650: nine Z[g,h] vs |theta/eta|^2 < 1%, S measured at N^{-4}, T exact as the Z6 Dehn shift, deck^3 = -1 as exact Z6 trace arithmetic), assembly (v651: H^2(Z3,U(1)) = 0 enumerated, exactly one seam assembly B, T kills the naive Z3 exactly), Arf/GSO hardening (v652: the 6^6 classification leaves {B, C3}, the one-Fock-space pin forces B, the mu6 ladder measures the C3-excised states directly), and the bond-defect premise as a lattice theorem (v653); the [C] remainder is the continuum CFT statement + the seam-identification premise; no marker moves.]",
      "status": "Exact [E] weights/crossing + [E-float] two-/four-point and OPE + [O] typed open items + [X-slice] honest RP negative (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "TWIST-OPE-COHERENT-RP-OPEN: the interacting slice stands at the abelian vertex level (weights 1/36, exponent 2/9, crossing, closed four-point, OPE data); RP validated in the real Z2 class, the complex Z3 pairing typed open (parafermionic Klein twist); GATE.QGEO does not move.",
      "dependencies": "QGEO.ORBCAS.01; QGEO.COVERSEAM.01; QGEO.SEAMID.01; v519; v622; v623; v628",
      "scripts": [
        "v639_twist_ope.py"
      ],
      "script": "v639_twist_ope.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 648
    },
    {
      "id": "QGEO.KLEINRP.01",
      "claim": "Parafermionic Klein-twist reflection positivity of the Z3 twist sector (one module, 17 checks, ~30 s; discovery probe parafermion_klein_rp_probe.py 17/17, verdict KLEIN-RP-POSITIVE).  The named open item of QGEO.TWISTOPE.01 (the naive antiunitary Z3 pairing fails positivity N-stably at -4.5e-3) is RESOLVED at the lattice OS level: the positivity-carrying pairing composes the mirror with CHARGE CONJUGATION -- Theta_K(sigma_k(x)) = eta_k sigma_{3-k}(r(x)) with omega^{k q} zero-mode phases (q = half-position mod 3).  MACHINERY [E]: the dressed-string reality identity (particle-hole I - C = DCD at half filling) makes every Klein Gram entry EXACTLY real (1.4e-15 over 40 mixed configs); for lam = pi the naive and Klein pairings coincide identically (the v639 Z2 control could not distinguish them); the naive glue pins a spurious 2 lam = 4 pi/3 defect on the axis bond, the Klein glue is defect-free.  POSITIVITY [E-float, central]: every charged Klein block is real symmetric AND PSD (worst min/max 2.9e-12), k = 1, 2, BOTH diameter axes, N = 48/96/192/384 (N-stable, no drift); charge blocks coincide exactly.  UNIQUENESS [E]: over the 36 pairs (eta_1, eta_2) in {+-1, +-w, +-w^2}^2 exactly ONE satisfies consistency + Hermiticity + PSD: eta = (1, 1); no phase rescues the naive pairing (the fix is structural).  MUST-FAIL [E]: the naive pairing reproduces the v639 violation (-4.4e-3..-4.5e-3 N-stable); eta = -1 loses PSD, eta = omega breaks Hermiticity.  THE LATTICE OS STATEMENT [E-float]: the FULL mixed Gram over {1, sigma_1, sigma_2, dipoles} with NO sector projection is real symmetric and PSD at N = 48/96 (min/max -5e-17), stable at N = 192.  HONEST TYPING [C]: continuum OS reconstruction, R/fermion-parity sectors (deck^3 = -1), modular invariance, chiral factorization, non-abelian content stay NAMED, not claimed; GATE.QGEO does not move. [2026-08-02 XLV: the modularity/assembly successors land -- v650 (S measured with a clean N^{-4} rate, T exact), v651 (the seam assembly is B, matching this row's Klein phase eta = (1,1)), v652 (Arf hardening; the half-twist tau sector is Klein-PSD too), v653 (bond defect = lattice theorem); the orbifold front is complete at the lattice level; GATE.QGEO does not move.]",
      "status": "Exact [E] reality/uniqueness/must-fail structure + [E-float] PSD Grams (N-stable) + [C] typed continuum remainder (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "KLEIN-RP-POSITIVE: the Z3 twist sector is reflection positive on the seam-double lattice under the parafermionic Klein-twist pairing (mirror x charge conjugation, omega^{k q} phases, eta = (1,1) unique among 36), N-stable 48..384, both axes, including the full mixed OS Gram; the naive pairing keeps its N-stable violation (must-fail); continuum OS/modularity typed open; GATE.QGEO does not move.",
      "dependencies": "QGEO.TWISTOPE.01; QGEO.ORBCAS.01; QGEO.COVERSEAM.01; v519; v622; v623; v628; v639",
      "scripts": [
        "v645_klein_rp.py"
      ],
      "script": "v645_klein_rp.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 649
    },
    {
      "id": "E8.RM13REV.01",
      "claim": "The RM(1,3) reverse reading -- are compiler numbers CODE operations? (one module, 20 checks, ~1 s; discovery probe rm13_reverse_probe.py 20/20; exact integer arithmetic).  An honestly TWO-PART answer.  THE ANCHOR IDENTITY [E, central]: the sigma-fixed syndromes of the equivariant placement C* are EXACTLY {0, s(e6), s(e7), q} with coset-leader weights (0, 1, 1, 2) and norm-square sum 6 = (1,1,2).(1,1,2); the anchor is the family sum in the quotient (P3 = P0+P1+P2); ALL four pair sectors carry (1,1,2)/norm 6 but ONLY the anchor pair is sigma-stable (pointwise fixed) -- the sigma action SELECTS the anchor copy.  HONEST SPLIT [E]: the (1,1,2)/norm-6 weights per pair sector are GENERIC for any [8,4,4] placement (R3.2: the naive placement carries them too); equivariance-specific are ONLY q-universality, the sigma-fixing of the anchor sector, and anchor = family sum (R3.1: all die for the naive placement).  THE BYTECODE NEGATIVE [E, honest]: the 42-entry invariant battery lands the structural anchors exactly (240 = #roots, 8 = n, 248 = roots + n, 60 = roots/4, 12 = #sigma-fixed roots, 6 = anchor norm = #pair-unions) -- but the p-sequence (4, 6, 10, 18) of p_n = 2 + 2^n has NO code reading (0/11 natural families), 27 and 81 have no hit at all, 18 no hit, 10 only as a C*-specific Z6-orbit count: the bingo is buried.  KILL CONTROL [E]: fires selectively -- the orbit censuses (6, 10) and the sigma/anchor selection die for non-equivariant placements, the pure-code invariants and even the sigma-fixed-root COUNT 12 do not (R3.6 honest correction: naive also has 12; only the mod-2 fibration is C*-structure).  No marker moves.",
      "status": "Exact [E] anchor identity + honest [E] generic/specific split + honest negative (no code reading of the p-sequence) + selective kill control (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "RM13-REVERSE-TWO-PART: the anchor decomposition (0,1,1,2)/norm 6 is exact with the sigma action selecting the anchor sector (selection equivariance-specific, weights generic); the bytecode p-sequence has NO code reading (0/11 families, 18/27/81 no hit) -- honest negative, route closed; kill control selective; no marker moves.",
      "dependencies": "E8.CODESEM.01; E8.CODE.01; EXTREV.LATTICE.01; v624; v626; v638",
      "scripts": [
        "v646_rm13_reverse.py"
      ],
      "script": "v646_rm13_reverse.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 650
    },
    {
      "id": "E8.ST31DEG.01",
      "claim": "The ST31 degree-24/-20 round (one module, 25 checks, ~4 s; discovery probe st31_degree24_probe.py 25/25; exact Fraction/Z[i] arithmetic).  THE REGULAR 24-CLOCK [E, structural]: an explicit w with free census {24:10} and |C_G31(w)| = 24 (Springer); on the 60 lines it reads as the free 6-clock {6:10}; EXACT center relations w^12 = -1 and w^6 = J^3 -- the mu4 center is a POWER of the regular 24-clock (mu4 = <w^6>, <w>/mu4 = Z6 = the line 6-clock); chi_w = x^4 + i x^2 - 1 EXACT (Gaussian-integral quartic factor of Phi_24; divides x^24 - 1 and no proper x^d - 1: all four eigenvalues PRIMITIVE zeta24); the half-clock w^2 is the REGULAR 12-element ({12:20}, |C| = 288), NOT the compiler clock.  THE CLOCK KILL [E, parity theorem]: c = w^2 is IMPOSSIBLE for any order-24 w, even in S_240 -- the 12-cycles of any square come in PAIRS from 24-cycles (4-cycles in pairs from 8-cycles) and census(c) = 19 x 12 + 3 x 4 has ODD counts; the full power-census scan over all 46080 elements confirms: the only (census, k) with census(x^k) = census(c) are order-12 elements with census(c) itself and k coprime to 12.  THE 20-DEGREE [E]: regular u with census {20:12}, |C(u)| = 20; u^5 = J^3 CENTRAL (stronger than J-class); u^4 of order 5 acts FREELY with census {5:48} = 240/5 EXACTLY, lines {5:12}, |C(u^4)| = 20; chi_u primitive-zeta20 exact.  HONEST TYPING [C]: the equality 48 = seam sites stays an OBSERVATION (number coincidence, no structural bridge in this module); g_car = 5 has only the census side; no marker moves. [2026-08-02 XLV: v654 completes the degree census (order-8 landscape 10560/4800/480, the unified d/4-theorem, 'free => regular' killed at d = 8 and already at order 4); LEAN ROUND 3: the order statements of the regular words are machine-proved (G31WordOrders.lean: chi_w | X^24 - 1 and chi_u | X^20 - 1 via the conjugate-quartic witnesses chi x conj = Phi_24 / Phi_20, and NO proper-divisor X^d - 1 by companion-matrix kernel decide -- all eigenvalues primitive; W^24 = U^20 = 1), and the combinatorial kernel of the c = w^2 parity kill is certified at the multiset level (SquareParity.lean: the exact count law #{L-cycles of rho^2} = 2 #{2L-cycles of rho} for even L + census_not_square; the permutation-level bridge honestly typed as not formalized); lake build green, 3383 jobs, no sorry.]",
      "status": "Exact [E] structural clock relations + [E] parity-theorem kill + [C] typed observations (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ST31-DEGREE-ANCHORED: the mu4 center is a power of both regular clocks (w^6 = +-J, chi_w = x^4 + i x^2 - 1 primitive-zeta24; u^5 = +-J central); c = w^2 killed by the parity theorem (19/3 odd -- proof-grade); u^4 free with census {5:48}, 48 = 240/5; 48 = seam sites stays a typed observation; no marker moves.",
      "dependencies": "E8.ST31.01; E8.ZIQUOTIENT.01; E8.INCIDENCE.01; v629; v633; v634",
      "scripts": [
        "v647_st31_degree24.py"
      ],
      "script": "v647_st31_degree24.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 651
    },
    {
      "id": "META.DISCIPLINE.01",
      "claim": "The discipline audit (one module, 17 checks, ~200 s; discovery probe discipline_audit_probe.py 17/17, verdict DISCIPLINE-MEASURED).  The suite parses ITSELF and freezes its process-discipline metrics as minimum bars.  [E] D1 LEDGER CENSUS: 715 rows / 710 active (display classes E/C/O/X/AXIOM/OTHER = 169/222/282/37/2/3 at freeze), and 214 rows documenting kills / honest negatives / ill-posed retirements / retypes (keyword census over claim + canonical_status) -- a project that documents its own buried hypotheses is not a confirmation-bias generator.  [E] D2 MUST-FAIL CENSUS: 643 modules (every one carries checks), 6040 static check() call sites (clears the published 5452 counter), 1469 must-fail/kill/scramble keyword occurrences across 233 modules, typing marks [E] x 3579 vs [C] x 1323 -- negative controls are corpus-wide, not decorative.  [E] D3 PRE-REGISTRATION CENSUS: 47 contract rows in the ledger (29 naming kill criteria BEFORE execution; 20 mentions in the contracts paper); executed chains verified in the ledger: FTR.PGL2.01 (v533 -> executed v632), PRIME.WEIL.OPERATOR.01 (kill tests K1-K3 named -> W1 theorem-closed v643), REG.FREEZE.01 (registry frozen 2026-06-09, re-derived each run by v84).  [E] D4 REPRODUCIBILITY: deterministic replay 12/12 on the DECLARED sample v55/v91/v108/v205/v305/v405/v505/v555/v600/v634/v643/v648 (two fresh subprocess runs each, identical output up to timing lines); 396 modules sympy-exact, 77 with explicit seeds.  [E] D5 EXTERNAL ANCHORS: five classical results recomputed here against the literature the suite quotes -- Jacobi 1834 four-square counts (exact, n = 1..40), Construction A [8,4,4] -> E8 with 240 minimal vectors, the Shephard-Todd G31 degrees (8,12,20,24), the Suzuki archimedean constants psi(1/4)/L/A to 25+ digits, and the completed-zeta functional equation (worst rel dev 3e-31).  [E] D6 LEAN LAYER: 54 committed TfptCarrier proof modules (git ls-files; uncommitted sandbox WIP excluded); lake build honestly NOT re-run in-suite (last green build quoted: 3380 jobs, no sorry).  [C] D7 THE LIMIT, honest: this module measures PROCESS discipline, NOT the truth of the physics; typed [C] observations stay observations; the open gates stay open (278 class-O rows incl. SEAM.THEOREM.01, PRIME.WEIL.OPERATOR.01 W2-W4, the QGEO premise family and the GATE.METRIC family); a discipline certificate is a necessary, not a sufficient, condition against numerology.  Read-only; no marker moves.",
      "status": "Exact [E] machine census + deterministic replay + external anchor recomputes + [C] typed limit (read-only meta-module; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "DISCIPLINE-CERTIFIED: frozen minimum bars -- 214 kill/negative rows, 1469 negative-control occurrences in 233 of 643 modules, 47 contract rows (29 with named kill criteria; FTR.PGL2 and the W1 chain verified as executed), replay 12/12 deterministic, 5 classical anchors recomputed, 54 committed Lean modules; the limit typed [C]: process discipline, not physics truth -- the open gates stay open.",
      "dependencies": "REG.FREEZE.01; FTR.PGL2.01; PRIME.WEIL.OPERATOR.01; AUDIT.NULLMC.01; v84; v100",
      "scripts": [
        "v649_discipline_audit.py"
      ],
      "script": "v649_discipline_audit.py",
      "location": "tfpt_3",
      "externalData": "classical literature values (Jacobi 1834; Shephard-Todd 1954; Suzuki arXiv:2606.09096)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 652
    },
    {
      "id": "QGEO.ORBMOD.01",
      "claim": "The modular data of the Z3 orbifold of the seam on the lattice (one module, 17 checks, ~25 s; discovery probe orbifold_modular_probe.py 17/17, verdict ORB-MODULAR-COHERENT).  SECTORS [E]: all nine twisted-sector partition functions Z[g,h] of the seam double (free complex fermion, N x L cylinder) are real positive with exact conjugation degeneracies and match the continuum |theta[g/3,h/3]/eta|^2 at < 1% on ln Z (N = 48; tau = i/2, i, 2i); the Casimir slopes reproduce the v628 coefficients per deck class (C_g/4 = (-1/24, 1/72, 1/72)).  S MEASURED [E-float, central]: modular S-covariance converges with a clean common N^{-4} rate (4.06-4.32 over N = 24/48/96) while the single-lattice deviation falls at N^{-2}: the leading lattice artifact is itself S-covariant and CANCELS in the S-relation.  T EXACT [E]: T is the discrete Dehn twist -- the Z6 sector map (g,h) -> (g, g+h) x (-1)^F at machine precision (3.6e-13); continuum T-phases not resolvable in the real-positive trace convention (honest scope).  CHARACTERS [E]: h_sigma = 1/36 (rel 5.6e-4), parafermionic 1/6 gap (multiplicity 2), omega^{k q} charge weights.  SPIN STRUCTURE [E]: deck^3 = (-1)^F closes exactly as Z6 trace arithmetic (1.4e-13) incl. the cover factorization; positive bookkeeping = Z6 charge projection, ground class c = 0 in every sector.  TYPED [C]: chiral phases, continuum T-phases, assembly (-> v651), continuum OS; GATE.QGEO does not move.",
      "status": "Exact [E] sectors/characters/spin structure + [E-float] measured S-covariance (N^{-4}) + [C] typed continuum remainder (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ORB-MODULAR-COHERENT: nine sectors match |theta/eta|^2 at < 1%, S measured with a clean N^{-4} rate (the N^{-2} artifact is itself S-covariant and cancels), T exact as the Z6 Dehn shift, h_sigma = 1/36 with the 1/6 gap, deck^3 = -1 exact as Z6 trace arithmetic; chiral phases/continuum OS typed open; GATE.QGEO does not move.",
      "dependencies": "QGEO.TWISTOPE.01; QGEO.KLEINRP.01; QGEO.ORBCAS.01; v622; v623; v628; v639; v645",
      "scripts": [
        "v650_orbifold_modular.py"
      ],
      "script": "v650_orbifold_modular.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 653
    },
    {
      "id": "QGEO.ORBASM.01",
      "claim": "The full Z3-orbifold assembly of the seam (one module, 17 checks, ~35 s; discovery probe orbifold_assembly_probe.py 17/17, verdict ORB-ASSEMBLY-SEAM).  COHOMOLOGY [E]: H^2(Z3, U(1)) = 0 by machine enumeration (27 mu3-valued 2-cocycles = 9 coboundaries x 3 classes, every cocycle symmetric) -- NO discrete torsion for a pure Z3 orbifold; residual phase tables are exact charge-class relabelings.  GRID [E]: all 35 nonvanishing Z6 sectors match |theta[gt/6,ht/6]/eta|^2 at < 1% (beyond v650: the odd deck powers).  THE CENSUS [E, central]: of SIX assembly candidates the survivors of {integer spectrum, unique vacuum, S measured clean, T exact} are EXACTLY {B, C3} -- A (the naive Z3) dies by an exact T violation 3.5e-3 (= its charge-3 content: T kills the naive Z3 exactly), C1 by S stalling at 1.7e-3, C2 has no vacuum, A' by exact T violation 4.6e-1 AND the wrong first twisted exponent (5/18, not 1/9).  THE PICK [E/C]: the seam data discriminate B from C3 -- B keeps every measured c = 0 sector ground and the Klein phase eta = (1,1) (v645); C3 excises all odd-sector grounds: THE SEAM ASSEMBLY IS B, the Z6 deck quotient with uniform charge-0 projection (data-backed [C] here; hardened structurally by v652).  MEASURED [E]: the chiral phase e^{-2 pi i a~ b} (spectral-flow-reduced branch).  GATE.QGEO does not move.",
      "status": "Machine [E] cohomology/census + [E-float] S rates and exact T + data-backed [C] B-vs-C3 pick (hardened by v652) (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ORB-ASSEMBLY-SEAM: H^2(Z3,U(1)) = 0 enumerated (no discrete torsion), six candidates -> exactly {B, C3} survive {integer spectrum, vacuum, S, T} (T kills the naive Z3 exactly at 3.5e-3), the seam data pick B (measured c = 0 grounds + Klein eta = (1,1)); chiral phase measured; GATE.QGEO does not move.",
      "dependencies": "QGEO.ORBMOD.01; QGEO.KLEINRP.01; QGEO.TWISTOPE.01; v622; v623; v628; v639; v645",
      "scripts": [
        "v651_orbifold_assembly.py"
      ],
      "script": "v651_orbifold_assembly.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 654
    },
    {
      "id": "QGEO.ORBARF.01",
      "claim": "Arf/GSO hardening of the B-vs-C3 discriminator (one module, 15 checks, ~30 s; discovery probe orbifold_arf_probe.py 15/15, verdict ARF-B-STRUCTURAL).  THE LADDER [E]: the mu6 defect ladder on the seam double measures the C3-excised states DIRECTLY -- exponents 2 Delta = (1/18, 2/9, 1/2) at rel devs < 8e-4 (N = 96..1536 and within-ring), the half-twist (tau) Klein Gram exactly real symmetric and PSD, exact charge-resolved minima (0, 1, 4, 9, 4, 1)/36 = the measured sector grounds.  THE EXCISION VERDICT [E, central]: the measured exponents hit B's c = 0 minima in every sector (< 3%); C3's odd-sector predictions 85/18 and 81/18 exceed the measurements by factors 85.0 and 9.0 -- the defect string is occupation-diagonal, hence exactly charge-neutral: C3 excises states that exist on the lattice seam with positive Klein norm and measured dimension.  THE PIN [E]: the cover circle is NS (sympy closed form); among ALL 6^6 = 46656 sector-character wirings exactly {B, C3} survive the exact lattice degeneracies, and the single-Fock-space requirement forces B.  HONEST: the earlier v639 2/9 read alone is B-vs-C3-BLIND (B and C3 coincide exactly on the even sectors); the new tau datum: OPE exponent 1/18 (rel 2.9e-3).  TYPED [C]: the bond-defect premise (-> v653); GATE.QGEO does not move.",
      "status": "Exact [E] classification/pin/excision data + [C] typed premise remainder (resolved by v653) (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ARF-B-STRUCTURAL: B vs C3 decided structurally -- the 6^6 classification leaves {B, C3}, the one-Fock-space pin forces B, and the mu6 ladder measures the C3-excised states directly (factors 85.0/9.0); the v639 2/9 read is B-vs-C3-blind; new tau datum 1/18; GATE.QGEO does not move.",
      "dependencies": "QGEO.ORBASM.01; QGEO.ORBMOD.01; QGEO.KLEINRP.01; QGEO.TWISTOPE.01; v622; v623; v628; v639",
      "scripts": [
        "v652_orbifold_arf.py"
      ],
      "script": "v652_orbifold_arf.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 655
    },
    {
      "id": "QGEO.BONDDEF.01",
      "claim": "The bond-defect premise upgraded to a lattice THEOREM (one module, 18 checks, ~40 s; discovery probe bond_defect_premise_probe.py 18/18, verdict BOND-DEFECT-CANONICAL).  THE THEOREM [E]: the deck-character blocks of the covered seam ARE the one-bond-defect chains, unitarily and exactly (A_r = sqrt(3) R P_r on all modes, exact in Q(zeta_6), at 16/48 AND 48/144; defect phase = deck character): twist sector = deck boundary condition = bond defect, canonically; D_can = L^4 and D_can^3 = -1 as exact operator identities (the quarter clock fixes the canonical section lift); NS-vs-R bookkeeping exact.  GAUGE + PIN [E]: the defect position is pure gauge (symbolic telescoping theorem in all 16 free bond phases; all positions exactly isospectral); the geometry pins the gauge class -- sheet reduction = one uniform twist unit per mark (cocycle support = the v622 mark set), window boundary at a crossing <=> defect at a mark, clock covariance selects the mark-uniform field.  EXCLUSION CENSUS [E + honest]: every unit-phase configuration with the sector holonomy is gauge-equivalent to the bond defect; the base chain has NO other Z3; site potentials/amplitude defects killed by exact traces, sign defect = the R sector (v628 kill), alternating weights trivialize (v623 kill); an isospectral real NN chain EXISTS (7.8e-16) -- honestly documented: purely spectral data cannot exclude it, but it is no Z3-twist realization and the measured string/correlator data discriminate it (0.26/0.066); the charge-string bridge makes the measured defect ladder bond-defect spectroscopy.  TYPED [C]: the continuum/CFT-level statement + the v622/v623 seam identification; GATE.QGEO does not move.",
      "status": "Lattice theorem [E] (exact unitary equivalence + gauge pin + exclusion census, honest isospectral documentation) + [C] typed continuum remainder (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "BOND-DEFECT-CANONICAL: twist = deck = bond defect as a lattice theorem in Q(zeta_6) (exact unitary equivalence, D_can = L^4, D_can^3 = -1 operator identities), position = gauge + mark pin, alternatives census complete incl. the honest isospectral chain (string data discriminate); continuum CFT + seam identification stay [C]; GATE.QGEO does not move.",
      "dependencies": "QGEO.ORBARF.01; QGEO.ORBASM.01; QGEO.TWISTOPE.01; v622; v623; v628; v639",
      "scripts": [
        "v653_bond_defect.py"
      ],
      "script": "v653_bond_defect.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 656
    },
    {
      "id": "E8.ST31D8.01",
      "claim": "The ST31 degree-8 round + the unified d/4-theorem (one module, 30 checks, ~10 s; discovery probe st31_degree8_probe.py 30/30; exact Fraction/Z[i] arithmetic).  ORDER-8 LANDSCAPE [E]: 10560 order-8 elements, 4800 with free census {8:30}, only 480 with x^2 central; the regular witness has |C| = 192 (Springer), chi_v = (x^2 + i)^2 EXACT (all eigenvalues primitive zeta8, doubled), v^2 a GENERATOR of mu4; HONEST KILL: 'free => regular' is FALSE at d = 8 (4320 free elements with x^2 noncentral; witness chi = Phi_8, |C| = 32) -- and already at order 4 (510 noncentral free 4-elements, none regular).  THE d/4-THEOREM [E, central]: for every degree d in {8, 12, 20, 24} the Springer-regular d-clocks satisfy x^{d/4} = +-i Id, a GENERATOR of mu4, never +-1 (one formula: all exponents of G31 are 3 mod 4); regular sets = 2 Galois classes each with sign split J vs J^3; THE CONVERSE is exact at d in {8, 20, 24} (extras 0) with the UNIQUE exception at d = 12: exactly 1280 extra elements = the classes of the COMPILER CLOCK c and its Galois twin -- chi_c = (x - i)^2 (x^2 + ix - 1) is half the regular-12 Springer block plus a doubled i, all eigenvalues satisfy lambda^3 = -i (exactly why c^3 = J^3 is central): the compiler-clock classes are the ONLY non-regular clocks in all of G31 whose d/4-th power generates the center.  CONTROLS [E]: full order census (max 24); free orders = the divisors > 1 of the degrees; degrees = free orders divisible by 4 above 4 (honest: NOT the maximal free orders).  No marker moves.",
      "status": "Exact [E] census/theorem/controls + honest kill ('free => regular' false) (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "D4-THEOREM: every Springer-regular d-clock has x^{d/4} = +-i Id (generator of mu4; exponents 3 mod 4); converse exact at d in {8, 20, 24}, unique exception at d = 12 = the compiler-clock class (chi_c = (x-i)^2(x^2+ix-1), c^3 = J^3); order-8 landscape complete; 'free => regular' killed; no marker moves.",
      "dependencies": "E8.ST31.01; E8.ST31DEG.01; E8.ZIQUOTIENT.01; v629; v633; v634; v647",
      "scripts": [
        "v654_st31_degree8.py"
      ],
      "script": "v654_st31_degree8.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 657
    },
    {
      "id": "META.LOOKELSEWHERE.01",
      "claim": "The look-elsewhere audit -- the bingo budget of the typed [C] numeric coincidences, quantified (one module, 14 checks, ~2 s; discovery probe look_elsewhere_probe.py 14/14, verdict LOOKELSEWHERE-QUANTIFIED; seed 20260802, 100k MC, exact Poisson-binomial tails).  THE BUDGET [E]: N = 42 structure slots, K = 19 hits against the compiler target list FROZEN before the computation; the 6 killed readings enter as 0-hit trials.  GLOBAL [E]: family-corrected p = 5.08e-3 in the most conservative of three null models (uniform 7.5e-13, divisor-weighted 8.4e-13, size-windowed 5.08e-3); empirical control: 0.2%/0.3% of 2000 size-matched pseudo-target lists reach the real hit count / the real p.  THE TRIAGE [E, the point]: NO single observation is significant on its own -- suggestive (0.01..0.1): 60 = D_start, 12 sigma-fixed roots = 12 marks, the G31 degrees, 48 = 240/5 = Omega_adm; unremarkable (p > 0.1): 6 = p_2(a) (p = 0.169) and the CODESEM number battery (p = 0.593); FORCED: 14 = A_4, 16 = |C| (their value is the theorem, not the number).  RE-TYPES executed as dated ledger notes (EXTREV.LATTICE.01, E8.CODESEM.01, E8.CODE.01).  The substance of the rounds lies in the [E] structure theorems, not in the number bingos; kills confirm the discipline (6 of 12 readings killed).",
      "status": "Exact [E] budget census + MC quantification + executed re-types (read-only meta-module; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "LOOKELSEWHERE-QUANTIFIED: 42 slots / 19 hits / 6 kills, global p = 5.08e-3 conservative (ensemble significant), pseudo-theory control 0.2%/0.3%, NO single observation significant; re-types executed (6 = p_2(a) p = 0.169; CODESEM battery p = 0.593; 14/16 FORCED); the substance is the [E] theorems.",
      "dependencies": "META.DISCIPLINE.01; EXTREV.LATTICE.01; E8.CODE.01; E8.CODESEM.01; E8.INCIDENCE.01; E8.ST31.01; v100; v624; v626; v629; v633; v634; v638; v646; v647",
      "scripts": [
        "v664_look_elsewhere.py"
      ],
      "script": "v664_look_elsewhere.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 658
    },
    {
      "id": "QGEO.ORBOS.01",
      "claim": "The first continuum-OS slice of the seam orbifold B -- the Osterwalder-Schrader reconstruction data measured at the CONVERGENCE level (one module, 19 checks, ~3 s; discovery probe orbifold_continuum_os_probe.py 19/19, verdict ORB-OS-CONTINUUM-SLICE).  MACHINERY [machine]: Toeplitz == mode-sum covariance (1.2e-15); zero-mode-dressed Klein reality for the full mu6 angle family (5.4e-15); the NEW Euclidean-time machinery validated (NS bond Hamiltonian == eps6 at 6.7e-16; G(tau) via thin-QR-stabilized determinant ratio: == 1 identically at lam = 0 and at tau = 0, matches the naive complex determinant at 1.7e-13, det(V+ U_D V) == covariance-route string at 6.0e-15); ONE fitted constant (the non-universal amplitude A^2, N = 1536 vs 768 consistent at 5.7e-6).  CONVERGENCE [E, central]: at fixed continuum data over N = 48..384 all six sigma-channel observables share the uniform raw rate 0.659 (spread 0.020) = the predicted Fisher-Hartwig branch exponent rho_sigma = 2/3; the tau-channel contrast measures 1.298 vs rho_tau = 4/3 (the channel law rho = 2(1-beta)^2 - 2 beta^2: the rates are UNDERSTOOD, not fitted; the honest answer to 'all rates >= 1?' is NO -- the leading artifact is the smooth FH branch, the same for every observable); after removing the ONE known FH term the residuals sit at the N^{-4/3} level; the extrapolated limits hit the CFT chord/cross-ratio values at < 0.05% (devs 3.9e-4 .. 5.1e-5); the epsilon/current channel is EXACT on the lattice (8.9e-15 -- no artifact at half filling).  RP IN THE LIMIT [central]: the correlation-normalized Klein Grams are PSD at every N (worst floors 2.9e-12 / 4.3e-13 / 1.3e-12) AND the resolvable spectra converge with lambda_min extrapolations bounded away from zero -- sigma 1.357e-3 (margin 7x the uncertainty band), tau 2.800e-4 (margin 358x), mixed OS Gram {1, sigma_1, sigma_2} 1.337e-3 (margin 31x; the dipole-extended mixed Gram PSD too); must-fail: eta = -1 flips the sigma Klein Gram negative definite (floor -1.00).  CLUSTER [E]: space -- the free-exponent chord fit at N = 1536 gives 2 Delta = 0.222220 vs 2/9 at 8.1e-6 (the chord/sin form beats the plain power law by 5 orders: the compact two-point IS the tau = 0 slice of the cylinder cosh form), and the N-extrapolated connected ratio equals the exact CFT law (1-w)^{-2/9} - 1 pointwise < 2%, falling like w^1.017; Euclidean time -- the connected C(tau) decays exponentially with the fitted gap = the exact transfer-matrix neutral gap at < 0.5% on BOTH lattices (Delta_exch = 0.99933 vs 1: the epsilon/current level IS the gap), the cosh/cylinder form holds pointwise (FH-extrapolated ratio 1 within 2.2e-3), the amplitude FH-extrapolates to 0.9977x the closed form.  CHARACTERS [E]: h_sigma -> 1/36 and Delta_sigma -> 1/9 at 1.7e-12, the charged level 5/18 at 6.8e-13, the neutral level 4/9 at 3.9e-13 -- exact mode sums with the measured N^{-2} rate (2.001/2.000/2.000).  TYPED [C]: OS axioms E0 (temperedness), E2 (reflection positivity), E3 (hermiticity), E4 (cluster) are now MEASURED at the convergence level; E1 (Euclidean invariance) is exact only for the discrete lattice symmetries -- full rotation invariance is EMERGENT (cross-ratio collapse + the N^{-4} S-covariance of v650); WHAT REMAINS for the formal limit: uniform bounds over ALL point configurations, tightness/equicontinuity, the GNS limit Hilbert space, operator (not Gram-entry) convergence, the R/parity sectors in the OS frame, and the assembled orbifold-B correlators beyond the twist blocks -- named, not claimed; GATE.QGEO does NOT move, no marker change.",
      "status": "Machine-exact wiring + Exact [E] convergence/cluster/character slices + [MEASURED] RP-limit extrapolations with must-fail control + [C] honest typing of the formal-limit remainder (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ORB-OS-CONTINUUM-SLICE: correlator convergence with understood FH rates (2/3 sigma, 4/3 tau channel law; limits < 0.05% at CFT after one FH term), RP survives the limit (lambda_min extrapolations bounded away from zero, margins 7x..358x, must-fail flips), the cluster property exact in space and Euclidean time (2 Delta = 0.222220 vs 2/9 at 8e-6; time gap = the epsilon level), characters at 1e-12 with the measured N^-2 rate; E0/E2/E3/E4 measured at the convergence level, E1 discrete; the formal limit (GNS, uniform bounds, operator convergence) is the named remainder; GATE.QGEO does not move.",
      "dependencies": "QGEO.ORBMOD.01; QGEO.ORBASM.01; QGEO.ORBARF.01; QGEO.BONDDEF.01; QGEO.KLEINRP.01; QGEO.TWISTOPE.01; GATE.QGEO.01; v622; v623; v639; v645; v650",
      "scripts": [
        "v679_orbifold_continuum_os.py"
      ],
      "script": "v679_orbifold_continuum_os.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 659
    },
    {
      "id": "E8.GAUSSCODE.01",
      "claim": "The Gaussian code bridge (review contract E8.GAUSSIAN.CODE.01) -- E8(Z[i])/(1+i) IS the information space of RM(1,3): the mu4 complex structure J makes E8 a rank-4 Z[i]-lattice L; reduction modulo the ramified Gaussian prime (1+i) gives L/(1+i)L = F2^4 EXACTLY (one module, 26 checks, ~3 s, exact integer/Fraction arithmetic, sympy only for the Smith normal form; discovery probe gaussian_code_bridge_probe.py 26/26, verdict GAUSSIAN-CODE-BRIDGE-EXACT).  THE QUOTIENT [E]: SNF exact; standard-model cross-check: basis |det| = 256 = 2^8 (doubled coordinates), SNF [1,1,1,1,2,2,2,2] -- presentation-independent.  THE CENSUS [E]: 240 roots fall 15 x 16 over the nontrivial classes (the zero class PROVABLY EMPTY); each class = a union of 4 of the 60 G31 lines (60 = 15 x 4).  THE SEMANTICS [E]: sigma = c^4 acts on the quotient as the RM(1,3) family permutation (3-cycle F1 -> F2 -> F3 + fixed anchor, exactly the v638 info-bit action); residual gauge = the order-18 centralizer of the family 3-cycle in GL(4, F2); the coordinate block = the sum-of-families label F1 + F2 + F3.  MUST-FAIL CONTROLS [E, kill]: the naive placement keeps the 15 x 16 census (a property of (E8, J) alone) but the sigma-bit semantics becomes UNDEFINED (witness codeword lift off the lattice); a non-piJ placement has NO Z[i]-module; Z[i]^4 trivializes (16 minimal vectors, 11 classes empty).  No marker move.  [2026-08-03 LI: formally verified -- the algebraic core (Z[i]-module + SNF certificate with unimodular P/Q both directions, the 240-root census 15 x 16 with the zero class provably empty, mu4 stability, the sigma = order-3 family action with coordinate block = F1+F2+F3) is kernel-checked in Lean 4 (experiments/lean4-carrier-rigidity/TfptCarrier/GaussianCodeBridge.lean, lake build green, no sorry / native_decide).]",
      "status": "Exact [E] SNF/census/semantics identities + [E, kill] three must-fail controls (exact arithmetic; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "GAUSSIAN-CODE-BRIDGE-EXACT: E8(Z[i])/(1+i) = F2^4 (SNF exact, presentation-independent), 240 = 15 x 16 census (zero class provably empty), 60 = 15 x 4, sigma = RM(1,3) family permutation (18 equivariant gauges), coordinate block = F1+F2+F3; all must-fail controls fire (naive placement, non-piJ placement, Z[i]^4).",
      "dependencies": "E8.CODE.01; E8.CODESEM.01; E8.ST31.01; E8.ZIQUOTIENT.01; v626; v634; v638",
      "scripts": [
        "v689_gaussian_code_bridge.py"
      ],
      "script": "v689_gaussian_code_bridge.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 660
    },
    {
      "id": "E8.QUARTICHALF.01",
      "claim": "The quartic invariant restriction (review contract E8.QUARTIC.HALF.01) -- is G31 the holomorphic mu4 shadow of the E8 Coxeter compiler?  ALIVE, with the honest split (one module, 22 checks, ~1 s, exact Gaussian-integer/Fraction arithmetic, no floats; discovery probe quartic_half_probe.py 22/22, verdict QUARTIC-HALF-ALIVE).  THE VANISHING HALF [PROOF, typed honest]: {2,14,18,30} is TRUE but TRIVIAL -- W(J alpha) = -i W(alpha) for all 240 roots => every J-orbit contributes t^d (1 + (-i)^d + (-1)^d + i^d) = 0 for every d != 0 mod 4 (holds for 6, 10, 22, 26 too: mu4 orbit factor, NOT degree-list-specific); degrees 4/6 vanish on ANY q-isotropic subspace (f_4 = 576 q^2, f_6 = 1920 q^3 symbolically); F_4| has all 35 monomial coefficients = 0 exactly (the first G31-consistency point: mod-4 alone would allow F_4 != 0).  THE LOAD-BEARING CONTENT [E, central]: F_8|, F_12|, F_20|, F_24| are G31-invariant (60/60 hermitian reflections + clock generators, exact), ALGEBRAICALLY INDEPENDENT (exact Jacobian != 0 off the mirrors), degrees (8,12,20,24) Molien-unique with product 46080 = |G31| => by Chevalley [cited] a system of BASIC INVARIANTS: the holomorphic restriction of the W(E8) invariant ring generates the FULL G31 invariant ring; the Jacobian carries the 60 mirror lines (deg Jac = 60 = sum(d_i - 1), on-mirror vanishing point-verified 60/60; the full factorization = cited standard theorem, NAMED residual).  MUST-FAIL CONTROLS [E, kill]: J' = RJR^T (orthogonal, NOT root-preserving) breaks the informative half {14,18,22,26,30} while the trivial isotropy layers persist; the root-preserving J'' keeps the full pattern (the theorem is about the W(E8)-conjugacy class of compiler-compatible J); a random non-isotropic 4-space breaks even F_2; honest negative: signed-permutation J-conjugates canNOT serve as controls (they stay in W(D8)).  No marker move.  [2026-08-03 LI: formally verified -- the mu4-orbit vanishing mechanism (orbit factor 1 + (-i)^d + (-1)^d + i^d = 0 iff d != 0 mod 4, per-pair weight law, the full vanishing half incl. the trivial extras) and the quartic power sum Sum <alpha,x>^4 = 576 (Sum x_i^2)^2 over the explicit 240 roots (+ holomorphic isotropy) are kernel-checked in Lean 4 (experiments/lean4-carrier-rigidity/TfptCarrier/QuarticHalf.lean, lake build green, no sorry / native_decide); the Chevalley basic-invariant step remains CITED, not formalized.]",
      "status": "Symbolic [PROOF] mu4-orbit vanishing + [E] G31 invariance/independence/mirror census + [E, kill] complex-structure controls + [cited] Chevalley basic-invariant theorem (exact arithmetic; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "QUARTIC-HALF-ALIVE: the vanishing half {2,14,18,30} proven symbolically but TRIVIAL (mu4 orbit factor, honest typing); the substance: F8/F12/F20/F24 algebraically independent G31 basic invariants (holomorphic restriction generates the FULL G31 ring, Chevalley cited); Jacobian carries the 60 mirrors (60/60 point-verified); controls fire exactly as demanded.",
      "dependencies": "E8.ST31.01; E8.ST31DEG.01; v634; v647; v654",
      "scripts": [
        "v690_quartic_half.py"
      ],
      "script": "v690_quartic_half.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "Chevalley Amer. J. Math. 77 (1955) (basic invariants, cited); Molien series (v634)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 661
    },
    {
      "id": "E8.GAUSSIAN.CODE.01",
      "claim": "Review contract (external review, theorem candidate, registered 2026-08-03): E8(Z[i])/(1+i) is the information space of RM(1,3) -- the mu4 complex structure J makes E8 a rank-4 Z[i]-lattice; the (1+i)-quotient is F2^4; the 240 roots fall 15 x 16; sigma acts as the RM(1,3) family permutation.  FIRST SLICE EXECUTED by v689 (26/26, GAUSSIAN-CODE-BRIDGE-EXACT): the full 6-step census verbatim, exact arithmetic, all must-fail controls firing.  REMAINING: the paper-level theorem writeup (the census IS the proof skeleton; the residual gauge order 18 documented).  NOTE (2026-08-04, v736/v737 -- marker UNCHANGED): the ORBIT PACKET lemma is machine-closed [E] (Burnside 7 = (15+3+3)/3, the canonical 5 x 48 split, the (1,1,2) anchor from the fixed-syndrome leader weights (0,1,1,2), the q = 16 normal form 240 = 16*15 / 248 = 240+8; P1/P2 typed as EXACT-INTERNAL-RECONSTRUCTION behind the circularity fence; companion Lean module OrbitPacket.lean, 34 theorems kernel-checked), and the seam-48 intertwiner sharpening is DEAD (v737: incidence distribution 0:96/1:64/2:64/3:16 instead of == 1, representative-independent over all 2304 order-5 elements) -- 240 = 48 x 5 remains a factorization WITHOUT coordinatization.  NOTE (2026-08-05, v782/v783 -- marker UNCHANGED): the code bridge acquires its operational identification -- the 60 Gaussian lines ARE the 60 two-qubit stabilizer rays entry-exactly over Z[i] with U G31 U^-1 an index-2 subgroup of the Clifford matrix group (classes = the 15 Pauli contexts, spreads = the MUB pentads; the index bit is the sqrt2/Hadamard Galois bit, NOT J <-> -J; v783), and the 240 = 16 x 15 transition reading is settled at TORSOR level (each class fiber a canonical v_perp pair-torsor; the frozen predicate reproduces all 57600 inner products; NO deck-equivariant 16-state labeling exists -- proof-grade obstruction; v782).",
      "status": "Review contract; first slice machine-closed [E] (v689); remaining: theorem writeup.",
      "canonicalStatus": "EXECUTED (first slice): v689 closes the census + semantics exactly (26/26); remaining: theorem writeup.",
      "dependencies": "E8.GAUSSCODE.01; E8.CODESEM.01; E8.CODE.01",
      "scripts": [],
      "script": "-",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 662
    },
    {
      "id": "E8.QUARTIC.HALF.01",
      "claim": "Review contract (external review, theorem candidate, registered 2026-08-03): restricting the W(E8) invariant root power sums to the holomorphic mu4 eigenspace V^{1,0} kills exactly the half {2,14,18,30} and keeps {8,12,20,24} = Deg(G31) -- 'G31 is the holomorphic mu4 shadow of the E8 compiler'.  FIRST SLICE EXECUTED by v690 (22/22, QUARTIC-HALF-ALIVE) WITH the honest correction: the vanishing half is TRUE but TRIVIAL (mu4 orbit factor -- holds for all d != 0 mod 4, not degree-list-specific); the substance is F8/F12/F20/F24 as a system of basic G31 invariants (Chevalley cited; the holomorphic restriction generates the FULL G31 ring).  REMAINING: the symbolic degree-60 Jacobian factorization (cited standard theorem, named residual) and the paper-level writeup.",
      "status": "Review contract; first slice machine-closed [E]/[PROOF] with honest typing correction (v690); remaining: Jacobian factorization symbolics + writeup.",
      "canonicalStatus": "EXECUTED (first slice): v690 proves the vanishing half (trivial, honestly typed) and establishes the basic-invariant substance; remaining: symbolic Jacobian factorization + writeup.",
      "dependencies": "E8.QUARTICHALF.01; E8.ST31.01",
      "scripts": [],
      "script": "-",
      "location": "tfpt_research_contracts",
      "externalData": "Chevalley Amer. J. Math. 77 (1955)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 663
    },
    {
      "id": "E8.ORBIT60.01",
      "claim": "The ORBIT60 census -- the 60 mu4-lines of E8 as a combinatorial object: quotient graph, sigma/clock/Coxeter dynamics on the ORBIT SPACE, and a STRICT park test for the cascade 60 -> 6 (one module, 18 checks, ~1 s; discovery probe orbit60_census_probe.py 18/18, verdict ORBIT60-PARKED).  THE FIVE ROUTES, all adjudicated: (i) degree rules DEAD (all quotient graphs regular -- no distinguished pair sequence from incidence degrees); (ii) sigma structure DEAD (sigma-orbit sizes {3,1} on the lines, no 2-blocks); (iii) THE INVOLUTION WITNESS DOES NOT EXIST: the full involution census of the line group LG (order 11520; 375 involutions, 21 conjugacy classes) shows NO element of cycle type 2^27 1^6 (fixed-line census {0: 120, 4: 180, 12: 15, 16: 60}) -- the one-shot group-action carrier of the cascade is absent; (iv) reflection/Coxeter DEAD: W(E8) Coxeter dynamics does not descend (reflection cycle types on the lines all equal, no distinguished element with 27 disjoint 2-blocks); (v) the MUST-FAIL numerology control fires as designed: rule-free lexicographic pair removal reproduces the counting 60 - 2n in exactly 27 steps and > 10^70 equally 'successful' sequences exist -- the counting carries no structural information (v633 Q8.2 reproduced).  A5 one-liner (as instructed, no expedition): order-5 elements exist (2304) but <sigma, sample order-5> has order 960 -- no visible natural A5 action attached to the compiler data.  PARKED per the review protocol; the cascade stays a numerological observation, not a structure.  Deterministic (fixed seed for the generating-set search only).",
      "status": "Honest [E, park] five-route adjudication with the involution-witness census and the must-fail numerology control (exact group computations; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ORBIT60-PARKED: all canonical cascade routes dead; no involution of cycle type 2^27 1^6 in the line group (order 11520; 375 involutions, 21 classes); W(E8) Coxeter does not descend; no A5 signature (order-5 elements exist but nothing natural); the must-fail numerology control fires -- parked per the review protocol.",
      "dependencies": "E8.ST31.01; v629; v633; v634",
      "scripts": [
        "v700_orbit60_census.py"
      ],
      "script": "v700_orbit60_census.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 664
    },
    {
      "id": "QGEO.DECKSECTOR.01",
      "claim": "TFPT scope of original v705: only D1 NS-circle clock/deck finite algebra, 48 modes with three 16-mode sectors. The prime/digamma application is maintained in Prime Front.",
      "status": "Exact [E] deck-sector trace identity (3.4e-16, scalar forced 1, must-fail x82); deterministic (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "The three digamma channels ARE the three deck sectors of the 48-site NS lift (exact tower traces, dev 3.4e-16, scalar forced 1, must-fail x82) -- the geometry anchor of the chain arc; GATE.QGEO untouched; no RH statement.",
      "dependencies": "PRIME.KEYSTONE.01; QGEO.ORBIFOLD.01; v623; v628",
      "scripts": [
        "v705_chain_deck_sector.py"
      ],
      "script": "v705_chain_deck_sector.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 665
    },
    {
      "id": "QGEO.CARLIMIT.01",
      "claim": "QGEO-CAR -- the quasi-free CAR route to the formal orbifold continuum limit (GATE.QGEO), first slice (one module, 22 checks, ~10 s; discovery probe qgeo_car_continuum_probe.py 22/22, verdict QGEO-CAR-RATES-SUMMABLE).  THE STRATEGY EXECUTED: not measure compactness -- the orbifold correlators are DETERMINANTS of a one-particle covariance, so the limit runs through SCHATTEN-NORM control plus Araki's quasi-free CAR theory.  THE MEASUREMENTS: all 6+1 deck/twist sector covariances AND all dressed twist channels are Schatten/shadow-norm CAUCHY on eps-separated configurations (N up to 3072 / 6144), with geometrically summable rates; the FH renormalization is FIT-FREE LATTICE-DERIVED (Barnes G, deviation 8.8e-7); the RP cone margins do NOT degenerate along the ladder; all four kill criteria are clear.  THE REMAINDER NAMED: the majorant lemma (L1)-(L3) is stated with constants and the CLASSICAL proof route -- named, not claimed: GATE.QGEO does NOT move.  Deterministic (no RNG); self-contained.  No marker move.",
      "status": "[MEASURED] Schatten-norm Cauchy rates (6+1 channels, geometrically summable) + [E] lattice-derived FH renormalization (Barnes G, 8.8e-7) + named (L1)-(L3) majorant lemma; deterministic (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "QGEO-CAR-RATES-SUMMABLE: all 6+1 channels Schatten-norm Cauchy (N up to 3072/6144), FH renormalization fit-free lattice-derived (Barnes G 8.8e-7), RP cone stable, kill criteria clear; majorant lemma (L1)-(L3) named with constants + classical proof route; GATE.QGEO does NOT move.",
      "dependencies": "GATE.QGEO.01; QGEO.CONEDYN.01",
      "scripts": [
        "v715_qgeo_car_continuum.py"
      ],
      "script": "v715_qgeo_car_continuum.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 666
    },
    {
      "id": "GNET.RAMIFIED.01",
      "claim": "PHYS moonshot slice (T2 / G_net) -- is the Ramond projection (1+i)-adic? (one module, 9 checks, ~1 s; discovery probe phys_ramified_ns_r_probe.py 9/9, verdict RAMIFIED-SEES-NSR).  THE THEOREM CHECK: on the explicit even-coordinate E8 with the integral complex structure J (pair rotation, J^2 = -I, orthogonal, lattice-stable -- a concrete witness of the v634/v689 Gaussian E8), (1+i)L is CONTAINED in the even/NS sublattice (verified on a Z-basis), so the SO(16) coordinate-parity functional (E8 roots = 112 integer/NS + 128 spinor/R, the GATE.METRIC.10 census) FACTORS through the quotient E8(Z[i])/(1+i) = F2^4 (v689).  PURITY: the 240 roots fall into exactly 15 classes x 16 (re-derived in this frame), every class purely NS or purely R (7 NS + 8 R), all classes mu4-stable, and parity is a HOMOMORPHISM F2^4 -> F2: the NS/R grading IS the parity character of the ramified quotient -- the Ramond projection is (1+i)-adic, the first exact witness for the groupoid reading of G_net (the ONE ramified edge of the Z[i]-E8 Hecke tower is the Gaussian prime (1+i)).  CONTROLS FIRE: the trivialization Z[i]^4 = Z^8 does NOT show the 15x16 pattern (4 classes); scrambled NS/R labels BREAK purity.  THE CONTRACT (stated with v726): the seam-Calderon inclusion A = B^{mu4} in B (index 4, the GATE.METRIC.08 Q-system) is the ramified-place structure of the groupoid -- kills: Frobenius mismatch Q-system vs (1+i)-correspondence convolution; index != 4 on the state-preserving ladder; dressing incompatible with the sigma descent.  Exact integer arithmetic, no fit, no external tables.  No marker move (GATE.METRIC.10 stays as typed).",
      "status": "Exact [E] integer-arithmetic slice ((1+i)L in the even sublattice on a Z-basis; 15x16 purity census; parity homomorphism) with both controls firing; contract half typed OPEN (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "RAMIFIED-SEES-NSR: the NS/R grading IS the parity character of E8(Z[i])/(1+i) = F2^4 (exact, controls fire); the Ramond projection is (1+i)-adic -- first exact witness for the groupoid reading of G_net; the Q-system identification stays the OPEN contract half (kills registered); no status move.",
      "dependencies": "GATE.METRIC.10; GATE.METRIC.08; E8.GAUSSCODE.01; PRIME.MOONSHOT.01",
      "scripts": [
        "v722_phys_ramified_ns_r.py"
      ],
      "script": "v722_phys_ramified_ns_r.py",
      "location": "tfpt_3;tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 667
    },
    {
      "id": "FTRANSFER.STT.01",
      "claim": "PHYS moonshot slice (T3 / F_transfer) -- can thermal time replace the external clock contract?  The conjugacy-class KILL test (one module, 11 checks, ~1 s; discovery probe phys_modular_clock_probe.py 11/11, verdict STT-KILLED + INTERNAL-PAIR-CONFIRMED).  STRONG THERMAL TIME (Connes-style: ONE modular flow sigma_t and four CONSTANT clock conversions t_k = c_k t) requires all four native time-1 maps in modular-exponential conjugacy classes -- a modular generator is (an image of) a self-adjoint operator, hence DIAGONALIZABLE, never parabolic.  THE KILL: the F_QCD 1-loop clock map u -> u + 2 b0 (u = 1/g^2) is PARABOLIC and NOT diagonalizable (single Jordan block), and the symbolic 2x2 lemma 'exp of a real-diagonalizable matrix is diagonalizable' closes the door: no constant-rescaled modular flow produces the QCD clock -- STT is dead on the declared clock exports.  THE CONFIRMATION: the F_pole/Koide multiplier at the attractor is EXACTLY (2/3)^6 = 64/729 (hyperbolic, dimensionless => internal; F_Boltzmann the same element by the v425 single-flow theorem, cited); F_relic is the identity (content = theta_i, internalization vacuous).  THE TYPED CONSEQUENCE: the external clock contract is CONFIRMED and sharpened to the anchor census {one unit = Lambda_QCD (v_geo-class by No-Unit v153), one angle theta_i, one lattice O(1) C_p} -- T3's clock wall and T1's unit wall are THE SAME WALL, counted once.  P5 syntax observation (seam rates {6 ln(3/2), 6 ln 3} in Z<log 2, log 3>, the two shortest groupoid orbit lengths; exact integer coefficients) typed as suggestive syntax, NOT evidence.  sympy-exact, flows recomputed from the native ODEs (not read off v632).  No marker move.",
      "status": "Exact [E] symbolic kill (parabolic + no-diagonalizable-log lemma) + [E] internal-pair confirmation ((2/3)^6 exact); the external clock contract confirmed and sharpened (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "STT-KILLED + INTERNAL-PAIR-CONFIRMED: strong thermal time is dead on the declared clock exports (QCD clock parabolic, not reachable by any modular flow); the pole/Boltzmann pair is internal ((2/3)^6 exact, v425); the external clock contract STANDS, sharpened to the typed census {one unit, theta_i, C_p}; no status move.",
      "dependencies": "FR.POLE.SOLVE.01; FIRSTPRINCIPLES.BOUNDARY.01; ANCHOR.VGEO.02; v425; v153; v632",
      "scripts": [
        "v723_phys_modular_clock.py"
      ],
      "script": "v723_phys_modular_clock.py",
      "location": "tfpt_3;tfpt_2;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 668
    },
    {
      "id": "FTRANSFER.COCYCLE.01",
      "claim": "T3-B -- conditioned modular flows from ONE internal mother state, the second thermal-time KILL (one module, 12 mechanical checks, ~1 s; discovery probe phys_t3b_conditioned_modular_flows.py 12/12, verdict T3B-DEAD).  THE CONSTRUCTION (frozen before any target test): the sheet geodesic M(1,t), t = -2..1, delivers the four intertwiners J, K, C, F and the target-independent tangent S = dM/dt; S builds the chiral mother operator D_Sigma = [[0, S], [S^T, 0]] and -- with the ONE global KMS scale Delta = 6 log(3/2) -- the mother covariance C_Sigma; each intertwiner canonically yields its graph projector / CP compression channel Phi_X (NO solver data, NO fit parameters anywhere); conditioned covariances C_X = Phi_X(C_Sigma) and modular generators K_X = log((1-C_X)/C_X) follow per v258; the N-ladder uses the target-independent CAR sine regularization f_N with verified N^-2 convergence.  THE KILL: the four conditioned flows come out LOXODROMIC/COMPLEX on the P1 shadow -- the frozen v578/v632 solver classes (hyperbolic / parabolic / degenerate) do NOT arise; the hard class gate honestly FAILS (0/4 matches) and the probe correctly REFUSES the NEEDS-LINDBLAD escape (the v238 GKSL variant is also all-loxodromic; a Lindblad verdict is only allowed on a full class match).  POSITIVES that survive: the Connes cocycle composition along J-K-C-F is EXACT (max triple residual 1.6e-15) and exactly ONE global KMS scale fits all four interfaces (Delta = 2.4328 each).  With v723 (FTRANSFER.STT.01) this closes BOTH thermal-time routes: the external clock contract of F_transfer stands.  Kill audit printed (projectors target-free, one scalar/one time, QCD-without-hand-setting FAIL as the kill, no external mother-state parameters, N-ladder).  No marker move.",
      "status": "[E] exact cocycle composition + [X] honest class-gate kill (0/4 matches, Lindblad escape refused); the second thermal-time route closed (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "T3B-DEAD: the four solver classes do NOT arise from conditioned modular flows of one internal mother state (all four loxodromic/complex; cocycle composition exact; one KMS scale); with FTRANSFER.STT.01 both thermal-time routes are closed -- the external clock contract stands; no status move.",
      "dependencies": "FTRANSFER.STT.01; FIRSTPRINCIPLES.BOUNDARY.01; v258; v578; v632",
      "scripts": [
        "v724_phys_t3b_modular_flows.py"
      ],
      "script": "v724_phys_t3b_modular_flows.py",
      "location": "tfpt_3;tfpt_2;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 669
    },
    {
      "id": "VGEO.TORSOR.01",
      "claim": "T1 CLOSURE -- the calibration/scale-torsor audit of the v_geo interface (one module, 21 checks, ~1 s; discovery probe phys_vgeo_torsor_audit_probe.py 21/21, verdict TORSOR-SCHLIESSUNG PASS).  THE STRUCTURE THEOREM (machine-audited on the corpus, sources read-only: v153/v152/v274/v401/v364/v75/v68/v60/v20/v272/v3/rt_E/v714): the v_geo interface is a well-defined R+ SCALE TORSOR -- (1) the corpus exports ALL dimensional quantities as O_i = r_i * v_geo^{d_i} with machine-derived dimensionless r_i (complete export table: Mbar, 1/G, Lambda, U_point, m/mu at d = 1, 2, 4, 1, 0); (2) the dimension matrix has RANK 1 -- exactly ONE independent scale (conditional [A] on the flavor block via the declared seam=Planck identification v68 + the rt_E EW-matching flag, NAMED not hidden); (3) ALL consistency conditions are homogeneous under L -> lambda L -- no internal absolute anchor (the machine form of the No-Unit theorem v153); (4) leave-one-anchor-out cross-calibrations compatible (gravitation G vs cosmology rho_Lambda: the same Mbar to 0.11%, v274, conditional on the Lambda branch [C]; flavor fiber exact); (5) the CALIBRATION THEOREM stated in torsor language: one external reference selects the point in the torsor, everything else is prediction.  FENCE: a STRUCTURE closure of the interface -- NO absolute-scale derivation (impossible per No-Unit; that is the theorem, not the defect).  THE MOONSHOT NAMED (not executed): H_EW = ln(Mbar/v_EW) = 37.1776 -- the ONE remaining dimensionless scale ratio whose derivation No-Unit permits (a success would turn the flavor-block [A] identification into a derivation); naive c3/phi0 ladders honestly excluded (11.53 / 12.67, not integer; v184 firewall respected).  Status moves only as evidenced: the interface contract closes in CALIBRATION FORM; the No-Unit typing of v_geo is unchanged.",
      "status": "Structural [E] torsor audit (export table, rank 1, lambda-homogeneity, leave-one-anchor-out 0.11%) with conditionalities named; closure in calibration form, NO scale derivation (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "TORSOR CLOSURE PASS: the v_geo interface is structurally closed as an R+ scale torsor (export table complete, dimension matrix rank 1, all conditions lambda-homogeneous, two-anchor consistency 0.11%); calibration theorem stated; FENCE: no scale derivation (No-Unit); moonshot named: H_EW = 37.1776.",
      "dependencies": "ANCHOR.VGEO.01; ANCHOR.VGEO.02; VGEO.SHARPEN.01; SCALE.OVERDET.01; METROLOGY.CLOSURE.01; FIRSTPRINCIPLES.BOUNDARY.01",
      "scripts": [
        "v725_phys_vgeo_torsor_audit.py"
      ],
      "script": "v725_phys_vgeo_torsor_audit.py",
      "location": "tfpt_3;tfpt_2;tfpt_research_contracts",
      "externalData": "CODATA Mbar; Planck-2018 rho_Lambda^{1/4} = 2.24 meV; PDG lepton masses (corpus-documented references, read-only)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 670
    },
    {
      "id": "GNET.PPINDEX.01",
      "claim": "PHYS T2 main slice -- Pimsner-Popa / Watatani index 4 on the CAR ladder, with the mu4 action DERIVED from the clock and the Ramond zero-mode test (one module, 57 checks, ~7 s; discovery probe phys_car_pp_index_probe.py 57/57, verdict T2-SLICE-GO (PP-INDEX-4-ON-LADDER)).  THE DERIVATION (circularity rule): mu4 is NOT inserted -- the clock L = S^{N/12} of the v679/v716 frame gives the half-turn H = L^6 whose order 4 is FORCED by the NS spin structure (H^2 = S^N = -1, H^4 = 1); the deck Dk = L^4 commutes and is unused in E.  THE INDEX: on windows W_k of the N = 48/96/192/384 ladder the mu4-average conditional expectation E = (1/4) sum_j U^j . U^-j (U = Gamma(H_W), vacuum-fixing phase, no convention freedom) has optimal Pimsner-Popa constant EXACTLY 1/4 (explicit rank-one minimizer + pinching theorem; HONEST SPEC CORRECTION documented: charge-homogeneous elements give lambda = 1 identically -- the bound lives on maximally sector-mixing elements) and Watatani index EXACTLY 4 via an explicit quasi-basis (sum v v* == 4 * 1; weights FORCED by the reconstruction identity -- uniform weights BREAK scalarness (kill K4), the Z2-only average gives lambda = 1/2, index 2: mu4 distinguished from mu2).  THE RAMOND HEALING: undressed, H_R = S^{N/2} has order 2 and the census DEGENERATES (m = 2, lambda = 1/2, index 2 -- measured); the v679 Klein zero-mode dressing U' = Gamma(H_R) e^{i pi/2 n_0} restores order 4 and index 4 STATE-PRESERVINGLY ([rho, U'] = 0 exactly; the zero mode maximally mixed, c_0 = 1/2) -- kill K2 does not fire.  THE (1+i) DICTIONARY (with v722): NS momenta half-integer <-> every window mode carries H-eigenvalue +-i (chi = 1); R momenta integer <-> chi = 0 with the dressed zero mode the unique chi = 1 carrier; the sector-dim anomaly N_0 - N_2 = 2^k sits ONLY in ker(chi) -- the ramified defect lives in ker(chi) on BOTH sides.  Outerness witnesses w_q -> 1/4 and |omega(U^j)| -> 0 typed as witnesses, not proof (kill K3 stays a witness question).  GNET.RAMIFIED.01 contract candidate printed with kills.  No marker move (GATE.METRIC.08/10 stay as typed). [2026-08-03 LIII, Lean round 5: the exact algebraic index-4 core is now FORMALLY VERIFIED in Lean 4 (WatataniIndexFour.lean, lake build green, kernel-checked, no sorry): the clock-order shadow (S^12 = -1 => H^4 = 1, mu4 order forced by the NS spin structure), the explicit 13-element quasi-basis with sum v v* = 4*1 EXACTLY = N(1+i)^2, the sharp Pimsner-Popa constant lambda* = 1/4 via an SOS certificate at the mixing minimizer, and the mu2/three-sector controls (index 2/3).  Honest scope: the smallest faithful matrix model M4(Z[i]) -- the CAR/Fock layer, Gaussian states, the k-window sector dimensions, the Ramond dressing and the ladder witnesses stay numeric (v726).]  [2026-08-05 LXII (v779): the l = 1 three-sector index-3 anomaly is now EXPLAINED by the exact dim law d_c(l) = 4^(l-1) + 2^(l-1) cos(pi c/2) (d_2(1) = 0); the C[Z4] 4-element quasi-basis is obstructed at every finite l with relative gap exactly 2^-l (asymptotically restored); the index-4 structure transports literally along the Haar tower with uniform sector-resolved split bounds -- the measured finite precursor of index continuity in the GNS limit.]",
      "status": "Exact [E] index results (PP = 1/4 optimal, Watatani = 4 with forced quasi-basis weights; mu4 derived, not inserted) + [E] state-preserving Ramond healing + [M] outerness witnesses (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "T2-SLICE-GO (PP-INDEX-4-ON-LADDER): optimal Pimsner-Popa constant exactly 1/4 and Watatani index exactly 4 on the CAR ladder with mu4 DERIVED from the clock; Ramond healed state-preservingly by the Klein dressing; (1+i) dictionary closes with v722; Q-system identification = the open contract half; no status move.",
      "dependencies": "GATE.METRIC.08; GATE.METRIC.10; GNET.RAMIFIED.01; v679",
      "scripts": [
        "v726_phys_car_pp_index.py"
      ],
      "script": "v726_phys_car_pp_index.py",
      "location": "tfpt_3;tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 671
    },
    {
      "id": "E8.ORBITPACKET.01",
      "claim": "The Gaussian ORBIT PACKET lemma (one module, 26 checks, ~1 s; discovery probe orbit_packet_probe.py 26/26, verdict ORBIT-PACKET-EXACT).  BURNSIDE EXACT: the sigma orbit census on the 15 nonzero Gaussian classes gives N_orbit = (15 + 3 + 3)/3 = 7 (3 fixed classes + 4 moved orbits).  THE CANONICAL PACKET SPLIT: 240 = 5 x 48 -- one fixed block + 4 moved blocks of 12 Gaussian lines (48 roots) each, pairwise disjoint, label-free canonical (the 15-class partition agrees with the intrinsic equivalence x ~ y <=> (1-J)(x-y)/2 in L; the only residual freedom is the ordering of the 4 moved blocks); packets = 1 fixed + 4 moved = 5 = g_car and 240 = g_car x 48.  ANCHOR: the fixed-syndrome leader weights (0,1,1,2) reproduce the (1,1,2) anchor, leader pair e6/e7 = the v638 anchor pair.  Q NORMAL FORM: 240 = 16 x 15 and 248 = 240 + 8.  CIRCULARITY FENCE: P1 (c3 = 1/(8 pi)) and P2 (g_car = 5) come back as EXACT-INTERNAL-RECONSTRUCTION -- the packet law reconstructs the axioms from the code, it does NOT derive them.  SIGMA-CLASS SPECIFICITY: the 160 free order-3 elements of G31 give Burnside 5, not 7 -- the 1 + 4 packet law is sigma-class-specific, not generic.  Companion Lean module TfptCarrier/OrbitPacket.lean (34 theorems, kernel decide/norm_num only, no sorry, no native_decide).  No marker move.",
      "status": "Exact [E] integer/finite-group census; P1/P2 typed as reconstruction behind the circularity fence, NOT derivation (Python-only per GATE.WOLFRAM.02; companion Lean module kernel-checked).",
      "canonicalStatus": "ORBIT-PACKET-EXACT: Burnside 7, the canonical 5 x 48 packet split (1 fixed + 4 moved = g_car), the (1,1,2) anchor from the leader weights, the q = 16 normal form; P1/P2 come back as exact internal reconstruction (circularity fence: reconstruction, not derivation); sigma-class-specific (free order-3 census gives Burnside 5); Lean companion OrbitPacket.lean.",
      "dependencies": "E8.GAUSSCODE.01; E8.GAUSSIAN.CODE.01; v638; v623",
      "scripts": [
        "v736_orbit_packet.py"
      ],
      "script": "v736_orbit_packet.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 672
    },
    {
      "id": "E8.SEAM48.01",
      "claim": "The seam-48 intertwiner test -- the sharpest structural kill of the review morning (one module, 24 machinery checks, ~4 s; discovery probe seam48_intertwiner_probe.py 24/24 machinery, verdict SEAM48-DEAD).  THE QUESTION (external review, preregistered kill): do the five canonical 48-blocks (from the proven sigma orbit structure of the 15 x 16 Gaussian classes) and the 48 regular five-cycles of the CERTIFIED regular order-5 operator form an intertwiner -- i.e. is the 48 x 5 incidence matrix identically 1?  THE KILL FIRES: the incidence entry distribution is 0:96 / 1:64 / 2:64 / 3:16 instead of == 1 -- representative-independent (0 of 2304 order-5 elements of G31 have the incidence-1 property against the fixed blocks) and the must-fail control fires (0 of 25 random regular operators pass; the test measures something).  VALUABLE RESIDUES: the 4 moved blocks carry the v623 character table EXACTLY, the fixed block deviates exactly by the 12 lifted marks; the 5-period lemma u^5 = +/-J holds exactly.  THE HONEST READING: 240 = 48 x 5 remains a factorization WITHOUT coordinatization -- no canonical bijection between the five 48-blocks and the five-cycles.  No marker move.  [2026-08-04 LIX, the v737 resolution -- v752 (PROJ-HAMMING-EXACT): the incidence distribution 0:96/1:64/2:64/3:16 is DERIVED exactly (48 x 5 = 16 x the 3 x 5 label table; five-cycles see ubar-orbits, not sigma packets), AND the 5 x 48 coordinatization EXISTS via exactly ONE ubar-invariant spread of the 56 (incidence identically 1 on the root level, zero triple overlap with the sigma packets, not sigma-stable) -- the kill is EXPLAINED, not revised: the sigma packets remain a non-spread.]",
      "status": "Exact [E] machinery; the preregistered kill fires -- honest structural negative (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "SEAM48-DEAD: the 48 x 5 incidence matrix is not identically 1 (0:96/1:64/2:64/3:16), representative-independent, must-fail control fires; residues: the 4 moved blocks carry the v623 character table exactly, the fixed block deviates by the 12 lifted marks, u^5 = +/-J exact; 240 = 48 x 5 stays a factorization without coordinatization.",
      "dependencies": "E8.GAUSSCODE.01; E8.ORBITPACKET.01; v623; v736",
      "scripts": [
        "v737_seam48_intertwiner.py"
      ],
      "script": "v737_seam48_intertwiner.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 673
    },
    {
      "id": "QGEO.CARL1.01",
      "claim": "The (L1) sector lemma of the CAR continuum majorant, proven modulo elementary steps with explicit constants (one module, 11 checks, ~1 s; discovery probe qgeo_car_l1_sector_lemma_probe.py 11/11, verdict L1-SECTOR-PROVEN-MODULO-ELEMENTARY).  THE LEMMA: || K_2N^(q) - K_N^(q) ||_{S1(Lambda_eps)} <= B_q(eps)/N for all 6 undressed sectors on the fixed grid -- the occupied-arc mode sums are EXACT geometric sums (no Riemann-sum error; exact summation replaces Euler-Maclaurin), the entire 1/N structure is the sublattice embedding offset, the leading term is the exact matrix (iu/2MN) f_q'(s) (ratio -> 1 at 6e-4), gamma = 2 is DERIVED and consistent with the measured collision exponents, the Taylor-Lagrange envelope covers every entry and every trace norm with 39744 + 108 inequalities at 0 violations, and r1 = 1 is sharp.  STATUS TYPED: (L1, undressed sectors, fixed grid) = PROVEN-MODULO-ELEMENTARY with explicit constants; the named residues -- (a) grid-sup [elementary], (b) L2 uniform FH/RH [hard-technical], (c) L3 uniformity [medium] -- are named, not claimed.  GATE.QGEO does not move.  No marker move.  [2026-08-05 LXII (v778): residue (a) is FULLY CLOSED in the corrected (1 + ln M) form -- (a1) the continuum entrywise supremum, (a2) the configuration-S1 closure with explicit B_conf(eps) (packing + zeta(2)/zeta(3)), and (a3) the refinement remainder with RHO(eps) via exact csc^3 integrals are closed elementarily with zero census violations; (a4) the leading part is honestly LOG-divergent under refinement (jump-driven, C^1-taper-controlled), so the literal v715 clause 'one B(eps) for ALL grid refinements' is FALSE at r1 = 1 with a fixed B(eps) and is REPAIRED to ||D_N^(M)||_1 <= [B0(q,eps) + (J(eps)/2) ln(M+3)]/N, certified at theorem level by the grid Fourier log-sum inequality ||F_q^(M)(eps)||_1 <= c0(q,eps) + (J(eps)/4) ln(M+3), J = 2|f_q'(eps)| (exact nu-twisted-circulant diagonalisation + two exact cyclic summation-by-parts + harmonic comparison; 162-case census to M = 6144, zero violations in all seven clauses; repaired clause verified on all 252 transfer cases; Araki/Powers-Stormer summability survives with explicit closed forms).  Honest: the measured slope is J/pi^2 (spike interference) -- the certified J/4 and named J/pi are valid upper bounds, NOT sharp.  Residues (b) and (c) stay OPEN; GATE.QGEO does not move.]",
      "status": "Proof-near [E] sector lemma (exact geometric sums, derived gamma = 2, 39744+108 inequalities at 0 violations) with the named residues typed (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "L1-SECTOR-PROVEN-MODULO-ELEMENTARY: (L1) proven modulo elementary steps for all 6 sectors -- exact summation instead of EM, leading term at 6e-4, gamma = 2 derived, 39744+108 inequalities 0 violations, r1 = 1 sharp; residues (a) elementary / (b) hard-technical / (c) medium named, not claimed; GATE.QGEO does not move; 2026-08-05 (v778): residue (a) fully closed in the corrected (1 + ln M) form (the literal v715 all-refinements clause repaired; log coefficient certified, measured slope J/pi^2); (b)/(c) stay open.",
      "dependencies": "QGEO.CARLIMIT.01; GATE.QGEO.01; v715",
      "scripts": [
        "v745_qgeo_car_l1_sector_lemma.py"
      ],
      "script": "v745_qgeo_car_l1_sector_lemma.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 674
    },
    {
      "id": "GNET.LOCALNET.01",
      "claim": "The G_net local net functor -- net axioms exact, Watatani index locally exactly 4, Ramond as half-line/bond defect (one module, 16 checks, ~2 s; discovery probe phys_gnet_local_functor_probe.py 16/16, verdict GNET-LOCAL-FUNCTOR-ALIVE).  THE NET AXIOMS ARE EXACT: isotony, graded locality and clock covariance hold exactly for the interval assignment on the quotient circle.  THE INDEX IS LOCAL: the Watatani index of the local fixed-point inclusion is EXACTLY 4 for every interval ell >= 2 -- the [D, A] index statement is interval-local, not merely global.  RAMOND IS SOLITONIC: the R sector is a half-line/bond defect with local footprint sqrt(2/N) leakage -- exactly the DHR/solitonic structure the seam-Calderon identification needs.  4 MUST-FAIL CONTROLS FIRE.  MISSING CONTINUUM THEOREMS NAMED, NOT CLAIMED: GNS limit net (tightness; the v679 formal residual), index continuity along the inductive limit (Pimsner-Popa/Longo heredity), identification with the Longo Q-system extension of GATE.METRIC.11, solitonic sector construction (twisted DHR/Longo-Roberts), Haag duality/split for the quotient-circle net.  No marker move.  [2026-08-05 LXII (v779): the finite-level precursors of theorems (1)/(2)/(3)/(5) are MEASURED COHERENT (GNS-PRECURSORS-COHERENT, 27/27) -- the Haar scaling isometry tower intertwines exactly (<= 2.2e-16, window data literally level-independent), the index-4 expectation squares commute exactly (the finite half of the Q-system identification), the canonical chiral-sea state is Cauchy with geometric contraction ~1/4 per doubling, the index census is machine-exact at every level with zero degradation, the split witness stabilizes and decreases with separation, and the finite Haag-duality census gives twisted-duality defect 0 with dim B'/dim A' = 4 (l = 1: 3); a 10-section continuum-paper skeleton is on record; theorem (4)'s finite witness stays the v746 bond-defect identity.  The exact dim law d_c(l) = 4^(l-1) + 2^(l-1) cos(pi c/2) types the C[Z4] obstruction at exactly 2^-l and EXPLAINS the l = 1 three-sector anomaly.  Every measurement stays typed finite evidence, never a continuum proof.]",
      "status": "Exact [E] finite net axioms + exact local index 4 + [C] the named continuum theorems (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "GNET-LOCAL-FUNCTOR-ALIVE: net axioms exact (isotony/graded locality/clock covariance), Watatani index LOCALLY exactly 4 for every interval ell >= 2, Ramond = half-line/bond defect (solitonic; leakage sqrt(2/N)); 4 must-fails fire; the five continuum theorems named, not claimed.",
      "dependencies": "QGEO.BONDDEF.01; GATE.METRIC.11; QGEO.DECKSECTOR.01; v722",
      "scripts": [
        "v746_phys_gnet_local_functor.py"
      ],
      "script": "v746_phys_gnet_local_functor.py",
      "location": "tfpt_3;tfpt_2",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 675
    },
    {
      "id": "FTRANSFER.CLOCKS.01",
      "claim": "Preregistered theory-contract search target (frozen 2026-08-04, experiments/ftransfer-clocks/hypotheses/ftransfer_clocks_v1.yaml; EXECUTED 2026-08-05, v777): the external-clock contract for the F_transfer jets.  THE FOUR FROZEN CLOCKS (exactly one physical clock per solver, fixed in advance): F_pole = proper time tau (on-shell/pole-mass chart; one v99 seam unit = one proper-phase e-fold), F_Boltzmann = inverse temperature beta (FIRAS-anchored adiabatic chart; one washout seam unit = one beta e-fold; the z-linear alternative recorded as excluded), F_QCD = RG time t = ln(mu/M_Z) (MS-bar, 1-loop b0 = 7 window [m_t, 1 TeV]; identity gauge), F_relic = cosmological e-folds ln a (Planck 2018 background; flow frozen, jets degenerate -- scored only on kills and dictionary validity).  THE PREREGISTERED PREDICTION (FTC.P1): under the frozen affine dictionaries (slopes -1, -1, +1, -1) ONE constant-coefficient Riccati equation on sigma = ln(mu/M_Z) carries all non-degenerate transported jets -- equivalently the transported Schwarzians of F_pole/F_Boltzmann/F_QCD are EQUAL (the seam channels transport to -Delta^2/2 = -2.959235170076978 exactly; the prediction forces the QCD value onto it).  THREE KILLS: K1 NO-COMMON-CONNECTION (symbolic inequality of transported Schwarzians; executor must exhibit the forbidden reparametrization per failing pair), K2 FREE-REPARAM-NEEDED (any post-freeze change of gauge/dictionary; byte-frozen YAML sha256 recorded), K3 SECOND-CLOCK-NEEDED (any channel export needing two time variables).  HONEST PRIOR (preregistered): EXPECTED = K1 KILL -- affine transport scales a Schwarzian by slope^2 and cannot move 0 (QCD) onto -Delta^2/2 (seam); a K1 kill would finalize the v578 diagnosis at the physical-clock level and close the jet route of CONTRACT.F.01; ONE-CLASS would be the surprise.  EXECUTOR: PENDING -- v752_ftransfer_clock_jets.py-pattern module at a future round (sympy-exact kernel; the FTC.01 data interfaces need the frozen g_*s tabulation and PDG alpha_s(Q) overlay data curated first).  FIREWALL: no marker move, nothing upgrades CONTRACT.F.01; the four observables stay [C]/[O]; this design is typed [X].  NO RH claim (unrelated surface; discipline note).  [2026-08-05 LXII, EXECUTED + PROMOTED (v777_ftransfer_clock_jets.py, one module from two probes, 24 + 14 checks, 0 failures): VERDICT NO-COMMON-CONNECTION -- the K1 kill fires EXACTLY as the frozen prior expected.  Freeze guards verified at run time (YAML sha256 880224f7..., g_*s table 8ae2a4fe..., alpha_s table 2ca6332d...; both data tables curated + byte-frozen BEFORE the first executor run).  All four dictionary-validity legs PASS (FTC.01.K: q_exp dev 6.6e-6 <= 3 sigma_q, scheme check 571x -- the on-shell chart is right and the running chart fails it; FTC.01.B: washout entropy-chart drift 1.15e-4, N_eff certificate 0.32 sigma; FTC.01.Q: frozen chain reproduced exactly, 1-loop overlay max dev 0.83% on [m_t, 1 TeV]; FTC.01.R: onset T_osc = 76.68 GeV, slope dev 0.0602).  The one-class prediction FTC.P1 dies sympy-EXACTLY: transported Schwarzians {-Delta^2/2, -Delta^2/2, 0}, symbolic difference exactly nonzero, no tolerance involved; K1 duty discharged (the pass-enabling reparametrization h = Moebius(e^{+-Delta sigma}) is exhibited and is NON-affine -- exactly the freedom the freeze forbids); K2/K3 quiet.  COMPANION (same module): the fibered-groupoid refinement CARRIES (FTRANSFER-FIBERED-CARRIES, 14/14) -- see the CONTRACT.F.01 dated note; the groupoid's first-run float64 spot-check artifact is disclosed in the probe docstring, the frozen 1e-12 bar untouched.  The four observables stay [C]/[O]; no marker move.]  NOTE (2026-08-06, v792 -- marker UNCHANGED): the coset assignment of the executed contract's clock charts is now a THEOREM from KMS/modular typing (S = -(ln lambda)^2/2 from the time-1 PGL2 holonomy; blind protocol, exact unblinding, controls incl. the Moebius-clock positive case) -- the three-probe chain executor (v777) -> fibered groupoid -> coset theorem is whole; the fiber classification is derived, not case-inspected.",
      "status": "Executed preregistered search target (frozen 2026-08-04 [X]; executed + promoted 2026-08-05, v777): the K1 kill fired exactly as the frozen prior expected (NO-COMMON-CONNECTION); the fibered-groupoid refinement carries; the four observables stay [C]/[O]; no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "EXECUTED 2026-08-05 (v777): the K1 kill fired sympy-exactly as the frozen prior expected -- NO-COMMON-CONNECTION ({-Delta^2/2, -Delta^2/2, 0}); all four dictionary legs construction-valid (571x / 1.15e-4 / 0.83% / 0.0602); K1 duty discharged (the needed reparametrization is the forbidden non-affine Moebius clock); the fibered-groupoid refinement carries (constant seam cocycle {0, +-Delta^2/2}); nothing upgrades CONTRACT.F.01; no marker move.",
      "dependencies": "CONTRACT.F.01; FTR.JETS.01; FTR.PGL2.01; FTRANSFER.STT.01; FTRANSFER.COCYCLE.01; v578; v632; v723; v724",
      "scripts": [
        "v777_ftransfer_clock_jets.py"
      ],
      "script": "v777_ftransfer_clock_jets.py",
      "location": "tfpt_research_contracts;tfpt_3;tfpt_4",
      "externalData": "PDG 2024 (pole masses, alpha_s), FIRAS T0, Planck 2018 background, Saikawa-Shirai 2018 g_*s tabulation (curated + SHA-frozen data tables in experiments/ftransfer-clocks/data/, byte-verified at every run; executed 2026-08-05)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 676
    },
    {
      "id": "E8.PROJHAMMING.01",
      "claim": "The projective Hamming incidence lemma on V = L/(1+i)L -- the symplectic F2^4 of the unimodular hermitian E8 lattice, and the v737 resolution (one module, 36 checks, ~5 s; discovery probe projective_hamming_incidence_probe.py 36/36, verdict PROJ-HAMMING-EXACT).  THE LEMMA FROM THE CONCRETE LATTICE: h = H/4 is Z[i]-valued with hermitian Gram determinant 1 (the UNIMODULAR hermitian E8 lattice over Z[i]); doubly-even => hbar alternating, well-defined on V, NONDEGENERATE: (V, hbar) is symplectic over F2 with explicit Darboux basis.  THE CORE IDENTITY: |H_y| = 7 for all 15 points, pair intersections 3 for all 105 pairs; B B^T = 4 I + 3 J EXACT with characteristic polynomial (x-7)(x-2)^9(x+2)^5 -- the eigenvalue multiplicities (1, 9, 5) are the SYMPLECTIC signature (honest P7.2 finding: B B^T = 4I + 3J alone holds for ANY nondegenerate form; the symplectic content is the projective part -- diagonal 1, tr B = 15, multiplicities).  THE CHANNEL: K = B/7 with K^2 = (4/49) I + (45/49) Pi0 exactly (K itself honestly NOT psd).  NS/R AS GEOMETRY: chi_NSR = hbar(., y0) for the UNIQUE sigma-fixed point y0 = FSUM (the anchor bit); 112 NS + 128 R, +8 Cartan: 248 = 120 + 128 (SO(16)).  THE v737 RESOLUTION: the seam-48 distribution 0:96/1:64/2:64/3:16 is DERIVED exactly (48 x 5 = 16 x the 3 x 5 label table), AND the 5 x 48 coordinatization EXISTS via exactly ONE ubar-invariant spread of the 56 -- the v737 kill is EXPLAINED (wrong blocks), not revised.  Two honest corrections to the review derivation documented (P6.5d, P7.2).  Companion Lean module TfptCarrier/ProjectiveHamming.lean (32 theorems, kernel decide/norm_num only, lake build green).  No marker move, NO RH claim.",
      "status": "Exact [E] finite lemma (integer/Fraction arithmetic; Lean companion module; Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "PROJ-HAMMING-EXACT: (V, hbar) is the symplectic F2^4 of the unimodular hermitian E8 lattice; B B^T = 4I + 3J exact, eigenvalues 7^1 (+2)^9 (-2)^5, K^2 = (4/49) I + (45/49) Pi0; NS/R = hbar(., FSUM) = the anchor bit; 248 = 120 + 128; the v737 distribution derived and the 5 x 48 coordinatization exists via exactly one ubar-invariant spread (kill explained, not revised); no RH statement.",
      "dependencies": "E8.GAUSSCODE.01; E8.ORBITPACKET.01; E8.SEAM48.01; v689; v634; v722; v736; v737; v738",
      "scripts": [
        "v752_projective_hamming_incidence.py"
      ],
      "script": "v752_projective_hamming_incidence.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 677
    },
    {
      "id": "QGEO.FANODISC7.01",
      "claim": "The preregistered Fano -> discriminant(-7) bridge -- the three-cycle half is real, the parity half dies arithmetically: the seven-correlation is typed decorative for good (one module, 26 checks, ~1 s, ONE preregistered [FAIL] line C2.d = the COND2 break; discovery probe fano_disc7_bridge_probe.py, verdict FANO-DISC7-DECORATIVE).  THE FOUR-CONDITION TEST breaks exactly at condition 2: the NS/R parity bit has NO arithmetic carrier in O_{-7} -- 2 SPLITS in Q(sqrt(-7)) (-7 == 1 mod 8, O tensor Z2 = Z2 x Z2), so every 2-adic ideal quotient has 2-rank <= 2 < 3 (complete census of all norm-8/16 ideals): the code F2-structures are arithmetically UNREALIZABLE; candidates 2a (norm parity does not factor through the ramified fiber) and 2b (chi_7 pullback nonconstant) die too.  THE REAL HALF (COND1, exact): equivariant lifts EXIST for both canonical difference-set Fano models (3 each), the canonical order-3 action is x -> 2x (the difference-set multiplier, 2 = the transfer step), and every lift sends the unique sigma-bar fixed point of the NS Fano plane to 0 = red(alpha_0) = the C corner (norm 14 = dim G2) -- forced.  Controls fire (x -> 3x zero lifts; broken incidence zero lifts; the ideal 7-set refuses the 3-cycle).  No marker move, NO RH claim.",
      "status": "Exact [E] machinery; the preregistered break fires -- honest structural negative, correlation typed decorative (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "FANO-DISC7-DECORATIVE: preregistered four-condition test, break exactly at condition 2 -- 2 splits in Q(sqrt(-7)) => 2-rank <= 2 < 3, the code F2-structures are arithmetically unrealizable; the three-cycle half is real (Fano sigma-bar = x2 multiplier, C corner = fixed point); the seven-correlation is permanently typed decorative; no RH statement.",
      "dependencies": "E8.PROJHAMMING.01; PRIME.HECKEMODRAM.01; QGEO.CAR.01; v533; v571; v632; v722; v738",
      "scripts": [
        "v757_fano_disc7_bridge.py"
      ],
      "script": "v757_fano_disc7_bridge.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 678
    },
    {
      "id": "ARF.SPINORCOMPILER.01",
      "claim": "The Arf/spinor compiler on V = L/(1+i)L = F2^4 -- Priority 1 of the Arf compiler programme, the exact finite-algebra layer (one module, 46 checks, ~2 s; discovery probe arf_spinor_compiler_probe.py 46/46, verdict ARF-SPINOR-EXACT, zero kills, both must-fail controls fire).  THE COMPILER, EXACT END TO END: hbar rebuilt from the concrete v752 lattice (Gram J - I in the family/anchor basis, census 240 = 15 x 16, sigma symplectic; standard-doubled-model cross-check: chi coefficients (0,0,0,1), y0 = FSUM); brute force over all 2^16 functions: EXACTLY 16 quadratic refinements; Arf census 6 + 10 with |Sp(4,2)| = 720 (full 65536-matrix census) and a faithful+surjective S6 action on the six Arf-1 forms; the frozen selector (sigma-invariance census 4 -> q(A) = 1: 2 -> q(F_Sigma) = 0: 1) picks a UNIQUE q*; the parity lift iota: V ~ C_even(5) satisfies beta = hbar in all 256 cells (crosslink 1) and the weight form q_wt IS q* (offset 0; all 16 refinements = q* + hbar(., c) bijectively); Stab(q*) of order 120 acts as S5 with orbits {1, 5, 10} on refinements AND words: 16 = 1 + 5bar + 10 as Arf geometry; the K6 duad model v <-> D(v) is a bijection onto E(K6), Sp/sigma-equivariant, vertex split 15 = 5 + 10 at q*; B = I + A_KG(6,2) ENTRYWISE in all 225 cells, charpoly (x-7)(x-2)^9(x+2)^5, ovoid eigenvector B(1_O - (1/3)1) = -2(1_O - (1/3)1) EXACT (the six ovoid indicators span the (-2)-eigenspace, rank 5); all 6 fully isotropic spreads carry q*-signature (1, 2) per block; the hypercharge table X = (-2,-2,-2,3,3) reproduces the v310/v14 dictionary (values AND multiplicities); code polynomial moments (16, 0, 120, 0) with the chain 120/3 = 40, +1 = 41 = 10 b1, 2*120 = 240, +8 = 248 (the v2 master moment Tr X^2 = 120 = 5!); Pati--Salam 16 = 8 + 8 with the A bit as isospin index and B-L = 1 - 2 n_c/3 exact; chi_NSR = hbar(., F_Sigma) = the anchor bit on all 16 words.  SEMANTIC SEPARATION (naming discipline, no physics claim): the SAME sigma acts as the family 3-cycle A3^Fam on the message bits AND as the 3+2 slot split A3^PS on the carrier legs -- two registers, never to be conflated.  HONESTY FENCE (binding): the PHYSICAL code->matter reading is NOT claimed -- it is KILLED at root level by v775 (ROOTCLASS-MIXED), incl. the 'empty mailbox = right-handed neutrino' reading.  Companion Lean module TfptCarrier/ArfSpinorCompiler.lean (23 kernel theorems, decide/norm_num only, no sorry, no native_decide, lake build green).  No marker move, NO RH claim.  NOTE (2026-08-05, v782/v784 -- marker UNCHANGED): the certified spread + q* selector become load-bearing in the transition-bus spread layer 240 = 5 x 3 x 16 (v782), and the q* parity acquires a sharp NEGATIVE boundary: the seam curve's canonical deck-invariant theta refinement is EVEN (Arf 0, decomposition 1+6+9) while q* is ODD (Arf 1, 1+5+10) -- the curve/code 2-torsion transport dies at exactly this layer alongside the syntheme-vs-duad sigma layer (v784, joint census 0 with positive control 6).",
      "status": "Exact [E] finite algebra (integer/Fraction/F2 arithmetic; Lean companion ArfSpinorCompiler.lean, 23 kernel theorems; Python-only per GATE.WOLFRAM.02); the physical code->matter reading FENCED (killed by v775/ARF.ROOTCLASS.01).",
      "canonicalStatus": "ARF-SPINOR-EXACT: unique sigma-selected Arf refinement q* (census 4 -> 2 -> 1; q_wt IS q*), parity lift iota ~ C_even(5) with beta = hbar exact, 16 = 1 + 5bar + 10 as Stab(q*) ~ S5 Arf geometry, K6 duad bijection, B = I + A_KG(6,2) with exact ovoid eigenvectors, spreads (1,2), hypercharge table -> v310/v14 dictionary, moments (16,0,120,0) with chain 40/41/240/248, Pati--Salam 8+8 with B-L exact, chi_NSR = anchor bit; A3^Fam != A3^PS discipline; matter reading fenced (v775); Lean companion 23 theorems; no RH statement.",
      "dependencies": "E8.PROJHAMMING.01; PRIME.HECKEPOLARITY.01; CAR.SM.01; AX.P2.01; v752; v753; v2; v310; v14",
      "scripts": [
        "v774_arf_spinor_compiler.py"
      ],
      "script": "v774_arf_spinor_compiler.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 679
    },
    {
      "id": "ARF.ROOTCLASS.01",
      "claim": "THE FIRED PHYSICAL KILL of the Arf compiler programme -- gaussian_class_d5_weight_purity, Priority 2, preregistered (one module, 18 checks, ~2 s; discovery probe gaussian_class_d5_purity_probe.py 18/18, verdict ROOTCLASS-MIXED, all 3 must-fail controls fire; FROZEN_SPEC SHA-256 58d9be9b935d92bbc3ddb838092d0cb7f8c1e5dad09e06ec28e850116ee3ef25, hashed BEFORE any root data).  THE QUESTION: does each nonzero Gaussian class of V = L/(1+i)L carry exactly ONE SU(5) weight state of the D5 half-spinor 16, per the frozen F2-linear prediction table (F_i -> {c_i, w1}, A -> {w1, w2}, X = (-2,-2,-2,3,3))?  THE ANSWER IS NO, FOR EVERY DEFENSIBLE CONVENTION: the embedding-INDEPENDENT counting bound kills the strict gate for EVERY D5(+)A3 c E8 (coordinate or not) -- every embedding has exactly 128 spinor-side roots, 15 x 16 = 240 > 128, so at most 8 of 15 classes can be all-spinor and >= 7 classes ALWAYS contain adjoint-side roots; the exhaustive scan (56 coordinate splits x 240 slot assignments x 2 chiralities) finds ZERO conventions with even one pure set of 15; under the canonical sigma-stable split all 8 spinor classes mix 8 distinct 16-weight pairs each; best-case match of the frozen dictionary over ALL assignments: 8/128 spinor roots; X-only weakened purity (secondary, not the gate): 0/15.  CONTROLS: wrong charge vectors break the frozen X column and the root identity (C1); the LCG-scrambled class map destroys NS/R purity and +-pair symmetry (C2); of the 28 nondegenerate alternating forms every wrong form breaks the canonical pairing package (C3).  CONSEQUENCE (binding fence): the code->matter reading must NOT enter any main claim -- including the 'empty mailbox = right-handed neutrino' reading; the Arf structure of v774 keeps its mathematics (census, symplectic form, polarity, NS/R point untouched).  No marker move, NO RH claim.  NOTE (2026-08-05, v781/v782/v783/v784/v786/v787/v790 -- marker UNCHANGED): the fence HELD through the strategic and prime-channel promotion round -- all new census/identification modules cite ROOTCLASS-MIXED as binding and none makes a code -> matter assignment; the fired kill stays the boundary of every main claim.",
      "status": "Exact [E] finite census kill -- the fired preregistered physical kill; the fence is binding on all main claims (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ROOTCLASS-MIXED: no D5(+)A3/SU(5) convention makes even one full set of 15 classes pure -- counting bound (128 spinor roots, >= 7 adjoint-side classes for EVERY embedding), 0/56 splits pass, all 8 spinor classes mix 8 weight pairs; the code->matter table must not enter any main claim (fence incl. the right-handed-neutrino mailbox reading); no RH statement.",
      "dependencies": "ARF.SPINORCOMPILER.01; E8.PROJHAMMING.01; PRIME.HECKEPOLARITY.01; v752; v1; v47; v310",
      "scripts": [
        "v775_gaussian_class_d5_purity.py"
      ],
      "script": "v775_gaussian_class_d5_purity.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 680
    },
    {
      "id": "ARF.BOUNDARY.CODE.01",
      "claim": "Boundary code uniqueness -- the non-circular selection chain boundary -> code -> E8 -> F2^4 -> 5 slots EXISTS with g_car = 5 as OUTPUT, and the residual premise is named (one module, 31 checks, ~2 s; discovery probe boundary_hamming_uniqueness_probe.py 31/31, verdict BOUNDARY-CODE-UNIQUE; frozen design SHA-256 c1893641...).  THE CHAIN, REBUILT WITHOUT HAMMING/E8 INPUT: direct enumeration gives 135 self-dual binary codes of length 8 (mass formula), 30 doubly even; J-compatibility leaves 14; sigma-equivariance (deck 3-cycle pair-coherence) leaves 2, swapped by the anchor transposition (67) -- which IS boundary gauge (centralizer order 12) until the flag datum; the flag/side-orientation datum selects EXACTLY ONE code, identically under all three predeclared formalizations (FF1/FF2/FF3), and the decisive content is ONE deck-invariant Z2 = the product of the four side orientations (all 16 assignments decide by sum(s) mod 2 alone).  DERIVED CHAIN EXACT: Construction A of the selected code IS E8 (240/2160 theta, Cartan det 1, arms 1-2-4; weight-4 supports == the deployed v638 C* list verbatim, post-selection cross-check); L/(1+i)L = F2^4 with the canonical symplectic bbar; the Arf selector with the BOUNDARY-canonical anchor naming (A = class of the flag-transversal root) picks a unique q*; sigma-orbit packets output g_car = 1 + 4 = 5 (240 = 5 x 48, 60 = 5 x 12).  Controls fire (drop-J / drop-flag / twisted deck / illegal deck / all 105 non-doubly-even codes fail Construction-A evenness).  HONEST RESIDUALS (typed): R1 the UNIVERSE premise -- 'the boundary realizes a doubly-even self-dual binary code on its 8 sides' is the new named premise (the circularity is RELOCATED and SHRUNK, not removed); R2 deck-placement blindness -- WHICH pair is the anchor is invisible to the selection (Aut(C*) contains every straight pair 3-cycle); the placement is SEMANTIC, not selective (the wrong-deck control does not fire, typed as an honest deviation); R3 anchor naming -- bbar(F_Sigma, A) = 1, so A vs A + F_Sigma changes q*; the flag-transversal naming is boundary-canonical but part of the frozen selector, not a theorem.  NO P2-REMOVAL CLAIM.  OPEN CONTRACT FORM (registered here [O]): derive evenness/self-duality of the mark bookkeeping from the seam axioms (R1), plus the R2/R3 residuals; kills: a second boundary-legal code surviving the flagged selection, or a seam-axiom derivation forcing a non-doubly-even universe.  No marker move, NO RH claim.  NOTE (2026-08-06, v798/v799 -- marker UNCHANGED): all three honest residuals move -- R1 is REPLACED by R1' (v799 TYPEII-FORCED: exhaustively at length 8, mutual locality <=> self-orthogonal, integer conformal spin <=> doubly even, holomorphy/index one <=> self-dual; the universe premise reduces to the three physical axioms A1-A3, each a standard physical demand, not a coding-theory premise); R2 is SOLVED semantically (v798: the deck J = iI IS the metaplectic anomaly (T3 K3)^3 of the total symplectic Fourier at tau = i, and zeta8 = (1+J)/sqrt2 the real half-deck swapping the seam pair); R3 is NAMED, not fixed (v798: the strict census over the 46080 coset elements is EMPTY -- no Theta-real metaplectic involution fixes q* -- but the sigma-orbit of lifts has Arf defects exactly {A + F_Sigma + F_i} whose XOR is the anchor A itself).  NO P2 marker move anywhere.  NOTE (2026-08-06, v803 -- marker UNCHANGED): the attack surface is SHARPENED by the affine NS/R reconstruction -- the [8,4,4] Hamming code re-emerges as the evaluation code of affine functions on the 8-point NS/R torsor A = chi_NSR^{-1}(1), with Stab(chi_NSR) = AGL(3,2) = Aut(H8) carrying the same natural 8-point action (1344 equivalences, exact census); HONEST TYPING: the route to V still passes through Gaussian E8, so this is an exact INTERNAL reconstruction, not a P2 removal and not an independent boundary-code derivation -- residual R1' (the three physical axioms, v799) unchanged.",
      "status": "Exact [E] finite selection chain (census / Construction A / Arf selector, all exact arithmetic) + OPEN research contract [O]: derive the doubly-even self-dual universe premise from the seam axioms; NO P2-removal claim (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "BOUNDARY-CODE-UNIQUE: census 135 -> 30 -> 14 -> 2 -> 1 without Hamming/E8 input; flag datum = one deck-invariant Z2; Construction A -> E8 exact; q* unique; g_car = 1 + 4 = 5 as OUTPUT; residuals R1 (universe premise, the relocated+shrunk circularity), R2 (deck placement semantic), R3 (anchor naming) typed; open contract: derive evenness/self-duality from the seam axioms; NO P2-removal claim; no RH statement.",
      "dependencies": "ARF.SPINORCOMPILER.01; AX.P2.01; E8.GAUSSCODE.01; E8.PROJHAMMING.01; v689; v736; v638; v752",
      "scripts": [
        "v776_boundary_hamming_uniqueness.py"
      ],
      "script": "v776_boundary_hamming_uniqueness.py",
      "location": "tfpt_1;tfpt_2;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 681
    },
    {
      "id": "QGEO.GRIDSUP.01",
      "claim": "Remainder (a) of the (L1) majorant lemma FULLY closed in the corrected (1 + ln M) form -- one module from two probes (19 + 19 checks, ~40 s; discovery probes qgeo_grid_supremum_probe.py 19/19 GRIDSUP-SPLIT-CLOSED-LOG-LAW and qgeo_fourier_logsum_probe.py 19/19 FOURIER-LOGSUM-CLOSED, both 2026-08-05).  THE FOUR-WAY SPLIT of the v745 residue (a): (a1) the continuum entrywise supremum CLOSED -- Taylor constant 5/8 symbolic, envelopes symbolic, 1-Lipschitz circ; ~1.26e6 census inequalities, ZERO violations (the fixed-grid restriction removed at entry level); (a2) the configuration-S1 closure CLOSED with explicit B_conf(eps) = floor(1/eps)[(5pi/6) zeta(2) eps^-2 + (5/192)(68pi^2/9) zeta(3) eps^-3] (packing + m-th-neighbour pigeonhole; zero violations); (a3) the remainder part under refinement CLOSED uniformly over ALL M <= N/2 with RHO(eps) via exact csc^3 integrals (zero violations); (a4) the leading part honestly LOG-DIVERGENT -- ||F_q^(M)(eps)||_1 grows like a + b ln M with b = (0.10..0.17)|f_q'(eps)|, the growth IS the eps-cut jump (C^1-taper control saturates), and it transfers: the literal v715 clause 'one B(eps) for ALL grid refinements' is FALSE at r1 = 1 with a fixed B(eps).  THE SURVIVING INEQUALITY CERTIFIED AT THEOREM LEVEL: exact nu-twisted-circulant diagonalisation (F normal, ||F||_1 = sum |mu_k| exactly) + two exact cyclic summation-by-parts (two lattice spikes of exact mass (1/2M)|f_q'(eps)|; smooth part with the F3 = C3/sin^4 envelope, C3 = 314pi^3/27) + harmonic comparison: ||F_q^(M)(eps)||_1 <= c0(q,eps) + (J(eps)/4) ln(M+3) <= c0' + (J/pi) ln M a fortiori, J = 2|f_q'(eps)|, all constants exact; 162-case census (18 pairs x 9 rungs to M = 6144), ZERO violations in all seven clauses; the REPAIRED (L1) clause ||D_N^(M)||_1 <= [B0(q,eps) + (J(eps)/2) ln(M+3)]/N verified on all 252 transfer cases; Araki/Powers-Stormer summability survives with explicit closed forms (sum_l (A + Bl) 2^-l = 2A + 2B exact; even the extreme joint scaling M = N/2 stays summable).  HONEST: the measured slope is J/pi^2 (spike interference) -- the certified J/4 and named J/pi are valid upper bounds, NOT sharp (the average-interference sharpening is not claimed); the constant part of the bound is loose, the log coefficient is the load-bearing content.  Residues (b) L2 uniform FH/RH and (c) L3 uniformity of v745 stay UNTOUCHED and OPEN; GATE.QGEO does NOT move.  No marker move, NO RH claim.",
      "status": "Theorem-level [E] closure of residue (a) in the corrected (1 + ln M) form (exact twisted-circulant + summation-by-parts + census, zero violations); the v715 literal clause corrected; residues (b)/(c) open; GATE.QGEO does not move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "GRIDSUP-SPLIT-CLOSED-LOG-LAW + FOURIER-LOGSUM-CLOSED: (a1)-(a3) closed elementarily with explicit constants (zero census violations), (a4) closed by the certified grid Fourier log-sum theorem ||F||_1 <= c0 + (J/4) ln(M+3); repaired (L1) clause [B0 + (J/2) ln(M+3)]/N verified; Araki summability unaffected; measured slope J/pi^2 (bound valid, not sharp); (b)/(c) open; GATE.QGEO does not move; no RH statement.",
      "dependencies": "QGEO.CARL1.01; QGEO.CARLIMIT.01; v745; v715",
      "scripts": [
        "v778_qgeo_gridsup_logsum.py"
      ],
      "script": "v778_qgeo_gridsup_logsum.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 682
    },
    {
      "id": "GNET.GNSLIMIT.01",
      "claim": "The G_net continuum strand -- finite-level precursors of the GNS limit net COHERENT, the exact C[Z4] obstruction dim law, and the honest Arf negative; one module from two probes (27 + 27 checks, ~50 s; discovery probes gnet_gns_limit_probe.py 27/27 GNS-PRECURSORS-COHERENT and gnet_arf_qsystem_probe.py 27/27 GNET-ARF-NO-CORRESPONDENCE, both 2026-08-05).  THE PRECURSORS (typed finite evidence FOR the five v746 continuum theorems, never a continuum proof): Haar scaling isometry tower intertwines EXACTLY (V isometric, V H_N = H_2N V, V L_N = L_2N V <= 2.2e-16; window data literally level-independent; Fock coherence <= 1.5e-14); the index-4 expectation squares COMMUTE exactly (restriction coherence, [E_I, E_J] = 0, joint idempotency -- the finite half of the Longo Q-system identification); the canonical chiral-sea state is Cauchy with geometric contraction ~1/4 per doubling (d5 = 2.4e-4; parity-odd exactly 0); the index census is machine-exact at EVERY level with zero degradation (lambda* = 1/4, index 4*1, l = 1 anomaly stable at 3, PP margins clean); the split witness stabilizes and decreases with separation; the finite Haag-duality census gives twisted-duality DEFECT = 0 (dim A' = 4^{n_c} exhausted by the twisted complement) and dim B'/dim A' = 4 = the local Watatani index (l = 1: 3); a 10-section continuum-paper skeleton is the work product.  THE EXACT DIM LAW: d_c(l) = 4^(l-1) + 2^(l-1) cos(pi c/2) for l = 1..6 -- the C[Z4]/Longo 4-element quasi-basis is OBSTRUCTED at every finite l with relative gap exactly 2^-l -> 0 (the crossed product is exactly asymptotic; the finite precursor of the GATE.METRIC.08/11 identification), and d_2(1) = 0 IS the l = 1 three-sector index-3 anomaly of v746; the explicit 185-element Watatani quasi-basis at l = 2 has sum v v* = 4*1 exact; sector-resolved split bounds are uniform along the tower; the sigma register exists internally on the Z6-quotient window W6 (deck + half-turn joint, dims (20,16,12,16), lambda* = 1/4) -- but dim Fock(W6) = 64 != 16.  THE HONEST NEGATIVE, EXHAUSTIVE: the 16 = 1 + 5 + 10 counting echo is REAL and exact but NOT geometric -- F2-rank certificate 3 vs 4 (linear bijection impossible), GL(4,2) census 0/20160 for EVERY Arf-1 form's 5-set, no nontrivial internal order-3 symmetry (6144-element census); K_matter does NOT govern the tower weight transport (x5 = (0.414, 0.586) vs pi = (1/3, 2/3); the degenerate-decimation caution preregistered and confirmed by control C4).  BINDING HONESTY RULE: index 4 = |mu4| and the channel eigenvalue 4/49 = (-2/7)^2 are different numbers from different registers -- nothing equates them.  All controls fire (bond scramble, Z2-only, scrambled sea, decimation, wrong Arf form).  Spec corrections carried verbatim (G4b spanning basis; S4 bar anchor).  No marker move, NO RH claim.",
      "status": "Exact [E] finite coherence data + exact [E] dim law + measured [MEASURED] precursors and the honest negative, all typed finite evidence for the named continuum theorems -- never the theorems (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "GNS-PRECURSORS-COHERENT + GNET-ARF-NO-CORRESPONDENCE: exact Haar tower intertwining, commuting index-4 expectation squares, geometric state Cauchy ~1/4, machine-exact index census with zero degradation, twisted-duality defect 0 with ratio 4; dim law d_c = 4^(l-1) + 2^(l-1) cos(pi c/2) => C[Z4] obstruction exactly 2^-l (explains the l = 1 anomaly); the 1+5+10 echo exact but non-geometric (rank 3 vs 4, GL census 0/20160); K_matter does not govern transport; the two fours stay in their registers; no RH statement.",
      "dependencies": "GNET.LOCALNET.01; GNET.PPINDEX.01; GATE.METRIC.08; GATE.METRIC.11; ARF.SPINORCOMPILER.01; v746; v726; v754",
      "scripts": [
        "v779_gnet_gns_arf.py"
      ],
      "script": "v779_gnet_gns_arf.py",
      "location": "tfpt_1;tfpt_2;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 683
    },
    {
      "id": "E8.ONEOBJECT.01",
      "claim": "The d = 4 forcing census + the 60-line clock synchronization kill, one module from two probes (21/21 + 23/23 checks, ~5 s; discovery probes one_object_reduction_probe.py ONE-OBJECT-PARTIAL and clock_fiber_product_probe.py CLOCK-FIBER-DEAD, both 2026-08-05).  THE FORCING CENSUS (exact Fraction arithmetic, d = 2..12): the TFPT glue pattern (cyclic isotropic glue, evenness q = 0 mod 2Z, unimodularity |H|^2 = 4d) admits an even unimodular DIAGONAL glue of A_{d-1} (+) D_{d+1} iff d = 4 -- at d = 4 the census reproduces v92 EXACTLY (precisely two even Lagrangian glues, the graphs of the anti-isometries k -> k and k -> 3k), the explicit glued lattice is E8 (Gram det 1, exactly 240 norm-2 vectors, all integral over the glued basis), d = 9 (the only other perfect-square candidate from |H|^2 = 4d) dies at isotropy (0 even Lagrangians in Z9 (+) Z2 x Z2), and both relaxation controls fire (dropping evenness opens d != 4; dropping unimodularity opens d != 4).  COROLLARIES AT THE FORCED d (verified arithmetic): g_car = d+1 = 5, N_fam = d-1 = 3, rank = 2d = 8, glue index sqrt(4d) = 4 = d = |mu4| -- the boundary object with four marked points reads P2 and the family count as the two functor readouts of ONE object.  HONESTY GATE (binding): c_3 = 1/(2 pi d)|_{d=4} = 1/(8 pi) is a corpus-legitimate REWRITING (c_3 = 1/(2 e_1(a) pi) with e_1(a) = 4 = |mu4|, tfpt_1 anchor-first refinement / v23), NOT a corpus derivation of P1 -- ONE-OBJECT-FORCED was predeclared unreachable and P1 stays a primitive postulate.  THE CLOCK KILL (part 2): the hypothesized fiber-product action L_60 = Z12 x_{Z6} Z30 of the seam clock (order 12, c = J o sigma, v629/v634) and the Coxeter clock (order 30, Kostant census {30: 8}, w^15 = -1) on the 60 Gaussian lines is DEAD at three levels -- the Coxeter clock does not preserve the line partition; the FULL line-compatible group (the 11520 line-reflection image extended by conjugation) has order 23040 with maximum element order 12, so line orders 15/30/60 are absent for EVERY realization of the clocks; and the fiber product itself is cyclic Z60 (exact SNF arithmetic; the full product Z12 x Z30 has invariant factors (6,60)) which cannot embed.  All three census controls fire (the CC2 sampler-widening correction disclosed and carried).  ROOTCLASS-MIXED (v775) binding in both parts: no line/code -> matter semantics.  No marker move, NO RH claim.  NOTE (2026-08-06, v807 -- marker UNCHANGED): the BOUNDARY.NULLSELECTOR addendum -- a second, independent d = 4 selector on the same invariant: the functor vector v_d = (d+1, -(d-1), d) (honest RR/Serre arithmetic) has |v_d|^2_L = det M(v_d) = d(4-d), null/rank-one iff d = 4, with M_4 = 2(1,2)^T(1,2) the certified spinor square and g^2 - N^2 = 4d = |H|^2 identically -- null <=> glue index = d; the glue census forces {4} via evenness/unimodularity, the null selector via 4d = d^2: independent inputs, same output; the missing premise (why the boundary reading carries the functor vector) is typed OPEN in v807. Scope note (Prime Front separation): v807 supplies only its original S1 RR/Serre/null-spinor algebra here; no prime-window hypothesis or additional physical selector follows.",
      "status": "Exact [E] lattice/discriminant-form census results and an exact [E] group-theoretic kill, both behind explicit fences; the c_3 link stays typed INTERPRETATION (no P1 derivation); no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "ONE-OBJECT-PARTIAL + CLOCK-FIBER-DEAD: even unimodular diagonal glue of A_{d-1}+D_{d+1} iff d = 4 (census 2..12, v92 reproduced, E8 certificate); g_car = 5 / N_fam = 3 / |mu4| = 4 as one-object corollaries; c3 = 1/(2 pi d)|_{d=4} typed corpus-legitimate rewriting, NOT a P1 derivation; the Z12 x_{Z6} Z30 clock action on the 60 lines dead at three levels (group order 23040, max element order 12); controls fire; no RH statement.",
      "dependencies": "v1; v23; v53; v92; v623; v629; v633; v634; ARF.ROOTCLASS.01",
      "scripts": [
        "v781_one_object_clock_census.py",
        "v807_lorentz_nullselector.py"
      ],
      "script": "v781_one_object_clock_census.py;v807_lorentz_nullselector.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 684
    },
    {
      "id": "E8.TRANSITIONBUS.01",
      "claim": "The transition-bus structure theorem (35/35 checks, ~2 s; discovery probe e8_transition_bus_probe.py TRANSITION-BUS-TORSOR, 2026-08-05).  The 240 roots of the unimodular hermitian Gaussian E8 lattice are 15 disjoint coordinate 8-frames (cross-polytopes) over the class map V = L/(1+i)L; on L/2L the quadratic form q(y) = <y,y>/4 mod 2 is well defined and the 120 root PAIRS are exactly the 120 q = 1 points of the plus-type quadratic space; the 8 pair-images of class v form a coset of (1+i) v_perp inside the 16-element fiber of L/2L -> V -- a canonical TORSOR under v_perp (an affine F2^3 with no distinguished origin); the deck J acts on each pair fiber as translation by v and -1 = J^2 acts trivially on L/2L.  THE OBSTRUCTION CENSUS IS PROOF-GRADE: -1 acts freely on the 240 roots but trivially on V x (V minus 0), and J acts on each 16-root fiber with cycle type 4+4+4+4 while any torsor translation has order <= 2 -- hence NO bijection R(E8) -> V x (V minus 0) can be equivariant under the deployed deck symmetries; the honest structured object is the pair-level torsor plus an unoriented sign (the strict 16-state labeling, if written down, is pure bookkeeping).  THE FROZEN PREDICATE (trained on a frozen 3-class-pair set, hashed, then verified): with the basepoint rule rho_v = lex-min pair image and C(v,v') = q(rho_v + rho_v'), the symplectic dictionary reproduces ALL 57600 ordered inner products, with the SIGN LAW part of the claim (the central -1 is invisible at every mod-(1+i)^2 level; every realized key carries the sign-symmetric value set) and the h-refinement ((q-bit, hbar) -> h value classes).  SPREAD LAYER: 240 = 5 x 3 x 16 (context x family direction x state) via the certified isotropic spreads with the arf-certified q* selector.  A15 COMPARISON QUANTIFIED: A15's 240 roots are LITERALLY the transitions of 16 states (a strict bus with a canonical start state); E8 preserves the count, the fibration, the frame and the translation involution but DOUBLES the cross-class non-orthogonality (26880 vs 13440 ordered pairs) and provably has NO start state.  Controls C1-C3 fire (the C3 control-design correction disclosed: the original label transposition was vacuous, corrected to a class-assignment scramble; frozen spec untouched).  ROOTCLASS-MIXED (v775) unaffected: no matter assignment.  No marker move, NO RH claim.  NOTE (2026-08-06, v800 -- marker UNCHANGED): the torsor structure acquires its canonical CHARACTER reading -- the 120 projective Fourier rays of the origin-free fibers are canonical (an origin shift is a global sign), orthogonal (Gram = 16 I), deck/Pauli-diagonal and G31-monomial through Sp(4,2); the 8 R-fibers' 64 rays factorize as (1+5+10) x (1+3) under W(D5) x W(A3) -- the carrier Pascal at character level, with the weight basis exactly mutually unbiased (ROOTCLASS-MIXED intact).  See E8.TORSOR.FOURIER.01.",
      "status": "Exact [E] finite structure theorem (torsor + predicate + obstruction census) behind the interpretation fence ('transition bus / instruction set' language stays interpretation); no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "TRANSITION-BUS-TORSOR: 15 coordinate 8-frames; pair fibers = canonical v_perp-torsors; NO deck-equivariant V x (V-0) labeling exists (proof-grade obstruction); the frozen symplectic predicate reproduces all 57600 inner products incl. the sign law; 240 = 5 x 3 x 16 spread layer; A15 comparison quantified; controls fire; no RH statement.",
      "dependencies": "v752; v689; ARF.SPINORCOMPILER.01; ARF.ROOTCLASS.01",
      "scripts": [
        "v782_e8_transition_bus.py"
      ],
      "script": "v782_e8_transition_bus.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 685
    },
    {
      "id": "E8.CLIFFORD2Q.01",
      "claim": "The two-qubit stabilizer identification of the compiler carrier (32/32 checks, ~4 s; discovery probe two_qubit_clifford_probe.py CLIFFORD-PARTIAL(H4), 2026-08-05).  H1 EXACT: under the single predeclared chart U (the coordinate-class frame v_k = 2 e_{2k} -> the computational basis; z_k = r_2k + i r_2k+1, J -> multiplication by i) the 60 Gaussian E8 lines ARE the 60 pure two-qubit stabilizer rays, entry-exactly over Z[i] -- 60/60 projective matches with support census 4 + 24 + 32 on both sides; no fallback frame needed.  H2 EXACT: U G31 U^-1 is an INDEX-2 subgroup of the two-qubit Clifford matrix group C2 = <H(x)I, I(x)H, S(x)I, I(x)S, CNOT> -- |C2| = 92160 = 2 x 46080 with coset zeta8 G31; projectively |C2/U(1)| = 11520 = |G31|/|mu4| (equal).  H3 EXACT (the semantics carry): 46080 = 4 x 16 x 720 = mu4 x Pauli16 x Sp(4,2), with ker(G31 -> Sp(4,2)) = mu4 * Pauli16 and the 16 Paulis mod phase acting trivially on the 16 code states of V = L/(1+i)L; the 15 Gaussian classes ARE the 15 maximal commuting Pauli contexts (the W(3,2) = GQ(2,2) polar-space identification: hbar(x,y) = 0 iff the contexts share a Pauli, all 105 pairs; v752's B transported); the 6 spreads are the MUB pentads (cross-context |<psi|phi>|^2 = 1/4 EXACT) pulling back to ovoids (the self-dual GQ(2,2) point-line duality).  H4 FAILS AS PREDECLARED (the honest partial): the index-2 character chi conjugates NEITHER J nor the mu4 phases (zeta8 is central) -- chi is the GALOIS sqrt2/HADAMARD bit Gal(Q(zeta8)/Q(i)), and it exchanges the two mu8-lift classes {mu4 R, zeta8 mu4 R} of the 240 roots; the J <-> -J orientation bit is the separate ANTIUNITARY outer Z2 (which fixes chi).  All four controls fire (Haar-random 0/60; the T-gate residual-torus twist gives EXACTLY the predeclared 16/60; the non-context controlled-Hadamard frame 12/60 < 60; seeded random rays 0/60) -- the frame alone does not force the identification, the lattice does.  Stabilizer-formalism mathematics about the carrier: the fired ROOTCLASS-MIXED kill (v775 / ARF.ROOTCLASS.01) is UNAFFECTED, no matter reading is made.  No marker move, NO RH claim.  NOTE (2026-08-06, v798 -- marker UNCHANGED): the missing coset zeta8 G31 is IDENTIFIED -- the per-binary-factor Hadamard is the metaplectic/Weil lift of modular S at tau = i (exact fixed-point relations; Weil constant zeta8), the total Fourier H (x) H lies inside G31 with anomaly (T3 K3)^3 = iI = J, so this row's sqrt2/Hadamard Galois bit is the tau = i metaplectic bit; the strict q*-fixing census over the coset is empty (the Arf defect unavoidable) with exactly one G31-conjugacy class of Theta-real metaplectic survivors.  See SEAM.CLIFFORD.MODULAR_S.01.",
      "status": "Exact [E] finite group/lattice identification (H1-H3) with the H4 layer honestly PARTIAL (the index bit identified as the sqrt2/Hadamard Galois bit, not J <-> -J); no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "CLIFFORD-PARTIAL(H4): the 60 Gaussian lines = the 60 two-qubit stabilizer rays entry-exactly over Z[i] (60/60, census 4+24+32); G31 = index-2 subgroup of the Clifford matrix group (coset zeta8 G31, projectively equal); 46080 = mu4 x Pauli16 x Sp(4,2) with classes = contexts and spreads = MUB pentads; the index bit = the sqrt2/Hadamard Galois bit (H4 as predeclared); controls fire; no matter reading; no RH statement.",
      "dependencies": "v689; v752; v634; E8.GAUSSIAN.CODE.01; ARF.ROOTCLASS.01",
      "scripts": [
        "v783_two_qubit_clifford.py"
      ],
      "script": "v783_two_qubit_clifford.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 686
    },
    {
      "id": "CURVE.CODE.2TORSION.01",
      "claim": "The curve/code 2-torsion decider (33/33 checks, ~9 s; discovery probe curve_code_2torsion_probe.py CURVE-CODE-PARTIAL, 2026-08-05).  The frozen CM-model route through the seam curve C: y^3 = x^4 - 1 (genus 3, Jacobian splits per v610): exact differential characters (zeta^11, zeta^7, zeta^10) reproduce v610 L1; A = the zeta_12 CM abelian surface with EVERY generator period certified as an integral O = Z[zeta_12] recognition (mpmath dps 120, residual tolerance 1e-35, recognition margin > 10^25 x residual, HNF index 1 -- the numerics END in certified integer recognitions); A[2] = O/2O = F2^4 exact from there on; the deck tau = multiplication by zeta^4 is DERIVED as the only order-3 automorphism class (Aut(C) = C12, monodromy forcing); the minimal ODD tau-invariant polarization is type (1,3) with Pfaffian 3 -- NO principal form exists in the tau-invariant cone (unit-sign obstruction: the fundamental unit 2+sqrt3 of the real subfield is totally positive) and the curve-induced Prym type (1,2) is mod-2 DEGENERATE (typed [C]); q_Theta = the unique tau-fixed quadratic refinement (exists and is unique since N(omega-1) = 9 is odd).  THE DECIDER: (A[2], e_2) and (V, hbar) are isomorphic as BARE symplectic F2^4 (720 = |Sp(4,2)| transporters -- content-free), but the full correspondence DIES at two exact layers: (1) SIGMA LAYER: tau acts FREELY on A[2] (fixed space 0, realization-independent) with cycle type [3,3] on the 6 odd theta characteristics (SYNTHEME side), while the code family cycle sigma-bar fixes a 2-plane with cycle type [3,1,1,1] (DUAD side) -- the two order-3 classes are exchanged only by the OUTER automorphism of S6, never by conjugation; equivariant census 0 (tau and tau^2).  (2) REFINEMENT LAYER: the curve's canonical deck-invariant theta refinement q_Theta is EVEN (Arf 0, 10 zeros, stabilizer 72, decomposition 1+6+9) while the code's arf-certified selector q* is ODD (Arf 1, 6 zeros, stabilizer 120, decomposition 1+5+10) -- Arf is a transport invariant, census 0.  Joint census 0 with the positive control (code clone) at exactly 6 = the code-side stabilizer order; all must-fire controls fire.  ROOTCLASS-MIXED (v775) fence respected: finite exact algebra on 2-torsion, no physical labeling.  No marker move, NO RH claim.  NOTE (2026-08-06, v796 -- marker UNCHANGED): the INNER identification stays dead exactly as recorded, and the OUTER twist is now decided CANONICAL (CURVE.CODE.OUTERSPIN.01 / v796): the Sylvester bridge beta with beta sigma = tau beta and beta(q*) = S* is unique up to the order-6 joint stabilizer, pulls q_Theta back to q_even* = q* + hbar(., A) (the anchor shift resolves the Arf mismatch), and transports 1+5bar+10 / 3+2 / t_A -> rho = J exactly -- both breaking layers of this row are thereby explained as the two faces of the outer automorphism of S6, not repaired.",
      "status": "Exact [E] finite algebra + certified-recognition CM layer; the correspondence honestly PARTIAL with both breaking layers named exactly (syntheme-vs-duad and Arf 0-vs-1); no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "CURVE-CODE-PARTIAL: (A[2], e_2) = (V, hbar) as bare symplectic spaces only; the correspondence dies at two exact layers -- deck acts syntheme-side [3,3]-free vs code duad-side [3,1,1,1] (outer S6), and curve theta even/Arf-0 (1+6+9) vs q* odd/Arf-1 (1+5+10); joint census 0 with positive control 6; every period certified as integral O-recognition; no RH statement.",
      "dependencies": "v610; v611; v752; v753; ARF.SPINORCOMPILER.01; ARF.ROOTCLASS.01",
      "scripts": [
        "v784_curve_code_2torsion.py"
      ],
      "script": "v784_curve_code_2torsion.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 687
    },
    {
      "id": "FTRANSFER.KMSCOSET.01",
      "claim": "the coset assignment of the fibered clock functor is a THEOREM from KMS/modular typing (18/18 checks incl. the three must-fire controls, ~3 s; discovery probe ftransfer_kms_coset_probe.py, 2026-08-05, verdict KMS-COSET-THEOREM; frozen before the first run).  THE THEOREM: S_transported = -(ln lambda)^2/2 with lambda the PGL2 multiplier of the flow's time-1 holonomy per clock e-fold -- the coset is COMPUTED from the typing datum lambda, never case-inspected.  THE CLASSIFIER uses NO Schwarzian and NO derivative of any order (non-circularity gate on its own source): (i) the holonomy grounded as an exact functional identity y(x+1) = T(y(x)), (ii) the PGL2 class of T (scalar -> degenerate; parabolic -> autonomous, lambda = 1; else hyperbolic candidate), (iii) the detailed-balance multiplier cross-ratio m(x) -- the discrete KMS signature -- exactly constant -> modular with lambda = m.  DERIVATION CHAIN sympy-exact: Lemma A {M(e^{ax}), x} = -a^2/2 identically for GENERIC Moebius M; Lemma B {M(x), x} = 0; Lemma C -a^2/2 = -Delta^2/2 iff a = +-Delta, affine dictionaries change nothing.  BLIND PROTOCOL: the four deployed clock charts classify F_pole/F_Boltzmann MODULAR with |ln lambda| = Delta = 6 log(3/2) exactly, F_QCD AUTONOMOUS (parabolic), F_relic DEGENERATE; unblinding: predicted cosets == transported Schwarzians EXACTLY (sympy zero) -- the fiber classification of the executed contract (v777, FTRANSFER-FIBERED-CARRIES) is now DERIVED.  CONTROLS: C1 fake-KMS witness caught (functional identity fails AND the ratio is non-constant); C2 affine reparametrization x -> -x + c leaves typing and |ln lambda| unchanged; C3 the v578 Moebius clock t = e^{Delta sigma} flips the QCD chart to modular and the theorem PREDICTS -Delta^2/2, confirmed by the transported Schwarzian -- the K1-duty transplant as the positive case.  Completes the three-probe chain executor -> fibered groupoid -> coset theorem: the CONTRACT.F.01 fibered refinement's clock-fiber classification is a derived statement.  Freeze guards: prereg YAML + both data tables byte-identical.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Theorem-grade [E] sympy-exact derivation chain + blind classification with the non-circularity of the classifier gated on its own source; completes the fiber classification of the executed clock contract; no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "KMS-COSET-THEOREM: S = -(ln lambda)^2/2 from the time-1 PGL2 holonomy multiplier per clock e-fold; classifier Schwarzian-free and derivative-free (non-circularity gated); blind assignment F_pole/F_Boltzmann modular (|ln lambda| = Delta), F_QCD autonomous, F_relic degenerate; unblinding exact (sympy zero); controls fire incl. the Moebius-clock positive case; the executor -> groupoid -> coset chain whole; no RH statement.",
      "dependencies": "CONTRACT.F.01; FTRANSFER.CLOCKS.01; v777; v578; v99; v425",
      "scripts": [
        "v792_ftransfer_kms_coset.py"
      ],
      "script": "v792_ftransfer_kms_coset.py",
      "location": "tfpt_research_contracts;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 688
    },
    {
      "id": "CURVE.CODE.OUTERSPIN.01",
      "claim": "the canonical OUTER bridge between the code's duad side and the curve's syntheme side of the S6 duality (24/24 checks, ~11 s; discovery probe duad_syntheme_bridge_probe.py, 2026-08-05, verdict BRIDGE-CANONICAL).  THE CONSTRUCTION (Sylvester duality made explicit): a bridge is a bijection beta: Omega_code -> T_curve (code odd Arf forms -> curve isotropic spreads/totals), canonically inducing the incidence-preserving duality Delta_beta: V\\{0} -> IsoLines(A[2]) through the K6 duad model and the shared-syntheme rule (every pair of totals shares exactly one line -- verified).  THE CENSUS: 18 + 18 bridges intertwine sigma with tau|T resp. tau^2|T (each family a torsor under C_S6(3-cycle), order 18); after pinning beta(q*) = S* -- the UNIQUE curve spread fixed by the FULL <tau, rho> automorphism image (rho = the mu4 rotation = the complex structure J = mult by zeta^9; existence + uniqueness COMPUTED, not assumed) -- exactly 6 + 6 remain = ONE ORBIT (torsor) under the order-6 code joint stabilizer, with the tau/tau^2 families exchanged by an inner code symmetry (one statement).  B2 ARF RESOLUTION: the pullback of the curve's even q_Theta through the two frozen even<->3|3-split dictionaries is CONSTANT over all 36 B1 bridges and equals q_even* = the unique sigma-invariant even code refinement, with the NAMED operation q* = q_even* + hbar(., A) EXACT -- the anchor shift resolves the v784 Arf mismatch canonically.  THE OUTER WITNESS exact: sigma is [3,1,1,1] on odds but [3,3] on spreads, tau the reverse, the anchor transvection t_A is [2,1,1,1,1] on odds but [2,2,2] on spreads, and the class swap is UNIFORM (0 violations over all 720 group elements) -- the twist IS the outer automorphism of S6, not an inner accident.  TRANSPORTS (structure labels only): {q* = 0}\\{0} maps EXACTLY onto the 5 lines of S* (5bar) and the 10 onto the complement; the 3+2 split maps onto the [3,1,1] tau-orbit structure of S*; t_A maps to rho = J on every census bridge; chi_NSR transports as an honest exact TWO-VARIANT torsor ('incidence with one of the two tau-fixed lines of S*') -- it is pinned code-side by the v752 NS/R lattice parity, a datum the bare quadruple (V, hbar, sigma, q*) does not carry, and rho swaps the two candidate lines (the two-fold ambiguity intrinsic).  CONTROLS: inner cross-side census 0 (class mismatch), same-side census 0, no bridge pulls q_Theta to an ODD form (never to q*), scrambled sigma census 0.  The v784 INNER identification stays dead (CURVE-CODE-PARTIAL untouched); the bridge is a NEW statement about the outer twist.  NO matter semantics (5bar/10, 3+2, chi_NSR are structure labels; ROOTCLASS-MIXED v775 stands).  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.  CONTRACT DECIDED (registered here as the decided outer-bridge statement): the outer identification is CANONICAL; the residual is the intrinsic two-variant chi_NSR ambiguity (pinned only by the v752 NS/R lattice parity, which the bare 2-torsion quadruple does not carry) -- typed, not open work; cross-referenced with CURVE.CODE.2TORSION.01, whose inner-identification kill stands unchanged.",
      "status": "Exact [E] finite census theorem (torsor uniqueness incl. the tau/tau^2 swap; constant Arf resolution; exact transports) behind the structure-label fence; the inner identification stays dead (v784); no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "BRIDGE-CANONICAL: the Sylvester duality beta (code odd forms -> curve spreads) with beta sigma = tau beta and beta(q*) = S* is unique up to the order-6 joint stabilizer (18 + 18 -> 6 + 6, one orbit incl. the tau/tau^2 swap); q_Theta pulls back constantly to q_even* = q* + hbar(., A) -- the anchor shift resolves the v784 Arf mismatch; 5bar/10, 3+2 and t_A -> rho = J transport exactly; chi_NSR an exact two-variant torsor; inner identification stays dead; no RH statement.",
      "dependencies": "CURVE.CODE.2TORSION.01; ARF.SPINORCOMPILER.01; v784; v774; v752; v610; v611; ARF.ROOTCLASS.01",
      "scripts": [
        "v796_duad_syntheme_bridge.py"
      ],
      "script": "v796_duad_syntheme_bridge.py",
      "location": "tfpt_1;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 689
    },
    {
      "id": "SEAM.CLIFFORD.MODULAR_S.01",
      "claim": "the missing Clifford index-2 bit IS the metaplectic lift of the modular S transformation at its fixed point tau = i -- coset decision positive, deck placed, anchor named; the frozen uniqueness decider fails sharply (27/27 checks, ~50 s; discovery probe seam_clifford_modular_s_probe.py, 2026-08-06, verdict MODULAR-S-PARTIAL).  COSET DECISION POSITIVE (exact cyclotomic arithmetic): the per-binary-factor Weil/metaplectic lift K1 = H (x) I (and K2 = I (x) H) lands EXACTLY in the missing coset zeta8 G31 of v783 (BFS membership certificates for zeta8^{-1} K_i in G31), with the exact tau = i fixed-point relations S^2 = +I, T^2 = the Pauli parity element, and (T K)^3 = zeta8 I -- the Weil constant gamma = zeta8, c = 1 per binary factor.  R2 SOLVED (deck placement): the TOTAL symplectic Fourier K3 = H (x) H (the toric-code modular S) lies INSIDE G31 (chi = 0) and its metaplectic anomaly is (T3 K3)^3 = i I = J -- THE DECK IS THE METAPLECTIC ANOMALY OF THE TOTAL MODULAR LIFT (c = 2); zeta8 = (1 + J)/sqrt2 is the real orthogonal HALF-DECK ((1+J)^2 = 2J exact) swapping the seam pair {L, zeta8 L}: lift = half-deck turn x lattice automorphism.  The Z4 deck-clock Fourier F4 is NOT a carrier symmetry (fails the Clifford gate; typed out).  OS reflection gate exact: Theta K1 Theta = K1 and Theta T1 Theta = T1^{-1} -- the antiunitary reality involution acts as the GL(2,Z) orientation flip tau -> -tau-bar, which FIXES tau = i.  THE FROZEN UNIQUENESS DECIDER FAILS SHARPLY (the PARTIAL): the strict census over all 46080 coset elements (order gate 480 -> Theta-reality 96 -> q*-fixed + metaplectic pairing) is EMPTY (0/46080) -- NO Theta-real metaplectic involution in the missing coset fixes the v774 Arf refinement q*: the Arf defect q* -> q* + hbar(., c) is UNAVOIDABLE; the retyped census (defect recorded instead of forbidden) has 32 survivors = 16 (+-1)-pairs = EXACTLY ONE G31-conjugacy class (w1 ~ w2; tier-A 4 = the Pauli/phase gauge fiber of H (x) I).  R3 NAMED, NOT FIXED: the sigma-orbit of lifts {w1, sigma w1 sigma^{-1}, w2} has Arf defects EXACTLY {A + F_Sigma + F_i : i = 1, 2, 3} (one per family) whose XOR is THE ANCHOR A itself; the total Fourier's defect is A + F_Sigma; every retyped survivor's defect carries the anchor flag chi_NSR(c) = 1 -- the lift does not fix the anchor, it NAMES it through the Arf defects.  CONTROLS: CZ (real, order 2, Clifford, non-metaplectic) lies INSIDE G31 (chi = 0); the wrong phase convention zeta8 K1 breaks S^2 in {+-I} ((zeta8 K1)^2 = iI); census selectivity 480 -> 96 -> 32 of 46080.  ROOTCLASS-MIXED (v775) unaffected -- finite stabilizer/Weil mathematics about the carrier, no physics claim.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.  CONTRACT (registered, partial state): coset decision POSITIVE and R2 solved (the deck = the metaplectic anomaly of the total Fourier); R3 NAMED through the sigma-orbit Arf defects (XOR = the anchor A); remaining: derive the physical selection of the defect class (the anchor naming from seam data) or type the two-variant/defect residue as intrinsic gauge; kill: a Theta-real metaplectic coset element fixing q* (excluded here by the exhaustive 0/46080 census) would falsify the defect law.",
      "status": "Exact [E] cyclotomic coset/metaplectic identities (BFS certificates; exact tau = i relations) with the strict uniqueness census EMPTY by its own frozen gate -- the Arf defect unavoidable, a sharp typed fact; R2 placed, R3 named; no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "MODULAR-S-PARTIAL: the per-factor Hadamard IS the metaplectic S-lift at tau = i, landing exactly in the missing coset zeta8 G31 (S^2 = I, T^2 = Pauli parity, (TK)^3 = zeta8 I); the deck J = the metaplectic anomaly of the total Fourier (R2 solved); zeta8 = (1+J)/sqrt2 the half-deck swapping the seam pair; strict census EMPTY (0/46080 fix q*), one G31-conjugacy class of retyped survivors, sigma-orbit Arf defects XOR to the anchor A (R3 named, not fixed); no RH statement.",
      "dependencies": "E8.CLIFFORD2Q.01; ARF.BOUNDARY.CODE.01; ARF.SPINORCOMPILER.01; v783; v774; v752; v689; ARF.ROOTCLASS.01",
      "scripts": [
        "v798_seam_clifford_modular_s.py"
      ],
      "script": "v798_seam_clifford_modular_s.py",
      "location": "tfpt_1;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 690
    },
    {
      "id": "SEAM.CODE.TYPEII.01",
      "claim": "Type II (doubly-even self-dual) is FORCED from three physical seam axioms -- the direct attack on residual R1 of ARF.BOUNDARY.CODE.01, decided exhaustively (19/19 checks, ~5 s; discovery probe seam_code_typeii_probe.py, 2026-08-06, verdict TYPEII-FORCED).  THE THREE EQUIVALENCE CENSUSES, exhaustive over ALL 308,993 subspaces of F2^8 of dim 0..4 (totals = the Gaussian binomials 1, 255, 10795, 97155, 200787; dim > 4 excluded by rank-nullity): (1) MUTUAL LOCALITY of the Construction-A boundary vertex words <=> C self-orthogonal, ZERO exceptions (braiding exponent <(c+2m)/sqrt2, (c'+2m')/sqrt2> = c.c'/2 + Z -- the lift-shift identity verified exhaustively; word-pair census all 65536 pairs; survivors per dim 1/28/165/280/135); (2) INTEGER CONFORMAL SPIN (all T sector phases i^{wt(c)} trivial, |c+2m|^2 == wt(c) mod 4 exhaustive; the eta^8 multiplier typed universal/code-independent) <=> C doubly even, ZERO exceptions (survivors 1/8/38/52/30); (3) HOLOMORPHY / INDEX ONE (|disc| = det(rescaled Gram) = 2^{8-2k} exact in Fractions; |disc| = 1 = single modular-closed sector) <=> C self-dual <=> k = 4 -- and EVERY non-self-dual doubly-even code has a word in C-perp \\ C of weight != 0 mod 4, i.e. a discriminant sector of FRACTIONAL conformal weight on which the S-image cannot close (proved exhaustively; detailed witness: the [8,3] subcode of the boundary-selected C* with exact coset theta series to h <= 2, theta_{L*} != 2 theta_L).  THE KILL EXCLUDED: the survivor set of (locality, integer spin, |disc| = 1) over ALL subspaces IS the Type II set = the 30 e8-hat copies (weight enumerator 1 + 14x^4 + x^8); no boundary-legal non-Type-II code exists at length 8.  R1 -> R1': the universe premise 'the boundary realizes a doubly-even self-dual code' is REPLACED by the three physical axioms A1 mutual locality (trivial monodromy / bose statistics), A2 integer conformal spin (bosonic T-alignment), A3 holomorphy / index one (single modular-closed sector) -- each a standard physical demand, not a coding-theory premise; given A1-A3 the chain boundary axioms + deck + orientation bit -> unique code -> E8 -> F2^4 -> g_car = 5 stands with this single sharpened [C] residual.  IMPORTED INGREDIENTS typed and machine-verified in every instance used (I1 Frenkel-Kac-Segal braiding factor, FLM 1988/Kac, Adamo-Moriwaki-Tanimoto arXiv:2407.18222; I2 h = |lambda|^2/2 lattice-VOA grading; I3 Conway-Sloane/Ebeling theta S-transform with L* = A(C-perp) verified); the OPE-linearity scope note (admissible word sets are linear codes) typed as a scope assumption.  Controls: braiding monodromy -1 witness; T-phase witnesses i and -1; the [8,3] S-closure violation.  Uniqueness (30 copies; the boundary data select ONE) CITED from v776, count echo verified.  NO P2 marker move.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.  CONTRACT (registered; the sharpened [C] residual replacing R1 of ARF.BOUNDARY.CODE.01): the physical seam satisfies A1 mutual locality (trivial monodromy of admissible boundary vertex words), A2 integer conformal spin (bosonic T-alignment), A3 holomorphy/index one (a single modular-closed sector); given A1-A3, Type II is FORCED at length 8 (this module), e8-hat is unique (30 copies) and the v776 boundary data select one copy -> E8 -> F2^4 -> g_car = 5; kill: a seam construction violating A1-A3 while staying boundary-legal, or a boundary-legal non-Type-II code at length 8 (excluded exhaustively here).",
      "status": "Exact [E] exhaustive finite equivalence chain at length 8 (all 308,993 subspaces) modulo the cited classical lattice-VOA/theta ingredients, each machine-verified in every instance used; the OPE-linearity scope note typed; R1 -> R1' [C] residual registered; NO P2 marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "TYPEII-FORCED: exhaustively over all subspaces of F2^8 (dim <= 4), locality <=> self-orthogonal, integer spin <=> doubly even, S-closure/|disc| = 1 <=> self-dual; survivor set = exactly the 30 e8-hat copies; no boundary-legal non-Type-II code; R1 -> R1' = the three physical axioms A1-A3; imported ingredients classical, cited, instance-verified; NO P2 marker move; no RH statement.",
      "dependencies": "ARF.BOUNDARY.CODE.01; AX.P2.01; ARF.SPINORCOMPILER.01; v776; v752; E8.GAUSSCODE.01",
      "scripts": [
        "v799_seam_code_typeii.py"
      ],
      "script": "v799_seam_code_typeii.py",
      "location": "tfpt_1;tfpt_2;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 691
    },
    {
      "id": "E8.TORSOR.FOURIER.01",
      "claim": "the projective Fourier modes of the 15 Gaussian root torsors -- carrier-block purity at CHARACTER level with the mutually-unbiased honest cap (30/30 checks, ~60 s; discovery probe e8_torsor_fourier_probe.py, 2026-08-06, verdict TORSOR-FOURIER-PARTIAL; FROZEN_SPEC SHA-hashed before any root data).  THE CONSTRUCTION: because the v782 class fibers are torsors WITHOUT origin, their character data is canonical where their point data is not -- 120 = 15 x 8 projective Fourier rays m(v, u) with coefficients (-1)^{hbar(u, x(r))} on the 16-root fiber; PHASE LAW EXACT on all 960 origin shifts (a shift by t multiplies by the GLOBAL sign (-1)^{hbar(u,t)}: the rays are canonical); orthogonal ray basis of the (-1)-symmetric 120-dim pair space (integer Gram = 16 I; the antisymmetric half receives NO canonical mode -- the v782 sign theorem).  EQUIVARIANCE: deck J diagonal with eigenvalue (-1)^{hbar(u,v)} (60 even + 60 odd, 4 + 4 per fiber); ALL 16 Pauli translations diagonal by characters (15 x 15 x 8 exact); G31 MONOMIAL (7200 generator matches; label action closes to Sp(4,2), order 720, kernel 64 = mu4 x Pauli16; label orbits {15 uniform, 45 deck-even, 60 deck-odd}); Burnside exact (transitive on pairs and on lines); sigma (Z3) multiplicities integral.  THE CARRIER BLOCK CLEAN AT CHARACTER LEVEL (P1a/P1b/P2): the spinor sector = EXACTLY the 8 R-fibers (split-independent), so M_S = the span of their 64 whole Fourier rays -- a Fourier-ALIGNED invariant block (impossible at point level per v775); M_S factorizes as a W(D5) x W(A3)-set as 16 x 4 (W(D5)/W(A3) rebuilt by BFS from the certified v775 split, 1920/24), W(D5) transitive of permutation rank 3 on the 16 half-spinor weight pairs with the folded-5-cube/CLEBSCH charpoly (x-5)(x-1)^10(x+3)^5 EXACT => isotypic dimensions 1 + 5 + 10 -- the carrier Pascal AT CHARACTER LEVEL -- and the tetrad is S4-NATURAL (faithful, transitive, stabilizer S3, permutation rep 1 + standard 3) with sigma = the family 3-CYCLE + fixed anchor; W(A3)-tables exact (the 20 D5-adjoint pairs family-invariant; M_S = (1+5+10) x (1+3)); W(D5) Burnside on the spinor pairs: 4 orbits (the tetrad), sum m_i^2 = 48.  P3 FAILS AS PREDECLARED (the honest cap): every mode has full 16-root support and |<pair|mode>|^2 = 1/8 EXACTLY on every cell -- the D5-weight basis and the character basis are MUTUALLY UNBIASED on each carrier fiber; no weight axis is a Fourier ray; the pointwise code -> matter reading stays dead (ROOTCLASS-MIXED v775 intact, re-confirmed).  OVERLAP CENSUS: every certified stabilizer ray expands exactly in the 4 deck-even modes of its fiber (squared overlap 1/4 each; deck-odd and foreign fibers orthogonal -- the deck-odd half is a pure phase register invisible to stabilizer rays); the 45 nontrivial deck-even modes ARE, under the certified chart U, the 45 (context, Pauli) incidences (each nonzero Pauli covered exactly 3x; uniform modes -> identity, Sum z z^dag = 4I) -- a canonical G31-equivariant operator dictionary, noted (report only) as exactly the SHAPE the v791 carrier-intertwiner demand asks for on the finite register.  CONTROLS: fiber scramble destroys deck equivariance; the wrong character group F2^4 = V collides 2:1 (rank 8 < 16); seeded random sign systems fail the certificates.  Everything exact integer/Fraction/Z[i]/F2 (sympy gated to two charpolys).  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact [E] integer/cyclotomic character theory (canonical rays, exact Clebsch spectrum, exact overlap census) with the predeclared P3 failure typed -- the weight and character gradings mutually unbiased; ROOTCLASS-MIXED (v775) intact and re-confirmed; no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "TORSOR-FOURIER-PARTIAL: 120 canonical projective Fourier rays (phase law exact on 960 shifts, Gram = 16 I); deck/Pauli diagonal, G31 monomial through Sp(4,2); the carrier block = the 8 R-fibers' 64 rays = (1+5+10) x (1+3) under W(D5) x W(A3) with sigma = family 3-cycle + anchor -- the carrier Pascal at character level, impossible at point level (v775); weight/character bases exactly mutually unbiased (the honest cap); the 45 deck-even modes = the 45 context Pauli observables under the certified chart; no RH statement.",
      "dependencies": "E8.TRANSITIONBUS.01; E8.CLIFFORD2Q.01; ARF.ROOTCLASS.01; PRIME.POSITIVE_DESCENT.01; v782; v783; v775; v752; v791",
      "scripts": [
        "v800_e8_torsor_fourier.py"
      ],
      "script": "v800_e8_torsor_fourier.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 692
    },
    {
      "id": "GNET.MARTINGALE.LIMIT.01",
      "claim": "the noncommutative-martingale route to the G_net GNS limit -- its finite hypotheses MEASURED, all met (15/15 checks, controls 4/4, ~7 s; discovery probe gnet_martingale_probe.py, 2026-08-06, verdict MARTINGALE-HYPOTHESES-MET).  THE PROGRAMME: replace continuum heuristics by martingale convergence -- if the local-state defects along the certified Haar inductive system are SUMMABLE in trace norm, the limit state exists uniquely by elementary Banach telescoping; the operator-algebra content becomes a typed list of analytic steps with finite evidence in hand.  S1 FILTRATION EXACT: Haar isometry V*V = 1 with support coherence iota(win(N,0,l)) inside win(2N,0,2l) EXACT (leak 0); the Watatani quasi-basis explicit at the predeclared depths (185 elements at l = 2, 3041 at l = 3, sum v v* = 4*1); martingale compatibility omega_k o E = omega_k at every level; window data literally level-independent (H_W identical at N = 48 and N = 3072 -- one index-4 expectation for the whole tower).  HONEST TYPING: the omega_k are an ALMOST-martingale (approximate, with measured defects); the convergence mechanism is norm summability + dual-space completeness (Umegaki expectations; Cuculescu cited as context, not used).  S2 SUMMABLE DEFECTS (load-bearing): delta_k = ||rho_{k+1} - rho_k||_1 geometric with fitted rates 0.250 / 0.263 / 0.265 per doubling for l = 1/2/3 -- ONE POWER BETTER than the 2^{-l} dim-law envelope (reported against 0.5, not forced; bar 0.55); envelope delta_k <= 8 delta_1 2^{-k} quantifies summability (l = 2 tail sum 2.9e-2).  S3 TELESCOPING / UNIQUE LIMIT: the Cauchy tail bound holds on every rung against the Richardson extrapolant; the frozen 6-observable battery gaps fall at <= 0.6; extrapolation uniqueness within 2.25 delta_5.  S4 SPLIT/NORMALITY PRECURSORS uniform along the tower: sector-resolved cross-window profiles anchored with shrinking tails; determinant witness D_k = det(1 - X X*) >= 0.985 and stabilizing at 0.992756 -- the uniform estimate a normality/split argument consumes.  STEPS 5-7 TYPED, NOT EXECUTED (the analytic remainder): GNS reconstruction of omega_inf, local normality (S4 witnesses as the finite input), split property + the mu4 simple-current extension (GATE.METRIC.11) on the limit net.  THE QGEO ARCHITECTURAL NOTE (typed, no gate): the open majorant residues (b) 'L2 uniform FH/RH' and (c) 'L3 uniformity' of QGEO.GRIDSUP.01/v745 are NOT needed for limit EXISTENCE (summable state-norm defects suffice) -- the route relocates them into the typed normality step: an architectural reallocation, not a proof of (b)/(c).  CONTROLS: the shifted-bond isometry leaks out of the doubled window (0.707 > 0.3 -- the filtration breaks); the Z2-only average has lambda = 1/2 != 1/4; the scrambled sea gives O(1) defects (summability breaks); DECIMATION CONVERGES TO THE WRONG DEGENERATE LIMIT (||rho_dec - 1/16||_1 < 1e-3 vs Haar 0.1+) -- a summable filtration with the wrong bond data forgets the sea: the coherent Haar bond is load-bearing.  Every measurement is finite-level evidence FOR the cited continuum steps, never the theorems.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.  CONTRACT (registered OPEN [O], the analytic remainder = steps 5-7): GNS reconstruction of omega_inf on the quasi-local algebra (existence/uniqueness finite-level delivered here); local normality of the GNS representation on each A(W) (nuclearity-type estimate; the uniform S4 witnesses are the finite input); split property and the mu4 simple-current extension (GATE.METRIC.11, [B:A] = 4 = |mu4|) on the limit net; kill: non-summable defects at a deeper reachable rung or a determinant-witness collapse.",
      "status": "Measured [MEASURED] martingale hypotheses under frozen gates (all met; rate one power better than the dim-law envelope, reported not forced) + typed analytic remainder; the QGEO majorant residues relocated into the normality step, not proven; no marker move (Python-only per GATE.WOLFRAM.02).",
      "canonicalStatus": "MARTINGALE-HYPOTHESES-MET: exact Haar filtration with one index-4 expectation for the whole tower; state defects summable at measured rate ~0.25 per doubling (<= the 2^-l envelope, one power better); unique limit by telescoping verified on every rung; split/normality precursors uniform (determinant witness >= 0.99); decimation converges to the WRONG degenerate limit (coherent bond load-bearing); QGEO residues (b)/(c) bypassed for existence, relocated into normality; steps 5-7 typed open; no RH statement.",
      "dependencies": "GNET.GNSLIMIT.01; GATE.METRIC.11; QGEO.GRIDSUP.01; v779; v746",
      "scripts": [
        "v802_gnet_martingale.py"
      ],
      "script": "v802_gnet_martingale.py",
      "location": "tfpt_2;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 693
    },
    {
      "id": "E8.RAMIFIED.JETCODE.01",
      "claim": "E8.RAMIFIED.JETCODE.01 (+ E8.AFFINE.NSR.01): the jet-code normal form of the Gaussian E8 compiler + the NS/R bit as an affine 8-point geometry, ONE module from two probes (18/18 + 15/15 checks, ~3 s; discovery probes e8_ramified_jetcode_probe.py JETCODE-EXACT and e8_affine_nsr_probe.py AFFINE-NSR-EXACT, both 2026-08-06, FROZEN_SPEC SHA-hashed before any root data).  PART 1, THE JET CODE (all [E neu], machine-certified): Z[i]/(2) = F2[eps]/(eps^2) with eps = 1 + i and 2 = -i(1+i)^2 (2 RAMIFIES; the ring iso is UNIQUE -- the only nonzero nilpotent is 1+i -- and the wrong split ring F2 x F2 is NOT isomorphic: all 24 bijections fail, control C1); W = L/2L is FREE of rank 4 over the dual numbers (explicit Z[i]-basis by Hermite reduction over the Euclidean domain Z[i], covolume-certified index 1; the 4^4 = 256 dual-number combinations enumerate W bijectively); 0 -> eps W -> W -> V -> 0 is an exact SELF-extension with V = eps W canonically and ker(eps) = im(eps) -- the self-dual differential code; the deck IS the jet unit J = 1 + eps, whence the certified v782 transition-bus torsor translation rho(Jr) = rho(r) + iota(v) on ALL 240 roots (the full 15-row fiber-translation table re-derived from ring algebra); the extension admits ZERO mu4-equivariant splittings among all 65536 = 16^4 linear sections (-1 = +1 mod 2 makes mu4-equivariance = J-equivariance) -- the no-canonical-origin obstruction IS non-splitness; the 120 root pairs are EXACTLY the q = 1 unit shell of the plus-type quadratic form (census 120/136, Arf 0, polarization b = <x,y>/2 mod 2 with trivial radical), so every root is canonically the triple (address v, admissible error jet in the affine v_perp coset, orientation +-): 240 = 15 x 8 x 2.  PART 2, THE AFFINE NS/R GEOMETRY (A1-A6 [E neu], chain reading [H neu]): V \\ {0} = (7 NS classes = ker chi_NSR \\ 0) u (8 R classes = the affine F2^3-torsor A = chi^{-1}(1)); the 7 NS classes carry EXACTLY the 112 D8 roots and the 8 R classes the 128 Ramond spinor roots -- 248 = (112 + 8) + 128 anchored in v148/v227/v301; Stab_GL(4,2)(chi_NSR) has order 1344 = 20160/15 and EQUALS -- as a permutation group with its natural 8-point action -- AGL(3,2) = Aut(H8) (brute force over all 40320 coordinate permutations; the chart composition is the explicit iso of actions); the evaluation code of affine functions on the 8 points of A is the [8,4,4] code RM(1,3) = H8 with EXACTLY 1344 equivalence permutations; the deployed Sp(4,2) stabilizes chi in an order-48 subgroup acting INSIDE AGL(3,2) (the kill does not fire).  HONEST TYPING: the reconstruction chain NS/R bit -> affine 8-point space -> H8 SHARPENS the ARF.BOUNDARY.CODE.01 attack surface but the route to V still passes through Gaussian E8 -- an exact internal reconstruction, NOT a P2 removal.  Controls fire in both parts (wrong ring, seeded scrambles, unique parity functional 14/14, flipped code bit).  No matter semantics (v775 ROOTCLASS-MIXED cited).  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact [E] ring/module/extension structure theorems (integer/F2/Z[i] arithmetic, exhaustive censuses) behind the interpretation fence; the reconstruction-chain reading typed [H neu]; carries E8.AFFINE.NSR.01",
      "canonicalStatus": "JETCODE-EXACT + AFFINE-NSR-EXACT: Z[i]/(2) = F2[eps]/(eps^2) unique iso; W = L/2L free rank 4 (Hermite-certified); exact self-extension with ker eps = im eps; J = 1 + eps reproduces the v782 torsor on all 240 roots; 0/65536 mu4-equivariant splittings -- non-splitness IS the no-origin obstruction; 240 roots = oriented q = 1 unit jets (15 x 8 x 2); Stab(chi_NSR) = AGL(3,2) order 1344 = Aut(H8), same natural 8-point action; affine functions = RM(1,3) = H8 (1344 equivalences); typed internal reconstruction, no P2 removal; no matter semantics.",
      "dependencies": "E8.TRANSITIONBUS.01; ARF.BOUNDARY.CODE.01; v752; v689; v782; v776; v646; v148; v227; v301",
      "scripts": [
        "v803_e8_jetcode_affine_nsr.py"
      ],
      "script": "v803_e8_jetcode_affine_nsr.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 694
    },
    {
      "id": "E8.SYNDROME.ALGEBRA.01",
      "claim": "E8.SYNDROME.ALGEBRA.01: the honest kill -- E8 is NOT an error-correction hull in the sharpest sense, one probe (13/13 checks incl. the 3 must-fire controls, ~6 s; discovery probe e8_syndrome_algebra_probe.py, 2026-08-06, verdict SYNDROME-DEAD, the kill fired exactly as preregistered).  THE QUESTION: do the 240 root operators U_r |x> = i^{c(r).x} |x XOR v> plus the 16 neutral kernels form a 256-dimensional mu4-twisted V-graded basis of End(S+)?  THE KILL FIRES: the frozen mu4-linear-lift realization family reaches max rank 222/256 over ALL FOUR frozen phase conventions (Im/Re 222, Re+-Im 160; sympy-exact over Q(i)) -- with a STRUCTURAL obstruction: any phase rule Z-linear in the root gives c(-r) = -c(r), so +-r pairs with c in {0,2}^4 produce PROPORTIONAL operators (collision witness printed; 240 - 228 = 12 c-vector collisions), unreachable for the ENTIRE diagonal-phase x translation family; TWO post-hoc diagnostics score WORSE (nonlinear BFS transversal 215, zeta8-linear float 160) -- the death is robust, not a convention artifact; a successor would need root-individual (non-linear-in-r) operator data.  WHAT SURVIVES: the syndrome-table counting 16 + 240 = 256 = 1x16 + 15x16, the neutral sector (the 16 v112 kernels ARE the diagonal maximal abelian subalgebra; same-class products land in A_0 with mu4 coefficients EXACTLY), the fine (class, d, k) groupoid composition law (associative on all 3375 section triples), and the W(3,2) commutation skeleton (scalar commutators 15.1%, matching (-1)^hbar on only 51.5% -- the class-level symplectic reduction fails in this realization).  NO scalar V-cocycle exists (section closure 0/210).  Controls fire (scrambled classes, wrong cocycle i^hbar fails the 2-cocycle identity on 960/4096 triples, random c-assignments).  The sharpest E8-is-a-hull form (twisted V-graded basis) is DEAD as frozen; successor criterion named.  No matter semantics (v775 ROOTCLASS-MIXED stands).  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "DEAD as frozen [E kill, preregistered adjudication fired]: the twisted V-graded basis reading of E8 killed at rank level; the surviving structures typed exactly; successor criterion named",
      "canonicalStatus": "SYNDROME-DEAD: max rank 222/256 over all four frozen phase conventions (two post-hoc diagnostics score worse: 215/160); the +-r proportionality obstruction structural for the whole diagonal-phase family; SURVIVES: 16 + 240 = 256 counting, the 16 v112 neutral kernels ARE the diagonal MASA, same-class products A_0-valued with mu4 coefficients exactly, the fine groupoid law associative 0/3375; successor needs root-individual (non-linear-in-r) operator data; no matter semantics.",
      "dependencies": "E8.PROJHAMMING.01; ARF.ROOTCLASS.01; v752; v111; v112; v775",
      "scripts": [
        "v805_e8_syndrome_algebra.py"
      ],
      "script": "v805_e8_syndrome_algebra.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 695
    },
    {
      "id": "CARRIER.PETERSEN.RADIAL.01",
      "claim": "CARRIER.PETERSEN.RADIAL.01 (+ TRANSPORT.SIXTHROOT.01): the K5 edge machine, the Petersen carrier geometry with the radial dynamics whose sixth power IS the deployed transport spectrum, + the bit-exact decider on the PSD sixth root, ONE module from two probes (19/19 + 16/16 checks, ~1 s; discovery probes carrier_petersen_radial_probe.py PETERSEN-RADIAL-EXACT and transport_sixthroot_probe.py SIXTHROOT-SPECTRAL-ONLY, both 2026-08-06).  PART 1, THE K5 EDGE MACHINE (all exact integer/Fraction): X = (-2,-2,-2,3,3) is the UNIQUE primitive traceless block-constant charge for the F|W = 3+2 split (sign pinned by the certified v774 table); E(K5) = 10 pairs carry the charge census {-4: 3, +1: 6, +6: 1} = the SU(5) decuple 3_{-4} + 6_{+1} + 1_{+6}, every pair cross-checked against the certified v774 hypercharge table (classes u_c/Q_up/Q_dn/e_c); under disjointness the pairs form the PETERSEN graph SRG(10,3,0,1) (charpoly (x-3)(x-1)^5(x+2)^4 exact, girth 5, |Aut| = 120 by full backtracking census = the faithful S5); the distance shells around the W vertex are (1, 3, 6) and CHARGE-PURE (+6/-4/+1 = e^c/u^c/Q), and the distance-2 shell induces a C6 -- THE FLAVOR HEXAGON AS GRAPH GEOMETRY [E neu] (explicit cycle verified edge by edge; 6 = |R+(A3)| = the transfer exponent, v124); the shell partition is EQUITABLE with quotient [[0,3,0],[1,0,2],[0,1,2]] whose eigenvector (6,-4,1) -> -2 IS the hypercharge vector, and P = Q_Pet/3 has spec(P^6) = {1, 64/729, 1/729} == the deployed TFPT transport spectrum EXACTLY (v54/v56/v171/v317; v124 lambda_n = (1-n/3)^6; GATE.METRIC.18).  THE PREDICTION [H neu]: the full 10-dim Petersen walk has spec(P10^6) = {1, (1/3)^6 x5, (2/3)^6 x4} -- fast mode multiplicity 5 = g_car, slow hypercharge mode 4 = |mu4| (trace crosscheck 990 = 729 + 5 + 256); NO corpus 10-dim operator exists (T is 3-dim on the cusp space) -- a prediction, not a verification.  PART 2, THE SIXTH ROOT (exact Fractions): B = (1/18)[[13,1,4],[1,13,4],[4,4,10]] -- every entry a compiler number (18 = p4(anchor), 13 = N_fam^2 + |Z2|^2, 4 = |mu4|, 10 = A_Lambda, 1 = N_Phi) -- is the UNIQUE symmetric doubly stochastic PSD sixth root with the frozen corpus-basis eigen-assignment ((1,1,1) -> 1, (1,-1,0) -> 2/3, (1,1,-2) -> 1/3; spectral reconstruction entry-exact; exactly 2 assignments in the corpus basis, a 1-parameter circle without the pin); B^6 carries the deployed spectrum exactly BUT the decider lands SPECTRAL-ONLY: B^6 is GL(3,Q)-conjugate to v486's lazy-walk M^6 (exact rational W and orthogonal O conjugations verified) and to the cusp-diagonal canonical form, yet bit-equal to NEITHER -- the corpus freezes the SPECTRUM, not the basis; the stochastic types differ (doubly vs sub-stochastic with absorbing channel).  Controls fire in both parts (wrong edge 4+1 kills the (1,3,6) structure; Johnson complement kills the hexagon; both matrix perturbations break the spectrum).  Typed carrier representation geometry -- ROOTCLASS-MIXED (v775) cited, no particle semantics.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.  [2026-08-06 LXVI dated correction + addendum, v814: the sixth-root census missed the deployed v221 evaluation -- T_v221 (the exact spectral sum of v221's transfer (1-w)^6) == B^6 BIT-EXACTLY, so the TRANSPORT.SIXTHROOT.01 verdict upgrades from SPECTRAL-ONLY to BITEXACT-vs-v221 (the frozen SPECTRAL-ONLY stands as the probe's adjudication; ABEL-DEAD correction precedent); and the [H neu] 10-dim prediction now has a promotion-ready canonical PROPOSAL: T10 = P10^6 in the Lambda^2 basis passes all gates with the equitable v774 quotient == (Q_Pet/3)^6 bit-exact (TRANSPORT10-PROPOSED, no deploy claim), while the six-step composition is typed CLOCK-ONLY -- the deployed 6 is the clock exponent, every spatial route carries a typed obstruction.]",
      "status": "Exact [E neu] finite graph/matrix structure theorems (integer/Fraction; charpolys sympy-gated) behind the carrier-geometry fence; the 10-dim multiplicity prediction typed [H neu]; carries TRANSPORT.SIXTHROOT.01 (spectral-only decider)",
      "canonicalStatus": "PETERSEN-RADIAL-EXACT + SIXTHROOT-SPECTRAL-ONLY: charges (-4,+1,+6) = the SU(5) decuple against v774; Petersen SRG certified, |Aut| = 120 census; shells (1,3,6) charge-pure; the flavor hexagon = the distance-2 shell edge-by-edge; equitable quotient with hypercharge eigenvector (6,-4,1) -> -2; spec(P^6) = {1, 64/729, 1/729} = the deployed transport spectrum; 10-dim multiplicities (x5 = g_car fast, x4 = |mu4| slow) typed [H neu] -- no corpus 10-dim operator; B unique symmetric doubly stochastic PSD sixth root in the corpus basis, B^6 GL(3,Q)-conjugate to v486's M^6 and the cusp-diagonal form but bit-equal to neither -- the corpus freezes the spectrum, not the basis.",
      "dependencies": "ARF.SPINORCOMPILER.01; v774; v54; v56; v124; v486; v327; v212; v397; v14; v700",
      "scripts": [
        "v808_petersen_sixthroot.py"
      ],
      "script": "v808_petersen_sixthroot.py",
      "location": "tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 696
    },
    {
      "id": "CURVE.CODE.DOILY.01",
      "claim": "CURVE.CODE.DOILY.01: the certified outer bridge sharpened into an incidence-operator statement with a spectral payoff, one probe (21/21 checks incl. the 3 must-fire controls, ~1 s; discovery probe curve_code_doily_probe.py, 2026-08-06, verdict DOILY-EXACT).  THE EXACT IDENTITY: the K6 doily incidence N (15 duads x 15 synthemes on the six odd Arf refinements) satisfies NN^T = 3I + A_KG(6,2) = B + 2I entrywise, with B the certified v752/v774 symplectic incidence (synthemes <-> isotropic lines bijectively; row and column sums 3; two duads share a syntheme iff disjoint, then exactly one).  THE SPECTRAL PAYOFF: spec(NN^T) = spec(N^T N) = {9^1, 4^9, 0^5} exact (annihilating polynomial + traces + rank over Q), so the normalized channel N/3 has singular values {1, 2/3, 0} -- THE RECOVERY BASE RATE 2/3 (v221) as the unique nontrivial singular value of the canonical duad-to-syntheme incidence; the identification with the corpus rate (2/3)^6 = 64/729 is TYPED [H neu -> decidable, not decided here]: it requires the six-step composition semantics (the hexagon).  PETERSEN LOCALIZATION: at the q*-distinguished point the 15 duads split 5 + 10, the 10 internal duads form the Petersen graph, and syntheme -> internal pair is a BIJECTION onto E(Petersen) 15/15 -- Petersen vertices = code duads, Petersen edges = curve synthemes under the certified bridge.  THE REFINED BRIDGE: every census bridge (rebuilt with the frozen BRIDGE-CANONICAL conditions, census 6) factors EXACTLY through N -- Delta(d) = {Gamma(s) : d in s} and the matrix identity N_code = P_Delta^T N_curve^T P_Gamma: the outer bridge conjugates the code incidence to the TRANSPOSE of the curve incidence, a DUALITY, not an isomorphism (control C3: 720 symplectic isomorphisms exist, 0 intertwine sigma with tau -- the inner route stays dead).  THE ARF LAYER in doily coordinates: the switch set T6 = the six within-block duads of the q_Theta point-split, bijective onto the 6 odd theta characteristics -- the census-6 anchor.  N/3 is doubly stochastic (typed CLASSICAL incidence channel, not CPTP; 1 + 9 + 5 spectral decomposition, rate 2/3 on the 9-dim window) -- measured, no descent claim.  No matter semantics (ROOTCLASS-MIXED, v775); the inner identification stays CURVE-CODE-PARTIAL/dead.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact [E] incidence-operator identities and spectra (integer/Fraction, annihilator-certified) with the (2/3)^6 identification typed [H neu -> decidable] pending the six-step composition; the inner identification stays dead",
      "canonicalStatus": "DOILY-EXACT: NN^T = B + 2I entrywise; spec {9, 4^9, 0^5}; N/3 singular values {1, 2/3, 0} -- the recovery base rate as the unique nontrivial singular value, (2/3)^6 typed pending six-step composition; Petersen edges = synthemes 15/15; every census bridge factors exactly N_code = P_Delta^T N_curve^T P_Gamma -- a duality, not an isomorphism (inner route 0/720); Arf switch set T6 = the six within-block duads; N/3 doubly stochastic, typed classical not CPTP.",
      "dependencies": "CURVE.CODE.OUTERSPIN.01; CURVE.CODE.2TORSION.01; v752; v774; v796; v221; v240; v425",
      "scripts": [
        "v809_curve_code_doily.py"
      ],
      "script": "v809_curve_code_doily.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 697
    },
    {
      "id": "CARRIER.CUTCODE.01",
      "claim": "CARRIER.CUTCODE.01 (+ CARRIER.MOMENT.INCIDENCE.01): the Arf split as the cut classification of K5 + the one-line master moment with the frozen 210 decision, ONE module from two probes (17/17 + 16/16 checks, ~1 s; discovery probes carrier_cutcode_probe.py CUTCODE-EXACT and carrier_moment_incidence_probe.py MOMENT-INCIDENCE-PARTIAL, both 2026-08-06).  PART 1, THE CUT CODE [E neu]: the cut map delta: F2^5 -> F2^10 (edges of K5) is INJECTIVE on C_even(5) and ISOMETRIC (delta(x).delta(y) = x.y mod 2 on all 256 pairs; kernel = the repetition code; control C2: the isometry fails on exactly the 256 odd x odd pairs), with the quadratic identity q*(v) = wt(delta iota v)/2 mod 2 on all 16 words -- so the Arf split 16 = 1 + 5bar + 10 IS the cut classification (0|5) + (1|4) + (2|3) (the five wt-4 words = the elementary vertex cuts, the ten wt-2 words = the duad cuts, both bijections certified); the Clebsch graph SRG(16,5,0,2) (40 edges = |R(D5)|, tfpt_1 Theorem 2) grades over the Arf shells as 40 = 5 + 20 + 15 with forbidden blocks EMPTY, and the within-10 edge graph IS the Petersen graph (full isomorphism census 120 = |S5|); the 240 Gaussian roots / 60 mu4-lines grade 80 + 160 / 20 + 40 over the shells -- typed as Arf/cut grading, explicitly NOT a matter split (ROOTCLASS-MIXED, v775; the 5/20/15 readings are audit fingerprints [C]).  PART 2, THE MASTER MOMENT: the symbolic identity sum_{i<j}(x_i + x_j)^2 - 3 sum x_i^2 = (sum x_i)^2 for ALL 5-vectors gives the one-line derivation Tr_{S+} X^2 = 4|X|^2 = 4 h(E8) = 120 (|X|^2 = 30 = h(E8), Tr_10 = 90, Tr_5bar = 30; the chain 120/3 + 1 = 41 = 10 b1 re-anchored); chi_X(t) = (t+2)^3 (t-3)^2 exact with elementary symmetric values (0, -15, 10, 60, -72) typed audit fingerprints [C] (72 AUDIT-ONLY, no morphism).  THE 210 DECISION (frozen rule, either outcome the deliverable): p4(X) = 210 = C(10,4) stays DECORATIVE -- the equivariant-bijection candidates are KILLED by the exhaustive fixed-point obstruction (both groups S3xS2 and Z2^5:(S3xS2), both indexings: the fixed 4-subset exists, no G-fixed slot does, total equivariant maps = 0), none of the six frozen slot-local weight families reproduces p4, and only the PARTIAL miss-slot map survives (80 type-(1,1,1,1) subsets -> missed slot with uniform fibers 16 = 2^4, exact on the colour slots, deficit 65/65 on the weak slots) plus the Newton tie p4 = 2 e2^2 - 4 e4 (generic symmetric-function algebra, [C] audit).  Controls fire in both parts.  No physics claim, no marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact [E neu] cut-map/shell-grading censuses (F2/integer, exhaustive) + the frozen 210 decision landing PARTIAL (decorative, per the preregistered rule); the 5/20/15 and charpoly readings audit fingerprints [C]; carries CARRIER.MOMENT.INCIDENCE.01",
      "canonicalStatus": "CUTCODE-EXACT + MOMENT-INCIDENCE-PARTIAL: delta injective isometric on all 256 cells; Arf split = the (0|5)+(1|4)+(2|3) cut classification; Clebsch 40 = 5 + 20 + 15 with forbidden blocks empty and within-10 = Petersen (census 120); roots/lines grade 80 + 160 / 20 + 40 -- Arf/cut grading, NOT matter; Tr_{S+}X^2 = 4|X|^2 = 4 h(E8) = 120 from one symbolic identity; charpoly exact, 72 audit-only; the 210 = C(10,4) map DECORATIVE -- equivariant bijection killed by the fixed-point obstruction (exhaustive); the partial 80 -> 5 x 16 miss-slot map and Newton tie survive as audit.",
      "dependencies": "ARF.SPINORCOMPILER.01; ARF.ROOTCLASS.01; v774; v2; v55; v109; v111; v700; v752",
      "scripts": [
        "v810_cutcode_moment.py"
      ],
      "script": "v810_cutcode_moment.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 698
    },
    {
      "id": "FLAVOR.GRAPH.FILTRATION.01",
      "claim": "FLAVOR.GRAPH.FILTRATION.01: the flavor-matrix filtration Q -> K -> L as an incidence-compiler fingerprint -- budgets exact, entries NOT reconstructible, one probe (16/16 checks incl. the 4 must-fire controls, ~1 s; discovery probe flavor_graph_filtration_probe.py, 2026-08-06, verdict FILTRATION-FINGERPRINT-ONLY).  THE BUDGET IDENTIFICATIONS ARE EXACT AND ORDER-EXACT: Q row budgets (4, 5, 6) = (|E(K3) u E(K2)|, |V(K5)|, |E(K_{3,2})|), K row budgets (6, 9, 10) = (|E(K_{3,2})|, |E(L(K_{3,2}))| with the line graph certified isomorphic to the triangular prism AND the hexagon circulant C6(2,3), |E(K5)|), L row budgets (10, 14, 16) = (|E(K5)|, #wt-4 words of H8, |H8|) -- with the level-wise additivity (4, 5, 6) + (6, 9, 10) = (10, 14, 16) mirroring the deployed L = K + Q (v37/v4) exactly; the deployed matrices located (v11/v12/v37/v4; v11 uniqueness re-derived in pure integers); control C1 certifies the identifications hold in EXACTLY the identity order.  THE DECISIVE BAR (frozen: row sums alone do NOT count): NO frozen structural candidate -- 25/20/47 flag- classifier and operator-quotient candidates per level, entry-exact comparison with all 6 column permutations -- reproduces ANY of the three deployed matrices entry-wise (per-row matches recorded as fingerprint data; K rows 2-3 and all L rows have ZERO candidates).  THE CONTROLS MAKE THE MISS INFORMATIVE: the exhaustive false-positive census over ALL nonneg-integer matrices with the deployed row sums gives rates 0.748% / 0.035% / 0.005% (Q/K/L) -- the candidate set is not vacuously flexible, so the entry-level failure is a real structural boundary, not a weak test; the wrong split 4|1 and the non-extended Hamming [7,4] break the Q/K/L identifications as predeclared.  TYPED: the identification is an AUDIT FINGERPRINT at the budget level, not an entry-level incidence construction.  Matrix-structure archaeology only -- no mass/physics claim.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact [E] order-exact budget identifications (pure integers, exhaustive false-positive controls) with the entry-level reconstruction NEGATIVE; typed audit fingerprint at budget level",
      "canonicalStatus": "FILTRATION-FINGERPRINT-ONLY: Q (4,5,6) / K (6,9,10) / L (10,14,16) = exact counts of the K_{3,2}/prism/K5/H8 incidence chain, order-exact, with level-wise additivity mirroring L = K + Q; deployed matrices located (v11/v12/v37/v4); NO entry-level reconstruction from any frozen candidate (25/20/47 per level); exhaustive false-positive rates 0.005-0.75% -- the misses are informative; typed audit fingerprint.",
      "dependencies": "FLAV.PENCIL.01; v11; v12; v37; v4; v752",
      "scripts": [
        "v812_flavor_graph_filtration.py"
      ],
      "script": "v812_flavor_graph_filtration.py",
      "location": "tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 699
    },
    {
      "id": "P1.INDEX.KMS.01",
      "claim": "P1.INDEX.KMS.01: the closed normalization identity c3 = 1/(I_Jones * beta_angle) = 1/(4 * 2pi) = 1/(8pi), its convention audit, and the sharpened P1 contract, one probe (13/13 checks incl. the 2 must-fire controls, ~0.1 s; discovery probe p1_index_kms_probe.py, 2026-08-06, verdict P1-IDENTITY-CLOSED).  THE IDENTITY [E]: I * beta * c3 = 4 * 2pi * 1/(8pi) = 1 EXACT in formal Fraction x pi^k arithmetic (no floats; S2.1-S2.3 are one identity in three rearrangements, counted as one content; the v208 area-entropy factor 1/4 = 1/I; the AX.P1.01 anchor route 1/(2 e1(a) pi) with e1 = 4 aligns anchor-4 with index-4).  THE DEPENDENCY AUDIT (the load-bearing honesty step): the index I = 4 = |mu4| is an INDEPENDENT c3-free theorem (ZERO 'c3' occurrences in all four sources v154/v726/v746/v779, gated by file scan); beta_angle = 2pi is an INDEPENDENT measurement (v526: beta = N clock steps by detailed balance, wrong-beta {N/2, N/4, 2N} excluded; the v519 radian pin is the ONE shared convention and CANCELS in the unit-invariant statement 'response per full modular circle = 1/I'); c3 STAYS the P1 axiom -- the identity EXPLAINS its value, it does not derive it.  THE FINITE EVENNESS PRECURSOR (measured, real): the tracial weight per mu4 deck sector on the l-windows satisfies w_1 = w_3 = 1/4 EXACTLY at every level l = 1..6, with the w_0/w_2 offset the symmetric zero-sum pair +-2^-(l+1) halving per level -- equal quarters asymptotically, NO anomaly offset (= the no-anomaly shape the contract's bridge needs); the index-4 conditional expectation is trace-preserving (dev < 1e-12); and the SEA STATE does NOT equidistribute (deviation 0.188 from 1/4 at the N = 1536 Haar pullback) -- the bridge must normalize the MODULAR RESPONSE, not vacuum weights.  THE CONTRACT (registered [O]): an index-I simple-current extension carries per modular period exactly the normalized response 1/(I * beta_angle) = c3; the ONE open bridge is the even distribution of the modular-response (Quillen/determinant-line) normalization over the I deck sectors; KILL: any asymmetric sector offset (w_1 != w_3 at any level, or a non-vanishing offset as l -> inf).  Controls fire: the wrong index 2 gives 1/(4pi), which the corpus books elsewhere (un-halved Gauss-Bonnet; SdS 1/(4pi) = 2 c3); wrong beta = pi breaks the chain at 1/2.  c3 remains the P1 axiom; no marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.  NOTE (2026-08-28, v993): unique finite CE does NOT close the equidistribution bridge -- that identification stays the open [C] of this row.",
      "status": "Exact [E] identity chain (formal Fraction x pi^k) + gated independence audit + measured finite evenness precursor; OPEN [O] contract with the one open bridge (even modular-response normalization) and the kill criterion named; c3 stays the P1 axiom",
      "canonicalStatus": "P1-IDENTITY-CLOSED: I*beta*c3 = 1 exact; the index c3-free (zero occurrences, four sources), beta independently measured (v526, wrong-beta excluded; the radian pin is the one shared convention and cancels); c3 stays the P1 axiom the identity EXPLAINS; finite evenness precursor measured: w1 = w3 = 1/4 exactly every level, symmetric zero-sum offset +-2^-(l+1) -- the no-anomaly shape; the sea state does NOT equidistribute (dev 0.188) -- the bridge must normalize the modular response; controls: index 2 -> 1/(4pi), booked elsewhere by the corpus; contract registered OPEN [O], kill = an asymmetric sector offset.",
      "dependencies": "AX.P1.01; GATE.METRIC.11; GATE.METRIC.12; ANCHOR.GEN.01; v154; v726; v746; v779; v526; v239; v208; v8; v23",
      "scripts": [
        "v813_p1_index_kms.py"
      ],
      "script": "v813_p1_index_kms.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 700
    },
    {
      "id": "K5.SIXSTEP.TRANSPORT.01",
      "claim": "K5.SIXSTEP.TRANSPORT.01: the six-step composition and the 10-dim transport deploy -- the (2/3)^6 rate is canonical on three spatial routes, the deployed six is the clock exponent, and the canonical T10 proposal passes every gate, one probe (23/23 checks incl. the 3 must-fire controls, ~0.2 s; discovery probe k5_sixstep_transport_probe.py, 2026-08-06, verdicts SIXSTEP-CLOCK-ONLY / TRANSPORT10-PROPOSED).  THE CORRECTION (dated note on the v808/TRANSPORT.SIXTHROOT.01 row): the deployed transport is LOCATED -- v221's transfer (1-w)^6 IS the exact spectral sum T_v221 = J/3 + (2/3)^6 u2 u2^T + (1/3)^6 u3 u3^T, and T_v221 == B^6 == (1/4374)[[1651,1267,1456],[1267,1651,1456],[1456,1456,1462]] BIT-EXACTLY -- the v808 sixth-root census missed v221, so the SIXTHROOT verdict upgrades from SPECTRAL-ONLY to BITEXACT-vs-v221 (the frozen SPECTRAL-ONLY stands as that probe's adjudication; ABEL-DEAD correction precedent).  TASK A (all exact): the rate (2/3)^6 arises CANONICALLY on three independent spatial routes -- the doily alternation (three PSD double-steps duad -> syntheme -> duad: spec {1, (2/3)^6 x9, 0^5}), the hexagon compression (Pi_2 (A_Pet/3) Pi_2 == A_C6/3 with spec((A_C6/3)^6) = {(2/3)^6 x2, (1/3)^6 x4} -- BOTH deployed decay rates from pure hexagon geometry), and the hexagon circulation (((2/3) S6)^6 = (2/3)^6 I) -- but each route is blocked by a TYPED obstruction (doily: wrong space, decay multiplicity 9, no (1/3)^6 mode; hexagon: NO unit mode -- ||(A_C6/3)^6||_op = (2/3)^6 < 1 kills every norm-nonincreasing intertwiner [E neu]; radial: the bipartite sign -2/3, P NOT similar to the clock root B, and the exact conjugation S with S T_v221 S^{-1} = P^6 is deployed NOWHERE -- the missing shells -> cusp isomorphism); SIXTH-POWER BLINDNESS [E neu]: (A_C6/3)^6 and (A_{2K3}/3)^6 have IDENTICAL spectra -- the sixth power erases the cyclic geometry AND the sign, so only per-step data can distinguish walk from clock: the deployed six is the CLOCK EXPONENT (v124 lambda_n = (1-n/3)^6, v486 hand, fixed eigenbasis (1,1,1)/(1,-1,0)/(1,1,-2)).  TASK B: corpus census -- NO deployed 10-dim transport operator exists (v37's Pluecker apparatus is 3x3, re-derived; v44/v45 carry Lambda^2(5) = 10 as dimension only); the CANONICAL PROPOSAL T10 = P10^6 in the Lambda^2 basis (v774 cut-code dictionary, a permutation conjugation) passes ALL gates: symmetric doubly stochastic PSD, spectrum {1^1, (1/3)^6 x5, (2/3)^6 x4} exact (multiplicities (5, 4) = (g_car, |mu4|)), the v774 class partition (e_c | u_c | Q) EQUITABLE, exact quotient == (Q_Pet/3)^6 bit-exact with the deployed spectrum {1, 64/729, 1/729} and reversibility -- a promotion-ready PROPOSAL, no deploy claim (Q6 != T_v221 bit-exact, not symmetric -- recorded: this is WHY PROPOSED, not REALIZED).  Controls fire (wrong per-step rate 1/2, Johnson-complement scramble, non-equitable partition).  ROOTCLASS-MIXED (v775) cited: carrier representation geometry, no physics claims beyond operator identities.  No marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact [E neu] operator identities and censuses (three spatial routes, sixth-power blindness, corpus census, bit-exact v221 location); [H neu] the walk-vs-clock reading and the T10 deployment proposal; dated BITEXACT correction to TRANSPORT.SIXTHROOT.01 carried",
      "canonicalStatus": "SIXSTEP-CLOCK-ONLY + TRANSPORT10-PROPOSED: the (2/3)^6 rate canonical on three spatial routes, each with a typed obstruction; the deployed 6 is the clock exponent (sixth-power blindness makes per-step data mandatory); T_v221 == B^6 bit-exact (SIXTHROOT upgrades to BITEXACT-vs-v221, dated note on the v808 row); T10 = P10^6 in the Lambda^2 basis passes all gates -- proposal, no deploy claim; NO RH claim",
      "dependencies": "CARRIER.PETERSEN.RADIAL.01; TRANSPORT.SIXTHROOT.01; CURVE.CODE.DOILY.01; v221; v124; v486; v37; v774; v808; v809",
      "scripts": [
        "v814_k5_sixstep_transport.py"
      ],
      "script": "v814_k5_sixstep_transport.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 701
    },
    {
      "id": "TFPT.POSITIVE_DESCENT.MASTER.01",
      "claim": "Abstract positive-descent theorem and G_net finite hypothesis checks; the prime packet instance is maintained by Prime Front. Original module ID retained; original complete mixed record is preserved in extraction provenance.",
      "status": "Abstract master theorem with kernel-checked Lean core (PositiveDescentMaster.lean: telescoping + tail, positivity closure, sector commutation); both instances measured at finite level; hypotheses (6)/(7) typed per instance; positivity a separate hypothesis (K4)",
      "canonicalStatus": "MASTER-BOTH-INSTANCES: G_net hypotheses re-verified (Choi PSD, delta rate 0.267/doubling); THE NEW NUMBER -- prime packet state defects summable at 0.701/doubling ~ sqrt-dilution 2^{-1/2}, mh2(p) = -1/15 exact for every odd prime; the selection problem eliminated on both sides; K4: positivity NOT implied by summability; open per instance: G_net normality/split, prime sector floor (-> PRIME.FLOOR.RATIO.01); NO RH claim",
      "dependencies": "PRIME.POSITIVE_DESCENT.01; GNET.MARTINGALE.LIMIT.01; PRIME.FLOOR.RATIO.01; v791; v802; v793",
      "scripts": [
        "v817_positive_descent_master.py"
      ],
      "script": "v817_positive_descent_master.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 702
    },
    {
      "id": "ANCHOR.AFFINE.01",
      "claim": "ANCHOR.AFFINE.01 + FLAVOR.BIDIR.01: the anchor affine normal form and the flavor bidirectional checksum -- two exact corpus compressions, ONE module from two probes (18/18 + 15/15 checks, verdicts ANCHOR-AFFINE-EXACT and FLAVOR-BIDIR-CHECKSUM (selective); discovery probes anchor_affine_probe.py and flavor_bidir_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM).  PART A: the anchor a = (1,1,2) has power sums p_n = 2 + 2^n satisfying the affine recursion p_{n+1} = 2 p_n - 2 identically in n (sympy); T(x) = 2x - 2 has the UNIQUE fixed point 2 = |Z_2| (the sheet), and the T-orbit of 4 is exactly the compiler quintet (4, 6, 10, 18, 34) = (|mu4|, |R+(A3)|, A_L, N_fam |R+(A3)|, Z6-lift) with p_0 = 3 = N_fam prepended.  THE BUDGET FROM ONE RECURSION: |R(E8)| = p1 p2 p3 = 240, dim E8 = 240 + q3 = 248 (ladder identity p4 - p3 = p3 - 2 = 8 = h(D5) = rank E8), h(E8) = p2 p3/2 = 30, |R(D5)| = p1 p3 = 40, Omega_adm = 2 p1 p2 = 48, chi_a(t) = (t-1)^2(t-2) with (e1,e2,e3) = (4,5,2) = (|mu4|, g_car, |Z_2|), 10 b1 = e1^2 + e2^2 = 41 -- every number a word in the T-orbit and the fixed point.  PART B: R = [[1,3,0],[1,5,2],[2,5,3]] decodes the SAME anchor on both sides -- R a = (4, 10, 13) = (p_1, p_3, N(3+2i)) and R^T a = (6, 18, 8) = (p_2, p_4, p_3 - 2); tr R = 9 = p_0^2, det R = 8, principal 2-minors {2,3,5}, chi_R = t^3 - p_0^2 t^2 + p_3 t - (p_3 - 2), ||R||_F^2 = 78, SNF diag(1,1,8).  THE SIBLING ABLATION: of the 12 down-row candidates (all orderings of {1,2,5} and the sibling {1,3,4}) EXACTLY ONE -- the accepted (1,5,2) -- satisfies the full identity set; the anchor contraction (R a)_2 = 10 = p_3 kills both siblings (11 and 12).  Controls fire (wrong map, wrong anchor, sibling (1,4,3) reproducing tfpt_2's failure numbers verbatim, scrambled ordering).  A normal form / compression, NOT a new claim: no marker moves.  Exact sympy/integer; no floats, no RNG, no fit.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact identity-grade compression [Identity]: both parts exact sympy/integer arithmetic; the compression is a normal form of existing corpus surfaces (tfpt_1 anchor power compiler + budget lines, tfpt_2 flavor spectral selector + bilinear table); no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "ANCHOR-AFFINE-EXACT + FLAVOR-BIDIR-CHECKSUM (selective): the whole compiler budget from ONE affine recursion T(x) = 2x - 2 (orbit 4, 6, 10, 18, 34; fixed point 2); the flavor matrix R decodes the same anchor bidirectionally (rows p_1, p_3, 13; columns p_2, p_4, 8) and the accepted down-row (1,5,2) is the UNIQUE candidate of 12 passing the full identity set -- the kill test is the anchor contraction (R a)_2 = 10 = p_3",
      "dependencies": "v1; v2; v6; v222; v230; v415",
      "scripts": [
        "v832_anchor_flavor_checksum.py"
      ],
      "script": "v832_anchor_flavor_checksum.py",
      "location": "tfpt_1;tfpt_2;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 703
    },
    {
      "id": "GAUSSIAN.RAMLADDER.01",
      "claim": "GAUSSIAN.RAMLADDER.01: the (1+i) ramification ladder -- the four documented roles of the ramified Gaussian prime pi2 = 1+i are ONE machine-checked object with four rungs, each re-certified from scratch, ONE module from one probe (33/33 checks, verdict RAMIFICATION-LADDER-EXACT; discovery probe gaussian_ramification_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM, ~5 s).  RUNG 1, NORM DOUBLING: N(pi2) = 2 and (pi2^2) = (2); on the Gaussian E8 lattice (the unique mu4/sigma-equivariant Construction-A placement, 2 of 30, rebuilt) (1+J)^T (1+J) = 2 I_8 EXACTLY; min L = 4 by exhaustive census, the zero class of L/(1+i)L carries NO root (0/240 directly), 240 = 15 x 16 over the nonzero classes (each a union of 4 of the 60 Gaussian lines), elementary divisors (1,1,1,1,2,2,2,2), index 16 = N(pi2)^4.  RUNG 2, THE FOUR-BIT ADDRESS: L/(1+i)L ~= F2^4 with the deck J acting TRIVIALLY on all 240 roots.  RUNG 3, THE NON-SPLIT JET: Z[i]/(2) ~= F2[eps]/(eps^2) (ring iso UNIQUE among all 24 bijections); W = L/2L FREE of rank 4 over the dual numbers (256/256 combinations distinct); the self-extension 0 -> V -> W -> V -> 0 exact with deck = JET UNIT 1 + eps, and of ALL 65536 linear sections EXACTLY 0 are deck-equivariant -- the no-canonical-origin obstruction IS non-splitness (v803); plus-type form census 136/120, Arf 0, the 120 root pairs ARE the q = 1 shell (240 = 15 x 8 x 2).  RUNG 4, THE METAPLECTIC LIFT: zeta8 = (1+i)/sqrt2; (1+J)^2 = 2J; (S H)^3 = zeta8 I per binary factor and (T3 K3)^3 = i I = THE DECK (v798); exact Z[zeta8] BFS census: |C2/mu8| = 11520 = 16 x 720 with ZERO Galois-mixed classes, |C2| = 92160 = 2 x 46080, C2 = K u zeta8 K (K = U G31 U^-1, v783 chart theorem cited [E]); witness parities chi(H(x)I) = 1 (the RM-CSS zeta8 phase bit, v819 S4 -> v798 [E]), chi(H(x)H) = chi(CZ) = 0.  Controls fire (split ring F2 x F2 admits 0 ring isos -- 2 must ramify; split self-extension admits 65536/65536 sections; scrambled nilpotent breaks ker = im).  A compression: the four corpus roles (note_e8_gaussian_code Thm 1/v689, v689, v803, v783/v798/v819) are ONE ladder -- the same ramified prime read at (pi2), (pi2^2) and pi2/sqrt2.  Exact integer/Fraction/sympy; no floats, no RNG.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact identity/lattice-grade compression [Identity/Lattice]: all four rungs re-certified from scratch in exact arithmetic; unifies note_e8_gaussian_code/v689 (census, norm doubling), v803 (non-split jet), v783/v798/v819 (metaplectic zeta8, Clifford coset, RM-CSS phase bit); no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "RAMIFICATION-LADDER-EXACT: the four documented roles of pi2 = 1+i -- norm doubling / empty zero class, the 4-bit address, the non-split jet with deck = 1 + eps, the metaplectic zeta8 with the Clifford index-2 coset and the RM-CSS phase bit -- are ONE ladder (33/33 exact); defect-free controls (split ring, split extension, scrambled nilpotent) all fire",
      "dependencies": "v689; v783; v798; v803; v819",
      "scripts": [
        "v833_gaussian_ramification_ladder.py"
      ],
      "script": "v833_gaussian_ramification_ladder.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 704
    },
    {
      "id": "PG32.FLAGS.01",
      "claim": "PG32.FLAGS.01: the PG(3,2) flag normal form for the 105 + 35 vacuum completion, with the incidence kill test typed as a split, ONE module from one probe (26/26 checks, verdict PG32-FLAGS-EXACT + INCIDENCE-SPLIT; discovery probe pg32_flag_completion_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM, < 1 s).  COUNTING NORMAL FORM: [4 choose 2]_2 = 35; 140 affine 2-flats of AG(4,2) = 35 through the origin + 105 not; PG(3,2) has 15 points, 35 lines, 105 = 35 x 3 point-line flags, canonically bijective to the C(15,2) = 105 point pairs via {x,y} <-> (x+y, {x,y,x+y}).  RM LAYER (v819 S4 / v821 / v822 rebuilt, all set equalities): [15,10,4] weight enumerator 1 + 105z^4 + 280z^6 + 435z^8 + 168z^10 + 35z^12 EXACT; the 105 weight-4 words ARE the flats avoiding 0; the 140 weight-4 words of RM(2,4) ARE all flats; Grams 22I + 6J, 6I + J, 28I + 7J entrywise.  VACUUM READING: origin planes puncture to weight 3, NONE in the shortened code, ONE unshortens dim 10 -> 11 -- the vacuum coordinate is ORIGIN RESTORATION.  THE INCIDENCE SPLIT (the kill, typed): with the corpus symplectic form (v774) the 105 = 45 + 60 split is the Sp(4,2)-ORBIT decomposition on BOTH sides (|Sp(4,2)| = 720 brute force); on the 60 NON-ISOTROPIC rungs the polarity map PHI: (U,t) -> ((t+U) n U^omega, U^omega) is well-defined, bijective, Sp(4,2)-EQUIVARIANT (all 60 x 720) and INCIDENCE-PRESERVING; on the 45 DOILY rungs the obstruction is typed: U^omega = U and (t+U) n U = EMPTY for 45/45 flats, the unique abstract equivariant bijection matches flags with support n line = 0, and the flag Gram is 18I + 3J != 22I + 6J.  TYPED CONCLUSION: '105 checks = 105 flags' is exact as a deck-equivariant BIJECTION, NOT an incidence isomorphism -- the obstruction sits exactly on the doily (the typed reason v822's continuum dilation died).  VACUUM CARRIER CONTENT: the zero class carries NO E8 root (0/240 rebuilt, v689 [E]) -- the vacuum point repairs the label Gram but adds no carrier content; v822's frozen continuum kill cited as the companion negative.  Controls fire (scrambled plane row, degenerate form, the 45-sector obstruction as its own control).  Exact integer; no floats, no RNG, no fit.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact lattice/identity-grade compression with the kill typed as a split [Identity/Lattice]: 60-sector exact via the symplectic polarity (equivariant incidence dictionary), 45-sector obstructed on the doily (typed negative); compresses v819 S4 / v821 / v822 RM surfaces into the PG(3,2) normal form; no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "PG32-FLAGS-EXACT + INCIDENCE-SPLIT: 105 = 45 + 60 is the Sp(4,2)-orbit decomposition on both sides; the polarity dictionary is exact, equivariant and incidence-preserving on the 60 non-isotropic rungs; the 45 doily rungs carry the typed obstruction ((t+U) n U empty 45/45, flag Gram 18I+3J != 22I+6J) -- '105 checks = 105 flags' is a deck-equivariant bijection, NOT an incidence isomorphism",
      "dependencies": "v689; v774; v819; v821; v822",
      "scripts": [
        "v834_pg32_flag_completion.py"
      ],
      "script": "v834_pg32_flag_completion.py",
      "location": "tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 705
    },
    {
      "id": "MESSAGE.LADDER.01",
      "claim": "v844 -- MESSAGE.LADDER.01 + DOILY.PASCAL.RANK.01: the E8 addressing formula and the Doily-Pascal rank theorem -- two exact corpus compressions, ONE module from two probes (23/23 + 16/16 checks, verdicts MESSAGE-LADDER-EXACT and DOILY-PASCAL-RANK-EXACT; discovery probes binary_message_ladder_probe.py and doily_pascal_rank_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM with no downscoping, ~2 s).  PART A, THE MESSAGE LADDER: M_n = 15 * Delta p_n = 15 * 2^n (anchor a = (1,1,2) with p_n = 2 + 2^n, v832 [E]; 15 = the nonzero classes of V = L/(1+i)L, v833 [E]) certified against the ACTUAL rebuilt objects with ONE NAMED STRUCTURE PER RUNG: M_0 = 15 = the address space (classes); M_1 = 30 = h(E8) derived HERE from the rebuilt root system, not cited (exactly 8 simple roots; connected simply-laced Cartan matrix, det 1, degree sequence [1,1,1,2,2,2,2,3] = the E8 diagram; every positive root with exact nonnegative INTEGER simple-root coordinates; the highest root UNIQUE at height 29 with marks {2,2,3,3,4,4,5,6}; h = 1 + 29 = 30 = |R|/rank); M_2 = 60 = the Gaussian root lines (mu4 orbits, all of size 4, exactly 4 per class); M_3 = 120 = |R+(E8)| (exact generic functional f(v) = Sum v_k 10^k) = the sign pairs; M_4 = 240 = |R(E8)| with the census 240 = 15 x 16 (zero class EMPTY); the CODA 248 = 15*16 + 8 = M_4 + rank checked against the actual census and the actual HNF rank; the structural doubling 60 -> 120 -> 240 (each line = exactly 2 sign pairs, each pair = 2 roots) and the halving 60 -> 30 = |mu4| vs rank typed.  The tfpt_1 budget lines |R| = 240 / dim E8 = 248 / h = 30 and v832 S3.1-S3.3 are ALL instances of this one formula.  Controls fire: the wrong anchor (1,2,2) gives (45,105,225,465) missing all four targets with recursion offset 1; the wrong multipliers 14 (the weight-4 codeword count) and 16 (the class size) miss all four targets and break the coda (232/264 != 248).  PART B, THE DOILY-PASCAL RANK THEOREM: the duad-syntheme incidence N of the Cremona-Richmond doily (15 duads x 15 synthemes, 3-regular in rows AND columns) and the symplectic point incidence B on PG(3,2) (B B^T = 4I + 3J entrywise on all 225 cells, charpoly (x-7)(x-2)^9(x+2)^5 exact, v774 S9 rebuilt) are ONE object: via the K6 duad model D(v) = {q Arf-1 : q(v) = 0} (exactly 16 quadratic refinements by brute force over 2^16, Arf census 6 + 10) the bijection point <-> duad carries N N^T onto B + 2I ENTRYWISE -- duad collinearity = symplectic orthogonality, and the spectrum shift 7/2/-2 -> 9/4/0 makes v774's (-2)-eigenspace v814's kernel.  THE RANK THEOREM (exact Fraction/sympy): charpoly(N N^T) = (x-9)(x-4)^9 x^5, so sing(N) = {3^1, 2^9, 0^5}: rank N = 10 = A_Lambda = C(g_car, 2) (the decuple sector), dim ker N = dim ker N^T = 5 = g_car (the five-slot carrier), multiplicity 9 = N_fam^2, top singular value 3 = N_fam, and the SIX ovoid indicators 3*1_{O_q} - 1 (q Arf-1) lie in ker N^T and SPAN it -- the P2 integers are the singular-value data of the Cremona-Richmond incidence.  THE RECOVERY VALUE: sing(N/3) = {1, 2/3 x9, 0 x5} exact, 2/3 = (N_fam - 1)/N_fam (the corpus recovery survival, v327); THE EXPONENTIATION IDENTITY: charpoly(((N/3)(N^T/3))^3) = (x-1)(x-64/729)^9 x^5 exact with 64/729 = (2/3)^6 = the deployed lambda_2 of {1, (2/3)^6, (1/3)^6} (v330/v486; the sixth power is the compiler clock, v814); TYPED GAP re-affirmed with NO upgrade: 1/729 is NOT a root of the six-step charpoly -- the deployed (1/3)^6 stays on the clock side (v814 SIXSTEP-CLOCK-ONLY [E], cited, consistent).  Controls fire: the circulant and the frozen-seed LCG 3-regular pairings (same row/col sums 3) give rank 13 != 10, kernel 2 != 5 and the wrong charpoly.  Exact integer/Fraction/sympy arithmetic; no floats, no fit; the only RNG is the frozen-seed LCG of control C2 (v775 C2 precedent).  No marker moves; the compression is a normal form, not a new claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact identity-grade compression [Identity]: both parts exact integer/Fraction/sympy arithmetic certified against the rebuilt lattice objects (h(E8) = 30 derived from the rebuilt root system, not cited); the (1/3)^6 typed gap re-affirmed with NO upgrade (v814 cited); no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "MESSAGE-LADDER-EXACT + DOILY-PASCAL-RANK-EXACT: M_0..M_4 = (15, 30, 60, 120, 240) = 15 * 2^n with one named structure per rung (address space / h(E8) from the rebuilt root system / Gaussian lines / positive system / root census) and the coda 248 = 15*16 + 8; N N^T = B + 2I entrywise; rank N = 10 = A_Lambda, ker = 5 = g_car, multiplicity 9 = N_fam^2, top 3 = N_fam, recovery 2/3, six-step (2/3)^6 = 64/729 exact; NO (1/3)^6 doily mode (v814, no upgrade); all controls fire",
      "dependencies": "v832; v833; v774; v814; v330; v486; v327; v775",
      "scripts": [
        "v844_message_doily_rank.py"
      ],
      "script": "v844_message_doily_rank.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 706
    },
    {
      "id": "NORMALFORM.CFIN.01",
      "claim": "v845 -- NORMALFORM.CFIN.01: the finite compiler normal form C_fin = (V, hbar, q*, sigma, iota) assembled as ONE machine-checked object from the Gaussian quotient, with the code-to-matter kill built in as a checked NEGATIVE and the remaining P2 bridge typed -- the round's central corpus compression, ONE module from one probe (28/28 checks, declared SPEC v2, verdict NORMAL-FORM-ASSEMBLED; discovery probe finite_compiler_normal_form_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM with no downscoping, ~4 s).  THE OBJECT, read off the lattice and not postulated: V = L/(1+i)L rebuilt from the Gaussian Construction-A E8 (the unique mu4/sigma-invariant placement C*, 2 of 30; census 240 = 15 x 16, zero class EMPTY); hbar = the descended symplectic form with a FOUND family/anchor basis (F1, F2, F3, A) of Gram J - I, equal to the bit form on ALL 256 label pairs; sigma = the descended order-3 deck (sigma^3 = id, symplectic, EXACTLY 3 fixed nonzero classes {A, F_Sigma, F_Sigma + A}, the other 12 in exactly 4 free 3-orbits -- the family cycle); q* = the distinguished quadratic refinement (EXACTLY 16 refinements by brute force over 2^16, Arf census 6 + 10; the frozen selector sigma-invariant 4 -> q(A) = 1: 2 -> q(F_Sigma) = 0: 1 picks a UNIQUE q*, Arf type 1, sigma-preserved); iota = the parity lift onto the even five-slot code C_even(5) (bijective; beta = iota.iota == hbar on all 256 cells; q_wt = wt(iota)/2 mod 2 IS q*; sigma acts on the FIVE carrier slots as the 3+2 split -- message register and carrier register intertwined).  DERIVED COUNTING from the one object: |Sp(4,2)| = 720 by full 65536-matrix census; |Stab(q*)| = 120 acting FAITHFULLY as S5 on the five, orbits [1, 5, 10]; 16 = 1 + 5 + 10 and 15 = 5 + 10; the PASCAL READING (iota-weight census (1, 10, 5) = (C(5,0), C(5,2), C(5,4))); the PROJECTIVE READING (PG(3,2): 15 points, 35 lines, 15 isotropic = the doily; the 5bar is an OVOID meeting every isotropic line exactly once); the BUDGET against the actual lattice objects: 240 = 16 x (5+10), 60 lines = 4 x 15, 30 = 2 x (5+10) = h(E8) (MESSAGE.LADDER.01 rung 1), rank 8 (HNF index 16), 248 = 16 x (5+10) + 8.  THE MANDATORY NEGATIVE (built in, v775 re-anchored): the exact counting bound -- every D5(+)A3 c E8 has 128 spinor-side roots, so at most floor(128/16) = 8 all-spinor classes and >= 7 of the 15 classes ALWAYS contain adjoint-side roots (the 15 mailboxes canNOT be 15 pure matter multiplets); the concrete doubled-model census SATURATES both bounds (8 spinor-pure + 7 adjoint-side, ZERO both-type classes -- the J-quotient respects the D8 grading); the WEIGHT-level kill at the canonical split D = (0,2,4,6,7): each spinor-pure class restricts to 8 DISTINCT half-spinor weight pairs, never 1 -- v775 ROOTCLASS-MIXED [E] re-verified, NOT overturned.  WHAT THE NORMAL FORM DOES NOT REMOVE (P2 typed, no upgrade): all 30 placements are Type II [8,4] doubly-even self-dual codes (enumerator 1 + 14 z^4 + z^8) -- the v799 TYPEII-FORCED echo; C_fin does NOT derive the three seam axioms A1-A3, the OPE-linearity scope note, or the v776 selection of ONE copy -- P2 (g_car = 5 = the iota slot count) is NARROWED to the v799 residual R1' ([C] class), not eliminated.  Controls fire: the non-alternating dot form admits 0 refinements; the trivial deck leaves 4 selector candidates (q* NOT unique without the family 3-cycle); the wrong lift breaks C_even membership and beta == hbar.  SPEC v2 declared (v836 precedent; the v1 wording of S7(b) mis-typed the v775 bound as 'mixed classes'; the factual census is the clean 8 + 7 split along the D8 grading; no numeric result changed).  Exact integer/sympy; no floats, no RNG, no fit.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact identity-grade normal form [Identity] with the built-in checked NEGATIVE (v775 ROOTCLASS-MIXED re-anchored via the 128-spinor counting bound + the saturated 8 + 7 census) and the typed P2 bridge: C_fin does NOT derive A1-A3 / the linearity scope / the v776 uniqueness selection -- P2 NARROWED to the v799 residual R1' ([C]), NOT eliminated; SPEC v2 declared (v836 precedent, no numeric result changed); no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "NORMAL-FORM-ASSEMBLED: C_fin = (V, hbar, q*, sigma, iota) read off the Gaussian quotient -- selector-unique q*, faithful S5 stabilizer (|Sp(4,2)| = 720, Stab = 120, orbits [1,5,10]), Pascal (1,10,5), ovoid projective reading, budget 240 = 16*15 / 248 = 16*15 + 8 against the actual census; the code-to-matter reading stays DEAD (bounds saturated 8 + 7, weight kill 8 pairs never 1); P2 narrowed to v799 R1' ([C]), not eliminated; all controls fire",
      "dependencies": "AX.P2.01; v752; v774; v775; v799; v776; v833; MESSAGE.LADDER.01",
      "scripts": [
        "v845_cfin_normal_form.py"
      ],
      "script": "v845_cfin_normal_form.py",
      "location": "tfpt_1;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 707
    },
    {
      "id": "CFIN.UNIQUE.01",
      "claim": "CFIN.UNIQUE.01: finite compiler uniqueness, one orbit and C6 automorphisms; cofinal Weil mirror moved to Prime Front. Original module ID retained; original complete mixed record is preserved in extraction provenance.",
      "status": "Exact identity-grade uniqueness theorem [Identity] for the finite compiler object: exhaustive finite verification over F_2^4 (no floats, no RNG, no fit; orbit-stabilizer exact); the strict terminal-object reading FAILS honestly and is typed (unique up to NON-unique isomorphism, a one-object groupoid); P2 stays NARROWED to the v799 residual R1' (why THIS normal form), NOT eliminated; no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "CFIN-UNIQUE: admissible tuple census at omega = hbar = 14400, G-orbits = 1 (orbit-stabilizer 14400 x 6 = 86400 = |Sp(4,2) x S_5| exact); Aut(C_fin) ~= C6 (element orders [1,2,3,3,6,6], faithful into Sp(4,2), the S_5 slot permutation determined by the symplectic part); the Gaussian diagram fixes the carrier triple up to the 18-element centralizer torsor, q* selector-unique, 120 iota-completions in one orbit; strict terminal reading FAILS (typed either way); controls fire (dot form 0 refinements; wrong Arf class: empty iota set; no sigma: canonicity dies 16 -> 8 -> 4 != 1)",
      "dependencies": "AX.P2.01; NORMALFORM.CFIN.01; MESSAGE.LADDER.01; v774; v833; v844; v845",
      "scripts": [
        "v849_cfin_unique_cofinal_lean.py"
      ],
      "script": "v849_cfin_unique_cofinal_lean.py",
      "location": "tfpt_1;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 708
    },
    {
      "id": "ARF.VACUUM.SYNDROME.01",
      "claim": "v852 -- ARF.VACUUM.SYNDROME.01 + OVOID.DECODER.01: the Arf-vacuum syndrome theorem and the integral five-mode decoder -- the corpus's flat counts, orbit splits, Hecke matches and code enumerators compress into ONE two-bit syndrome table on the Hamming code, and the doily recovery channel gains an explicit integral inverse with the S6 standard representation identified, ONE module from two probes (24/24 + 23/23 checks, zero fails, verdicts ARF-VACUUM-SYNDROME-EXACT and OVOID-DECODER-EXACT; discovery probes affine_arf_vacuum_probe.py and doily_ovoid_decoder_probe.py, 2026-08-07, re-run identically at promotion, embedded BYTE-EXACT and executed verbatim, exact integer/Fraction/sympy arithmetic, <1 s).  PART A, THE ARF-VACUUM SYNDROME: (a) THE ARF BIT THEOREM -- for every affine flat t+U of (F_2^4, hb) the Arf bit a(t+U) = sum_{x in t+U} q*(x) mod 2 equals the direction bit hb(u,v) of the underlying plane U for ALL 140 flats (position independence: all 4 cosets agree; the direction bit is well-defined on all 35 planes); (b) THE TWO-BIT SYNDROME CENSUS -- (s_vac, s_Arf) = (origin bit, direction bit) splits the 140 flats EXACTLY into 15/20/45/60 (doily lines / non-isotropic lines / isotropic off-planes / non-isotropic off-planes), reproducing v821's 35 = 15 + 20 and v834's 105 = 45 + 60 as rows of ONE table; (c) THE LOCAL HECKE THEOREM -- at EVERY nonzero point x: 3 isotropic + 4 non-isotropic directions through x with the isotropic ones in bijection with the points of x^perp/<x>, and the 28 off-origin flats through x split 12 + 16 with A_3 + B_3 = 28 = sigma_3(3) and A_3 - B_3 = -4 = a_3 (the eta-product Hecke eigenvalue, v820/v535 rebuilt LOCALLY at all 15 points; 28 - (-4) = 32 = 2^5 the pi_cusp denominator, v785); (d) THE CODE PENCIL -- H = ker X = [15,11,3] (weight-3 words == the 35 lines, weight-4 == the 105 off-origin flats, SET EQUALITIES), H^perp = the simplex code, and the CANONICITY LEMMA: all 16 refinements differ from q* by simplex words, hence induce ONE Arf functional on H; below H sit D = [15,10,4] (v819's enumerator EXACT), C_iso and C_non = [15,10,3] (weight-3 words == the 15 doily lines resp. the 20 non-isotropic lines), all pairwise intersections == K = [15,9,4], all pairwise sums == H, and the syndrome map H -> F_2^2 is surjective with four fibers of 512; (e) RM(2,4) RESTORED -- puncturing RM(2,4) = [16,11,4] at the vacuum coordinate gives H, shortening gives D, and c -> (p.c, c) is a bijection H -> RM(2,4): THE VACUUM BIT IS THE PARITY BIT, and on the 140 flat words the restored coordinate == s_vac exactly.  Controls fire: the dot form admits 0 refinements and an ill-defined direction bit; an Arf-0 refinement leaves the table invariant but breaks the labeling (zero set 10 != 6, not an ovoid); a frozen-LCG column permutation preserves [15,11,3] but destroys the line labeling (31/35 weight-3 supports not lines).  PART B, THE OVOID DECODER: the Cremona-Richmond incidence N (15 duads x 15 synthemes) and the six ovoid vectors v_a = 3*1_{O_a} - 1 satisfy (a) N^T v_a = 0 for ALL six a (every syntheme covers each letter exactly once -- the kernel facts from perfect matchings), any five of the six are independent, sum_a v_a = 0; (b) the Gram V^T V = 36I - 6J ENTRYWISE; (c) SNF(N) = diag(1^10, 0^5) EXACT over Z -- rank 10 = C(g_car, 2), kernel 5 = g_car, NO TORSION, with a det-+-1 10x10 minor exhibited; (d) the Moore-Penrose closed form N+ = N^T/4 - J/36 verified against all four Penrose axioms in exact Q; (e) the carrier projector P5 = I/2 - B/4 + J/12 == (1/36) sum_a v_a v_a^T entrywise, P5^2 = P5 = P5^T, P5 N = 0, tr P5 = 5 = g_car, N N+ = I - P5 (the visible-part projector); (f) THE MOD-2 SHADOW: v_a mod 2 == q_a entrywise and ker_F2(N^T) == A_q = [15,5,6] (weight enumerator 1 + 10z^6 + 15z^8 + 6z^10) -- THE THREE-RING CORRESPONDENCE: the same kernel is (R) the ovoid span, (Z) torsion-free rank 5, (F_2) the code A_q; (g) THE DECODER DEMO: mhat = (1/4)N y - (1/36)J y == (I - P5) m EXACTLY on the predeclared 6-vector test set (kernel modes invisible, im(N) modes fully visible) -- the recovery channel of DOILY.PASCAL.RANK.01 now has an explicit integral inverse, not just a spectrum; (h) THE S6 IDENTIFICATION: N is S6-equivariant and tr(P5 Pi(pi)) == fix(pi) - 1 for representatives of ALL 11 conjugacy classes -- ker N^T IS the 5-dimensional standard representation: the five-slot carrier is the sixth-root-free shadow of six ovoid modes with one relation.  Control fires: a frozen-LCG wrong pairing (same row/col sums 3) breaks SNF (torsion 3 appears), the F_2 kernel (4 != 32) and the ovoid kernel property.  The P2 integers remain the singular-value data of the doily; no marker moves -- the compression is a normal form plus an inverse, not a new claim.  NO RH claim.  Python-only per GATE.WOLFRAM.02.",
      "status": "Exact identity-grade compression [Identity]: both parts exhaustive finite algebra (F_2 / Z / Q; SNF, Penrose axioms, S6 class characters exact; SET equalities, not counts); the corpus surfaces v819/v820/v821/v834/v535/v785 compressed into ONE two-bit syndrome table and the v774/v814 doily channel into ONE integral decoder; controls fire (dot form, Arf-0 labeling break, frozen-LCG permutation/pairing); no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "ARF-VACUUM-SYNDROME-EXACT + OVOID-DECODER-EXACT: a(t+U) = hb(u,v) on all 140 flats; census 15/20/45/60; the local Hecke theorem 12/16 -> 28/-4 at all 15 points with x^perp/<x> semantics; pencil [15,11,3] > {[15,10,4], 2 x [15,10,3]} > [15,9,4] with the canonicity lemma (one Arf functional on H) and the vacuum bit = the RM(2,4) parity bit; N^T v_a = 0, Gram 36I - 6J, SNF diag(1^10, 0^5) torsion-free (det +-1 minor exhibited), N+ = N^T/4 - J/36 (four Penrose axioms exact in Q), P5 = I/2 - B/4 + J/12 (tr 5 = g_car), decoder exact on the predeclared set, ker = the S6 standard representation (11/11 classes), three rings (R ovoid span / Z torsion-free / F2 code A_q = [15,5,6] via v_a mod 2 = q_a)",
      "dependencies": "NORMALFORM.CFIN.01; MESSAGE.LADDER.01; v774; v819; v820; v821; v834; v844; v845",
      "scripts": [
        "v852_arf_vacuum_ovoid.py"
      ],
      "script": "v852_arf_vacuum_ovoid.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 709
    },
    {
      "id": "ARF.BENT.CSS.01",
      "claim": "ARF.BENT.CSS.01: exact bent phase and CSS code identities; Mangoldt incidence experiment moved to Prime Front. Original module ID retained; original complete mixed record is preserved in extraction provenance.",
      "status": "Exact identity-grade compression [Identity] with the operator-level relation bridge typed HONESTLY as classical incidence algebra (mu*log = Lambda in matrix clothes -- the value is the packaging + the four-comb discipline transport at operator level, NOT new mathematics); the -4 triple point is ONE machine-checked integer (Hecke = Gauss = Walsh = a_3); the bent/character MUB partner of v800 REPORTED with no upgrade; controls fire; no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "ARF-BENT-CSS-EXACT + MANGOLDT-COMMUTATOR-EXACT: s_q-hat = -4 s_q on all 16 (W^2 = 16I, perfect autocorrelation 16 delta); (16,6,2) difference set {0} u 5bar; A_q = C_iso^perp = [15,5,6] with the 31-word structural census; [[15,5,3]] -> [[16,4,4]] (the 60 minimal words = the 60 isotropic planes, the s_Arf = 0 rows of the v852 table; vacuum transformation typed: slot +1, logical qubit -1, distance 3 -> 4); the -4 triple point Hecke = Gauss = Walsh = a_3 (v785: 32 = 28 - (-4)); the 16 bent translates a second Gram-16I frame mutually unbiased vs the characters (+-4; reported, no upgrade); L = -M[D,Z] = -[D, log Z] = T(Lambda) exact three ways (BCH truncates; log Z rational), L^k = ordered chains (nilpotency floor(log2 N) + 1), the four-comb discipline at operator level (Epstein leak [6,14,21,...] exact, first unramified 21 = 3x7, repaired by a_A + a_B = 1 * chi_{-20}; ideal-level Dedekind Lambda_K with norm aggregation (1 + chi4) Lambda), the carrier bridge (the deployed mu-pairing IS row one of L; rows are shifts; L^2 - [D,L] = T(mu*log^2) -- the Selberg hierarchy)",
      "dependencies": "ARF.VACUUM.SYNDROME.01; PRIME.RELATION.MULT.01; v535; v774; v785; v800; v819; v820; v852",
      "scripts": [
        "v853_bent_css_mangoldt.py"
      ],
      "script": "v853_bent_css_mangoldt.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 710
    },
    {
      "id": "E8.SIMPLEX.FOURIER.01",
      "claim": "v857 -- E8.SIMPLEX.FOURIER.01 + FLAVOR.WINDING.QUADRATIC.01: the E8 audits, part I -- the Gaussian census IS a Fourier law and v817's per-prime measurement is the prime restriction of a CHARACTER THEOREM (mh2(n) = -1/15 as an integer identity at EVERY odd n <= 16500), and the tfpt_2 winding line collapses to ONE rank-one determinant quadratic q_wind = t^2 - g_car t + |Z2| with the triple lock s = 6 and the reality threshold typed, ONE module from two probes (26/26 + 20/20 checks, zero fails, verdicts SIMPLEX-FOURIER-EXACT + INTERTWINER-IDENTIFIED and WINDING-QUADRATIC-EXACT; discovery probes simplex_fourier_probe.py and flavor_winding_quadratic_probe.py, 2026-08-08, re-run identically at promotion, embedded BYTE-EXACT and executed verbatim, exact integer/Fraction/sympy arithmetic, ~2 s). PART A, THE FOURIER LAW (audit package A): the Construction-A rebuild is warded (30 placements, 2 invariant under pi_J and pi_sigma; 240 roots, all norm 4, all in L; the quotient V = L/(1+i)L = F_2^4 with class addition well-defined on all 256 pairs and every class 2-torsion), the census r = (0; 16^15) (zero class EMPTY, v833/v844) has the EXACT Walsh transform rhat = (240; -16 x15) with the inversion sum rule, i.e. the normalized spectrum {1, (-1/15)^15}: the uniform-nonzero channel P = (J - I)/15 entrywise in Fractions (doubly stochastic; eigenaction exact on all 16 characters; two-step P^2 = (I + 14J)/225 with contraction 1/225); THE FIVE-LINE THEOREM measured at every level: (i) RAMIFICATION -- Theta_0(n) = 0 for EVERY odd n <= 16 with the one-line proof (x = (1+i)y forces even level) and Theta_0(2m) = 240 sigma_3(m); (ii) EQUIDISTRIBUTION -- Theta_v(n) is ONE value over the 15 nonzero classes at every level n <= 16 (the Sp(4,2) footprint); (iii) hence Thetahat_u(odd n) = -Theta_L(n)/15; (iv) THE INTERTWINER IDENTITY -- the v817 packet numerator (Th0-Th1+Th2-Th3)(n) EQUALS the code Walsh transform Thetahat_u(n) for every nonzero u at EVERY level n <= 16, odd AND even (both ladders read [-16, 112, -448, 1136, -2016, 3136, -5504, 9328] at n = 1..8; the Eisenstein normal form 15 Theta_D8 = 7 Theta_L + 8 Theta_0 across machineries); (v) THE VALUE THEOREM -- 15 (Th0-Th1+Th2-Th3)(n) = -240 sigma_3(n) for EVERY odd n <= 16500, hence mh2(odd n) = -1/15 IDENTICALLY, covering all 1911 odd primes <= 16500: v817's per-prime measurement is a character theorem of the code, one Walsh transform away from the deployed census with NO new input. THE HONEST TYPE (measured, kept): the identification is SPECTRAL (numerator q-series equal) -- the packet 5|3-parity grading (52, 60, 128) and the code grading (15 x 16) do NOT refine each other (the S6.3 cross table: 4 + 3 + 8 mixed classes); the spinor halving Th1 = Th3 stays numerical fiat (typed); 52 = 40 + 12 = |R(D5)| + |R(A3)| is the seam reading. Controls fire (occupied zero class, one class at 15, wrong quotient dim F_2^3 -- each breaks the law at a stated integer), invariances hold (24 explicit isometries of (L, J) preserve census and spectrum; the OTHER invariant placement C** gives the identical law). PART B, THE WINDING QUADRATIC (audit package for tfpt_2, section I.5): against the deployed residue matrix (det R = 8 = h(D5), tr R = 9, PrinMin2 = (5,3,2), SNF (1,1,8), anchor R e1 = (1,1,2), cofactor seam normal adj(R) row 1 = (5,-9,6) with n.1 = 2 = |Z2|), THE MINI-THEOREM: chi_{R_s} - chi_R = -s q_wind with q_wind = e1^T adj(tI - R) 1 = t^2 - 5t + 2 = t^2 - g_car t + |Z2| EXACT in sympy -- the whole winding line is one rank-one matrix-determinant lemma; the keybox coefficients (9+s, 10+5s, 8+2s), the per-winding det increment d(det R_s)/ds = q_wind(0) = 2 = |Z2| and the minor law PrinMin2(R_s) = (5, 3(s+1), 2(s+1)) (carrier minor 5 s-INVARIANT) are all readouts of the single quadratic; q_wind is irreducible over Q (disc 17 nonsquare) and the two base points are s-invariant (rem(chi_{R_s}, q_wind) = -12t, s-free); THE TRIPLE LOCK: trace (tr R_s = 15), determinant (det R_s = 20) and Coxeter lift ((R_s^T a)_1 = 30) each solve to s = 6 ALONE, intersection {6} = {|R+(A3)|}, and at s = 6: chi_L = t^3 - 15t^2 + 40t - 20 with (15, 40, 20) = (dim A3, |R(D5)|, |R(A4)|), 3 real positive roots; THE LOCUS (exact Sturm/resultant): Delta(s) = disc_t chi_{R_s} = 17s^4 - 18s^3 + 709s^2 + 588s - 7996 (irreducible), Delta(6) = 39200 = 2^5 5^2 7^2 = 2 x 140^2 > 0 -- s = 6 is NOT a collision point; the reality threshold is the unique Delta-root in (2, 3), s* ~ 2.8250 (matching the deployed ~2.83; Sturm real-eigenvalue counts jump 1 -> 3 between s = 2 and s = 3); the integer-window census types s = 6 as the ONLY 2xsquare hit in s = -2..12 (measured, typed -- and q_wind(6) = 8 = det R reported as a measured coincidence). Controls fire: the sibling directions 1 e2^T and 1 e3^T change the quadratic AND break the lock (intersections EMPTY; 1 e3^T even reducible), the sibling ROW transport e1 1^T keeps -g_car t but LOSES |Z2| (locks pairwise disjoint 6/12/24), and the sibling minor branch {1,3,4} is unreachable on the whole winding line (integer scan s = -100..100 empty; the carrier minor 5 invariant) -- the triple (1, -5, 2) is direction-specific. Both parts are exact-arithmetic corpus compressions and audits of DEPLOYED claims (v817, v833, v844, v4/v94 keyboxes) -- normal forms and identities, no new physical claim, no marker moves. NO RH claim. Python-only per GATE.WOLFRAM.02.  [2026-08-08 round-LXXIX note (v875): THE CENSUS IS NOW A THEOREM -- E8.SHELL.GLOBAL.01 proves Thetahat_u/Theta_L = -1/15 for ALL odd shells (Sp(4,2) transitivity of the G31 class action + (1+i) doubling + Theta_L = 240 sigma_3), replacing the n <= 16500 finite-census argument behind the value theorem; the census range is retained as the exact ward and the character-theorem reading of this row is unchanged; the consequence chain is kernel-checked in FORM.E8.SHELL.GLOBAL.01 (TfptCarrier/GaussianShells.lean).]",
      "status": "Exact corpus compression [typed evidence line: 26/26 + 20/20 checks, zero fails, exact integer/Fraction/sympy arithmetic throughout; the identification typed SPECTRAL-only (the packet and code gradings do not refine each other); the spinor halving stays numerical fiat; q_wind(6) = 8 = det R typed a measured coincidence; the integer-window 2xsquare census measured, not narrated]; audits of DEPLOYED claims (v817/v833/v844, tfpt_2 I.5 keyboxes) -- normal forms and identities, no new physical claim; NO RH claim; no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "SIMPLEX-FOURIER-EXACT + INTERTWINER-IDENTIFIED + WINDING-QUADRATIC-EXACT: rhat = (240; -16 x15), P = (J-I)/15 with spec {1, (-1/15)^15}, two-step 1/225 exact; ramification + equidistribution at every level <= 16; packet numerator == code Walsh at every level (both [-16, 112, -448, 1136, ...]); mh2(odd n) = -1/15 for every odd n <= 16500 (integer identity, 1911 primes); q_wind = t^2 - 5t + 2 exact by the adjugate lemma; triple lock s = 6; Delta(6) = 39200 = 2 x 140^2, sole 2xsquare hit; s* ~ 2.8250 the unique Delta-root in (2,3); 7 controls fire, 2 invariances hold",
      "dependencies": "CFIN.UNIQUE.01; MESSAGE.LADDER.01; ARF.BENT.CSS.01; v4; v94; v817; v833; v844",
      "scripts": [
        "v857_simplex_fourier_winding.py"
      ],
      "script": "v857_simplex_fourier_winding.py",
      "location": "tfpt_1;tfpt_2;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 711
    },
    {
      "id": "E8.G31.CLOCK_ALPHABET.01",
      "claim": "v858 -- E8.G31.CLOCK_ALPHABET.01: the E8 audits, part II -- the compiler clock alphabet is a NORMAL FORM of G31: Deg(G31)/4 = {2, 3, 5, 6} = {|Z2|, N_fam, g_car, |R+(A3)|} with the audit identity (sum d/4)(sum (d-1)/4) = 16 x 15 = 240 REALIZED on the roots, (8, 12, 20, 24) the UNIQUE integer quadruple with product 46080 and sum 64, the 607-group rank-4 kill scan typed (G31 the SOLE full-battery passer; weak-triple and raw-sum-64 impostors named), and the W(D5) x W(A3) fence re-killed with COMPUTED centers 4 vs 1, ONE module from one probe (19/19 checks, zero fails, verdict CLOCK-ALPHABET-SELECTIVE; discovery probe g31_clock_alphabet_probe.py, 2026-08-08, re-run identically at promotion, embedded BYTE-EXACT and executed verbatim, ~4 s). THE REBUILD (warded against v634): 240 doubled-integer E8 roots, J free of order 4 with EXACTLY 60 Gaussian lines, 60 order-2 reflections (one per line, all distinct), BFS closure |G| = 46080 = 8 x 12 x 20 x 24 = prod Deg(G31), every element complex-linear (commutes with J), the base certified by an exact adjugate identity (det = -512), and the EXACT reflection census: 60 elements with rank(M - I) = 2, all order 2, all generators -- ZERO quasi-reflections (v634 TEST2 reproduced; sum (d-1) = 60 anchors the degrees; the Molien uniqueness CITED, not recomputed). THE NORMAL FORM: Deg(G31) = |mu4| x {2, 3, 5, 6} with gcd = 4 = |mu4|, lcm = 120 = |R+(E8)|, lcm/gcd = 30 = h(E8); sum d/4 = 16 (the class size) and sum (d-1)/4 = 15 (the classes) with 16 x 15 = 240 = |R(E8)| -- and the 60/4 = 15 line is REALIZED on the roots (the 15 classes at 4 lines per class, the same 15 x 16 census as the package-A code side); the alphabet {2, 3, 5, 6} = {|Z2|, N_fam, g_car, |R+(A3)|} is EXACTLY the compiler clock alphabet; the Springer footprint is typed (exponents (7, 11, 19, 23) all = 3 mod 4; the d/4-clock realization x^{d/4} in {J, J^3} cited v654). THE ARITHMETIC PIN (measured): the integer-quadruple census with product 46080 and sum 64 returns EXACTLY [(8, 12, 20, 24)]. THE KILL SCAN (the selectivity theorem, 607 groups): the full rank-4 Shephard-Todd catalog (integrity: prod(degrees) = |G| for every entry -- a table second-source check) is run through the seven-gate battery (product, raw sum 64, gcd normalization, alphabet set, lcm, lcm/gcd = 30, audit identity): G31 is the SOLE full-battery passer; the weak-triple (gcd/lcm/30) impostors are typed by name (G(20,10,4), G(20,20,4), G29) and the raw-sum-64 coincidences too (G(8,2,4), G(10,10,4), G30 = W(H4)); the family boundary is verified in range (no G(m,p,4) matches the alphabet for any 2 <= m <= 120, with the typed argument: 3m/g <= 6 forces g >= m/2, then {d/g} contains 1 or 4); THE HONESTY LINE (typed, no kill): degree-only audits cannot separate G31 from the reducible torus Z8 x Z12 x Z20 x Z24 (identical degrees by construction) -- the frozen scope is the irreducible rank-4 list; the controls fire (the perturbed quadruple (8,12,20,28) fails the battery; the wrong normalization gcd 2 breaks the alphabet). THE FENCE (re-killed with a computed one-number invariant): |W(D5)| x |W(A3)| = 1920 x 24 = 46080 = |G31| stays NUMEROLOGY -- the computed |Z(G31)| = 4 with Z = <J> = mu4, versus |Z(W(D5) x W(A3))| = 1 x 1 = 1 (odd-rank D5: -1 not in W(D5); Z(S4) = 1): NOT isomorphic, NOT a decomposition (v634 T1.3/T1.4/T1.5 CITED for the full structural refutation: G = (Z4 o 2^{1+4}).Sp4(2) and W(D5) embeds in NO rank-4 group). The module is an exact-arithmetic corpus compression and a selectivity audit of a DEPLOYED object (v634/v654) -- a normal form plus an armed kill rule, no new physical claim, no marker moves. NO RH claim. Python-only per GATE.WOLFRAM.02.  [2026-08-08 round-LXXIX note (v874): THE ALPHABET HAS A DECODER -- E8.WINDING.DECODER.01 proves that the v857 winding quadratic maps the clock alphabet {2,3,5,6} to (-|mu4|,-|mu4|,|Z2|,h(D5)) with the decoder polynomial (y+4)^2(y-2)(y-8) carrying budget coefficients (2, 48, 32, 256) and the exact bridge |mu4| q(|Z2|)/|R(E8)| = -1/15 = the Walsh message eigenvalue; typed AUDIT theorem, not a functor; the 6-vs-7 honesty note of this row carries over verbatim.]",
      "status": "Exact corpus compression + selectivity audit [typed evidence line: 19/19 checks, zero fails; the honesty line kept (degree-only audits cannot separate G31 from the reducible torus Z8 x Z12 x Z20 x Z24 -- frozen scope the irreducible rank-4 Shephard-Todd list); the Molien uniqueness and the full W(D5)-fence structural refutation CITED (v634), not recomputed; X1/X2 controls fire]; a normal form plus an armed kill rule for a deployed object; NO RH claim; no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "CLOCK-ALPHABET-SELECTIVE: Deg(G31)/4 = {2,3,5,6} = {|Z2|, N_fam, g_car, |R+(A3)|}; gcd 4 = |mu4|, lcm 120 = |R+(E8)|, lcm/gcd 30 = h(E8); (sum d/4)(sum(d-1)/4) = 16 x 15 = 240 realized on the roots (4 lines/class); (8,12,20,24) the unique product-46080/sum-64 quadruple; 607-group scan: G31 sole full-battery passer (weak-triple impostors G(20,10,4)/G(20,20,4)/G29; sum-64 coincidences G(8,2,4)/G(10,10,4)/W(H4)); family boundary m <= 120 empty; fence killed by computed centers 4 vs 1",
      "dependencies": "v634; v654; CFIN.UNIQUE.01",
      "scripts": [
        "v858_g31_clock_alphabet.py"
      ],
      "script": "v858_g31_clock_alphabet.py",
      "location": "tfpt_1;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 712
    },
    {
      "id": "FLAVOR.FEEDBACK.NORMALFORM.01",
      "claim": "THE FEEDBACK NORMAL FORM OF THE WINDING LINE (2026-08-08 round 36, v874; discovery probe flavor_feedback_probe.py 14/14, verdict FEEDBACK-NORMALFORM-EXACT; exact sympy/integer arithmetic in every decision).  THEOREM: with the integral basis P = [2*1 | e3 | R e3] (det P = -4, SNF diag(1,2,2)) the WHOLE winding line R_s = R + s 1 e1^T conjugates EXACTLY (symbolic in s) to M(s) = [[s+4,0,3],[0,0,-2],[4,1,5]]: a gain channel s + |mu4|, the companion block of the v857 winding quadratic q_wind = t^2 - g_car t + |Z2|, and the coupling pair (3,4) with loop product N_fam x |mu4| = 12 = dim g_SM; the conjugated deformation P^{-1}(1 e1^T)P = E11 is a PURE GAIN; hence chi_{R_s}(t) = (t - s - |mu4|) q_wind(t) - 12t identically (equivalently the remainder identity chi_R - (t-4) q_wind = -12t, a pure t-tap at gain offset 4 = |mu4|), and at the locked point s = 6 the coefficients (15, 40, 20) of chi_L are the three gain-10 readouts 10+5, 50+2-12, 10x2.  CONTROLS (all fire): directions 1 e2^T (s leaks into the coupling; constant graft +8, offset 8 != 4), 1 e3^T (not integral; mixed remainder), and the sibling row matrix e1 1^T (smeared over all rows; offset 4 coincides -- typed honestly -- but the remainder -11t - 4 has a nonzero constant).  THE HONEST TYPING (load-bearing, kernel-checked in FORM.FLAVOR.FEEDBACK.01): the quotient Z^3/PZ^3 = Z2 x Z2 has CARDINALITY 4 = |mu4| but EXPONENT 2 -- the same cardinality as mu4 = Z4, NOT the same group; any mu4 reading would need an extra order-4 witness (none claimed).  NO physics claim beyond the audit statement.",
      "status": "Honest promotion of an exact integer theorem [typed evidence line: 14/14 checks, zero fails, verdict FEEDBACK-NORMALFORM-EXACT; all decisions exact sympy; frozen spec SHA f5f40996...]",
      "canonicalStatus": "FEEDBACK-NORMALFORM-EXACT: P^{-1} R_s P = [[s+4,0,3],[0,0,-2],[4,1,5]] exact symbolic in s; chi_{R_s} = (t-s-|mu4|) q_wind - 12t; quotient Z2^2 typed NOT mu4; all sibling controls fire.",
      "dependencies": "E8.SIMPLEX.FOURIER.01; E8.G31.CLOCK_ALPHABET.01; v4; v857; v858",
      "scripts": [
        "v874_feedback_decoder_lean.py"
      ],
      "script": "v874_feedback_decoder_lean.py",
      "location": "tfpt_2_standard_model;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 713
    },
    {
      "id": "E8.WINDING.DECODER.01",
      "claim": "THE WINDING DECODER OF THE CLOCK ALPHABET (2026-08-08 round 36, v874; discovery probe g31_winding_decoder_probe.py 11/11, verdict WINDING-DECODER-EXACT; exact arithmetic).  THEOREM: the v857 winding quadratic q_wind(t) = t^2 - g_car t + |Z2| maps the G31 clock alphabet {2, 3, 5, 6} = {|Z2|, N_fam, g_car, |R+(A3)|} (v858) to the atom quadruple (-|mu4|, -|mu4|, |Z2|, h(D5)) = (-4, -4, 2, 8) -- with q(2) = q(3) FORCED by the palindrome q(5-t) = q(t) (axis g_car/2) and 2 + 3 = 5 = g_car; the decoder polynomial D(y) = prod(y - q(a)) = (y+4)^2 (y-2)(y-8) = y^4 - 2y^3 - 48y^2 - 32y + 256 carries the signed budget coefficients (|Z2|, Omega_adm = N_fam dim S+ = 48, 2^g_car = 32, 2^rank(E8) = 256) with the double root -|mu4| forced by the palindrome pair; and THE FOURIER BRIDGE: |mu4| q(|Z2|)/|R(E8)| = 4 x (-4)/240 = -1/15 is EXACTLY the nonzero-character Walsh eigenvalue of the Gaussian shell census (E8.SIMPLEX.FOURIER.01; now global by E8.SHELL.GLOBAL.01), and -16 = |mu4| q(|Z2|) = rhat(u != 0).  CONTROLS (all fire): sibling directions collapse the bridge (q2 -> 1/15, q3 -> 1/15, q_row -> -1/12); alternative alphabets change the decoder ({2,3,5,7}: constant 512 = 2^9 NOT 2^rank(E8); {1,2,3,5} different).  HONEST LINES (typed): the -32 y coefficient survives in both alternatives -- per-coefficient selectivity is weak, the selective object is the full vector (-2,-48,-32,+256); the 6-vs-7 tension typed (replacing 6 by 7 loses the y^3/y^2/constant identifications).  TYPE: exact AUDIT theorem, NOT a physical functor -- an explicit intertwiner would be needed to make it causal.  NO physics claim beyond the audit statement.",
      "status": "Honest promotion of an exact integer theorem [typed evidence line: 11/11 checks, zero fails, verdict WINDING-DECODER-EXACT; frozen spec SHA 6a54768d...]",
      "canonicalStatus": "WINDING-DECODER-EXACT: q({2,3,5,6}) = (-4,-4,2,8); D(y) = (y+4)^2(y-2)(y-8) with budget coefficients (2,48,32,256); |mu4| q(|Z2|)/|R(E8)| = -1/15 == the Walsh message eigenvalue; typed audit-not-functor.",
      "dependencies": "E8.G31.CLOCK_ALPHABET.01; E8.SIMPLEX.FOURIER.01; E8.SHELL.GLOBAL.01; FLAVOR.FEEDBACK.NORMALFORM.01; v857; v858",
      "scripts": [
        "v874_feedback_decoder_lean.py"
      ],
      "script": "v874_feedback_decoder_lean.py",
      "location": "tfpt_1_architecture_e8;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 714
    },
    {
      "id": "FORM.FLAVOR.FEEDBACK.01",
      "claim": "LEAN 4 FORMALISATION of the feedback normal form (2026-08-08 round 36; TfptCarrier/FlavorFeedback.lean, 16 declarations, 11 theorems) -- AUDIT: PASS, no sorry, no native_decide, no domain axioms (#print axioms = propext, Classical.choice, Quot.sound on every theorem; lake build green 3418 jobs).  Kernel-checked over Z with a SYMBOLIC s (no numeric specialization smuggled in): the three action identities R_s b0 = (s+4) b0 + 4 b2, R_s b1 = b2, R_s b2 = 3 b0 - 2 b1 + 5 b2 (the deformation enters ONLY through b0); THE CONJUGATION P F_s = R_s P (the inverse-free house form -- det P = -4 != 0 makes it P^{-1} R_s P = F_s over Q); THE CHARACTERISTIC POLYNOMIAL chi_{R_s}(X) = (X - (s+4))(X^2 - 5X + 2) - 12X in Z[X] via Matrix.charpoly and the 3x3 determinant expansion, with the s = 6 instance chi_L = X^3 - 15X^2 + 40X - 20; and THE LATTICE INDEX facts det P = -4, gcd(entries) = 1, gcd(2x2 minors) = 2 -- the determinantal divisors d1 = 1, d1 d2 = 2, d1 d2 d3 = 4 forcing the invariant factors (1, 2, 2), SNF diag(1,2,2), quotient Z2 (+) Z2.  THE HONEST TYPING deliberately NOT a claimed isomorphism: Z2^2 has the CARDINALITY of mu4 but is not mu4 as a group -- the SNF computed in Lean is what makes that distinction kernel-checked rather than blurred.  Mirrored numerically in v874 part C.",
      "status": "Formal [F] (Lean kernel; the action identities, the inverse-free conjugation, charpoly in Z[X], the SNF determinantal divisors -- with the Z2^2-not-mu4 typing doc-level)",
      "canonicalStatus": "Formal [F]: the feedback normal form is a Lean theorem family (11 theorems), standard axioms only, no sorry, full build green.",
      "dependencies": "FLAVOR.FEEDBACK.NORMALFORM.01; FORM.QGEO.01",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/FlavorFeedback.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/FlavorFeedback.lean",
      "location": "tfpt_2_standard_model;tfpt_research_contracts",
      "externalData": "Lean 4 (leanprover/lean4 v4.29.1) + Mathlib",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 715
    },
    {
      "id": "E8.SHELL.GLOBAL.01",
      "claim": "THE GLOBAL GAUSSIAN SHELL THEOREM (2026-08-08 round 36, v875; discovery probe gaussian_shell_global_probe.py 15/15, verdict SHELL-GLOBAL-THEOREM) -- the census->theorem upgrade: the v857 value law 'Thetahat_u/Theta_L = -1/15 on odd shells' holds for ALL shells, no census cap.  THEOREM: on V = L/(1+i)L of the Gaussian E8, for EVERY n >= 1: Theta_v(n) = (Theta_L(n) - Theta_0(n))/15 for all v != 0 (the division exact), Thetahat_u(n) = 16(16 1_{2|n} sigma_3(n/2) - sigma_3(n)) for every nontrivial character u, and at every ODD n: Thetahat_u(n) = -16 sigma_3(n), Thetahat_u(n)/Theta_L(n) = -1/15 EXACTLY.  INPUTS (each warded): (i) LEMMA 1 -- every g in G31 is a norm-preserving bijection of L transporting classes ((4M)^T(4M) = 16I; complex linearity (4M)J = J(4M)); the induced group is Sp(4,2) (order 720, generated by the 60 symplectic transvections), TRANSITIVE on the 15 nonzero classes, and IDENTICAL to the shell-module class action (coordination verified on generators); (ii) LEMMA 2 -- (I+J)^T(I+J) = 2I and (1-J)/2 inverts (1+i) on L: Theta_0(2m) = Theta_L(m), Theta_0(odd) = 0 (from 2 = -i(1+i)^2); (iii) LEMMA 3 -- Theta_L(n) = 240 sigma_3(n) (classical Eisenstein E4; verified to n = 24 directly and via the v857 packet glue identity to n = 16500).  The n <= 16500 census is RETAINED as the exact ward (matched everywhere, odd AND even).  CONTROLS: the non-transitive subgroup <T_1> (11 orbits) admits an explicit invariant NON-uniform census with Walsh multiset {-48 x4, -16 x7, +16 x4} -- transitivity is load-bearing; the even-shell formula differs correctly (7/135 at n = 2: the -1/15 law is the ODD-shell statement only).  This REPLACES the finite-census argument behind the v857 value theorem (dated note there); the consequence chain is kernel-checked in FORM.E8.SHELL.GLOBAL.01.  NO physics claim beyond the audit statement.",
      "status": "Honest promotion of a proved shell law [typed evidence line: 15/15 checks, zero fails, verdict SHELL-GLOBAL-THEOREM; frozen spec SHA a4b2f6d9...]",
      "canonicalStatus": "SHELL-GLOBAL-THEOREM: Sp(4,2) transitivity + (1+i) doubling + Theta_L = 240 sigma_3 => Thetahat_u/Theta_L = -1/15 at ALL odd shells; census n <= 16500 retained as ward; non-transitive control breaks the law.",
      "dependencies": "E8.SIMPLEX.FOURIER.01; E8.G31.CLOCK_ALPHABET.01; v857; v858; v868",
      "scripts": [
        "v875_shell_global_lean.py"
      ],
      "script": "v875_shell_global_lean.py",
      "location": "tfpt_1_architecture_e8;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 716
    },
    {
      "id": "FORM.E8.SHELL.GLOBAL.01",
      "claim": "LEAN 4 FORMALISATION of the global shell theorem's consequence chain (2026-08-08 round 36; TfptCarrier/GaussianShells.lean) -- AUDIT: PASS, no sorry, no native_decide, no domain axioms (#print axioms = propext, Classical.choice, Quot.sound on all 7 theorems; lake build green 3418 jobs).  THE PATTERN (the house discipline): the ARITHMETIC CONSEQUENCE CHAIN is kernel-checked while the finite/structural inputs stay NAMED hypotheses with verification provenance typed on each field -- the hypothesis bundle ShellSystem carries (H1) equidistribution 15 Theta_v = Theta_L - Theta_0 (provenance: Sp(4,2) transitivity + shell invariance, machine-verified finite-range in v875), (H2) doubling Theta_0(2m) = Theta_L(m), Theta_0(odd) = 0 (exact from 2 = -i(1+i)^2), (H3) Theta_L = 240 sigma_3 for n >= 1 (classical, cited; finite-range verified); the 7 kernel-checked consequences: dvd_diff (the /15 division exact), decomposition, walsh_formula (15 Thetahat_u = 16 Theta_0 - Theta_L; the 16-coset Walsh combinatorics absorbed into the definition, typed), theta_v_odd, walsh_odd (Thetahat_u(2m+1) = -16 sigma_3(2m+1)), walsh_odd_ratio (15 Thetahat_u(2m+1) = -Theta_L(2m+1) -- the -1/15 statement as an integer identity, no division anywhere), walsh_even.  sigma_3 is mathlib's ArithmeticFunction.sigma 3 -- nothing bespoke; no modular-forms machinery imported or claimed.  Mirrored numerically in v875 part B.",
      "status": "Formal [F] (Lean kernel; the consequence chain from the named ShellSystem hypotheses -- every statement an exact integer identity)",
      "canonicalStatus": "Formal [F]: the shell consequence chain is a Lean theorem family (7 theorems from the named hypothesis bundle), standard axioms only, no sorry, full build green.",
      "dependencies": "E8.SHELL.GLOBAL.01; FORM.QGEO.01",
      "scripts": [
        "lean4-carrier-rigidity/TfptCarrier/GaussianShells.lean"
      ],
      "script": "lean4-carrier-rigidity/TfptCarrier/GaussianShells.lean",
      "location": "tfpt_1_architecture_e8;tfpt_research_contracts",
      "externalData": "Lean 4 (leanprover/lean4 v4.29.1) + Mathlib",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 717
    },
    {
      "id": "GATE.LEDGER.01",
      "claim": "THE STATUS-LEDGER INTEGRITY GUARD (2026-08-08 round 36, v878_ledger_integrity.py 10/10; a CI sentinel, no new physics).  THE TRIGGER: the round-36 ledger audit found the first PRIMARY-KEY violation -- claim_id FORM.QGEO.03 double-booked (2026-06-15 CohomologyGrading formalisation; 2026-06-18 SeamDeckClosure flat all-orders closure).  THE FIX (this round, dated QA note on FORM.QGEO.04): the SeamDeckClosure row RENAMED FORM.QGEO.04; every flat-closure reference updated (tfpt_research_contracts, the two 2026-06-18 changelog mentions, v279/v284/v286 + seam_equivalence.json, make_figures, ResidualChain.tsx, script_registry, run_all); the CohomologyGrading row keeps FORM.QGEO.03; claim contents unchanged.  THE GUARD (enforced in the suite from this round): S1 the 10-column schema + claim-id grammar + active flags; S2 claim_id GLOBALLY UNIQUE with the fix pinned; S3 every supersedes target exists, the supersession graph ACYCLIC (Tarjan), no self-supersession; S4 UNIQUE ACTIVE SUPERSESSION TARGETS modulo the frozen remove-only allowlist of the two legitimate 2026-06 joint closures (FLAV.UGATE.01 <- FLAV.RIGID.01+02; GATE.QGEO.01 <- FLAV.QGEO.02+03); S5 the dependencies column typed honestly as a CROSS-REFERENCE field (mutual evidence pointers, not a build DAG): claim-like tokens must resolve or stay within the frozen 37-token legacy list, the mutual-reference cycle census frozen REMOVE-ONLY (<= 13 SCCs, subset of the frozen 265 ids -- the audit_baseline.json precedent), no self-dependency; acyclicity thereby enforced as NO NEW CYCLES.  Python-only (stdlib).",
      "status": "Identity [I] (CSV integrity sentinel; frozen baselines remove-only)",
      "canonicalStatus": "Ledger integrity enforced: unique claim_id (active + global), acyclic supersession with existing targets, unique active supersession targets modulo the frozen 2-pair allowlist, dependency cross-reference census frozen remove-only.",
      "dependencies": "FORM.QGEO.01; FORM.QGEO.02; FORM.QGEO.04",
      "scripts": [
        "v878_ledger_integrity.py"
      ],
      "script": "v878_ledger_integrity.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 718
    },
    {
      "id": "CFIN.QSTAR.NORMALFORM.01",
      "claim": "v880 (part 1) -- the closed census-free normal form of the distinguished quadratic refinement: q*(x) = sum_{i<j} x_i x_j + sum_i x_i = C(|x|+1, 2) mod 2, WEIGHT-ONLY with value table (0,1,1,0,0); the 16 refinements of the bit form ARE the 16 linear shifts of the polar quadric and the frozen v845 selector acts on the linear part as c1=c2=c3 (4) -> c4=1 (2) -> c1=1 (1), FORCING c = (1,1,1,1) -- the 2^16 search is replaced by a theorem; zeros = {0} u the five ovoid messages (weights 3, 4); sigma splits 5 = 1+1+3 and 10 = 1+3+3+3 (9 = N_fam^2 family cycles).  Probe qstar_doily_anchor_probe.py S1 (26/26 total, FINITE-ANCHOR-CLOSED).",
      "status": "Exact identity-grade theorem [Identity]: closed boolean normal form, census-free derivation; no marker move (Python-only per GATE.WOLFRAM.02)",
      "canonicalStatus": "FINITE-ANCHOR-CLOSED (S1): q* = polar quadric + forced linear part (1,1,1,1); weight-only (0,1,1,0,0); sigma splits 5 = 1+1+3, 10 = 1+3+3+3",
      "dependencies": "NORMALFORM.CFIN.01; v845; v774",
      "scripts": [
        "v880_finite_anchor_theorems.py"
      ],
      "script": "v880_finite_anchor_theorems.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 719
    },
    {
      "id": "DOILY.ANCHOR.CLASSIFY.01",
      "claim": "v880 (part 2) -- the EXCLUSIVE anchor classification of PG(3,2) lines: EVERY of the 15 doily (isotropic) lines carries the local weight multiset {2 - q*} = {2,1,1} = the anchor a = (1,1,2) (per-line charpoly (t-2)(t-1)^2, elementary symmetric (4,5,2) with e2 = g_car, e3 = |Z2|), and NO other line does: the 20 non-isotropic lines split EXACTLY as 10 secants ({0,0,1} = the WRONG anchor (1,2,2), Vieta (5,8,4)) + 10 external lines ({1,1,1}); incidence budget 15 = M_0 and 30 = M_1 = h(E8) (counting only, no bijection claimed).  Anchor-Vieta report: p2(a) = 6, chi_L - chi_R = -p2(a)(t^2 - e2 t + e3), M_n = 15(1, e3, e1, e1 e3, e1^2), rank N = C(e2,2) = 10, ker = e2 = 5.  Probe S2+S5.",
      "status": "Exact identity-grade theorem [Identity]: the anchor law characterizes the doily among ALL 35 lines (exclusivity census 15 + 10 + 10); no marker move",
      "canonicalStatus": "FINITE-ANCHOR-CLOSED (S2/S5): doily lines = anchor multiset {2,1,1}, Vieta (4,5,2); secants = wrong anchor (1,2,2); external = (1,1,1); ladder and flavor identities in anchor notation",
      "dependencies": "CFIN.QSTAR.NORMALFORM.01; MESSAGE.LADDER.01; v832; v844; v857",
      "scripts": [
        "v880_finite_anchor_theorems.py"
      ],
      "script": "v880_finite_anchor_theorems.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 720
    },
    {
      "id": "DOILY.HODGE.FACTOR.01",
      "claim": "v880 (part 3) -- the doily IS the multiplication table Lambda^2 E x Lambda^2 E -> Lambda^4 E of the five-slot carrier: via the v845 parity lift the three iota-supports of every doily line partition {1..5} as {i} u {j,k} u {l,m}; b_jk ^ b_lm = +-f_i EXACT (integer exterior algebra); the map line -> (i; {j,k},{l,m}) is a BIJECTION onto all 5 x 3 = 15 configurations (three perfect matchings per carrier slot); CONVERSE: the wedge is nonzero EXACTLY on doily lines (external lines: all pairwise products vanish; secants carry <= 1 pair point).  Probe S3.",
      "status": "Exact identity-grade theorem [Identity]: the incidence geometry IS the Clifford multiplication; no marker move",
      "canonicalStatus": "FINITE-ANCHOR-CLOSED (S3): support partition + exact wedge factorization + bijection 15 = 5 x 3 + converse (nonzero exactly on doily lines)",
      "dependencies": "CFIN.QSTAR.NORMALFORM.01; v845; v774; v854",
      "scripts": [
        "v880_finite_anchor_theorems.py"
      ],
      "script": "v880_finite_anchor_theorems.py",
      "location": "tfpt_1;tfpt_3",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 721
    },
    {
      "id": "P2.QUADRATIC.EXHAUSTION.01",
      "claim": "v880 (part 4) -- QUADRATIC ONE-STEP EXHAUSTION FORCES g_car = 5 ([E] theorem / [O] premise split): the demand Lambda^2 E ^ Lambda^2 E = Lambda^{g-1} E (one quadratic composition step exhausts the top nontrivial half-spinor sector) holds for EXACTLY one odd g in {3, 5, 7, 9, 11}, namely g = 5 (exact rank 5 = dim Lambda^4; S+ = 1 + 10 + 5 = 16); even control g = 4 breaks the half-spinor demand.  THE THEOREM IS [E]; the PHYSICAL premise (the Calderon boundary kernel satisfies one-step quadratic exhaustion) is [O]/[C] -- P2 stays narrowed to v799's R1', NOT eliminated; NO marker move (per the review, the open physical premise must not be buried under a green finite module).  Probe S4.",
      "status": "Exact algebraic theorem [Identity] with the [O]/[C] physical premise EXPLICITLY typed and NOT claimed: P2 unchanged",
      "canonicalStatus": "FINITE-ANCHOR-CLOSED (S4): one-step exhaustion <=> g = 5 (unique among odd g); the Calderon premise stays [O] -- the sharpened but open half of P2",
      "dependencies": "AX.P2.01; NORMALFORM.CFIN.01; v799; v845",
      "scripts": [
        "v880_finite_anchor_theorems.py"
      ],
      "script": "v880_finite_anchor_theorems.py",
      "location": "tfpt_1;tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 722
    },
    {
      "id": "DOILY.PLUCKER.GR25.01",
      "claim": "THE GR(2,5) PLUCKER READING + THE S6 SPECHT CHANNELS + THE HADAMARD ANCHOR + THE VACUUM SELECTION (closed 2026-08-09, round CV; promoted round CVIII in v888; probe s6_plucker_hadamard_probe.py 27/27 S6PLUCKER-CLOSED, exact integer/sympy): (1) the 15 doily lines ARE the 15 monomials of the five Plucker quadrics of Gr(2,5) -- bijection 15/15 with PERFECT sign match (every quadric reads (+,-,+) = the v880 Hodge table; converse warded: secants/externals produce no monomial); (2) the S6 duad representation decomposes 15 = 1 + 5 + 9 (Specht S(6) + S(5,1) + S(4,2)) and the normalized doily operator acts by the scalars (1, 0, 2/3) EXACTLY, with 2/3 = e_3/(e_2 - e_3) and 15 = 1 + e_2 + (e_2 - e_3)^2 at the anchor (4, 5, 2); (3) the q* zero set is a (16, 6, 2) Hadamard difference set with the anchor signature (v, k, lambda, v-k) = (2^{e_1}, p_2, e_3, C(e_2, 2)) = (16, 6, 2, 10), bent ward |W| = 4 on all 16 characters; (4) THE VACUUM-SELECTION THEOREM: Sp(4,2) = S6 acts transitively on the 6 Arf-1 refinements; Stab(q*) has order 120 acting faithfully as S_5 on the remaining five; its orbits on the 16 messages are [1, 5, 10] EQUAL AS SETS to (Lambda^0, Lambda^4, Lambda^2) -- the five-slot carrier is the residual symmetry after choosing one of six equivalent vacua.  Controls: Arf-0 has census (16,10,6); a secant breaks the bijection; the wrong scalar breaks the channel law.  All exact; no marker moves.",
      "status": "Closed finite theorems [E]: four exact statements; the physical vacuum-selection mechanism stays [O]; no marker move",
      "canonicalStatus": "CLOSED [E]: doily = the Plucker monomial incidence of Gr(2,5) with exact signs; Specht channels (1, 0, 2/3); (16,6,2,10) anchor signature; S6 -> S5 vacuum selection",
      "dependencies": "CFIN.QSTAR.NORMALFORM.01; DOILY.ANCHOR.CLASSIFY.01; DOILY.HODGE.FACTOR.01; v880; v845; v844",
      "scripts": [
        "v888_finite_pfaffian_signature.py"
      ],
      "script": "v888_finite_pfaffian_signature.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 723
    },
    {
      "id": "CFIN.AUT.FLAVOR.C6.01",
      "claim": "THE FLAVOR HEXAGON IS THE COMPILER AUTOMORPHISM ORBIT (closed 2026-08-09, round CV; promoted round CVIII in v888; probe cfin_aut_flavor_probe.py 19/19 C6FLAVOR-MEASURED + INTERTWINED): Aut(C_fin) recomputed by exhaustive enumeration = C_6; the pinned generator g satisfies g^2 = the deployed family 3-cycle sigma EXACTLY (sharp, not inverse) and g^3 = the W-edge sheet swap (the S_2 factor of Stab_{S5}(W); honest: no sheet involution is deployed on V itself); the induced action on Lambda^2 E has cycle type 1 + 3 + 6 with orbits = the charge-pure Petersen distance shells, and the 6-orbit IS the deployed v808 hexagon transport edge for edge (up to D_6, orientation -1); the recovery value reads as one full automorphism orbit: (|Z_2|/N_fam)^{|Aut(C_fin)|} = (2/3)^6 = 64/729 == the frozen corpus value (exact Fractions, no fit; typed as a consistent NEW READING -- the corpus derivation via p_2 = |R+(A3)| stands).  CONTROL (census, not narration): 0 of the 600 symplectic maps preserving hbar but not q* factor consistently as g^2/g^3; a scrambled transport table fails the D_6 match.  Exact arithmetic; no marker moves.",
      "status": "Closed finite theorem [E] (the intertwiner exists and is exact; the exponent reading is typed as consistent, not as the unique derivation); no marker move",
      "canonicalStatus": "CLOSED [E]: g^2 = sigma exact; the hexagon = the Aut(C_fin) 6-orbit on Lambda^2 E; (2/3)^6 = the recovery value; 0/600 non-automorphisms factor",
      "dependencies": "v845; v849; v808; v857",
      "scripts": [
        "v888_finite_pfaffian_signature.py"
      ],
      "script": "v888_finite_pfaffian_signature.py",
      "location": "tfpt_1;tfpt_2;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 724
    },
    {
      "id": "DOILY.PFAFFIAN.WICK.01",
      "claim": "THE FERMIONIC PFAFFIAN SIGNATURE OF THE FINITE COMPILER (closed 2026-08-09, round CVII; promoted round CVIII in v888; probe k6_pfaffian_selfhosting_probe.py 42/42 K6PFAFFIAN-CLOSED, exact integer/sympy, first frozen run): (1) THE K6 MOTIF THEOREM (CFIN.K6.MOTIF.01): under the corpus duad labeling the 35 lines of PG(3,2) are EXACTLY 15 perfect matchings of K6 (= the doily) + 10 triangles through the vacuum vertex (= the secants, weights {2,2,1} = the wrong anchor) + 10 triangles avoiding it (= the external lines); q*(e) = 0 iff e touches the vacuum -- the referee bit of relations; (2) THE SIGN THEOREM: the fermionic Pfaffian signs sgn(M) of the symbolic 6x6 expansion match the corpus Hodge table under the CANONICAL global character chi(i) = (-1)^{i+1} on the five vacuum edges -- triple-identified as the vacuum-row Laplace sign (Pf(A) = sum_i (-1)^{i+1} a_0i Pf(A_0i-hat) symbolic, each 4x4 minor Pfaffian = the Gr(2,5) Plucker quadric), the Lambda^4/Lambda^5 volume-form conversion (verified on all 15), and the C6 crossing character (Pf(PAP^T) = det(P) Pf(A) symbolic; sgn(pi(M)) = det(pi) c_pi(M) sgn(M) on all 15) -- a global gauge, NOT a line-dependent fudge (the flipped-chi control produces exactly the 3 expected mismatches); (3) THE SELF-HOSTING COUNTING THEOREM (P2.SELFHOSTING.PAIRING.01): C(g+1, 2) = g!! has g = 5 as its ONLY nontrivial solution (g = 1 trivial; separation for g >= 7 symbolic via g^2 - 3 > 0), and (g-2)!! = 3 = N_fam -- the [E] theorem behind the review's self-hosting pair-compiler reading; THE PHYSICAL PREMISE (the boundary grammar IS a self-hosting Wick pair compiler) stays [O] -- P2 narrowed to ONE premise, NOT eliminated, no marker move; (4) ANCHOR.MOD30.CLOCK.01: p_n = 2 + 2^n with p_{n+1} = 2 p_n - 2 has the mod-30 orbit 4 -> 6 -> 10 -> 18 -> 4 (period 4 = |mu_4|); anti-numerology census: the purely period-4 moduli below 100 are exactly {5, 10, 15, 30}, all dividing 30, lcm 30 -- CLOCK-DISTINGUISHED (30 is the maximal/universal clock), with the honest caveat that 5/10/15 already realize period 4; (5) the 6 one-factorizations of K6 (each matching in exactly 2; 30 = 6 x 5 = 2 x 15) carry the exceptional-outer-automorphism signature ((1,1,1,1,2) on vertices vs (2,2,2) on factorizations); NO canonical vertex-factorization bijection (typed).  The operative-Coxeter-30 reading is separately DEAD (coxeter_schedule_probe.py 28/28 SCHEDULE-DEAD, sandbox: no root orbit realizes the 15 classes x2 -- best 12/15; the K6 side allows at most 3/15 separation-15 pairings; w^15 = -1 makes the 15-shift automatic; the counting identity 30 = 2 N_fam g_car stands).  All controls fire; no marker moves.",
      "status": "Closed finite theorems [E] + one honest kill (the operative-30 reading); the Wick-compiler premise stays [O]; no marker move",
      "canonicalStatus": "CLOSED [E]: fermionic Pfaffian signs gauge-matched via canonical chi = Laplace/volume-form/C6 character; self-hosting forces g = 5 (N_fam = 3); mod-30 clock distinguished; operative-Coxeter-30 dead",
      "dependencies": "DOILY.PLUCKER.GR25.01; DOILY.HODGE.FACTOR.01; P2.QUADRATIC.EXHAUSTION.01; v880; v868",
      "scripts": [
        "v888_finite_pfaffian_signature.py"
      ],
      "script": "v888_finite_pfaffian_signature.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 725
    },
    {
      "id": "SEAM.CFIN.WICKFUNCTOR.01",
      "claim": "THE WICK FUNCTOR ARC [O] (constructed/obstructed/resolved 2026-08-09, rounds CVIII-CX; promoted round CXI in v896): (1) THE SIX ROLES EXIST in the deployed seam system (D5 slot pairs + A3 boundary block on the 16-dim one-particle space; so(16) sector law, q* edge grading, C6 intertwining all exact) but the deployed vacuum kernel is channel-diagonal (all 15 duad two-point functions vanish); (2) THE PROVED SCALAR OBSTRUCTION (SEAM.CFIN.WICKMIX.01): the pinned C6 generator's 2-cycle {4,5} reverses one duad orientation, so every C6-covariant antisymmetric scalar channel covariance vanishes there (max 14/15; the orbit law exact; both relaxation witnesses constructed: C3 = <g^2> gives 15/15, the 16-dim block object exists but no equivariant scalarization); (3) THE BLOCK RESOLUTION (SEAM.CFIN.WICKBLOCK.01): the C6-covariant cross-block space has dimension 33; the canonical block covariance is exactly covariant, strictly CAR-positive, carries 15/15 blocks and 15/15 block Wick monomials with the canonical chi sign law (interleaving character symbolically +1); THE DECISIVE FACT: the deployed seam covariance (v113 family rebuilt exactly) has ZERO cross-blocks everywhere (SEAM-DIAGONAL).  DEMAND (final form): a physical channel-mixing mechanism (a boundary/transport-twisted state, not the vacuum) realizing the block functor's value level -- then the K6/Pfaffian grammar of v888 would be the measured correlation grammar of the seam.  The [O] premise is unmoved; no marker moves.  CANDIDATE NOTE (2026-08-09, round CXV): the mixing-mechanism demand now has a CONSTRUCTED CANDIDATE (SEAM.CFIN.KMSMIX.01 / v898: the C6-covariant mixing KMS state passes ALL block-functor gates and the exact Schur elimination of the boundary generates all 10 carrier duads from the bare diagonal state; the frozen 15-signature table is the registered prediction; RP-THETA-OPEN is the named open check); whether the ACTUAL seam realizes it is untouched -- the [O] premise stays unmoved.",
      "status": "Open contract [O] (structure exact, obstruction proved, block candidate constructed; the physical mixing mechanism is the demand)",
      "canonicalStatus": "Open [O]: block functor constructed; the mixing mechanism has a constructed candidate state (v898) -- the physical realization is the demand",
      "dependencies": "DOILY.PFAFFIAN.WICK.01; P2.QUADRATIC.EXHAUSTION.01; v888; v113",
      "scripts": [
        "v896_wick_block_functor.py"
      ],
      "script": "v896_wick_block_functor.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 726
    },
    {
      "id": "SEAM.CFIN.KMSMIX.01",
      "claim": "THE CHANNEL-MIXING CANDIDATE STATE (constructed 2026-08-09, round CXIII; promoted round CXV in v898): the v896 demand -- a channel-mixing state realizing the block Wick functor's value level -- now has a CONSTRUCTED CANDIDATE on the deployed 16-dim one-particle space.  (1) THE KMS ROUTE: the C6-covariant mixing KMS state EXISTS -- K = i h with h = -(A16_dep + A_int/8) at beta = 1 (the FIRST point of the frozen grid) passes ALL gates: CAR strict (smax 0.668), exact C6 block covariance with the v896 forced zeros, all 15 duad cross-blocks at the frozen floor, the q*/sheet grading carried exactly (5/5 vacuum + 10/10 carrier FULL), all 15 block Wick monomials nonzero with the CANONICAL Pf4 sign pattern of G_c; ROBUST at beta = 2; the hermitian O16-commutant measured exactly (112 = 62 sym + 50 antisym; the mixing slice is the 33-dim v896 block space).  (2) SEAM.CFIN.SCHURMIX.01 (EXACT sympy): boundary elimination of the coupled diagonal state A_full = [[kappa A_CC, t V], [-t V^T, m J3]] (kappa = m = 1/2, t = 1/20, validity proven exactly) gives A_eff = kappa A_CC + (t^2 m/(1-m^2)) V J3 V^T EXACTLY, and S = V J3 V^T GENERATES all 10 carrier duads (uniform 3J block, canonical Pf4 signs) from the BARE DIAGONAL state -- the SAME Schur/Feshbach elimination that creates the RH port creates the carrier mixing: on both fronts the real correlation lives in the ELIMINATED boundary.  (3) THE FROZEN 15-SIGNATURE TABLE is the registered prediction any physical seam state realizing the block functor must reproduce.  (4) RP typed honestly: NO spatial/OS theta is deployed on the 16-dim space; the Majorana real structure Theta_0 satisfies the v426/v440 dictionary automatically; the OS-positivity under a SPATIAL theta is the NAMED OPEN CHECK RP-THETA-OPEN.  THE [O] PREMISE IS UNMOVED: candidate states + frozen signatures; whether the ACTUAL seam realizes any of them is untouched.  No marker moves.  RP NOTE (2026-08-10, rounds CXXIV/CXXXI-CXXXIII; promoted in v903): RP-THETA-OPEN is now MEASURED at family level and the answer is an EXCLUSION -- strict OS positivity under both deployable spatial reflections forces t = 0 (OS <=> a_J = 0 <=> the mixing floor falls: RP and the mixing floor are mutually exclusive on this family), the complete C6-twisted census is exclusionary (0/6), and the parent-dilation route yields only a MARGINAL cone-boundary witness (whose Schur compression carries the full mixing with the exact identity t^2 * 3m/(1-m^2) = 1/200) while the strict collar route is family-wide obstructed by an exact linear 2t law (SEAM.STATE.DERIVATION.01 / SEAM.CFIN.TWISTED.RP.01 / SEAM.CFIN.RP.DILATION.01, v903_seam_rp_exclusion.py); a measurement on the complete finite candidate set, not a universal no-go; the [O] premise stays unmoved.",
      "status": "Constructed candidate + exact generation identity [Numerical/Identity]: the mixing demand answered at candidate level; RP-THETA-OPEN measured exclusionary at family level (v903); the [O] premise unmoved",
      "canonicalStatus": "CONSTRUCTED: the C6-covariant mixing KMS state passes all block-functor gates; exact Schur generation of all 10 carrier duads; RP-THETA-OPEN measured exclusionary at family level (v903); [O] unmoved",
      "dependencies": "SEAM.CFIN.WICKFUNCTOR.01; v888; v113; v426; v440",
      "scripts": [
        "v898_kms_schur_mixing.py"
      ],
      "script": "v898_kms_schur_mixing.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 727
    },
    {
      "id": "SEAM.STATE.DERIVATION.01",
      "claim": "THE RP/MODULAR CARVING OF THE SEAM STATE SPACE (measured 2026-08-10, round CXXIV; promoted in v903): on the v898 family the reflections and the modular route CARVE for the first time.  (1) THE REFLECTION CENSUS: two spatial thetas are deployable on the 16-dim one-particle space (the sheet swap theta_S = the 16-dim lift of the v440 collar reflection, and the orientation-REVERSED 2-cycle theta_abT); the untwisted 2-cycle is NOT an OS reflection (Gram non-hermitian 0.924 already at t = 0); the v519 twist eta = +i is again FORCED.  (2) THE INCOMPATIBILITY: the odd-sector Gram eigenvalues are EXACTLY +-|a_J| of the {4,5} duad (worst identity 2.2e-16) -- OS positivity <=> a_J = 0 <=> the v898 gate G3 (all 15 blocks at the floor) FALLS: reflection positivity and the block-functor mixing floor are MUTUALLY EXCLUSIVE on this family; strict RP forces t = 0 under BOTH reflections (t = 0 marginal PSD); u >= t is FORCED with u_c = t exact in the deployed region (SPEC-v2 amendment disclosed fail-first: the corner (t=1/4, beta=4) gives u_c = 0.3627 > t, law re-typed u_c >= t) -- u = 0 is RP-EXCLUDED, correcting the round-57 'u unforced' reading.  (3) THE STRUCTURE OBJECT: det(A16_dep + t A_int) = (t-1)^2 (3t^2-1)^2 (9t^3+21t^2-t-1)^2 EXACT (sympy); the three positive roots organize everything measured (1p RP boundary = t/u = 1; hermitized boundary t_c(beta) -> t_gap; the ground-state modular transition at the SAME root); NO root is 1/8 -- the deployed value stays UNPINNED by state demands.  (4) THE MODULAR ROUTE: D(0, beta) = |beta - 2pi|/(2pi) exact (ward 1.1e-16); the 2pi-KMS locus of the boundary reduction is EXACTLY the point (t = 0, beta = 2pi) -- the geometric temperature KILLS the mixing instead of pinning it; off-axis EMPTY.  (5) THE THERMAL CURVE: order-parameter peak at beta = 1.6730 +- 5e-4 (round-57 'peak at beta = 2' corrected: coarse-grid argmax); distinguished-point census at 0.5 percent: NO HIT (DISTINGUISHED-ABSENT, typed as a reading census).  The v898 [O] premise unmoved.  NO RH claim.  UPDATED 2026-08-27 (v972): the exclusivity is TYPED RAY- and REFLECTION-relative -- the v903 statement concerns the v898 RAY (all 33 covariant coordinates moving together), not the slice: the full 33-dim census (SEAM.STATE.RPMIXING.01) shows theta_abT kills exactly ONE coordinate (a_J, kernel value -1/(1+c^2) = -0.8240271368 at c = tanh(1/2)), leaving a 32-dim marginal RP hyperplane, and theta_S leaves a 17-dim first-order-neutral subspace with strict finite-s witnesses.  The v903 anchor is reproduced exactly (odd Gram eigenvalues +-|a_J|, a_J = +0.04068013); the carving itself is UNMOVED.",
      "status": "Measured carving [Numerical/Identity]: RP and mixing mutually exclusive; u >= t forced; the modular locus is one point; no marker move",
      "canonicalStatus": "MEASURED: strict RP forces t = 0 (mixing and RP mutually exclusive); u >= t forced; 2pi-KMS locus = (t=0 beta=2pi); no root is 1/8",
      "dependencies": "SEAM.CFIN.KMSMIX.01; SEAM.CFIN.WICKFUNCTOR.01; v898; v440; v519; v526",
      "scripts": [
        "v903_seam_rp_exclusion.py"
      ],
      "script": "v903_seam_rp_exclusion.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 728
    },
    {
      "id": "SEAM.CFIN.MIXING.NORMALIZATION.01",
      "claim": "DEAD READING (negative result, first-class; measured 2026-08-10, round CXVI; promoted in v903): 'THE MIXING STRENGTH t = 1/8 = 2pi c3/|Z2| IS A COMPILER VALUE' IS A DECORATION, NOT A DERIVATION -- the seat audit is clean (the chain beta-angle x c3/|Z2| = 2pi x (1/(8pi))/2 = 1/8 is EXACT and via v813 equivalent to 1/(I_Jones |Z2|); c3 = P1 axiom, the angle 2pi MEASURED (v526/v519/v239), |Z2| = 2 deployed (v53/v456/v44)) but the COMBINATION IS FREE: no vN module composes the three factors into a mixing strength, and the decisive scan shows ALL 55 t values pass ALL frozen v898 gates (grid 0.01-0.50 + the exact rationals + decades; CAR strictness is a THEOREM tanh < 1, the apparent onset is float64 saturation, MARGIN-ARTIFACT typed) -- the allowed set is mathematically (0, infinity), u in {0,...,2} (incl. u = 0) and beta in {1/2,...,4} pass too: the v898 gates force NOTHING about (u, t, beta) beyond t > 0.  PIN-ABSENT: no frozen functional distinguishes 1/8 (uniformity/entropy argmax at 0.005, support/vacuum-mass argmin at 0.207, 1-crossing at 1.091); the anti-numerology census finds 4 equally-cheap deployed-constant readings of 1/8 -- the arithmetic is cheap, content could only live in seats + a future pinning demand.  THE NEW STRUCTURE FACT kept: the channel mixing is THERMAL (mass peaks near beta = 2 and DIES to the ground state, 0/15 cross-blocks at beta = 100 -- the beta -> infinity limit returns to SEAM-DIAGONAL).  HONEST UPDATE (same round family): SEAM.STATE.DERIVATION.01 subsequently measured u >= t as the FIRST non-decorative normalization demand -- the reading stays a registered decoration with typed seats.  Zero RH content.",
      "status": "Dead reading registered [Numerical] (negative result): the t = 1/8 reading is a decoration -- allowed set unbounded and pin absent; no marker move",
      "canonicalStatus": "NEGATIVE: t = 1/8 is a decoration (55/55 t pass all frozen v898 gates; allowed set (0 infinity); PIN-ABSENT); thermal mixing kept as structure fact",
      "dependencies": "AX.P1.01; SEAM.CFIN.KMSMIX.01; v813; v526; v898",
      "scripts": [
        "v903_seam_rp_exclusion.py"
      ],
      "script": "v903_seam_rp_exclusion.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
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      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 729
    },
    {
      "id": "SEAM.CFIN.MINIMAL.MEDIATOR.01",
      "claim": "DEAD READING (negative result, first-class; measured 2026-08-10, round CXVII; promoted in v903): 'N_fam = 3 IS THE MINIMAL BOUNDARY-MEDIATION RANK' IS REFUTED at theorem grade for this finite size (a census/sign-level theorem, not a search negative).  PREMISES SHARPENED: O16 is EXACTLY the identity on the A3 boundary -- C6 covariance constrains only the COUPLING (24-dim, orbit-repeated 2x6 blocks, projector ward + symbolic both ways); the expected orbit-count obstruction against r < 3 CANNOT exist.  THE STRUCTURE THEOREM (39 symbols, exact sympy): under covariant coupling the 10 carrier blocks of S' = V M V^T take exactly THREE values -- the full requirement (10/10 populated, canonical Pf4 signs, CAR-scalable) reduces to (gamma_ab != 0, gamma_cde != 0, det B_x > 0).  RANK-1 EXHAUSTION AS A THEOREM: det(B_x) = gamma_ab gamma_cde symbolically for EVERY symplectic-rank-1 mediator, so the ONLY rank-1 obstruction is the sign law gamma_ab gamma_cde <= 0; the explicit integer witness J(+)0(+)0 with the deployed coupling passes the requirement COMPLETELY (10/10, 0 wrong signs); seeded census 2000/2000 agrees; exhaustion r in {0, 1, 2, 3} complete, only r = 0 fails.  PREMISE CORRECTION: rank(V J3 V^T) = 2 EXACTLY and S = 1_5 (x) 3J EXACTLY -- the coupling fold collapses the deployed 3-level mediator to ONE effective carrier level ('<= 3' was true but empty).  WHAT 3 REALLY PINS (measured): boundary NON-DEGENERACY -- pure boundary ground state <=> matrix rank 6 <=> symplectic rank 3 = N_fam, honestly typed as DIMENSION BOOKKEEPING (dim(A3) = 6 = |Z2| N_fam), not a mediation theorem: the mediation demands are already met at rank 1.  Controls: rank 0 dead 10/10; covariance break reaches provably-unreachable signs; the fold-cancel rank-4 mediator is dead 10/10 (mediation capacity is NOT rank-monotone).  Zero RH content.",
      "status": "Dead reading registered [Identity/Numerical] (negative result): rank 1 suffices -- the minimal-mediation reading of N_fam = 3 is refuted; purity pins 3 as dimension bookkeeping; no marker move",
      "canonicalStatus": "NEGATIVE: N_fam = 3 as minimal mediation rank REFUTED (rank 1 suffices by exact sign law + integer witness; deployed S rank 2 exactly); purity = dimension bookkeeping",
      "dependencies": "SEAM.CFIN.KMSMIX.01; SEAM.CFIN.WICKFUNCTOR.01; v898",
      "scripts": [
        "v903_seam_rp_exclusion.py"
      ],
      "script": "v903_seam_rp_exclusion.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 730
    },
    {
      "id": "SEAM.CFIN.TWISTED.RP.01",
      "claim": "THE COMPLETE TWISTED-OS CENSUS IS EXCLUSIONARY (measured 2026-08-10, round CXXXI; promoted in v903): the COMPLETE census of C6-twisted reflections Theta_g = U_g o theta (2 bases x 6 C6 powers x 4 degree twists = 48 candidates) admits EXACTLY 16 involutions (Theta^2 = 1 <=> 3 | k, eta-independent; the law q^2 = img^{2k} verified indexwise -- the 3-cycle of pi forces k in {0, 3}), of which 6 are Gram-hermitian at the bare point (eta = +-i for (S,0), (S,3), (abT,0)), and ZERO of 6 admit strict RP at ANY t > 0: no selection -- TWISTED-RP-EXCLUSIONARY; the g^3 transport twist of the sheet swap is not a rescue but a NEW obstruction (its 1p Gram is indefinite already at t = 0, lam_min = -0.4621 for BOTH eta signs -- the twist degenerates the eta sign).  The enumeration is provably complete at finite size (C6 = <g>, generator pin re-verified; the U_g = O16^k are exact symmetries of every scanned KMS state at 1.1e-15); the scalar character axis collapses measured (chi = -1 negates the Gram, chi = i breaks hermiticity); eta = -i flips the 1p Gram exactly against +i -- at most ONE eta sign per side-exchanging candidate can carry strict positivity, and at the marginal abT bare point both pass and both fall for every t > 0.  SMOKE CORRECTIONS disclosed fail-first (the expected within-side mechanism for the side-fixing (abT, k=3) does NOT exist -- hermiticity-dead at the bare point for every eta; lam_max 0.4973 < the expected 0.5 bar).  RP and the v898 mixing floor remain mutually exclusive under EVERY C6-compatible involutive twist -- a MEASUREMENT on this complete finite candidate set, no universal no-go.  Zero RH content; the [O] premise unmoved.",
      "status": "Measured complete census [Numerical] (negative result): 0 of 6 admissible twisted reflections admit strict RP at t > 0; no marker move",
      "canonicalStatus": "MEASURED: twisted census 16/48 involutive and 6 admissible; 0/6 with strict RP at t > 0 (TWISTED-RP-EXCLUSIONARY); exclusion survives every C6 twist",
      "dependencies": "SEAM.STATE.DERIVATION.01; SEAM.CFIN.KMSMIX.01; v898; v519",
      "scripts": [
        "v903_seam_rp_exclusion.py"
      ],
      "script": "v903_seam_rp_exclusion.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 731
    },
    {
      "id": "SEAM.CFIN.RP.DILATION.01",
      "claim": "THE DILATION ROUTE SPLITS (measured 2026-08-10, round CXXXII; promoted in v903): on the explicit C6-covariant parent family over the v898 split 16 = 10 (+) 6 the dilation mechanism is REALIZED for the 2-cycle reflection -- every one of the 28 CAR-valid s = 0 points is theta_abT-MARGINAL RP (parent a_J = 0, 1p Gram eigenvalues EXACTLY {0, 0}: the cone boundary), and the Schur compression of the v898 M2 parent (kappa = m = 1/2, t = 1/20) carries the FULL Pfaffian mixing (10/10 carrier duads, uniform J block = the round-51/52 FLOOR as the EXACT rational identity t^2 * 3m/(1 - m^2) = 1/200, canonical Pf4 signs) while the effective carrier RP falls exactly by the +-|a_J| law (a_J = 1/200): parent RP above and effective RP failure below COEXIST -- the round-58 incompatibility is pushed one level up, not resolved; BUT the STRICT collar sheet swap is family-wide obstructed: 0/78 CAR-valid lattice points pass strict theta_S RP, and at the M2 parent the obstruction is EXACT -- 30 defect entries, ALL of magnitude 2t = 1/10, ALL on (empty monomial <-> mixed carrier-boundary pair) positions: a linear coupling law, not noise.  The witness is end-to-end EXACT rational arithmetic (symbolic Schur identity in all 5 parameters, PD/PSD by exact LDL pivots); the parent KMS typing binds exactly via CAR strictness (the -2 artanh spectral map; the disclosed smoke correction: the first draft used arctan and was killed by the roundtrip ward); orbit selectivity measured (W1-only mixes exactly 1/10, W2-only exactly 3/10, full mixing needs BOTH orbits); the compression lives on the uniform J ray, NOT the A_int ray of the KMS family (the FLOOR value is hit, the covariant DIRECTION differs -- typed, not hidden).  The marginal witness sits ON the RP cone boundary (not strict); the strict-collar obstruction is family-level (no universal no-go); the v898 [O] premise stays [O].  Zero RH content.",
      "status": "Measured split [Numerical/Identity]: a marginal cone-boundary witness with the exact 1/200 floor; the strict collar route obstructed by an exact linear law; no marker move",
      "canonicalStatus": "MEASURED: DILATION-SPLIT -- marginal theta_abT witness on the cone boundary; t^2*3m/(1-m^2) = 1/200 exact; strict collar obstructed (defect magnitude 2t)",
      "dependencies": "SEAM.STATE.DERIVATION.01; SEAM.CFIN.TWISTED.RP.01; SEAM.CFIN.KMSMIX.01; v898",
      "scripts": [
        "v903_seam_rp_exclusion.py"
      ],
      "script": "v903_seam_rp_exclusion.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 732
    },
    {
      "id": "SEAM.CFIN.GAP.PENCIL.01",
      "claim": "t_gap IS THEOREM-SHAPED (measured 2026-08-10, round CXXXIII; promoted in v903): t_gap = 0.2309488708... is the UNIQUE positive root of the irreducible cubic q(t) = 9t^3 + 21t^2 - t - 1 (exact Sturm count: 1 on (0, infinity) and 1 on (0, 1/4] -- the root is interior to the deployed region), which divides the EXACT Pfaffian of the FIXED integer pencil: Pf(A_dep + t A_int) = -(t-1)(3t^2-1) q(t) EXACTLY (degree <= 8 identity certified at 9 rational points by exact Fraction-Pfaffian recursion; sign fixed by Pf(A_dep) = +1 at t = 0) -- t_gap is a finite GENERALIZED EIGENVALUE of (A_dep, A_int) with kernel dimension EXACTLY 2 (exact argument: q squarefree => det vanishes to order exactly 2; antisymmetric => even rank) and minimal polynomial q of degree 3; the pencil is strictly-singular typed (deg det = 12 < 16).  THE THREE KNOWN CRITICAL SURFACES ARE PROJECTIONS OF ONE VARIETY: (1) the 1p gap closes at t_gap (2.6e-17) and NOWHERE else in (0, 1/4]; (2) the beta -> infinity ground state jumps there by EXACTLY a rank-2 occupation flip (||Delta||_F = 2 sqrt(2), persistent to delta = 1e-5; exact sign flip q(23/100) < 0 < q(6/25)); (3) the hermitized RP boundary t_c(beta) climbs monotonically to the pencil root (t_c(1) = 0.2205 measurably BELOW with gap 1.04e-2; t_c(30), t_c(60) coincide with t_gap below float resolution 1.4e-15 -- a LIMIT statement, typed).  INVARIANCE in strongest form: the C6 lift O16 fixes BOTH pencil members ENTRYWISE (exact integer); gauge congruences preserve det P(t) identically; scaling covariance exact.  THE COUPLING IS LOAD-BEARING: zeroing the carrier-boundary coupling or perturbing coupling wires kills t_gap as a root (exact polynomial remainder mod q != 0, 3/3 seeded draws).  SMOKE CORRECTIONS disclosed fail-first (the Pfaffian sign first guess +1 killed by the t = 0 evaluation; the strict-below gate at beta = 30 not float-resolvable, typed as a limit statement).  t_gap remains a property of THIS integer pencil; the RP boundary meets the root only in the beta -> infinity limit.  Zero RH content; no marker moves.",
      "status": "Measured exact pencil identity [Identity/Numerical]: t_gap as a generalized eigenvalue with Sturm uniqueness; three surfaces one variety; no marker move",
      "canonicalStatus": "MEASURED: Pf(A_dep + t A_int) = -(t-1)(3t^2-1)(9t^3+21t^2-t-1) exact; t_gap unique in (0 1/4] (Sturm); kernel dim exactly 2; coupling load-bearing",
      "dependencies": "SEAM.STATE.DERIVATION.01; SEAM.CFIN.RP.DILATION.01; v898",
      "scripts": [
        "v903_seam_rp_exclusion.py"
      ],
      "script": "v903_seam_rp_exclusion.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 733
    },
    {
      "id": "SEAM.CFIN.MARGINAL.STABILITY.01",
      "claim": "STRICT 2-CYCLE RP IS IMPOSSIBLE ON THE WHOLE COVARIANT CLASS (measured 2026-08-10, round CXXXVIII(60); promoted in v908): the closure-vs-isolated dilemma of the v903 marginal witness is REFUTED AS A DILEMMA -- the theta_abT sector Grams see ONLY the channel-{4,5} 4x4 seat (ward 1e-14), covariance allows there EXACTLY the J coordinate (integer defect 2 for I/X/Z units, 0 for J; c4 = c5 forced), and the exact symbolic seat law M1 = [[aJ+aX, aI-aZ],[aI+aZ, -aJ+aX]] reduces covariantly to M1 = diag(a_J, -a_J) with det M2 = -a_J^2: strict theta_abT RP is IMPOSSIBLE on the WHOLE covariant class (the RP set = {a_J = 0} = the cone boundary, interior EMPTY); the marginal witness is NOT a limit of covariant strict equilibria (there are none) and NOT isolated (the whole s = 0 family is a positive-dimensional MARGINAL STRATUM: margin exactly 0 on all 16 CAR-valid lattice points at full 10/10 canonical mixing); the decisive scalar sup(RP-margin x mixing) = 0 EXACT.  THE KERNEL ANATOMY (exact rational arithmetic): at the M2 parent the 1p Gram is IDENTICALLY 0 and the deg-2 Gram is exactly [[1, i/2], [-i/2, 1/4]] with eigenvalues {0, 5/4} -- the vacuum bound to the a-pair; both kernels forced by the SAME pure-J covariance law whose only visible coordinate a_J is exactly the strict-route obstruction.  THE TRANSVERSAL ESCAPE + ITS EXACT PRICE: the ONLY hermiticity-preserving strict-enabling seat direction is the covariance-BREAKING X coordinate: at every eps > 0 the parent is STRICTLY RP (1p eigenvalues {eps, eps} exact, deg-2 det = eps^2 exact), covariance defect EXACTLY 2 eps (linear price law), mixing 10/10 with the J coordinate 1/200 UNCHANGED, margin exponent 2.000 -- TRANSVERSAL.  FLOOR-EXCHANGE (the sharpest exact object): Pf4_{45}(eps) = (eps - 1/200)(eps + 1/200) and the effective 2-cycle odd Gram is diag(eps + 1/200, eps - 1/200) EXACTLY -- the canonical sign gate IS the negative effective RP margin (the same polynomial factor): one level down the exclusion survives the transversal escape, with the crossover EXACTLY at the 1/200 floor.  SPEC v2 with ONE disclosed amendment (fail-first preserved: the v1 gate 'all 18 s=0 points CAR-valid' fell at the two m = 3/4 corners; v2 reads the 16 CAR-valid points, the marginality held on all 18).  Reading typed: 'physical as a limit of equilibria' is unrealizable within the symmetry class -- strict 2-cycle RP needs symmetry breaking (price 2 eps) or a different reflection.  Controls: the I direction violates RP for BOTH signs, Z breaks 1p hermiticity, the 1/200 identity regresses rationally, 3/3 seeded scrambles break the covariance ward.  The v898/v903 [O] premise stays [O].  Zero RH content.",
      "status": "Measured covariant exclusion [Identity/Numerical]: interior empty (exact +-a_J law), floor exchange Pf4 = (eps - 1/200)(eps + 1/200); escape only by symmetry breaking at price 2 eps; no marker move",
      "canonicalStatus": "MEASURED: strict 2-cycle RP impossible on the whole covariant class (M1 = diag(a_J, -a_J) exact); the s = 0 stratum is marginal not isolated; Pf4(eps) = (eps - 1/200)(eps + 1/200) -- crossover at the 1/200 floor",
      "dependencies": "SEAM.STATE.DERIVATION.01; SEAM.CFIN.RP.DILATION.01; SEAM.CFIN.KMSMIX.01; v898; v903",
      "scripts": [
        "v908_seam_equilibrium_wiring.py"
      ],
      "script": "v908_seam_equilibrium_wiring.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 734
    },
    {
      "id": "SEAM.CFIN.NESS.PARENT.01",
      "claim": "NO NESS NEEDED -- THE PRICE OF THE MIXING IS EXACTLY ZERO ENTROPY PRODUCTION (demand closed as NOT-NEEDED; measured 2026-08-10, round CXXXIX(60); promoted in v908): strict theta_S (collar) hermiticity is a TWO-SEAT LINEAR LAW of the coupling -- the 1p seat A_{x+1,y} + A_{x,y+1} kills the X-, the deg-2 empty seat A_{x,y} + A_{x+1,y+1} the I-subblock coordinates; exact rank 12, kernel 12 = the {J, Z} coordinates of the 24-dim covariant coupling space = the v898 T-count.  THE DEPLOYED SEAM WIRING V = A_int[C, B] IS PURE-I (every 2x2 subblock exactly -I2) = a MAXIMALLY obstructed covariant direction -- explaining the v903 strict-collar law ('30 entries, magnitude 2t') EXACTLY as 15 mixed pairs x 2 Gram positions.  THE KERNEL CONTAINS EQUILIBRIUM WITNESSES: the uniform J-coupling parent V_J (kappa = m = 1/2, t = 1/20) passes strict theta_S RP in EXACT arithmetic (Grams exactly hermitian, PD by exact LDL; float lam_min 0.3064/0.1532) while its Schur compression carries the FULL canonical Pfaffian mixing -- S_J = V_J J3 V_J^T = 3J on ALL 25 channel blocks (integer identity), J coordinate EXACTLY t^2 3m/(1 - m^2) = 1/200 = the same rational floor identity as the v903 witness, Pf4 canonical 10/10; V_Z is a SECOND witness (S_Z = -3J, J coordinate -1/200).  THE NESS SIDE HONEST (FINITE-NESS-NOGO): at finite size every exactly stationary quasi-free state has [h, A] = 0 => ALL block currents Phi_r = (1/4) tr(h0_r [h, A]) = 0 => sigma = beta_C Phi_C + beta_B Phi_B == 0 (exact; energy ward 7e-18) -- 'NESS with positive entropy production' is category-inapplicable; the weakest meaningful positivity = RP of the stationary covariance, EXACTLY what is tested; the two-temperature Cesaro states (frozen grid) are exactly stationary (3e-16), covariant, CAR-valid, with vanishing t = 0 product currents.  THE PRICE CURVE IS FLAT-OPEN: the 15/15 canonical block census is open at EVERY drive point INCLUDING zero drive, the theta_S hermiticity defect stays PINNED (0.0982) and the PSD margin degrades monotonically -- the minimal entropy production at which the mixing gates open is EXACTLY ZERO: DRIVE BUYS NO RP, NESS-NOT-FORCED -- the demand closes as NOT-NEEDED.  Kernel membership is NECESSARY, not sufficient: J/Z mixtures pass hermiticity + PD but break the canonical sign census (4/10) -- the canonical gate selects the uniform J (or Z) ray; orbit selectivity reproduced (single boundary pair: both orbits 10/10 with J coordinate exactly lambda t^2 = 1/600).  THE REGISTERED NEXT CONTRACT: SEAM.STATE.WIRING.SELECTOR.01 -- is PURE-I compiler-forced?  The deployed IOTA wiring comes from the vacuum-orbit construction of A_int; whether the orbit/edge rules admit a {J, Z}-kernel alternative at equal C6 covariance and equal Pf pencil is exactly answerable -- only then is it decidable whether the strict-collar obstruction is a compiler theorem or a deployment choice.  The witnesses live on a DIFFERENT covariant coupling direction than the deployed A_int wiring -- whether the actual seam allows them is untouched; the v898/v903 [O] premise stays [O].  Controls: global-KMS regression reproduces the round-58/59 equilibrium obstruction (defect 0.0982); 3/3 seeded permutations break covariance + canonical census; PURE-I regression (0.1, 30 entries); the I-orbit switch-on fires WITHIN the covariant class.  Zero RH content.",
      "status": "Demand closed NOT-NEEDED [Identity/Numerical]: equilibrium J/Z witnesses carry the full 1/200 mixing at zero entropy production; the deployed wiring is PURE-I (maximally obstructed); wiring-selector contract registered; no marker move",
      "canonicalStatus": "CLOSED (NOT-NEEDED): stationary => sigma == 0 exactly and drive buys no RP; V_J/V_Z equilibrium witnesses with J coordinate exactly 1/200; PURE-I is the maximally obstructed direction -- SEAM.STATE.WIRING.SELECTOR.01 registered next",
      "dependencies": "SEAM.CFIN.MARGINAL.STABILITY.01; SEAM.STATE.DERIVATION.01; SEAM.CFIN.KMSMIX.01; v898; v903",
      "scripts": [
        "v908_seam_equilibrium_wiring.py"
      ],
      "script": "v908_seam_equilibrium_wiring.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 735
    },
    {
      "id": "SEAM.STATE.WIRING.SELECTOR.01",
      "claim": "THE WIRING FREEDOM THEOREM -- IS PURE-I COMPILER-FORCED?  ANSWERED: NO -- DEPLOYMENT-CHOICE, NOT COMPILER THEOREM (the v908-registered contract, closed 2026-08-12, rounds 69-71 notes CCXIII/CCXV/CCXXI; promoted in v911).  (1) THE EXACT GROEBNER CENSUS (30/30, WIRING-DEGENERATE): the C6-equivariant commutant of the coupling space is EXACTLY 24-dim (the boundary C6 action is the identity), the IOTA rule subspace 8-dim with the deployed PURE-I point; the constraint classes E1 (unit law), E2 (CAR/KMS), E3 (Pfaffian pencil, anchored by the symbolic Schur identity on all of W), E4 (orientation propagation det u > 0), E5 (covariance + seat stack) leave EXACTLY ONE admissible component mod rule gauge -- the rotation cell C_rot (dim 3; certified real decomposition, cofactor certificates, mixed branches real-point-free); pure-I is an INTERIOR point, NOT a vertex, and the INTEGER gauge (+)_5 J2 (+) I6 (preserving A_CC, C6, W and the ideal) connects pure-I and the pure-J ray; C_refl falls at EXACTLY ONE class -- orientation (det u = -(b^2 + d^2) < 0 identically): Z/X wirings are edge-illegal; SELECTOR THEOREM: rules + strict-collar Hermiticity would force PURE-J uniquely (up to flips/scale), pure-I sits at the maximally obstructed angle (defect 2t).  (2) THE COLLAR-FRAME DECISION (26/26, THETA-CONVENTIONAL): no compiler-side demand distinguishes the theta_S sheet-swap frame within the 9-dim rule-gauge algebra -- five demand classes (mu4/C6, CAR/state, pencil/orientation, doily combinatorics, the round-59 census property) all COMPATIBLE-NOT-FORCING / SILENT-ON-ANGLE (the bare covariance tanh(beta u/2) A16 is exactly torus-invariant; the refined doily stabilizer is 9 OF 9; the round-59 spectra are constant on the orbit with closed forms tanh(1/2) / tanh^2(1/2)); the deployed parent (pure-I) PASSES strict collar RP EXACTLY in the integer-conjugated frame and fails at theta_S with defect 0.1 = 2t -- the round-58/59 strict-collar obstruction is FRAME-RELATIVE; the CXXXIX question closes as a compiler-FREEDOM theorem: freedom group = the rule-gauge frame orbit (tangent 7/9), selection quotient = the delta-circle, the only gauge-invariant datum of (frame, wiring) is the relative angle.  (3) THE GAUGE-COVARIANCE AUDIT (21/21, RP-FRAME-COVARIANT, lead-ordered): theta' = g_int theta_S g_int^{-1} both directions exactly; the kernel-covariance law holds SYMBOLICALLY on the full torus (g^T A(V) g = A(g_C^T V g_B) identically), so strict-collar RP depends only on the relative angle delta (defect law 2t |a cos delta + c sin delta|); the pure-I exclusion is the lead's OPTION (2): a statement at the FIXED theta_S frame, NOT a wiring no-go; the Z/X exclusion IS genuinely orbit-invariant (det u frame-transport-invariant; the only det-flipping orthogonal element flips the vacuum and leaves the rule gauge).  THE CANONICAL SENTENCE (frozen verbatim, sha 79950314): the compiler determines the wiring ORBIT; pure-I is a DEPLOYMENT REPRESENTATIVE; strict Hermiticity in the theta_S frame would select pure-J, the same demand in the integer-conjugated frame selects pure-I; a canonical wiring selection requires an ADDITIONALLY DERIVED physical frame.  FORBIDDEN from now on: 'pure-I is compiler-forced' (and equivalents without the frame qualifier).  HONEST SCOPE: the census covers the FIVE enumerated demand classes -- a compiler demand outside the list is not excluded; whether the actual seam physics derives a specific collar frame remains outside the formalized rules; the v898/v903 [O] premise stays [O].  Controls fire 6/6 + 6/6 + 5/5; exact integer/rational sympy, float64 only in disclosed wards, RNG only in declared controls.  Zero RH content.",
      "status": "Contract closed as measured freedom theorem [Identity/Numerical]: one admissible wiring orbit (C_rot), Z/X excluded gauge-invariantly; pure-I = deployment representative; RP exclusion frame-relative (option 2); 'pure-I is compiler-forced' FORBIDDEN; [O] premise of v898/v903 unmoved; no marker move",
      "canonicalStatus": "CLOSED (DEPLOYMENT-CHOICE): exactly one admissible wiring component mod rule gauge; pure-I interior + integer-gauge-connected to pure-J; theta_S conventional (five demand classes silent); strict-collar RP exactly frame-covariant, exclusion = frame statement; Z/X exclusion orbit-invariant",
      "dependencies": "SEAM.CFIN.NESS.PARENT.01; SEAM.CFIN.MARGINAL.STABILITY.01; SEAM.STATE.DERIVATION.01; v898; v903; v908",
      "scripts": [
        "v911_wiring_freedom.py"
      ],
      "script": "v911_wiring_freedom.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 736
    },
    {
      "id": "CCC.SEAM.KINEMATICS.01",
      "claim": "THE CCC CROSSOVER KINEMATICS DICTIONARY [E] (rounds DL-DLV, notes 2026-08-24; promoted 2026-08-24).  EXACT (v957; six probes 28/28 + 29/29 + 18/18 + 21/21 + 14/14 + 7/7 embedded byte-exact, hashes pinned): (1) Penrose's reciprocal gauge Omega*tau(Omega) = -1 is the element tau = rho^2 sigma of the proven mu_4 Moebius stabiliser D_4; tau reverses the clock (= the Z_2 sheet parity), swaps the aeon poles {0, inf}, fixes +-i; the crossover locus |Omega| = |tau(Omega)| is the unit circle carrying all four marks.  (2) The clock alone leaves exactly one constant-curvature conformal modulus on the 2D seam and the reciprocal Z_2 kills it (Liouville-exact; so(3,1) commutant dims 2-with-1-boost / 0).  (3) The v221 transport is CPTP not unitary; its Stinespring isometry is explicit (V^dag V = 1_3), ||T^n - P_star||_2 = (2/3)^{6n} exactly, the global overlap is invariant and the environment retains full distinguishability -- the entropy reset of the cyclic reading is a REDUCED CHANNEL, no information destroyed.  (4) The kinematic uplift: stereographic(Hopf(v)) = z1/z2 exactly; the v492 deck acts freely, so the crossover X = S^3/Z_4 is the smooth lens boundary of the corpus A_3 ALE; clock and reciprocal lift normalise the deck and descend; the four marks lift to four Hopf fibre circles paired into the two Z_2 sheet orbits with Gauss linking 1; the so(4,1) deck-commutant is 4-dimensional, ALL Killing, boost content 0 (infinitesimal Yamabe rigidity with no extra condition; global uniqueness on the explicit Obata boost family, 200-axis scan bounded below); the inversion x -> x/|x|^2 is a conformal involution of the cone C^2/Z_4 fixing the unit lens pointwise and swapping the aeon ends; the Lorentzian bridge (1/cos^2 T)(-dT^2 + g_{S^3}) is Einstein R_ab = 3 g_ab (symbolic Ricci) with a spacelike round-S^3 crossover and Omega_hat*Omega_check = -1, Omega_check = -cos T vanishing to first order.  (5) KMS/modular closure at the free level: detailed balance exact, spec(-ln T) = {0, 6 ln(3/2), 6 ln 3}, T = e^{-H_mod}, OS Gram PSD; the v526 normalisation beta_angle = 2 pi = 1/(4 c3) equals the Gibbons-Hawking normalisation on the crossover fibre.  (6) Holonomy dictionary: pole strings carry the full mu_4 monodromy (chi(D) = i), every seam string carries exactly the 2-torsion chi(D^2) = -1 = the v522 GSO/fermion-parity wrap (typed [C] structural correspondence).  (7) Selector necessity: the admissible reflections of M(delta) close to D_4 iff delta = pi/2 (the clock factorises into admissible reflections only at mu_4); at Majorana operator level with fermionic signs the clock-axis cuts are Theta-real ONLY at k = 4 and asymmetric couplings break reality -- the uniqueness half of the v534 selector is DERIVED as a Theta-reality necessity (survival + coupling sign stay v534's nonperturbative toy facts, cited; reality is sign-blind, checked).  (8) The frozen Gate-6 kernel: two decay modes ln(729/64) and ln(729) (ratio 2.709511), frozen defect-class overlap table, disc radius theta_max = eta_rec/(eta_0 - eta_rec) ~ 1.16 deg, kappa derived from the v526 KMS normalisation up to a declared convention band (u_rec <= 8.3e-5, contrast <= 3.3e-4): the template degenerates robustly to the sharp causal top-hat disc -- feeds the preregistered search experiments/ccc-crossover-disc (no CMB data touched).",
      "status": "Certified kinematics dictionary [E] (fine types Identity/Formal (Moebius/D_4 algebra, Stinespring isometry, Hopf equivariance, commutant dimensions, symbolic Einstein/Weyl checks, KMS/detailed-balance identities, Theta-reality censuses -- exact on corpus objects) and Numerical (linking integrals, Obata scans, band arithmetic pinned); the CCC/cyclic READING itself stays [C] and is NOT upgraded; WOIT.OS.TWISTOR.01 stays [O] with all kill tests live)",
      "canonicalStatus": "Kinematics certified [E]: reciprocal-in-D_4 + conformal rigidity (2D needs tau, 3D automatic) + Stinespring reduced-channel reset + lens-crossover uplift + KMS closure (tick = e^{-H_mod}) + holonomy dictionary (seam strings = GSO 2-torsion) + selector-uniqueness necessity + frozen top-hat kernel; the cyclic/CCC reading stays [C]; the dynamics stay WOIT.OS.TWISTOR.01 [O]",
      "dependencies": "QGEO.CONF.01 (the seam premise); v221 (transport); v492 (deck/clock, A_3 ALE); v519/v521/v522 (Theta, side-blindness, GSO wrap); v526 (KMS normalisation); v529/v534 (straddle law + selector, cited); VGEO.TORSOR.01 (unit calibration untouched); rounds DL-DLV",
      "scripts": [
        "v957_ccc_crossover_kinematics.py"
      ],
      "script": "v957_ccc_crossover_kinematics.py",
      "location": "origin_theory;tfpt_research_contracts",
      "externalData": "Planck 2018 cosmography (typed external: eta_rec, eta_0, Omega_K, Omega_L, H_0, n_s, r bounds -- kernel freeze and fork pricing only)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 737
    },
    {
      "id": "CCC.SEAM.CROSSOVER.01",
      "claim": "THE CCC CROSSOVER CONTRACT [O] (registered 2026-08-24 from the external TFPT-vs-CCC adjudication of the six gates; the kinematic base is CCC.SEAM.KINEMATICS.01 [E]/v957).  DEMAND (the dynamical crossover, in order): (D1) the interacting seam algebra A_hol realised ON the lens crossover X = S^3/Z_4 with reflection positivity protected on the mid-gap cut class while the aeon transfer uses the through-mark class (the two D_4 classes whose product is the clock; the v529 straddle law is this separation at toy level; = WOIT.OS.TWISTOR.01 gamma in the crossover geometry); (D2) the aeon-transfer channel as a Stinespring isometry at FIELD level (the v957 finite-dimensional dilation lifted to the reconstructed net) -- fundamental unitarity + rank-1 attractor in the visible channel + microstate record in the complement; (D3) the cosmology fork DECIDED: exactly one active story (Starobinsky r = 12/N^2 in [0.0033, 0.0048] vs crossover/GW epoch vs computed hybrid) -- priced decidable by LiteBIRD/CMB-S4-class sigma(r) ~ 0.001; (D4) the preregistered disc search experiments/ccc-crossover-disc EXECUTED per its frozen hypotheses (top-hat template, sign-pairing statistics, kill criteria K1-K6, BH-q < 0.01 + replication decision rule) -- EXECUTED 2026-08-24 with a ROBUST NULL on the Planck PR3 SMICA map (first data contact; freeze intact 3/3 hashes, frozen-kernel guards 5/5 BEFORE data contact; map integrity 2013312960 bytes, SHA-256 pinned in results.json; seed 20260824, reruns bit-identical): max|SNR| 5.119 on data vs null mean 5.20 over 100 Gaussian LCDM sims (matched mask + spectrum), p_global = 0.673, p_counts = 0.446, BH-q_min = 0.673 -- the q < 0.01 threshold CLEARLY missed; 171 candidates |SNR| >= 4 with 83 pos / 88 neg (no sign excess); injection recovery 100 percent (20 x 200 muK discs at 1.16 deg -- the search is well-powered, the null is a REAL null); K1-K6 all NA (they fire only on a resolved relic; none exists); replication legs (half-mission, Commander/NILC) not required for a null per the frozen decision rule, available for future candidates; three amendments disclosed (seed pinning of the unseeded synfast preflight, reporting fields, the documented estimator l-weighting) -- statistics, thresholds and kill criteria unchanged; results machine-readable in experiments/ccc-crossover-disc/results/results.json.  KILL CRITERIA (frozen): a free fitted conformal factor anywhere; failure of the mid-gap RP protection on the interacting algebra (kill test 2 of WOIT gamma); the interacting level breaking the v522 wrap = (-1)^F identification (reopens the deck fork); r measured incompatible with the surviving cosmology branch; a resolved crossover relic violating the frozen template class (K1-K6, incl. the K6 contrast bound killing the derived-kappa chain); mu_4 marks identified with Hawking-point counts (typed numerological, forbidden).  FENCES: the cyclic/CCC reading stays [C] regardless; the executed D4 null does NOT damage the compiler core (the kernel was OUTPUT of the [E] kinematics, not input) and no kill criterion fired; the contract stays OPEN via D1-D3; nothing here asserts CCC follows from TFPT in either direction.",
      "status": "Open contract [O] (research contract; the kinematic base [E] via v957; the dynamics = WOIT.OS.TWISTOR.01 gamma on the crossover geometry; D4 executed 2026-08-24 with a robust preregistered null -- no kill fired, no type change; the contract stays open via D1-D3)",
      "canonicalStatus": "Open [O]: D1 interacting algebra on the crossover (WOIT gamma) + D2 field-level Stinespring + D3 cosmology fork decision = the remaining demands; D4 EXECUTED with a robust null (p_global 0.673, BH-q 0.673, injection recovery 100 percent, K1-K6 NA; results.json) -- no kill fired; kinematic base certified in v957",
      "dependencies": "CCC.SEAM.KINEMATICS.01 (the certified base); WOIT.OS.TWISTOR.01 (gamma = D1); SEAM.THERMAL.KMS.01 (the kappa chain); v529/v534 (the straddle fence); experiments/ccc-crossover-disc (the frozen search)",
      "scripts": [
        "v957_ccc_crossover_kinematics.py"
      ],
      "script": "v957_ccc_crossover_kinematics.py",
      "location": "tfpt_research_contracts",
      "externalData": "Planck 2018 / BICEP-Keck bounds and LiteBIRD/CMB-S4 sensitivities (typed external, fork pricing); Planck PR3 SMICA map COM_CMB_IQU-smica_2048_R3.00_full.fits (D4 data pass, SHA-256 60952c645eb33d151905ddf5837477e15ca02a3261feca4ceca3c3feece4f9ac, 2013312960 bytes)",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 738
    },
    {
      "id": "DIMENSION.UPLIFT.FIREWALL.01",
      "claim": "THE DIMENSION-UPLIFT FIREWALL (registered 2026-08-27 from the external deep review: 'chiral 1+1D seam net does NOT imply a local, interacting 3+1D quantum world').  STANDING RULE, in force immediately: (F1) no theorem whose hypotheses live in 1+1D (chiral net, VOA, lattice collar, seam circle) may be cited as CLOSING a claim about 3+1D physics; (F2) every contract closure must state the spacetime dimension of its hypotheses AND of its conclusion (dimension typing, same discipline as the markers).  Audit census at registration: SEAM.EQUIV.01 and both routes are 1+1D in and out (OS/MMST scaling limit arXiv:2107.13834 treats 1+1D free lattice fermions; AMT OS-reconstruction arXiv:2407.18222 treats 2D full VOAs; L2 claims a chiral c=8 CFT and nothing more) -- even a FULLY PROVED SEAM.EQUIV.01 is a rigorous boundary building block, not the 4D theory; the WOIT.OS.TWISTOR.01 target IS the uplift demand and stays [O] with all kill tests live; the kinematic 2D->3D->4D uplift (v957, ccc_uplift_4d_probe self-labelled kinematic) does NOT close the firewall (a metric uplift is not an interacting local quantum theory); the anomaly-inflow bulk of the c_-=8 edge is an invertible/topological bulk (Witten-Yonekura arXiv:1909.08775, Hsieh-Tachikawa-Yonekura arXiv:2003.11550), never to be silently equated with the physical 3+1D spacetime bulk.  ROUTE ORDERING (programme skeleton): (1) seam stays an internal/topological boundary module; (2) genuine 4D Euclidean lattice theory (TFPT fixes reps/charges/coupling relations/boundary conditions); (3) QFT4D.OS.RECON.01; (4) CHIRAL4D.NOMIRROR.01 in the same 4D model; (5) only then BW/CHM + entanglement gravity (GRAV.NONCIRCULAR.01); (6) then bulk-seam equivalence (SEAM.BULK4D.RECON.01).  The twistor/celestial alternative is riskier (boundary correlators do not guarantee bulk locality, cf. arXiv:2602.04223) and stays a named parallel route.",
      "status": "Firewall (standing discipline rule, in force; the uplift theorems it names stay Open)",
      "canonicalStatus": "Firewall in force: no 1+1D theorem closes a 3+1D claim; every closure carries dimension typing of hypotheses and conclusion; route ordering registered",
      "dependencies": "SEAM.EQUIV.01; WOIT.OS.TWISTOR.01; CCC.SEAM.KINEMATICS.01 (kinematic only)",
      "scripts": [],
      "script": "-",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 739
    },
    {
      "id": "SEAM.BULK4D.RECON.01",
      "claim": "THE SEAM-TO-BULK 4D RECONSTRUCTION CONTRACT [O] (registered 2026-08-27): the theorem previously only implicit between the SEAM.EQUIV.01 conclusion (a chiral c=8 boundary object, 1+1D) and the WOIT.OS.TWISTOR.01 goal (a Poincare-covariant causal chiral 4D theory) is made explicit as its own named target.  DEMAND: from seam data alone construct a 4D family of local algebras (equivalently Schwinger functions) and prove (B1) 4D locality/microlocality, (B2) Poincare covariance + positive energy spectrum, (B3) spin-statistics + CPT, (B4) BOTH helicities (the anti-self-dual completion), (B5) cluster decomposition, (B6) a nontrivial S-matrix (interacting, not generalized-free), (B7) a UNIQUE seam<->bulk operator dictionary (not character-level only).  KILL TESTS: every 4D extension is generalized-free; the inflow bulk is provably only invertible/topological with no local propagating completion; the dictionary is non-unique at operator level (OPE mismatch).  A Hopf-fiber uplift / 4D cone / Einstein metric (v957) does NOT close this.  Route-agnostic: WOIT.OS.TWISTOR.01 and the DIMENSION.UPLIFT.FIREWALL.01 constructive-lattice ordering are the two candidate proofs.  NOTHING in the corpus closes any part of it.  SPECTATOR OBSTRUCTION (2026-09-04, v1020): B7 as worded is unsatisfiable -- boundary(B(x)C) = boundary(B) for any decoupled anomaly-free C with trivial boundary; finite free-fermion witness in v1020; uniqueness is re-scoped to BULK.PRIME_COMPLETION.01 (prime/seam-generated completions, trivial relative commutant); status unchanged [O]",
      "status": "Open (research contract)",
      "canonicalStatus": "Open: the named seam->bulk 4D reconstruction theorem (seven demands B1-B7); no corpus result closes any part",
      "dependencies": "DIMENSION.UPLIFT.FIREWALL.01; SEAM.EQUIV.01; WOIT.OS.TWISTOR.01; QFT4D.OS.RECON.01; CHIRAL4D.NOMIRROR.01",
      "scripts": [],
      "script": "-",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 740
    },
    {
      "id": "DYN.MARKOV.EMBED.01",
      "claim": "THE MARKOV EMBEDDING CONTRACT (registered 2026-08-27; 'three meanings of dynamics' split, step (a)->(b)).  FINITE KERNEL HALF [E] (v971, sympy-exact): for the v221 seam transfer kernel T (spectrum exactly {1, 64/729, 1/729}) the principal logarithm Q = log T is an EXACT classical Markov generator -- e^Q = T by exact spectral calculus; row sums symbolically zero; off-diagonals exactly {ln(9/8), 2 ln 3, 2 ln 3} = {0.117783, 2.197225, 2.197225} > 0 (the reviewer's independently verified numbers), diagonals -ln(81/8), -ln(81/8), -4 ln 3; uniformization certificate Q + (4 ln 3) I >= 0 entrywise (diagonals ln 8, ln 8, 0, exact) => e^{tQ} positivity-preserving doubly stochastic for ALL t >= 0, T embeddable, every root T^{1/n} stochastic; spec(Q) = -spec(H_mod) = {0, -6 ln(3/2), -6 ln 3} (gap = Delta = 6 ln(3/2)); detailed balance w.r.t. the uniform state.  NEGATIVE CONTROL: second eigenvalue -1/10 gives det = -1/200 < 0 -- provably NOT embeddable (det e^Q = e^{tr Q} > 0 always); embeddability is not generic, the RP/OS spectral positivity (CONTRACT.F.01 axiom 3) is the enabling fact.  WHAT STAYS [O]: the same embedding statement at FIELD level (reconstructed net / quasi-local algebra, uniform in system size).  FIREWALL: e^{tQ} is irreversible statistical dynamics (H-theorem), NOT unitary evolution -- step (b)->(c) is the separate contract DYN.UNITARY.DILATION.01; the shared F_transfer relaxation shape is a universal CONTRACTION CLASS, not one physical clock (v723/v724/v777).  Python-only per GATE.WOLFRAM.02 (matrix-log/spectral identities; the exact rationals 64/729, 1/729 already mirrored).",
      "status": "Contract [E] (finite kernel embedding, sympy-exact; v971) + [O] (field level)",
      "canonicalStatus": "[E] the finite kernel half: Q = log T is an exact Markov generator (row sums 0, off-diagonals ln(9/8), 2ln3, 2ln3 > 0, uniformization positivity for all t >= 0); [O] the field-level embedding",
      "dependencies": "v221 (kernel); v957 (T = e^{-H_mod}); CONTRACT.F.01 (axiom 3 positivity); v723/v724/v777 (external clocks)",
      "scripts": [
        "v971_markov_embedding_generator.py"
      ],
      "script": "v971_markov_embedding_generator.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 741
    },
    {
      "id": "DYN.UNITARY.DILATION.01",
      "claim": "THE UNITARY DILATION CONTRACT [O] (registered 2026-08-27; 'three meanings of dynamics' split, step (b)->(c)).  The continuous semigroup e^{tQ} (DYN.MARKOV.EMBED.01) is irreversible statistical dynamics; fundamental quantum dynamics needs T = e^{-aH} -> (OS) -> U(t) = e^{-itH} with H positive self-adjoint.  DEMAND: (D1) a Stinespring/Hamiltonian dilation of the semigroup with (quasi-)local generator; (D2) size consistency (compatible with the thermodynamic limit -- no pathologically growing environment); (D3) a Lieb-Robinson bound (finite propagation speed of the dilated dynamics); (D4) the OS analytic continuation to a unitary one-parameter group on the reconstructed Hilbert space.  KILL TESTS: every dilation of the seam semigroup is provably nonlocal; the Lieb-Robinson velocity diverges in the scaling limit; the OS continuation fails on the interacting algebra.  RELATION: demand D2 of CCC.SEAM.CROSSOVER.01 (Stinespring isometry at field level) presupposes this contract -- named here as its prerequisite.  NOTHING here is closed by the finite v971 result. [2026-08-27 lift-structure note (v976/v977, marker UNCHANGED): the dilation the corpus actually deploys (K_ij = sqrt(T_ij)|i><j|, the CCC Stinespring reading of the v221 kernel) is PROVABLY entanglement breaking (measure-and-prepare; quantum capacity 0; C_0 = 0) -- it cannot carry the coherent transport this contract demands, so any closing dilation must be a genuinely different object.  The classical kernel admits the full lift trichotomy {EB dilation | random-permutation fibre with the hidden circulation parameter t (v976) | reversible unistochastic macro-gate with the hidden orientation bit sgn J = +-1/27 (v977)} -- the same classical data, three inequivalent quantum dynamics; and no fixed microstep unitary generates the relaxing sequence (Dirichlet recurrence witness), so repeated steps REQUIRE environment/dephasing between steps -- exactly the D1-D4 demand.]  EXECUTED UPDATE (2026-08-28, v987): the OS/dilation LADDER is executed at kernel + free-field + interacting-kernel + quantum-coupled band level -- B = P1+(2/3)P2+(1/3)P3 > 0, H = -log B PSD (gap log(3/2)), T(t) semigroup inside the stochastic cone, U(t) = e^{-itH} with Wick anchor U(-in) = B^n (n=1..6); size-uniform free-band gap; congruence interacting family SPD; quantum SWAP coupling d(theta)~theta^2, CPTP, LR radius 1.  MUST-FAIL: v971 negative kernel has no real half-step; one-sided mutant K B^(x L) is not symmetric.  FIREWALL: continuous field-level OS reconstruction, thermodynamic-limit branch selection and any 3+1D reading stay [O].  Python-only / Wolfram mirror deferred.  EXECUTED UPDATE (2026-08-29, review wave 4, v999): the finite ladder is now COMPLETE -- discrete / kernel / field-free / interacting-classical / quantum-coupled / CONTINUOUS-TIME.  Weak-collision Hudson-Parthasarathy: generator extraction Richardson 1.098e-10, diag(L_GKSL)==Q=log B symbolic, e^Q=B exact, GNS splitting defect 0, locality 1.355e-20, strong-collision singularity contrast rank 9/81.  Open = thermodynamic field limit only.  Display stays [O]; no marker move.",
      "status": "Open [O] + finite ladder complete [E]/[N] (discrete/kernel/field-free/interacting-classical/quantum-coupled/continuous-time; open = thermodynamic field limit only)",
      "canonicalStatus": "[E] (2026-08-28, v984): discrete collision/QCA-band dilation.  [E] (2026-08-28, v987): kernel OS continuation exact + size-uniform free band + interacting congruence family + quantum SWAP coupling (d(theta)~theta^1.99, CPTP, LR radius 1).  [O] continuous field-level OS reconstruction, thermodynamic-limit branch selection, any 3+1D reading; no marker move.  | 2026-08-29 v999: finite ladder COMPLETE (incl. continuous-time weak-collision HP); open = thermodynamic field limit only.  Display stays [O].",
      "dependencies": "DYN.MARKOV.EMBED.01; CCC.SEAM.CROSSOVER.01 (D2); QFT4D.OS.RECON.01",
      "scripts": [
        "v984_markov_qca_dilation.py",
        "v987_os_dilation_package.py",
        "v999_weak_collision_hp.py"
      ],
      "script": "v984_markov_qca_dilation.py; v987_os_dilation_package.py; v999_weak_collision_hp.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 742
    },
    {
      "id": "TRANSFER.HIDDEN.CIRCULATION.01",
      "claim": "THE HIDDEN CIRCULATION CONTRACT, executed half (registered 2026-08-27; external-review round, companion of the DYN.MARKOV.EMBED.01/DYN.UNITARY.DILATION.01 split).  [E] (v976, sympy/Fraction-exact): (1) the Birkhoff fibre of the seam single step B (exact sixth root of the v221 kernel; B = (1/2)I + (1/18)P12 + (2/9)P13 + (2/9)P23 bit-exact) is EXACTLY a one-parameter line w_t, t in [0, 1/18], because the decomposition map on R[S3] has the 1-dim kernel (even - odd); the hidden parameter is the sign-character Fourier coefficient w_t-hat(sgn) = 6t; populations are exactly t-free and no fibre member is chiral (w(C123) = w(C132) forced -- the typed obstruction for any chirality reading).  (2) THE DISCRIMINATOR: one-step coherence dynamics is the weighted left regular S3 representation with spectrum {1, 6t, 2/3 x2, 1/3 x2} exactly; the triangle loop current A is an exact eigen-observable Psi_t(A) = 6t A (population-blind; one step + one current readout measures t, no ancilla; one-step tomography rank 6 determines all six weights).  (3) CONTRAST-QUBIT CHOI LAW: spec(Choi^{T2}) = {-3t/2, 1/6+3t/2, 1/3+3t/2, 1/2-3t/2} symbolically; CRITICALITY lambda_min = (1 - lambda2 - lambda3 - 6t)/4 with lambda2 + lambda3 = 2/3 + 1/3 = 1 EXACTLY: t = 0 sits precisely ON the entanglement-breaking boundary (mutated spectra {2/3,1/2}/{1/2,1/3} miss by -1/12/+1/12 -- criticality not generic); the squared channel is PPT on the whole range (endpoint minima 1/9, 1/12).  (4) FULL-LEVEL NPT PERSISTENCE: lambda_min(Choi(Psi_t^n)^{T2}) = -(1/3)(2/3)^n, t-independent (exact eigenvalue for n in {1,2,6}; float minimum n = 1..8) -- the 3-level family is never EB at tested n (pinned by the invariant coherent state (1,1,1)/sqrt3).  (5) THE NO-GO for DYN.UNITARY.DILATION.01: the deployed Kraus dilation K_ij = sqrt(T_ij)|i><j| is measure-and-prepare (one step kills every coherence, fully symbolic), its Choi is diagonal => entanglement breaking => quantum capacity 0; C_0(T) = 0 exactly (T entrywise > 0); the Kraus channel is NOT in the Birkhoff fibre.  (6) BINARY MESSAGE (KKT, 50 digits): p* = (1/2, 1/2, 0) exactly capacity-achieving for BOTH T and B; C(T) = 0.00835801913781 bit, C(B) = [13 log2(13/7) - log2 7]/18 = 0.4890415237 bit, ratio 1.709%.  [C] literature-typed: PPT<=>separable in 2x2 (Peres-Horodecki), separable Choi <=> EB (Horodecki-Shor-Ruskai quant-ph/0302031), EB index (Lami-Giovannetti 1411.2517) => qubit n_EB = 1 (t = 0) / 2 (t > 0).  [C] VALUE COINCIDENCE (observation, NOT a theorem): the t = 0 weights (9,1,4,4)/18 = (3^2,1^2,2^2,2^2)/18 hit the v652 c = 0 defect-ladder values 2*Delta = g^2/18 including w(I) = 1/2 = 2*Delta(gt=3); multiplicity matches only at g = 2; the forcing functor (CFT defect dimension -> Birkhoff weight) is the OPEN half, named TRANSFER.DEFECT.BIRKHOFF.01.  OPEN: the physical selection of a fibre point t; any 4D/flavor reading.  Python-only per GATE.WOLFRAM.02 (channel/Choi/Birkhoff algebra; the exact seam rationals already mirrored).",
      "status": "Identity [E] (executed half: fibre line, discriminator, criticality, EB no-go, KKT) + literature [C]; fibre-point selection open",
      "canonicalStatus": "[E] the Birkhoff fibre of B is the sign-character line t in [0,1/18]; loop-current discriminator Psi_t(A) = 6t A; qubit Choi law -3t/2 with EB criticality 2/3 + 1/3 = 1; full-level NPT -(1/3)(2/3)^n; the deployed Kraus dilation is EB (Q = 0, C_0 = 0); binary-message KKT certificates; [C] defect-Birkhoff value coincidence (forcing functor OPEN as TRANSFER.DEFECT.BIRKHOFF.01)",
      "dependencies": "DYN.UNITARY.DILATION.01; DYN.MARKOV.EMBED.01; QEC.SEAM.01; v221_seam_qecc.py; v652_orbifold_arf.py",
      "scripts": [
        "v976_seam_lift_birkhoff.py"
      ],
      "script": "v976_seam_lift_birkhoff.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 743
    },
    {
      "id": "TRANSFER.COHERENT.WILSON.01",
      "claim": "THE COHERENT WILSON CONTRACT, executed half (registered 2026-08-27; external-review round, companion of TRANSFER.HIDDEN.CIRCULATION.01).  [E] (v977, sympy-exact): (1) UNISTOCHASTICITY: the seam single step B equals |U_ij|^2 entrywise for an exact U in SU(3); the angles are forced by B (sin^2 th13 = 2/9, sin^2 th12 = 1/14, sin^2 th23 = 2/7), cos delta = -4/sqrt(65) is uniquely solved, so the complete lift set up to rephasing is the two-branch family {U, U*} with e^{i delta} = (-4 +- 7i)/sqrt(65).  (2) THE WILSON DATA ARE CLASSICAL UP TO ONE SIGN: J = 1/27 exact, Re of every plaquette fixed by B through unitarity (Re Q_{12;12} = -5/324, J^2 = 1/729 from B alone) -- the classically invisible datum is the Z2 orientation bit sgn J (U* lifts the same B with J = -1/27); minimal observable = the plaquette interference readout Im(U11 U22 U12* U21*).  (3) THE 5-12-13 CODE: Q_{12;12} = (-5+12i)/324, normalized (-5+12i)/13 with 5^2 + 12^2 = 13^2 (atoms 5 = g_car, 12 = dim g_SM, 13 = Delta_Q = N(3+2i)); all nine plaquettes close on four Gaussian phase types with norms exclusively {10, 13, 130, 169} = {2*5, 13, 2*5*13, 13^2}; every plaquette has Im = +-1/27; the four fundamental plaquettes generate all nine via exact B-weighted composition.  (4) THE CENTER BRIDGE: disc chi_A = 65 = det(A-I)^2 + (det A/2)^2 = 4^2 + 7^2 for the v530 atom matrix A = [[8,2],[5,3]], and e^{i delta} = (-det(A-I) +- i det A/2)/sqrt(disc chi_A) equals the transport-lift phase EXACTLY; -4+7i = (1+2i)(2+3i) with Gaussian norms 5, 13 (the v738 tower primes); NEGATIVE controls: the v530 siblings A_K (disc 48 vs identity 13) and A_Q (disc 13 vs 13/4) FAIL the identity -- the bridge selects the atom quotient A.  (5) POWER LAW + RATIONALITY SELECTION: J_n^2 = (1-b)^2(1+2b-3a^2)/108 (a = (2/3)^n, b = (1/3)^n), every B^n unistochastic with its own U_n (U_6 built, |U_6|^2 = T exact); |J_6| = 0.0951088904 = 0.9884006 J_max; the uniform limit lifts to the Fourier matrix F3 at J_max = 1/(6 sqrt 3) exactly; J_n is RATIONAL ONLY at n = 1 (the deployed step hits 3(1+2b-3a^2) = 1 exactly, so J = 1/27) -- the Gaussian-Pythagorean code selects the single step.  (6) NO FIXED MICROSTEP: B^2 != |U^2|^2 exact, and the Dirichlet recurrence witness n* = 967909 has min diag |U^{n*}|^2 = 0.999999 while B^{n*} is uniform to 5.6e-17 -- no single closed unitary generates the relaxing sequence; repeated steps require dephasing between steps, the six-step macro-gate U_6 is NOT U_1^6 (the coherent sharpening of the k5 clock-vs-walk verdict).  (7) SELECTIVITY (hard kill passed): the obstruction certificate J^2 = -1/64 < 0 fires on the classic non-unistochastic control; mutated spectra give irrational J; census over 238 valid family points: 4 rational-J, EXACTLY ONE full 5-12-13 code point = the deployed (2/3, 1/3) (0.42% << the preregistered 10% kill).  (8) PMNS NEAR-MISS stays NEGATIVE: branch phase 240.2551 deg near the deployed 240 deg (4pi/3, v270) but J_tr = -1/27 = -0.037037 != J_PMNS = -0.02965 and all three angles differ -- NO PMNS derivation.  [C] literature: coherification by a reversible unitary iff unistochastic (Dunkl-Zyczkowski 0909.0116; Korzekwa et al. 1710.04228); a unitary lift carries quantum capacity log2 3.  OPEN: the physical selection of sgn J; the A -> angles intertwiner (the bridge fixes the PHASE from A-invariants, the ANGLES come from B alone); any CP/time-orientation reading.  Python-only per GATE.WOLFRAM.02 (SU(3)/plaquette/center-bridge symbolic algebra with nested radicals).",
      "status": "Identity [E] (executed half: unistochastic lift, Wilson code, center bridge, selectivity census) + literature [C]; branch selection and intertwiner open",
      "canonicalStatus": "[E] B unistochastic with exact SU(3) lift {U, U*}, J = +-1/27; 5-12-13 plaquette code (norms {10,13,130,169}); center bridge disc chi_A = 65 = det(A-I)^2 + (det A/2)^2 with sibling negatives; rationality selection (J_n rational only at n = 1); census unique full-code point = deployed (2/3, 1/3); no fixed microstep (Dirichlet witness); [C] coherification literature; PMNS near-miss stays negative; sgn J and the intertwiner OPEN",
      "dependencies": "TRANSFER.HIDDEN.CIRCULATION.01; DYN.UNITARY.DILATION.01; v530_center_quotient_compiler.py; v738_hecke_mod_ramified.py; v270_pmns_jarlskog_assembly.py",
      "scripts": [
        "v977_transfer_unistochastic_wilson.py"
      ],
      "script": "v977_transfer_unistochastic_wilson.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 744
    },
    {
      "id": "QFT4D.OS.RECON.01",
      "claim": "THE 4D OSTERWALDER-SCHRADER PACKAGE CONTRACT [O] (registered 2026-08-27; step (3) of the DIMENSION.UPLIFT.FIREWALL.01 route ordering).  INPUT: TFPT fixes the discrete data (representation content, charge lattice, coupling relations, boundary conditions -- the seam as internal/topological boundary module); the genuine 4D Euclidean lattice carries spacetime, gauge fields, Higgs, fermions.  DEMAND: (O1) reflection positivity of the lattice measure; (O2) Euclidean covariance restoration (hypercubic -> full Euclidean group); (O3) clustering; (O4) uniform bounds (uniform in lattice spacing and volume); (O5) continuum limit; (O6) OS reconstruction of the Wightman/Haag-Kastler theory.  BINDING PHRASING RULE (the perturbative firewall): QFT4D.EG.ALLORDER.01 (v381) is a LOCAL PERTURBATIVE CONSTRUCTION UNDER THE STATED EG/BRST HYPOTHESES and no more -- the weak adiabatic limit for massless particles is the cited Duch theorem (arXiv:1801.10147) whose hypotheses must be verified for the concrete TFPT model; a seam-transfer gap does not substitute for the IR structure of massless photons/gluons; even full EG/BRST success yields NONE of: confinement/hadron spectrum, Yang-Mills mass gap, asymptotic charged states in QED, nonperturbative Higgs vacuum, controlled full S-matrix.",
      "status": "Open (research contract; binding phrasing rule in force for the perturbative leg)",
      "canonicalStatus": "Open: the nonperturbative 4D OS package (RP, Euclidean covariance, clustering, uniform bounds, continuum limit, reconstruction); the all-orders EG/BRST leg is fenced as local-perturbative-under-hypotheses",
      "dependencies": "DIMENSION.UPLIFT.FIREWALL.01; QFT4D.EG.ALLORDER.01 (fenced perturbative leg)",
      "scripts": [],
      "script": "-",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 745
    },
    {
      "id": "CHIRAL4D.NOMIRROR.01",
      "claim": "THE CHIRAL 4D LATTICE MATTER CONTRACT [O] (registered 2026-08-27; step (4) of the firewall route ordering).  The seam's c_- = 8 and Chern/inflow structure are a boundary-ANOMALY signal, not yet 4D matter.  DEMAND: an explicit chiral lattice operator (Ginsparg-Wilson/overlap type) for the TFPT representation content with proofs of (C1) a local gauge-invariant fermion measure; (C2) perturbative AND global anomaly freedom; (C3) no Witten SU(2) anomaly; (C4) no mirror generation in the physical spectrum (mirror decoupling surviving the continuum limit); (C5) correct index and charges (TFPT charge lattice, N_fam generations); (C6) a stable continuum limit compatible with QFT4D.OS.RECON.01.  CALIBRATION (state of the art): Mastropietro arXiv:2303.02790 -- weakly coupled 4D U(1) with a MASSIVE gauge field; the TFPT target (chiral SU(3)xSU(2)xU(1), three generations, massless nonabelian fields, EWSB and confinement downstream) is far stronger.  KILL TESTS: mirror fails to decouple (WOIT kill test 3 at lattice level); Witten anomaly obstructs the carrier SU(2); every gauge-invariant measure is provably nonlocal.  CONSISTENCY (ledger wins): the celestial-branch scope fence honestly lists interacting RP, mirror decoupling, internal SU(2), both helicities, local matter fields, EWSB, confinement as NOT closed; the contradictory 'zero open internal mechanisms' phrase in docs/OPEN_PROBLEMS.md is corrected in the same change.  EXECUTED UPDATE (2026-08-28, v989): chiral Gauss census on a two-species Z4 Hamiltonian (factorized 16 x 256, 4096 never materialised) -- single-exponent Gauss iff q_L = q_R; two-exponent Gauss always (lattice Dirac doubling); V^4 = I; mutant (g_L,g_R)=(1,0) on vectorlike fails.  NOT the TFPT representation content and not 3+1D.  Contract stays [O].  Python-only / Wolfram mirror deferred.  EXECUTED UPDATE (2026-08-28, completeness wave, v997): FK SMG toy theorem -- N=8 unique Spin(7)-singlet ground state, Schmidt rank 1, volume-independent gap 14 at L=1,2,3; N=4 contrast 2^L-degenerate; physical chiral edge keeps its pi/L spectrum exactly. SMG mechanism toy-proved [E]; 4D Spin(10) theorem stays [O]. Contract stays [O]. Python-only / Wolfram mirror deferred.  WAVE-5 KILL (2026-08-29, v1002): number-preserving Casimir projector route excluded -- singlet census one 16 = {N=0:1, N=16:1} (none at N=8); k=2 half-filling multiplicity 27; k=3: 1463 (exact Weyl alternation, |W(D5)|=1920); uniqueness fails at every cluster size.  The DET16 number-violating projector is a SEPARATE candidate (CHIRAL4D.MIRROR.DET16.01 [C]). EXECUTED UPDATE (2026-09-05, v1023): child CHIRAL4D.MIRROR.DFILTER.BLOCK.01 proves the finite derivative-filtered 2x2 singular-value block, exact ker(D_GW) preservation and the adiabatically mirror-continuated high branch. The bare mirror variable is not invariant, the low branch is quadratic (z=2), and no bare-mirror response gap, Weyl cone, DET16/DET32 realization, interacting parent, local measure/anomaly/index control or 3+1D continuum theorem follows. Parent status stays Open [O]. ROUND-4 CURRENT: v1027 repairs canonical DET excitations under the singlet premise and preserves signed z=1 dispersion for fixed c-number gauge fields. It changes flatDeltaQ to additiveDeltaNa. No hard bare-mirror gap, quantum-link CAR diagonalization, chiral SM or continuum closure; [O] unchanged.",
      "status": "Open (research contract; 1+1D chiral Gauss census executed; SMG mechanism toy-proved [E]; Nielsen-Ninomiya as Gauss statement; 4D Spin(10) theorem stays [O]; no 3+1D closure) | Round-3 v1023 adds an exact finite block lemma [E]/conditional GW reading [C]; NOMIRROR remains Open [O].",
      "canonicalStatus": "Open: explicit GW/overlap chiral lattice operator for the TFPT content.  EXECUTED (v989): 1+1D Gauss census (single-exponent iff vectorlike; two-exponent always).  No 3+1D closure.  EXECUTED (v997): SMG mechanism toy-proved [E] (N=8 unique Spin(7) singlet, gap 14 volume-independent; N=4 2^L contrast; physical edge pi/L exact).  4D Spin(10) theorem stays [O].  No 3+1D closure.  WAVE-5 (2026-08-29, v1002): number-preserving Casimir projector route excluded (census N=8:0, k=2:27, k=3:1463, |W(D5)|=1920).  Display unmoved Open. | ROUND3 2026-09-05: finite derivative-filtered block only; no invariant mirror-response gap, Weyl cone, many-body realization or 4D closure. Display stays [O]. | ROUND-4 CURRENT: v1027 repairs canonical DET excitations under the singlet premise and preserves signed z=1 dispersion for fixed c-number gauge fields. It changes flatDeltaQ to additiveDeltaNa. No hard bare-mirror gap, quantum-link CAR diagonalization, chiral SM or continuum closure; [O] unchanged.",
      "dependencies": "DIMENSION.UPLIFT.FIREWALL.01; QFT4D.OS.RECON.01; WOIT.OS.TWISTOR.01 (kill test 3)",
      "scripts": [
        "v989_lattice_gate_battery.py",
        "v997_spin2_smg_witnesses.py",
        "v1002_spin10_mirror_projector.py",
        "v1023_derivative_filtered_wall.py"
      ],
      "script": "v989_lattice_gate_battery.py; v997_spin2_smg_witnesses.py; v1002_spin10_mirror_projector.py; v1023_derivative_filtered_wall.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 746
    },
    {
      "id": "GRAV.NONCIRCULAR.01",
      "claim": "THE BISOGNANO-WICHMANN DIRECTION FIREWALL (registered 2026-08-27, with audit census).  BW is a theorem WITHIN an already-local Poincare-covariant QFT with suitable vacuum (wedge modular flow = boost); it does NOT generate Lorentz spacetime from an arbitrary modular flow (Poincare-covariant nets violating BW exist: Mund arXiv:hep-th/0101227, Morinelli arXiv:1703.06831).  Correct order: 4D net + locality + vacuum + spectrum condition => BW => geometric modular time.  NEVER: KMS/modular spectrum => Lorentz space => BW.  RULE: every BW/CHM use must declare ASSUMPTION (imported; result conditional on QFT4D.OS.RECON.01-class existence) or RESULT (derived intrinsically).  AUDIT CENSUS at registration: entanglement-equilibrium Einstein derivation (v323/v358/v359) = ASSUMPTION (CHM ball + BW boost are inputs; conditional on the 4D package; never citable as evidence FOR it); QG.AMB.01 redundancy (v369) = ASSUMPTION declared ('conditional on SEAM.EQUIV.01 + BW'); seam KMS normalisation chain (v259/v526, beta_angle = 2 pi) = ASSUMPTION dimension-typed (Tomita-Takesaki on 1+1D/2D seam objects, a normalisation, not a spacetime derivation); intrinsic-BW programme (v438/v440/v446, HSMI/Borchers route) = RESULT-DIRECTION, open (corpus already records the v215 circularity warning); v478 CHM-parabola flow = EVIDENCE not derivation; origin_theory 'BW downstream of the seam being a chiral net' = CONDITIONAL declared.  No marker moves; binds all future uses.",
      "status": "Firewall (standing discipline rule, in force; audit census recorded)",
      "canonicalStatus": "Firewall in force: every BW/CHM use declares assumption-vs-result; Einstein-derivation chain typed conditional on the 4D package; census of all current usages recorded",
      "dependencies": "QFT4D.OS.RECON.01; QGEO.SYM.01; QG.AMB.01; SEAM.EQUIV.01",
      "scripts": [],
      "script": "-",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 747
    },
    {
      "id": "OBS.TRANSDUCTION.01",
      "claim": "THE TRANSDUCTION PRINCIPLE CONTRACT (registered 2026-08-27; elevates the S15 typing rule to a named principle).  PRINCIPLE: observable signature = internal eigenvalue [compiler] + nonvanishing projection B P_r != 0 [transduction] + physical coupling operator [physics].  WITHOUT THE MIDDLE ARROW, NEITHER A NON-OBSERVATION FALSIFIES NOR AN OBSERVATION CONFIRMS.  Minimal model machine-checked at exploration level (experiments/theory-contracts/fo01_transduction_invisibility.py): the uniform measurement functional sees the seam kernel exactly not at all (1^T (T^n p - pi) = 0 to 1e-14; burst-train statistics KS-indistinguishable from a scrambled control), while a character-resolved channel reconstructs 64/729 to 1e-10 -- invisibility is a property of the readout B, not of the channel.  DEMAND [O]: for each empirical bundle (FRB/GW/pulsar nulls, the axion/birefringence family, any future search) prove or refute B P_r != 0 for the declared readout BEFORE scoring the null or the hit.  BINDING APPLICATION: ACT+Planck isotropic birefringence close to beta_TFPT = phi_0/(4 pi) (arXiv:2608.06480) is an interesting coincidence, NOT proof, until the shared normalized axion action + transduction chain are proven (the axion bundle beta, f_a, m_a, g_agg, Omega_a, EDM from ONE action with ONE proven coupling).  EXECUTED UPDATE (2026-08-28, v990): the first executed W[J] transduction shadow -- mixed d2W/dJ dtheta nonzero on the seam-coupled link (~-9.3e-3) and EXACTLY 0 on the decoupled e^c control (frozen companion: Q-staggered transduces, e^c stays 0).  Per-bundle empirical transduction proofs stay [O].  Python-only / Wolfram mirror deferred.",
      "status": "Contract (principle registered; finite W[J] transduction shadow executed; per-bundle proofs stay Open)",
      "canonicalStatus": "Principle registered: eigenvalue + projection + coupling = signature.  EXECUTED (v990): finite W[J] mixed-derivative transduction shadow (link transduces, e^c control = 0).  Per-bundle middle-arrow proofs stay Open.",
      "dependencies": "v221 (kernel); FO.01 (exploration minimal model); FRB/GW scorecard rows (readout typing)",
      "scripts": [
        "v990_wj_transduction_mechanism.py"
      ],
      "script": "v990_wj_transduction_mechanism.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 748
    },
    {
      "id": "PRED.JOINTLIKELIHOOD.01",
      "claim": "THE JOINT LIKELIHOOD CONTRACT [O] (registered 2026-08-27; from hit-counting to overdetermined relations).  The 27+ individually matched values descend from the same few atoms (phi_0, c_3, g_car, mu_4) and are statistically CORRELATED: counting hits overstates independent evidence.  DEMAND: reorganize the evidence into overdetermined relation bundles, each carried by one proof object -- (i) the flavor invariant sum rule (one spurion/operator proof yielding theta_12, theta_13, delta_PMNS, J_PMNS, mass splittings and the CKM texture TOGETHER); (ii) the inflation consistency relation r = 3(1 - n_s)^2, alpha_s = -r/6 (eliminating N_star); (iii) the axion/birefringence bundle from ONE normalized axion action with the OBS.TRANSDUCTION.01 chain; (iv) the character-resolved transfer signature.  STATISTICAL DEMAND: a joint likelihood over the frozen registry with the full atom-induced covariance matrix, the EFFECTIVE number of degrees of freedom (not the raw hit count), a prespecified null space and formula-selection rule, and a genuine holdout dataset (future measurements only).  HONEST READING of the existing null test: the '13/13, < 1e-30' Monte-Carlo statement (v100) quantifies the look-elsewhere burden UNDER ITS DECLARED NULL MODEL and is only interpretable as evidence to the extent that null space, formula grammar and development data were prespecified -- which for historically known observables they were not and cannot retroactively be; the 2026-06-09 freeze (v84) genuinely protects FUTURE measurements, it does not turn historically known data into blind predictions.  docs/FALSIFICATION.md wording adjusted in the same change (honest, not weakened: freeze and kill tests stand).  EXECUTED UPDATE (2026-08-28, v992): v1 joint likelihood over nine frozen pull-matched scorecard triplets -- chi2 = 11.80 / dof 9 (p = 0.225), nu_eff = rank(J_atom) = 1, TFPT at 0th percentile of 200 scrambled decoders, LOO stable.  Holdouts declared not consumed: JUNO Delta m^2, DESI Sigma m_nu, DUNE delta_CP, LiteBIRD r.  v1 declarations: Sigma_exp diagonal (NuFIT/Planck covariances UNMODELLED, flagged); Sigma_bridge = Sigma_th = 0.  PROMOTION CONDITIONS for full closure: off-diagonal Sigma, formula-grammar census, genuine future holdouts.  Contract stays [O].  Python-only / Wolfram mirror deferred.",
      "status": "Open (statistics-construction contract; v1 executed [N]; promotion conditions typed)",
      "canonicalStatus": "Open: joint likelihood with atom-induced covariance, effective dof, prespecified null + formula grammar, real holdout.  EXECUTED [N] (v992): v1 chi2=11.80/9, p=0.225, nu_eff=1, 0th percentile of 200 scrambled decoders, LOO stable; promotion conditions typed.  Display marker stays Open.",
      "dependencies": "v84 (frozen registry); v100 (null MC, reinterpreted); v375 (scorecard); OBS.TRANSDUCTION.01",
      "scripts": [
        "v992_joint_likelihood_v1.py"
      ],
      "script": "v992_joint_likelihood_v1.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 749
    },
    {
      "id": "SEAM.INT.ODDSPLIT.01",
      "claim": "THE ODD-SPLIT LAW (measured 2026-08-27; promoted in v972): the v529 straddle law refined on the 16-Majorana FK seam toy via the Jaffe-Pedrocchi Majorana-RP class made DECIDABLE -- the crossing coefficient matrix B is UNIQUE (left-monomial pairs biject onto crossing monomials), so 'B Hermitian AND -B PSD' is a finite check.  (1) THE INSTRUMENT: the checker certifies the free FK Hamiltonian on all 8 bond cuts (twist eta = -i forced); confusion matrix vs measured RP over 35 cells PERFECT (23 cert&PD / 12 uncert&fail / 0 FP / 0 FN).  (2) THE EXCLUSION: the mu4-covariant quartic space has dim 464 and ZERO basis directions (and 0 of 64 seeded combos) are JP-certifiable on all cuts; the support-level exclusion extends to the full 1820-dim quartic space (CLASS_EMPTY_ON_BASIS).  (3) THE DISCOVERY: JP predicted RP survival on CENTER-straddled cuts (m = 4 model, axes 7/15, 2+2 mirror split) and direct measurement CONFIRMED ((37,0,0) at g = 0.5/1/2) -- a straddled cut kills RP iff the split is ODD (1+3); the v529 anchor ladder reproduced exactly ((37,0,0),(33,4,0),(33,4,0),(31,6,0),(29,8,0)).  FENCE INHERITED VERBATIM: one toy, one interaction class, [C] flat-band parent; JP is SUFFICIENT-ONLY (the emptiness is a statement about the certificate, not a no-go for measured RP).",
      "status": "Measured/Exact [E] (toy level; JP sufficient-only; v529 fence inherited); no marker moves",
      "canonicalStatus": "[E] JP made decidable (unique B) + free FK certified on all 8 cuts + confusion 23/12/0/0 + quartic class EMPTY on the basis => the straddle law refined to the ODD-SPLIT LAW (JP-predicted, measurement-confirmed); toy level, fence inherited",
      "dependencies": "SEAM.INT.FKTOY.01; WOIT.OS.TWISTOR.01; v529; v903",
      "scripts": [
        "v972_seam_interaction_front.py"
      ],
      "script": "v972_seam_interaction_front.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "Jaffe-Pedrocchi arXiv:1406.1384; Wei-Li-Xiang arXiv:1601.01994 -- literature framing, no data",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 750
    },
    {
      "id": "SEAM.INT.OSAVOID.01",
      "claim": "OS VIABILITY ON THE QUARTET-AVOIDING FAMILY (measured 2026-08-27; promoted in v972): the v529 kill-test-2 counterweight -- (1) the straddle census 24/24 reproduced; (2) the clock-invariant m = 4 member's quartet-avoiding cut family generates the FULL even seam algebra (rank 15, dim 32768 = 2^15; the single-cut members m = 2/6 carry a typed deficit of EXACTLY 1); (3) OS/GNS on the avoiding cut k = 3 at g = 1 is POSITIVE (min eig +4.4338e-5, GNS 37/37, PD clock-transfer shadow, the low spectrum deforming continuously from g = 0, distances 1.044/1.386/1.650 monotone); (4) the straddled control GENUINELY FAILS (-6.936e-2) -- same machinery, honest negative; (5) the mu4 clock maps the avoiding pair {3, 11} onto itself EXACTLY at operator level.  READING: the interacting toy state IS OS-positive on the cut family its own clock geometry selects -- the kill-test-2 threat is NARROWED (an OS-viable geometry exists), NOT removed (every A_hol must still protect RP against the odd-split mechanism).  FENCE INHERITED: one toy, one interaction class, [C] flat-band parent; toy statement, not a reconstruction of an interacting continuum theory.",
      "status": "Measured [E] (toy level; v529 fence inherited); no marker moves",
      "canonicalStatus": "[E] the avoiding family generates the full even algebra (rank 15) and carries a positive OS/GNS structure at g = 1 while the straddled control fails; kill-test-2 counterweight narrowed, not removed; toy level",
      "dependencies": "SEAM.INT.FKTOY.01; SEAM.INT.ODDSPLIT.01; WOIT.OS.TWISTOR.01; v529",
      "scripts": [
        "v972_seam_interaction_front.py"
      ],
      "script": "v972_seam_interaction_front.py",
      "location": "tfpt_3;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 751
    },
    {
      "id": "SEAM.STATE.RPMIXING.01",
      "claim": "THE RP MIXING CENSUS (measured 2026-08-27; promoted in v972): the v903 exclusivity is RAY- and REFLECTION-relative, NOT slice-wide.  The full 33-dim C6-covariant mixing slice censused under BOTH deployable reflections: (1) theta_abT sees exactly ONE coordinate -- a_J, killed 1st order odd / 2nd order even, kernel value -1/(1+c^2) = -0.8240271368 at c = tanh(1/2) -- and 32/33 directions are identically invisible => a 32-dim MARGINAL RP hyperplane {a_J = 0}; (2) the theta_S Hermiticity-defect map has rank 16 => a 17-dim first-order-neutral subspace, NO second-order kill, with STRICT finite-s witnesses (W_S: pure a_J at s = 1/8, lambda_min 0.1558 -- the very direction theta_abT kills); (3) the v903 anchor reproduced (odd Gram eigenvalues == +-|a_J|, defect 1.7e-16, a_J = +0.04068013).  ANSWER TO THE ROUND QUESTION: RP + C6-covariance do NOT force the channel-diagonal point; each reflection carves only the coordinates its half-side Gram can SEE, and the two reflections see complementary windows -- the v903 statement is about the v898 RAY, not the slice.  TYPED: sector-truncated census (deg <= 2 Grams + deg <= 4 spots, not the full 256-monomial algebra); theta_abT invisibility = RP-SILENT, not certified positivity; the v898/v903 [O] premise UNMOVED.  EXECUTED UPDATE (2026-08-28, completeness wave, v995): ALL 33 v972 mixing directions are killed by KMS ALONE on the finite CAR algebra (CAR_8 = M_256); covariance is provably redundant under full KMS (COVARIANCE_DROP_REDUNDANT_UNDER_FULL_KMS); unique E-invariant crossed-product state extension (196608 -> 0). Type-III/continuum KMS uniqueness stays [O]. Display of the census stays [E]; no marker move on the open continuum residual. Python-only / Wolfram mirror deferred.",
      "status": "Measured census [E] (sector-truncated; float64 exploration grade warded by exact identities) + finite KMS rigidity [E] (v995); type-III/continuum stays [O]; no marker moves",
      "canonicalStatus": "[E] census: theta_abT kills exactly one coordinate (32-dim marginal hyperplane {a_J = 0}); theta_S leaves a 17-dim neutral subspace with strict witnesses; the v903 exclusivity typed ray-/reflection-relative; premise unmoved  EXECUTED UPDATE (2026-08-28, v995): finite KMS rigidity [E] -- all 33 mixing directions killed; covariance redundant; 196608 -> 0. Type-III/continuum stays [O].  WAVE-5 (2026-08-29): continuum leg listed as a child of SEAM.MMST.TYPEIII.CHARGED.01 [O]; finite KMS rigidity unmoved.  Display unmoved.",
      "dependencies": "SEAM.STATE.DERIVATION.01; SEAM.CFIN.KMSMIX.01; v903; v898",
      "scripts": [
        "v972_seam_interaction_front.py",
        "v995_kms_subnet_rigidity.py"
      ],
      "script": "v972_seam_interaction_front.py; v995_kms_subnet_rigidity.py",
      "location": "tfpt_1;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 752
    },
    {
      "id": "SEAM.EQUIV.SKELETON.01",
      "claim": "THE CROSS-ROUTE OS SKELETON (2026-08-27, v973): the two continuum routes of the seam equivalence target -- the MMST scaling-limit route (the p+ip collar edge) and the twistor route's Z4 lattice object -- share the EXECUTABLE OS skeleton: all FIVE WOIT kill-test main invariants (alpha: forced eta = +i, Theta^2 = +1, bond RP; beta1: GSO 2-torsion; beta2: OS quotient exists + KMS expansion lambda_max > 1; beta3: perp descent Theta^2 = +1; gamma: mirror flip, multiplicity-free clock, mark incidence) hold VERBATIM on the collar edge, 5/5; ALL divergences between the two route objects are control-typed and trace to ONE mechanism -- the twistor lattice's exact chiral checkerboard C(even) = 0 (a bipartiteness consequence, re-derived exactly in v973 S0c: S H S = -H forces every odd function of H to vanish on even distances) vs the collar kernel's K(even) up to 0.1736 -- independently confirming v519-R4's typing of the site-cut RP failure as a lattice-placement artifact, now from the second route; FIRST_DIVERGENCE = ALPHA.SITECUT.CHECKERBOARD (control-typed). BONUS: the collar's top edge realizes the anti-chiral state EXACTLY (max |K_top + K_bot| = 7.5e-16; v973 S0c re-derives K_top = -K_bot as an exact symmetry consequence P H P = -H on a strip); the M = 3 trivial collar loses the skeleton FIRST at GAMMA.MARK.INCIDENCE (mark Grams lose rank (13,0,3) vs (16,0,0)) -- the skeleton tracks the topological phase, not the lattice bookkeeping. HONEST BOUNDARY: an executable-level statement about the two lattice route OBJECTS, typed FULL/PARTIAL per test -- NOT an equivalence of the routes and NOT a continuum statement; it narrows the seam question (both routes carry the same OS/RP/GSO/mark structure wherever it is computable) but closes neither SEAM.EQUIV.MMST.01 nor SEAM.EQUIV.TWISTOR.01, which stay [O].",
      "status": "Exact [E] (executable skeleton; fine types Lattice/Numerical for the collar battery, Identity for the S0 mechanism gates)",
      "canonicalStatus": "[E] the executable OS skeleton is shared 5/5 with all divergences control-typed to the checkerboard mechanism; an executable-level narrowing -- does NOT close either route or the parent",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.MMST.01; SEAM.EQUIV.TWISTOR.01; WOIT.OS.TWISTOR.01",
      "scripts": [
        "v973_seam_route_narrowing.py"
      ],
      "script": "v973_seam_route_narrowing.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 753
    },
    {
      "id": "ALPHA.QUILLEN.FACES.01",
      "claim": "THE TWO NAMED FACES OF ALPHA.QUILLEN.EXACT.01 CARRY COMPUTED WITNESSES (2026-08-27, seam/alpha promotion wave; probes seam_bfk_conical_det_probe 21/21 SPEC d06b3bbec70e7efc / laughlin_pump_em1_probe 20/20 SPEC f9b5cb0bf20d0293 / quillen_rigidity_chain_probe 18/18 SPEC b4ee0dd0141da6ab, embedded byte-exact in v974). [E] BFK FACE EXECUTED: the Mayer-Vietoris determinant gluing on the seam circle at the four mu4 marks -- C_glue = 2^-N EXACT (N = 1..4 symbolic, general lengths), det'_zeta(Delta_S1) = l^2 reassembled, R(m) = (G_m|_marks)^-1; at KMS R_0 = 16 c3 circ(2,-1,0,-1), det'R_0 = 2^16 c3^3, glued product (2 pi)^2; vs v484 channel-wise EXACT (same mu4 eigenbasis, |spec| {0,2,2,4}, doublet ratio 4/ln2 with c3 cancelling, Tr B^k = 4^k + 2 2^k vs Tr C^k = 4^k + 2(-2)^k) but NOT globally proportional -- the kernel swap (gauge zero mode vs sign rep) is the sharpened diagonal-zeta face. [E] EM1 FACE COMPUTED: pumped charge == edge spectral flow == Streda == Chern == 1 on the v367 collar, single complex copy, sign-coherent conjugation mutant; 16-copy bookkeeping exact (c_- = 8, kY = 5/3, (1/kY) 41/6 = 41/10 = b1). [E] QUILLEN IDENTITY AT THE FINITE SHADOW: Berry integral == Poincare-Lelong divisor integral == +1 two-sided, difference an exact form, transition winding == Chern; anchor re-derived (alpha^-1 = 137.03599921684071, split residual 6.4e-58 at 50 digits, 8 b1 = 164/5 exact). [C] TYPED CONDITIONAL: IF the seam state is quasi-free (the rigidity premise, measured not forced in the strong form by SEAM.STATE.RPMIXING.01) THEN the identity applies literally to the seam det line. HONEST SCOPE: closes NOTHING on ALPHA.QUILLEN.EXACT.01 -- remaining content {rigidity premise} + {channel-swap question} + {continuum Bismut-Freed identification}; the BFK face is Delta = |D|^2, not a |D|-level operator identity; the pump is a single-copy statement.",
      "status": "Exact/computed witnesses [E] + typed conditional [C]; the target row stays [O]",
      "canonicalStatus": "Exact witnesses for both faces; target stays open",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; SEAM.CONTACT.UNIT.01; SEAM.CONTACT.UNIT.02; EM.WARD.01; ALPHA.QUILLEN.INFLOW.01; SEAM.STATE.RPMIXING.01",
      "scripts": [
        "v974_alpha_faces_computed.py"
      ],
      "script": "v974_alpha_faces_computed.py",
      "location": "tfpt_1;origin_theory;tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 754
    },
    {
      "id": "DIMENSION.SELECTOR.4D.01",
      "claim": "THE CONDITIONAL 4D DIMENSION SELECTOR [C] (registered 2026-08-27 from the external bird's-eye review, the master-route wave; machine-checked in v975, 16/16 sympy-exact).  STATEMENT: under the named TFPT-compatible physics axiom set A1 (d > 2: propagating local gauge fields, d-2 >= 1 transverse polarizations), A2 (dimensionless Yang-Mills coupling), A3 (real self-dual/anti-self-dual 2-form sectors: the Euclidean Hodge star is an involutive endomorphism of 2-forms), A4 (chiral Weyl representations: even d), the spacetime dimension d = 4 is UNIQUE and MINIMAL on the exact scan window d = 2..12, and the selector is OVERDETERMINED: A2 alone = {4} ([g_YM] = (4-d)/2 forced symbolically from dimensionless-action power counting; unique zero d = 4) and A3 alone = {4} (* : Omega^2 -> Omega^{d-2} is a self-map iff d = 4; the explicit 6x6 star on Lambda^2(R^4) has *^2 = I with eigenvalues +-1 at multiplicities 3/3 -- the real SD/ASD split su(2)+ x su(2)-), while A1 & A4 alone give {4, 6, 8, 10, 12} -- chirality does NOT select 4 by itself (honest ablation ledger).  Weyl chirality certified by exact Clifford algebras d = 2..7 (even d: gamma_* anticommutes with every gamma_mu, gamma_*^2 = I, tr = 0; odd d: the full product is CENTRAL and scalar -- no grading).  Signature typed honestly: Lorentzian *^2 = -1 on 2-forms in d = 4 -- the real split is a EUCLIDEAN/OS-side statement.  MUTANTS caught: the wrong power counting [g] = (6-d)/2 selects {6}; the 'A4 alone selects 4' overclaim is REFUSED.  FIREWALL (binding): mu4 and g_car enter NOWHERE -- 'mu4 => 4D' stays forbidden per DIMENSION.UPLIFT.FIREWALL.01; this is a CONDITIONAL selector, not a derivation: AXIOM PROVENANCE IS THE OPEN HALF (kill criterion: if A1-A4 cannot themselves be forced from the compiler primitives {c3, g_car}, the selector is a consistency statement, not a prediction); no 4D construction contract is advanced by this row.",
      "status": "Conditional [C] (exact selector under the named axiom set A1-A4; axiom provenance from the compiler open)",
      "canonicalStatus": "Conditional [C]: d = 4 unique + minimal + overdetermined under {d > 2, [g_YM] = 0, real SD/ASD 2-form split, Weyl chirality} on d = 2..12; A2 and A3 each pin {4} alone, A1 & A4 alone = {4, 6, 8, 10, 12}; mu4 enters nowhere; axiom provenance from {c3, g_car} is the open half",
      "dependencies": "DIMENSION.UPLIFT.FIREWALL.01; SEAM.BULK4D.RECON.01; QFT4D.OS.RECON.01; CHIRAL4D.NOMIRROR.01",
      "scripts": [
        "v975_dimension_selector_4d.py"
      ],
      "script": "v975_dimension_selector_4d.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 755
    },
    {
      "id": "SEAM.SIMPLECURRENT.GENERATOR.01",
      "claim": "THE SIMPLE-CURRENT GENERATOR CONTRACT [O] (registered 2026-08-27 from the external bird's-eye review; the seam-gap compression conjecture).  CONTEXT: v973 carved the cited-theorem coverage of the seam target -- the so(16)_1 fermion-bilinear half is COVERED (strong convergence, Virasoro/correlator control, polynomial errors), the NAMED residual is N1 (the mu4/GSO extension SO(16)_1 -> (E8)_1 as a NET statement: the 128 spinor currents are not bilinears), N2 (the twist/spin-field sector), N3 (the one-sided chiral edge), N4 (the holomorphy selector det K = 1 vs 4).  DEMAND (the compression): construct ONE normalized simple-current/spin-field intertwiner -- the generator of the index-4 Q-system extension (v125/v154) -- with proofs of (G1) convergence on the finite-particle core with energy bounds, (G2) locality relative to the bilinear net, (G3) the correct braiding/monodromy relation, (G4) Q-system closure, so that symmetry and fusion generate the ENTIRE 128-dimensional extension sector from the single generator -- collapsing N1 + N2 (and, via uniqueness of the extension, the N4 selector) onto ONE convergence theorem instead of 128 separate current controls.  KILL TESTS: the generator is provably nonlocal relative to the bilinear net; the energy bounds fail on the core; the braiding closes on the WRONG extension (SO(16)_1 stays unextended); the convergence rate is incompatible with the measured bilinear collar rate (v973: p = 3.48 inside the preregistered band).  DIMENSION TYPING (binding, per DIMENSION.UPLIFT.FIREWALL.01): hypotheses AND conclusion live in 1+1D -- closing this contract does NOT close SEAM.BULK4D.RECON.01 or any 3+1D claim.  EXECUTED UPDATE (2026-08-28, v988): S3 crossed-product/tower exchange EXACT (naturality; non-equivariant mutant fails); S1 window convergence + sector collapse measured (rates ~N^{-1.85} on the large collar; qualitative law on a reduced collar); S2 quantitative: windowed HS bounded (Shale ~0.376) + global HS^2 = 3.605+0.203 log N (topological) vs saturation (trivial); matrix-element Cauchy on the phase-fixed band; energy bounds f_n < C^{n+1}(n!)^2 (probe C=2.86).  VERDICT: the remaining lemma REDUCED to cited quasi-free theorems (Shale-Stinespring/Araki/Ruijsenaars) + the MMST identification -- REDUCTION, NOT CLOSURE.  Open rest = the MMST identification.  Python-only / Wolfram mirror deferred.  EXECUTED UPDATE (2026-08-28, v993): [lambda]^2 = [v] EXACTLY -- the order-4 glue class squares to the fermion-parity / D5-vector class (lattice shadow of U^2 = (-1)^F); non-split (vector glue norm 7/4 not in 2Z).  The FULL rank-8 ADE census selects (D5, A3) uniquely, so the D+A architecture is an OUTPUT of (rank 8 + cyclic Z4).  Python-only / Wolfram mirror deferred.  EXECUTED UPDATE (2026-08-28, completeness wave, v994): 5 of 7 MMST identification criteria executed [E-measured] (c = 8.005, conformal spectrum, k = 1, det D8=4 -> det E8=1, (E8)_1 vacuum character [1,0,248,0,4124,0,34752] three orders deep). C6 is the written-out quasi-free convergence theorem (articles/2026-08-28/psi_lambda_convergence_theorem_en.tex; 2/pi^2 = 0.20264 vs measured 0.203, 0.18%) -- this IS the G1/C6 hand-off. C7 modular invariance stays cited/open. Display stays [O]. Python-only / Wolfram mirror deferred.  EXECUTED UPDATE (2026-08-29, review wave 4, v998): the C7 arithmetic shadow of the MMST identification (Theta_E8=E4 exact, character four orders, two-edge HS 2.11) is now machine-checked; net-level C7 / G2/G3/G4 stay cited.  Display stays [O].",
      "status": "Open [O] + lattice shadow [E] (v983) + quantitative S1/S2 skeleton [E-measured] + reduction-to-cited-class [C conditional] + MMST criteria 5/7 executed [E-measured] (v994) + C6 written-out [C]; open rest = C7 modular invariance + net-level G2/G3/G4; no closure",
      "canonicalStatus": "[E] (v983): lattice-arithmetic generator lambda = (omega_s, omega_f).  [E-measured] (v988): S3 exact + S1/S2 quantitative skeleton on toy and v367 QWZ collar.  [C conditional] (v988): lemma reduced to cited quasi-free class + MMST identification.  [O] open rest = C7 modular invariance + net-level G2/G3/G4; no closure; 1+1D in and out.  [E] (v993): [lambda]^2 = [v] parity shadow of U^2 = (-1)^F; rank-8 ADE census unique (D5, A3) -- D+A architecture is an OUTPUT.  [E-measured] (v994): 5/7 MMST identification criteria (c=8.005, character [1,0,248,0,4124,0,34752], k=1, det D8=4->det E8=1).  [C] (v994): C6 written-out in psi_lambda_convergence_theorem_en.tex (2/pi^2 vs 0.203).| 2026-08-29 v998: C7 arithmetic shadow exact (shared with SEAM.EQUIV.MMST.01); net-level C7 stays cited.  Display unmoved.  WAVE-5 (2026-08-29): parent scaling-limit theorem SEAM.MMST.TYPEIII.CHARGED.01 [O] named; G1-G4 analytic half still [O].  Display unmoved.",
      "dependencies": "SEAM.EQUIV.01; SEAM.EQUIV.MMST.01 (the v973 N1-N4 carve); v125; v154; DIMENSION.UPLIFT.FIREWALL.01; v993",
      "scripts": [
        "v983_simple_current_generator.py",
        "v988_psi_lambda_reduction.py",
        "v993_minimal_defect_selector.py",
        "v994_mmst_criteria_battery.py",
        "v998_seam_modular_closure.py"
      ],
      "script": "v983_simple_current_generator.py; v988_psi_lambda_reduction.py; v993_minimal_defect_selector.py; v994_mmst_criteria_battery.py; v998_seam_modular_closure.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 756
    },
    {
      "id": "TFPT4D.LATTICE.ACTION.01",
      "claim": "THE EXPLICIT 4D LATTICE ACTION CONTRACT [O] (registered 2026-08-27 from the external bird's-eye review; route step (2) of the DIMENSION.UPLIFT.FIREWALL.01 ordering made concrete -- THE single object on which nearly all open 4D questions become simultaneously testable).  DEMAND: exhibit ONE fully explicit finite family of 4D Euclidean lattice theories Z_a[J] = int DU Dpsi DPhi exp(-S_a[U, psi, Phi] + J O) with S_a = S_Wilson[U] + psibar D_GW[U, Phi] psi + S_Phi[U, Phi] + S_seam/top, where U are the fundamental holonomies (TFPT fixes representation content, charge lattice, coupling relations, boundary conditions), D_GW is a Ginsparg-Wilson/overlap operator for the TFPT chiral representation content (the CHIRAL4D.NOMIRROR.01 operator), Phi is the ONLY allowed relevant scalar/link-mixing sector (EWSB downstream), and S_seam/top is the boundary/topological module transporting the seam data into the bulk -- FIRST at finite lattice spacing a, deliberately WITHOUT continuum claims.  FINITE GATES (all machine-checkable at finite size; the promotion criteria for any candidate): (T1) exact gauge invariance; (T2) a positive transfer matrix (RP at finite a); (T3) the seam boundary restriction reproduces the TFPT seam data EXACTLY (not merely dimensions or group labels); (T4) anomaly bookkeeping including the global SU(2) anomaly; (T5) the index/generation count; (T6) the mirror spectrum separated; (T7) at least one NONVANISHING connected four-point function (not free / generalized-free).  KILL TESTS: no local reflection-positive action with the correct seam restriction exists (master route dead); every candidate is generalized-free (T7 unreachable); a hidden freely tunable dimensionless parameter is required (the strong parameter-freeness claim is refuted); the uniform twistor flat direction (v973: the sector-0 Okubo admixture) shifts NORMALIZED observables rather than only the vacuum normalization of Z (a genuine undetermined modulus).  RELATION: the continuum limit is the SEPARATE contract QFT4D.OS.RECON.01; the unitary dilation is DYN.UNITARY.DILATION.01; d = 4 is typed by DIMENSION.SELECTOR.4D.01 [C]; the generating-functional collapse of the four transfer bridges is FTRANSFER.GENERATING.01.  NOTHING in the corpus closes any gate.  EXECUTED UPDATE (2026-08-28, v989): T4 exact SM one-generation anomaly bookkeeping (drop-L mutant fires); T5/T6 2D overlap index=q and GW circle (N=8); T7 Z2-cube kappa_4 != 0 at beta=0.4 (free control 0).  EUCLIDEAN T2 KILL (Z2 N_t-exact): overlap |det D_ov|^2 violation ABSENT at N_t=2, APPEARS at N_t=4 (det-only -0.249); Wilson control PSD everywhere -- pinned on overlap time nonlocality.  HAMILTONIAN ROUTE CLEARS T2 by construction (Gauss-law Hilbert space, H=H^dag, T=e^{-aH} SPD; seam intertwiner; background-field mu4 split k=1,3 degenerate; determinantal-law Gaussianity test).  NOTHING closes the 4D contract.  Python-only / Wolfram mirror deferred.  See also QFT4D.LATTICE.FUNDAMENTAL.01 (registered route/typing).",
      "status": "Open (research contract; T1-T7 harness + Euclidean T2 kill + Hamiltonian-route T2 clearance executed at toy level; no gate closed)",
      "canonicalStatus": "Open: one explicit finite 4D lattice family with seven finite gates T1-T7.  EXECUTED (v989, toy level): T4 exact, T5/T6 2D mechanism, T7 toy, Euclidean T2 kill (N_t-exact), Hamiltonian route clears T2 by construction.  No gate of the 4D contract closed; display marker stays Open.",
      "dependencies": "DIMENSION.UPLIFT.FIREWALL.01; QFT4D.OS.RECON.01; CHIRAL4D.NOMIRROR.01; DYN.UNITARY.DILATION.01; DIMENSION.SELECTOR.4D.01; SEAM.BULK4D.RECON.01",
      "scripts": [
        "v989_lattice_gate_battery.py"
      ],
      "script": "v989_lattice_gate_battery.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 757
    },
    {
      "id": "SEAM.DETLINE.UNIFICATION.01",
      "claim": "THE DETERMINANT-LINE UNIFICATION HYPOTHESIS [O] (registered 2026-08-27 from the external bird's-eye review; a precise research hypothesis, NOT a result).  OBSERVATION: three separately open structures may be projections of ONE geometric object -- (1) the seam extension is described by a Q-system/simple current (SEAM.SIMPLECURRENT.GENERATOR.01), (2) the chiral fermion measure lives on a determinant line (CHIRAL4D.NOMIRROR.01 demand C1), (3) anomalies and CP phases appear as holonomies of that determinant line (the finite Quillen identity is already TWO-SIDED at finite level -- v974/ALPHA.QUILLEN.FACES.01: Berry integral == Poincare-Lelong divisor integral == +1 -- and the continuum Bismut-Freed identification is the named missing step of ALPHA.QUILLEN.EXACT.01).  DEMAND (the compatibility theorem): Res_seam det D_4D ~= det D_seam as line bundles WITH connection, AND the holonomy of the bulk determinant line equals the discrete TFPT orientation phase (the classically invisible sgn J orientation bit exhibited at exploration level, round DCXCVII -- experiments only, not promoted).  IF such a theorem exists, seam extension, chiral anomaly freedom, the generation index and the discrete CP orientation become different projections of the same object.  KILL TESTS: the two determinant lines differ by a nontrivial local factor no counterterm removes; the bulk holonomy disagrees with the discrete orientation phase on a computable family.  HONEST BOUNDARY: the continuum Quillen/Bismut-Freed identification is exactly the critical missing step; nothing in the corpus proves any part of the 4D side; the sgn J bit is exploration-level input, cited as MOTIVATION only.  EXECUTED UPDATE (2026-08-28, v991): finite bulk-edge determinant-line correspondence on the v367 QWZ collar -- occupied-frame FHS Chern = +1 (curvature 2 pi * (+1)); seam holonomy winding = +1 (diff 6e-16); M=3 control zero both sides; conjugation mutant flips both signs coherently.  FIREWALL: 2+1D/1D finite lattice shadow; continuum Bismut-Freed and the 4D chiral-measure unification stay [O].  Python-only / Wolfram mirror deferred.  LATE EVENING (2026-08-30, v1014): finite restriction isomorphism Res_Sigma L_bulk isomorphic to L_Sigma verified on the QWZ collar (next finite check after v991); orientation orbit 2 -> 1 only after a U(1) anchor (A0).  Continuum Bismut-Freed and the 4D unification stay [O].",
      "status": "Open [O] + finite bulk-edge shadow [E] (2+1D/1D QWZ; continuum Bismut-Freed stays Open)",
      "canonicalStatus": "Open: Res_seam det D_4D ~= det D_seam with connection + bulk holonomy == discrete orientation.  [E] (v991): finite 2+1D/1D QWZ bulk-edge correspondence (C_bulk = W_seam = +1, M=3 control 0=0, conjugation mutant).  Continuum Bismut-Freed stays [O]; no 4D unification. | LATE EVENING (2026-08-30 v1014): finite restriction isomorphism verified; orientation fixes conjugate branch; residual U(1) is the A0 anchor.  Continuum stays [O].",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.FACES.01; CHIRAL4D.NOMIRROR.01; SEAM.SIMPLECURRENT.GENERATOR.01; TFPT4D.LATTICE.ACTION.01",
      "scripts": [
        "v991_detline_bulk_edge.py",
        "v1014_bridge_refinements.py"
      ],
      "script": "v991_detline_bulk_edge.py; v1014_bridge_refinements.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 758
    },
    {
      "id": "FTRANSFER.GENERATING.01",
      "claim": "THE GENERATING FUNCTIONAL CONTRACT [O] (registered 2026-08-27 from the external bird's-eye review; the constructive successor named by the FR.TRANSFER.01 guard).  The four continuous-transfer interfaces (F_pole: source masses -> pole, Koide v183; F_Boltzmann: CP source -> eta_B, v169/v184; F_relic: (f_a, m_a, theta_i, N_DW) -> Omega_a, v185; F_QCD: (alpha_s, b3, Lambda_QCD, lattice) -> m_p/m_e, v164) are currently four SEPARATE bridges.  DEMAND: IF the 4D theory Z_a[J] of TFPT4D.LATTICE.ACTION.01 exists with a controlled continuum limit (QFT4D.OS.RECON.01), the four bridges MUST arise from ONE generating functional W[J] = log Z[J] -- lepton/quark masses as poles of two-point functions (zeros of Gamma^(2)), m_p/m_e from the baryon and electron correlator poles of the SAME theory, eta_B from the real-time Schwinger-Keldysh dynamics of the SAME theory (CP violation, B violation, nonequilibrium), the axion relic from the cosmological solution of the SAME normalized action.  F_transfer is then not another 3x3 matrix; it is the full W[J].  BINDING (the guard stays in force): the four frontier observables remain [C]/[A]/[O] under FR.TRANSFER.01 UNTIL W[J] exists and each bridge is DERIVED from it -- this contract does NOT permit any marker upgrade; it names the single object whose existence would collapse the four transfer problems into computations within one theory.  KILL TEST: the four bridges provably require mutually inconsistent normalizations of the same W[J].  EXECUTED UPDATE (2026-08-28, v990): Z6-centre 6Y table + the first executed W[J] at dim 96 -- dW/dJ = <O> (6e-11), Hessian = Kubo (1e-7), transduction shadow d2W/dJ dtheta: seam-coupled -9.3e-3, decoupled e^c control exactly 0.  Finite-level mechanism, not the 4D generating functional.  FR.TRANSFER.01 guard stays binding.  Python-only / Wolfram mirror deferred.",
      "status": "Open (research contract; finite W[J] derivative identities + transduction shadow executed at 1+1D Z6; no upgrade permitted)",
      "canonicalStatus": "Open: one generating functional W[J] = log Z[J] from which all four F_transfer bridges derive.  EXECUTED (v990): finite 1+1D Z6 W[J] derivative identities + transduction shadow.  No upgrade permitted by this row.",
      "dependencies": "FR.TRANSFER.01; TFPT4D.LATTICE.ACTION.01; QFT4D.OS.RECON.01",
      "scripts": [
        "v990_wj_transduction_mechanism.py"
      ],
      "script": "v990_wj_transduction_mechanism.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 759
    },
    {
      "id": "QFT4D.LATTICE.FUNDAMENTAL.01",
      "claim": "THE LATTICE-FUNDAMENTAL READING (registered 2026-08-28 as a typed DECISION, not an [E] claim and not a marker move).  DECISION: TFPT's PHYSICAL completeness criterion is the finite, local, unitary lattice quantum theory -- Hamiltonian route: Gauss-law Hilbert space, hermitian local H, T = e^{-aH} positive by construction, Lieb-Robinson cone -- while the exact continuum Osterwalder-Schrader limit (QFT4D.OS.RECON.01) is retyped as MATHEMATICAL REINFORCEMENT, a named strengthening program, no longer the physical bottleneck.  Poincare invariance is an IR fixed-point property with experimentally bounded irrelevant Lorentz-violating operators.  JUSTIFICATION ANCHORS (v989): today's Euclidean T2 kill of the overlap shortcut (exact; N_t-exact confirmation: violation ABSENT at N_t=2, APPEARS at N_t=4 det-only -0.249; Wilson control PSD everywhere -- pinned on overlap time nonlocality) and the Hamiltonian route clearing T2 by construction (interior Gauss exact, seam-sector intertwiner, H = H^dag).  DISPLAY: nothing closes; QFT4D.OS.RECON.01 UNCHANGED as the math program; TFPT4D.LATTICE.ACTION.01 stays [O] as the constructive 4D lattice contract.  AMENDMENT (2026-08-29, review wave 4): the reviewer's precision is accepted -- 'a single finite box is not a world'.  The fundamental object is the QUASILOCAL CONSISTENT FAMILY {H_Lambda} with the thermodynamic-limit dynamics tau_t = lim e^{it H_Lambda} A e^{-it H_Lambda}.  Binding phrasing: CONTINUUM (a -> 0) OPTIONAL (mathematical reinforcement); THERMODYNAMIC LIMIT (|Lambda| -> infinity) AND CONTROLLED IR UNIVERSALITY MANDATORY.  Decision typing UNCHANGED; no [E] claim, no marker move.  WAVE-7 (2026-08-30, v1008): 2+1D master-object scaffold-coherent -- combinatorial K=0 (plaquette exponents cancel), frozen-link q^2~4, unique-contractive selector |R'|<1, QWZ wall + DET4 mirror, Z6/Z4 four-block assembly.  L=3 262144 / matter 129024 out of suite.  Decision typing UNCHANGED; no [E] claim, no marker move.  LATE EVENING (2026-08-30, v1013): MANDATORY-DYNAMICS LEG CLOSED at Hamiltonian-class level (theorem memorandum articles/2026-08-30/tfpt_thermodynamic_dynamics_en.tex/pdf).  Uniform Lieb-Robinson bound J=12/5, R=2, z=4, kappa=16043/450, v_LR=32086 e/225 ~ 387.64; norm-convergent thermodynamic-limit dynamics tau_t; gauge-invariant quasilocal algebra A^G (Gauss caveat resolved via the invariant subalgebra); existence of ground and beta-KMS states.  Numeric twin: measured v=2.0 << proved bound; convergence bound dominates nested differences at buffers 8/16/32.  Remaining: phase uniqueness, limit gap, IR universality.  Decision typing UNCHANGED; no [E] claim, no marker move.",
      "status": "Decision (registered route/typing; no [E] claim, no marker move; QFT4D.OS.RECON.01 unchanged as the math program)",
      "canonicalStatus": "Decision: physical completeness = finite local unitary lattice QFT (Hamiltonian route); continuum OS limit retyped as mathematical reinforcement.  No [E] claim, no marker move; QFT4D.OS.RECON.01 unchanged as the math program.  | 2026-08-29: quasilocal family {H_Lambda} is the fundamental object; continuum (a->0) optional; thermodynamic limit + controlled IR universality mandatory.  Decision typing UNCHANGED. | WAVE-7 (2026-08-30, v1008): 2+1D scaffold-coherent (K=0 theorem, contractive-root selector, wall/mirror, Z6 assembly); no marker move. | AFTERNOON (2026-08-30 v1011): 3+1D minimal object viable; fully coupled link-wall dynamics demonstrated at minimal scale; Kronecker limitation removed.  Decision typing UNCHANGED. | LATE EVENING (2026-08-30 v1013): mandatory-dynamics leg CLOSED at class level (LR + tau_t + A^G + state existence proved); remaining phase uniqueness, limit gap, IR universality.  Decision typing UNCHANGED.",
      "dependencies": "TFPT4D.LATTICE.ACTION.01; QFT4D.OS.RECON.01; CHIRAL4D.NOMIRROR.01; DYN.UNITARY.DILATION.01",
      "scripts": [
        "v989_lattice_gate_battery.py",
        "v1008_master_assembly_scaffold.py",
        "v1011_3p1d_ladder_battery.py",
        "v1013_thermodynamic_dynamics.py"
      ],
      "script": "v989_lattice_gate_battery.py; v1008_master_assembly_scaffold.py; v1011_3p1d_ladder_battery.py; v1013_thermodynamic_dynamics.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 760
    },
    {
      "id": "GAUGE.DETLINE.FIXPOINT.01",
      "claim": "THE GAUGE-COUPLING FIXPOINT CONTRACT [O] (registered 2026-08-28, review wave 3).  DEMAND: all three gauge couplings (g1, g2, g3 / equivalently alpha, sin^2 theta_W, alpha_s) arise as stationary points of ONE nonabelian zeta-det / inflow functional on the determinant line -- the nonabelian upgrade of the U(1) face ALPHA.QUILLEN.EXACT.01.  ACCEPTANCE: unique stationary points, positive Hessian, alpha_s(M_Z) without measured input, RG from the matching scale reproduces the data, mutants fail.  HONEST: alpha_s(M_Z) is currently an EXTERNAL INPUT (the QCD matching / lattice input of F_QCD); this contract names the missing first-principles replacement, it does not claim the number is derived today.  RELATION: generalizes ALPHA.QUILLEN.EXACT.01.  KILL: a unique stationary point fails to exist; Hessian not positive; mutants (wrong level, wrong inflow multiplicity) still pass; RG from the functional fails to land on the observed couplings.  EXECUTED UPDATE (2026-08-28, completeness wave, v996): the simplest-extension H0 of the U(1) alpha grammar to SU(2)/SU(3) is a STRUCTURAL KILL -- negative b2/b3 flip the stationary-root signs (SU(2) inverse -151.2, SU(3) -115.9); 64 conventions 0 hits; U(1) control 137.0359992168 survives all mutants. Exponential detline-curvature convergence to 2pi with rate 0.6936 ~ ln 2 is a numerical Bismut-Freed shadow. Naive grammar extension excluded [E]; genuinely nonabelian structure required -- typed. Contract stays [O]; alpha_s(M_Z) remains an external input. Python-only / Wolfram mirror deferred.  WAVE-7 (2026-08-30, v1008): the 2+1D plaquette scaffold is now executed -- combinatorial K=0, frozen-link q^2 content law, unique-contractive selector |R'|<1 (L2 0.578 -> L3 0.936 harvest pins; L=3 Lanczos out of suite).  The 1+1D no-plaquette hurdle is lifted at the scaffold level; thermodynamic 4D fixpoint stays [O].  Display unmoved Open.",
      "status": "Open (research contract; naive grammar extension excluded [E]; genuinely nonabelian structure required -- typed; alpha_s(M_Z) currently an external input)",
      "canonicalStatus": "Open [O] + H0 grammar kill [E] (v996): naive census-only extension of the U(1) alpha grammar to SU(2)/SU(3) excluded (sign kill, 64 conventions 0 hits); genuinely nonabelian Casimir/instanton structure required -- typed. Honest: alpha_s(M_Z) is currently an external input. ln-2 detline rate is a numerical Bismut-Freed shadow.  WAVE-5 (2026-08-29, exploration probes gauge_stiffness_fixpoint/stepscaling; NOT a new module): mechanism executed at FIXED VOLUME BUT thermodynamic step-scaling DIVERGENT; typed hurdle >= 2+1D.  WAVE-7 (2026-08-30, v1008): 2+1D scaffold-coherent (K=0, contractive selector, q^2 law); 4D fixpoint stays [O]. Display unmoved Open. | AFTERNOON (2026-08-30 v1011): BINDING constraint -- holonomy stiffness collapses in the confined phase; the matching point of the coupling functional must be defined in the deconfined/weak regime; charged channels enhance (~2.014x at weak g) but do not prevent collapse at minimal volume.  Display stays [O].",
      "dependencies": "ALPHA.QUILLEN.EXACT.01; SEAM.DETLINE.UNIFICATION.01; EM.FP.01",
      "scripts": [
        "v996_gauge_grammar_kill.py",
        "v1008_master_assembly_scaffold.py",
        "v1011_3p1d_ladder_battery.py"
      ],
      "script": "v996_gauge_grammar_kill.py; v1008_master_assembly_scaffold.py; v1011_3p1d_ladder_battery.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 761
    },
    {
      "id": "GRAV.SPIN2.EMERGENCE.01",
      "claim": "THE SPIN-2 EMERGENCE CONTRACT [O] (registered 2026-08-28, review wave 3).  DEMAND: from the SAME spectral determinant that carries the gauge fixpoints, a massless transversal spin-2 pole with two helicities, universal T_munu coupling, diffeomorphism Ward identities, and positive H / RP transfer.  RELATION: the entanglement-equilibrium Einstein equation (v358/v359) stays a DOWNSTREAM / CONDITIONAL readout -- it presupposes a local QFT and can never be cited as producing one; GRAV.NONCIRCULAR.01 is BINDING (BW/CHM uses remain assumption-declared).  KILL: a massive or longitudinal extra mode; non-universal coupling; failed diffeomorphism Ward; negative Hessian / RP violation on the spin-2 sector.  EXECUTED UPDATE (2026-08-28, completeness wave, v997): exact Barnes-Rivers quadratic decomposition -- R+R^2 clean (one massless TT pole, two helicities, scalaron at M_scal exact); the LOCAL a4 Weyl^2 truncation necessarily carries an opposite-residue spin-2 ghost -- typed. The contract's content is pinned on the untruncated form factor a(Box)=e^{-Box/M^2}. Einstein eq stays downstream/conditional. Display stays [O]. Python-only / Wolfram mirror deferred.  EXECUTED UPDATE (2026-08-29, review wave 4, v1000): quadratic Hamiltonian half executed -- positivity exact, 2 TT helicities at all 5852 nonzero momenta (N=6..12), omega = k - k^3/24 exact (lattice q_i = 2 sin(k_i/2)).  Spectral-action Hessian identification, nonlinear Ward and universal coupling stay [O].  Display stays [O].  WAVE-6 (2026-08-30): GRAV.WEINBERG.WITTEN.01 registered as the BINDING firewall BEFORE any spin-2 marker move (microscopic Hamiltonian family must NOT possess a fundamental Lorentz-covariant local conserved T_munu; Lorentz, diffeomorphism Ward identities and the spin-2 mode must emerge in the SAME IR limit). Display stays [O].  EVENING (2026-08-30, v1012): interacting TT half-step executed -- conserved interaction-complete lattice stress ~1e-15, TT spectral positivity exact, sum rules ~1e-15, dominant positive-Z pole persists under phi^4 (gapped, as expected for massive generic matter); Z trend k1 suppressed 0.958 / k2 enhanced 1.033.  Missing: TFPT content, k->0 volume scaling, Ward identities.  Display stays [O]. EXECUTED UPDATE (2026-09-05, v1024): GRAV.PARENT.GAP.NO_GO.01 proves that a uniform positive full-parent gap excludes a massless spin-2 pole and that the exact TT projector has a direction-dependent k->0 limit, hence cannot be a strictly finite-range direct microscopic projector. A critical physical sequence with local tensor gauge constraints remains the viable open route. No TFPT stress operator, massless pole, Ward identity or universal coupling is constructed. Parent status stays Open [O]. ROUND-4 CURRENT: v1029 gives a local quadratic first-class constraint target with two z=1 nonzero-momentum modes. The full periodic target is unbounded below at k=0; an additional global zero-mode removal is indispensable. No common-parent embedding/nonlinear universal coupling; [O] unchanged.",
      "status": "Open (research contract; quadratic witnesses [E] + quadratic Hamiltonian half executed; content pinned on the untruncated form factor; Einstein eq stays a downstream/conditional readout) | Round-3 v1024 supplies finite gap/TT locality no-gos [E]; spin-2 emergence remains Open [O].",
      "canonicalStatus": "Open [O] + quadratic witnesses [E] (v997): R+R^2 clean (massless TT pole, two helicities, scalaron at M_scal exact); local a4 Weyl^2 truncation necessarily carries an opposite-residue spin-2 ghost -- typed. Content pinned on the untruncated form factor. Einstein equation stays downstream/conditional; GRAV.NONCIRCULAR.01 binding.  | 2026-08-29 v1000: quadratic Hamiltonian half executed (positivity, 2 helicities at 5852 momenta, omega=k-k^3/24).  Display stays [O]. | WAVE-6 (2026-08-30): GRAV.WEINBERG.WITTEN.01 is the binding firewall before any spin-2 marker move; display stays [O] | EVENING (2026-08-30 v1012): interacting TT half-step: conservation/positivity/sum-rules/Z-persistence executed; missing TFPT content, k->0 volume scaling, Ward identities.  Display stays [O]. | ROUND3 2026-09-05: uniform full gap and direct finite-range TT are excluded mechanisms; critical gauge-constrained escape remains open. No graviton claim. | ROUND-4 CURRENT: v1029 gives a local quadratic first-class constraint target with two z=1 nonzero-momentum modes. The full periodic target is unbounded below at k=0; an additional global zero-mode removal is indispensable. No common-parent embedding/nonlinear universal coupling; [O] unchanged.",
      "dependencies": "GRAV.NONCIRCULAR.01; GAUGE.DETLINE.FIXPOINT.01; QFT4D.OS.RECON.01; QG.G2.01",
      "scripts": [
        "v997_spin2_smg_witnesses.py",
        "v1000_spin2_sk_witnesses.py",
        "v1012_toe_gate_witnesses.py",
        "v1024_parent_internal_boundaries.py"
      ],
      "script": "v997_spin2_smg_witnesses.py; v1000_spin2_sk_witnesses.py; v1012_toe_gate_witnesses.py; v1024_parent_internal_boundaries.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 762
    },
    {
      "id": "FTRANSFER.SK.RHO0.01",
      "claim": "THE SCHWINGER-KELDYSH INITIAL-STATE CONTRACT [O] (registered 2026-08-28, review wave 3; mechanism executed 2026-08-29).  DEMAND: cosmological transfers (F_relic, F_Boltzmann, and any nonequilibrium 4D readout) need the pair (S, rho_0), not W[J] alone -- the Euclidean-cap / KMS initial state.  TYPED CANDIDATE: the theta_i = 3 pi/5 saddle S_init = -kappa cos(5 theta - 3 pi) as the selection.  ARITHMETIC: stationary points theta = (3+k)pi/5, the mu4 orientation picks k = 0.  RELATION: FTRANSFER.GENERATING.01 covers EQUILIBRIUM correlators of W[J] only; this contract is the SK completion.  KILL: the (S, rho_0) pair is inconsistent with the W[J] of TFPT4D.LATTICE.ACTION.01; the 3pi/5 saddle is not selected by mu4 orientation (a different k is forced); the Euclidean cap fails KMS.  EXECUTED UPDATE (2026-08-29, review wave 4, v1000): mechanism executed -- KMS/FDT ~1e-17 on the exact Bohr grid, rho_0-dependence contrast 3.550e-3 vs static 1e-16, saddle 5-fold degenerate with 3pi/5 unique in the chosen mu4 lift [C].  Display stays [O]; no cosmological solve.  EXECUTED UPDATE (2026-08-30, review wave 7, v1009): finite uniqueness closed -- constrained minimizer uniquely the normalized KMS compression (Klein), Hessian spectrum [3.76, 12.6], center dim 1, compression identity ~8e-17.  TYPED: V1000_CONSEQUENCE_MISMATCH (compression kills the SK response, RMS ~1.5e-31 vs frozen contrast 3.550e-3) -- the proposed constraint set is too restrictive; CENTER_FLATNESS_PREMISE_FALSE (Umegaki keeps uniqueness with a nontrivial centre).  Canonical formula needs a weaker admissible set.  Display stays [O].  EVENING (2026-08-30, v1012): 8-candidate table ALL unique.  DOUBLE DIAGNOSIS: hard sectors kill the SK response (~1e-31); ALL response-alive sets return exactly rho_KMS even WITHOUT mu4 -- the entropic-proximity selection principle is VACUOUS at finite level.  Verdict NO_SET_WORKS.  BINDING: the canonical state requires a different principle (e.g. Euclidean-cap / orientation-branch).  Display stays [O].  MONDAY MORNING (2026-08-31, v1016): R4 seam-modular cap is the unique finite-level non-entropic selector (D_tr(KMS)=0.1434, SK response alive 3.038e-3, mu4-invariant, 3pi/5 saddle preserved).  R1 vacuous, R2 response-dead, orientation-free mutant loses saddles.  Caveat: the difference lives in the neutral spectator factor -- finite proxy.  Display stays [O]. EXECUTED UPDATE (2026-09-05, v1024): FTRANSFER.PARENT.Z5CAP.01 constructs a finite 12D internal cap with gap 0.197423715893, but the five-cycle is chosen, the cap breaks D4 (residual 1.094700) and it vanishes on a large complement, so it is not a global rho_0 selector. The Z5 Weyl divisibility result is scoped to the unprojected dimension; the Gauss-projected physical dimension is uncounted. The relative commutant is a prime/irreducibility obstruction, not a dynamic-coupling no-go. Parent status stays Open [O]. ROUND-4 CURRENT: v1028 constructs a symmetric gap-one internal cap, but D4 leaves a2D invariant-vector space and the vacuum-preserving extension selects no unique full-parent state. Order-five unitary is not a Weyl pair or derived selector; [O] unchanged.",
      "status": "Open (research contract; mechanism executed [N]/[C]; FTRANSFER.GENERATING.01 covers equilibrium only) | Round-3 v1024 gives a finite internal candidate and algebraic boundaries [E/N]; global rho_0 and SK uniqueness remain Open [O].",
      "canonicalStatus": "Open [O] + mechanism executed (v1000): KMS/FDT ~1e-17; rho_0 contrast 3.5e-3 vs static 1e-16; 3pi/5 unique in the mu4 lift [C].  WAVE-7 (2026-08-30, v1009): finite uniqueness closed [E-finite]; V1000_CONSEQUENCE_MISMATCH + CENTER_FLATNESS_PREMISE_FALSE typed; weaker admissible set needed.  Display stays [O]. | EVENING (2026-08-30 v1012): double diagnosis -- hard sectors response-dead, KMS-affine sets selection-vacuous; canonical state requires a different principle (e.g. Euclidean-cap/orientation-branch).  Display stays [O]. | MONDAY MORNING (2026-08-31 v1016): R4 first viable non-entropic selector (D_tr 0.143 + response alive + saddle preserved; finite-proxy caveat).  Display stays [O]. | ROUND3 2026-09-05: 12D cap is arbitrary and D4-breaking, not global rho_0; divisibility/commutant scopes explicit. Display stays [O]. | ROUND-4 CURRENT: v1028 constructs a symmetric gap-one internal cap, but D4 leaves a2D invariant-vector space and the vacuum-preserving extension selects no unique full-parent state. Order-five unitary is not a Weyl pair or derived selector; [O] unchanged.",
      "dependencies": "FTRANSFER.GENERATING.01; FR.TRANSFER.01; TFPT4D.LATTICE.ACTION.01",
      "scripts": [
        "v1000_spin2_sk_witnesses.py",
        "v1009_rho0_minimizer.py",
        "v1012_toe_gate_witnesses.py",
        "v1016_state_gap_batteries.py",
        "v1024_parent_internal_boundaries.py"
      ],
      "script": "v1000_spin2_sk_witnesses.py; v1009_rho0_minimizer.py; v1012_toe_gate_witnesses.py; v1016_state_gap_batteries.py; v1024_parent_internal_boundaries.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 763
    },
    {
      "id": "TFPT.TOE.COMPLETE.01",
      "claim": "THE MASTER TOE-COMPLETENESS CONTRACT [O] (registered 2026-08-29, review wave 4).  Eight gates, all named; NOTHING claimed closed.  T1 structure postulate -> P1/P2/compiler (AX.P1.01, AX.P2.01, TFPT.IRREDUCIBLE.01, DIMENSION.SELECTOR.4D.01).  T2 seam = (E8)_1 proven (SEAM.EQUIV.01, SEAM.EQUIV.MMST.01, SEAM.EQUIV.TWISTOR.01, SEAM.SIMPLECURRENT.GENERATOR.01, SEAM.DETLINE.UNIFICATION.01, SEAM.STATE.RPMIXING.01).  T3 quasilocal local unitary 3+1D Hamiltonian family (QFT4D.LATTICE.FUNDAMENTAL.01, TFPT4D.LATTICE.ACTION.01, DYN.UNITARY.DILATION.01, DYN.MARKOV.EMBED.01).  T4 chiral SM with local measure + uniform mirror gap (CHIRAL4D.NOMIRROR.01).  T5 IR Lorentz + confinement + clustering + nontrivial scattering (QFT4D.OS.RECON.01, SEAM.BULK4D.RECON.01).  T6 all three gauge couplings + neutrino texture internally fixed (ALPHA.QUILLEN.EXACT.01, GAUGE.DETLINE.FIXPOINT.01, FLAV.NUSCALE.01-.06).  T7 quantum massless spin-2 with universal coupling (GRAV.SPIN2.EMERGENCE.01, GRAV.NONCIRCULAR.01, GRAV.WEINBERG.WITTEN.01).  T8 unique initial state + one SK functional generating all readouts (FTRANSFER.SK.RHO0.01, FTRANSFER.GENERATING.01, OBS.TRANSDUCTION.01, PRED.JOINTLIKELIHOOD.01).  SEPARATION: TOE_Complete := AND(T1..T8); TOE_Validated := Complete AND independent holdouts passed.  Dual rest: compiler Rest = v_geo oplus G_net oplus F_transfer sits beside Rest_TOE; this row is the named AND of the eight gates, not a closure.  Children listed one-way here; children do not list this row back.  EVENING (2026-08-30, v1012): gate-by-gate wave -- T5 four IR witnesses executed + the propagating-mode precondition (common-c needs dynamical gauge excitations); T7 interacting stress mechanism executed; T8 selection-principle vacuity typed (needs a non-entropic selection); T6/nu scaffold-level triple bound complete.  Display stays [O].  LATE EVENING (2026-08-30, v1013/v1014): T3 mandatory-dynamics leg CLOSED at class level (LR + tau_t + A^G + state existence).  Remaining axiom-core and continuum gaps now run through ONE externalized theorem (MMST/Quillen: TEL-B/ALG-EXH/P1 + character-blind det-response + Bismut-Freed/A0).  Display stays [O].  MONDAY MORNING (2026-08-31, v1015): T1 structure-postulate route is now fully premise-supported at finite level (both W-bridge premises derived; axiom-core remainder 0 modulo externalized MMST).  Display stays [O]. EXECUTED UPDATE (2026-09-05, v1022-v1025; no marker moves): T1 retains finite axiom-core/W-bridge support only; unconditional structure-postulate provenance and the dimension selector remain open. T2 closes C1, C2a and the interval-TV input and executes the analytic scalar C2b budget, while CF, DG and FE-GEN/ALG-EXH remain open and the old ALG2 Vandermonde implication is refuted. T3 remains Hamiltonian-class only. T4 gains a finite derivative-filtered block but no invariant mirror response gap, Weyl cone, DET realization, measure or continuum. T5 retains finite IR witnesses and the v1020 spectator obstruction but lacks prime completion, OS reconstruction, confinement and scattering. T6 gains a parent-internal intertwiner with projective full-Fock lift but no mass texture. T7 gains gap/TT no-gos but no graviton. T8 gains an arbitrary D4-breaking internal cap but no global rho_0. The historical one-external-theorem wording is restricted to the finite compiler/W-bridge lane. A single shared 3+1D parent satisfying T3-T8 is still missing. All T1-T8 gates and TOE remain Open [O]. ROUND-4 CURRENT (v1026-v1030): uniform relaxed TEL-B norm proved; canonical DET-CAR and signed z=1 wall only for fixed classical gauge fields with an explicitly changed additive parent; full Gauss site/cube dimensions3/0mod5; symmetric internal D4 cap noncanonical; local tensor target needs global zero-mode removal; ALG2 frame theorem conditional. All T1-T8 remain [O]; no shared complete3+1D parent.",
      "status": "Open (research contract; named AND of eight gates; no closure, no marker move) | Round-3 v1022-v1025 adds certified subgates and no-gos only; Round-4 v1026-v1030 adds narrow proved and conditional subresults only; all T1-T8 gates and TOE remain Open [O].",
      "canonicalStatus": "Open [O]: TOE_Complete := AND(T1..T8); TOE_Validated := Complete AND independent holdouts passed.  Named AND of existing 4D/dilation/detline/SK/flavor/gravity/SK contracts; not a closure.  Compiler Rest sits beside Rest_TOE.  WAVE-5 (2026-08-29): T2 master scaling-limit theorem named SEAM.MMST.TYPEIII.CHARGED.01 [O] (children listed there, not back).  Display unmoved.  WAVE-6 (2026-08-30): T7 child GRAV.WEINBERG.WITTEN.01 [O] (Weinberg-Witten firewall before any spin-2 marker move). Display unmoved. | EVENING (2026-08-30 v1012): T5 four witnesses + propagating-mode precondition; T7 interacting stress mechanism; T8 selection-principle vacuity typed (needs a non-entropic selection); T6/nu scaffold-level triple bound complete.  Display unmoved. | LATE EVENING (2026-08-30 v1013/v1014): T3 mandatory-dynamics leg closed at class level; remaining axiom-core and continuum gaps run through one externalized theorem (MMST/Quillen).  Display unmoved. | MONDAY MORNING (2026-08-31 v1015): T1 structure-postulate route fully premise-supported at finite level.  Display unmoved. | ROUND3 2026-09-05: current T1-T8 matrix recorded; no marker moves; shared 3+1D parent absent. The one-theorem statement is compiler/W-bridge scope only. | ROUND-4 CURRENT (v1026-v1030): uniform relaxed TEL-B norm proved; canonical DET-CAR and signed z=1 wall only for fixed classical gauge fields with an explicitly changed additive parent; full Gauss site/cube dimensions3/0mod5; symmetric internal D4 cap noncanonical; local tensor target needs global zero-mode removal; ALG2 frame theorem conditional. All T1-T8 remain [O]; no shared complete3+1D parent.",
      "dependencies": "AX.P1.01; AX.P2.01; TFPT.IRREDUCIBLE.01; DIMENSION.SELECTOR.4D.01; SEAM.EQUIV.01; SEAM.EQUIV.MMST.01; SEAM.EQUIV.TWISTOR.01; SEAM.SIMPLECURRENT.GENERATOR.01; SEAM.DETLINE.UNIFICATION.01; SEAM.STATE.RPMIXING.01; QFT4D.LATTICE.FUNDAMENTAL.01; TFPT4D.LATTICE.ACTION.01; DYN.UNITARY.DILATION.01; DYN.MARKOV.EMBED.01; CHIRAL4D.NOMIRROR.01; QFT4D.OS.RECON.01; SEAM.BULK4D.RECON.01; ALPHA.QUILLEN.EXACT.01; GAUGE.DETLINE.FIXPOINT.01; FLAV.NUSCALE.01; FLAV.NUSCALE.04; FLAV.NUSCALE.03; FLAV.NUSCALE.02; FLAV.NUSCALE.05; FLAV.NUSCALE.06; GRAV.SPIN2.EMERGENCE.01; GRAV.NONCIRCULAR.01; GRAV.WEINBERG.WITTEN.01; FTRANSFER.SK.RHO0.01; FTRANSFER.GENERATING.01; OBS.TRANSDUCTION.01; PRED.JOINTLIKELIHOOD.01",
      "scripts": [
        "v1012_toe_gate_witnesses.py",
        "v1013_thermodynamic_dynamics.py",
        "v1014_bridge_refinements.py",
        "v1015_axiom_core_closure.py",
        "v1022_telb_round3_certificates.py",
        "v1023_derivative_filtered_wall.py",
        "v1024_parent_internal_boundaries.py",
        "v1025_telb_c1_c2a_certificates.py",
        "v1026_telb_round4_closure.py",
        "v1027_signed_det_car_wall.py",
        "v1028_gauss_d4_cap.py",
        "v1029_local_tensor_constraints.py",
        "v1030_alg2_sector_frame.py"
      ],
      "script": "v1012_toe_gate_witnesses.py; v1013_thermodynamic_dynamics.py; v1014_bridge_refinements.py; v1015_axiom_core_closure.py; v1022_telb_round3_certificates.py; v1023_derivative_filtered_wall.py; v1024_parent_internal_boundaries.py; v1025_telb_c1_c2a_certificates.py; v1026_telb_round4_closure.py; v1027_signed_det_car_wall.py; v1028_gauss_d4_cap.py; v1029_local_tensor_constraints.py; v1030_alg2_sector_frame.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier;introduction",
      "externalData": "-",
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    {
      "id": "FLAV.NUSCALE.06",
      "claim": "THE PENTAGON-CLASS MISALIGNMENT CANDIDATE (registered 2026-08-29, review wave 4, v1001).  Candidate [C]/[N]; mechanism = the Q_+-to-flavor operator [O]; NO closure; FLAV.NUSCALE.05 unmoved.  [E] U_e = I inventory theorem: no charged-lepton mass matrix in the compiler; companion algebra G-self-adjoint with eigenvalues (16/7, 4/3, 7/6).  [E] misalignment U = U_v9 R13(theta, phi) solves the 13/23 pair analytically (held-out theta12 inside 1 sigma of NuFIT).  [N] pentagon double hit: phi = 288 deg = 4(2pi/5) frozen => all three measured angles <= 0.56 sigma (honest max pull 0.557) at theta = 2pi/35; that theta is the unique LEE survivor of nine pre-declared candidates (LEE expectation 2.7%).  [N] v3 chain SHA-16 a4c28732fa687620: Sigma = 0.0599 eV, m_beta = 9.0 meV, m_bb in [1.5, 3.8] meV, delta_CP = 287.66 deg.  [N] v270-theta23 tension 1.85 sigma TYPED (TFPT-target refit pull; the pentagon phase does not recover the assembled v270 angle set).  [X] census nulls: matrix 0/1607, ratio grammar null, circularity kills -- frozen 3x3 inventory insufficient.  Kills: DESI floor on Sigma, DUNE delta_CP discriminates 287.7 vs 240, JUNO.  y2 is DATA_CALIBRATED; no seesaw closure.",
      "status": "Candidate [C]/[N] + mechanism [O] (pentagon-class misalignment; no seesaw closure; FLAV.NUSCALE.05 unmoved)",
      "canonicalStatus": "[C]/[N] pentagon-class misalignment candidate (v1001): phi=288 deg, theta=2pi/35, LEE 2.7%, v3 SHA a4c28732fa687620, Sigma=0.0599 eV, m_beta=9.0 meV, m_bb [1.5,3.8] meV, delta_CP=287.66 deg; v270-theta23 tension 1.85 sigma TYPED; census 0/1607.  Mechanism Q_+-to-flavor [O].  No closure.  WAVE-5 (2026-08-29, v1003): both structural nulls (D4-even Schur; scale cancellation 1.885e-4) are binding on FLAV.NU.TEXTURE.MECHANISM.01 [O].  Pentagon-class candidate unmoved; no seesaw closure.",
      "dependencies": "FLAV.NUSCALE.05; FLAV.NUSCALE.02; FLAV.NUSCALE.01; AX.P2.01",
      "scripts": [
        "v1001_nu_pentagon_chain.py",
        "v1003_nu_structural_nulls.py"
      ],
      "script": "v1001_nu_pentagon_chain.py; v1003_nu_structural_nulls.py",
      "location": "tfpt_2_standard_model;tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
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    },
    {
      "id": "CHIRAL4D.MIRROR.DET16.01",
      "claim": "THE DET16 RANK-ONE MIRROR PROJECTOR CANDIDATE (registered 2026-08-29, review wave 5, v1002).  Candidate [C]: exact finite-cluster gapping mechanism on the full 2^16 chiral Spin(10) Fock space.  [E-finite] Omega^dag Omega = P_16, Omega Omega^dag = P_0, defect 0; P_phi hermitian idempotent (2.8e-18), rank 1; h_mir spectrum {0^1, 1^65535}, gap exactly 1; all 45 so(10) Fock commutators = 0 (det rho_16 = 1); parity commutator 0; |0> and |F> singlets.  [N] hopping stability on the 4-mode analogue: gap 0.90 (2 clusters) / 0.86 (3 clusters) at t = 0.2; spectators untouched.  OPEN: dynamical gauge fields, domain-wall geometry, 4D volume theorem.  The number-preserving Casimir projector route is a SEPARATE kill recorded on CHIRAL4D.NOMIRROR.01 (not this candidate).  Display Candidate [C]; no upgrade of NOMIRROR.  EXECUTED UPDATE (2026-08-30, review wave 7, v1007): T1/T2 proved; T3 Michalakis-Zwolak cited-verified (LTQO, t*>0, gap>=1/2).  [N] counting line n=2,3; n=2 closing scan through t=4 (no closing); dynamical Z2/Z4 uniquely gapped (full-probe min 0.728854416 pinned; compressed grid coarser).  SW 1-alpha t^2 + O(t^4); c_geo/L bounded.  1/192 remains the uncertified counting threshold (ordinary hopping fails the relative-form block hypothesis).  BDL supplies no volume-uniform gap remainder (boxed).  Display stays Candidate [C]; NOMIRROR unmoved.  MONDAY MORNING (2026-08-31, v1016): Froehlich-Pizzo hypotheses cited-verified for Z2/Z4 DET open chains -- volume-uniform spectral gap >= Delta/2 in the parametric window tau_FP (a=0.0233, a/16=0.00146).  Dynamical-link leg closed at that finite-group level.  Remaining: rotor links, wall mixing, 3+1D placement.  Display stays Candidate [C]. EXECUTED UPDATE (2026-09-05, v1023): the separate derivative-filtered wall block preserves ker(D_GW) in a finite quasi-free singular-value model, but it does not realize the DET16/DET32 interaction. The bare mirror response is not gapped and the low branch is quadratic. DET16 remains Candidate [C]; NOMIRROR/T4 remain Open [O].",
      "status": "Candidate [C] (exact finite-cluster gapping; open: dynamical gauge fields, domain-wall geometry, 4D volume theorem) | Round-3 v1023 is a separate finite block lemma and does not upgrade DET16.",
      "canonicalStatus": "Candidate [C] (v1002/v1007): T1/T2 proved; T3 MZ cited-verified (t*>0, gap>=1/2); dynamical [N] + boxed BDL/1/192.  Open: compact-group dynamical links, domain-wall geometry, 4D volume theorem.  No marker upgrade of NOMIRROR.  WAVE-7 2026-08-30. | MONDAY MORNING (2026-08-31 v1016): DFP volume-uniform gap cited-verified for finite-group open chains; dynamical-link leg closed at that level; remaining rotor/wall/3+1D.  Display stays [C]. | ROUND3 2026-09-05: derivative-filtered witness is not a DET16/DET32 realization and supplies no 4D theorem. Candidate stays [C].",
      "dependencies": "CHIRAL4D.NOMIRROR.01; DIMENSION.UPLIFT.FIREWALL.01",
      "scripts": [
        "v1002_spin10_mirror_projector.py",
        "v1007_det16_stability_battery.py",
        "v1016_state_gap_batteries.py",
        "v1023_derivative_filtered_wall.py"
      ],
      "script": "v1002_spin10_mirror_projector.py; v1007_det16_stability_battery.py; v1016_state_gap_batteries.py; v1023_derivative_filtered_wall.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "articles/2026-08-30/det16_mirror_gap_theorem_en.tex (12 pp; T1/T2 proved, T3 MZ cited-verified, 1/192 boxed, BDL boxed)",
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      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 766
    },
    {
      "id": "SEAM.MMST.TYPEIII.CHARGED.01",
      "claim": "THE MASTER SCALING-LIMIT THEOREM FOR THE CHARGED/TYPE-III IDENTIFICATION [O] (registered 2026-08-29, review wave 5).  DEMAND: ONE scaling-limit theorem Psi_lambda,N -> Psi_lambda strongly on the finite-energy core + crossed-product continuity, from which B ≅ (E8)_1, mu = 1, type III_1, unique rotational KMS, unique conditional expectation, deck equidistribution, and modular invariance all follow as corollaries.  CHILDREN (listed here, not listing back): SEAM.SIMPLECURRENT.GENERATOR.01, SEAM.STATE.RPMIXING.01 continuum leg, ALPHA.QUILLEN.EXACT.01 rigidity premise, AX.P1.01 reading.  CITED DOCUMENT: articles/2026-08-29/holomorphic_kms_extension_en.tex (13 pp): unconditional rotational-KMS uniqueness via KLM + Longo-Tanimoto GIVEN the identification; P1 = theorem-conditional on seam identification + beta_angle = 2pi.  KILL: the scaling limit fails to exist; uniqueness of rotational KMS fails; type III_1 fails; a competing conditional expectation exists.  Status: Open [O].  No marker move on children.  EXECUTED UPDATE (2026-08-30, review wave 7, v1006): five lemmas proved in-house -- L3 crossed-product, UGF K_G=pi^2/4 (measured plateau ~1.023), TEL-a N^{-2} with exact isometries (A_TEL=5120(1+pi/2), measured envelope ~3.42), integer D5+A3 pairings, Z4 outerness in the limit + finite-inner correction (256-dim gauge unitary is inner).  Residual boxed: TEL-B-EXTERNAL (Hankel C_R; HS max 2.105079358 / fit 2.107565497 / v998 bound 2.11) + ALG-EXH (two Buchholz-Verch estimates).  Display stays [O]; no marker move.  MONDAY MORNING (2026-08-31, v1016): TEL-B third (certified-tail) route BLOCKED -- N^2 D_N grows 530 -> 19032, remainder UV-supported.  TEL-B-EXTERNAL (2026-09-04 reduction): triangle bound via cover identity + sawtooth split; (A) lattice piece certified numerically for all N, ||R^sm_N|| <= 1.7833 (all-mode BV, one float64 TV constant); (B) explicit piece reduced to Sigma|Res_N|^2 <= 0.2769 (measured 0.1550), open items C1 (C* <= 0.1525 enclosure), C2a (|phi_N| <= 0.51/N), C2b (cell oscillation of r); numerically 2.57 < 3, conditional analytic 2.9495 < 3. Status unchanged [O].  N-uniform CAR nuclearity PROVED (nu_N <= exp(16 C_beta); C_beta = (67.97, 29.73, 12.15) for beta = (0.5, 1, 2)) as ALG specialist compactness input; ALG-EXH projector-identification line UNCHANGED.  Display stays [O]. EXECUTED UPDATE (2026-09-05, v1022/v1025): TEL-B C1 is interval-certified at C_star<=0.1520453048...<0.1525; C2a is analytically proved with N sup|phi_N|<0.509 for every even N>=16; the native 256-bit/5812-cell TV certificate gives TV_p(F'')<=15.817695962988924...<15.928674 and K3_TV<2.517464500835. The analytic C2b scalar reduction plus Arb arithmetic gives c_potential=0.399224424139106...<0.400. Full C2b remains open exactly at CF<=0.139/sqrt(N) and DG<=0.060/sqrt(N). The old ALG2 Vandermonde-to-generated-algebra implication is refuted by an exact M2 countermodel; FE-GEN or a precise gauge/code-projected algebra is required and ALG1 is conditional on the same generated-sector completeness. MMST/type-III identification remains Open [O]. ROUND-4 CURRENT: v1026 proves the relaxed uniform norm ||R_N||HS<2.995906<3 for every evenN>=16 (CF/DG/CROSS plus validated Fourier aliases). Historical2.11 and finite1.7833 are not promoted. The residual is microscopic one-edge FE-GEN, actual-word/joint-adjoint tails, ALG-EXH and MMST identification; v1030 is conditional. Aggregate [O] unchanged.",
      "status": "Open (research contract; master scaling-limit theorem; children listed here not back) | Round-3: C1/C2a/TV and scalar C2b subclaims proved [E]; ALG2 implication refuted [X]; CF/DG and ALG-EXH remain Open [O].",
      "canonicalStatus": "Open [O] + five lemmas in-house (v1006): L3, UGF K_G=pi^2/4, TEL-a N^{-2} isometries, integer D5+A3 pairings, Z4 outerness in the limit.  Residual boxed TEL-B-EXTERNAL + ALG-EXH.  WAVE-7 2026-08-30.  Display stays [O]. | MONDAY MORNING (2026-08-31 v1016): TEL-B narrowed to scalar dyadic A_R<43 (factor-350 headroom; UV-supported remainder); uniform CAR nuclearity proved as specialist input; ALG-EXH unchanged.  Display stays [O]. | ROUND3 2026-09-05: TEL-B reduced to CF/DG; C1/C2a/TV closed; ALG2 correction binding. Charged/type-III parent stays [O]. | ROUND-4 CURRENT: v1026 proves the relaxed uniform norm ||R_N||HS<2.995906<3 for every evenN>=16 (CF/DG/CROSS plus validated Fourier aliases). Historical2.11 and finite1.7833 are not promoted. The residual is microscopic one-edge FE-GEN, actual-word/joint-adjoint tails, ALG-EXH and MMST identification; v1030 is conditional. Aggregate [O] unchanged.",
      "dependencies": "SEAM.SIMPLECURRENT.GENERATOR.01; SEAM.STATE.RPMIXING.01; ALPHA.QUILLEN.EXACT.01; AX.P1.01",
      "scripts": [
        "v1006_mmst_lemma_battery.py",
        "v1016_state_gap_batteries.py",
        "v1022_telb_round3_certificates.py",
        "v1025_telb_c1_c2a_certificates.py"
      ],
      "script": "v1006_mmst_lemma_battery.py; v1016_state_gap_batteries.py; v1022_telb_round3_certificates.py; v1025_telb_c1_c2a_certificates.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "articles/2026-08-29/holomorphic_kms_extension_en.tex (KLM; Longo-Tanimoto) -- cited theorems, no data; articles/2026-08-30/mmst_charged_scaling_limit_en.tex (21 pp; five lemmas in-house; residual TEL-B-EXTERNAL + ALG-EXH); articles/2026-08-30/externalization_mmst_handoff_v2_en.html",
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      "sourceLine": 767
    },
    {
      "id": "FLAV.NU.TEXTURE.MECHANISM.01",
      "claim": "THE NEUTRINO-TEXTURE MECHANISM CONTRACT [O] (registered 2026-08-29, review wave 5, v1003).  Both structural nulls are BINDING CONSTRAINTS.  [E] charged-sector data h=(1,1,1), d=(64,60,64) (coset min norms all 2).  [X] Schur-texture NULL: all 8 preregistered (K,B) combinations fail; D4 reflection forces lambda_1=lambda_3; hierarchy (eps, 2eps, 3) structurally unreachable from D4-even seam data; required K spectrum [3536.998, 1768.499, 1/3] matches none of 9 comparators.  [E] orientation-doubling structure theorem: the sgn-J odd sector generically lifts the degeneracy (exact rational witness).  [X] scale NULL: 12 declared trials all fail; required cancellation 1-|c/a|=1.885e-4 not supplied by any frozen constant; eta-branches isospectral.  CONSEQUENCE: the mechanism must be D4-odd AND supply a ~2e-4 even-odd cancellation; pure seam data excluded [X-typed].  Does NOT close FLAV.NUSCALE.05/.06; no seesaw closure.  Status: Open [O].  EVENING (2026-08-30, v1012): kappa six-fold NULL completes the scaffold bound -- six pre-declared D4-odd candidates, all parity-checked, all NULL (five at 1e-14; K4 = 3.55e-3 is 18.8x too large).  The 1.885e-4 cancellation scale requires the true 4D functional.  Mechanism scale strictly beyond frozen finite objects.  Display stays [O]. EXECUTED UPDATE (2026-09-05, v1024): FLAV.NU.PARENT.INTERTWINER.01 constructs a positive, even, gauge-neutral non-additive finite parent-internal A3-D4 coupling with conditional gap 0.25 and mixed resolvent 0.104712041885. Ordinary D4 acts only on the selected pair operators; the full Fock lift is projective, with C_f^8=1, C_f^4=(-1)^(N1+N3) != (-1)^F and J_f C_f J_f=(-1)^F C_f^-1. It supplies no neutrino mass matrix or 2e-4 cancellation and is not a 4D functional. Parent status stays Open [O]. ROUND-4 CURRENT: v1028 exactly counts full Gauss dimensions and constructs a D4-symmetric cap only on the selected even pair subspace. The invariant-vector space is2D; the actual symmetric fermion lift forbids all Majorana bilinears. No mass texture; [O] unchanged.",
      "status": "Open (research contract; both structural nulls are binding constraints; NUSCALE.05/.06 unmoved) | Round-3 v1024 provides a finite parent-internal witness [E/N]; the physical texture/scale remains Open [O].",
      "canonicalStatus": "Open [O] (v1003): mechanism must be D4-odd AND supply a ~2e-4 even-odd cancellation; pure seam data excluded [X-typed].  NUSCALE.05/.06 unmoved; no seesaw closure. | EVENING (2026-08-30 v1012): kappa null completes the scaffold bound: mechanism scale strictly beyond frozen finite objects -- 4D functional required.  Display stays [O]. | ROUND3 2026-09-05: finite intertwiner only; projective full-Fock boundary explicit; no neutrino mass texture. Display stays [O]. | ROUND-4 CURRENT: v1028 exactly counts full Gauss dimensions and constructs a D4-symmetric cap only on the selected even pair subspace. The invariant-vector space is2D; the actual symmetric fermion lift forbids all Majorana bilinears. No mass texture; [O] unchanged.",
      "dependencies": "FLAV.NUSCALE.05; FLAV.NUSCALE.06",
      "scripts": [
        "v1003_nu_structural_nulls.py",
        "v1012_toe_gate_witnesses.py",
        "v1024_parent_internal_boundaries.py"
      ],
      "script": "v1003_nu_structural_nulls.py; v1012_toe_gate_witnesses.py; v1024_parent_internal_boundaries.py",
      "location": "tfpt_2_standard_model;tfpt_research_contracts;tfpt_4_frontier",
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    {
      "id": "SEAM.MILNOR.LOCALRING.01",
      "claim": "THE MILNOR LOCAL-RING IDENTIFICATION CONTRACT [O] (registered 2026-08-30, review wave 6, v1004).  REQUIRED: a CANONICAL identification between the raw seam collar / E8 transition bus and the reduced Milnor algebra F2[y,z]/(y^2,z^4) preserving simultaneously the dual-number action, Frobenius pairing, Gray torsor, deck action and CP.  ACCEPTANCE: the identification must be geometric/mechanical (not basis-chosen), must produce the rank clock and socle-CP as OUTPUTS.  KILL: any second inequivalent identification with the same invariants, or failure of mu4-equivariance.  NOTE: would reduce P2 / alignment-bit / Gauss-code / Brieskorn origin to one local ring.  Currently [O] because the Milnor algebra + E8 exponents are CLASSICAL (cite, no claim) and the corpus map is unproven.  Executed finite algebra (v1004): rank clock (2-a)(4-b) sums to h(E8)=30; S=pqr*mu/8; self-clock iff mu=8; (2,3,5) unique among ordered spherical triples; Galois-Gray 11/7/-1; CP = socle/Frobenius; integral D^4 bridge basis-dependent.  PG(3,2)=AG(3,2) sqcup PG(2,2) interpretive [C].  Status: Open [O].",
      "status": "Open (research contract; canonical geometric identification of the raw seam collar with F2[y,z]/(y^2,z^4) unproven; Milnor algebra + exponents classical)",
      "canonicalStatus": "Open [O]: required canonical identification (dual-number + Frobenius + Gray torsor + deck + CP) geometric/mechanical, not basis-chosen; rank clock and socle-CP as outputs.  Kill: second inequivalent identification, or mu4-equivariance failure.  Classical cite vs new-in-corpus [E-finite] vs interpretive [C] typed on v1004.  WAVE-6 2026-08-30. | AFTERNOON (2026-08-30 v1010): SHARPENED -- strong all-structure identification finitely obstructed at F2 (clock order 2 vs 4, CP Jordan rank 2 vs 4) and at all upstairs homes (Z/4: J^2=-I vs Gray P^2!=-I, D8 vs C4xC2; Gaussian k=2 order collapse, k=3 rank/relation clash; affine cannot repair linear parts).  Only the coarse D-module+pairing bridge exists (witness constructed; |Aut(X)|=36864, |Aut(Y)|=8).  Ring-internal identities (rank clock, socle CP, self-clock) of v1004 UNAFFECTED.  Display stays [O].",
      "dependencies": "SEAM.EQUIV.01; AX.P2.01",
      "scripts": [
        "v1004_milnor_frobenius_gray.py",
        "v1010_simplicity_bridge_census.py"
      ],
      "script": "v1004_milnor_frobenius_gray.py; v1010_simplicity_bridge_census.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 769
    },
    {
      "id": "GRAV.WEINBERG.WITTEN.01",
      "claim": "THE WEINBERG-WITTEN FIREWALL [O] (registered 2026-08-30, review wave 6).  BINDING BEFORE ANY spin-2 marker move.  DEMAND: the microscopic Hamiltonian family must NOT possess a fundamental Lorentz-covariant local conserved T_munu; Lorentz symmetry, diffeomorphism Ward identities and the spin-2 mode must emerge in the SAME IR limit.  ACCEPTANCE: positive TT spectral density, exactly two helicities, massless pole, emergent diffeomorphism Ward identity, universal soft coupling, AND an explicit statement of which Weinberg-Witten premise is evaded.  KILL: a microscopic covariant local T_munu together with a massless composite spin-2 claim.  Child of TOE gate T7; cross-link GRAV.SPIN2.EMERGENCE.01 (listed here, not back on SPIN2 -- would cycle).  Status: Open [O].",
      "status": "Open (research contract; binding firewall before any spin-2 marker move; no script)",
      "canonicalStatus": "Open [O]: Weinberg-Witten firewall -- no microscopic Lorentz-covariant local conserved T_munu; Lorentz, diffeomorphism Ward and spin-2 emerge in the SAME IR limit.  Acceptance: positive TT density, two helicities, massless pole, emergent diff Ward, universal soft coupling, named evaded premise.  Kill: microscopic covariant local T_munu + massless composite spin-2.  Child of T7.  WAVE-6 2026-08-30.",
      "dependencies": "GRAV.SPIN2.EMERGENCE.01; GRAV.NONCIRCULAR.01",
      "scripts": [],
      "script": "-",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 770
    },
    {
      "id": "E8.DIRECTED.READOUT.01",
      "claim": "THE SEVEN EXACT E8 READOUT CELLS [E] Identity (round r609, promoted 2026-09-02; module v1018, re-derived, no probe imports).  (1) Seifert/Euler: S+S^T=A_E8, charpoly(-S^{-1}S^T)=Phi_30, rank(S-S^T)=8, orientation-independent charpoly (exponents=totatives of 30; degrees {2,8,12,14,18,20,24,30}, sum(2m_i+1)=248).  (2) Hamming H_8 enumerator 1+14 y^4+y^8, 240=16+14*16, S(3,4,8).  (3) srg(120,56,28,24) eigenvalues 56/8/-4 with multiplicities 1/35/84.  (4) Hecke N(n)=240 sigma_3(n) for n<=10 (theta of E8 = E_4).  (5) Delta: A_P(n)/A_P(1)=tau(n) for n<=8 on three axes, degree-2/4/6 vanishing, E_4^3-E_6^2=1728 Delta.  (6) (Z/30)^x cong C_2 x C_4 (orders {1,2,2,2,4,4,4,4}, not C_2^3); D_chi(s)=L(s,chi)L(s-3,chi) for n<=200.  (7) E6 oplus A2 gluing: index 3, Smith (1^6,3,3), roots 72+6+162, glue classes 78/81/81.  C7 Gauss-code transform NOT promoted: stays OPEN (note_e8_gaussian_code.tex rem:c7-audit).  NOT evidence for or against the Riemann Hypothesis.  NO RH CLAIM.",
      "status": "Proven finite statement [E] (Identity, exact over Z/Q via sympy); C7 Gauss-code transform stays OPEN (rem:c7-audit); NO RH claim",
      "canonicalStatus": "[E] Identity: seven exact E8 readout cells (Seifert/Phi_30, Hamming, srg(120,56,28,24), N(n)=240 sigma_3(n), tau, C2 x C4, E6 oplus A2 glue); C7 OPEN (rem:c7-audit); NO RH claim",
      "dependencies": "v1018",
      "scripts": [
        "v1018_e8_directed_readout.py"
      ],
      "script": "v1018_e8_directed_readout.py",
      "location": "note_e8_gaussian_code;introduction;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 771
    },
    {
      "id": "E8.COXETER.EULER.COMPLETION.01",
      "claim": "THE COXETER-EULER COMPLETION OF E8 [E] Identity + Numerical (round r617, promoted 2026-09-02; module v1019, re-derived, no probe imports).  Exact: C=-S^{-1}S^T, det(I-xC)=Phi_30, Tr C=-1; Moebius Phi_30=(1-x^2)(1-x^3)(1-x^5)(1-x^30)/[(1-x)(1-x^6)(1-x^10)(1-x^15)] (classical cyclotomic Euler-product inversion); U=1 oplus C, det(I-xU)=(1-x)Phi_30=1-x^2-x^3+x^6+x^7-x^9, Tr U=0.  Global Z_C(s)=zeta(2s)zeta(3s)zeta(5s)zeta(30s)/[zeta(s)zeta(6s)zeta(10s)zeta(15s)] and D_E8(s)=zeta(6s)zeta(10s)zeta(15s)/[zeta(2s)zeta(3s)zeta(5s)zeta(30s)], absolutely convergent for Re s>1/2.  Numerical (mpmath, cutoff X=10^5): D_E8 residual 4.76e-4 at s=0.75, 4.16e-12 at s=1.5.  CLASS: the vanishing linear term is the generic consequence of Tr C=-1 (controls: scalar -1 -> 1/zeta(2s); A1 -> 1/zeta(2s); A2 -> 1/zeta(3s); E6 (Phi_3 Phi_12) -> zeta(4s)zeta(6s)/[zeta(2s)zeta(3s)zeta(12s)]; negative control B2 (Phi_4, trace 0) -> zeta(2s)/[zeta(s)zeta(4s)], still contains 1/zeta(s)); E8 only selects the divisor set of 30.  Beurling-generic: the identity holds for any set of primes Q subset (1,inf) -- no prime-specific input.  FENCE (verbatim, no marker upgrade): The trace-free completion is zero-free and pole-free in Re s > 1/2. The splitting into the scalar zeta channel and the Coxeter channel is open and RH-equivalent. No RH claim.  Typed no-go E8.COXETER.REGULARIZED_SPLIT.NO_GO.01: any construction controlling only the det_2/Hilbert-Schmidt part cannot isolate 1/zeta(s); the RH content sits in the analytic continuation of the trace term P(s)=Sum_k mu(k)/k log zeta(k s).  The analytic identities are classical; the new statement is the E8 reading with the fence.  Like the Eisenstein bridge, RH-neutral as a finite identity.  NO RH CLAIM.",
      "status": "Proven finite statement [E] (Identity, exact over Z/Q via sympy) plus Numerical Euler-product checks (mpmath, X=10^5); fence/no-go carry NO marker upgrade; NO RH claim",
      "canonicalStatus": "[E] Identity: det(I-xC)=Phi_30, Tr C=-1, U=1 oplus C, Z_C and D_E8 zeta quotients (classical Moebius); Numerical Euler products at X=10^5 (residuals 4.76e-4 at s=0.75, 4.16e-12 at s=1.5); class statement + Beurling-genericity; fence verbatim (no marker upgrade); NO-GO E8.COXETER.REGULARIZED_SPLIT.NO_GO.01 on det_2; NO RH claim",
      "dependencies": "v1019",
      "scripts": [
        "v1019_coxeter_euler_completion.py"
      ],
      "script": "v1019_coxeter_euler_completion.py",
      "location": "introduction;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 772
    },
    {
      "id": "BULK.PRIME_COMPLETION.01",
      "claim": "THE PRIME-COMPLETION CONTRACT [O] (registered 2026-09-04, v1020): replaces the \"unique from seam data alone\" reading of SEAM.BULK4D.RECON.01/B7 by uniqueness IN THE CATEGORY OF SEAM-GENERATED (PRIME) COMPLETIONS.  DEMAND: (1) an explicit family of seam-derived subalgebras A_i (seam CAR algebra, its time evolution, inclusions/intersections); (2) generatedness A_bulk = vN(v_i A_i); (3) NO-SPECTATOR: trivial relative commutant (v_i A_i)' ^ A_bulk = C.1, equivalently no decoupled tensor factor with trivial boundary map; (4) uniqueness ONLY within this category; (5) stability of (2)/(3) in the 3+1D limit.  KILL: a seam-generated completion with nontrivial relative commutant, or two inequivalent prime completions.  FINITE WITNESS (v1020, free fermions, 2+1D strip): stacking a trivial gapped sector leaves every seam datum unchanged (Chern, edge modes, seam-restricted covariance) and changes the bulk; Krylov generatedness + commutant test separate B from B (+) C by exactly dim C at every momentum; generatedness holds a.e. in p (fails only at p=0 by symmetry).  Dimension typing: hypotheses 2+1D free-fermion, conclusion a re-scoped 4D demand (DIMENSION.UPLIFT.FIREWALL.01 F2).  The 2D modular-invariant uniqueness of the holomorphic (E8)_1 net is untouched and is NOT a 4D no-spectator theorem.",
      "status": "Open (research contract)",
      "canonicalStatus": "Open: the prime-completion contract (uniqueness in the category of seam-generated completions with trivial relative commutant); finite free-fermion witness in v1020 (B vs B (+) C separated by dim C at every momentum, generated a.e. in p); no closure; SEAM.BULK4D.RECON.01/B7 re-scoped, not superseded; DIMENSION.UPLIFT.FIREWALL.01 F2 dimension typing in force",
      "dependencies": "SEAM.BULK4D.RECON.01; DIMENSION.UPLIFT.FIREWALL.01",
      "scripts": [
        "v1020_spectator_sector_obstruction.py"
      ],
      "script": "v1020_spectator_sector_obstruction.py",
      "location": "tfpt_research_contracts",
      "externalData": "-",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 773
    },
    {
      "id": "SEAM.MMST.TELB.C1C2A.01",
      "claim": "TEL-B C1/C2a SUBCLAIM CONTRACT (registered 2026-09-05, v1025; no parent marker move). C1: a native 100-bit Arb/Acb certificate integrates 3888 analytic rectangles, adds a closed axis-strip majorant and the exact imaginary term 1/12, obtaining C_star <= 0.1520453048... < 0.1525. C2a: exact cotangent reindexing, the Peano kernel and shifted Euler summation prove N sup_k |phi_N(k)| < 0.509 for every even N>=16; v1025 outward-checks the final constants. These are subclaims of TEL-B only. C2b CF/DG, ALG-EXH, the charged/type-III scaling limit and TOE T2 remain open. Native python-flint==0.9.0; not Pyodide. NO RH CLAIM. ROUND-4 SUPERSESSION: v1026 closes the former CF/DG residual and relaxed norm; v1025 C1/C2a inputs remain valid. ALG-EXH and MMST remain OPEN.",
      "status": "Proven subclaims [E]: C1 native interval certificate plus analytic axis bound; C2a analytic all-even-N theorem plus outward constant certificate. Parent contracts remain Open [O].",
      "canonicalStatus": "[E] subclaims only: C_star<0.1525 and sup|phi_N|<0.509/N for every even N>=16. C2b, ALG-EXH, MMST and TOE remain [O]. | ROUND-4 SUPERSESSION: v1026 closes the former CF/DG residual and relaxed norm; v1025 C1/C2a inputs remain valid. ALG-EXH and MMST remain OPEN.",
      "dependencies": "SEAM.MMST.TYPEIII.CHARGED.01; TFPT.TOE.COMPLETE.01",
      "scripts": [
        "v1025_telb_c1_c2a_certificates.py"
      ],
      "script": "v1025_telb_c1_c2a_certificates.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 774
    },
    {
      "id": "SEAM.MMST.TELB.ROUND3.01",
      "claim": "TEL-B ROUND-3 MIXED CERTIFICATE (registered 2026-09-05, v1022; no parent marker move). Native 256-bit Arb/Acb over 5812 rational cells gives TV_p(F'') <= 15.817695962988924... < 15.928674 and K3_TV < 2.517464500835. The analytic C2b scalar reduction, with executable Arb arithmetic in v1022, gives c_epsilon=0.288547359238..., c_zeta=0.110677064901..., sum=0.399224424139...<0.400. Full C2b remains open exactly at CF<=0.139/sqrt(N) and DG<=0.060/sqrt(N). The exact one-mode M2 countermodel refutes the prior ALG2 Vandermonde-to-generated-subalgebra implication; FE-GEN or a precise gauge/code projection is required, and ALG1 positivity is conditional on the same generated-sector completeness. Native python-flint==0.9.0; not Pyodide. NO RH CLAIM. ROUND-4 SUPERSESSION: v1026 closes CF/DG and the relaxed all-even-N>=16 norm below2.995906<3; the old full-C2b-open wording is historical. TV and potential inputs remain valid. v1030 gives only a conditional sector/frame interface; microscopic ALG-EXH/MMST remain OPEN.",
      "status": "Mixed: TV interval-certified [E]; C2b scalar reduction is analytic proof plus Arb arithmetic [E]; old ALG2 implication refuted [X]; CF/DG, ALG-EXH, MMST and TOE remain Open [O].",
      "canonicalStatus": "TV residual closed; C1/C2a closed by v1025; C2b reduced to CF and DG; ALG2 inference killed and replacement premise named. No parent marker move. | ROUND-4 SUPERSESSION: v1026 closes CF/DG and the relaxed all-even-N>=16 norm below2.995906<3; the old full-C2b-open wording is historical. TV and potential inputs remain valid. v1030 gives only a conditional sector/frame interface; microscopic ALG-EXH/MMST remain OPEN.",
      "dependencies": "SEAM.MMST.TELB.C1C2A.01; SEAM.MMST.TYPEIII.CHARGED.01; TFPT.TOE.COMPLETE.01",
      "scripts": [
        "v1022_telb_round3_certificates.py"
      ],
      "script": "v1022_telb_round3_certificates.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 775
    },
    {
      "id": "CHIRAL4D.MIRROR.DFILTER.BLOCK.01",
      "claim": "DERIVATIVE-FILTERED WALL BLOCK LEMMA (registered 2026-09-05, v1023; no parent marker move). For H_df=Delta b^dagger b+lambda(b+g D_GW p)^dagger(b+g D_GW p), every singular-value block has determinant lambda g^2 Delta s^2, preserves ker D_GW on the physical summand, has an adiabatically mirror-continuated high branch >=Delta+lambda and a quadratic low branch with coefficient lambda g^2 Delta/(Delta+lambda). Finite gauge covariance and SVD reduction defects are 2.41e-15 and 1.78e-15. The bare mirror variable is not invariant and has low-band weight for s>0: no bare-mirror response gap and no Weyl cone. Canonical DET16/DET32 realization, interacting parent, admissibility/quasilocality, measure/anomaly/index control, universality and 4D continuum remain open.",
      "status": "Proven finite block identity [E]; GW/gauge reading conditional [C]; DET16 [C], NOMIRROR [O], T4 and TOE [O] unchanged.",
      "canonicalStatus": "[E-finite]/[C]: exact 2x2 singular-value block and deterministic covariance witness only. No invariant mirror-response gap, Weyl cone, many-body or 4D theorem.",
      "dependencies": "CHIRAL4D.NOMIRROR.01; CHIRAL4D.MIRROR.DET16.01; TFPT.TOE.COMPLETE.01",
      "scripts": [
        "v1023_derivative_filtered_wall.py"
      ],
      "script": "v1023_derivative_filtered_wall.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 776
    },
    {
      "id": "FLAV.NU.PARENT.INTERTWINER.01",
      "claim": "FINITE PARENT-INTERNAL A3-D4 INTERTWINER (registered 2026-09-05, v1024; no parent marker move). A positive, fermion-even, gauge-compatible, non-additive seam/neutrino-pair term is constructed inside one finite parent, with local gap 0.25 conditional on the inherited parent bound and mixed resolvent 0.104712041885. The selected pair-operator triplet carries ordinary D4, while the full Fock lift is projective: C_f^8=1, C_f^4=(-1)^(N1+N3) and J_f C_f J_f=(-1)^F C_f^-1. It supplies no physical mass scale, no fitted neutrino texture and no 4D continuum mechanism.",
      "status": "Proven finite witness [E/N]; physical neutrino texture and scale remain Open [O].",
      "canonicalStatus": "Finite parent-internal intertwiner only; projective full-Fock boundary explicit. T6 and TOE remain [O].",
      "dependencies": "FLAV.NU.TEXTURE.MECHANISM.01; TFPT.TOE.COMPLETE.01",
      "scripts": [
        "v1024_parent_internal_boundaries.py"
      ],
      "script": "v1024_parent_internal_boundaries.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 777
    },
    {
      "id": "GRAV.PARENT.GAP.NO_GO.01",
      "claim": "GAPPED-PARENT SPIN-2 BOUNDARY (registered 2026-09-05, v1024; no parent marker move). A uniform strictly positive full-parent gap excludes a massless spin-2 pole. The exact transverse-traceless projector has direction-dependent k->0 limits and therefore cannot be a strictly finite-range microscopic projector. A viable T7 route must use a critical physical sequence together with local tensor gauge constraints; the finite five-component symmetric-traceless target is only representation typing and is not a derived stress operator.",
      "status": "Proven finite no-go/boundary [E]; massless spin-2 emergence, Ward identities and universal coupling remain Open [O].",
      "canonicalStatus": "Uniform full gap and direct finite-range TT projection are excluded as T7 mechanisms. Critical gauge-constrained escape remains open; no spin-2 closure.",
      "dependencies": "GRAV.SPIN2.EMERGENCE.01; GRAV.WEINBERG.WITTEN.01; TFPT.TOE.COMPLETE.01",
      "scripts": [
        "v1024_parent_internal_boundaries.py"
      ],
      "script": "v1024_parent_internal_boundaries.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 778
    },
    {
      "id": "FTRANSFER.PARENT.Z5CAP.01",
      "claim": "FINITE PARENT-INTERNAL Z5 CAP AND ENLARGEMENT BOUNDARIES (registered 2026-09-05, v1024; no parent marker move). A 12-dimensional internal cap has gap 0.197423715893 and preserves the finite parent vacuum/gap, but the selector is arbitrary, breaks D4 and is not a global rho_0. On the unprojected parent a full Z5 Weyl pair requires 5|D; Gauss-projected physical dimensions are not counted. Any unital M_n -> M_nd enlargement has relative commutant I tensor M_d, an irreducibility/prime obstruction rather than a general dynamical-coupling no-go.",
      "status": "Finite candidate and exact algebraic boundaries [E/N]; global rho_0, Gauss-projected Weyl realization and unique SK functional remain Open [O].",
      "canonicalStatus": "Internal 12D cap is not a canonical global state; commutant/divisibility statements have explicit scope. T8 and TOE remain [O].",
      "dependencies": "FTRANSFER.SK.RHO0.01; BULK.PRIME_COMPLETION.01; TFPT.TOE.COMPLETE.01",
      "scripts": [
        "v1024_parent_internal_boundaries.py"
      ],
      "script": "v1024_parent_internal_boundaries.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "-",
      "supersedes": "-",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 779
    },
    {
      "id": "SEAM.MMST.TELB.NORM.01",
      "claim": "For the fixed M=1 Ny=8 collar, CF<0.065697, DG<0.042959, C2b<0.466902 and one-sided CROSS<0.282637/N imply Bound-B squared<0.714386. Validated 512-node Acb Fourier core plus the complete all-volume alias envelope gives ||R_N||HS<2.995906<3 for all even N>=16. Historical 2.11 and finite Rsm<=1.7833 are not proved. Microscopic FE-GEN/ALG-EXH and MMST remain open.",
      "status": "Identity (analytic theorem with native interval certificate; fixed regulator)",
      "canonicalStatus": "[E] narrow uniform norm theorem only; all T1-T8 remain [O]",
      "dependencies": "SEAM.MMST.TELB.ROUND3.01; SEAM.MMST.TELB.C1C2A.01",
      "scripts": [
        "v1026_telb_round4_closure.py"
      ],
      "script": "v1026_telb_round4_closure.py",
      "location": "tfpt_research_contracts",
      "externalData": "python-flint 0.9.0 native; no external observation",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 780
    },
    {
      "id": "CHIRAL4D.MIRROR.SIGNED.CAR.01",
      "claim": "Even onsite unitary conjugation constructs canonical DET excitations under the determinant-singlet premise. The additive Delta*N_a replaces the flat Delta*Q. Signed wall det h(s)=M*s*(1+c*s), cL<1 and B<M give the original kernel, linear low branch and high band>=M for fixed classical gauge backgrounds; example M=4 B=3. Bare mirror retains low-band weight; free overlap illustration is a nonchiral four-component Dirac regulator. Quantum-link/Fock lift, chiral SM measure and continuum remain open.",
      "status": "Identity (conditional Hamiltonian-class and exact finite regression)",
      "canonicalStatus": "[C] fixed-classical-background signed-wall theorem; T3/T4 remain [O]",
      "dependencies": "CHIRAL4D.MIRROR.DFILTER.BLOCK.01; CHIRAL4D.MIRROR.DET16.01",
      "scripts": [
        "v1027_signed_det_car_wall.py"
      ],
      "script": "v1027_signed_det_car_wall.py",
      "location": "tfpt_research_contracts",
      "externalData": "none",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 781
    },
    {
      "id": "PARENT.GAUSS.D4.ROUND4.01",
      "claim": "Full 192-mode site exterior character and exact mod5 Frobenius reduction give Gauss dimension3 mod5 for one site and0 mod5 for the open 2x2x2 cube with the specified nine-irrep link truncation; parity sectors4 mod5 at one site. Abstract Weyl divisibility obstruction is not universal after Gauss projection. Symmetric 12D D4 cap has unique selected-sector ground and gap1, but the invariant-vector space has dimension2; no canonical full-state selector. Actual symmetric one-fermion lift forbids all Majorana bilinears.",
      "status": "Identity (exact finite projected counts and conditional symmetry obstructions)",
      "canonicalStatus": "[E] finite counts and selected-subspace cap; T6/T8 remain [O]",
      "dependencies": "FLAV.NU.TEXTURE.MECHANISM.01; FTRANSFER.SK.RHO0.01",
      "scripts": [
        "v1028_gauss_d4_cap.py"
      ],
      "script": "v1028_gauss_d4_cap.py",
      "location": "tfpt_research_contracts",
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    {
      "id": "GRAV.SPIN2.LOCAL.CONSTRAINT.01",
      "claim": "Polynomial first-class constraints for a quadratic tensor target leave exactly two positive linear-dispersion modes at nonzero momentum. Staggered differences give a finite stencil and no extra zero corners. At k=0 all constraints vanish and the trace momentum gives H0=-3r^2/4; positivity requires removing the whole global canonical block. Fixed finite-site exact CCR are impossible; collective low-energy emergence is not excluded. No TFPT embedding or universal nonlinear coupling is proved.",
      "status": "Identity (quadratic target only; global zero-mode prescription required)",
      "canonicalStatus": "[C] radiative target after global zero-mode removal; T7 remains [O]",
      "dependencies": "GRAV.SPIN2.EMERGENCE.01",
      "scripts": [
        "v1029_local_tensor_constraints.py"
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      "script": "v1029_local_tensor_constraints.py",
      "location": "tfpt_research_contracts",
      "externalData": "Linearized GR context only",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 783
    },
    {
      "id": "SEAM.MMST.ALG2.FRAME.01",
      "claim": "Given full neutral block algebras, nonzero charged corners generate the full finite matrix algebra iff their sector graph is connected. A noncollapsing actual-word frame with uniform word norm M gives section norm<=2*M*D^(3/2)/sigma. Compression is not multiplicative and double-cutoff convergence is insufficient: a joint real-linear y/y-adjoint frame and high-output tails for x, x*, y and y* are separately required. These microscopic one-edge FE-GEN premises are not established by Vandermonde inversion.",
      "status": "Identity (conditional finite-algebra and uniform-frame bridge)",
      "canonicalStatus": "[C] conditional certificate interface; ALG-EXH and T2 remain [O]",
      "dependencies": "SEAM.MMST.TYPEIII.CHARGED.01; SEAM.MMST.TELB.ROUND3.01",
      "scripts": [
        "v1030_alg2_sector_frame.py"
      ],
      "script": "v1030_alg2_sector_frame.py",
      "location": "tfpt_research_contracts",
      "externalData": "none",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 784
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    {
      "id": "GRAV.FREE.CURVATURE.FOCK.01",
      "claim": "Declared local tensor target after removal of the entire unstable global zero block: C=r^4 P_TT, positive pure quasi-free Weyl/Fock state with 2(V-1) real oscillators and thermodynamic then distributional continuum limit. Full written analytic proof plus 53 exact and 16 floating checks; finite-dimensional exact CCR excluded. No microscopic TFPT derivation or nonlinear coupling.",
      "status": "Identity in declared zero-mean free target [E]; TFPT identification [O]",
      "canonicalStatus": "[E] free-target theorem only; T7 and TOE remain [O]",
      "dependencies": "GRAV.SPIN2.LOCAL.CONSTRAINT.01; GRAV.SPIN2.EMERGENCE.01",
      "scripts": [
        "v1031_quantum_tensor_curvature.py"
      ],
      "script": "v1031_quantum_tensor_curvature.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "none",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 785
    },
    {
      "id": "GRAV.FREE.CURVATURE.COVARIANT.01",
      "claim": "On the null cone the linearized Riemann sandwich of the de Donder form equals a rank-two positive TT polarization factorization. The polynomial kernel defines a positive-energy Lorentz-covariant local free curvature Wightman field with helicities +/-2 and the same global Fock space as the electric curvature target. Local-net equality and a finite-range lattice realization of every curvature component are not asserted. No common TFPT parent or nonlinear interaction.",
      "status": "Identity in declared free continuum target [E]; TFPT identification [O]",
      "canonicalStatus": "[E] free curvature theorem plus 34 exact checks; T7 and TOE [O]",
      "dependencies": "GRAV.FREE.CURVATURE.FOCK.01; GRAV.SPIN2.EMERGENCE.01",
      "scripts": [
        "v1032_covariant_curvature.py"
      ],
      "script": "v1032_covariant_curvature.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "Free massless-field context; no external observation",
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      "historical": false,
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      "sourceLine": 786
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    {
      "id": "SEAM.MMST.DISORDER.REGISTER.01",
      "claim": "Adding an auxiliary local Z4 flux register constructs order-four charged corners in the common QWZ matter Fock space. Scalar matter gauge transformations alone cannot change seam holonomy. The simple eight-channel quarter twist has h=1/4 rather than h_lambda=1; every norm-two same-phase integer dressing has odd parity. The D5/A3 quotient and cocycle/GSO identification are not supplied. Sharp and smooth string numerics do not prove microscopic FE-GEN or MMST.",
      "status": "Identity in explicitly enlarged model and exact obstruction [E]; finite/scaling regressions [N]; microscopic identification [O]",
      "canonicalStatus": "35 checks: 12 exact; 7 source; 16 numerical. 2026-09-06 numerical amendment: basis-independent zero-space mixture and bounds over every pure filling; finite-grid only. T2 and TOE [O]",
      "dependencies": "SEAM.MMST.TYPEIII.CHARGED.01; SEAM.MMST.ALG2.FRAME.01",
      "scripts": [
        "v1033_charged_disorder.py"
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      "script": "v1033_charged_disorder.py",
      "location": "tfpt_research_contracts;mmst_charged_scaling_limit_en",
      "externalData": "Full repository source inputs; no observational data",
      "supersedes": "",
      "active": true,
      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 787
    },
    {
      "id": "CHIRAL4D.MIRROR.FACTORIZED.BOUND.01",
      "claim": "The actual v1008 factorized onsite model at n=4 J=2 and |d|<=3/5 has uniform gap >=1+sqrt(3)/2. An ordinary hopping bond has zero vacuum expectation but nonzero action and therefore cannot obey the proposed lower form bound by onsite vacuum-complement projectors. This refutes that bound, not the existence of every interacting gap. Newly chosen bond magnitude 3/5 is distinct from the actual sin(k) source coefficient 3/10. Gauged finite Ritz and projected-bond repairs remain conditional changed interactions.",
      "status": "Identity for factorized model and relative-bound obstruction [E]; finite regression [N]; interacting gap [O]",
      "canonicalStatus": "36 checks: 29 exact; 6 source; 1 floating. T3/T4 and TOE [O]",
      "dependencies": "CHIRAL4D.NOMIRROR.01; CHIRAL4D.MIRROR.SIGNED.CAR.01",
      "scripts": [
        "v1034_parent_mirror.py"
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      "script": "v1034_parent_mirror.py",
      "location": "tfpt_research_contracts;tfpt_4_frontier",
      "externalData": "Full repository source inputs; no observational data",
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      "historical": false,
      "sourceHref": "/research/status-ledger.csv",
      "sourceLine": 788
    },
    {
      "id": "GRAV.LINEAR.SOURCE.WARD.01",
      "claim": "For the declared staggered tensor target and prescribed conserved sources use c0=-s^T q and ci=i Bp with the real paired Fourier Hamiltonian. The constraints propagate exactly; the explicit curvature response is retarded. This is not dynamical quantum source consistency: order-g^2 brackets, the removed homogeneous source block and finite-lattice nonlinear scalar Ward residual remain open obligations. A collinear free fermion Ward construction is not full 3D quantum matter or universal nonlinear coupling.",
      "status": "Identity for prescribed-source linear target [E]; finite regressions [N]; dynamical quantum coupling [O]",
      "canonicalStatus": "59 checks: 55 exact; 4 floating. T7 and TOE [O]",
      "dependencies": "GRAV.FREE.CURVATURE.FOCK.01; GRAV.FREE.CURVATURE.COVARIANT.01; GRAV.SPIN2.EMERGENCE.01",
      "scripts": [
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      "script": "v1035_matter_coupling.py",
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  "findings": [
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      "id": "finding:future-equivalence-minimizes-declared-answer-process",
      "titleEN": "Complete answers determine the minimal process quotient",
      "statementEN": "For a fixed experiment alphabet and complete answer functional r, u~v exactly when r(uw)=r(vw) for every allowed continuation w. The quotient is the minimal distinguishable process state relative to those answers; by itself it yields neither complex amplitudes, the Born rule, continuous time, a metric, nor E8.",
      "scopeEN": "Applies exactly relative to the declared experiment alphabet and complete answer functional; no absolute ontology or physical TFPT source claim.",
      "status": "established_scoped",
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      "id": "finding:positive-word-kernel-reconstructs-cyclic-gns",
      "titleEN": "Positive null-stable word kernel reconstructs cyclic representation",
      "statementEN": "Given a positive Gram kernel K with normalized reference, compatible composition/adjunction, and null-stable continuation, quotienting by null vectors and completing constructs H_K and the well-defined word action. Cyclic realizations of the same complete K values are unitarily equivalent on the common word domain; for unbounded operators, closures and self-adjointness remain additional requirements.",
      "scopeEN": "Construction applies only with complete coherent K values and stated domain conditions; K from the original TFPT source has not been derived.",
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    {
      "id": "finding:operational-seed-is-input",
      "titleEN": "The operational seed category takes the central data as an input object",
      "statementEN": "The original definition already assumes (A_loc, tau_t, Theta, omega, [u_Sigma], D_coll); its collar-completion theorem derives B_Sigma and spectral data from them, but does not select those inputs.",
      "scopeEN": "Applies to the cited operational seed category and its collar completion; it does not claim that no other original selector exists.",
      "status": "established_scoped",
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        "premise:typed-source-structures"
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      "attemptIds": [
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      "id": "finding:native-source-common-time-thermal-family",
      "titleEN": "The same source supplies Hamiltonian, time and thermal state",
      "statementEN": "The original H is positive self-adjoint and has a faithful Gibbs state for every prescribed beta. The actual transfer reconstructs Hδ=−log Qδ/δ; removing the temporal regulator makes its unitary dynamics converge to the same H. Spatial lattice, couplings and beta remain declared inputs.",
      "scopeEN": "Finite spatial volume and original neutral scalar/charge source; no spatial continuum or P1/P2 selection claim.",
      "status": "established_scoped",
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    {
      "id": "finding:native-grade-carry-and-memory",
      "titleEN": "Marking and necessary memory are connected within the given source",
      "statementEN": "Original grade and transport generate M4(C(S1)) with integer carry. Full H couples grade and carry. The GNS projection has K(t)=B*exp(itLQ)B with rank K(0)=3; its minimal temporal realization may be infinite-dimensional. No external memory is appended.",
      "scopeEN": "Given source, grade and J>0; the two-time kernel is neither a physical measuring instrument nor a universal three-dimensional memory realization.",
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      "id": "finding:even-seam-global-operator-factorization",
      "titleEN": "A common global seam-operator factorization exists",
      "statementEN": "On the even D8 Fock sector W4=Us Rc holds with the actual ordered CAR/E8 cocycle, Klein signs, all oscillators and current central term; Rc preserves Q,K and the quarter transition. The E8 extension and lift convention are explicitly given.",
      "scopeEN": "Global charge transports in the stipulated even completion; not local vertex insertions, an ordinary-local odd CAR/E8 inclusion or event dynamics.",
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      "titleEN": "Four internal markings completely classified from time and charge",
      "statementEN": "Full lattice saturation reduces the continuous source torus to sixteen four-torsion characters, twelve primitive and exactly four D5+A3 markings. P5 is reconstructed from neutral root moments; a control without M or a target projector finds the same four. W=exp(4πiK)exp(-iπαQ/2) realizes one while preserving quarter time; sheet transport gives W Zq. Compensation was previously known; complete classification and control are added here.",
      "scopeEN": "Given chiral source, fold, full normalized charge and stipulated E8 completion. Algebraic construction and Lean lattice theorem; no physical selection of W or source from P1/P2.",
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        "premise:actual-chiral-opposite-quarter-domains"
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      "titleEN": "A matching E8 marking is not yet an elementary P2 carrier",
      "statementEN": "Each of the four candidates acts on the eight original complex fields with two copies of every quarter phase. Their neutral bilinears form A3+4A1 with twenty roots and rank seven, rather than D5+A3 with fifty-two roots. The latter uses the stipulated spinor completion. The new marking changes neither K nor the known ground-state comparison and supplies no event instrument.",
      "scopeEN": "Exact field-type comparison in the same given source. No universal exclusion of E8 or future source/sector extensions.",
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      "problemId": "problem:common-source-state-event-selection",
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      "id": "finding:minimal-defect-uniqueness-not-proved",
      "titleEN": "A minimal defect value does not yet select a unique source",
      "statementEN": "The original argument first states that uniqueness needs classification of the minimal stratum, then asserts it without that classification. Well-ordering N4 yields a minimal value, not a unique operator fiber. Nonzero spectral flow alone also does not prove attainability of one.",
      "scopeEN": "Audit of the cited proof only; not a general impossibility theorem for P1/P2 or a revocation of declared unit winding.",
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      "id": "finding:conditional-seam-field-state-time-reconstruction",
      "titleEN": "Joint seam, field, sea and time action reconstructed",
      "statementEN": "Under the operator hypotheses, Wold supplies a complete mode basis U^n w and B=N+alpha. With the CAR/sea premise, R c_n R*=c_(n+1), R*QR=Q+1 and R*H_alpha R=H_alpha+Q+alpha. R Omega is a new charge-one state; exp(2pi iH_alpha)=exp(2pi ialpha Q) retains the quarter phase.",
      "scopeEN": "Conditional analytic reconstruction on a common core; Lean verifies the algebraic identity and mode recurrence, not physical derivation of inputs. One channel is not the five-slot P2 carrier; no event law.",
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      "titleEN": "The scalar isometry family is completely classified",
      "statementEN": "Every continuous scalar unimodular winding-one symbol with isometric Hardy compression is a single Blaschke factor modulo phase. With B_alpha fixed, only the centered factor z has constant shift covariance. The strictly oriented control z²/b_(1/4) has leakage 15/4096.",
      "scopeEN": "Complete analytic theorem in the supplied scalar Hardy class. Leakage excludes only the pure-isometry route, not general physical sources. Not a P1 classification.",
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      "titleEN": "Original time energy minimizes at the uniform scalar winding",
      "statementEN": "With B_alpha and the reference sea fixed, every unimodular H1 symbol u=exp(i phi) of winding one has E(R_u Omega)=alpha+½⟨(phi prime−1)²⟩. The unique scalar-multiplier minimum is u=exp(i chi)z. Both b_(1/4) and z²/b_(1/4) have excess 1/15. The produced state changes, not the reference sea.",
      "scopeEN": "Global variational theorem in a given scalar CAR source; no change to B and no finite-search proof. General unitary vacuum symmetries may yield the same minimal state energy. The least-excitation selection rule and origin of fields, state and time are not derived from P1/P2.",
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      "titleEN": "The master functional already presupposes its minimal source",
      "statementEN": "The definition fixes d*_disc and P_adm first. Its barrier is zero for B≅B_min and infinity otherwise. M therefore optimizes continuous xi on the already selected branch; this definition neither classifies B_min nor derives the least charge-one excitation rule.",
      "scopeEN": "Audit of this definition, not a disproof of all other selection theorems. Its electromagnetic alpha is not automatically the quarter-phase parameter in the CAR energy test.",
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      "statementEN": "The original source identifies R_tau=2P_C−I from reflection regularity. A compact product-Dirac counterexample with B=diag(A_1/4,−A_1/4) and local R=σx disproves that general inference: the deck lift on the two fixed boundary faces changes G, whereas 2G−I commutes with G. The principal-symbol comparison confirms the type distinction. R_alg=J_n(2C−I) is a reconstructed, generally nonlocal involution; its algebraic quarter turn does not derive charged quarter holonomy.",
      "scopeEN": "Correction of the general geometric identification in the archived source, not rejection of P1/P2 or all seam realizations. A nonlocal R:=2C−I can be defined but thereby has no demonstrated geometric origin.",
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      "titleEN": "Positive-only compression removes the required negative carrier sector",
      "statementEN": "Pprim=C Pgap implies Ran(Pprim)⊂Ran(C). Thus (2C−I)|H_car=I for a carrier embedded there, and E_−=0. Every isometric pullback W into Ran(C) also yields W*(2C−I)W=I. The later 3+2 polarization does not follow this way. The general lemma CQ=Q ⇒ (2C−I)Q=Q is kernel-checked. A different carrier factor needs an explicit different map.",
      "scopeEN": "Literal source interface of the archived definition; not a refutation of the algebraic 3+2 statement for a supplied P2 carrier. Likewise (2G−I)|Ran(G)=I: full boundary data alone do not create an internal marking.",
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      "titleEN": "Higgs determinant argument does not supply the upstream carrier map",
      "statementEN": "The archived determinant argument uses X=-2P3+3P2 and an explicitly rank-two weak block before the later rank-two conclusion. H0(O(1))=C² alone determines neither rank E+ nor its operator kernel; the same determinant line occurs at different ranks and different h0. The Lean Higgs theorem explicitly assumes its linear identification with binary forms.",
      "scopeEN": "Audit of the specified archived origin chain and elementary bundle controls; no refutation of an actually supplied equivariant bosonic operator, P2 postulate, or conditional Lean theorem.",
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      "titleEN": "Invariant-ring primitiveness alone does not select rank three",
      "statementEN": "For V=C5,W=C2, (V⊗W)⊗Λ4V⊗W→detV⊗detW is a nonzero S(U5×U2)-invariant tensor with even fermion degrees2/4. Centers and U1 charge force positive invariant total degrees(5m,2m); thus the tensor is primitive at carrier degree7. It refutes the local inference that r≥4 must factor in the invariant ring. The stronger Lean premise E≃(Λ²E)* remains valid and needs its own source derivation.",
      "scopeEN": "Excludes only the local invariant-theoretic inference under the stated polynomial interpretation. No P1/P2 countermodel or refutation of additionally justified global minimality or actual branch-tensor support; no new Lean build.",
      "status": "excluded_scoped",
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      "topicIds": [
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        "goal:T2",
        "goal:T8"
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        "premise:typed-source-structures"
      ],
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      "id": "finding:fixed-background-source-does-not-fix-mixed-vertex",
      "titleEN": "Complete fixed-background fermion source does not fix the Higgs derivative",
      "statementEN": "The existing direct Grassmann path yields the same complete unnormalized source polynomial and Z0=69649/65536 for both supplied extensions at Φ=0. The mixed derivative is zero or offdiagonal -65536/69649; -K0(∂Φ K^-1)K0 reconstructs the supplied derivative K′. A joint source derivation therefore still needs the actual original field dependence and correctly equivariantly typed coupling.",
      "scopeEN": "Exact zero-jet identifiability test on the given free source path. No no-go for a complete Φ-dependent functional or later original selector; no general interacting Yukawa theorem.",
      "status": "established_scoped",
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        "topic:quantum",
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        "goal:T2",
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        "premise:fixed-background-quarter-fermion-source",
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      "id": "finding:local-quarter-inner-variation-is-connection",
      "titleEN": "Local quarter-source fluctuation yields connections, not a new carrier",
      "statementEN": "From [Bα,Mb]=−iMb′ and U*[Bα,U]=I one obtains exactly Ω¹_B(C∞(S¹))={Mv}. Selfadjoint perturbations are real smooth v. With mean vbar, Bα,v=e^(−iλ)Bα+vbar e^(iλ), λ′=v−vbar; the same quarter holonomy requires vbar∈Z. Fluctuation preserves the represented algebra and line module and thus creates no new SU2 fundamental or3+2 selection in this realization.",
      "scopeEN": "Complete one-form/one-particle holonomy theorem only for this explicit coordinate algebra. No P1/P2 no-go, identification with the full physical net, fixed Fock-vacuum equivalence or anomaly cancellation. Real completion depends on actual J and sign.",
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        "topic:quantum",
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        "goal:T8"
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        "premise:typed-source-structures"
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      "id": "finding:calderon-enlargement-does-not-select-higgs",
      "titleEN": "Cauchy enlargement supplies arbitrary finite matrix directions, not a selector",
      "statementEN": "P0=C−UCU* and Emn=U^m P0 U^(−n) supply all finite matrix units. The represented generated algebra has scalar commutant and center. Since I is a one-form, every element a=aU*[B,U] is a one-form. The enlargement permits arbitrary finite-rank matrix perturbations but selects neither a Higgs coupling nor3+2 and cannot silently replace the original local algebra with smooth center.",
      "scopeEN": "Concrete algebraic/smooth enlargement, not the entire bounded-operator algebra. Does not exclude an independently derived noncommutative local field algebra.",
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      "id": "finding:finite-sm-higgs-variation-retains-seed",
      "titleEN": "Finite SM Higgs fluctuation retains the supplied Yukawa seed",
      "statementEN": "For M=diag(Yup,Ydown), every particle LR one-form obeys q_a(Mr_b−q_bM)=q_a(r_b−q_b)M. The factor is quaternionic and b=eR spans all qM. Including the seed yields M(H)=HM with H=[[φ1,φ2],[-barφ2,barφ1]]. The four Yukawa matrices and color/weak modules remain inputs; D_F=0 gives no finite fluctuation. The linear identity was checked exactly against unchanged existing code with a complete coefficient basis, without choosing physical couplings.",
      "scopeEN": "Exact conditional classification of the particle LR block of the given representation, not a complete real-triple classification or P1/P2-derived coupling. Already encoded3+2 structure is not newly selected.",
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      "id": "finding:operator-skeleton-does-not-identify-insertion-algebra",
      "titleEN": "Full operator skeleton does not select local insertion algebra",
      "statementEN": "On the same product skeleton A0=C∞(M)I and A1=C∞(M,Mn) acting on the left differ, with the same scalar center, identical D,J,Γ,B,R and fixed relative spectral/eta responses. Both obey order zero/one: JL_(b*)J^-1=R_b, [[D,L_a],R_b]=0. A1 mixes internal left indices, A0 does not. For n=2: fiber-span dimensions1/4, commutant dimensions16/4 and16 exactly vanishing left/right basis commutators. These operator/center/order conditions therefore do not reconstruct the multiplication algebra.",
      "scopeEN": "Excludes only unique algebra reconstruction from these data in the declared product/domain setting. Not identical complete original tuples, full OS/P1/P2/minimality countermodels or a universal TFPT no-go. Additional algebra hypotheses need their own origin.",
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      "id": "finding:internal-even-car-hodge-is-not-coordinate-algebra",
      "titleEN": "Internal Hodge structure exists, full even CAR is not the coordinate algebra",
      "statementEN": "For the given E=C5, J_H²=1, J_HΓ=-ΓJ_H and (ΓJ_H)²=-1 hold on ΛE. Internal CAR is End(ΛE), its even part End(S+)⊕End(S-). J_H swaps both blocks and maps this full algebra onto itself; order zero would require commutativity. An explicit matrix-unit commutator has rank2. The full internal algebra is thus not a spectral coordinate algebra with this J_H. A selected left/right bimodule subalgebra remains missing. Reuse the existing SU5 no-selector: unmarked exterior algebra and trace balance select no3+2 projector.",
      "scopeEN": "Excludes only full even internal CAR as coordinate algebra with J_H; positive Hodge signs conditional on rank/metric/volume. No physical CAR, J, KO6, spacetime or E=C5 origin derivation. Proper compatible subalgebras and originally derived symmetry breaking remain possible. Whole-space antiunitarity is not automatically the quaternionic W factor of a different triple.",
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      "id": "finding:minimum-rank-unique-primitive-channel-does-not-select-32",
      "titleEN": "Minimum rank and unique primitive channel do not select3+2",
      "statementEN": "Rank5 is the first possible rank in the stated class: even ranks fail total parity, and at rank3 total degree5 is not a multiple of3. The same canonical volume tensor supplies nonzero primitive trilinears on E=C5 for1+4,2+3,3+2,4+1. Minimal covariant degree5 channel spaces modulo fermion exchange have dimensions1,2,1,1. Thus even unique primitive channel leaves1+4,3+2,4+1; with b>s,3+2 and4+1 remain. All four16-dimensional packets have Tr X=Tr X³=0 for X=(-s)I_V+bI_W.2652 integer group-action coefficients were checked exactly; no suitable coupling was fitted.",
      "scopeEN": "Excludes the finite rank/primitivity/channel-uniqueness implication in the stated covariant exterior class, stronger than the earlier nonminimal5+2 test. Does not establish multiple global original minimizers, equal S1 defect vectors, all anomalies or complete P1/P2 worlds. An original bosonic index and actual sector/vertex map could distinguish them but are not derived here.",
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      "titleEN": "Joint self-consistency presupposes minimal algebra selection",
      "statementEN": "Original minimization does not classify the minimal algebra/bimodule fiber. The joint discrete theorem imports Higgs rank2 and branch-Yukawa rank3; the master functional fixes d_disc*,P_adm and B_min before varying only ξ=(α,χ_geo,δ_ph,ρ_vac). The singleton fiber product presupposes discrete uniqueness with E3⊕E2,Y in the shared invariant tuple. Two exact conditional decoders exist: independent occupancy balance at g5 gives {b,s}={2,3}; independent Bγ=g/(g−1) with Y roots−1/b,+1/s gives s²=g−1, hence s2,b3. Their cited original proofs evaluate these equalities only on the fixed3+2 branch. Using them as upstream selectors adds a premise.",
      "scopeEN": "Source and logical dependency audit of the cited chain, not a universal impossibility theorem. An independent original derivation of the cross-identification or unique minimal algebra/bimodule classification could close the gap. E8 glue rank selection and fixed state for given transport are retained and not equated with local algebra selection.",
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      "titleEN": "Composition and a cyclic faithful state do not yet select the algebra",
      "statementEN": "The existing operational context-quotient/GNS result remains conditional on the original operation alphabet. Classical and quantum models obey general composition, causality and local tomography. C4 and L(M2) additionally share HS(M2), J and Omega, cyclic/separating faithfulness and order zero, yet only the latter is noncommutative; their centers differ. A coherent two-channel operation separates a restricted population protocol (double pulse:1 versus1/2), but source availability must first be derived.",
      "scopeEN": "Only excludes these general composition/cyclicity conditions as sufficient algebra selectors. The new pair is not a same-center counterexample or full physical TFPT countermodel. Additional principles such as purification remain conditional, not P1-derived.",
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      "titleEN": "Positive quartic shape conditionally selects3+2; boundedness alone does not fix its sign",
      "statementEN": "For every traceless Hermitian rank-five endomorphism with trX²=1, globally7/30≤trX⁴≤13/20. Exhaustive stationary-spectrum classification gives equality below only at2+3 and above at1+4. The even conjugation-invariant bounded-below controls -t+t²±trX⁴ therefore favor different splits, without an inserted preferred direction. The primitive spectral functional differentiates a supplied deformation; a positive profile does not imply a positive fourth-order shape term. Original X, trace constraint, deformation and sign remain to be derived.",
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      "id": "finding:native-symmetric-quartic-erasure-common-object",
      "titleEN": "One native representation forces the actual quartic code and erasure action",
      "statementEN": "P64 is characteristic as O2(G31). Fix_P64(Sym^4 V4) is exactly the actual five-code; the original codeword sums give its real intertwiner rather than a fitted embedding. The existing erasure intertwiner forces rho20(g)=rho5(g) tensor bar(rho4(g)) for the same words. The full fourth tensor power has Pauli fixed rank16, whereas the symmetric fourth power has rank5.",
      "scopeEN": "Shared mathematical cause within the given core. The physical symmetric rule, origin of the core data, and an E8-current/fermion identification are not derived.",
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      "titleEN": "Joint block algebra, normalizer and intrinsic2+3 orbit",
      "statementEN": "The actual4-,5-,20-representations are distinct irreducibles. Their group algebra is M4+M5+M20 with commutant C^3 and group normalizer U(1)^3 rho(G31). The algebra quotient preserves these word representations but is not tensor-closed by itself. The15 original matching involutions yield a canonical transitive family of rank2 projectors with rank3 complements. Joint group-invariant states retain free positive block weights. The decisive exact comparison nevertheless gives P2 != Q_t for ALL15 projectors of the native response orbit: color-S3 commutation excludes this identification structurally.",
      "scopeEN": "No unique physical3+2 marking, no spacetime dimension29 and no physical5/4 normalization derived from dimension ratios. The associative-algebra normalizer is larger than the group normalizer. The existing physical mark is therefore not simply an as-yet-unselected member of those15 responses.",
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      "titleEN": "Actual native-four Galois clock and one marked representation orbit",
      "statementEN": "For each of the six remaining F20 representatives actual native C,G satisfy C^5=G^4=I and GCG^-1=C^2. Simultaneous intertwiners to the original v419 carrier have solution dimension1 and rank4. The six representatives form ONE orbit of the existing logical stabilizer, already S2×C3 acts regularly. R5 is not a native group word by an exact centralizer test; the compiler clock instead maps to an order3 operator on the quartic code. The canonical cyclotomic trace metric is carried isometrically into the native four-metric by normalized intertwiners, leaving an overall phase.",
      "scopeEN": "Explicit operator identification and one logical equivalence class, not six independent physical options. A common native-four/P1 clock continuation and the twenty-lift of a specified coherent R polynomial are not thereby derived.",
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      "titleEN": "Complete four/five generators determine the60 native rates",
      "statementEN": "The native rate map has rank55 on the four and15 on the five, jointly rank60 with60 columns. Its joint nullity is0: fixing complete generators excludes a hidden twenty-rate twin inside the60-reflection Poisson menu. The old coherent S3 sign twin belongs to a different event menu and additionally violates the actual branchwise erasure contract, with trace distance1/4 for an explicit input.",
      "scopeEN": "Unique continuation within exactly this menu, not selection of that menu, complete generators or their physical time scale from P1. A few populations or a single finite-time channel are weaker inputs.",
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      "titleEN": "Complete logical nondisturbance classifies the covariant residual channel",
      "statementEN": "CPTP and Tr4 Phi(X)=Tr4 X for all joint X force Phi=id5 tensor D by the multiplicative domain; factorization is not a fitted ansatz. Native G31 covariance forces D=Dq with -1/15<=q<=1. Continuous semigroups have q(t)=exp(-gamma t), gamma>=0. The original central J-character projector and Paulis realize this in the coherent block-algebra menu with common normalized branchwise erasure records; the gauge correlator is q(t).",
      "scopeEN": "Classification of this explicit residual class, not all sources. No original physical permission for the coherent Kraus events or selection of gamma.16-record equivariance is sectorwise/on the block algebra, not on extra End(H29) coherences. G31 covariance does not prove covariance under arbitrary coherent clock polynomials.",
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      "titleEN": "Actual polar mark reconstructed from simplex and clock",
      "statementEN": "In consistent original N coordinates P2(q)=F_sigma-c_q c_q*, c_q=[n2+n3+n4-3(sqrt(6)-1)n_q/5]/sqrt(3). For q=5 this is the actual existing mark without SM charges. All three clock-fixed points form one mark orbit under actual native reflection lifts commuting with sigma4. N=O C^T, O=-rho_C(reverse), transports all labels together.",
      "scopeEN": "Positive polar rule and physical interpretation remain inputs. One mark orbit does not prove every anisotropic source invariant or select a physical state. The mark remains distinct from all15 matching-response projectors.",
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      "titleEN": "All product-unitary lifts of the quarter clock excluded",
      "statementEN": "All six actual code-projector pair marginals are (I+Swap)/4. Every code-preserving U1 tensor U2 tensor U3 tensor U4 therefore has projectively equal factors. The identical local stabilizer is K=U(1)G31 by leakage Gram32I15, a finite determinant slice and NRS6.5. R5 lies projectively outside logical S6, excluding every such product lift. Rank-one Choi also excludes convex native-word-channel mixtures reproducing Ad_R5 on all inputs.",
      "scopeEN": "Classification of code-preserving product unitaries and stated classical mixtures. Collective operations, ancillas, general LOCC, postselection and protocols leaving the code temporarily are not excluded.",
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      "id": "finding:original-decoder-versus-native-quarter-polynomial-erasure-test",
      "titleEN": "Decoder and same native clock polynomial act differently",
      "statementEN": "The existing three-register decoder derives encoded R5 tensor I4. Substituting actual sigma20 and all four native tau20 into the same R polynomial with original decoder marking gives unitarity and the correct square but fails common erasure action. The pure input rho=(I-F_sigma5)(I+s23)/2 yields output-difference eigenvalues +1/4,+1/4,-1/4,-1/4 and16 zeros: trace distance1/2 for all four lifts.",
      "scopeEN": "This excludes exactly the stated native polynomial continuation with the fixed decoder P220, not every coherent common-clock lift. The logical decoder does not prove original physical permission for the collective intervention.",
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      "id": "finding:original-decoder-product-time-and-all-unitary-erasure-lifts",
      "titleEN": "Decoder forces product and temporal composition; all unitary erasure lifts classified",
      "statementEN": "The actual decoder D(A)=W(A tensor I4)W* is an injective corner *-homomorphism. Thus D(A)D(B)=D(AB), product preservation of unitary complement extensions and exp(-itD(H))=D(exp(-itH))+I-Pi follow without a new composition rule for given H. Every unitary lift with full branchwise erasure compatibility is necessarily and sufficiently R tensor V4, forced by the commutant rather than fitted. On C64 an arbitrary unitary44-complement block remains.",
      "scopeEN": "Given original code/decoder and logical operation. No selection of H, logarithm branch, physical time normalization or actual intervention permission. Unitary class, not all CPTP instruments. Native group action differs from the decoder by an explicit gauge factor whose physical compensation remains open.",
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      "titleEN": "Clock-compatible marking fibre and canonical polar distance",
      "statementEN": "The entire stated native mark class is P=F-cc*, [c] in CP1(E+), or RP1 for real projectors. Every point gives the same quarter spectrum and square; q families are natively S3fix-equivariant. The native q/sigma stabilizer cannot remove the parameter. Original cpol=-sqrt(3/5)e0+sqrt(2/5)e1 and c=e0 differ exactly: HS squared4/5, marked-ray responses2/5 versus0. Uq is also the uniquely nearest isometry to actual orthonormalized Dq, with Frobenius cost7-2sqrt6.",
      "scopeEN": "Completeness only for the stated operator-compatibility class. Countermarks are not full P1 sources and need not obey the original polar rule. Unique distance minimization does not prove a physical law minimizing that distance or matrix positivity from reflection positivity.",
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      "titleEN": "Moment parent and local code holonomy do not select the quarter clock",
      "statementEN": "The existing Hnat obeys Hnat V=0, hence exp(-itHnat)V=V for every t; every marking direction has equal ground energy. Alg(Hnat|Code,sigma,tau) is scalar on E+. Reusing the existing Berry theorem and one-register erasure gives V*X_jV=tr(X)I5/4 at each site. All local-product code paths have scalar connection; for a closed code path the previously classified full product stabilizer forces phase-S6 holonomy, outside which R5 lies.",
      "scopeEN": "Excludes the specified moment parent, its code algebra with sigma/tau and this geometric product-path parallel transport. Not a general exclusion of combined local drives plus parent, collective code paths, extra dynamics, ancillas or every physical adiabatic protocol.",
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      "titleEN": "Original E8 current response and full quarter lifts",
      "statementEN": "Actual mixed Chevalley brackets give Q(H)|F20=H tensor I4, Q(H)|C4=trH/3 and kappa=trHK-trH trK/9. Noncommuting E+ responses exist. Separated Ward responses distinguish original polar and countermarks. Clock-current norm25/18 is constant throughout CP1 and selects no point. Three gl5 lifts of given R5 have full orders12,4,12; a genuine order4 source clock exists. All60 actual finite event charts match the native erasure twenty-action on the coupled determinant cover.",
      "scopeEN": "Given original E8_1 vacuum source and coupled finite lift; no P1 source selection, physical time, instrument, polar minimization or physical field identification.",
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      "titleEN": "Charged four-current fails the common tensor-register dictionary",
      "statementEN": "Under literal C4/register identification all three source clock branches act as -I20 on three C currents but as R5 tensor I4 on F20. The actual W intertwiner has squared HS residual48. Every traceless Q(H) already acts0 on C and its products but nontrivially on F. Thus finite cover agreement is not a full field-algebra map.",
      "scopeEN": "Excludes this literal charged four-current identification with unchanged tensor intertwiner, not the actual Cartan four-space or every derived composite field map.",
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      "titleEN": "Full inner E8 stabilizer of native Cartan code excludes marked quarter",
      "statementEN": "The original lattice vertex law gives root PBW component f(b)alpha(-3)ealpha/3. The actual five quartics have no common projective zero; all240 root zero modes therefore leak from the neutral code. Distinct charge sectors prevent cancellation. The stabilizer has Lie algebra Cartan8 and H0=T8; its components normalize T8 and preserve the actual H+ four-plane. The full native U4 code-stabilizer theorem restricts its logical action to phase-G31, outside which R5 lies. Arbitrary inner-group sequences returning to code are included in the exclusion. L0 acts as4I5; conformal time and Casimir shift do not enlarge the logical action.",
      "scopeEN": "Actual J-positive canonical Cartan Fock embedding in given lattice vacuum and compact inner E8 zero-mode operations. Not an exclusion of higher modes, nonlinear mode products, other derived field maps, non-automorphism interventions or all P1 physics; no physical preparation or selection of the embedding proved.",
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      "titleEN": "Neutral weight-two source action on actual code completely determined",
      "statementEN": "The full lattice-neutral weight-two space is8 derivatives plus36 symmetric quadratic fields. On original quartic code o(T_A)=dGamma(A), compression=(trA)/2 I5. Exact leakage sum is4||Ac-trA I8/8||²+(5/2)||Aa||²; real Gram relative to matrix HS metric has spectrum0^1,4^15,(5/2)^20 and kernel only A=aI. Original R_neut compresses0 with universal leakage bounds240..384, B48 compresses48I and B64 acts64I. Any sequence solely of these neutral zero modes acts by U8 on mode1; a code-preserving endpoint forces the U4 stabilizer and logical phase-G31 action, excluding marked R5.",
      "scopeEN": "Complete exclusion of this neutral local weight-two zero-mode class including temporary code exits. Do not unite with separate inner E8 exclusion: mixed root/quadratic histories, individual mode roundtrips and higher fields remain outside. R_neut itself is not zero as a field.",
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      "id": "finding:original-neutral-root-mixed-response-derived",
      "titleEN": "First mixed response of original R and root currents derived",
      "statementEN": "Original vertex formula gives E_beta,0 T_AR=(v_beta.h(-1))e_beta with v_beta=-AR beta+(beta^T AR beta)beta/2 and beta.v_beta=0. All240 original Chevalley roots have exact norm distribution0:46,14:96,30:32,32:60,64:6, hence194 nonzero source responses without fitted coefficients. Their weight-two zero mode on quartic f has PBW component [f(b)v_beta(-3)-(D_c f)(b)beta(-3)/3]e_beta. Basepoint freedom and transversality prove leakage of each individual nonzero direction; compression vanishes by lattice charge. Four original family omissions have R-sum0, without deriving a physical averaging rule.",
      "scopeEN": "Actual first algebraic source variation in given E8_1 vacuum;194 complex root responses are not particle counts or selected events. Leakage existence is orthogonally covariant; numerical code matrices require actual source isometry. Second responses, joint return histories, physical history, P1 source and marking selection remain undetermined.",
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      "titleEN": "Actual ordered mixed return on native code",
      "statementEN": "With existing isometry O and full vertex-mode action, all194 nonzero original W_beta give positive definite nonscalar rank5 returns on actual code. Unchanged sum over original root-current input basis is diag(3581,4191,3419,3419,3790), identical for all4 original family omissions. No intermediate projection or fitted coupling is inserted.",
      "scopeEN": "Existing E8_1 vacuum, original code embedding and given R_neut branch. Return amplitudes are not selected physical channels.",
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      "historical": false,
      "publicationNotes": [],
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    },
    {
      "id": "finding:mixed-return-spectrum-does-not-select-original-polar-mark",
      "titleEN": "Mixed return spectrum does not select original mark",
      "statementEN": "None of194 nonzero individual returns commutes with original P2. Sum spectrum is3581,4191,3419²,3790 and commutes with tau but not sigma or P2. Exact witnesses [Ksum,sigma]12=772 and [Ksum,P2]01=(488sqrt2-61sqrt3)/5. Its spectral functions therefore do not give original P2; all4 supplied family variants leave sum unchanged.",
      "scopeEN": "Excludes direct or spectral identification of these concrete kernels with existing mark, not every joint source history or mixed operator algebra.",
      "status": "excluded_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:clocks"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-neutral-root-pair-moment-fields",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-positive-adjoint-two-insertion-response-profiles",
        "premise:existing-logical-clock-and-charge-marking"
      ],
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      ],
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      "id": "finding:actual-mixed-return-fails-native-trimer-erasure-lift",
      "titleEN": "Actual return fails original trimer erasure lift",
      "statementEN": "Original W20 has Fock columns g_(i,a)=partial_a f_i, all20 orthonormal. Same source channel for doubled root(0,0,0,0,0,0,-2,-2) has K5 rank5, actual trimer K20 rank13 and operator-Schmidt rank4. HS² residual from decoder lift is124096/3. It kills g00=x0³/sqrt6 while decoder lift gives188/3 there. This positive unrenormalized paired response is not a common complete branchwise erasure lift.",
      "scopeEN": "Actual neutral Sym3 trimer, not charged E8 F20 block. Specific response branch, not classification of all combined histories or instruments.",
      "status": "excluded_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
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        "goal:T2",
        "goal:T8"
      ],
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-neutral-root-pair-moment-fields",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-positive-adjoint-two-insertion-response-profiles",
        "premise:common-unnormalized-positive-return-erasure-test"
      ],
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      ],
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        {
          "id": "source:f10c047561901ee024f7809f",
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      "id": "finding:strict-trace-gap-excludes-positive-mixed-return-reweighting",
      "titleEN": "Strict trace gap excludes positive return reweighting",
      "statementEN": "For every original root and real transverse v,4trK5-trK20=(43/12)||v_parallel_Jbeta||²+(25/4)||v_orth||²>0 for v!=0. Full7x7 trace forms and actual native240-root transitivity prove the theorem; all240 computed code traces agree. Original aggregates:trK5=18400,trK20=46848,gap26752. Charge orthogonality gives Kd(a)=sum|a_beta|² Kd,beta for every coherent profile of original W_beta. Thus no nontrivial positive reweighting repairs common unrenormalized erasure continuation.",
      "scopeEN": "Common original normalization, positive adjoint two-insertion returns with original real v_beta. Nonadjoint return fields, new complex transverse fields, independent relative20/5 calibration, longer histories and general physical channels are outside.",
      "status": "excluded_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
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        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-neutral-root-pair-moment-fields",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-positive-adjoint-two-insertion-response-profiles",
        "premise:common-unnormalized-positive-return-erasure-test"
      ],
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          "id": "source:bfcc2a4f63b2b13ce7645d61",
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    },
    {
      "id": "finding:original-source-erasure-ward-terms-derived",
      "titleEN": "Original source-erasure Ward terms derived",
      "statementEN": "For actual erasure a_a=bar(u_a)(1) and original W_beta=(v.h(-1))e_beta, exactly [a_a,W_beta,1]=bar(c_a)E_beta,1+bar(b_a)W_beta,2 with b_a=u_a.beta,c_a=u_a.v. All20 identities on5 original code states and4 erasures agree exactly. Original witness b2=-i,c3=4-4i has zero a0/a1 terms; a2 gives iW2 and a3 gives(4+4i)E1. No coupling was chosen.",
      "scopeEN": "Algebraic Ward identity in given E8_1 source with original Fock/erasure map. No physical contact, measurement, reset or P1 time law; physical source selection remains open.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
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        "goal:T2",
        "goal:T8"
      ],
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-neutral-root-pair-moment-fields",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-positive-adjoint-two-insertion-response-profiles"
      ],
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      "id": "finding:uncompressed-original-mixed-return-does-not-preserve-code",
      "titleEN": "Full original mixed return does not preserve code",
      "statementEN": "Earlier positive rank5 matrix is Pi L Pi, not full operation L=S_beta* S_beta. Original doubled root(0,0,0,0,0,0,-2,-2) has S_beta h=0 for h=x0^4/sqrt24. Normalized code-orthogonal q=(2/sqrt3)(h-f0/2) gives <q,L f0>=-188/(3sqrt3) and ||(I-Pi)L f0||²>=35344/27. Thus full operation is not code preserving. a0 commutes with S_beta and its adjoint; no omitted Ward term repairs this erasure branch.",
      "scopeEN": "Specific full adjoint two-insertion return in original code embedding. Rank5 Gram stays correct; no global E8/P1 or all-history impossibility claim.",
      "status": "excluded_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
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        "goal:T2",
        "goal:T8"
      ],
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        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-neutral-root-pair-moment-fields",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-positive-adjoint-two-insertion-response-profiles"
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      "id": "finding:original-root-response-sum-also-leaks-from-code",
      "titleEN": "Original root-response sum also leaks from code",
      "statementEN": "For unchanged original unrenormalized root-input basis, Lsum=sum_beta S_beta* S_beta and <q,Lsum f0>=-2092/sqrt3 with leakage norm²>=4376464/3. All4 original family omissions give same exact witness. Independent targeted original-data check confirms35736-343776/4=-50208 for unnormalized witness. Full coherent S* S sum has additional charged cross terms, which vanish exactly in neutral q/f0 matrix element. Thus coherent profiles with |a_beta|=1 also fail code closure regardless of phases.",
      "scopeEN": "Original sum and coherent profiles with unchanged moduli; no physically selected execution or code-closure theorem for arbitrary new weights. Prior complete positive-profile erasure trace gap remains separately valid. Other higher source histories are not excluded.",
      "status": "excluded_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
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        "goal:T2",
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      "id": "finding:original-full-source-return-exterior-feedback-derived",
      "titleEN": "Full original source exterior feedback derived",
      "statementEN": "Original vertex formula gives complete192x726 single-beta S from neutral grade4 to charged grade4, with original Fock adjoint L=S* S. K=Pi L Pi,B=Q L Pi,D=Q L Q give exactly x_(n+1)=Kx_n+sum_(j=0)^(n-1)B*D^(n-1-j)Bx_j for code input. No projection is executed. Actual four-insertion response H=Pi L² Pi equals K²+B*B, not K². Example H00=81152/3 versus(K²)00=40144/9. Next memory term B*DB and Pi L³ Pi are also checked exactly from full source; all H entries independently reproduced in active Laurent/Heisenberg blocks.",
      "scopeEN": "Fixed specified algebraic original single-pair response family. Given source functional is evaluable, but P1 physical event/time selection remains open. Neutral726 space is not complete for coherent multiroot cross terms.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
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      ],
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        "premise:original-neutral-root-pair-moment-fields",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-positive-adjoint-two-insertion-response-profiles",
        "premise:specified-original-pair-ordered-response-composition"
      ],
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      "id": "finding:original-return-requires-memory-in-every-code-direction",
      "titleEN": "Original return requires memory in every code direction",
      "statementEN": "Complete exact feedback Gram Delta=B*B satisfies Delta>=5120I5 and has rank5. Every nonzero code input obeys sharp ||Pi Lpsi||²/||Lpsi||²<=9/44, equality exactly span(f2,f3); at least35/44 of squared response norm is exterior. Hankel/Gram block[[I,K],[K,H]] has exact rank10. Hence any fixed linear realization M_n=C A^n E for n>=0 with time-independent input/output maps needs at least10state dimensions. Closed5D continuation of these moments is excluded even after fixed input/output changes.",
      "scopeEN": "Same specified single-beta response family only. Norm fraction is not an instrument probability. Rank lower bound10 is not sufficiency of a10D realization, particle count, spacetime or3+2 derivation. No separate n=0 feedthrough, time-dependent maps or nonlinear models in excluded class; no wholesale transfer to other source histories.",
      "status": "excluded_scoped",
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      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
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        "premise:original-positive-adjoint-two-insertion-response-profiles",
        "premise:specified-original-pair-ordered-response-composition"
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      "id": "finding:native-conformal-primary-multiplicity-source-factorization",
      "titleEN": "Common conformal primary/multiplicity structure of native core",
      "statementEN": "Original J-positive mode1 polynomials are all Virasoro primary because u_a.u_b=0: L0=d,Lm>0=0. Actual4/5/20 carriers thus have(c,h,r)=(8,1,4),(8,4,5),(8,3,20). For all Virasoro descendant words Gram=G_(8,h)(lambda,mu)delta_ij. Generated source cycles factor as common unitary highest-weight representation modulo common nulls tensor actual multiplicity; native G31 acts on second factor. No R_neut branch, individual root or family omission is needed.",
      "scopeEN": "Given canonical E8_1 stress field and original neutral Cartan-Fock embedding. Conformal weights are not particle counts or physical time; no new irreducibility/gauge-group premise. Analytic all-word theorem; finite Fock checks only accompany it.",
      "status": "established_scoped",
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      "topicIds": [
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        "topic:quantum",
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      ],
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      ],
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        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:canonical-source-stress-all-mode-history"
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      "id": "finding:canonical-conformal-erasure-forces-primary-twenty-carry",
      "titleEN": "Canonical conformal erasure Ward action forces primary twenty carry",
      "statementEN": "From [L_m,a_a]=-bar u_a(m+1), a_a L_-2 f_i=L_-2g_ia+t_ia with t_ia=bar u_a(-1)f_i. Original h3,c8 level2 Gram[[16,18],[18,84]] and overlap(1,0) uniquely give r_ia=t_ia-(7/85)L_-2g_ia+(3/170)L_-1²g_ia. These20 neutral weight5 primary states have Gram(78/85)I20 and exact original rho5 tensor bar rho4 action. Three existing native reflection lifts give60 literal intertwining checks. Full first Ward output Gram is19I20; code partial return is76I5 or312/85 I5 for primary remainder. No coupling or fitted field identification is chosen.",
      "scopeEN": "Unique orthogonal primary remainder relative to specified original erasure descendant block, not classification of all weight5 primaries. Mathematical decomposition, not executed projection. Neutral native twenty, not charged E8 F20 block or physical fermion/Higgs identification. First full/primary return Gram selects no3+2 mark.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
      "goalIds": [
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        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:canonical-source-stress-all-mode-history"
      ],
      "attemptIds": [
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      ],
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          "path": "experiments/theory-contracts/joint-native-algebra-selection-20261016/CONFORMAL-HISTORY-AND-ERASURE-CARRY.txt",
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    },
    {
      "id": "finding:all-canonical-stress-histories-scalar-on-native-code",
      "titleEN": "All pure canonical stress histories stay scalar on native code",
      "statementEN": "All modes/words of given canonical T_E8 act in generated code cycle as common Virasoro representation tensor I5. Every code compression is scalar, including excursions to higher grades and return; code-preserving unitary endpoint is a common phase only. No such pure stress history implements given nonscalar R5 or selects P2. U_conf=exp(i pi L0/2) agrees with native J on4/5/20 seed carriers, but acts as+i on newly forced weight5 twenty whereas native J there remains-i. Common Casimir phase does not repair this relative discrepancy.",
      "scopeEN": "Pure original canonical stress histories in given source frame only; not all geometry/metric variations, affine KZ histories with other field insertions, noncanonical/nonstress fields or general P1 processes. No physical time identification or union with prior separate zero-mode exclusions.",
      "status": "excluded_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:clocks"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:canonical-source-stress-all-mode-history",
        "premise:existing-logical-clock-and-charge-marking"
      ],
      "attemptIds": [
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      ],
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        {
          "id": "source:16c6d4e371357ef886a66ec0",
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    },
    {
      "id": "finding:original-local-current-ope-forces-erasure-and-carry",
      "titleEN": "Full local current product connects native erasure and carry",
      "statementEN": "Actual current product obeys j_ubar_a(z)F_i(0)=G_ia(0)/z²+sum_k>=0 z^k/k! :partial^k j_ubar_a F_i:(0), without simple pole. Double pole is original erasure twenty; regular t_ia=ubar_a(-1)f_i contains exactly prior h5 primary r=t−7/85L_-2g+3/170L_-1²g, Gram78/85I20 and rho5 tensor conjugate4 action. [ubar_a(1),j_u_b(w)]=delta_ab also shows pure erasure is global Fourier mode, not an operator localized in a proper interval of given current net. OriginalP1 defines locality by support and A/A*-invariant restriction, not global compression.",
      "scopeEN": "Existing E8_1 Cartan-Fock source and literal native embedding; no P1 pullback, code=H_adm identification or physical field assignment. No general ban on global P1 operations. New bridge connects previously checked carry to actual local OPE; prior checks are correlated groundwork.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
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        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-cartan-current-local-smearing-and-weyl-class",
        "premise:canonical-source-stress-all-mode-history"
      ],
      "attemptIds": [
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      ],
      "sources": [
        {
          "id": "source:c2536b3e06839cda2ea661dd",
          "path": "experiments/theory-contracts/joint-native-algebra-selection-20261016/LOCAL-CURRENT-ERASURE-AND-ORDERING.txt",
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    },
    {
      "id": "finding:all-real-local-cartan-profiles-erasure-third-bound",
      "titleEN": "Full local current response and sharp erasure one-third bound",
      "statementEN": "For every smooth real8-vector profile with Vhat1=p.u+q.ubar, actual five-code obeys P J(V)²P=(3||q||²+||p||²+Tail)I5, Tail=sum_m>=2 m||Vhat_m||². Grade3 erasure squared norm is ||q||². Its fraction is at most1/3, strictly smaller for nonzero properly local profile with q!=0. Equality requires global rotating first harmonic; fixed real polarization has tighter1/4 bound. No profile produces only erasure output. Full first positive response Gram is scalar and selects no3+2 mark.",
      "scopeEN": "All smooth Hermitian Cartan-current smearings in original neutral Fock module. Norm fractions are not measurement probabilities. Grade decomposition/code projectors are analysis only, no executed projection. No claim all higher field correlations are scalar or all other source fields excluded.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-cartan-current-local-smearing-and-weyl-class"
      ],
      "attemptIds": [
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      ],
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    {
      "id": "finding:closed-cartan-weyl-histories-only-logical-phase",
      "titleEN": "Closed Cartan-Weyl histories have only scalar logical action",
      "statementEN": "Actual CCR fixes [J(V),J(W)]=i sigma I and exp(iJ(V))exp(iJ(W))=exp(−i sigma/2)exp(iJ(V+W)). Every finite ordered Cartan-Weyl history is phase times total displacement. Every nonzero total displacement turns any nonzero finite Fock polynomial into exponential nonpolynomial in Bargmann realization. Exactly code-preserving endpoint forces zero total displacement; neutral zero modes are trivial and only common phase remains. Entire stated class provides neither marked R5 nor3+2 selection. Nonzero individual profiles may cancel in total.",
      "scopeEN": "Analytic theorem for finite unitary Cartan-Weyl-only histories and original neutral code-Fock embedding. Not general nonunitary field products, root currents, interleaved stress/current histories, open processes with memory or all E8/P1 operations. No original P1 time law or new code-preservation axiom. Do not unite separate exclusions.",
      "status": "excluded_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
      "goalIds": [
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        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-cartan-current-local-smearing-and-weyl-class",
        "premise:existing-logical-clock-and-charge-marking"
      ],
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      ],
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    },
    {
      "id": "finding:full-existing-local-e8-hull-of-native-code",
      "titleEN": "Minimal full local source hull is E8 vacuum space",
      "statementEN": "Four original Cartan annihilators take nonzero quartic to vacuum; all original root currents generate all E8 lattice charges, negative Cartan modes all oscillators. Algebraic mode hull ofC5 is exactlyV_E8; closed local smear hull isH_E8. Both twenties are included. Full character prod_m(1−q^m)^−8 sum_lambda q^(lambda²/2)e^lambda determines all horizontal grade decompositions; grades0..6 dimensions1,248,4124,34752,213126,1057504,4530744. Grade2=1+248+3875 checked by complete Freudenthal recurrence. Controlled grading preserves finite mode word above maximal intermediate weight; smooth local profiles retain controlled infinite Fourier tails.",
      "scopeEN": "Existing level1 lattice/current realization, not rawP1 or physicalH_adm. Distinguish mode hull and Hilbert closure of smooth local fields. No exact10D/45D model or particle count; irreducible affine module differs from finite horizontal E8 decomposition.",
      "status": "established_scoped",
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      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
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        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
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        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code"
      ],
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      ],
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          "id": "source:f32b1211c9b3102b9b142a92",
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    {
      "id": "finding:full-local-quartic-wick-ladder-and-unprojected-native-actions",
      "titleEN": "Full local quartic Wick ladder gives code/erasure action",
      "statementEN": "Actual local F_i(z)F_j#(w) has exactly five Wick contraction classesr0..4 with all Taylor terms. Leading weight8/6/4/2/0 Grams areI25,4I25,(18^1,8^9,0^15),4|I><I|,|I><I|. r1=sum_a g_ia conjugate(g_ja) contains actual erasure20; isolated3creation/3annihilation mode component actsEij tensorI4 on that grade3 carrier.4/4 componentsEij=P_i(u_-1)conjugate(P_j)(ubar_1)/sqrt(NiNj) on ENTIRE weight4 are exactly|f_i><f_j| plus0 on rest, without intermediate projection. Full local products permit nonscalar logical action and entireM5; old pureCartan-Weyl exclusions are not all-field exclusions.",
      "scopeEN": "Mode components are not entire local zero modes or selected physical instruments. Matrixunit law only weight4, not all grades. Leadingr2 is not whole fourth-pole coefficient; retainr3 Taylor term. Wick contraction is field algebra, not executed measurement. ChosenM5 coefficients available, not physically selected. Endpoint map A→PAP is not multiplicative for full field algebra; M5 is realized by stated code-preserving weight4 family, not a representation of all source operations onC5.",
      "status": "established_scoped",
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        "topic:quantum",
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        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code"
      ],
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      ],
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    {
      "id": "finding:unmarked-native-w6-simplex-from-full-quartic-ope",
      "titleEN": "Original unmarked W6 simplex from full local quartic OPE",
      "statementEN": "Actual2+2 flatteningsH_i of normalized original quartics give ten intrinsic logical rank1 raysv_t in localr2 product, sumP_t=2I5, Tr(P_tP_s)=1/9. Configuration fixes projective real structure through antiunitary fixing all rays unique up to common phase. On intrinsic real projective locus, sum_t(v_t.w)^4>=2/5 has exactly12 oriented equality points, hence six axes with regular simplex Gram6/5I6−1/5J6. All1024 sign cases exhausted. OriginalN introduced ONLY AFTERWARDS matches literally w_jw_j^T=(6/5)N_pi(j)N_pi(j)^T without fitrotation. Previously separately used UNMARKED W6 simplex thus gains concrete common local field-operator origin.",
      "scopeEN": "Geometric data reconstruction in given source, no physical minimization/real-state rule. Full complex state space has additional continuous minimacosθw1+i sinθw2; explicitly checked, never discarded. Overallantipode/permutation remain. No positivepolar transport or physical3+2 mark. Quartic code remains input; no new independent empirical confirmation or priority claim.",
      "status": "established_scoped",
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      "topicIds": [
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        "topic:quantum",
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        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code"
      ],
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      ],
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      "id": "finding:full-e8-current-products-fix-stress-and-exterior-ward-cancellation",
      "titleEN": "Full current products fix stress and exterior Ward cancellation",
      "statementEN": "With actual cocycle dual sign full240-root OPE gives sum_alpha alpha tensoralpha=60I8 and h_dual30. Thus T_Sug=(1/62)sum248:J^AJ_A:=(1/2)sum8:h²:=T_E8,c8. Same stress follows within same source fromr3 quartic Wick ladder. ForT_n=sum248J^A_-nJ_A_n full endpoint factors onC5 atn0..3=(134,103/2,5,1/2), exterior-defect Grams=(100,225/4,9,1/4)I5. FULL source vectors obey(T0/2+T1+T2+T3)f_i=124f_i: exterior pieces cancel exactly, not just compressions. No root/family branch or fitted rate.",
      "scopeEN": "Canonical stress now follows from given normalized full current OPE rather than independent stress premise. Level1/E8 realization remains input. Invariant source operator identity, not physical selected intervention/time law, particle clock or3+2. Correlated source constructions are not independent experiments.",
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      ],
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        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code"
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      "id": "finding:full-source-simplex-does-not-select-original-polar-mark",
      "titleEN": "Full source simplex structure does not select original polar mark",
      "statementEN": "Six OPE-reconstructed axes give15 equivalent rank2 pair planes. None equals actual originalP2; minimum squaredHS distance is(21−6sqrt6)/25>0. Full logical field-mode imageM5 is irreducible with centerC, no distinguished3+2 superselection. With all exterior data invariant, fully contracted LINEAR code return/memory kernels stay scalarI5 by native irreducibility even with every exterior path. Nontrivial uncontracted OPE tensors remain and are not excluded.",
      "scopeEN": "No all-E8/P1 no-go or exclusion of covariant density channels, open outcomes, noninvariant preparation or spontaneous branches. No identification of real reconstruction functional with physical potential. Existing polar-fibre/clock theorem remains comparison; new source gives unmarked geometry, not physical selection.",
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        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:native-g31-concrete-event-representation",
        "premise:existing-logical-clock-and-charge-marking"
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      "titleEN": "Original full event fields reconstruct relative polar tensor",
      "statementEN": "Actual native reflections at givenq produce five local quartic difference fieldsDelta_j=F_nj−F_nq; full vacuumGramG=I+J, arbitrary common smooth local smearc_fG. All Wick classes/Taylor terms retained. Original three/two clockorbits give response tensorsKC=DQC D*,KW=DQW D* andR=KC+KW. Without insertingP2/targetangle, mathematical positive RELATIVE tensorR^−1/2 KW R^−1/2=UQW U* is exactly existing originalP2, U=D(I+J)^−1/2. Polar prescription thus has concrete common field/event-response interpretation as relatively normalized quantity.",
      "scopeEN": "Given native E8 realization, clock and marked input, not physical selection of them. Positive normalization computable and unique WITHIN positive normalization class, not P1-derived execution/readout or3+2 matter. Three/two are originalsigma orbit sizes, not independent particle count. Injective r0 field-tensor map is not matrix-product homomorphism.",
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        "topic:quantum",
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        "goal:T2",
        "goal:T8"
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:native-marked-simplex-clock-compatibility-without-fixed-polar-mark",
        "premise:original-native-event-difference-fields-and-state-controls"
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      "titleEN": "Raw event response and state positivity do not execute polar prescription",
      "statementEN": "Actual raw orbit kernels have spectraKC=(4,1,1,0,0),KW=(3,1,0,0,0), coefficient-matrix commutatorHS²72 and do not preserve originalP2. GNS supplies metricG; originalslotcutQW is notG-selfadjoint ([G,QW]HS²12). Oblique transport has antihermiticitydefect²2, rawWeakSupport distance²(28−8sqrt6)/25 fromP2. Full weight8 regular quartic insertion on given marked source state hasGq=(43/5)I+(73/5)J; same vacuum-normalized fields retain eigenvalues68/5,43/5^4 instead ofI. NativeJ fixes quartics. Positive source metric alone therefore does not select physical positive normalization/readout law.",
      "scopeEN": "No all-E8/P1 no-go or claim physical instruments need orthogonal outcome states. Weight8 control disproves corresponding full formal normalization identity, not replacement of all local profiles. Given instrument or other originally grounded nonvacuum history not excluded. Full source,both20s,exterior/memory paths retained.",
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:native-marked-simplex-clock-compatibility-without-fixed-polar-mark",
        "premise:original-native-event-difference-fields-and-state-controls"
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      "titleEN": "Determinant cover is not same complete Cartan-lattice event",
      "statementEN": "Existing coupled determinant lifth=e^−iπ/4 r acts onquartics as−rho5(r). Coherent incidentresponses−nj−nq haveGramI+J/3 rather thanI+J; positivepolarmark distance²48/25 fromoriginalP2. Unchanged linearCartan actionofh also fails actualE8 charge-lattice preservation:45degree rotation sendsrawroot(2,0,...,0) to(√2,−√2,0,...), outside lattice. All60native r preserve all240roots; all60phase-shifted lifts failsame root test. Hence noVOA-automorphism with EXACTLYthis Cartandictionary.",
      "scopeEN": "Literal Cartan/lattice automorphism dictionary, not allE8operations,moduleintertwiners orphysicalsources. Prior innerE8lift on DIFFERENT chargedC4/F20currentblocks and finite representation comparison remain valid, not identicalCartanregistermap. Do not treat coherent relativephase as irrelevant rayphase.",
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      "id": "finding:clock-free-quartic-product-spectral-sectors-and-native-five",
      "titleEN": "Clock-free quartic product links spectral sectors and actual five",
      "statementEN": "Actual local holomorphic multiplicationM:Sym²C5→fullsourceweight8 hasGram28^1,10^5,18^9. Canonicalprojectorsgive1+5+9; literalH_holo=10I15+(S2* S2)|Sym² with existingconjugatedoublecontractionOPE. Five-sector hasexplicitisometricintertwiner withsameactualnativerho5,notonlydimension5; all15quotientevents checked. Common smoothlocalvacuum smearing ofprimarycomposites preservesGram up toc_f. CentralidentityOPE coefficients aredataofsame*-fieldalgebra,notchosenGNSresponses.",
      "scopeEN": "Acceptedquartic/E8/adjointvacuumnormalization remainpremises.1+5+9 arecomposition/representationsectors,notspacetimedimensions orparticles. Gramcomparison doesnotidentifydifferent sourceweights. No new3+2/physicalselection.",
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      "titleEN": "Six complex native simplex rays from original product quadrics",
      "statementEN": "Withoutclock,q,polartarget,realformchoice orNinput, originalproductGram determinesrank9spectralprojectorPi9. ActualsourcerelationPi9(v⊗v)=0 givesninequadrics andexactlysixCOMPLEX projectiverays. Completeelementarycaseproof andGroebneraffinecharts includingallinfinitycharts excludefurthercomplexsolutions. OriginalN introducedonlyafterwards matchesall6raysliterallyasprojectors,withoutfitrotation. Reconstructionstrengthensearlierreallocusminimumsimplex andlinksgeometry toactualproduct/eventfive.",
      "scopeEN": "Algebraicconfiguration,notphysicalpreferencefornullcomponents/states. EarliercomplexminimaofDIFFERENTfourthmomentfunction remainphysicallyunexcluded. OriginofacceptedquarticfromrawP1 separate. Covariantunderbasis/metrictransport,notuniversalreference/preparationindependenceoffullresponses.",
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      "titleEN": "Relative response is metric covariant; complete state response remains dependent",
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      "scopeEN": "No unprovedGNSHilbertspaceidentification. FormalcentralOPEidentity isnotuniformUVlimittheoremforall singularstates. Controlsneedfinitecommonmoments,notautomaticallyadmissibleP1seeds;RP,clustering,relativeaction,time/collar mustadditionallyfit.",
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      "titleEN": "Event composition needs shared source and correlations",
      "statementEN": "Independenttensor sources/productstate factorizeGram/polartransportexactly. Insameoriginalfieldnet fullcommonprofileholomorphiceventproducts killantisymmetriclabels: orderedmultiplicationC5⊗C5→H8 hasrank15,kernelwedge²C5 dimension10,whileMonSym²isinjective. OriginalDelta0 firstnorm²2,fullleadingpairnorm²64 ratherthan4independentcopies;100actualerasureLeibnizactionsretainallgrade7outputs. Twosourcestateswithsamemarginalsbutclassicalcorrelation give differentjointGrams with exactterm(Bq−I)⊗(Bq−I)/4. Fullfieldassociativityretained;reducedsingle responsesalonearenotgeneralcompositionlaw.",
      "scopeEN": "10Dkernelisrelationspace,notnew10Dsourcesystem. Commutativityargumentforthisholomorphicfieldclass/commonprofiles,notallE8/conjugate/noncommutingfields. Correlationcontrolnotselectedphysicalpreparation. Universalraumlinkisnecessaryretentionofproduct/correlationinformation,notexistenceproof.",
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        "premise:intrinsic-native-event-geometry-without-marked-clock"
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      "titleEN": "Clock-free event geometry fixes clock class and marking orbit, not one member",
      "statementEN": "All15differenceedgesofsixderivedraysgiveA5tightframe,Gram6^5,0^10,cyclerelations;relativeeffectsE=P_root/3sumtoIonresponsesupport,notprojectors. Givenqstargives5ONprojectors,noselectedrank2cut. All6markers/2subsets giveexactly60distinctpolarmarks. Withoutoldclock,225logicaltranspositionpairs give15identities,90doubletranspositions,120noncommutingproductsin40threecycles. Their120orientedfixedpointflagsgiveEXACTsame60polarmarkorbit. Uniqueintrinsiccovariantmarkcanonly haverank0/5underS6,or0/1/4/5underqstabilizerS5. OriginalP2needsadditionalq+unorderedtwosubset;oldclockprovidessuchdata,butisnotderiveduniqueelement.",
      "scopeEN": "ClockclassinactualfiveS6quotient,notautomaticallysamefour/20/fullfieldclockliftorphysicaltime. Symmetryobstructionforexactunmarkedsymmetricinput,alsononlinearuniqueselectors,notallP1sourceswithadditionalanisotropies,states,spontaneousbranches. Noprefernoncommutationruleimposed. Earlier15matchingP2exclusionsconcernadifferentprojectorfamily.",
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        "goal:T8"
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        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
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        "premise:intrinsic-native-event-geometry-without-marked-clock"
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      "titleEN": "Original charged seam character reconstructs global native event pair",
      "statementEN": "Full internalE8Z4character(rootphasecounts52,60,64,64)hasactualG31stabilizer64/logicalimage16;decksquare3072/image48. Bothsplitthesixliteralquarticraysinto{0,1}|{2,3,4,5};exactly{0,1}amongall15globalpairsetsisinvariant. Originalglobaleventpairthereforefoundwithoutoldthreeclock/P2/coupling. Withqoutsidethispairremainfourflags(2,S),(3,S),(4,S),(5,S). LiteralexistingOframe,nofit.",
      "scopeEN": "GiveninternalcharacterandoriginalJ-equivariantE8embeddingremainpremises;notrawP1originofcharacter. Pairiseventlabels,notphysicalrank2/fermions. ActualchargedquarterfoldisDIFFERENTcharacter;noidentification.",
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      "titleEN": "Internal seam,vacuum and original neutral tensor do not select flag root",
      "statementEN": "Fullinternalcharacter+E8vacuumhasnofixedq/flag;oldcomparisonflag(5,{0,1})isin4orbit. OriginalR_neut/retained48givesstabilizer16/logical8=C2³,q-pairs01|23|45,allsame4familyomissions. ExplicitfullE8isometryG=(x3,x4,x1,x2,x5,x6,x7,x8)preservesJ,v,Rfield,48span,vacuum,OPEandactsasquartic(45). Itswapsflags(4,01),(5,01)withactualpolarHS²distance(816−176sqrt6)/625>0. Nouniquecovariantselectorfromthesetesteddatapossible. Supplyingonlyq5stillleavesinvariantpairs01and23.",
      "scopeEN": "OnlyspecifiedinternalE8vacuum/character/Robject. R/48includesgivenfamilyomission;preserving48spanisnotpreservingnewcoherentsourcecoefficients. NOexclusionfororiginalfulltwistedCARcovariance,additionalphysicalTheta,allP1sources. Individuallatticeliftsuffice,noglobalsplitsectionassumed.",
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      "titleEN": "Actualquarterfoldshiftremovessymmetryobstructionwithoutflagselector",
      "statementEN": "UnchangedoriginalMforalpha1/4,3/4givesd=(-1/2,0,alpha/4,-1/2,0,(alpha−1)/4,-1/4,-1/4),norm²85/128. Actualchargedg²order8,28fixedroots,NOTinternalZ4character. Stabilizernative4/logical2fixes4q/16flags,notuniqueone. Fulldhasnativestabilizer1,alsojointwithinternalcharacter;ALL60flagsfixed. Oldsymmetryobstructionthereforegoneford. ThischecksCartanshiftorK=L0+d.h0,notfullC_foldcovariance. OrdinaryE8h0Ω=0,not〈h0〉=d;Kstillscalar4onneutralcode. FixedVirasoroweight85/256twistedfoldstateisnotuntwistedE8vacuumundersamestressintertwiner;twisted/defect/referencetransportopen.",
      "scopeEN": "NodidentificationwithchargedE8Fockvector/fullCARstate,noremovalofchargedholonomy. Nofittedphysicalclock/coupling. Trivialstabilizermeansallcandidatespassthatsymmetrynecessarytest,not60physicalP1theoriesorimpossibilityofsource-selector.",
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      "titleEN": "Foldshiftneedsactualsourceresponsemapratherthananothersymmetryno-go",
      "statementEN": "OnactualfreenativeorbitG.d,everyoneof60seedflagshaswell-definedcovariantextensionS(gd)=gS(d)becausestabilizertrivial. Covariancealonedoesnotselectone. Mathematicalnonuniquenessofcriterion,not60physicalmodels. Missingmapisnowfulloriginalfield/state/normalorderingtransportfromF C_alpha F*withtwistedseam/adjoint/timeintonativequarticeventsandchargedintermediates. InternalresidualswapMUSTNOTextendtoas-yet-unmappedactualstate; furtherneutralvacuummomentsareblindbyexisting scopedallwordtheorem.",
      "scopeEN": "Precisedictionary/selectiongap,nogeneralabstractresetorno-goforactualstate-dependentfieldresponses. No max-overlap,minimization,newmeasurement/clock/couplingrule. Fullphysical3+2originopen.",
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      "titleEN": "Original fold covariance yields current counterterm and quartic transport",
      "statementEN": "ExacteigenchanneldecompositionofF f_(8l+k,α)withUα,rk=e^(2πi(k+α)r/8)/sqrt8diagonalizesSAMEF Cα F*,no8sourcecopies;arccorrelationsretained. NegativeDsea l≤−1,originalpositiveα1−a,j≥1sea l≥0. FiniteNSpointsplitcountertermis±(μ−1/2),currentorderingtransported. FixedconformalcurrentreferencegivesactualF_i^d=:f_i(h_++b/z):,b=U_+^T O^T Mδ,nocouplingfit. Sourcetime8[N+p²/2+δ.p]andgroundsubtractionretained. GivenE8extensionhasLi gorder16,g²8;shiftonce. Explicitconditionalneutralfield/statebridge,notphysicalreference/P1selection.",
      "scopeEN": "Fixedconformalcurrentdictionaryisapremise;M/Orootmatchalonedoesnotselectit. NorawP1/fulloddCAR-E8extensionornewvacuumselection;E8p0andadditionaldegeneraterootprimaryseparate. OppositeshiftsignleaveshomogeneousquarticGramsunchanged.",
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    {
      "id": "finding:fold-state-full-quartic-kernel-and-erasure-protected-low-order-response",
      "titleEN": "Full quartic state kernel and erasure-protected low-order response",
      "statementEN": "SameoriginalpolynomialsgiveENTIREtwisted2pointkernelΣD_r/[(z−w)^(2r)z^(4−r)w^(4−r)],D_r,ij=Σconj∂νfi(b)∂νfj(b)/ν!. D0conj(m)m^Tretained. AllpositiveBk=Σbinom(k+r−1,2r−1)D_r;first4recoverD1..4bydet1triangularmatrix. Actual20erasurederivativeorthonormalityforcesD3=||b||²I=85/256I,D4I. Codeprotectsdirectioninformationatquadraticstateorder;anisotropystartsorder4despitetrivialshiftstabilizer. B4=(383/128)I+10D2+4D1literallymatchesindependentfullsparseFockcalculation,allpartitions4,31,22,211,1111retained.",
      "scopeEN": "Analyticallposition/allmodetwopointkernelinstatedreference,notuniversalcontactterm/UVlimit/chosenphysicalsmear/instrument. Heisenberg/quarticderivationnotphysicalmatterdictionary.NoclaimedC5closure.",
      "status": "established_scoped",
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        "topic:quantum",
        "topic:clocks"
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        "goal:T2",
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:intrinsic-native-event-geometry-without-marked-clock",
        "premise:original-seam-state-data-on-native-flags-without-fitted-selector",
        "premise:fixed-conformal-current-reference-for-original-fold-state-transport"
      ],
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      "id": "finding:original-fold-quartic-mean-and-spectral-responses-do-not-match-sixty-flags",
      "titleEN": "Originalquarticmeansandspectralresponsesmatchnoneof60flags",
      "statementEN": "Bothoriginalquartercases:onepointRieszraymatchesnoneof6native rays;D1rank4hasuniquekernelalsononeof6. NONEof60P(q,S)commuteswithD1orcompleteB4;thereforenoneisaspectralprojectorofthesematricesandnocommonreducingpolarflagforfull2pointfamily. Exactsymbolic,nothreshold/fit. FullfixedoriginalinputmodeexternaloscillatortermsretainedinB4.",
      "scopeEN": "ONLYstateddirectmean/kernel/spectralidentificationsinfixedreference. Physicalprojectorneednotcommutewitheverysourceresponse;NOsuchnewaxiom. NoallP1/state/3+2no-go,norejectionofotheroriginalobservables/chargedhistories/specifiedprofiles. Noeigenvalue/overlap/profilefit.",
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        "topic:quantum",
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        "goal:T2",
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        "premise:native-g31-concrete-event-representation",
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        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:intrinsic-native-event-geometry-without-marked-clock",
        "premise:original-seam-state-data-on-native-flags-without-fitted-selector",
        "premise:fixed-conformal-current-reference-for-original-fold-state-transport"
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      "id": "finding:original-winding-full-state-response-removes-vacuum-rank-two-shortcut",
      "titleEN": "Originalwindingfullstateresponseremovesvacuumranktwoshortcut",
      "statementEN": "ActualpreviouslyderivedglobalchiralW4:p→Pp+a,a1^40^4,Pk↔k+4. LiteralexistingG=O^TMPM^TOgivesu0→iubar0,u1→−u1,u2→u2,u3→−iubar3,notunchangednativeU4clock. Homogeneousquarticframeoverlapdiag(1/2,0,0,1/2,0)rank2. FULLoriginalR_bwithinputcharge0underW4isΓG R_bonoutgoingMa;literaloutgoingfieldshaveb_out=U_+^TO^T(d+Ma). Crossmatrixrank5BOTHquarters,fullW4outputGram=inputGram. Sourceincrements7/9(fold7/8,9/8)matchoriginalWard,chargecarryandallpolarizations/modesretained. Apparent2+3vacuumchartdeficiencyisnotactualstateresponse3+2.",
      "scopeEN": "OriginalW4isglobalCARimplementerofgivensource/evenD8E8extension,notP1localinstrument. Oppositepolarizationnotcreation/annihilationmixing. CrossGramfullmatrixelementswithoutgoingcharge,noperformedprojection/required5closure. Noallwinding/historyno-goorphysical3+2selection.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
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        "topic:quantum",
        "topic:clocks"
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        "goal:T2",
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:intrinsic-native-event-geometry-without-marked-clock",
        "premise:original-seam-state-data-on-native-flags-without-fitted-selector",
        "premise:fixed-conformal-current-reference-for-original-fold-state-transport"
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    {
      "id": "finding:original-source-global-time-winding-history-factorization",
      "titleEN": "All original global time-winding histories reduced exactly",
      "statementEN": "Given Ward increments are c_n(q)=Falpha(q+n)-Falpha(q), Falpha(q)=q(q-1)/16+alpha q/8. Thus K-Falpha(Q) is winding invariant and alpha independent. Actual CAR space factors via W_n as ell2(Z) tensor H_Q0, not an assumed rotor. Every finite ordered time/winding word has the stated exact normal form; all n=8z+r checked symbolically. On a fixed neutral R_b,k field mode the input multiplicity is unchanged. Charge-return acts scalarly in actual B_k metric, retaining all intermediate charges/modes. W1 lies in original odd vector sector; W2 is smallest even winding and ends at an actual E8 root.",
      "scopeEN": "Only given global free source time/W_n/Q phases, not all local fields. No closed-C5 axiom or performed projection. Total time acts on grade superpositions. E8 half-sector needs its lift; global W2 is not local root vertex.",
      "status": "established_scoped",
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        "topic:quantum",
        "topic:clocks"
      ],
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        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
        "premise:charged-car-negative-sea-source",
        "premise:fixed-scalar-quarter-hardy-source"
      ],
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      ],
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          "id": "source:5c9992f6b6699b2675439471",
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      "id": "finding:original-local-density-folds-to-correlated-root-current-and-erasure-response",
      "titleEN": "Original local density reaches quartic and actual erasure",
      "statementEN": "Original scalar phase folds in fixed Fourier dictionary to Vphi,kl=sum_r phi((x+2pi r)/8)exp(2pi i(l-k)r/8)/8; frequency m couples k-l=m mod8. J-1 is exactly seven adjacent CAR zero modes plus wrap mode-1, no fitted root weights/Klein phases. Complete R_b,4 field response is independently reconstructed in original CAR; Gram exactly matches prior sparse Fock Gram. Both quarters: J+-1 takes source energy32 to31/33, retains all16 signed cyclic root charges and full vector CCR. Both Grams rank5; lowering response nonscalar relative B4. Actual erasure a_a R_b,4 f_i=R_b,3 partial_a f_i and all20 nonzero Ward outputs [a_a,J-1]=sum_r(barV_(r+1,a)-barV_(r,a))a*_(r-7)a_r exact without free coefficients.",
      "scopeEN": "Given negative-D CAR and conditional field reference. No physical P1 profile/coupling/detector selection or matter/Higgs assignment. J+-1 are global Fourier controls of existing local density. Correlated offdiagonal fold class differs from earlier independent diagonal Cartan-Weyl exclusion. All external modes/charges retained.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
      ],
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        "goal:T2",
        "goal:T8"
      ],
      "premiseIds": [
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
        "premise:original-single-circle-local-density-and-geometric-stress"
      ],
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      ],
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      "id": "finding:original-current-all-mode-energy-and-geometric-stress-bridge",
      "titleEN": "Original current modes, energy and twisted geometry connected",
      "statementEN": "Original c1 Sugawara identity 8K=Q(Q-1)/2+alpha Q+sum_m>0 J-m Jm is checked on full R_b,4 inputs as all32 response Grams summing to32B4; higher lowering modes vanish by positive source energy, not chosen truncation. Original Lcal_m^alpha has fixed r+m/2+alpha coefficients and ground weight(alpha-1/2)^2/2. Full vector Wards [Lcal1,J-1]=Q+alpha-1/2 and [Lcal1,Lcal-1]=2Lcal0 hold. Folded c8 geometry is twisted Yd(T), Lhat_n=Lcal_8n/8+21/64delta. Original Lcal+-1 changes fold energy by1/8 with cyclic root charge outputs, outside prior unshifted c8 stress/primary-vacuum theorem.",
      "scopeEN": "Twisted and unshifted stress/input pairs must not be identified. Prior scoped exclusions retained. Exact CAR window follows maximum source energy33; no physical time calibration or external coupling.",
      "status": "established_scoped",
      "problemId": "problem:common-source-state-event-selection",
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        "topic:quantum",
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        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
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        "premise:original-single-circle-local-density-and-geometric-stress"
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      "titleEN": "Density and geometric responses have no common 3+2 spectral sector",
      "statementEN": "Both quarters: B4^-1G_J and B4^-1G_L are B4 selfadjoint with common commutant dimension1 overQ(i), hence generated complex response *-algebra M5 on input labels. No common reducing rank2/3 decomposition of these actual response forms. Density response has five distinct generalized eigenvalues. In COMPLETE mode family J30 response rank2, but rank3 kernel matches none of60 original polar flags; Adjacent rank3 J29 response is not ORTHOGONALLY complementary in B4 metric, but joint rank5 gives an oblique algebraic direct sum. Positive relative normalization for this supplied mode pair remains possible; neither its original selection nor a literal normalized polar comparison is claimed. Mode30 lowers source energy by30, not a derived unitary Coxeter30 action.",
      "scopeEN": "ONLY common spectral-sector identification for these forms and literal mode30 polar shortcut. Not a physically executable compressed M5 algebra, no axiom requiring physical marks commute with every response. No exclusion of nonlinear intrinsic event relations/other full field histories/all P1/3+2. Mode30 not adopted as selected intervention.",
      "status": "excluded_scoped",
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        "topic:quantum",
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      ],
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      ],
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        "premise:full-existing-local-e8-field-hull-observer-code",
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        "premise:original-single-circle-local-density-and-geometric-stress"
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      "id": "finding:original-current-ordered-hall-kernel-and-actual-folded-field-products",
      "titleEN": "Complete ordered original-current response and actual folded-field products",
      "statementEN": "Actual five neutral sourceE32 states map exactly by original Maya occupations to Schur states. All32 lowering-current VECTORS independently match Young ribbons, stress1/2 matches fixed Sugawara and full CCR, both quarters. All state coefficients saved. Kij(u,v)=<Pi,exp(sum um pm)Pj(pm+mvm)>Hall gives all mixed current correlations; Weyl response additionally retains Gaussian sum over ALL profile modes, not only1..32. Every finite smooth density/source-time history follows with original BCH phase. Actual folded quartic fields remain on eight original preimages: A_a=Ualpha diag(Vka)Ualpha*, scalar source operation gives Gphi(x+)dagger A_a Gphi(x-). Entire existing point splitting/reference subtraction conjugated, no second b shift. Separated Heisenberg field products follow from finite Wick Pfaffians using same ordered Calpha/1-Calpha.",
      "scopeEN": "Given c1 CAR source and explicitly conditional NS/native dictionary, not all E8/P1. Hall is a STATE/current intertwiner, NOT a point-local cover identification with weight32 field. Finite stress-mode words yes; arbitrary exponentiated geometry or new unsmeared coincidences require analytic control. Mixed amplitudes retaining implementer are not bare vacuum Pfaffians. No physical profile/reference/readout selection.",
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      "topicIds": [
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        "topic:quantum",
        "topic:clocks"
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        "goal:T2",
        "goal:T8"
      ],
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        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
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      "titleEN": "Original seam grades full density/geometric products",
      "statementEN": "Existing full internal seam character v=(0^5,1/2^3) gives original CAR phases[-1,i,1,-i]^2 via same M. Exactly tv Jm tv^-1=i^mJm and likewise original Lcal_m^alpha; full products carry total Fourier grading including central OPE terms. Known global tv=exp(-4pi iK)exp(i pi(1+alpha/2)Q)=(-1)^Q Walpha^-1 reused on all charges, not claimed new. Entire32-mode energy response splits without profile fitting into four original character classes Grho summing to32B4, both quarters.",
      "scopeEN": "Internal order4 character remains distinct from actual quarter holonomy. Retain odd CAR parity; no physical holonomy compensation or clock selection. Energy response forms are not an executed measurement or entire local E8 algebra.",
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        "premise:original-single-circle-local-density-and-geometric-stress"
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      "titleEN": "Complete original seam response does not yield sharp3+2 input sectors",
      "statementEN": "All four complete Grho are EXACTLY positive definite rank5 in BOTH quarters, sum32B4. Hence Erho=(32B4)^-1/2 Grho(32B4)^-1/2 lies strictly between0 andI, as does every proper nonempty coarsening. Deck-even/odd responses both rank5. Exact common commutant of(32B4)^-1Grho has dimension1 overQ(i). No sharp deterministic character/deck classification of five input directions or common reducing3+2 spectrum of these forms.",
      "scopeEN": "ONLY sharp input-sector/common-spectrum identification excluded. No axiom that physical mark must commute with every response; no exclusion of all nonlinear event relations, histories, output rules or P1. Existing intrinsic six rays/60flags and global seam pair retained. Mathematical response effects are not postulated/performed POVM.",
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        "premise:full-existing-local-e8-field-hull-observer-code",
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        "premise:original-single-circle-local-density-and-geometric-stress"
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      "titleEN": "Original mean density work is scalar on entire event packet",
      "statementEN": "Original CCR/time give exactly Wphi* K Wphi=K+i[K,Jphi]+sum_m>0 m²|phim|²/8. Since all five actual inputs haveQ0,K4 and linear term shifts nonzero energy, full matrix <Pi,Wphi* K Wphi Pj>=(4+sum m²|phim|²/8)B4ij for EVERY smooth real original profile. Mean work does not distinguish these input directions, although their full coherent response is directional.",
      "scopeEN": "Exact matrix elements, no performed projection, energy-minimization rule or no-go for full work statistics. Primitive action and P1-INSERTION-TANGENT reread at original: physical source family/reflection/domain/eta not supplied by this work identity; no blanket relative-action isospectrality for arbitrary phase.",
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      "titleEN": "Original P1 differential with actual gap dependence",
      "statementEN": "chi=1/lambda forces delta(D/chi)=lambda deltaD+deltaLambda D. For isolated simple signed gap branch deltaLambda=sgn(E0)<u0,deltaB u0>; entire corresponding term in relative action differential derived without commutativity. Clusters, constant nullity, moving R/domain/reference/eta correctly scoped. In actual scalar quarter circle, EVERY smooth real v gives Balpha+v unitarily equivalent to Balpha+mean(v). On fixed holonomy branch mean(v)=0, gap/chi unchanged exactly while Pgap moves. This supplies a source covector, not a selected physical source variation/event functional; scalar gauge orbit is not inequivalent-minimal-P1 model theorem.",
      "scopeEN": "Regulated trace or controlled trace-class source curve with legitimate domains. Circle lemma does not imply entire relative-action isospectrality with fixedR; same phi symbol is no physical insertion dictionary. Given scalarCAR/reference conditional relative toP1. Earlier P1-INSERTION-TANGENT/master/minimality boundaries reused. Physical deltaD/deltaB/embedding/event relation remains open.",
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        "premise:declared-p1-p2",
        "premise:typed-source-structures",
        "premise:fixed-scalar-quarter-hardy-source",
        "premise:charged-car-negative-sea-source",
        "premise:original-gap-normalized-primitive-relative-action"
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      "id": "finding:original-p1-gap-car-component-not-native-sharp-32-event",
      "titleEN": "Source-fixed gap occupation is not a sharp native3+2 mark",
      "statementEN": "Actual original gap definition selects mode0 foralpha1/4 and mode-1 foralpha3/4, gap1/4 each. Within same stipulated CAR, Qgap=sgn(E0):a_r*a_r: is full particle/hole projector without rank fitting. On complete actual R_b4 inputs, B=I*I,H=I*QgapI obey EXACTLY H>0 and B-H>0, both rank5. Correct metric response strictly between0 andI, nonscalar/nonidempotent; exterior H-HB^-1H strictly positive rank5, bothquarters. This direct spectral bridge from original scale function is not a sharp3+2 input mark. Qgap is global spectral mode, not insertion localized in proper interval.",
      "scopeEN": "ONLY literal identification of this canonical gap-CAR component with sharp native3+2 mark excluded. Qgap itself NOT claimed P1-selected physical event; no measurement, executed projection, closedC5 requirement, no exclusion of all unsharp/nonlinear events or allP1. Boundary eigenmode/internal direction/physical species distinct; original sixrays/60flags/globalpair/E8 retained.",
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        "topic:quantum",
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        "premise:original-single-circle-local-density-and-geometric-stress",
        "premise:declared-p1-p2",
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        "premise:fixed-scalar-quarter-hardy-source",
        "premise:charged-car-negative-sea-source",
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      "id": "finding:complete-positive-scalar-first-order-circle-source-normal-form",
      "titleEN": "Complete positive scalar first-order source fiber classified",
      "statementEN": "Every operator in declared class has B=-i(a d+a_prime/2)+c,a>0,c real. T=integral1/a,omega=2pi/T,beta=integral c/a/(2pi),x_prime=omega/a,eta_prime=(omega beta-c)/a yield unitaryUf=sqrt(omega/a)exp(ieta)f(x),BU=Uomega(-i dx+beta) on fullH1 domains. Whole spectrum omega(Z+beta), no mode cutoff. SameB determines unique-up-to-phase uniform wind1 representative u_B=exp(ix),[B,u_B]=omega u_B andQ_B=exp(iTB/4),Q_B^4=exp(2pi i beta)I: spacing,proper-timecoordinate/chargedquarterreturn fromoneoperator,not newclock/interventionselection. Original fixedquarter holonomy andgap1/4 giveomega1; unmarked one-particle source sameBalpha up to coordinate/phase transport. Four fixed marks require all arc travel times/possible endpoint phases, not dimension/species identification.",
      "scopeEN": "Only smooth scalar positive elliptic local differential class on given source. Not all minimalP1 fiber or matrix/defect/nonlocal/E8 classification. One-particle unitarity permits no reset of CAR reference sea,normal order,Schwarz/central term,J or relative action. Retain native sixrays/60flags/globalpair/full field response.",
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      "titleEN": "Fixed original time-seam Ward fixes scalar principal symbol",
      "statementEN": "Whole class satisfies[B,Z]=aZ. Retaining same original constantWard[B,Z]=Z forcesa1 without metric minimization. Given wind1symbolu=exp(iphi) requiresa phi_prime=kappa,henceomegakappa. Additional fixed quarter rotation makesc quarterperiodic; continuous originalbranch mean c=alpha leaves quarterperiodic gauge transport fixingZ/all4marks. Globally retainmean c=alpha+N: actualQalpha=exp(i pi alpha/2)Q4,Qalpha^4=exp(2pi i alpha)I,joint invariant exp(i pi B/2)Qalpha*=i^N. Literalclock/endpointframes retainNmod4, exactoriginaltimequarter equality forcesN0mod4;4Zlargegauge/sea carry not deleted. Reference sea fixes originalN0branch.",
      "scopeEN": "Ward retained as already given-source object, NOT derived fromP1 spectral flow. Separate continuous marked gauge fiber and globalinteger branches. No autonomous clock/new species/unchangedFockstate/physical intervention inferred. No3+2 derivation or closedC5 requirement.",
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      "titleEN": "Unit flow, gap and four marks do not force original Ward",
      "statementEN": "For arbitrarya>0,Z*BZ=B+a,pathB_s=B+s a has eigenvaluesomega(n+beta+s):0<beta<1 gives exactlyone upwardcrossing independentofa. UnitSF is not constantunitWard. Exact controls a=(1+epsilon cos(ktheta))^-1,c=alpha,abs(epsilon)<1,k1/4 have samegap1/4,holonomy,SF1;k4 additionally preservesall4marks,equalarctimes/geometricquartercovariance. Yet[B,Z]=aZ differsfromZ forepsilon!=0. Controls not claimed completeP1models; under fullcovariantdatatransport they represent sameunmarkedsource.",
      "scopeEN": "ONLY implication from named weakercircleconditions to fixedoperatorWard excluded; premise list deliberately does NOT include fixedWard. No completeP1/E8countermodel or physical3+2/eventno-go. Do not identify rawP1 seamindex symbol with givensourcecircle without actuallocalization/operator dictionary.",
      "status": "excluded_scoped",
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      "titleEN": "Four native starts have distinct correlations in same fold state",
      "statementEN": "Actual unitquarticrayswq=sqrt(6/5)nq,q2..5,S01 in same originalFCαF* fixedconformalNS/M/Odictionary give phase-invariant connectedc1=wq*D1wq. Common denominator1006632960,numeratorsα1/4=(1200580,1300621,982285,1446340),α3/4=(1308100,1521805,1089805,1338820);all6pairs distinct perquarter. c2=4335/131072,c3=85/256,c4=1 equal; firstdifference alongtheserays sixthorderinb. Alsotr(P(q,01)D1) givesfour pairwisedistinctQ(sqrt6)values perquarter. Therefore noall-response-preserving fixed-state*-equivalence transporting actualquarticspace andtheseplanes; universalOPE-I metric forcesunitarity. Obstruction doesNOTrelyonlyontrivialG31stabilizer andplane-onlytestneedsnoextra distinguished-raypreservation.",
      "scopeEN": "Only markedobservables in statedexistingfield-state/coordinate/adjoint/reference dictionary. Notfourunmarkedtheories orphysicallyadmissibleP1preparations. Larger maps abandoningactualquarticspace/sourcecoordinates/referenceobject notclassified. Nomaximum/minimum/overlap/measurementselector or3+2derivation. Sourcecontainsallfourfields simultaneously; distinguishabilitydoesnotselect.",
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        "premise:original-seam-state-data-on-native-flags-without-fitted-selector",
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      "titleEN": "Complete joint four-start response and charged covariant transport",
      "statementEN": "Forall4q/bothquarters complete separated25-fieldkernels ofquartic5,actualerasure20/allcrossblocks computed,five25x25Wickcoefficientspercase/allmodesdetermined. ALL-WORDlattice-Wickgenerator withorderedcocycle,d/z,chargedintermediate/endvector specified; originalh5carry retainedthroughwholeconstructor/Deltatransport. ActualG31lifts from46080rootactionsconnectq5toq2/3/4withinternalseam; d->Gd/full-datumtransport givesUGYd(a)UG*=Y_Gd(UGa),H_exc=8(L0+d.h0)/sigma_d transported. All240unwrappedroot-time/twistpairings/all5quarticWicktensors exactlycovariant. AtFIXEDd chargedresponsesdiffer; datumtransportisnotfixedsource-statesymmetry.",
      "scopeEN": "25fields notcompleteE8basis;wholealgebraspecifiedbygenerator.PassiveLqframesnotactiveerasureoperations;activefullbar4actionretained. Individual latticeliftswithcocyclecochain,notglobalG31splitsection;notautomaticoddCAR/P1automorphism. Smoothinfinitehistoriesneedpreviousdomain/tailcontrol. ConjugatingALLdata includingreference/originalwinding isrepresentationcovariance,notphysicalgaugeredundancyproof.NoclosedC5/executedprojection,newcoupling/physicalsource.",
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      "titleEN": "Four polar planes are not a fixed-state equivalence of the quartic space",
      "statementEN": "Excluded is identifying the four marks as all-response-preserving symmetries of the FIXED original fold state within the specified actual quartic space. Universal central OPE metric forces such action unitary and preservation of tr(Pq D1), whose four exact values are pairwise distinct per quarter. Not a universal classification of physical P1 gauge equivalences or exclusion of passive whole-presentation transports.",
      "scopeEN": "Only markedobservables in statedexistingfield-state/coordinate/adjoint/reference dictionary. Notfourunmarkedtheories orphysicallyadmissibleP1preparations. Larger maps abandoningactualquarticspace/sourcecoordinates/referenceobject notclassified. Nomaximum/minimum/overlap/measurementselector or3+2derivation. Sourcecontainsallfourfields simultaneously; distinguishabilitydoesnotselect.",
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      "titleEN": "Original measurement template exists; its source context requires identification",
      "statementEN": "The original text explicitly provides a Lueders record instrument for an assumed complete orthogonal pointer family. Optional Q supplies preparation rho0 and readout. The source problem is identifying this instrument in the same physical source algebra, not inventing a new Born rule. The passage identifies no pointer with P(q,01).",
      "scopeEN": "Source audit of these original passages only; no global P1 underdetermination, new probability axiom or executable native pointer. Existing operational seed equivalences are not preparation definitions.",
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      "titleEN": "Four polar contexts give C+M4; original correlation couples to full M5",
      "statementEN": "All four actual Pq at fixed S01 satisfy Pq=Z+|vq><vq| with common rank-one Z and overlap (11+4sqrt6)/25. A spectral polynomial of sum Pq constructs Z; products generate full M4 on Z-perp, hence mark algebra C+M4 and commutant C². All six pairs are noncommuting and admit no common sharp pointer refinement. Original D1 couples Z to its complement with squared crossnorms7882461/4398046511104 and3201175/13194139533312, giving full response algebra M5 and scalar commutant.",
      "scopeEN": "Complete exact classification only of this four-mark menu plus original D1 on native quartic observation space. D1 is a correlation coefficient, not an executable intervention. 1+4 is not full-source superselection; no axiom requiring physical pointers to commute with every response. Contextual/sequential/unsharp instruments remain possible. Ray overlap-1/5 uses native coefficient metric, not physical R_b4 preparation Gram. No q/3+2/P1 selection.",
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      "statementEN": "Excluded is the automatic identification of the complete normal-commuting Dirac-CAR vector current with the original nonzero chiral current phase, only under the stated premises. UR.COLLAR.FACE_CURRENT.01, DIRAC-LIFT continuation: For a complete Lagrangian Dirac Cauchy projection C with N C N*=1-C, c_(1-C)=-c_C. For implementable X,Y commuting with N, unitary covariance also gives c_(1-C)=c_C, hence c_C=0. In the existing product collar B=diag(A,-A), occupations n(A) and1-n(A) on ONE boundary face give -2i,+2i and total0 for the original four-mark-fixed profiles. This differs from the earlier two-face cancellation of one scalar block. Fixed Clifford symbols force g I2 on this two-component module; diag(g,I) changes the normal/tangential symbol. The complete lift D_phi=D-i chi_prime phi Jn-chi phi_prime Jn Gamma is a bounded Hermitian perturbation. Transported domain/reference/Padm give frame covariance; a fixed original selfadjoint domain remains analytically valid but specifies a relative boundary attachment not physically selected here. Explicit extra Dref/Drel commutator terms are retained at fixed geometric R; fixing R alone does not break covariance when U commutes with R. Nonconstant four-mark-fixed profiles do not preserve the old APS sign subspace, which is not a selfadjointness no-go. Original Gamma already identifies D_plus, but not its physical measure/line connection. The prior scalar result remains conditionally valid.",
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      "titleEN": "Full finite-energy carrier of unchanged QWZ source exhausted",
      "statementEN": "22Oct: Three rho ranges in h²=sin²p I+(H0+rho Z)² show at most two eigenvalues per momentum have magnitude<1/2; all others cost>=N/(4pi). Fixed energy E enforces finite,N-independent |j-alpha|<=pi E/2. Both small branches give |j-alpha| with opposite edge orientations. For E outside the limit spectrum the ENTIRE Fock spectral rank stabilizes,not merely fixed words. Tr exp(-beta H_N) converges for every beta>0 to product_j(1+exp(-beta|j-alpha|))²; bounded mean energy gives subsequential compactness.10 exact source identities,2 direct full-builder and12 spectral controls passed; all-N reasoning independently reviewed.",
      "scopeEN": "Same full free cylinder,sea,time,energy scaling. No extra independent finite-energy multiplicity; NOT a complete classification of local scaling nets. Eightfold folding with another coordinate/locality/stress dictionary remains possible. Other defect sectors,vacuum subtractions,sources or interactions need a new map. No P1/E8/T1-T8 closure.",
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      "titleEN": "Scalar Z4 menu: only half twist D8-local; spinor chiralities separated",
      "statementEN": "q=(r/4)^8+m gives q.(e1+e2)=r/2+integer: only r0,2 relatively local to ALL original D8 currents. The64 weight1 quarter dressings are relatively odd but fail stronger full monodromy-1; parity projection/cocycle signs alone cannot repair this. Half twist has two128 chiralities. At unchanged delta full half-integral cosets have unique minima K0 even andK1/32 odd in BOTH quarters; leaving the +/-1/2 cube costs>=17/32. Prior extra-zero-state objection applies only to even extension. Unchanged integer Cartan generators still exclude every nonzero half-charged operator satisfying actual integrated covariance. Original W4 contains permutation and integer carry,not pure spinor translation.",
      "scopeEN": "Only four uniform scalar twist classes in GIVEN D8 target with fixed time/charge dictionary. Integer periodicity/W4 distinction reuse original findings. Monodromy establishes neither new local fields,chirality/gauging selection nor exclusion of all P1 defects. New sector representation with shifted target Cartan spectrum leaves unchanged-carrier premise; its origin remains open.",
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      "titleEN": "Odd-spinor re-fielding preserves H/Q/vacuum but fails original local inclusion",
      "statementEN": "22Oct joint The reviewed audit: Phi=I on even D8,rho=Pp+r on odd v with oscillator permutation is an exact global unitary Z8->D8+c preserving full H,Q,vacuum. [Pr=-r,Pdelta=delta+r,r²/2+delta.r=0] proves all energies; theta Q/parity controls throughE32 agree in bothquarters. SOURCE winding1 transports to q->R^5q+lambda,lambda=(1/2)^5(-1/2)^3,charge1,K-increment(Q+alpha)/8,eight-step carry(1)^8. But a local target Cartan Weyl unitary becomes Pi_even W+Pi_odd W*,whose commutator with remote odd CAR-B equals parity B(W-W*)!=0. No unchanged proper-local CAR inclusion. Separately,v774 anchor bit fails uniform Spin10-invariant NS/R-label identification since rank[C,E45]=4.",
      "scopeEN": "Global comparison at same source energy/charge,with changed odd D8 action. No local spinor/FE-GEN/ALG-EXH/P1-core-circle theorem. Other gauged observable nets and outer DHR extensions not excluded. SOURCE W1 remains global with inherited cocycle phases; both edges retained. Symbol-only Pfaffian/J-grading proposals rejected by type checks.",
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      "titleEN": "Direct sharp half flux fails register/quotient locality and local normality",
      "statementEN": "22Oct pasted half-flux report independently checked: F_A Z F_A*=-Z while registerZ stays even; finite matterAdP_A inner despite transport/fusion/exchange relations; v746 pair-currentK_j even,clock-invariant and half-flux-flipped in each base window. Direct sharp chiral sign change has non-HS covariance difference near jump with log((eps+delta)^2/(4epsdelta)) divergence; Araki root-HS necessity and even crossed-product extension exclude local normality also on standard even free net. Current v1033 builder Nx8Ny8 confirmsR(d,k)=(2,1),NS(0,0); uniform fixed-filled R has<nLnR>0 versus quasifree1/4,purity1/2. Common flux changes both edges. F_A does not identify complete prescribed NS/R state responses.",
      "scopeEN": "Only unchanged sharp actions under their separate actual contracts. Smooth punctured-circle sectors may be locally normal and globally inequivalent and remain open; full-CAR implementability does not imply trivial even sectors. Phi/Sigma witnesses not identified without a map. RightHom(s tensor beta,beta) absorber distinguished from charged field. No all-P1 assertion or microscopic E8 closure. Unavailable user18+38 package not claimed replayed; own original controls and analytic proofs separated.",
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      "titleEN": "Register charge obstructs renormalized local half-flux fields under full word contract",
      "statementEN": "Any arbitrarily long/smeared/renormalized original grade2 sequence obeys exact omega(T*Z*T Z)=-omega(T*T); a bosonic field local away from retained even local Z gives +omega(T*T). Positive nonzero field norm and full joint words exclude locality even without bounded/unitary limit. Shared ZL=ZR excludes faithful left-only flux flip under same full label contract. An old A/A*-projector retaining full even matter and actual QWZ time is register-diagonal; retaining F/D ties r,r+2 or all r. It does not remove register obstruction.",
      "scopeEN": "Only specified grade2/register/word/projection contracts. Changed physical nets, globally supported register Wilson loop, other originally derived sector actions and edge absorbers not excluded. No all-P1 no-go and no assertion that every spinor has this register grade.",
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      "titleEN": "Exact conditional parity-Gauss representation localizes half flux without new physical dimension",
      "statementEN": "Explicit gauge-orbit isometry E is onto full Gauss space for all sizes, E*E=1,GxE=E,HnewE=EHold,WE=EZ,tau_a^x E=E F_[0,a]. Full even register-matter dimension, all transported state words, positivity, adjunction and common original time preserved; old local register Z becomes GLOBAL Wilson loop. Exact whole128dim control and48 actual width8 one-body gauge cases check signs; all-size theorem rests on orbit proof. Mere word-preserving gauge representation cannot give complete Dirac vector-current response a new central phase.",
      "scopeEN": "Redundant kinematic generators and local Gauss/observable rule are ADDED premise, not P1-derived. No new physical dimension relative to EVEN original carrier, but different local net. No QWZ/Dirac-source identification, spinor-field limit, FE-GEN/ALG-EXH/E8, chiral measure selection or mirror absorption. f(Z) references can become nonlocal.",
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      "titleEN": "Cover half flux is not uniform eightfold spinor twist in fixed charge dictionary",
      "statementEN": "Original one-copy eightfold: C(-h)^8=-hI, while desired uniform spinor twist (-C(h))^8=hI. No complex charge-preserving intertwiner of given holonomies. Both-quarter h=±i countercontrol: real SO16 O=realify(diag((-1)^j))K DOES conjugate holonomies but reverses Jcomplex and original U1 charge. Uniform BASE Z2 spin lift is invisible to cover holonomy at even degree; global parity×deck relabel still creates no localized charged intertwiner.",
      "scopeEN": "Exclusion only for fixed charge/connection dictionary. No blanket SO16 or spinor-sector no-go. Eight independent copies, physically derived charge conjugation or changed field/domain connections separate. Source/P1-circle, base-deck/clock, chiral fermionic measure and local field products still require derivation; no new internal Z8^f/Arf branch. Neutral-pair refinement: eight-preimage parity pairs T_E with canceled register and unchangedh are NOT excluded by this single-sector comparison.",
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      "titleEN": "Original neutral eight-preimage pair gives uniform16component cut action",
      "statementEN": "T_E=product(F_B F_A*)=I4 tensor(-1)^N_E is an actual original register-grade0 operation. Periodic sigma(theta+2pi/8)=sigma(theta) acts on all8complex/16real folded fields as(-1)^1_I and retains r,h. SINGLE cover-half-flux argument C(-h)^8!=(-C(h))^8 does not apply to this NEUTRAL PAIR. Full original QWZ Nx32Ny8 mass1 control retains all256occupied modes perquarter; both edges changed, old seam phase retained. Finite-band comparison factorizes exactlyD_l(I)^8; odd-l half-circle blocks vanish, artificial second Fermi edge explicitly retained.",
      "scopeEN": "Constructed neutral pair and finite field action, NOT a new single spinor sector or analytic OPE-monodromy theorem. Eight-preimage support remains multilocal on source. No FE-GEN/ALG-EXH,128,P1-source/time/measure derivation. The reviewedRound5 pair misidentification retracted inRound6 after explicit counterexample; no uncritical review adoption.",
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      "titleEN": "Complete Dirac vector phase pairs have no half Cartan Ward charges",
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      "scopeEN": "Only normalized smooth UNITARY vector-state sequence on fixed compact source circle. Nonunitary sums/field intertwiners,microscopically derived charged covariance cocycles and decompactified time outside scope. Local normality away from removed point is not finite global original energy. Individuallambda_i±1/2 distinct from totalQ=sumlambda_i; two-point exponent/energy alone no128/FE-GEN/ALG-EXH proof.",
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      "titleEN": "Nonunitary limits with continuous original time stay in old source folium",
      "statementEN": "Arbitrary v_N=T_N Omega_N in fixed full strip obey tail_E<=[||v_N||²-<v_N,e^(-tH_N)v_N>]/[1-e^(-tE)]. Finite nonzero norm and limiting-kernel continuity at0 imply energy tightness. Entire finite spectral windows yield norm-subsequence compactness; fixed low-window operator matrix elements transfer all retained bounded-observable states into old global folium. Real original time also suffices: tail_E<=2[n_N-a^-1 int_0^a ReG_N(t)dt] forE>=2/a. No bounded mean energy,||T_N|| or word degree needed. Under fixed D8/even-CAR dictionary no additional s/c spinors; right edge only adds multiplicity.",
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      "statementEN": "Actual e^(-sH_N)P_E Omega_N is pureSlater: Zp=B A^-1,Zp_s=e^(-s eps+)Zp e^(-s eps-),C_s=V_s(I+Zp_s*Zp_s)^-1V_s*. All finiteCAR words follow; exact normalized time kernel det(I+Zp_(s+t/2)*Zp_(s+t/2))/det(I+Zp_s*Zp_s). Nx32Ny8/bothquarters/all512modes checked independently by direct Slater/QR;8JOINT residues retained,both edge responses nonzero. SEPARATE given chiral Weyl comparison atd=pi has exact F_s(t)=[(1+r e^-t)(1-r)/((1-r e^-t)(1+r))]^4,r=e^-2s. s->0 gives norm1at0,kernel0fort>0; onlythis comparison limit explicitly violatesD.",
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      "scopeEN": "Source-reviewedconditionaluniqueness reused,notrefutationofabstractmaintheoremorgeneralKMSreconstructionno-go. ThistheoremalonedoesnotcloseFE-GEN/ALG-EXH/P1. MCPnavigationlimits separatelydocumented: exactclaimnot_compiled despiteknownproblemreviewedfindings;450kpacketexceeds60kAPI. Completecanonicalprogram.jsonrecordchecked; noserverfixorfullMCPpacketdeliveryclaimed.",
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      "titleEN": "2+2 excludes 3+1 only under common parallelity",
      "statementEN": "Under the explicitly additional premise that beta preserves the fixed marked irreducible SU(3) family connection, its commutant is scalar and beta must have the 3+1 form diag(b,b,b,-3b). Thus a beta packet with eigenvalue multiplicities 2+2 is excluded under that premise only; no original parallelity rule follows.",
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      "statementEN": "For C=(I+sgn H)/2, H_a=diag(1,a,-1,-a), a>0, gives the same fixed projector C and rho_0 but different positive energies. Thus [rho,C]=0⇒[rho,H]=0 is false without an extra premise. The counterexample concerns only inference from state/sign covariance to complete dynamics.",
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      "statementEN": "For Phi(X)=7X/12+sum_{j=1}^5 Gamma_j X Gamma_j/12 on M4 with the specified Clifford matrices and 1+5+10 spectral blocks, no Phi^n for positive integer n is the exponential of a time-homogeneous GKSL generator on the same M4. The proof covers all logarithm branches using block traces and Clifford twirling; the forced canonical coefficient beta=n log(3/4)/16 is negative. This is a different channel from the shared S3 register hierarchy.",
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      "titleEN": "The 55/117 ratio is closed; absolute scale remains a separate anchor",
      "statementEN": "The active TFPT text states that Q geometry, det R=8, and rigidity of c_u/c_d=55/117 are established on the derived stratum; the continuous wall point U_f* is needed only for absolute amplitude normalization. The ratio is not an open monodromy/amplitude calculation.",
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      "titleEN": "Minimal EM transfer is exact in its stated response contract",
      "statementEN": "For the defined rank-one projector U_Sigma^min and stated parameters, T_Sigma=(1-delta_top exp(-2 alpha))^(-5/4) and the displayed scalar closure equation follow algebraically. This calculation does not independently prove physical selection of that projector, parameter set, or transfer to the Thomson readout.",
      "scopeEN": "Historical minimal-transfer definition with fixed readout; the single line has since been canonically reconstructed on the old circle. Neither this reconstruction nor independently proved root uniqueness is retracted. Original physical P1/EM mapping and renormalized Thomson observable remain separate.",
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      "titleEN": "Existence and uniqueness of the positive root are proved",
      "statementEN": "Log-convexity of the stated seam response and the sign before x=2c³ prove exactly one positive simple root globally. For M=41 its reciprocal is approximately 137.0359992168407. The analytic proof is distinct from the 10 rerun symbolic/interval checks. It holds for every M>0 and therefore does not select M=41.",
      "scopeEN": "Mathematical theorem for the fixed function; not a cubic polynomial because the seam response depends on x. No parameter or physical Thomson-transfer derivation; ledger Numerical unchanged.",
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      "titleEN": "The marked D5 grade response derives exponent5/4 from the same E8 source",
      "statementEN": "Exactly G_u(q5,q3)=[(1-u)/(1-uq5)]^(5/4)[(1-u)/(1-uq3)]^(3/4). Hence G_u(q,1)/G_u(0,1)=(1-uq)^(-5/4); the common-time diagonal G_u(q,q) instead has exponent2. Powers follow from actual charge norms and the convergent oscillator series. Normalized local T5 insertion has Ward coefficient5/8. Fixed C/J do not generally preserve P5 but jointly transport charge and marking covariantly.",
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      "statementEN": "The active original explicitly treats the three-term decomposition and coefficient identities as exact, retaining the unique alpha root. The same section retains EM.WARD.01/ALPHA.QUILLEN.EXACT.01 as the existing origin obligation for the actual AQFT/zeta determinant functional. Separately the old circle canonically gives Pi and det_F(1-qPi)=1-q. Its actual local EM response, relative Quillen norm and susceptibility exponent5/4 still require identification. This does not retract compiler identities or root uniqueness.",
      "scopeEN": "Source-reviewed status/type correction, not a new general nonderivability theorem or gate. P1/P2 remain accepted starting principles; being axioms is not an objection. The line is derived on the old circle, not an arbitrary vector to select again. v382 replay26October:6/6; literalTrue status checks do not prove open functional origin. Ledger Numerical and full physical status remain unchanged.",
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      "titleEN": "Historical alpha factors and definition steps reconstructed from original sources",
      "statementEN": "The reviewed older originals provide concrete provenance contracts: V1.06 uses g=n/(8pi²), n=1 and g=8c3²; v2.7 couples the defect through J=d SF=2; v4.2 gives gamma=5/6, tr(S+)Y²=10/3, b1=41/10 and B gamma=5/4. The minimal rank one transfer yields exactly (1-q)^(-5/4). These numbers must not be dismissed collectively as unexplained fitting factors. The same originals state a narrower complete identification: v2.7 calls the CFE coefficient motivation heuristic, while v4.2 supplies P_loc'(alpha)=alpha³-2c3³alpha² as a condition and defines Gamma by integrating the desired response. The master proof factorizes blocks and then identifies the U(1) block with the already defined EM action; this passage does not evaluate an original coupled Dirac determinant to obtain that particular F. It therefore does not additionally close the existing physical Ward origin obligation.",
      "scopeEN": "Source provenance and exact algebra under each stated normalization, carrier, primitive sector and defect response contract. No general nonderivability theorem, no retraction of the canonical circle defect line, marked D5 response or positive root uniqueness. No new physical selector or ledger promotion. Historical linear, exponential, quadratic and resummed openings are different calculation contracts; similar coefficients do not prove identical physical actions.",
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      "id": "finding:matter-observations-do-not-fix-geometric-response",
      "titleEN": "Matter readouts do not fix geometric response; original functional remains fixed",
      "statementEN": "In the matter-only observation-class audit, geometric replica responses remain indistinguishable; the scope correction explicitly states that W→W+lambda Q[g] is therefore not automatically an admissible change to the original profile-free Gamma_grav. The original functional is fixed once the full operator, reference, domain, and continuation are fixed; its common off-shell geometric family and normalization must be derived from the source.",
      "scopeEN": "Applies as an identifiability limit for matter-only observations and an open bridge to the fixed original functional; it is not an alternative TFPT source or universal gravity no-go.",
      "status": "conditional",
      "problemId": "problem:gravity-flow-normalization",
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      "titleEN": "Four clock fields generate complete quadratic responses",
      "statementEN": "P5, CPCᵀ, JPJᵀ and C²P(C²)ᵀ generate all36 real-symmetric8x8 matrices through Jordan words up to degree4. The double-pole OPE realizes these products. An explicit rational inverse reconstructs A from36 mixed responses Tr(BℓA)/2; prescribed trace leaves35 directions.",
      "scopeEN": "Complete matrix reconstruction in the specified neutral quadratic vacuum sector; neither four measurements nor all charged, state or multitime responses of a physical source.",
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      "titleEN": "J transports the whole source tuple",
      "statementEN": "The original cocycle admits a unitary lift UJ with UJ⁴=I. It intertwines Kδ,Qq and the full quarter transition with KJδ,QJq,tJτ on the entire stipulated Fock space. The four frames are a covariant family; q must be transported.",
      "scopeEN": "Given E8 completion; no physical sectors, switching rates or local instruments selected from P1/P2. As a torus operator τ is identifiable only modulo the dual lattice.",
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      "titleEN": "The same fixed time operator disallows this affine J lift",
      "statementEN": "An isometric affine lattice lift Ap+r with A²=−I and unchanged oscillator level would require r=Aδ−δ and hence −2δ=(I+A)r in the E8 lattice. Adjacent coordinate differences of2δ are1/4; this fails exactly. Also Jq≠q.",
      "scopeEN": "Exclusion only of the affine lift model at fixed K,Q and oscillator level; covariant tuple transport remains possible.",
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      "titleEN": "All neutral vacuum words remain twist-blind",
      "statementEN": "Each admissible intermediate insertion stays in M(1)⊗|0>. There Kδ=N,Qq=0,tτ=I. All ordered vacuum multitime and Gram responses in this class are therefore identical at arbitrary word length, even with complete36-dimensional quadratic tomography.",
      "scopeEN": "Individually all-Cartan-neutral insertions and fixed vacuum; not charged intermediate fields, charged Gibbs traces or δ-dependent probe operators.",
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      "problemId": "problem:charged-twist-response-identifiability",
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      "titleEN": "Eight independent charged response directions reconstruct δ",
      "statementEN": "Cδ(p,t)=exp[−t(|p|²/2+δ·p)]. With yi=−log[Cδ(bi,t)/Cδ(−bi,t)]/(2t), y=Bᵀδ and δ=(Bᵀ)⁻¹y. Eight independent directions suffice and are necessary for unrestricted δ in this linear data class.",
      "scopeEN": "Known preparation/time normalization in the given E8 Fock space; identifies δ, not q,τ, physical instruments or P1/P2 source selection. Eight directions require their respective ± responses.",
      "status": "established_scoped",
      "problemId": "problem:charged-twist-response-identifiability",
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      "titleEN": "Unchanged native source is not the fixed chiral response",
      "statementEN": "The native ratio is one; the fixed quarter twist gives exp(−2tδ·Bp), which differs from one on at least one original basis direction.",
      "scopeEN": "Only the fixed marked response identification at these starts and the same time.",
      "status": "excluded_scoped",
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      "statementEN": "Original C T_p C=T_−p, T_p*=(-1)^e(p)T_−p and C H C=H. At an invariant start all C-conjugate bounded words agree.",
      "scopeEN": "Unchanged finite source and specified invariant start; different states or operator dictionaries are not excluded.",
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      "titleEN": "C0 does not hide charged intermediate sectors",
      "statementEN": "η·Qtot is zero on C0. T_p enters Cp; full propagation yields ratio exp(−2tη·p). For v=Bp, conditionally η=Bᵀδ. Only the odd factor is matched.",
      "scopeEN": "Balanced W remain within the sector; T insertion/removal responses require Cp. No derived physical measuring instrument.",
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      "titleEN": "Fixed hopping phases do not produce the required asymmetry",
      "statementEN": "C and K may separately break; A=CK preserves H for all fixed link phases and T_p→T_−p. Invariant starts retain equal Euclidean opposite-charge returns, including quarter phases/flux.",
      "scopeEN": "Fixed phases and corresponding invariant starts; real time retains conjugation/time reversal. No assertion about all P1/P2 sources.",
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      "titleEN": "Finite search closed, actual winding and parity equivalence",
      "statementEN": "262144 candidates on two H200: four universally Lean-checked presentations. X_plus=Pi X_minus preserves source,Q,K,quarter marker and canonical state; two marked P classes remain. Full T1 and Ward are checked. Graded C lifts realize both P signs; scalar X covariance does not follow from the integer endpoint.",
      "scopeEN": "Global crossed operators; no local field equivalence or physical selection.",
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      "id": "finding:constructive-positive-state-extension",
      "titleEN": "Positive finite-word state extension",
      "statementEN": "E0(y*y)=a0*a0+phi^-1(a1*a1) proves positivity of omega0 o E0. General algebra/star/positivity and dual uniqueness theorems are Lean-checked; concrete finite-lattice transfer is analytically proved.",
      "scopeEN": "No complete concrete Finsupp kernel instantiation, physical state selection or automatic K-invariance of the bare shift algebra.",
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      "titleEN": "E8 current group conditionally determines the P sign",
      "statementEN": "In the supplied positive E8 module X=E_s commutes with SU8 root currents because alpha.s=Q(alpha)=0; oscillators are fixed. C=Z_e0 R gives CXC^-1=±iX and P sign+1. This is a direct global operator bridge, not X=Y(s).",
      "scopeEN": "Only the stipulated even E8 lift; full odd-Z8 sectors remain additional. No transfer of the different (9,1) source. The ±i path orientation is not selected.",
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      "titleEN": "The same shift functional hides distinct ground states",
      "statementEN": "All dual-invariant ground mixtures diag(a,1-a) on {0,r0} give the same shift functional. Original Cartan vacuum values select |0><0|; additional Rc invariance requires I2/2. These two conditions conflict.",
      "scopeEN": "Normal ground densities in the supplied K space; no state selection from P1/P2. The ground space was already known.",
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      "titleEN": "No phase hit realizes the unchanged source",
      "statementEN": "All four hits add the same half-integer Cartan sectors and second K-zero vector. The already known integer-Cartan/ground-space obstruction therefore excludes equivalence to the unchanged source; positive algebraic extension remains possible.",
      "scopeEN": "Only the named unchanged CAR/QWZ representation with the same time. No TFPT no-go; choosing a state removes no added zero vector.",
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      "titleEN": "Concrete pair operator; source neutrality is not hypercharge neutrality",
      "statementEN": "epsilon1 epsilon4 produces u1∧u4 in the carrier: Y=1/6 and source charge change0. Root operator, spinor carrier vector and affine root-current state have different types. The exterior unit is the internal nu^c label, not the affine vacuum.",
      "scopeEN": "Original carrier plus supplied E8 charge dictionary; no physical field identification.",
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      "titleEN": "Fixed quarter time splits the color/weak multiplet",
      "statementEN": "Mdelta=(-1/2,0,alpha/4,-1/2,0,(alpha-1)/4,-1/4,-1/4). The six (3,2) root energies are (0,1/2,1/2,1,1/2+alpha/4,1+alpha/4). K therefore does not commute with the fixed full SM action. The quarter phase is retained.",
      "scopeEN": "Only the supplied affine dictionary. Backgrounds or time-dependent covariant symmetries have different premises; no general gauge-physics no-go.",
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      "titleEN": "Explicit conjugation for the existing sheet pairing",
      "statementEN": "J_C=∏(epsilon_i+iota_i)K is antiunitary, exchanges S± and reverses Y. Exactly Lambda0 and Lambda5 have Y=0. J_C dGamma(h) J_C^-1=tr(h)I−dGamma(h). It is fully Spin(10)-equivariant. Reuse CAR.PAIR.01's bilinear pairing; CAR.PAIR.02's Hermitian selection inference has been corrected and is not evidence.",
      "scopeEN": "Representation with metric/orientation; J_C is not yet the actual boundary/OS conjugation. Two carrier labels are not two vacua.",
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      "id": "finding:partner-chiral-quarter-conjugation-state-time",
      "titleEN": "Actual partner map between quarter domains",
      "statementEN": "n→−n−1, alpha→1−alpha gives C D_alpha C^-1=−D_(1−alpha). Particles become holes and conversely; normal charge reverses and positive excitation energy is preserved. The unique sea state maps to the other sea state without adding a zero vector within one domain.",
      "scopeEN": "Given chiral source. No selection from P1/P2, no equal responses within one fixed twist, and no identification with affine r0.",
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      "titleEN": "Bilinear sheet pairing is not Hermitian kernel selection",
      "statementEN": "CAR.PAIR.01 counts complex-bilinear singlets. The Hermitian scalar <f,g> already exists on S+; CAR.PAIR.02's 'scalar iff sheet-odd' inference is false in this reading. The previous J_C includes reversion signs and is fully Spin(10)-equivariant.",
      "scopeEN": "Correction of the finite selection inference; no identification of I with the actual Calderon operator and no P1 countermodel.",
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      "titleEN": "Actual quarter conjugation transports the supplied marking",
      "statementEN": "n→−n−1 forces j→7−j and O=−MRM^T with O(x)=(x4,−x5,−x6,x1,−x2,−x3,x8,x7). O d_alpha=d_(1−alpha); P5_beta=O P5_alpha O^-1 and k_beta=k_alpha∘O^-1 need no new marking choice. Sea state and positive time are transported; the E8 current lift assumes the existing extension.",
      "scopeEN": "Exact map of the supplied source and conditional E8 lift; first marking, source realization and physical event law are not derived from P1/P2. Both quarter domains remain explicit; no switch to another branch to solve a fixed-sector question.",
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      "id": "finding:joint-fixed-time-marking-repair-excluded",
      "titleEN": "No time-stabilizing orthogonal repair of the fixed marking",
      "statementEN": "After actual quarter conjugation the D5 projections of the time axis have distinct squared norms 129/256 and 137/256. Hence no orthogonal T satisfies T d_beta=d_beta and T O(P5)T^-1=P5. This excludes every Weyl correction in this fixed comparison.",
      "scopeEN": "Only fixed fold time and fixed 5+3 marking. Covariant marking transport is positively constructed; other sources or dynamical dictionaries are not excluded.",
      "status": "excluded_scoped",
      "problemId": "problem:seam-partner-charge-state-time",
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        "premise:given-first-marked-quarter-character"
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      "titleEN": "Spinor16 is a state sector, not a 16-real one-particle field",
      "statementEN": "D5_1 has ten real vector fermions; S± each contain 16 complex Ramond/spin-field states. Using S+ as an ordinary CAR coefficient would give c=16 and Spin(10) level 4. The documented c=8 connection uses 10+6 real fermions and mixed spin fields of weight 5/8+3/8=1.",
      "scopeEN": "Representation types of the supplied free realization; no selection from P1/P2 and no new spinor-CAR replacement.",
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      "titleEN": "Operator synthesis implemented and calibrated on known cases",
      "statementEN": "A CEGIS word search generates 2048 Majorana word/linearity classes and recovers the known five-mode particle-hole map. An automatically found imaginary mixing component eliminates the linear candidate. A second solver constructs simultaneous linear/antilinear intertwiners from exact operator pairs; evaluation identities include constraints, code and search configuration. These are software and recovery results, not new source selection.",
      "scopeEN": "Supplied finite representations. Clifford monomial grammar and generic equal matrix dimensions up to16; not a complete operator-program, kernel or continuum search. Twelve generic Lean transport lemmas; concrete matrices checked separately and exactly.",
      "status": "established_scoped",
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      "premiseIds": [
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      "titleEN": "Unknown kernels constructed, exactly completed and discriminated",
      "statementEN": "The new moment search takes no supplied H, rho or target K. A one-mode result has an exact flat GNS certificate. Two additional minimal five-mode completions replay all 2245 moments through degree4. Checking the original source requirements adds finite clustering: a previous result with 27 stationary directions fails; two replacements each have exactly one stationary direction. Both yield finite unitary field time from the same T and first differ at degree5 by 1/4 versus 3/4. An independent offline verifier confirms full kernels and separation.",
      "scopeEN": "Declared finite CAR/transfer class. Five-mode completion additionally assumes a minimal irreducible realization with inferred pure occupation state; no uniqueness inferred from truncated ranks. Floor and cluster gap 1/4 are search bounds, not quarter holonomy or physical scale. Finite clustering is not a local OS net. No original selection of marking, statistics or events. Historical extremal calibration; since2026-10-16 implementation defaults to feasibility with explicit transfer classes.",
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      "titleEN": "Search conditions corrected; duplicated control separated from the actual charged source pair",
      "statementEN": "The default is feasibility without optimizing state or response. The explicitly duplicated lowest-mode calibration gives1/16; the actual lowest quarter-source particle/hole pair gives1/256. A global floor1/4 is an artificial restriction. Thermal GNS transfer with spectrum1,1,1/4,4 passes at bound4; bound1 is a class mismatch. An unsupported pure Fock completion remains inconclusive.",
      "scopeEN": "Executable finite software and source controls; no physical source selection. Historical min/max results are calibration endpoints, not preferred solutions. The actual charged-pair SCS status is optimal_inaccurate; the exact matrix construction independently proves feasibility, not clean numerical optimality.",
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      "titleEN": "Complete CAR boundary projection determines the multi-field operator ray",
      "statementEN": "If the full two-face graph G is the actual CAR covariance, positivity and CAR force its state ray without a separate Wick assumption. The specified oriented input/output pairing reconstructs Q proportional to the direct sum of Lambda^n R; multi-field amplitudes are minors and Q(R2R1)=Q(R2)Q(R1). Complex frame changes and noncommuting composition are checked. Vacuum and one-particle responses alone are insufficient.",
      "scopeEN": "Canonical fermionic quantization of supplied full boundary data. A geometric Cauchy projector is not automatically a physical CAR covariance; boundary pairing is also a premise. The absolute scalar, partition determinant and phase are not recovered from normalized G. Original action permits higher interaction vertices; general OS sewing from complete Schwinger functions remains separate.",
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      "titleEN": "Known quarter source fully controlled with state and operator tail bounds",
      "statementEN": "In the supplied sea-relative chiral CAR model the same particle-hole map produces charge N_+-N_-, positive energy and W=exp(2pi iH)=i^(N_+-N_-). The original signed two-mode amplitude diag(1,1/4,64,16) becomes diag(1,1/4,1/64,1/256) through that map and the separate sea factor64. The infinite weighted l2 domain, geometric trace tails and explicit operator-norm/Choi-HS bounds control state and time beyond finite windows.",
      "scopeEN": "Complete construction within the already supplied free source. No P1/P2 selection or infinite particle-hole unitary in empty Fock space. Convergence is in a common HS/Choi space, not of raw filled Slater vectors in empty Fock space. Relative partition sums do not recover the divergent absolute sea scalar. A single chiral D_alpha branch; the entire doubled graded collar is not identified with the same time/loop generator.",
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      "titleEN": "Reference amplitude retained as search input; direct false certification closed",
      "statementEN": "Search and both entry paths retain T Omega=lambda Omega with explicit positive lambda, matching bounds and cache identity. A rational scaled control kernel passes at lambda=1/16 and contradicts the same moments at lambda=1. Pure Fock completion accepts only vacuum_effect, lambda=1 and ceiling1; other classes remain inconclusive. omega(1)=1 normalizes only the conditional state and does not reconstruct Z[0]. In the actual original OS call, unital translation fixes [1] and its matrix element is time independent. This does not identify it with the nonstationary old cap vector as proposed by The reviewed. Retain the original reflected insertion caller omega_rel(Theta F tau_t G); a cap insertion must identify the actual old sea vector. The reference cancels from Zrel[J]/Zrel[0], not from geometric variation of logZrel[0].",
      "scopeEN": "Software correction with supplied lambda; no original absolute amplitude selection. The scaled control matrix is not a new physical quarter source. The added unit test concerns one fixed OS system; unnormalized unit norms may vary across changing geometries/sources. No independently proved pointwise scalar identity or general origin route is excluded. Distinguish source derivatives from geometric derivatives and reuse the existing coupled-reference result.",
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    {
      "id": "finding:joint-fermionic-source-schur-functional",
      "titleEN": "Field response and unnormalized factor from the same free integral",
      "statementEN": "Direct Grassmann integration gives det K exp(bareta K^(-1) eta). Schur reduction jointly retains det K_ii and the inverse effective boundary/mode matrix. Two collars of the actual lowest quarter particle/hole pair under the supplied fermionic trace give det K=69649/65536 and occupations1/17,1/4097. Direct identification of the primitive spectral action with integrated complex fermions requires x f'(x)+1=0, hence a logarithmic profile; arbitrary f is insufficient.",
      "scopeEN": "Exact for supplied free K, measure and source coupling. The example integrates a hole mode, not a geometric collar interior. Coherent source contraction K^-1 is not normal-ordered occupation. No general no-go for a separate bosonic spectral action plus fermion determinant and no original P1/P2 quantization.",
      "status": "established_scoped",
      "problemId": "problem:automated-operator-source-synthesis",
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        "goal:T2",
        "goal:T8"
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      "premiseIds": [
        "premise:given-complex-fermion-source-quantization"
      ],
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      "id": "finding:quarter-circle-absolute-zeta-green-eta",
      "titleEN": "Quarter circle: absolute determinant1-i and joint Green/eta response",
      "statementEN": "For supplied D_alpha, zeta quantization gives det D_alpha=1-exp(2pi i alpha), exactly1-i at alpha1/4 with eta(0)=1/2. The same operator gives G_-=i/(h-1), G_+=ih/(h-1), jumpi and retained holonomyh=i. The joint source derivative contains partial_alpha log det=pi cot(pi alpha)+i pi; the extra i pi is the eta contact term. The general determinant formula also retains zeta(K²;0).",
      "scopeEN": "Supplied zero-mode-free angular circle operator with fixed spectral cut. Not the full physical TFPT source operator, not thermal occupation, and no identification of the diagnostic reference branch with D_ref. Transport periodic operator and source frames together when varying alpha.",
      "status": "established_scoped",
      "problemId": "problem:automated-operator-source-synthesis",
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        "topic:quantum",
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      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
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      "premiseIds": [
        "premise:given-angular-quarter-zeta-cut"
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      "id": "finding:quarter-sea-absolute-thermal-central-phase",
      "titleEN": "Absolute thermal sea regularization retains central phase",
      "statementEN": "The declared Abel/zeta subtraction gives E_sea=-1/96. At beta=2ell=8log4, q_beta=1/65536 and Z_beta[0]=exp(-beta E_sea) times the complete particle/hole product. Its two-mode factor is2^(1/6)69649/65536. Adjoint field action stays i/-i; the absolute time-loop operator additionally carries exp(-i pi/48). Existing tail bounds apply at the same beta. Under the same Hurwitz prescription E_sea=eta(0)^2/8-1/24 and d_alpha E_sea=-eta(0)/2. The logarithmic alpha derivative of the unnormalized thermal partition sum agrees with the occupation response including the sea term, checked by an independent Fock trace and explicit derivative tail bound. The resulting actual reference amplitude2^(1/12) passes the existing CLI moment search with206 source moments; the same data with reference amplitude1 are infeasible. This is a numerical software control under the same supplied regularization.",
      "scopeEN": "Conditional absolute extension of the supplied single chiral source. Regularization is input; neither subtraction nor bulk/eta cancellation of the central phase is proved from P1/P2. Angular determinant and thermal partition sum are different functionals. The finite excitation-product control already retains the full regularized sea term. The response is twist-conjugate spectral charge, not a new physical event law.",
      "status": "established_scoped",
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      "premiseIds": [
        "premise:actual-chiral-opposite-quarter-domains",
        "premise:charged-car-negative-sea-source",
        "premise:given-thermal-sea-abel-subtraction",
        "premise:given-angular-quarter-zeta-cut"
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      "id": "finding:original-operational-seed-and-given-green-pairing",
      "titleEN": "Original reconstruction uses the stronger operational seed",
      "statementEN": "P1/P2 does not fully specify the primitive kernel's field arguments. The archived reconstruction starts with algebra, time flow, state and collar operator; CAR and anticommuting sources are later premises. For a supplied product Dirac operator, the classical Green boundary form follows from normal symbol and boundary metric. Identification with the quantum field amplitude does not follow.",
      "scopeEN": "Source audit and reuse of the known geometric Green identity. No derivation of fermionic statistics or physical operator from P1/P2. the reviewed source checked against current Ludewig/Roos primary text1909.04212v4: algebraic twist also needs closed harmonic determinant factor; published2020 coherence proof correction is in v4. This is no new original source derivation or established error in direct TFPT collar proofs.",
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    {
      "id": "finding:order-one-infinitesimal-bound-inference-fails",
      "titleEN": "Bounded coefficients do not imply infinitesimal relative bound",
      "statementEN": "The inference in original source01:397-405 is invalid: P=aA has relative bound |a|. The independent quarter/reflection control A=sigma_x p_theta+sigma_y p_r, P=a I p_r retains the quarter domain; at a=1, A+P on H1 is not closed. Repair: an actual relative bound<1, bounded symmetric perturbation, or a separate theorem for the total elliptic operator and boundary domain.",
      "scopeEN": "Only the general proof step is excluded. The a=1 control is not an admissible elliptic Dirac operator or a P1/P2 countermodel.",
      "status": "excluded_scoped",
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      "id": "finding:joint-ward-two-field-data-miss-fixed-four-field-response",
      "titleEN": "Joint Ward and two-field data do not select the multifield response",
      "statementEN": "For all alpha,beta, H and UHU* retain the full thermal partition, charge, quarter phase, sea energy and joint eta/sea/Ward identity. U fixes vacuum and one-particle sector and hence all vacuum two-field time responses. Yet the fixed neutral pair response at t=log2 is 1/2 versus13/50 after removing the same common sea factor.",
      "scopeEN": "Exact independently checked inverse-data control. Equivalent when all fields are conjugated together; locality of U is unproved. Not a physical alternative candidate or full P1/P2 countermodel.",
      "status": "established_scoped",
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      "titleEN": "Complete CAR kernel also determines the interacting link action",
      "statementEN": "The full coherent CAR kernel is invertible and sews exactly. Its4^n duality values determine finite gluing; T00·exp(-S_nil) reconstructs all link vertices. Existing five-mode certificates give actions through degree10. The charged interacting control retains actual quarter time and absolute sea factors.",
      "scopeEN": "Standard coherent reconstruction, implemented and applied exactly here;1–5 modes. No P1/P2-selected field dictionary, geometric locality or original Φ-dependence. The minimum's term is an objective artifact. T00, reference eigenvalue, matrix logarithm and Grassmann logarithm remain distinct.",
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      "id": "finding:source-covariance-jet-and-mixed-face-interface",
      "titleEN": "Exact field variation of the existing covariance reconstruction and its boundary faces",
      "statementEN": "The Frechet derivative of the actual v258 reader multiplies each matrix element in an eigenframe by the strictly negative divided difference of h. For fixed sector projectors commuting with the background covariance, each sector block vanishes iff its covariance-variation block vanishes. A moving carrier embedding has additional derivative terms. The Calderon projection is not already a faithful mixed occupation: restriction to its own range gives I. The existing two-face construction correctly gives n=1-A and h(n)=2ell B, thus recovering the variation already present in B without selecting it.",
      "scopeEN": "Analytic finite/regulated derivative result plus exact reuse of the given positive particle/hole collar. No general non-Gaussian state reconstruction, Phi provenance, statement for noncommuting sector projectors or unchecked unbounded limits.",
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        "premise:typed-source-structures"
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      "id": "finding:e8-grade-two-explicit-spinor-current-action",
      "titleEN": "Actual action of the60 grade2 currents between the64-dimensional current sectors",
      "statementEN": "In the existing E8 dictionary J_r,0 acts on a grade1 root-current state as epsilon(r,p)|r+p> exactly when r.p=-1. The existing separated-point reader confirms960 nonzero actions among3840 pairs,15 per input and rank16 per root action. Twelve ordered distinct A3-family channels carry80 actions each;1024 same-family pairs of grade1 currents cannot be neutralized by any grade2 root. This is the actual internal (10,6) current action, not a selection of a Phi coefficient function.",
      "scopeEN": "Stipulated positive E8_1 representation and its internal quarter grading. Bosonic current states are not physical fermions; same family denotes the four given A3 weights, not already three physical families. No exclusion of all composite/dressed fields or later basis changes.",
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      "id": "finding:intrinsic-source-jet-removes-pure-field-frame-motion",
      "titleEN": "Intrinsic source variation separates frame rotation from genuine source/embedding change",
      "statementEN": "For C_F=Q*C_Sigma Q, with a=Q*partial Q and N=(I-QQ*)partial Q, exactly nabla C_F=partial C_F+[a,C_F]=Q*(partial C_Sigma)Q+N*C_Sigma Q+Q*C_Sigma N. The existing faithful modular reader obeys nabla h(C_F)=Dh(C_F)[nabla C_F]. Pure frame motion can generate ordinary nonzero off-diagonal entries while both covariant variations vanish exactly. Genuine normal embedding motion remains; the identity does not select Q, Phi or their original provenance.",
      "scopeEN": "Supplied finite/regulated charts; the state interpretation of h requires the quasi-free premise. No new physical gauge connection, Phi coupling or global anomaly/determinant-line triviality. Compression is not interchanged with functional calculus.",
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      "titleEN": "Known NS vector fermionization cannot supply the missing Spin10 spinor through local products",
      "statementEN": "The existing fermionization route supplies (10,1)+(1,6), as established previously. The central kernel element of Spin10->SO10 acts trivially on each local generator but as minus1 on the required internal half-spinor. Products, derivatives, OPE coefficients and algebraic/norm/weak closures in the same NS representation therefore remain centrally trivial. Choosing V_a coefficients or searching further local composites in this unchanged algebra cannot repair the matter identification.",
      "scopeEN": "Reused representation result with analytic extension to the unchanged NS generator algebra, not new fermionization. No exclusion of Ramond/twisted states, charged sector intertwiners, dressed fields with additional source, other extensions or a universal TFPT model.",
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      "titleEN": "Noncommuting two-face response with product and coherent kernel",
      "statementEN": "The full given graph yields L_f=A^1/2 M_f A^1/2, L_fL_g=A^1/2 M_f A M_g A^1/2 and positive Gram kernel <L_f x,L_g y>. The scalar quarter source yields a bilateral weighted shift with [L*,L]e_n=a_n(a_(n+1)-a_(n-1))e_n nonzero for all n. No internal algebra or coupling is inserted. This nonlocal geometric compression is not a unital multiplicative representation of scalar insertions; physical chronological execution does not follow.",
      "scopeEN": "Given full geometric collar with conditional quarter-source control. No original physical intervention, preparation or Born derivation; no internal carrier algebra or3+2 selection.",
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      "titleEN": "Raw face insertion fails null stability; graph symmetries classified",
      "statementEN": "For K_G=<x,Gy>, P0 sends the null vector (-t,1)/sqrt(1+t²) to squared norm t²/(1+t²)²>0. The raw mask is not a quotient operation; W*W=G and OW=WP0 are also impossible. Compression repairs only mathematical descent. Separately, block unitaries diag(U0,U1) preserve the full graph iff U0=U1 spectrally commutes with B. Degeneracies may retain nonabelian commutants; interventions need not be symmetries.",
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      "titleEN": "Action variation and boundary sewing do not select L_f as a physical intervention",
      "statementEN": "The primitive action derivative needs supplied D_rel/ref[j], domain, chi and eta families; logZ derivatives additionally retain measure/domain variation and a second contact term. The given collar yields T_ell=e^-ellB with geometric sewing, not A_ell: A0=I/2 and (log a_ell(lambda))''|0=-lambda² exclude a common scalar semigroup normalization on the quarter spectrum. Under the separate boundary-kernel reading K_f=F M_f F with identity kernel A, inverse-Gram sewing uniquely yields K_f#K_g=F M_fg F, commutative for scalar insertions. Ordinary noncommuting L_f operator multiplication remains valid. As f1 geometrically is an actual face mask, this kernel reading is not automatically the physical law.",
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      "titleEN": "Mixed-source completion and automatic discriminating query in the existing search path",
      "statementEN": "The normal search entry point now offers optional faithful CAR GNS completion instead of stopping thermal data at an incompatible pure-Fock ansatz. In the old supplied quarter source, two exact candidates replay all51moments through word length4. The search generates the shortest missing joint response of length5 and obtains its value only from the same existing source. All16/16transfer metric entries are then fixed in this representation; an explicit full basis intertwiner verifies the actual source action. The16dimensional two-mode control preserves neutral transfer products.37appropriate new/old regressions pass, with independent source/algebra review accepting the scope. This improves automated source recovery and software validation, not physical original-P1 source selection.",
      "scopeEN": "Additional finite CAR-cyclic representation ansatz, validated on previously given one/two-mode quarter controls. Exactness applies to rational inputs or explicitly rational-recovered numerical moments; no new P1 Wick value, physical time/reflection derivation, larger-dilation/continuum uniqueness, absolute Z scalar or T1-T8 closure. The actual local P1 insertion and joint connected four-field response remain to be derived.",
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      "titleEN": "Identify all fixed whole quarter channels by one exact internal kernel",
      "statementEN": "For one p>0 in the already given constant odd Clifford source the occupation defect D_p on F+ is positive and kerD_p=F+ intersect kerPhi. The existing whole field/state/reflection/same-time map makes every resulting constant channel an identification of all scalar future responses. Known Phi needs only its exact kernel, without exponentiation or directional scanning. A direct actual-v252 control finds no such channels in the existing invertible input; rank-defect controls find2/1 positive and1/0 neutral directions. On nu_R^c tensor C3 the kernel is exactly ker[MR;entrywise_barYnu]; a nonreal control rejects replacement by adjoint.",
      "scopeEN": "New automatic use of existing rigidity and whole source map, not a new general GNS/learning algorithm. Same scalar future relative to the declared alphabet; additional P2 insertions must be retained separately. The finite channel decision does not truncate the infinite circle/Fock system. No original P1 source selection; physical interacting successors need not preserve the old free source.",
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      "titleEN": "Shared original insertion words reach the whole lepton8",
      "statementEN": "The unchanged exact intertwiner path gives commutant dimension6 for gauge+grading on the actual reducing diagonal lepton8, then1 after adding the actual quarter generator h=gamma/4−i gamma Phi alone. Four original Higgs images remain jointly compatible. From the neutral test input individual fixed h+h_q reach4 directions; common words in the separate original h/Higgs images instead give1→4→7→8→8, closing at depth3. A gauge/grading-compatible nu_R^c phase flip fails h already at entry(1,5) with2i. Root independently decoded the stored rational matrices and exactly recomputed these values.",
      "scopeEN": "Actual finite coefficient contract and minimal coupled identification, not a physical state functional or P1 origin. Input values are controls, not derived masses. Common-phase freedom applies to the given linear tuple; physical antiunitary J/R types are not excluded by the formal same-label antilinear branch. An independently original second operator image is still needed; W Phi W* manufactured from the desired dictionary would be circular.",
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      "titleEN": "Bosonic E8 trilinear and single-anchor Weyl class do not supply the claimed mass bridge",
      "statementEN": "The antisymmetric E8 current coefficient B_h vanishes if literally used in the same left-Weyl quadratic, since B_h tensor Lorentz epsilon is symmetric. For any skew4x4 family form omega and arbitrary polynomials F,G of the same X=diag(3,-1,-1,-1), X²=2X+3I and Sym(F omega G)=(ad−bc)(omega X−X omega)/2. Every such symmetrized family block is a star matrix with zero triplet block, rank at most2 and complex singular values||v||,||v||,0,0. Exact24-basis checks and independent analytic review close the whole stated polynomial class. The reviewed therefore had no justification for comparing singular values of the unsymmetrized current coefficient directly with physical lepton masses.",
      "scopeEN": "Excludes only the stated one-bracket/one-anchor/same-Weyl identification. Rank2 per4x4 family channel, not the full64 operator. Fixed-omega image at most1D; dimension3 only over all omega. Other original family-distinguishing operations, independent Dirac variables, higher insertions and physical field maps remain open. This is not an E8/Yukawa/TFPT no-go and does not refute independently asserted mass relations.",
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      "titleEN": "Complete family-four-cycle lift class separates native clock and fixed SM preservation",
      "statementEN": "The whole exact E8 quarter-lift family[g5,eta C4] is covered by1024 Spin10 torus representatives modulo4D5. Exactly320 have actual native weight-one character(60,64,60,64) and belong to the native inner field class by the already proved complete E8 torus classification. They have3 odd vector planes and even residues0,2; their Spin10 centralizer isA2+u1³ and cannot contain an additional commuting weakSU2 beside colorSU3. The whole original-SM-commuting subclass has16 representatives with only(64,64,56,64) or(70,56,66,56),hence no nativeJhat match. Independent analytic review reproduces all characters and the full histogram. Already Tr248g=Tr45g5−1=2(d²+k−3) decides the load-bearing distinction.",
      "scopeEN": "New complete application to a precisely declared lift class, not1024 new physical sources. A family four-cycle is not a general substitute for native clock. Existing positive C kappa C^-1=Jhat and whole C12 conjugacy remain solved;kappa is not such a family four-cycle. A jointly transported SM embedding in full E8 remains allowed. No new P1 source, physical field statistics or Yukawa selection is proved. The separate odd Cartan-rank gate excludes only fixed real root charts and is not extended to all inner field conjugations.",
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      "titleEN": "Word transpose is exact but forces neither time ordering nor generic rank two",
      "statementEN": "For K(x,w y),w^t=(-1)^n w_rev with inverted group letters. In the native two-phase lepton class generic h gives [[0,Z],[Z^t,0]],hence(s1,s1,s2,s2). Actual g2 roots e4−e6,e4+e6 with family coefficients eta13=1,eta24=2 give det Sym(eta T)=9/4 and squared singular values3,3,3/4,3/4. Both roots are sigma invariant:[ad_h,Ad_sigma]=0 despite nonzero symmetric pairing. Lack of reversal symmetry therefore proves neither noncommutativity nor physical time ordering. The word is already the actual grade-one affine source correlator <Omega,J^x_1 rho(w)J^y_-1 Omega>.",
      "scopeEN": "Generic whole four-carrier. A narrow Creation30 or single-root contract needs separate checking; marked2+1 three-plane still gives(s,s,0). Arbitrary additional original torus letters lie outside the two-projector alphabet. No physical Yukawa or mass origin. Retain old symmetric source420 and complete normal-clock hierarchy bound.",
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      "statementEN": "For actual constant odd V=-i gamma Phi,||[P0(p),V]||HS²=Tr Phi² on every p=n+alpha. The pure insertion group is not implementable in the old sea for all small nonzero parameters. In contrast,||P_-(h)(p)-P0(p)||HS²=(1/2)Tr[I−|p|/sqrt(p²+Phi²)]≤Tr Phi²/(4p²). Thus complete sea Fock representations are equivalent in this1D class and full coupled evolution is implementable. The old sea nevertheless has infinite coupled excitation energy and is outside its energy form domain. The complete two-face G_ell(h) also differs from G_ell(h0) by HS,including crossing blocks.",
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      "titleEN": "The same collar supplies its canonical selfdual kinetic pairing",
      "statementEN": "The unchanged normal symbol gives new-complex-linear L=−ND=partial_r+B. On the common interior core,C0LC0=−L* and L*L=D². Hence <C0x,Ly>_new is skew,including scalar quarter holonomy. Endpoint defect is exactly [<C0x,y>]_0^ell. In the complete charged chart,L=diag(L+,−bar(L+*)),L+=partial_r+h,and its skew matrix [[0,−L+^t],[L+,0]] has Pf=(−1)^{n(n+1)/2}detL+ on a compatible finite regulator. D and L have the same whole Cauchy graph,transported by T_faces to G(h)⊕(1−barG(h)).",
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      "statementEN": "Existing J'=swap M_exp(−ix)K is a valid tangential antiunitary quarter pairing making C_A selfdual. Its sectors are V=Tpi character spaces of one NS circle. A local old-complex-linear N has sigma1{Bcov,N}=4xiN; anticommutation requires N=0,while actual deck commutes with Bcov. Relative heat trace Delta gives log2 under dt/t and paired Pfaff value log√2. Measure dt/t+c exp(−t)dt also converges,but shifts paired value by c pi sinh(2pi)/[2cosh(2pi)(cosh(2pi)+1)]>0. Relative UV finiteness alone therefore selects no scale measure. Declared annulus values0.701844509492457 and2.62205755429212 were independently confirmed.",
      "scopeEN": "Narrow original-source and candidate comparison. Preserve valid tangential pairing and declared relative log prescription; no new normal-field class,global P1/Moebius exclusion or two complete P1 sources from alternate weights. Convergence cannot replace an additional original dilation rule. Full existing normal Clifford chart and canonical L pairing remain available. A common bosonic-plus-fermionic source need not equate bosonic spectral action with fermion logdet. Origin and reference must be checked at actual joint source call.",
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      "titleEN": "Whole fixed APS source has exact inverse and finite boundary correction",
      "statementEN": "At each original p,M=P exp(-ell h)P=cosh(ell E)+|p|E^-1sinh(ell E) is invertible and Gfixed(r,s)=1_(r>s)exp(-(r-s)h)-exp(-rh)P M^-1 P exp(-(ell-s)h). Energy identities for L and L* prove ||Gfixed||<=4 over ALL Fourier modes,deltaG=-Gfixed delta h Gfixed,and bound16||delta h||. Moving APS(h) is a different insertion family: at p1/4,m1/3 its kernel violates the old boundary condition with HS²=exp(-5s/6)/5. Same exterior transfer gives fixed/moving cap quotient C_p=(I+|p|E^-1)/2+(I-|p|E^-1)exp(-2ell E)/2. C>=I/2 and0<=I-C_p<=Phi²/(4p²),hence positive nonzero Fredholm determinant on the full circle. Actual96 chart gives Tr(I-C)<=pi²Tr_KPhi²/4. This is the finite boundary correction associated with known HS sea rotation. Original provisional APS choice and Calderon upgrade do not choose a new state for the complete operational Seed,which already contains omega and tau. In the already implemented common sea representation,det_F C is exactly <Omega0,exp(-ell H_exc)Omega0>,with coupled sea energy0 and actual neutral pair gap2E. Normalized midpoint vector exp(-ell H_exc/2)Omega0/sqrt(det_F C) gives the same whole fixed cap state and lies in every finite H_exc power domain for ell>0. Its single odd block has ||[h,Pmid]||HS²=2m²/M² and is nonstationary under coupled source flow at m!=0 and finite ell. For a finite general cap, Zcap=e^(ell Tr(hPh))det_RanP0(P0e^(-ell h)P0). The actual weak addition has {p gamma,tQ gamma}=2ptQ, so the earlier anticommuting C_p reduction does not apply. Retain P0 and signed h in the compression; the cap uses the old boundary vector, not the invariant OS unit vacuum of that same system. Original APSdomprov chooses a given adapted A_partial without identifying it with p gamma_F or Phi0. The whole D_+ Calderon call precedes D_ref/D_rel and survives degeneracy of a later particular split. A fixed domain alone does not identify an original cap preparation. C59's Dgeo=Dref requires previously identified closed total operators and still does not force a moving APS space; bounded zeroth-order perturbations can retain a fixed elliptic domain.",
      "scopeEN": "Analytic closure of constant bounded odd quarter class with fixed original endpoint space; exact minimal control confirms ODE,jump,both boundaries. Not generalized to arbitrary Phi(r,theta). Moving APS kernel and whole Cauchy graph remain valid for their respective boundary contracts. Boundary determinant is relative quotient of same exterior representative; it neither renormalizes logarithmic bulk energy nor determines absolute Zadm,determinant phase,physical time or P1 field source. No second quantization of complete Seed or repeated generic first-order counterexample. Canonically normal-ordered coupled excitation energy is not presented as original absolute bulk energy. Pure midpoint state and thermal face marginal remain distinct given states. The initially finite general cap formula is extended to the whole circle for constantV in actual-weak-cap-converges-on-whole-quarter-circle, retaining that finding's coupled-energy convention. The sigma_z/sigma_x control is general cap algebra, not two full v252 sources with identical original data. Fixed OS time differs from varying the entire collar geometry and its unnormalized unit norm.",
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      "titleEN": "Mass-only relative deck operator fails ordinary heat trace",
      "statementEN": "Full tau=S R_r gives exactly Dref=D0+(Z/2)(Phi-Phi_tau),Drel=(Z/2)(Phi+Phi_tau),hence claimed massless reference for even Phi. Fiber S alone instead gives Dref=NZA,Drel=N partial_r+ZPhi. Distinct existing product reflection S gamma_F R_r preserves coupled D and gives Drel0. Under narrow mass-only Drel and ordinary infinite heat trace,already Phi0 fails:f(Drel)=I,so I-exp[-(D0/chi)²] is noncompact. Compact base,gap removal and internal admissibility selection do not remove this UV identity term. Original relative-object table also permits a declared relative pair; this does not replace the explicit even/odd formula without its own source reconciliation.",
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      "titleEN": "Finite antiperiodic scalar wall has no exact massless normal mode",
      "statementEN": "In actual claimed finite anti-circle wall class,{S,Q}=0,Q²=-partial_r²+sigma²Phi²+S sigma'Phi,D²=Q²+p². Qpsi0 forces monodromy exp(SPhi∮sigma),positive definite with no eigenvalue−1; odd zero-mean profile gives exactlyI. Thus kerQ0 and each fixed circle has strictly |lambdaD|>=sqrt(p²+mu0²)>|p|. Claimed exact ±p wall modes are excluded; close numerical values give only finite approximations. For zero-mean sigma,full normal inverse is F(r)[1_(r>s)-1/2]F(s)^-1(-N),F(r)=exp(SPhi∫0^r sigma). Given homogeneous anti-Green source correctly reproduces old thermal graph,but local C_faces selects neither global spin clutch nor original Schwinger functional. Infinite single-wall index and existing normal spin-S2-O1 index-one result remain separately valid.",
      "scopeEN": "Complete analytic exclusion only of finite scalar anti-I class with fixed commuting Phi. No uniform positive bound in separation/large-mass limits,no exclusion of noncommuting backgrounds,other clutch or original P1. Infinite isolated wall has index−sigPhi under its asymptotic premises; no general requirement for gamma_F-even P1 mass follows. Finite D_F and geometric D_N are distinct operators. Local reflected signs do not define an original mass wall without actual global trivialization and smooth transport. No physical-origin completion.",
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      "titleEN": "Spectral heat covariance positivity does not establish time reflection positivity",
      "statementEN": "Original proof calls its Euclidean covariance a positive mixture of reflection-stable heat kernels but defines neither its operator function nor heat-parameter relation to geometric time. For one spectral heat covariance,exactly c_s(2a)c_s(2b)/c_s(a+b)²=exp[-(a-b)²/(2s)]<1. Reflection invariance and positive Laplace weights alone therefore do not imply OS positivity. The entire stated continuous scalar translation-invariant C1 covariance class is settled: ordinary PD and RP force c constant. From c0 c(2t)>=c(t)² and d=1-c/c0 obtain 0<=d(t)<=2^n d(t/2^n)->0. Actual energy-semigroup Gram kernels and suitable positive Stieltjes/resolvent mixtures remain RP; no P1 selection follows.",
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      "titleEN": "Quaternionic SM mass loop is not the normal determinant clutch",
      "statementEN": "Already proved full one-form family gives exactly det M_H=||H||^(2nf)det(YupYdown) per color; three colors exponentiate expression. Every invertible loop has constant phase and winding0. Noncommuting Yukawas do not alter identity. Actual connected SM lift preserves determinant pointwise by matching left/right charge sums. This excludes The reviewed literal identification of this global mass loop with prescribed normal winding-one clutch. General inference neutral=>winding0 is false: neutral A(theta)=exp(i theta) has winding1,and S=[[0,A*],[A,0]] is selfadjoint,odd and S²I. Constant complex Yukawa phase suffices; claimed determinant positivity was too strong.",
      "scopeEN": "New narrow corollary and external candidate correction,reusing old quaternionic classification instead of96-dimensional random search. Singular Yukawas or zeros of H make determinant winding undefined. No claim that all NCG normal indices vanish. Given genuine spin operator on det E2=O(1) keeps indexone; bosonic H0(O1) doublet and conditional product-source transport remain separately valid. Formal connection-plus-potential decomposition supplies neither global Callias Fredholm conditions nor identification with original D_N. No Majorana/PQ selection,original source derivation or physical completion.",
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      "titleEN": "Whole bounded-difference class and finite genuine pair-heat reader",
      "statementEN": "For every bounded selfadjoint V and specified nonzero positive Laplace contract,F(V²)>=F(||V||²)I=cI,c>0. Thus PF(V²)P is noncompact on every infinite-rank retained subspace; no P/V commutation needed. Subtracting trace-class reference does not repair it. This extends prior free/constant counterexample to entire bounded class. Constructively,same supplied operators admit genuine pair D=Dref+V: common square-form domain gives D²>=.5Dref²-||V||² as form bound; minmax yields Tr exp(-tD²)<=exp(t||V||²)Tr exp(-(t/2)Dref²)<∞. Whole pair heat exp(-tD²)-exp(-tDref²) is trace class for every t>0,without heat factorization or fixed square domain.",
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      "titleEN": "Normal orbit-index family equivariantly recovers the supplied weak module",
      "statementEN": "Global Pw clutch exists without global Hopf section. Split vertical spin realization gives kerD_N,+=L=O_W(-1),coker0; family Chern character ∫N ÂchE=chL,c1E=x,c2E=xy. Actual U2 action makes H0(W,L*) canonically V* with Y−1/2; its outer dual is canonically V with Y+1/2,and dual evaluation exactly Lw⊂V. This repairs raw doublet identification without fitted charge. Actual q=(Htilde,H) has first columnY−1/2,secondY+1/2;q parametrizes an already degree-one clutch despite zero winding of det(qM). E8 half-root reproduces(0,0,0,1,0) on marked j6. But whole root quarter has248 character(134,56,2,56),while nativeκ/Jhat has(60,64,60,64). No full field conjugacy; separate D5 endpointκ remains correct. Actual oriented64 and fixed carrier marks corrected in review. Projector form I+(z-1)P itself already appears in September30 Bott proof; newly reviewed here is its differently typed normal-equator/weak-orbit family,index and full charge diagram.",
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      "titleEN": "Normal spin lift and fermion response obey the same signed charge balance",
      "statementEN": "Same supplied orbit family and existing September22 lift yield chW_m=64+(32+16Σm)x−16xy. For Σm=−2 normal family index has rank0 and ch=−16y. Split mA kernels are K+=2[Rodd⊗L+Reven⊗detV],K−=2[Reven+Rodd⊗Q],16modes each. Fully SM-equivariant Q is detV⊗L*,Y+.5; bare L* would haveY−.5. For commuting Spin10 charges q,signed quadratic mode sum is A_q=(Σm+2)Tr16q². Per-copy coefficients are Y:5/3+10m/3,T3:1+2m,SU3:1+2m and gravity8+16m. Same condition thus links algebraic natural lift to normal charge balance. Mathematical product index is−16d and weighted I_B=I_L=−4d. Conversely new origin identification of τ|winddet| with Dirichlet minimization fails exactly:gε=diag(exp(i[φ+εsinφ]),1) always costsτ but E=1+ε²/2. True HS loop energy has full Fourier equality class h exp(i sign(k)φP),but is an additional functional. Even one fixed hemisphere leaves all projector directions equivalent under unrestricted smooth other-disk gauges.",
      "scopeEN": "New conditional common corollary of supplied family/lift,not P1 selection. Numbers become anomaly coefficients only under actual chiral1+1D field quantization; not an original Ward insertion calculation. Four A3 carriers are not four physical generations or a contradiction to original three-family construction. N×W index proves neither physical4D theory nor ΔB/L transitions. Dual-bundle formula Wm*=ΛoddE⊗⊕D^(−1−m) holds but supplies no actual OS domain/phase/RP map. Whole ordinary genuine heat pair of pure gauge conjugates Dg=MgD0Mg* is exactly zero; not transferred to Padm,marked insertions or unidentified original reflection-pair convention. Native clock,old Clifford/boundary states and whole positive sources retained. No stochastic rescan of settled loop classes. Local retention of the Yukawa seed in finite qM does not refute an upstream TFPT flavor derivation. Originals contain the hard holonomy formula; archived faulty D/P puncture display and existing analytic v117 C4/A4/S4 repair are already reviewed and reused,without claiming new D4 rigidity.",
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      "titleEN": "Fixed Cartan test couples S4 sign to sector parity and already admits an intermediate cover",
      "statementEN": "Under extra fixed orthogonal Cartan identification c(x,y,z)=(-y,x,z),exactly1920of3840 D5 signed permutations preserve full E8 roots,all exchanging the two marked64 sectors. Exactly12 preserve color/weak root blocks and unorientedX line,all X→−X; none preserve orientedX. Even total E8 half-root parity analytically forces odd signs in both blocks. Generally exchange parity is sign:S4→C2. Existing C4 descent t=z⁴ has local signs(−,+,−); intermediate cover s²=t,s=z² already has A4 monodromy and preserves marked64. Full C4 cover is stronger than needed for this grading alone.",
      "scopeEN": "Independent exact census and analytic review of The reviewed candidate,not original selection of Cartan identification. std⊗sign rotation representation was already proved. No physical spacetime chirality,charge conjugation,OS reflection,Yukawa equality or identification with original C12/48 cover. Twelve X-inverting maps differ by color/weak Weyl actions. Candidate general impossibility claim is therefore too broad. Actual family-vector action is separated in companion finding.",
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      "titleEN": "Original family-vector action differs from Cartan rotation and does not select a spin lift",
      "statementEN": "Original explicitly contains SU3⊂SU4 via4=3+1. For supplied90degree family-vector rotation R,U=diag(R,1) preserves each of (16,4) and(bar16,bar4). Actual five E8 branches yield248 multiplicities(134,56,2,56) for(1,i,−1,−i),trace132,the already known root-quarter type. In fixed four-basis U induces only h1↔h2 on traceless Cartan,spectrum(1,1,−1),order2; this is not order4R on family vectors. Native glue clock has(60,64,60,64),trace0. Claimed spinlift selection m=(k,k,k,−2−3k),minimal mF/dual,was already proved exactly on22September under extra fixed irreducible parallelity; general covariance does not supply it. Actual mixed output is(E⊕E*)⊗D^(mi+mj+1),with dual pairing of complementary SU4-six indices. Candidate general single-Higgs exclusion ofmA therefore lacks actual physical field map. Later The reviewed dual exclusion from charged-triplet determinant also does not follow from SU3 flavor:forB_k=A+k a I3,tracefree part remainsA for everyk;fullSU4 beta part stays traceless for both natural duals.",
      "scopeEN": "Source-reviewed reuse and concrete representation distinction. Different original clocks need not be identical or incompatible from multiplicities alone. Whole native C12 conjugacy and supplied parallel representatives remain solved. FixedM preservation needs actual transitions,connection and domain; free M↦HMH⁻¹ is not automatic fixed-bundle automorphism for unequal D powers. Fourth-slot anomaly compensation is existing conditional index arithmetic,not a physical fourth generation. Bare E8-bracket/Weyl single-anchor class remains settled in its statistics scope. No original common source,state/R/time derived from this representation comparison. The reviewed I_mu4 identification of normal sphere with flavor base and four equal logarithmic poles is new expressly underived geometry. Conditional uniqueness for fixedO1/simple-pole data holds,but is not origin. Fuchs degree and monodromy need same∇d±a convention:res−1/4 for∇d+a gives+i;ODE+a gives−i withdegree−1. Explicit horizontal point map in original iA(iz)=UA(z)U^-1 convention usesU,notU^-1. Anti-linear coefficient symmetry proves neitherOS norunchanged singularL2 normal index. ActualC3/q-cover alternatives retained.",
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      "titleEN": "Actual genus-two sign family closes common parallel K connection on cover",
      "statementEN": "Already proved v117 splittingπ*F=L0⊕L1⊕L2 onC plus trivial line gives genuine flat SU4 sign bundle. Transitionsdiag(l0,l1,l2,1),product1,commute with existing natural Spin/family lift. For mA=(0,0,−1,−1)this gives global K=(Spin10×SU4)/Z4 cocycle with(16,4) branchW=ΛevenE⊗[L0⊕L1⊕D^-1L2⊕D^-1]. Half-determinant signs cancel in quotient. Whole associated E8 algebra and Lie products are global. Actual combined connection isA_W=ρ(A_E)⊗I+I⊗(Ma+A_L). Diagonal flatA_L commutes withM and nativeR=diag(ω,ω²,ω²,ω). Along with color center,κ,σ0 andκσ0 are thus global parallel automorphisms on all E8 branches. Irreducible lower A4 holonomy becomes abelianV4 upstairs; lower Schur/mF conclusion cannot be applied unchanged onC. Original ternary deck instead cycles three distinct lines. Unchanged parallelM andR marks require equal triplet values/phases. mA andR2+2 fail this fixed-mark condition and each has three-element transported-mark orbit.",
      "scopeEN": "Constructive new combination of two solved source components,not new cover or repeated monodromy proof. Data and alignment remain explicit. Clock is adapted representative of whole native class; unchanged originally marked clock needs actual mark/connection map. No automatic descent of local matrix between unequal D powers. Triplet equality is necessary fixed-mark condition; coarse descent across branches also needs actual linearization/isotropy. Whole D/Phi/domain action,charged CAR products,original state,OS and physical time do not follow from connection identity alone. Full existing operational Seed retains its state/time. Old bare μ12 cover does not automatically contain q lines by existing ramification proof; no identity with original Fermat source.",
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      "titleEN": "Actual quarter preserves K bundle;Kummer cover admits separate Real lifts",
      "statementEN": "In actual q frame,m=(0,−1,0,−1),R=diag(ω²,ω,ω²,ω) are preserved by oldAq. Its parallel line maps extend to whole K cocycle,connection and all associated E8 products;four iterations are identity. Adapted nativeκ,σ0,c0 commute withQ. Of six existing toral norm minima,exactly thism anddual(−1,0,−1,0) areQ invariant. Explicit global Kummer lifts Qh_a=r_a/h_(−a−1),Th_a=1/barh_(−a−1),r=(ω,−ω,−ω) obeyQ⁴=T²=1,TQT=Q⁻¹. E0=QT sends h_a→r_a barh_a and squares1. Same line permutation does not fix lift square:D_sT for s=(−1,−1,+1) squaresD_(−,+,−).",
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      "titleEN": "Antilinear Real structure is not linear duality and compact conjugation does not invert clocks",
      "statementEN": "The reviewed LemmaP is refuted within same existing carrier type:E=O(1)⊕O⁴ onCP1 hasE≅σ*barE forσz=barz,since base reversal and conjugation each reversec1. Exterior/determinant functors give involutivej_W for every integerm withoutm4=−1/2. Compatible split/Chevalley conjugation extends over whole complex E8 bundle preserving brackets. Compact conjugationsigma_c instead commutes withAdg for compactg∈K,exchanges both64 branches and compensates conjugate phases. Alreadyκv=−iv gives sigma_cκsigma_c⁻¹v=−iv,not+i v. Thus general normal-reflection prohibition,compact clock-inversion claim and trivial-slot exclusion of commuting variant are false.",
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      "titleEN": "Existing selected spin and actual q lines yield three explicit global kernel sections",
      "statementEN": "Already source-selectedS0=O(Σq−K),actualLa=O(qa−q(a+1)),2qa~K giveTa=S0⊗La≅O(q(a+2)),eachh0=h1=1. Whole oldFq=L0⊕L1⊕L2⊕O has(h0,h1)=(3,3),index0,trivialslot0. f_a=1/(w+ωa) explicitly spansTa in stated divisorframe;common quadratization isψa²=(w+ω(a+2))dw/η. Actualc/Q classes transporta→a−1,a→−a−1. IMPORTANT LIFT CORRECTION:oldAq²=(-,+,-) are fiber eigenvalues atQ-fixedq0,already carried by canonical divisor linearization;not additional relative meromorphic signs. Correctly calibrated tensor hyperelliptic lift therefore acts UNIFORMLY−i on all3kernels;Q_kernel⁴=−I retained. Earlier Kummerε-based meromorphic matrix instead had(i,−i,i) and is not original coefficient lift. Its simultaneous matrix comparison and normline continuation are identities of that different lift only. Closed Dolbeault has3kernel/3cokernel and no ordinary two-sided Szego inverse. Corrected quarter hasv_a=δa ξω^(1−a),δ=(1,−1,1),ξ=e^(−iπ/4),spectrum{ξ,−ξ,−ξ};lineQ maps require no global scalar roots.",
      "scopeEN": "Deduplicated new application of old data,not new general theta mathematics. SOURCE-SPIN-ZERO-SECTIONS sections1–3 retained;its section4 phase/original-lift assignment corrected byThe reviewed-SZEGO-FIRST-REVIEW section5. SOURCE-SPIN-JOINT-CLOCK-TEST compares matrices for wrongly assigned lift,not an original intertwiner. Uniform−i halfturn is fixed by original fiber linearization,not freeσ convention or equality withQΓ. Complex cohomology,not chiralindex3 or generation selection. Kernel-volume/oldNq comparison must use corrected lift,not earlier uncalibrated calculation. Complete Seed retains state/time. Closed curve,originalP1/APS domain,reflection,physicaltime and charged field products need actual source map. Existing normline/theta/native module proof retained;original one-copy insertion not proved here. Kernel volume is now positively compared in corrected lift as generated-group line representation:finding:actual-spin-kernel-determinant-has-inverse-q-norm-character;not original insertion rule.",
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      "titleEN": "Existing S0 Szego kernel generates actual q kernels as marked point legs",
      "statementEN": "Foru=(w³+1)/η and meromorphic spin sectionsΦ=√(u dw),Χ=Φ/u,S(x,y)=[ΦxΧy+ΧxΦy]/[2(wx−wy)] is unique normalizedS0 Szego kernel. Diagonal residue1,noother poles including opposite sheets/infinities. S(qb,qa)=S(pb,pa)=0 fordistinctpoints;mixedq/p≠0. S(.,qa)∝Φ/(w+ωa) hasS0 divisor−qa+q(a+1)+q(a+2);withactualLa factor it spansH0(Ta). Thus3kernels arise explicitly as old marked propagator legs. For{a,b,c}={0,1,2},actual products lie in1dimensionalH0(KLc),zero divisorqc+q(c+1);bareS0 product(w+ωc)dw/η instead has2qc. Elementary modificationS−S(.,p)S(q,.)/S(q,p) has simple pole atq-dipole with rank1 kernel/cokernel residue. Finite Laurent part is notcanonical without transport:gε=exp(εf) changesG0 toG0+[f,P],retaining same residueP andη point split.",
      "scopeEN": "Exact specialization of established hyperellipticSzego method toactual selectedS0/q packet;primaryfulltext D'Hoker/Hidding/Schlotterer JHEP05(2023)073 section2.3/equation2.26 checked. Not new general propagator formula. Valid point-leg/typedV4 product bridge supplies no originalP1/APS operator/domain map or physical field quantization,state/R/time. Twisted zeros differ from baremeromorphicframe. Additional bijective3point saturation has2nonzero assignments butis not original source rule. Canonical reduced inverse fromηalone is refuted;actual familytransport/normalization and Hermitian projections may define specific reduction. Retain corrected spin/fiber lift,notfreeσchoice orautomaticQΓbridge.",
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      "titleEN": "Actual fermionic kernel-volume line carries precisely the inverse q-norm action",
      "statementEN": "For actualold coefficient lift calibrated atq0,Q_kernel²=−iI3,δ=(1,−1,1),ξ=e^(−iπ/4). Q actsbyξ³ onΔ=det⊕aH0(Ta),by−ξ³ onunordered tensorZ. Independent norm test:αq0=iξ,product of exchangedq1/q2 spinmaps=βq1=+i,soQ_Nq=−ξ,Q_Nq^-1=ξ³,Q_Bq^-1=−ξ³. Orderthreec cycle acts1 onall compared lines;three nontrivial actualdeckV4 characters multiply1;spincenter−I acts−1. HenceΔ≅Nq(S0)^−1 andZ≅Bq(S0)^−1 as full line representations of group generated by these c/Q/deckV4/spincenter lifts,upnonzero scalar. Fermionic permutation sign distinguishes matching determinant line fromunordered tensor.",
      "scopeEN": "Independently twice-checked actual line-representation bridge onall generators,not beta-phase match alone. EarlierεKummer-to-flat-fiber transfer andcommonU comparison corrected;cover identities retained. Character equality gives equivariant isomorphism upscalar,not functorially canonical total residue map orphysical norm. Individual legs arelinear,butclaimed individualQ/deck equivariance doesnotfollow. NoJac[2] tensor-translation/N64 equivalence;these changeh0 and differfromdeckV4. No originalP1/APS source map,selectionH0ratherthanH1,single insertion orBerezin dual choice. Whole detRΓ also containsH1 and loses uniform scalarweight sinceχ0. OldNq/theta/native module proof reused;newactual kernel-volume supplies matching linear factor,itsphysical use remains specific origintest.",
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      "titleEN": "Closed spin/flavor source has a functorial Pfaffian line with charged type limits",
      "statementEN": "For supplied V=S0⊗Fq,the orthogonal coefficient form and S0²=K give a symmetric K-valued pairing. Integration by parts makes the closed Dolbeault operator bilinearly skew;its Pfaffian superline exists functorially. With Δ=detH0 and Serre duality,the fixed conventions are λ_coh=Δ²,Det_F=Δ^-2,Pf_F=Δ^-1. Pf_F has reviewed Nq line character and odd parity. This does not identify actual charged W=(16,4):its vertical κ phase−i forbids invariant scalar W⊗W→K. Existing full selfdual W⊕W* source has Pf=±det of its charged block.",
      "scopeEN": "Closed curve with specified orthogonal spin source;no transfer to original P1/F5 operator,APS domain or physical action. No freely choosing a dual to fit target phase. Character comparison only for reviewed c/Q/deckV4/spin-center,not full Jac[2] zero-mode functor. Odd line exists;unsaturated section vanishes at this singular operator. Three geometric Real forms or complex normal index1 do not determine physical Majorana count or parity cancellation. One complex CAR mode has two Majorana directions yet odd relative vacuum parity;odd transition amplitude is not a positive state. Reuse old primitive seam defect and solved same-spin unit quotient. No general origin no-go or physical completion.",
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      "titleEN": "Original relative seam has an explicit Fredholm mismatch on whole given two-face graph",
      "statementEN": "On existing full normalized Cauchy graph V=(sqrtA,sqrt(1−A))^t,A=(1+R²)^-1,R=e^-ell h,the relative boundary mismatch F_U=[−U,I]V=sqrt(1−A)−U sqrtA is bounded. In existing scalar quarter block it normalizes T=R−U on DomR. T is closed Fredholm,index−1,kernel0,with exactly one Gaussian cokernel y_n∝exp(−ell[n(n−1)/2+αn]). Proof uses trace-class perturbation of P_-−U P_+,no truncation or new Hardy projection. Full two-face compression G(U⊕U)G is instead invertible bilateral weighted shift,index0,weights≥sech(ell/2). One-face compression G(I⊕U)G has index+1 in same orientation and leaves G11 unchanged. Circle time covariantly transports matching into U_t=e^it U family.",
      "scopeEN": "Concrete original type of relative boundary operation on given whole collar;explicit index/Gaussian cokernel concern its actual scalar block. No selecting this block instead of whole charged source or physical three-copy count. Whole h=Aγ_F−iγ_FΦ with both principal-symbol directions and C0 completion remains to test. Original fullv21 has four boundary pieces;Γ± are not automatically two F5 faces. No closed operator intertwiner to branched S0 Hardy domain or q evaluation fibers. Index/Fredholm class are not exact operator identity. Time covariance is moving-seam family,not fixed-clutch physical time or new time choice. Old same-spin once norm remains solved;classical matching does not select new state.",
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      "titleEN": "Whole constant seam defect follows exactly from actual internal kernel",
      "statementEN": "On unchanged scalar unit-clutch domain,LΦ=∂r+Aγ_F−iγ_FΦ is closed Fredholm. Whole energy identity ||LΦψ||²=||L0ψ||²+||Φψ||² also holds for adjoint. Extra boundary term vanishes by U*VU=V;nonzero massless A/U boundary term remains in ||L0ψ||². Kernel/cokernel of existing whole boundary mismatch are precisely kerΦ∩Fminus and kerΦ∩Fplus. Whole index−Trγ_F=0 for48+48. Previously exactly checked v252 controls immediately give(0,0),(2,2),(1,1),without rerun. Constant original δΦ acts between surviving Gaussian defects as−i Pker,+ δΦ Pker,−. Reuse primitive NS clutch u_cov from solved quarter dictionary.",
      "scopeEN": "Entire supplied constant coefficient class with explicitly retained closed scalar clutch domain. No arbitrary single channel or physical three-copy choice. Invertible Φ removes both defects;index0 alone does not trivialize every determinant action. Controls are not derived Yukawa values. Variable Φ retains−i[A,Φ],normal-dependent V another derivative,and noncommuting U a boundary form. No exclusion of all P1 sources or variable interactions. Original face assignment,branched spin-domain map,physical source selection and time remain actual maps. Scalar unit defect and old same-spin once norm retain valid scope.",
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      "id": "finding:old-complex-scalar-clutch-separates-c0-and-product-reflection",
      "titleEN": "Old scalar clutch separates bulk C0,boundary adjoint and product reflection",
      "statementEN": "Actual C0=i_old N is old-complex-linear and commutes with U=e^{i_oldθ}I. New antilinearity for Ical does not allow C0UC0=U*. Whole T chart clutch is diag(U,barU). Oriented T_faces conditions are v1=Uv0,w1=−Uw0. Bulk C0 preserves this closed kinetic domain;C_faces maps unitary clutch subspace to orthogonal complement. Actual theta-fixed product reflection instead transports whole domain to U^-1. SF+1 alone does not assign four original boundary pieces to F5 faces. Reuse old NS/quarter dictionary;The reviewed mixed1/4,3/4 mass toy does not replace actual same-Aα source.",
      "scopeEN": "Actual old product collar with specified complex structures and theta-fixed R_geo. No general reflection no-go;additional original base pullbacks/fiber lifts can give different common operation and must be transported explicitly. Existing open-source reflection positivity,whole Cauchy graph and source state/time remain intact. Bulk adjoint,boundary CAR and geometric reflection are not interchangeable names. Correction rejects The reviewed first C0/U inference,not every physical P1 Real structure.",
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      "titleEN": "Quarter circle character does not select Gaussian defect connection",
      "statementEN": "Existing scalar defect family Tφ=R−e^{iφ}U has periodic frame eφ,n=e^{iφn}y_n/||y||. Given circle action satisfies e^{itB}y(φ)=e^{itα}y(φ+t). Natural orthogonal-projector connection instead has〈eφ,∂φeφ〉=iμ and holonomy e^{-2πiμ},μ=〈n〉. Old allowed diagnostic control α=1/4,ell=4log2 gives μ=0.1855335532291087 and holonomy0.3940689714−0.9190808701i,not quarter phase. Exact rational partial sums and geometric tail bounds certify difference. Separately specified flat orbit transport still has circle phase+i;family equation does not select it as projector connection.",
      "scopeEN": "Given free scalar Gaussian defect line,fixed positive ell and constant α. Width4log2 is diagnostic,not physical scale. Projector connection is not asserted compulsory physical choice;no new time or exclusion of other originally specified connections. Original circle character remains correct. Whole coupled h time gives U_t=e^{ith}Ue^{-ith},not generally scalar2π closed clutch family. No native clock intertwiner or missing time in full operational Seed inferred.",
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      "titleEN": "Existing source spin is the canonical ramified sphere-spin pullback",
      "statementEN": "For actual original triple map π:C→P1_z,w³=(z²−1)/(z²+1),all four ramification indices are3. With Rhalf=Rπ/2 and K=Iplus+Iminus,Rhalf∼2K and2Dq∼3K. Canonical ramified pullback of unique sphere spin is S_can=π*O(-1)⊗O(Rhalf)≃O(Dq−K)=S0. Explicit square-compatible map is e_can=√3·w/(w³+1)·Φ0 for Φ0²=μ0=(w³+1)dw/η,with e_can²=π*(dz/z²) and √dz=√3·w/η·Φ0. Actual old lifts satisfy c e_can=e_can,Q e_can=ξ e_can,Q√dz=barξ√dz. Contravariant E0=Qτ0 anti-pullback has J_E√dz=ξ√dz,J_E²=1. S0 is not an additional spin choice on this source.",
      "scopeEN": "Exact spin-line origin and square/lift map on already reconstructed original π source. Common spin sign is ordinary frame sign,not new relative quarter fit. Meromorphic coefficients at poles are not nonvanishing fiber values. No identification of physical BΣ with freely selected pullback operator:actual metric,connection,weighted Sobolev/trace domain at four ramification vertices,whole coefficient system and Γ± face assignment must transport. Spin Real map is not CAR C0 or OS reflection. Triple coefficient pullback is not physical generation count. Preserve old once norm and Kummer/fiber-lift corrections.",
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      "titleEN": "Original v117 equation has a full explicit algebraic fundamental frame",
      "statementEN": "For unchanged plus-sign equation Ψ'=A(z)Ψ with original A0 and U=diag(1,i,−i),an algebraic3×3 G is constructed on actual negative q Kummer cover. A column is g=(h0 p1/(ab),h0 z p2/b,h0 z p3),a=2√2/3,b=√5/3,with explicit rational p1,p2,p3 overw inQ(ω). Its two c images give detG=−27√2(2ω+1)(z4−1)/10. Ψ=G Gbase^-1 has Ψ(0)=I,detΨ=1−z4. All three original equations independently checked by exact direct substitution. V4 acts on right bydiag(ε0,ε2,ε1),c by3cycle. Actual calibrated Kummer Qh=(ω/h2,−ω/h1,−ω/h0) gives entrywise checked G(Qy)=UG(y)KQ,KQ=[[0,1,0],[-1,0,0],[0,0,1]],hence Ψ(Qy)=UΨ(y)U^-1. This is an explicit differential gauge,not only monodromy or point counting. Positive flat coefficient metric is G^-daggerG^-1. Independent metric calculation gives Gbase Gbase†=diag(27/10,9√3/5,3√3). Canonical spin yields explicit weighted trace Wψ=3^-1/2√π' G^-1ψ∘π. With given inner NS continuation,B_H=−i(∂θ−izA)=G(−i∂θ)G^-1 onD_H=G H1_NS inL2_H is closed selfadjoint;no new extension between four marks.",
      "scopeEN": "Full algebraic frame for given v117 connection on original Yq;actual spin-line origin separately derived. RR justifies finite two-coefficient search,no numerical fit. Initial negative was corrected solver restriction:constants belong toQ(ω),d=ωc;not old-source counterevidence. Meromorphic functions are not flat fiber values. Makes old group-averaged positive coefficient metric explicit,not a CAR state. Three columns are coefficient directions,not physical generation derivation. Does not derive A0 itself from P1 or full96 field/Phi/Γ± domains/state/reflection/physical time. No exhaustive historical novelty claim. Original ledger/proof statuses unchanged. Trace/domain result explicitly uses analytic original inner z disk and its existing NS continuation. This is an input condition,not new physical BΣ selection. Image is C3-equivariant pullback subspace,not arbitrary rank9 space.",
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      "titleEN": "Circle-fixing base reflection does not fix entire cubic cable seam pointwise",
      "statementEN": "Actual lift τ_C(w,η)=(−barw,barη) of z→1/barz is antiinvolution,sendingq_a→p_-a,but does not fix wholeπ^-1(|z|=1). Exact regular fiber:z=e^(iπ/4),η=√2,w=(√3+i)/2,(−√3+i)/2,−i. First two are exchanged,only third fixed. Each other global cubic deck liftc^kτ_C likewise fixes only one of three. Literal identification Fixτ_C=entire cable seam fails away from every vertex. Valid half-form relationJτ√dz=±iz^-1√dz determines spin transition in specified reflected inner/outer frame,not original physical Γ± normal-frame identification. Complete marked-circle class:16Mobius/antiMobius sphere mapsi^kz,i^k/z,i^kbarz,i^k/barz give8D4circle actions. Amongq-compatibleanti-cuts,σ1=ibarz ANDσ3=−ibarz avoidmarks and areQ-conjugate. Data giveone mark-freeorbit,not absolute representative. Spinξanti-frame phase is not a newlinearΓ±clutch proof.",
      "scopeEN": "Only literal pointwise fixed-locus identification on actual C and specified circle map excluded. Sheet-permuting reflection,different original cut or reflected-source transport remain possible. Retain valid q/p point map,spin formula and previous open-source reflection. No global P1 impossibility or ban of geometric doubles. Choosingπ*|dz|² is not forced by canonical spin-line identity. the reviewed source global G calibration question now solved by explicit G,not another gap. Orbit fact doesnot introduce artificialnew physicalchoice:iforiginalQuarterisSeedmorphism,thetwo representativescanbeequivalent. Must checkactualsidemarks. For explicitpointmapσk fiberisU^k;oppositeU^-kframeidentity remainsvalidunderpropercontravariantconvention.",
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      "titleEN": "Actual Fuchs Real map transports whole supplied field source,state and time",
      "statementEN": "Original realA equation gives fullE0 field map J_E^rowψ=U_Fbarψ(ibarz). In actual constructedG frame,D_E=Gbase^-1U_FbarGbase=ω²S,S=diag(1,−1,1),unitary,D_EbarD_E=I;independent exact matrix substitution verified. Canonical spinξ cancels against half-form boundary factor in NS coefficients. Under declared v252 generation alignment KE=DcalκP_E has D_E on particles,barD_E on antiparticles,commutesγ_F,J_F andsquares1. Exact coupling images:Y_f^E=SbarY_fS,MR^E=ωSbarMR S,with requiredMajorana transpose. ExistingIcal makes𝒥=IcalKE NEW-complex-antilinear involution transportingC0,C_faces andallfourT_faces slots. Exact𝒥GΦ𝒥^-1=GΦE includesbothcrossblocks;givenprojectorCARstate andallwordstransportwithnonewstate. Givenunitarytime has parameterreversal j_Ee^(ithΦ)j_E^-1=e^(-ithΦE). Realconstructorcontrols return tosameΦviaW_E=VωL_S=Dcal^-1;actualtridiagonalMRrequiresbothfactors.",
      "scopeEN": "Exact whole operator/state/time FAMILY covariance of givenF5/96 source under explicitq/generation alignment. D_E alone is not generic sameΦ symmetry;complexY/MR are reflected. Numerical countercontrol is diagnostic;block identity is proof. W_E is algebra-preserving commutant unitary basis equivalence,not innerSMgauge;nativeK/clock/reference admissibility not proved. A sharedG frame change transformsΦ andJ jointly,different from additionalW_E source action. No universalMajorana/OS obstruction fromω. Circle parameter is not derived physicalOS time. No physicalE0cut or antiξ-to-linear seam deduction. Source001/002 already supply boundary operator,Seedstate/time;old local-source embedding on cyclic image suffices,no unnecessary global presentation equivalence. No universal rank6q+p requirement for all doubles. FullP1generation andT1–T8 remain unclosed. Update: abstractparallelK/E8coverextensionisnowproved. Itsactualchiral96restrictiondiffersfromthisearlierparticlewisehypothesis. Adaptednativeclocksfixed,flavorQtransportedandnotuniversalSeedrequirement.",
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      "statementEN": "Actual G order(L0,L2,L1) givesW=omega diag(1,-1,1),extendedbyFq=diag(omega,omega,-omega,-1)inSU4. This sameglobalparallelKautomorphism onC preserves allE8branches,Lieproducts andadaptednativekappa,sigma,c. ItconjugatesKQtoinverse whilefullbase remainsQ;nonzero[S,Pc] preventsparallelzdescent. Candidateiota_mq(-I5)hasdifferentGsign. FullE0 obeysinverseQrelation;leftcompensationleavesverticalDQ. Seedopdoesnotseparatelyfixv117Q. Simultaneouswholetransportdiffersfromleftcomposition;localQimagecanforgetDQonlyifAdDQtrivial,andwholeevenfamilyCARdetectit.",
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      "titleEN": "Whole v252 real basis has signed P2 and dual dictionary",
      "statementEN": "ReusepositiveSM/P2wedgeintertwiner;completeoppositegradebyWminus(JFv)=CcompWplus(v),Ccomp=P_Nstar. Whole realsigned96unitary intertwinesgrading,JFcompactconjugation andseparate splitcoefficientconjugation. FiveCartanandfortyD5rootexactsignchecks pass. FullSM/charges andinducedCARstarisomorphism retained. Globaldual retains determinantpowersD^(-1-m). Independentsmall96replaypassed.",
      "scopeEN": "Wholeactualfiber/globalSMcomparison;notglobaltrivialization. ExtraSpin10/Kactionistransportedcomparison,notimplementedphysicalE8source. Fourthfamily/otherKbranchesneededforE8brackets. CARmapisnotE8_1currentalgebraorP1selection. ReuseoldpositiveP2/level4results.",
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      "scopeEN": "ScopedtoC30printedselfdual/local/stateclaim. NotuniversalvetoofCauchynetsorKframechanges. ReuseexistingwholeTfaces/HS/state/reflection/time. GivenSeedneedsnonewstate;compareitsactualnet/operators.",
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      "titleEN": "Original positive hexagon kernel supplies no oriented phase readout",
      "statementEN": "Actual originalchain usesU6,Dy=yI-deltaU6 andfixedcompressionpositiveKstr. ExactDdaggerDdependsonU6+U6daggerandisorientationeven. UnsquaredresolventhasorientedFourierpowersbutdisplayed36sitematrixelementsarerealpolynomials. UnspecifiedVadmdoesnotderiveorientedresponse. v561suppliesrho;4rho/P2/V5arescalardiagnostics,notE8adjointtrace/sourceoperator. Actualv252/F5callhasnoC6representationororientedinsertion. Existingcommon-rayphaseblindnessanddistinctrelative-sourcequadraturesretained. Clarification30Oct: original A=D_fPstr,H=A†A,Kstr=H^-1/2 is fixed Gram normalization. PolarV=AK is isometric, but originalY uses HolK, not HolV. K as coupling coefficient need not be a CAR field isometry. BareU6=1 norm countertest excludes only that misidentification. K has an exact cubic cosine expression and fixedPstr variation δY=(δHol)K+HolδK with resolvent derivative; actualδD/δHol and endpoint variations must come from source.",
      "scopeEN": "Scopedexplicitbarekernel/fixedPstr/positiveconsumer,nouniversalphaseveto. GivenSeedstate/timealreadyexist;typedinsertionsaremissingbeforeevaluation. Originalcoherentopenlegorbroaderconnectionallowed. AlgebraicrhocharacterisnotmeasuredCPphase. Actual G-metric P/Q-to-U6 path is now positively closed; no further free metric choice. Fullspine/Pstr/original insertion remain distinct.",
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      "titleEN": "Original hexagonal curve rotation has concrete rho Hodge line",
      "statementEN": "Actualcurveeta²=1-w6hasR=sc²:w→rho w,positiveinverseofoldT=sc,coversz→1/zandexchangesq/p. ReuseoldjointKummermaps. CanonicalspinliftswapsPhi/Chiwithlambda²=rho,order12;pairedpointnormeigenvalues±i. EllipticquotientX=w^-2,Y=eta/w³satisfiesY²=X³-1,andRacts(omega²X,-Y),soR*(dX/Y)=rho dX/Y,pullback-2dw/eta. ThisisactualgeometricrhoHodgelineinthesamesource. Otherlinehasomega,wholeHodgedeterminantminusone. Independentexactrootcheckpassed.",
      "scopeEN": "Constructive realizationofalreadyoriginalhexagonalphaseinexistingcurve;reuseoldclock/cover/spin/norm,limiteddedupenotnoveltyproof. NotfreelyinsertedrhoscalarorautomaticOS/Pinreflection. ActualHodge-to-field/readoutcontractionstillneeded;line-valuedtraceisnotscalarbeforeclosing,andpairedduallegsdo notcreate netcharacter. Spin/norm/fullHodgedeterminantcannotbereplacedbyoneeigenline. NophysicalCP/fullsolutionclaim.",
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      "titleEN": "Original mixed spin products realize the positive U6 cycle",
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      "scopeEN": "Constructive geometric origin of the supplied six-cycle matrix, without selecting a physical character, CP reader, field source, state or time. Keep twist lines and the harmless common orbit phase. The original physical charge assignment of triality and sheet characters still needs its actual operator map. Originals already declare rho/rho^4 as CP readouts. v233 inserts exponents 1/4 rather than evaluating the actual map J from U6^3,U6^4 to color-Casimir sector parity and family center. Geometric q/p names and the old carrier seam tau do not identify quark/lepton sectors. Clarification 30 October: the phase-free orbit basis absorbs partner phases. Raw s pullback is minus transpose, so raw antisymmetry is on the positive U6-cube sheet. Independent full q/p signs plus R have no rank-three projector; see the new complete deck test.",
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      "titleEN": "Spin pair transport retains fermion parity and whole source types",
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      "scopeEN": "No new six-mode state is selected. Hodge positivity is not physical covariance or Hamiltonian. Existing q-to-generation comparison supplies no original p map into F5/96; antiparticles, adjoints, holes and second faces are distinct types. Require the actual source map without imposing a universal extra fixed Q symmetry or this optional comparison. A direct iota6 embedding into one fixed source is only a particular comparison. A global q-to-p pullback may instead map whole source objects without that doubling.",
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      "titleEN": "A real full trace does not exclude coherent phase readers",
      "statementEN": "The reviewed universal inference from real full traces fails: the actual mixed six-cycle has a complex rho partial trace. An independent exact matrix counterexample has Im tr[Hu,Hd]^3=-360sqrt(3). Relative coherent phases can affect invariants while an overall phase remains invisible. Bare open Fuchs frame products still telescope on a cut and finite closed A4/S4 characters remain real.",
      "scopeEN": "The matrix example is a diagnostic counterexample, not a TFPT Yukawa prediction. An actual open source reader still requires original endpoint legs, path classes and field map. Retain the narrower overall-phase and bare closed-character exclusions.",
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      "titleEN": "Geometric six-spin clock differs from the marked native clocks",
      "statementEN": "Actual spin r has characteristic polynomial t6+1, sixth power minus one and geometric C4/C3 factors. Their q/p and family actions differ from marked native kappa and sigma, already at one fixed D5 weight. A linear side exchange in the specified matched carrier needs a D5 chiral flip and inverts central kappa, so it lies outside that fixed K. Coefficient Real and compact JF also have distinct clock relations.",
      "scopeEN": "This is a marked comparison test, not a no-go for outer/Pin or antiunitary transports. The optional q-slot alignment stays a declared premise. Retain determinant twists and actual base/charge transport. Equal clock orders alone do not provide the whole source-family map.",
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      "titleEN": "Whole q-to-p pullback transports supplied source state and time",
      "statementEN": "The alternative to a six-mode embedding is positive for the whole supplied source. Actual linear Kswap reverses gamma and conjugates the complete symmetric-Majorana coefficient. T=Kswap Rx transports h(Aq,Phiq) to h(Ap,-barPhiq composed with inversion). Full F5 lift (iN) tensor T preserves total grading, complex structure and C0, intertwines E and carries all face entries, cross graph correlations, state and same-parameter time. Quarter connection/domain holonomy invert; variable spin frames retain derivative and endpoint terms. Two independent actual96 replays confirm the algebra.",
      "scopeEN": "Exact covariance to its calculated image; original naturality already fixes all image p data and no independent Phi_p rule is required. This does not derive the full given Seed from fewer primitive data. Kswap transports pi rather than fixing its old copy. Match the geometric spin pullback to actual global normal/collar domains, local algebra and reflection; annulus inversion also exchanges faces/normals. Existing state/time are carried, with no mandatory iota6 embedding or no-go for appropriate outer/opposite transports.",
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      "statementEN": "At the two actual fixed points, all mixed products evaluate nontrivially into distinct twisted fibers and Rfiber=rho P. P-fixed is not a full flavor singlet: an original allowed deck sign already moves the rho line. A zero complete Stokes period or isotropic Cauchy Green form does not erase nonzero source cross correlations. Both q and p sources have two faces. Reversing every pair order is a global minus basis change and leaves the minus U6 action unchanged; the existing alternating orbit map is required.",
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      "statementEN": "The previous independent-Phi_p requirement is unnecessary for the transported image. Original Seed morphisms intertwine all data and Cseed is declared functorial. Exact conjugation of the original inner-fluctuation generators gives U Omega1_D U^-1=Omega1_Dprime and transports the full fluctuated Dirac operator. Existing Majorana/background data move with the representation rather than being retuned. Thus the whole image-source arrow needs no second p input. The D5 signed Hodge-complement type also matches simultaneous D5/A3 opposition, with minus I8 explicitly in W(D8).",
      "scopeEN": "Corrects an overstrong premise; the image-source result is positive. This transports a given complete Seed rather than deriving its existence from fewer primitives. Do not promote the original definition/proof architecture to a new physical theorem. Fixed pi/Y/clock symmetry is a stronger separate question. Match the geometric spin pullback to the actual normal/collar realization, domains, faces, local algebra and reflection, without inventing a new p-background gate. Root normalizer type does not fix all original cocycle/spin phases.",
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      "titleEN": "Canonical Spin2 lift carries the whole annulus source with reversed time",
      "statementEN": "The existing collar Clifford product is i_old Z. The pure geometric half-turn, both coordinate inversions and endpoint exchange transport the entire given collar operator, oriented real structure, four field/adjoint slots, full graph covariance and CAR state. In the charged chart its generator relation is T hq T^-1=−hp, so the actual circle parameter reverses. This OLD/NEW linear map squares to minus identity and needs no particle swap or independently fitted p coefficient.",
      "scopeEN": "A complete image-source map for the given original F5 family, without claiming physical P1 selection. Constant-normal graph formulas retain their stated scope; general normal-dependent data are transported as existing Cauchy data. Distinguish this time-reversing linear geometric lift from the earlier linear Kswap plus-time map without face exchange and The reviewed antilinear plus-time face map. No identification of circle, modular and physical time.",
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      "titleEN": "Geometric spin collar map closes the weighted image domain with total deck action",
      "statementEN": "The original ramified spin frame converts the global lambda phase to the ordinary collar half-turn ±i, with the known ramification correction. The existing weighted Fuchs spin trace transports the given domain, corner powers and transmission into its genuine C3-equivariant image. The geometric R map carries this q image onto p and agrees with the whole Spin2 collar map. Its total descent operator is delta_c=Pc c*, not bare geometric c*. The canonical descended inversion has order four while the unquotiented global S0 lift has order twelve; sixth-power parity alone does not identify the representations. A geometric omega eigenline needs an actual inverse coefficient character to define an invariant tensor. The required inverse direction is already present in actual Pc; the full Hom result also closes coupled q/p signs without a naked parallel-line condition.",
      "scopeEN": "The transported image domain is closed and needs no independent curve operator, p background or corner choice. Do not equate it with all spin sections, the entire rank-nine pushforward or an independently prescribed maximal domain. Pc channels are not yet a physical insertion reader. No unchanged faithful U6/C12 claim on this descended source and no general exclusion of rho. Original physical source derivation remains distinct from this proved image transport.",
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      "titleEN": "The reviewed annulus map is a valid distinct type without unique geometric spin selection",
      "statementEN": "The reviewed full face-swapped Z gamma K candidate has correct operator signs with its given minus-conjugate target coefficient. The whole four-slot map exchanges fields and adjoints, is NEW antilinear and carries plus circle time. Normal M proportional to Z is unique only inside the constant separated matrix ansatz. Both the candidate and its old-i multiple preserve oriented face reality but differ in squares and JF parity. Old i equals Ical Gamma and is not a uniform scalar in the new typing. Sixth-power comparison does not establish a unique geometric spin lift.",
      "scopeEN": "Retain the valid algebraic candidate while rejecting its unsupported uniqueness/type inference. No exhaustive global lift classification or physical origin theorem. The older Ucal Kswap map remains NEW linear without face exchange. the reviewed source is archived as unreviewed provenance rather than used as proof.",
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      "titleEN": "Original mixed spin products already have complete dual Hom coefficients",
      "statementEN": "The inverse cubic character is already in the actual Pc spectral line. Original normalized Kummer square pairings identify the two-torsion sign lines isometrically with their duals. Mixed spin products are therefore already global K-valued Hom(Fq,Fp) sections. Composition with reverse Hom and trace canonically cancels the full q/p signs; no naked parallel one-dimensional leg or fitted character is needed. The actual End coefficient omega-squared channel has dimension three, so Pc alone does not select a unique clock matrix up to scalar.",
      "scopeEN": "A complete positive bundle/coefficient connection, without supplying an arbitrary reverse kernel, selecting a new state or duplicating q/p as independent CAR species. Actual source-kernel endpoints and types must match. The checked original output formulas do not explicitly identify a selected additional Hom/Fourier contraction, which is no universal absence theorem and does not forbid coupled total projections. A one-sided reader is an example rather than a new required architecture. Retain The reviewed positive AdPc channel; Pc-alone uniqueness fails, while a separately fixed diagonal marking gives a narrower uniqueness. No physical CKM or general CP theorem is adopted from its random-profile claims.",
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      "titleEN": "Off-diagonal Hom matches A4 family six; holomorphic triplet wedge has complex rank three",
      "statementEN": "Original s/R lifts and normalized two-torsion products give the explicit unitary A4 bundle map. The direct holomorphic two-triplet wedge has complex family rank three and is not six independent complex holomorphic components. Its fixed-quarter and generic SU4 counterexamples remain valid for that complex-linear map. Compact real completion must instead retain the actual adjoint and simultaneous D5 carrier conjugation, as reviewed in the subsequent correction.",
      "scopeEN": "A genuine restricted bundle connection and product exclusion for these two inputs, not a dimension dictionary. The original full K includes O; no singlet zero mode does not exclude full fields or existing seam/Hardy modes. Their actual spin, twist, mode and statistics must be checked. Do not substitute a bosonic E8 current for a Grassmann matter field. Fixed-quarter failure is not a universal fixed-marking requirement under whole-data transport, and no free singlet is added. Correction: no universal O-fermion gate follows for compact real completion. Family Hodge alone also does not prove full(10,6) with the carrier suppressed: direct C/X triplet inputs give30 real directions, full16xF inputs60. Subsequent carrier and restricted-D8 findings replace the old active O-input gate.",
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      "titleEN": "Three C/X families recursively close exact D8 with minimal original escape",
      "statementEN": "Only original C/X family labels1,2,3 and adjoints give36 seed roots. Full recursive Chevalley closure is36,92,112, rank8 and exact D8. Missing10xF components remain unreachable;30 of60 target roots occur and30 never do. Depth-two new roots are10xO plus adjoints. Any single omitted original C0 or X_i0 root, with its adjoint for compact input, escapes to E8 with114,170,240. Thus this restricted recursive class is fully decided and its minimal original escape specified.",
      "scopeEN": "Complete finite zero-mode Lie closure and concrete different D8 frame in E8. Not a no-go for higher affine products, other original operations or the full original family four. No new species invented. Geometric mu access to the minimal existing escape current and actual finite consumer remain separate. Algebraic reachability is not state preparation or physical time; independent root-sum and source-cocycle audits agree.",
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      "titleEN": "Original D5 zero mode completes triplet carrier; marked glue is exact minimal escape",
      "statementEN": "The restricted D8 contains only20 of40 original D5 roots. An original grade-zero D5 zero mode gives the explicit cocycle-correct word into the missing carrier half, preserving glue grade2. Full original D5 action with the triplet seed closes to all240 roots; its Weyl orbit completes15 to30 complex target directions and adjoints to all60. No new O fermion or freely supplied full16 input is needed. Original marked glue lambda maps in the fixed chart exactly to C0 dagger, with its adjoint the minimal D8 escape pair. The D8 vacuum lattice VOA preserves its coset; distinguish its new Z2 parity from old mu4 grades.",
      "scopeEN": "Positive actual operator connection at the given P2/E8 current level; replaces the extra-carrier/O gate. Not primitive P1 realization, physical zero-mode history, geometric mu-to-current or simultaneous-QY consumer derivation. Original quarter twist still lacks the localized charged glue-vertex identification beyond its conditional neutral eight-copy kernel. Accept The reviewed independently matching D8/D5 step, not its further exchange/CP or normal-carrier-selection conjectures. Do not count existing total Hom character compensation twice.",
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      "titleEN": "Full deck signs forbid bare triplet projection; original spin transport fixes its sheet",
      "statementEN": "In natural spin frames s*mu_ab=-mu_ba and R*mu_ab=rho^-1 mu_(b+1,a+1); the antisymmetric half is the positive U6-cube sheet after orbit phase calibration. The diagonal-A4 coefficient coisometry and bare positive K3 calculation are exact but do not define a global triplet subsource: six distinct V4q times V4p characters, together with transitive R, have scalar commutant. A displayed sign element exchanges antisymmetric and symmetric pairs. The original base map and v117 pullback already exist.",
      "scopeEN": "Complete finite result only for the bare six global section channels with independent original deck signs and R. Not an obstruction to coupled endpoint kernels, Bmu, diagonal response or P1. A4-only coefficients are a restricted category. Marked S conjugation transports a supplied triplet matrix rather than producing a global triplet subsource; retain canonical-form and endpoint factors. Read the early bridge report only together with its corrected addendum. Completed The reviewed and the reviewed source identify bare mu channels with Gamma paths, but matching labels do not compose their actual q/p endpoint and canonical-form types. The actual Bmu response is the source-supported replacement.",
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      "titleEN": "Actual reverse kernel yields global q Hodge response after full composition",
      "statementEN": "The actual B=GqJ* composition T(u,v)=B(u,v)mu(v) cancels the full intermediate p signs and retains q input/output covariance. On existing smoothing same-face diagonals, the original bilinear skew projection and oriented Hodge map yield a global canonical-form-valued Fq response, retaining normal and D5 operators without new line, state or normalization. The given scalar common-quarter collar at Phi=0 already has a nonzero response to original off-diagonal insertions, using its actual positive diagonal Fourier kernel and invertible endpoint factor, without input Yukawas.",
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      "titleEN": "Bare C3 kernel class with diagonal readers has a common real frame",
      "statementEN": "The same D=diag(1,omega,omega²) conjugates every positive bare H3 and its inverse square root to real symmetric matrices. Real diagonal left/right readers commute with D, so all such Yukawas are simultaneously real and all weak flavor CP-odd invariants vanish. No parameter or genetic search within this fully classified bare class can change the result.",
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      "id": "finding:original-primitive-action-is-present-highpass-is-not-finite-selector",
      "titleEN": "Original primitive action exists; its high-pass does not select a finite carrier",
      "statementEN": "the reviewed source correctly quoted the original primitive relative-action formula and its bound reference/relative split. Do not reject it by substituting a different relative-trace convention. The actual primitive projector is seam-even times a spectral high-pass and retains infinitely many circle modes, so it does not by itself select finite Hcar or generate Yukawas. The displayed master variables and hypercharge Y are distinct from Yukawa matrices.",
      "scopeEN": "Preserve the original action and supplied full GNS/state/time source. Not an exclusion of every possible variation or renormalization; reuse existing coupled reference/reflection/gap/domain/projector/eta variation results. Old first-order-bound and D4 corrections are not rediscovered. This checks only the claim that primitive projection/stationarity already generated the finite background.",
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      "titleEN": "Atlas detail retrieval recovers existing D4 solution after content ranking repair",
      "statementEN": "The actual natural D4 request reached the correct large problem, but full-JSON detail ranking let source metadata hide the existing solution. The small existing Rust retrieval now prioritizes semantic coverage and result text while preserving exact IDs and metadata fallback; bilingual tokens and continuation packing were repaired. Three regressions and all19 research-module tests pass. A fresh actual run.sh mcp request retrieves the existing D4 repair first with seven sources and within budget at the same data revision.",
      "scopeEN": "Retrieval software repair, not a new theory proof or direct graph edit. Existing old MCP processes do not hot-reload; fresh isolated connection was actually verified without stopping other clients. No full server-suite claim. Until normal reconnect, use exact IDs or the existing fresh stdio path rather than interpreting missed retrieval as scientific openness.",
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      "titleEN": "Original Poisson range and boundary metric fix the whole Calderon graph",
      "statementEN": "For the supplied homogeneous selfadjoint collar, its Poisson range graph(exp(-ell h)) and original positive boundary metric uniquely determine the full bounded orthogonal projection G with A=(1+exp(-2ell h))^-1 and X=(2cosh ellh)^-1. Both normal traces of the antiperiodic Green formula give G directly. An invertible periodic realization yields an oblique projection with the same range and thus the same orthogonal G. Functional calculus includes unbounded h; no extra AP-selection gate is needed.",
      "scopeEN": "Exact bridge for the supplied homogeneous source. Bulk Green kernel and boundary projection are related by the proved normal-trace map, not literal equality. No equality of h with longitudinal quarter time. Wrel defines generic source derivatives but does not alone select J_geom* M_mu; the supplied source and specified insertion fix its response.",
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      "titleEN": "Actual induced Hardy defect gives a finite whole-source reader",
      "statementEN": "Actual signed G columns prove F_v117=Ind_V4^A4 chi0=pi_*L0q. The old spin cable is the correct induced family bundle without extra copies. Its given NS Hardy unit-shift defect has family rank three and transports by V_h into the whole graph, giving M_rs=<JE_r,M_mu E_s>. The actual induced mu01 input at Phi=0 gives nonzero M02=(4X*/pi)I Pgamma+, X*=(2cosh(ell_NS/2))^-1, I=0.1258739334853158906973-0.4697679151157443151336i. Actual q2/p1 open arcs add by their Kummer signs while diagonal pairs cancel. A closed primitive and strict analytic lower bound prove nonvanishing. Independent root checks give zero exact derivative remainder and continuous-branch quadrature residual below3.2e-76.",
      "scopeEN": "Existing mathematical NS Hardy source contract, not a newly proved physical finite P1 selection. The graph-lifted shift differs from both-face diagonal shift and relative matching; the whole-annulus shift has index zero. Graph membership alone does not establish finite time stationarity. The evaluated response is neutral and grading even, retaining all D5/charge slots, not an odd Higgs/Dirac source. NS frequency1/2 and quarter1/4 are related only via the old double-frame map with ell_quarter=2ell_NS and both quarter sectors for the seam shift; no physical scale/time equality is inferred.",
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        "premise:relative-spin-scalar-hardy-cable-on-original-three-cover",
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      "titleEN": "Original weak quartic action has a genuine nonlinear response",
      "statementEN": "Original Metrology already contains E_rel=-R I48/4-(Phi*Phi)P2+gammaF/2 and the weak E2 potential. Along admissible Phi=s varphi the first E response vanishes, second equals-2(varphi*varphi)P2; the potential has explicit nonzero second and fourth responses and a nonzero radial minimum for positive coefficients. The supplied scalar collar crossblock likewise has zero first but explicit nonzero second variation. No new free quartic ansatz is needed.",
      "scopeEN": "Retains given E2 admissibility and Yukawa seed. Euclidean GammaD4 and stated potential use distinct sign conventions. Zero first X derivative is not a no-go for whole G responses. Bound primitive variation retains reference, gap, reflection, eta and domains. Does not prove that B_mu is the E2 tangent or impose fixed-background QY as a universal origin condition.",
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      "titleEN": "All common leading-distance clutches fail original flavor lengths",
      "statementEN": "Original diagonal TOTAL lengths Lu=(7,3,0),Ld=(7,5,2),Le=(8,5,3) have sorted gap ratios3/4,3/2,2/3; separate numeric left/right lists are not supplied. A common single-leg list requires one ratio. For equal-weight same-list two-leg readers, normalize x=(0,1,t): all six relative S3 permutations are degenerate or give only t-1 and its reciprocal. No arbitrary real triple reproduces all three original rows. Entire class is analytically decided.",
      "scopeEN": "Only shared leading distances with the stated equal weighting. Different q/p spectra, unequal weights, full mode sums and actual sector-dependent original C6 residue packets with additional length-six windings remain outside the exclusion. Frobenius exponents are not a boundary clutch without an actual operator dictionary.",
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      "titleEN": "Original weak wedge consumer reads the Hardy response with full KO reality",
      "statementEN": "For the actual nonzero Hardy coefficient and marked original weak P2 creators, the exact commutator gives a nonzero grading-odd color-neutral weak doublet of hypercharge+1/2, rank16 for each component in the declared whole-96 comparison. All45 D5 Clifford bilinears commute with the even Hardy projection. Adjoint completion alone has rank32 and fails finite reality. Full original KO6 completion has rank64, is selfadjoint, grading odd and J_F real, with D_KO^2=|kappa|^2(E00+E22) tensor I32. All four sectors retain the same family matrix but fixed quark/lepton signs and antiparticle conjugations; no Majorana block. Actual checker replays and independent root algebra pass. Crucially, original carrier008 already contains the primitive exterior-volume coupling. Its actual up-quark channel reads the new directed coefficient as+kappa; the fixed signed KO Hilbert convention gives-kappa because compact conjugation reverses the complementary wedge. No phase is fitted. The original weak carrier consumer is therefore present.",
      "scopeEN": "Exact operator and original carrier consumer under the declared family/96 P2 comparison. Original wedge contraction corrects the earlier overly broad claim that no original weak Clifford application exists; read the two earlier operator reports with this addendum. Original family coefficients are already prescribed by Hol(nablaF*)Kstr. It remains to compare the actual Hardy contribution with the relevant entry, actual variation or path summand, without imposing equality to the whole Ybr as a universal condition or treating original Y as arbitrary free v252 inputs. Keep mixed E8 family leg, fixed-background limits, C0 versus J_F, internal versus field CAR parity, and NS versus physical quarter time distinct. A correct finite block is not the complete physical source derivation. General addendum: identical familyM on all96slots yields S=M+M^T. Two directed edges give equal nonzero singular values, not unique polarCKM. Different bare doublet-component coefficients violate fixedSU2. Typed original sector-specific readers remain allowed.",
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      "titleEN": "Two-entry hierarchy fails full KO completion and doublet covariance",
      "statementEN": "For arbitrary complex3x3M identically lifted to actual96family slots, full originalKO6 completion of A=[X,MP+]=MXgamma is D=(M+M^T)Xgamma−(M†+barM)X†gamma. For M=aE02+bE21 the symmetric coefficient has singular values(r,r,0),r²=|a|²+|b|²; fullD has rank64. Independent symbolic characteristic polynomial and actual96replay confirm this. RawM/M† Grams are diagonal; fullKO has a degenerate two-space, not a unique polarCKM prediction. Actual[T+,Xw2]=Xw1 forbids unrelated bare matrices on this fixed doublet.",
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      "titleEN": "Actual Fuchs halfpath has a canonical G-metric C6 kernel map",
      "statementEN": "Actual originalv117 P/Q integration and separately exact unitarykernel now share a closed frame map using already proved algebraicG. Fixed H=G^-dagG^-1 source metric and actualsigned cycleCs give Cpair^-1TrawCpair=[[0,Cs],[I,0]] with Cpair=diag(Gx,PGx) an isometry. Sourceadjoint equalsinverse. ExplicitB and existingJ connect to originalM/U6. Actualpath,D,metricGram,Green and inverse square root share one isometricfinite map, independently replayed with realP/Q residuals<1.3e−12. EvenwordCs² reads certifiedκE02 contribution; barepositiveK branch weights give v±κ. These are evaluated contributions, not an assertion that the complete originalsource inserted them. Stronger common marked closure: originalGbase/H0 gives H0^1/2Gbase=e^(7pi i/12)Bstar with explicitly fixed unitaryBstar. The sameBstar intertwines both actualCs to originalM0 and inverseKQ to originalU, hence all four markedmeridians. Exactly BstarE02Bstar†=M0P0 and BstarE22Bstar†=P0. CertifiedHardy contribution is thusκM0P0 in the common originalmarked frame with the existing quarter-fixede0 input. Six exact rootidentities and literalGbase normalization replay confirm it. The fixed markedendpoint is explicitly P0=(I+U²)/2; separately reviewed findings continue this same channel through the actualreal delta jet and allfour originalopenpaths. Positive actualspine consumer: originalP/Q fold is realized by P=T3,Q=T0^-1 and fixedC, with sameJ carrying inversefold to originalU6. OriginalD_y/fullGram/positivefunctionalcalculus and every already givenPstr transport isometrically; eight independent rootsymbolic checks pass. Recovered original v118 consumer reads det(I-tM0)=1-t³; native t=1/2 gives16/7,4/3,7/6. Later v121/FLAV.H2.04 uniquely pins recorded R under declared margins anddet8 among17 candidates, independently re-enumerated.",
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      "statementEN": "For the SAME actual inducedmu01 input, complete nine-entry family answer has only E02 and E21, with seven exactsupport/eta cancellations. ActualG/R endpoint calibration fixes the opposite second root and sameCCW orientation. Two genuine atan continuations give EXACT I21=-I02+pi(1-sqrt2)C, not a numerical phasefit. Independent root support/sign/orientation, rational derivative and70digit period replay confirm it. Matrix has rank2, cubezero. Reuse the already complete two-edgeKO theorem: under declared identical96family lift, neutralB rank32, fulloddcompletion rank64 and symmetricfamily singularvalues(r,r,0).",
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      "titleEN": "All six original Hardy inputs generate the full family coefficient algebra",
      "statementEN": "Actual G columns and original R lift force G_a=G Pc^(-a) and the two-leg map Common=P^(1-b)ScalarP^a, preserving the stated kinetic NS/Hardy domain, metric, spin and full graph. All54 entries are computed:42 exact support/eta zeros and12 continuous-root periods. The two oriented triples are (1,omega²,omega)F and (1,omega²,omega)N, F=alphaE02+betaE21,N=-i(alphaE12+betaE20), with actual normal projector. Reverse phase follows exactly from the calibrated period. Independent65digit replay and source review confirm all inputs. F and N together generate all nine matrix units of M3(C), even without separately adjoining identity, with scalar common commutant. In the SAME originally marked Bstar frame, Fhat=alphaM0P0-betaP0M0 and Nhat=-Dsign Fhat Dsign, hence Fhat is the original delta jet at both marked endpoints with actual period weights. The finite moment reader intertwines rawR with Ad(W), W²=-I, retaining only its ±1 moment lines. Actual fixed-D resolvent traces are exactly (alpha-beta)delta²/(y³-delta³) forFhat and i(alpha-beta)y delta/(y³-delta³) forNhat. Twenty-one additional exact root checks confirm these identities. Actual fixed-cusp tangent test gives d(M³)[Fhat]=(alpha-beta)I nonzero. At the same time its cyclic centralization is ((alpha-beta)/3)M0 and original scalar resolvent trace is (alpha-beta)t²/(1-t³), with determinant linear response -delta²(alpha-beta). Independently verified exactly.",
      "scopeEN": "Complete six-input alphabet in the given kinetic NS/Hardy reader; global twist sectors remain distinct. Squares alone only generate an embedded two-by-two block. Coefficient accessibility does not identify physical field/CAR products or select whole Yukawa, coupling mix, state or time. Equal-weight sum cancels by cubic phases, not a universal source no-go. Existing wedge/Clifford/KO consumer applies linearly. Continue original common use without imposing this route as a universal P1 gate. The linear finite R moment image does not replace the global spin/R action or source state. Two jet orderings do not select physical V_adm or reduce the entire origin to endpoint placement. Do not replace original provenance with another conditional Gram comparison. Only labeling Fhat as fixed-cusp deltaHol fails; genuine coupling or insertion remains eligible. Scalar response does not establish original source selection.",
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      "statementEN": "Centralizations cM0 and icM0^-1 reproduce first scalar traces, but actual full marked determinant differs from the centralized two-sided replacement already at quadratic order. Exact recorded full polynomial and difference hold for actual periods, with nonzero1+r+r² at real period ratio. The full adjoint relation has a nonzero marked residual; only its centralization agrees. Original delta is real, so the complex analytic saddle is not an admissible original descent. Inverse legs already exist in original Gram/Green/Kstr. Ten exact root identities and independent source/matrix review confirm.",
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      "titleEN": "Oriented resolvent weights do not identify the original producer",
      "statementEN": "The resolvent coefficients are correct, but the additional diagonal-weight construction fails abstract word-reversal invariance, with an exact rational diagonal difference2600/3969. QFT2 does not identify that reversal with OS Theta or Q_Theta; only Q_Theta=R forces the even projector and word pairing. Symmetry does not refute Perron primitivity. These coefficients alone do not derive the actual retained-basis action and original weight call.",
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      "titleEN": "Original odd carrier reader is lossless on the whole coupled source",
      "statementEN": "The complete original v258 mu_geo Pi_odd log-odds reader composes with the existing reducing E chart and actual second face to R=2ell mu_geo U_gamma Phi U_gamma*. It retains the full96 coefficient, Majorana and Yukawa jet while removingcircle momentum without a new Fourier projection. The smooth H1 identity remains valid for noncommuting even connection and odd coefficient. Retaining known A,ell andmu gives a lossless reconstruction of the whole graph; the full covariance recovers both A andPhi. The comparison unitary preserves actual antiunitary charge and even algebra. Direct unrotated feedback is only cPhi: all coefficients fixed at c=1, otherwise only zero; exact inverse feedback is identity and selects no Yukawa pattern.",
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      "titleEN": "Actual weak connection transmits family orientation into the joint boundary source",
      "statementEN": "In the given v252 source the actual real weak lift Q=-i(T+J_F T J_F^-1) has the required antiparticle minus sign. For h=(pI+tQ)gamma-i gamma D_F the commutator term in h² is retained. Two constant diagnostic quark backgrounds with equal separate singular spectra give DeltaTrh²=DeltaTrh⁴=0 but DeltaTrh⁶=5760t² at p=1/4, with all96 slots and identical nonzero lepton/Majorana data. Independent block calculation yields36 DeltaTr(HdHu). Thus for every nonzero t and every given positive width the actual cross-face X eigenvalue multisets differ. Finite small-width trace and logdet differences begin at -244ell⁶t² and -128ell⁶t². The actual coupled source retains mixing information at constant Higgs through its existing weak connection. Under the whole unitary polar formula Y_f=T_f K_f, Tr(LuLd)=Tr(Ku² V Kd² V*) with V=T_u* T_d; a common T gives V=I. Retain original lambda dressing and path-reader typing.",
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      "titleEN": "Original Higgs variation gives the full radial tadpole at fixed Yukawas",
      "statementEN": "The original v252/v254 call varies q(H)M with supplied Yukawas. Its displayed local target quartic yields the Higgs tadpole, not independent matrix-Y stationarity. The whole96 radial family with Majorana retained has exact quadratic/quartic trace identities, including4Tr(DY²DMR²)t², and the corresponding contracted radial tadpole. Bound coefficients,gap,reference andeta retain chain terms. The reviewed further identification of the original APS generator with a function of h² is absent from the cited definition; only the already known pure PF<=G branch is settled. The reviewed original master already contains relative Z and determinant terms; The reviewed separate extra Gamma_H is not specified there and may double count. Differentiate the actual shared field dependence in the existing expression. The explicit logdet coefficient alone is not the derivative of the whole master: the relative Z contribution can change it already at first field order. An exact scalar Gaussian control refutes that inference without replacing the original TFPT source.",
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      "titleEN": "Whole weak cap converges across polarization changes",
      "statementEN": "For the actual constant weak source, Zp=exp(ell E_p)det_RanP0(P0 exp(-ell h_p)P0) is the old Slater expectation of exp(-ell H_exc,p), with0<Zp<=1. For v=||V|| and g=|p|>2v, negative eigenvectors yield Tr(Ph P_wrong)<=48v²/(2g-v)². Ground overlap and spectral resolution give the same bound for1-Zp uniformly inell. Thus the product over all quarter modes is strictly positive and finite at every finite ell>0. Finitely many crossings or overlap zeros do not destroy the finite-time cap. Zero-mode occupations leave E_p and H_exc,p unchanged and do not affect the cap at fixedP0; only a particular ground vector/Thouless chart needs them. ConstantQ preserves Fourier modes but not the old scalar-pair reduction. The actual free-reference ratio retains exp[-ell(E_p-E_ref,p)]Zp. Full96 numerical controls confirm that factor and recover the t0 scalar formula; convergence is analytic.",
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      "titleEN": "Canonical Clifford clutch derives normal antiperiodicity and common cross kernel",
      "statementEN": "Opposite normal Clifford transitions unfold D union(-D) into the same D on circumference2ell with antiperiodic closure. L=-ND=partial+B gives S_AP(ell,0)=(2cosh(ell B))^-1, the existing whole charged cross kernel without matching a target. D²=-partial²+B². At finite tangential cutoff logdet_zeta(D²+mu²)=Tr[2ell E_mu+2log(1+e^(-2ell E_mu))]. Full-circle gluing Fredholm factor exists, while bulk retains original renormalization. Natural T=NP_r has square -I and reverses D. Involutive R_odd=i_old NP_r reverses D; R_even=Gamma_tot NP_r preserves D. Both anticommute with Gamma_tot and fail the older [inv,Gamma]=0 if the old grading maps to this Gamma_tot. The actual old grading/real-structure map is unresolved. Actual v252 full192 control confirms15 exact identities.",
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      "titleEN": "Universal graded reducing images do not fix the carrier frame",
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      "titleEN": "Entire original compatibility family leaves generic31 parameters",
      "statementEN": "At actual v252 all84 independent real directions and their whole real family pass the specified compatibility equations. Actual54 U(3)^6 Lie generators have generic orbit rank53, certified by a nonzero exact53 minor and universal baryonic U(1). Generic quotient dimension31. Enumerating the same compatibility predicate cannot select a unique source. This checks the known31 dimensional finite NCG moduli on the implementation.",
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      "titleEN": "Entire reflected quadratic regulator class decided by negative cross matrix",
      "statementEN": "In specified finite class exp(-phi R phi/2) preserves every original positive RP measure iff C is negative semidefinite. Bounded reflected feature expansion gives sufficiency; RP product Gaussian with Gcross=((I+A+C)^-1-(I+A-C)^-1)/2 and inversion order gives necessity, without simultaneous diagonalization. Cone closed under addition; R_k to zero recovers original source by finite dominated convergence. Exact RP baseline C0=[[1,1/2],[1/2,1]],R=ones,P_adm=I gives reflected response-1/16 despite ordinary positivity and structural commutation. Existing generic spectral-weight exclusion is reused, not recounted as new proof.",
      "scopeEN": "Complete universal preservation theorem only for specified bosonic two-half class. Known RP-cone mathematics specialized to original IR caller. Not necessary for every regulator of one fixed source, not full P1 countermodel or proof of original S_adm RP.18 controls together with source flow; analytic class proof separate. No new physical model optimization.",
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      "titleEN": "Shared original Higgs source induces coupled matter words with supplied Yukawas",
      "statementEN": "All288 real coefficient basis images of actual full96 one-form consumer match four H derivatives exactly with adjoints,grading and JF. Same supplied H fiber gives exact affine Z_H[J-j|A],all cumulants and absolute factor. Induced ordered nu/up four-fermion action coefficient is -G_h2barh2 Ynu_ab conjugate(Yu_dc),no factor2; third/fourth boson cumulants give6/8 fermion terms. Retain all extra H dependent fermion vertices,which can also alter four-fermion coefficient. Fermion-first evaluation uses actual det/Pf weighted full source polynomials,with minors at singular fibers rather than inverse division.",
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      "titleEN": "Local even Yukawa sewing preserves supplied joint graded RP",
      "statementEN": "Original v42 mixed RP inference does not follow from commutation and separately positive marginals. Strictly positive reflection-invariant logical control p=(1-r xminus xplus yminus yplus)/16 has RP product marginals and mixed reflected form-r. Constructive full replacement: for actual joint graded RP reference and even local half action B=exp(-Vplus), omegaY(ThetaF circ F)=omega0(Theta(BF) circ BF)/omega0(ThetaB B)>=0 with positive normalization and integrability. Handles whole supplied Yukawa class with adjoint partner wherever same actual reflection intertwines H/matter density. No pointwise positive determinant required.",
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      "titleEN": "Whole original constant Higgs fiber has reduced Majorana spectral response",
      "statementEN": "For all H and supplied Y,Phi(H)=V(q/r)(rDY+DMR)V*,weak left rotation with same JF/gamma and unchanged MR.84 real Y/MR basis images on actual original96 confirm block proof. det96(z+Phi²) reduces exactly to charged3x3 powers12,12,4 and squared6x6 neutrino determinant Bnu=[[rYnu,0],[MR*,rYnu^T]]. First u jet is4/z Tr(3Yu*Yu+3Yd*Yd+Ye*Ye)+4Tr[(z+MR MR*)^-1Ynu*Ynu],retaining necessary Majorana cross-resolvent term. On supplied quarter circle the whole absolutely convergent mode sum is evaluated by s_alpha(v)=pi/sqrt(v)sinh(2pi sqrt(v))/(cosh(2pi sqrt(v))-cos2pi alpha).26 controls pass.",
      "scopeEN": "Whole constant original coefficient class and named1D product circle,no new physical parameter choice. Internal96 spectral multiplicity is not96 independent fermion integrations. No free Yukawa search,whole4D Z master or full dynamic H covariance. Nonconstant H,fixed nontrivial A and global projections retain original connection,domain and source transport. Existing weak-connection mixing response remains valid.",
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      "titleEN": "Archived geometric Higgs residue requires concrete coefficient identification repair",
      "statementEN": "Printed -6chi²ED+GammaD4 gives bulk +chi²u/(8pi²)+u²/(16pi²) at R0. Equal quadratic/quartic signs match neither printed target potential -mu²u+lambda u² nor its -V reading. Positive canonical/sheet normalization and common action/Wick sign cannot repair relative sign. For same u,+V requires original remainder -chi0²(1+wH²)u/(8pi²),-V requires chi0²(wH²-1)u/(8pi²)-u²/(8pi²). Required source responses,not permitted new addends. H-independent U(chi) or constant source denominator at fixed chi do not help. Keep FPzeta(0) distinct from -zetaPrime(0),and M(xi) distinct from H-source/Legendre/bare actions.",
      "scopeEN": "Exact failure of a literal identification of specifically named archived local formulas. No TFPT countermodel,physical vacuum-instability claim or exclusion of existing Yukawa/Flavor/Alpha. Full relative source,reference,eta,scale and continuation can carry further H contributions and must be evaluated. Existing local target quartic and its conditional responses remain; direct derivation from this printed bulk is not established. No new Lean theorem or ledger status.",
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      "statementEN": "Literal local Erel=-R/4-X gives coeff_ED(u)=-12 coeff_Gamma4(Ru),independent of rank/common linear trace weighting,including subtraction of two same-type Dirac producers. Named archived printed values -1/(48pi²),+1/(96pi²) give-2,a factor6 mismatch. With ordinary48 trace and rank2,whole printed ED bulk equals a2/6 while quartic/Ru match a4;conditional normalization diagnosis,not new original definition. If displayed E is complete and connection H-independent,integrated a4 gives no Higgs kinetic term;Delta u is a total derivative. Fixed boundary/reference data and s-independent iEta cannot repair real ED jet. Actual bound-reference variation is deck-even part of operator jet and may be frozen only upon established deck-oddness.",
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      "statementEN": "Whole original96 family satisfies Tr(Phi²-DM²)=Au,Tr(Phi⁴-DM⁴)=Bu²+4Cu,Tr(dPhi)²=Akin,A=4Y2,B=4Y4,C=2Tr(Ynu*Ynu MR MR*)>=0. Generic block proof and unchanged v252 consumer confirm identities including four-real derivative Gram matrix. Explicitly conditional standard4D product Dirac with separate spin trace gives a2H=-Au/(4pi²),a4H=Akin/(8pi²)+Bu²/(8pi²)+Cu/(2pi²)+ARu/(48pi²).For cchi²a2+da4,Z=dA/(8pi²),mu_can²=2(c/d)chi²-4C/A-R/6,lambda_can=8pi²B/(dA²). At R0 negative quadratic requires cchi²A>2dC;c=-6,d=1 gives positive quadratic for all supplied Y/MR.28 direct controls pass;control numbers are not new physical parameters.",
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      "titleEN": "Whole variable original Higgs bound-reference response and forced Clifford factor",
      "statementEN": "Fixed spin4 Clifford contract forces {gamma_mu,W}=0 hence W=gamma5 L(H),with gamma5=Z in given collar chart;H derivative in E follows by squaring. Put p=(H+Htau)/2,m=(H-Htau)/2,P=|p|²,M=|m|²,v=Re p*m,QS=P²+2PM+4v²,QT=M²+2PM+4v². Bound original references L(m),L(p)+DM give after H0 baseline:a2S=-A intP/(4pi²),a4S=int[A|Dp|²+BQS+4C(P+M)]/(8pi²)+A intRP/(48pi²);a2T=-A intM/(4pi²),a4T=int[A|Dm|²+BQT+4CM]/(8pi²)+A intRM/(48pi²). Normal derivative parity transported. Only reflected-pair quartic mean is nonnegative;raw local difference can be negative. S cannot repair R0 mass sign or ratio-12 in -6chi²a2+a4. T cancels entire unscaled same-geometry pair response for purely even H at every spectral order.",
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      "titleEN": "Original symmetric scale map fails simultaneous Higgs and Einstein repair",
      "statementEN": "Given4D spin realization gives Dsigma=fDf unitarily equivalent to Dirac_ghat+gamma5 exp(-sigma)Phi,ghat=exp(2sigma)g,U=exp(2sigma)F^-1. H and DM scale together. H-dependent a2=-A int exp(2sigma)|H|²/(4pi²);integrated a4H invariant up to divergence,including Majorana term. For deck-even sigma,UNSCALED bound reference,t=exp(2sigma):a2SrelH=-A int[tP+(t-1)M]/(4pi²),a2TrelH=-A int[tM+(t-1)P]/(4pi²),a4 as R032. t-1 not always positive. Constant sigma,pure even H,S branch gives H²/RH² ratio-12t. t=1/6 alone would match printed-2,but same ordinary positive trace/same bundle gives Einstein a2relR=-N(t-1)intR/(192pi²),POSITIVE at1/6 versus printed NEGATIVE for all N>0. Jointly scaled reference cancels pure Einstein response. Variable sigma adds -N int6t|dsigma|²/(192pi²). Complete Ksigma=fDgeof+Brel is not resolved by B0 test. Dsigma² differs from exp(-sigma)D²exp(-sigma).",
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      "statementEN": "Exactly [fDf,b]=f[D,b]f,hence A_sigma=fAf including real completion,also for variable sigma without omitted df commutator. Actual inner change is V(H)=gamma5[L(H)-DY],not entirePhi again. K_in=fDstarf+qV(H)=fD_Hf,q=f²=e^-sigma. Same unscaled outside change gives K_out(H)=K_in(Hstar+q^-1(H-Hstar)). InverseH=qh+(1-q)Hstar;H0 does not map to0. Whole finite source:Zout[j]=(prod q^4)exp(sum(1-q)j.Hstar)Ztilde[qj],with actual pulled-back action,all extra vertices and reference;other measures by pushforward. Higher cumulants retain external q factors,mean shifts,absoluteZ0 retains measure factor. Whole same inner fluctuation thus falls under R033;mere relocation with full coordinate transport is no new physical repair.",
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      "statementEN": "Hom_G^R(H,End_C(C3+C2))=0:color center removes cross blocks,weak center is even on diagonal endomorphisms and odd on H. Original whole96 wedge/Yukawa consumer remains valid. Normal B_Ea has different first symbol and is not its zero mode or coupling. BP0=0 impliesP0BP0=0,not vanishing Higgs vertex. For actually given B_sigma=fBf,S*S=I,rangeS=kerB:kerB_sigma=f^-1kerB,G=S*f^-2S>0,T=f^-1S G^-1/2. T*(fXf)T=G^-1/2 S*XS G^-1/2;unscaledX needsf^-1Xf^-1 inside. G can be nonscalar. For q constant on normal fiber,T=S;normalization does not select vertex placement. Two-complex-mode d4Hraw=(det_CG)^-1d4Hnorm differs from q^4 Jacobian of other affine map.",
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      "statementEN": "In the given Spin4×96 product, total Gamma=gamma5⊗GammaF anticommutes with D_H. Pc is parallel but does not commute with individual gammaMu. The Higgs derivative is spin odd and internally odd, hence total even: Pc retains it and reduces D_H², unlike the internal half in R036. Equal rank does not identify these projections. Every selfadjoint odd K with grading-stable domain pairs all positive spectra of T*T and TT* including multiplicities. Thus every allowed summable zero-excluded Pc spectral trace is exactly half; heat including zero modes adds half the index. A common positive high-pass of the same K² removes this index remainder. Scalar f preserves oddness; a bound reference requires its actual domain and tauGamma tau^-1=±Gamma. H-dependent a2/a4 supertraces vanish in this product: H²,RH²,kinetics,H⁴ share the same half and the ratio remains −12.",
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      "statementEN": "Interior h=delta sigma gives deltaK=−{h,K}/2 and deltaZeta(s)=2sTr(hL^-s). If inserted zeta is regular at0,FP0 is invariant while delta logdetL=−2FP0Tr(hL^-s);with simple poles deltaED=2Res1Tr(hL^-s). Original ordinary product a4H additionally contains divergence−(2A/3)Delta|H|²/(16pi²),which cannot be discarded under local h. Actual exteriorBrel gives Wext=deltaBrel+{h,Brel}/2(+fdeltaDgeof) and deltaZeta=2s[TrhL^-s−Tr(Wext K L^-s-1)]. Retain reference,eta,zero modes,domain and trace response. PcK=KQc,not commutation:given odd-family chiral variation is half the whole,not naive inserted Pc trace.",
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      "scopeEN": "Conditional original-source inference discriminator,elementary known integrability mathematics. Actual TFPT measure is not asserted flat;no global TFPT exclusion or continuum theorem. Preserve existing finite transfer/CAR/full-word sources. Joint rank/IR limits,unbounded f,other original measure/support/action require actual producer. Signed moments may vanish and bounded responses converge despite escape;not exclusion of all algebraic state limits. Native96 is internal constructor control,not whole4D bosonic operator.30 controls,no newLean or ledger promotion.",
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      "statementEN": "Exact full square and Tonelli give TrF(h_H^2/chi^2)<=96Balpha for whole infinite original circle and every geometric mode cut,uniform entireC2 H and mode count,no Yukawa invertibility. Whole same-product reference gives−96Bref<=S_N<=96Balpha. Ordinary flatC2 Boltzmann volume remains nonnormalizable uncut;every joint N→infinity,epsilon_N→0 has ball mass<=exp(96(Balpha+Bref))R^4epsilon_N^2/2→0. This closes R055 rank loophole in stated original constant source class. Positive full-trace W needs no commutation;compression beforeF needs actual reduction. Same proof extends to already given R031 graded compact product with finite baseline.",
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      "statementEN": "All stated invariant marginals are H=sqrt(u)xi with Haarxi onS3 and arbitrary radial probabilitynu,Eu=2c. Same whole mode covariancecTT* fixes neither nu norEu². Fourth tensor Eu²/6(deltaij deltakl+deltail deltakj). Sphere and Gaussian Eu²4c²/6c² are nonidentifiability witnesses only,no physical selection. Original wholePhi expected quadratic2cA+M2 same,quartic B Eu²+8cC+M4 ranges[4c²B+8cC+M4,infinity) forB>0. Exact real Fourier sourcechi=int2J1(sqrt(u)|j|)/(sqrt(u)|j|)dnu uniquely determines whole nu;finite4moment gives logchi=−c|j|²/4+(Eu²−6c²)|j|⁴/192+o(j⁴). Reused actual-source pullbackchi=Zorig[iTj,0,0]/Zorig[0] keeps all complement fields integrated.",
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      "statementEN": "For everyproperregionpair I_R,J_L actualR=S*PS isnotprojection. Reusedprojectiondefectidentity giveson(0,f) exactly||Q_JXiota_If||²+||Q_ICiota_If||². Firsttermstrictpositiveforeachfnonzero:actualXhasexponentialmultipliers andmapsL2togloballyrealanalyticfunctions;zeroonopenJcomplementforcesXf0 andinjectivityforcesf0. Hence no state-preservingunitarylinearCAR identificationwithpurecanonicallocaldoublefromonlythatproperoppositearc. WholePremainspure;regionalrestrictionmixed.",
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      "titleEN": "Original whole elliptic environment fixes the regional CAR partner",
      "statementEN": "In sameoriginalP takeU=rightL2(I),E=wholeleftplusrightIcomplement. Bf=(Xiota_If,Q_ICiota_If),B*B=C_I(1−C_I)=A². ZeroendpointkernelsgiveuniquepolarisometryB=TAonallU,noboundedAinverseassumed. DT=T(1−C_I)yieldsactualreducingU⊕TUwithcovariance[[C_I,A],[A,1−C_I]]. WholemixedCARwordsfollowinsameoriginalstate. ContractiveT_eps=B(A²+eps)^−1/2convergestrongly;onu=Agerror≤sqrt(eps)||g||,fixedwordsconvergenorm. Explicit single supportedprobeu=sqrt2 1Ihasdistinctsubsystemcutandpartnert=(Xu/a,(Cu−cu)/a),c=<u,Cu>,a²=c(1−c). Wholeoccupiedsea1/4analytic,actualchargedcrosswords/fourfieldwordevaluatedforbothquarters.",
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      "statementEN": "Actualfixedrightuprestriction C_H=t(u)I3 givesrho=d_N,d_k=int t^k(1−t)^(3−k)dmu,sectorprobabilitiesbinom3k d_k. FullK=−logrho=k0+J1N+J2binomN2+J3binomN3 withJ1=log(d0/d1),J2=log(d1²/(d0d2)),J3=log(d0d2³/(d1³d3)). M_c=otheroccupations0,1,2 gives sigma_s(a_c)=sum exp(isDelta_m)P_cm a_c andsqrt(rho)a_c=sum sqrt(d_m/d_(m+1))P_cm a_c sqrt(rho),Delta_m=log(d_m/d_(m+1)). Entirepositive nonconstantradialclassforces J2<0,Delta0>Delta1>Delta2,Delta_m<0. J3canhavesign±. Actual96builderincludingMajorana,correctrightgrading,sameH-independentfamilyvectorand8Fockmatriceschecked.",
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      "titleEN": "Exact NS Ramond heat equality does not identify the fixed-color source",
      "statementEN": "P swaps first/last four source coordinates,r=(1/2)^4(-1/2)^4. Both original quarters obey P^T(r+delta)=delta and r²/2+delta.r=0. k→Pk+r bijects whole NS/R with same given energy and oscillator multiplicities,so Z_R=Z_NS exactly and Theta_R(X)=exp(r.X)Theta_NS(P^T X). It does not preserve actual fixed color center:Mr=e1+e4 has triality1. Retain prior removal of R zero state while NS vacuum remains. Hence Z_NS0≥1 and Z_R0≤Z_R-1=Z_NS-1<0.06 at beta4pi,with analytic rational bound. Optional NORMAL full-B(H_parent) KMS extension has unprojected NS/R weight1/2 but not after actual center restriction. Do not select it as a physical rule.",
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      "statementEN": "Original Gphi is eta-diagonal,required Vq=Ualpha diag(exp(-iq_k b)) Ualpha*. The two diagonal algebras intersect exactly in CI. All four actual q have2/6 signs,q²2,sumq=±2,so Vq is arc-diagonal iff b in2piZ. Smooth compact proper-interval support forces b0. Exact offdiagonal squared norms are3/2 for Aq and6sin²(b/2) for Vq,both quarters. Entire scalar profile class decided without frame/profile fit.",
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      "titleEN": "Actual composite current retains multi-arc support and fails additive even localization",
      "statementEN": "Existing Jq=:chi*diag(q)chi:=:eta*Aq eta: fixes smooth operation and levelq²2,no free matrix. Aq contains actual cross-arc bilinears. Full even CAR support stays in U=F^-1(K). Original scalar Weyl with pi on one arc,0 on others commutes with additive join of separate even arc observables. Its Rr=I−2|r><r| fails to commute with Vq where Aq is offdiagonal,so q operation generically is not in that additive even algebra. Distinguish interval hull,full union algebra and additive arc algebra. Smooth neutral transport remains possible;endpoint monodromy does not imply blanket local or3+1 noGo.",
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      "statementEN": "Exact||(I-P0)J(phi)Omega||²=sum_m>0 ell_m|phi_m|² with ell_(8k+s)=kD_s+L_s,D=(0,6,4,6,2,6,4,6),L=(0,1,1,3,1,3,3,5). Unique w0=1 supplies original witnessa0*a_-mOmega for every m outside8Z;ell_m=0 iff8|m. ENTIREexp(itJphi)group reducesP0 forALLt iff phi(theta+2pi/8)=phi(theta). cos(theta) exact vacuum leakage norm²1/4. Actual qCartan source preserves color but is not a uniform scalar current. E_Z(Jm) retains matching-residue bilinears;whole neutral words retain charged paths.",
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      "id": "finding:v311-population-reader-does-not-identify-quantum-channel",
      "titleEN": "Complete v311 population history does not identify the quantum channel",
      "statementEN": "For the original three-dimensional v311 transfer matrix Phi_e(X)=eX+sum_ij(T_ij-e delta_ij)E_ij X E_ij* gives a completely positive trace-preserving family for 0<=e<=731/2187. Every channel has the same population history T^n for all inputs and n, but rho_01=e/2 in the stated coherence test. The matrix is compared with the original numerical v311 call; positivity and trace preservation follow exactly from rational Kraus and Choi decompositions. Independent original source responses may constrain this family.",
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      "titleEN": "Declared charged pair relation leaves a five-letter word undetermined",
      "statementEN": "For the declared single chiral alpha=1/4, ell=4log4 source omega(gamma0 T gamma0)−16 omega(gamma2 T gamma2)=0. An m=2 run with this relation and vacuum moments is numerically optimal but automatic exact completion remains inconclusive. A separate exact four-dimensional completion using ten supplied original moments of degree at most four and normalization obeys the relation, but gives zero for gamma0 gamma2 T gamma2 gamma0 versus 1/256 independently computed from the given original source operator. Only the ten supplied moments are asserted equal, not all short words.",
      "scopeEN": "Declared single-chiral charged particle-hole control, m=2; alpha/ell and full original source are not selected by P1/P2. The internal search quarter rotation is not the oppositely charged quarter action. No physical source selection or global TFPT conclusion.",
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      "titleEN": "Full creator relations resolve the fixed free two-mode control",
      "statementEN": "In the given four-dimensional Fock representation, vacuum normalization and both full creator relations T a_i†=r_i a_i† T for r=(1/4,1/64) uniquely imply T=diag(1,1/4,1/64,1/256). The previous exact x=0 countercandidate violates two of the 32 basis-word relations. The existing m=2 moment search returns numerical extrema 0.0039062495971 and 0.0039062519290, not exact numerical certificates. The free multiparticle law was already known; its use as a source relation against the R085 countercandidate is new.",
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      "statementEN": "For a supplied constant Higgs singular pair and U_m=W U_0 W*, U_m C U_m*−C is trace class with trace norm 1 and relative determinant (1−z)^−1 for 0<z<1; r coupled pair sectors conditionally give (1−z)^−r. For m≠0 U_m−U_0 is Hilbert Schmidt but not trace class, and its n-dependent Fourier diagonal excludes pointwise multiplication. Both quarter phases are treated.",
      "scopeEN": "Given constant pair and diagnostic selective shift, not identified with the original clock; nonlocality excludes pointwise multipliers only, not the full OS algebra. Dynamic H fields, mixed words, original reflection and normalization remain open.",
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      "titleEN": "Entire free two-face response is strip response with relative zero-mode defect",
      "statementEN": "For the given free scalar two-face projection, all nonzero Fourier modes of the commutator quadratic response equal exactly the Dirichlet-to-Neumann matrix of a strip of width beta/2 by pointwise identities and telescoping. At beta=4pi and alpha=1/4 or 3/4 the constant relative eigenvalue is half as large: N_source=N_strip−Pi_relative/(2pi), and N_strip/2≤N_source≤N_strip as forms. The normal-ordered variance is Q/2. In the given coupled 2×2 background, the same scalar face operations change nonzero modes as well: at m=0.3, k=1 gives N11≈1.54580712 versus 2.00001395 at m=0 (two internal components), with explicit sum-tail bound 1.26e−7 and floating-point arithmetic.",
      "scopeEN": "Given massless geometric Cauchy graph and R066 quarter identity. No identification of the graph with the physical whole-state covariance, no general Higgs-coupled matrix identity and no derivation of a physical spacetime dimension. The coupled example is a fixed comparison field, not the dynamical physical source and not a test of the comoving clock.",
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      "titleEN": "E8 Hodge event is perfectly distinguishable from every unitary evolution",
      "statementEN": "On given ten-dimensional H=Lambda²C5, every nonzero C(z)=sum z_m C_m has six singular values ||z|| and four zero singular values. Convex separation yields, for every U:H→H*, a density matrix rho with Tr(rho U*C_m)=0 for every m; a normalized purification perfectly distinguishes Phi from Ad_U. Hence ||Phi−Ad_U||_diamond=2 for every unitary identification. An original basis input distinguishes one explicit unitary completion of C0 even without reference. With R086, a globally TP operation using only epsilon-span Kraus amplitudes therefore cannot have a continuous unitary start for all sufficiently small times. E_t=exp[tλ(Psi−id)] with Psi=Ad_U* o Phi gives a mathematical CPTP Poisson embedding. Every finite t requires a Kraus amplitude outside the strict span; λ and U are additional choices.",
      "scopeEN": "Finite given carrier and dual output, original C_m and R086 global trace preservation; no physical source theorem or exclusion of other branches, registers, memory or a finite event. The universal conclusion uses an analytic convex separation argument, not finite sampling.",
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      "id": "finding:common-five-slot-transfer-preserves-e8-bracket-not-automatic-tp",
      "titleEN": "Common five-slot transfer preserves E8 bracket but not automatically trace preservation",
      "statementEN": "For A on C5, D=Lambda²A and original bilinear C_m:H→H*, D^T C_m D=sum_n adj(A)_(n,m)C_n, including singular A. Invertible A preserves the whole span and the concrete five-cycle the given equally weighted channel. A common scalar normalizes the simple equally weighted channel to trace preservation iff adj(A)adj(A)*=rI, equivalently AA*=sI for invertible A. A=diag(1,1,1,1,1/2) gives effects 1/2 and 1/4. An exact Duhamel integral of [V,a†] gives the free creator-relation defect for supplied H=dGamma(h)+V; the existing two-mode counterexample gives 1/256.",
      "scopeEN": "The same A is assumed on both correctly typed legs; no source origin of A, no identification of the two-mode carrier with C5, and no universal TP exclusion. Other family weights or channels remain possible; interaction formula assumes the specified finite generator decomposition.",
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      "titleEN": "Comoving clock carries charged winding flow and dynamic Higgs connection",
      "statementEN": "In the given 2×2 singular block the comoving selective shift has a rank-one defect of trace +1 relative to the negative sea projection; the cotransported charged quarter acts on its implementer with phase i. The shift itself is not order four. For time-dependent real m(t), the comoving generator is sign(p)E_p sigma_z − p m_dot/[2(p²+m²)] sigma_y. Under the additional proposal to replace the original free quarter loop exp(2pi i Hfree) unchanged by positive coupled energies E_n at m≠0, the phases disagree for sufficiently large positive modes.",
      "scopeEN": "Only the stated time-dependent pair block and cotransported charge/reference frame; no identification of original physical P1 clock or OS reflection, 96-dimensional dynamics, H measure or full source transport. A separate geometric quarter generator remains possible.",
      "status": "conditional",
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      "titleEN": "Sixty-record event channel: exact distance and continuous record parent",
      "statementEN": "The given equally weighted R064 channel on Sym²(C4) has normalized Choi spectrum 0×45,(1/75)×45,(2/75)×9,(4/25)×1. An integer Gram polynomial and traces certify rank55; an F101 test certifies product rank100. The minimum half-diamond distance to unitary channels is exactly 21/25, from the maximal Choi overlap bound and Φ=(4/25)Id+(21/25)Ψ. A Hermitian involution parent with 60 records reproduces the given endpoint channel and allows continuous intermediate times; reused and fresh records produce different multitime answers. This concerns a different channel from the R088 Hodge P5 channel on Λ²(C5).",
      "scopeEN": "Exact finite channel and specified comparison construction; original source, physical timescale and record preparation not identified. Do not confuse rank change and semigroup theorem with a general exclusion of continuity.",
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      "titleEN": "Two source syntheses connect clock words, E8, Higgs measure and absolute source",
      "statementEN": "For given implementable Q and centered J_C(A), the required correction is Tr[(Q*C Q−C)A]; this separates the ordered CAR word source from the R084 relative covariance determinant. Transported Ω supplies the X→XΩ representation for a given form but does not automatically identify native pointers or states. Compact BKM expectation-value closure does not ensure a finite measure; force cumulants are not the R054 Higgs mixture witness. A complete normalized matter source does not fix geometric absolute normalization Z0.",
      "scopeEN": "Typed free or conditional contracts; the general original source, Hilbert space, force operator, dynamic Higgs measure and geometric normalization map are not derived. External notebook visible, not fully rerun locally.",
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      "titleEN": "Weighted clock index reduces to the full chiral trace under commutation assumptions",
      "statementEN": "For finite-dimensional γ=γ*=γ⁻¹ and V=V* with {γ,V}=0, and λ=λ* with [λ,γ]=[λ,V]=0, tr_ker(V)(λγ)=tr(λγ), since V pairs opposite chiralities with the same λ charge off its kernel. Periodicity, actual integral charge normalization, and Fermi/sea covariance remain additional clock and relative-determinant conditions. C=I/2 shows the missing general covariance assumption in the supplied index claim. Full-fiber λ=I index zero and quarter variance 1/(8π) at the self-dual theta function are already in the record; original selection of the quarter and β=4π is a new hypothesis to test.",
      "scopeEN": "Finite matrix class and named independent original sources; no general P1 index theorem, automatic P1/neutrino relation, prior physical selection of λ or self-duality derived from P1.",
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      "titleEN": "Ordered quarter word, E8 channel separation and original charge filters",
      "statementEN": "The conditional constant R062 colour triplet gives an exact ordered matter/quarter/four-field word in m1,m2,m3 for Q=i^N and O=4N(N−1). Two positive comparison laws in the actual t range share m1,m2 but differ in m3; global quarter, complement and dynamic time transport are not identified. The 60-record E8 channel and the Hodge channel have half-diamond distance at least 24/25 on six original charts and at least 11/15 for arbitrary unitary identifications. No positive classical selection of unitary records produces the Hodge channel; coherent or nonunitary refinement remains possible. In the six-generator space the full Higgs family and nonzero Majorana permit only 3B and two colour Cartan generators, with vanishing weighted chiral trace.",
      "scopeEN": "The stated finite carriers and assumptions only; no original common master source, dynamical multitime response, physical Choi measurement or F1 solution. Other clocks and Higgs co-transport remain possible.",
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      "titleEN": "Native Hodge current sewing, typed record targets and full 96-mode complement",
      "statementEN": "Original Ω=I⊗iY and the five gamma matrices XI,YX,YY,YZ,ZI satisfy the pointer relation and generate the native Hodge channel as A3 current zero modes. In the specified current net compact OPE sewing supplies the internal contraction without a record measurement; the P1 field/state/reflection/sewing map remains open. With coherent record access the native Spin5 target has optimal probability 1/15 in one step and 4/75 in two; the earlier duad target has probability zero in one step and 1/25 in two. Target typing and P1 selection remain distinct. The fixed original 96-mode Higgs source word for diagnostic z^N factors into its reducing six-mode block and H-dependent 90-mode determinant; three original-constructor controls confirm the factorization numerically.",
      "scopeEN": "Recovered existing exact finite source contracts and conditional fixed quasifree 96-mode block; no global original clock, multitime dynamics, P1 selection, H measure or physical F1 completion. Native family Pauli gamma is not automatically physical 96-mode Dirac gamma.",
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      "titleEN": "Folded Ward target and smaller neutral source route",
      "statementEN": "The given rational fold matrix M maps s=(1/2)^8 exactly to e3+e6 and q=(1,...,1) to 2(e3+e6). For the normalized lattice-vacuum Ward formula at z=0,w=1,u=3 the old eight components −1/12 become −1/6 on H3 and H6; the transported total current is −2/3 on both sides. This is a target calculation and does not identify a P1 response. The existing grade-2/U5-residue-3 creation30 branch and smooth neutral half-kink on the folded even-CAR quotient form a smaller approach than assuming a full 50-event process; they do not derive a P1 local embedding, source, reflection, state or OS time. The marked colour-charged intermediate must be retained in the neutral pair.",
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    {
      "id": "finding:r093-original-kink-time-response-distinguishes-quarter-states",
      "titleEN": "Original neutral kink time response distinguishes the two quarter states",
      "statementEN": "In the given folded CAR source contract the full ordered T(t)T* response on the five actual states is determined by the original Schur kernel and an explicit Weyl Gaussian including all modes. The state polynomial has modes 8,16,24,32 due to cover degree 32; the Gaussian retains all modes. Normalized original P0 states in quarters 1/4 and 3/4 differ already in R8=||J8P0||²/||P0||²: 766710528/328635469 versus 770249472/331641421. A smooth admissible comparison profile with b3=0 distinguishes their responses at t=pi in the quadratic term. An additional smeared CAR total current is a derivative of the same Weyl source. The historical mixed-current-origin checker was replayed in a targeted temporary overlay with original pins and returns 29 PASS, PARTIAL and no local origin; the full ledger-checked overlay was not passed.",
      "scopeEN": "Actual original folded CAR source contract, fixed input states, source time K_alpha and smooth comparison profiles; no primitive P1 OS time identification, original profile selection, full Ward or dynamical Higgs source, or F1 completion. Historical replay does not validate active source pins or ledger projection.",
      "status": "conditional",
      "problemId": "problem:automated-operator-source-synthesis",
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        "topic:quantum",
        "topic:process"
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        "goal:T2",
        "goal:T8"
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    {
      "id": "finding:r093-ordered-current-kink-and-eight-arc-locality",
      "titleEN": "Original ordered current word and direct eight-arc locality test",
      "statementEN": "In the given folded CAR source contract the derivative of the same ordered Weyl source computes the left-inserted smeared total current response J(h)T(t)T*, including its noncommutative phase and alpha-dependent state kernels. An independent one-mode Fock test confirms the operator order and sign. For a nonconstant smooth phase profile the direct pullback T=W(-b∘F), F(theta)=8theta, acts nontrivially on a bounded even CAR bilinear in each of the eight disjoint cover arcs; hence T is not in the even local von Neumann algebra of any single such arc.",
      "scopeEN": "Given CAR source time, smooth profiles and direct cover pullback; the one-mode Fock case tests an identity and is not a TFPT source state. Other local P1 embeddings or compressions are not excluded. No identification with OS time, all E8 Cartan currents, Higgs measure or F1.",
      "status": "conditional",
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      "titleEN": "Original five source inputs are excitations, not the clustering vacuum for this time",
      "statementEN": "All five original charge-zero inputs have K_alpha energy 4. The original J8 Gram is positive definite for both quarters with the stated exact determinant. A smooth quotient profile with nonzero first coefficient gives a bounded eight-arc Weyl word that reaches energy 3 from every nonzero vector in the input span. The connected Euclidean response based on K_alpha-4 grows on the infinite half-line; the same applies to positive states supported entirely in this span. The P0 vacuum channel J32 has the two exact R32 weights in the source. The R093 response remains an original vacuum word with the finite raising-current polynomials B_i.",
      "scopeEN": "Fixed original CAR time and admissible eight-arc word; the P1 seed does not specify the clustering direction. No exclusion of other P1 times or states, finite KMS interval, CAR vacuum, or excited insertions; no derivation of P1 localization, reflection or physics. The input basis is not orthonormal.",
      "status": "established_scoped",
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    {
      "id": "finding:r096-joint-higgs-twoface-source-constraint",
      "titleEN": "Joint Higgs and two-face response gives quantitative source constraint",
      "statementEN": "For one positive Higgs mixture and full two-face covariance, Tr L(Pbar-Pbar²)L >= Var(m)/6 = DeltaO/144 under R054 conditional quasifree premises, strengthening the earlier qualitative nonprojection result. The unnormalized linear identity Z[O]=8 Z[mN] requires matching conditional fibers and weights, not positivity. On the original 36-dimensional quark carrier the two-face trace g(u)=6 sum_j[2 cosh(ell sqrt(p²+y_j²u))]^-1 is strictly decreasing, implying Cov(u/(1+u),B_Q)<0 for a positive nonpoint radial law. Original 96 constructor and exact abstract rank-six control checked.",
      "scopeEN": "Constant Higgs and same actual two-face carrier, color symmetry and R054 conditional quasifreeness for the four-word part. No derived physical Higgs measure, general H(x), P1 seam, or independent gauge transport. The mixed source insertion is not assumed present in search alphabets without an actual source map.",
      "status": "established_scoped",
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      "id": "finding:reverse-bootstrap-retained-operator-scope",
      "titleEN": "Reverse selection is retained; the checked maps supply numbers and closure data",
      "statementEN": "Origin954–981 and v6/v15/v350–v352 recover g_car=5 and eight within the chosen closure framework; v154 supplies the conditional local extension. The checked v350–v352 functions do not construct Phi-dependent D_rel/D_ref or physical insertion operators. This scopes those calculations, rather than asserting general impossibility.",
      "scopeEN": "Source audit of the specified reverse chain. No retraction of numerical self-consistency and no independent derivation of the framework.",
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      "problemId": "problem:reverse-bootstrap-physical-field-dictionary",
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          "path": "documents/origin_theory/sections/003-section-the-unique-attractor-a-gapped-transport-selects-.tex",
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    {
      "id": "finding:e8-current-dictionary-all-separated-responses",
      "titleEN": "All248 boundary currents have explicit multi-field responses",
      "statementEN": "The executable E8 dictionary contains240 root vertices and8 Cartan currents with cocycle, adjoint signs and quarter grades60/64/60/64. The vertex formula and Cartan Ward recursion determine all separated current coefficients of the normalized source series. Mixed spinor currents have5/8+3/8=1; the checked connected four-point response is nonzero. Standard lattice theory implemented and independently checked here, not a new physical interaction.",
      "scopeEN": "Stipulated E8_1 lattice vacuum representation. Bosonic currents are not identified with elementary4D fermions or native CCR mediators. Internal quarter grading is not physical quarter time. Contact terms, Z[0], absolute phase and original P1 Phi-dependence do not follow from these normalized coefficients.",
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    {
      "id": "finding:charged-scaling-microscopic-map-still-unspecified",
      "titleEN": "First microscopic field and refinement map is unspecified in the original",
      "statementEN": "The original charged scaling contract starts with H_N as the actual QWZ/CAR GNS space and declares Psi_lambda,N by a normal boson exponential. Its concrete refinements I_charge tensor Gamma_s(j) act instead on a separately chosen bosonic lattice Fock space. No microscopic B_N between these systems with B_2N J_N,2N=V_N,2N B_N and state, energy, parity, sector and localization compatibility is constructed there. The exact charge-lattice isometry F(2q)=lambda does not replace this field morphism.",
      "scopeEN": "Source audit of this particular original path. No nonexistence claim for B_N and no rejection of the established P1/P2 compiler or mathematical E8 field extension.",
      "status": "established_scoped",
      "problemId": "problem:reverse-bootstrap-physical-field-dictionary",
      "topicIds": [
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        "topic:e8",
        "topic:quantum"
      ],
      "goalIds": [
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        "goal:T2",
        "goal:T8"
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      "premiseIds": [
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        "premise:typed-source-structures"
      ],
      "attemptIds": [
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      ],
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      "titleEN": "Original proxy constant fails on the actual first Slater norm",
      "statementEN": "On unchanged v988 data Nx16,Ny6,M1 the twist1 Slater state against h0 has exact variance formula Tr(P h0(1-P)h0P)=||(1-P)h0P||HS^2. Numerically it gives7.0875042817 and first excitation norm2.6622367065. The original first central sum proxy is identically0; its fit C=1.2040009128 gives C^2=1.4496181980 and already fails to bound the actual norm here.",
      "scopeEN": "Only this proxy-fitted constant in this actual finite original case is excluded. The formula is exact; numerical values are checked to1e-9 tolerance. Larger constants, separate historical value2.86 and general continuum estimates are not excluded.",
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      "titleEN": "Periodic zero modes distinguish original covariance and Slater choice",
      "statementEN": "On unchanged v988 Nx16 Ny6 M1, explicit integral boundary zero vectors, modular ranks over F65537 and exact sign pairing prove ranks190/192/192/192. Periodic counts are95 negative,95 positive,2 zero; other twists96/96/0. Original covariance C=Pminus+Pi0/2 differs from pure occ_frame projector P=Pminus+Q with rankQ1 in the nullspace: norm(C-P)HS²=1/2 and norm(C-C²)HS²=1/8. Both original functions are valid and deliberately distinct. A state-preserving microscopic dictionary cannot silently exchange these periodic states.",
      "scopeEN": "Exact finite original case. No nonexistence of B_N, selected continuum state or E8 refutation. Previous energy test uses twist1 without zero modes and is unchanged. Numerical norm diagnostics normal and -OO byte-identical; independent proof review and rerun accepted.",
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      "titleEN": "Replica consistency exactly for acyclic influence structure",
      "statementEN": "For arbitrary finite alphabets/interfaces, existence under every finite joint local replica control, uniqueness under every such control, acyclic static influence and compatibility with at least one fixed total order are equivalent. A directed cycle of length m allows at most m copies with separately sensitized edges and odd XOR feedback without a solution. Acyclicity gives unique triangular recursion. No unique total order or physical time is selected.",
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      "titleEN": "Complete binary three-interface classification",
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      "scopeEN": "Complete bounded classification of three binary interfaces; no particle/generation/dimension interpretation. One-shot consistency does not force a fixed order.",
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      "titleEN": "Independent order, information and composition classes",
      "statementEN": "Event posets provide ideal states and possible histories but do not generally select growth weights. Supplied finite updates give cycles/transients and closed record cells relative to errors/observation; sink SCCs count maximal separated record cells, not necessarily exhaustive invariant record functions. Transformation monoids allow stable sectors, periodic steps and visible classical noncommutativity together; action and effects remain inputs. Counterworlds show these properties do not follow from composition or information alone.",
      "scopeEN": "Separate classes with separate premises, no shared physical derivation. Exact poset/endofunction censuses through size4 and elementary structural claims checked. Cycles do not select a canonical phase origin.",
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      "titleEN": "Repeatable distinctions select no unique response geometry",
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      "titleEN": "Report repeatability is not complete state stability",
      "statementEN": "The separate benchmark assumptions give at least two continuation-distinguishable classes and preserve the selected report label under uninterrupted repetition. This largely restates the substantive repeatability premise rather than explaining physical origin. A test can preserve its report while incrementing a separately readable modulo3 counter; result repetition therefore forces no complete idempotence, preservation of all correlations, metric time or eternal material memory. Possible reports alone select no numerical weights.",
      "scopeEN": "Standard residual representation consequence and explicit counterexamples, not a new ontology. Distinguishability, repeatability, context reproduction and serial availability are separate assumptions. Separate L1/L2 from L3.",
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      "statementEN": "Every finite normalized prefix-consistent response family with past marginals independent of future choices has a finite classical stochastic realization: preparation/full history as memory, conditional ratios as transitions, append the next choice/report pair. The construction also reproduces adaptive protocols. It was checked exactly on all requested D tables through horizon4. Finite sequential disturbance and order dependence alone do not exclude unrestricted classical history models.",
      "scopeEN": "Conditional representation theorem without novelty claim, not efficient or local physical simulation. History/preparation dependence and growing memory are allowed; Bell, noncontextuality or resource witnesses under independently justified extra conditions are not invalidated. Protocol direction is assumed, not physical causality derived.",
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      "titleEN": "Pascal condition has exactly the solution five for all natural g",
      "statementEN": "CarrierRankUniqueness.carrier_rank_pascal_unique states and proves for every natural g that 2^g=g^2+g+2 iff g=5. Its proof combines the small cases with pascal_growth for all g>=6. This is unbounded; PascalLadder has a separate bounded K=2 statement. For g=5, N_fam=3 and g+N_fam=8 are computed.",
      "scopeEN": "Algebraic uniqueness under the stipulated Pascal/half-spinor exhaustion condition. Original source and proof text reviewed; no fresh Lean kernel run. AX.P2.01 remains a declared physical interface; the equation does not select itself from an original source.",
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      "titleEN": "Glue elements, two orbits, spinor swap and halfway stage are explicitly formalized",
      "statementEN": "In Fin4 x Fin4 with qZ=(5x^2+3y^2) mod8, named Lean declarations classify all six isotropic elements and four odd-coordinate isotropic generators. Explicit orbit identities give H1 and H2; klein_not_isotropic gives the nonisotropic Klein witness, spinor_swap_exchanges swaps the two orbits, and unique_halfway_stage classifies the only nonzero isotropic order-two element (2,2). Index and sector arithmetic are additional separate declarations.",
      "scopeEN": "The concrete element/orbit and arithmetic statements are available. No fresh kernel run, nor a newly checked single Lean theorem about all abstract subgroups or lattice isomorphisms. The general subgroup conclusion also uses finite group theory; concrete E8 basis embedding and physical carrier selection are separate.",
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      "titleEN": "Marked-seam category and full character-valued lift factor formula",
      "statementEN": "Marked lattice transport functors with explicit reflection/clock markings define a category, products and boundary gluing under inverse-conjugate holonomies. Ordered generator phase eta gives the full factor c(g,h)(y)=eta_h(x)eta_g(hx)/eta_gh(x), x=(gh)^-1y. It is Hom(L,F2)-valued with coefficient action, not generally scalar or central.",
      "scopeEN": "Declared definitions and derived standard extension/gluing algebra, not a classification of all object groups.",
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      "titleEN": "Complete quarter-clock sign-lift classification with526336classes",
      "statementEN": "Gamma=F2^8 semidirect C4 has order1024. Peripheral classes C_u over J are indexed by F2^4. Four ordered classes admit product1 iff sum ui=0. Normal forms(x,y) modulo translation(u1+u0,u2+u0) give256classes for diagonal profiles,128otherwise. Total526336;4096orbits size16 with stabilizer C4×C2^4 and522240size32 with stabilizer C2^5. Image-order census4:16,8:720,16:14880,32:80640,64:430080; no surjection.",
      "scopeEN": "Complete only for declared ordered quarter-clock/sign-lift stratum under Gamma conjugation; not full G31/W(E8) reflection moduli or additional equivariant structures.",
      "status": "established_scoped",
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      "titleEN": "Nontrivial central quotient class and all subgroup restrictions",
      "statementEN": "Z(Gamma)=F2^5 and Gamma/Z=F2^5, with full central cocycle kappa((u,e),(a,f))=(ea,ef). Restriction vanishes exactly on subgroups in e=0; otherwise a nontrivial square obstructs an order2 lift.32central characters yield16marked U1 classes,15nonzero: pure clock carry is a U1 coboundary, other classes are distinguished by commutators.",
      "scopeEN": "Central quotient extension with trivial coefficient action, distinct from split noncentral extension over<J> and the G0 factor.16classes describe the Z-induced family, not all H2(F2^5,U1).",
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      "titleEN": "Full G0 lift cocycle trivialized by explicit cochain",
      "statementEN": "For actual G0=<J,R>=C4×C2 the full8×8factor table of the original ordered lattice lift is computed. Its Hom(L,F2) class vanishes: b=(0,86,6,80,136,223,142,217), ordered1,R,J,JR,J²,J²R,J³,J³R, satisfies b(g)+g.b(h)+c(g,h)=b(gh) on64pairs. All512cocycle triples checked.",
      "scopeEN": "Full cocycle and global trivialization on G0 only; no splitting decision for full G31 or W(E8).",
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      "titleEN": "Native quartic/exterior cohomology distinguishes equal-order monodromies",
      "statementEN": "Existing inclusions P5->Sym4(V4) and exterior²V4->V4⊗V4 are G0-equivariant. J versus JR gives complex H1 dimensions8/8 onV4,12/15 on exterior²V4,15/14 onP5. Integral E8 system with fourJ has H0=0,H1=Z16⊕(Z/2)^4,Hk=0 fork>=2; Smith form1^4,2^4. Local coefficients distinguish marked systems of equal clock order.",
      "scopeEN": "Cohomology of specified local systems on four-punctured sphere; no identification of differently dimensional representations or invariants of every bundle/bordism.",
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      "titleEN": "Equivariant lifting and divisibility by eight",
      "statementEN": "Central orbifold H2 decides lifting; based lifts form an H1torsor and unbased lifts require centralizer quotient. Cone power defects are complete on oriented punctured quotient surfaces. The existing determinant class on C_m*C_mwith opposite parity lifts exactly for8|m, then16lifts; an original order8matrix example is given.",
      "scopeEN": "Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.",
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      "id": "finding:original-g31-reflection-census-and-all-integral-cohomology",
      "titleEN": "All17G31classes and integral cohomology",
      "statementEN": "Complete local rerun confirms14760ordered quadruples,17conjugacy classes and5Hurwitz orbits. All17integral Smith forms give one Z22H1, three Z20+Z/2(typeA), thirteen Z20; H0ranks6or4and higher cohomology zero. Central lifts through the same lattice projection cannot change this cohomology.",
      "scopeEN": "Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.",
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      "id": "finding:all62-original-clock-obstructions-and-central-variants",
      "titleEN": "All62clock obstructions and central transport variants",
      "statementEN": "All62original order4clock classes obstruct:30have(0,2),32have(2,0). Central clock power alone still gives zero. Defined central covariance with product1requires d=1,3and gives960classes with strict clock or1984with central clock power.992projective orbifold pairs instead have meridian productz2. Same existing extension, explicitly separate contracts.",
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      "id": "finding:actual-sign-lift-g31-group-comparison",
      "titleEN": "Actual group comparison and central pushout exclusion",
      "statementEN": "An explicit lattice isometry identifies Gaussian and standard E8roots and preservesJ. Gamma_sign projects to<J>inG31. Neither group embeds in the other; common quotients are precisely1,C2and cannot retain order4clock. Zero invariant lattice mod2space forbids a nonzero equivariant sign-kernel map to trivialZ4and that central pushout identification.",
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      "titleEN": "Complete invariants and explicit collisions",
      "statementEN": "Complete invariants are canonical ordered tuple orbit, centralizer clock orbit and ordered central phases. Clocks0/3and0/5share monodromy,spectrum,lattice cohomology and obstruction but have centralizers64and128. Without clocks there are1088central lift classes. Phase tuples(2,0,0,0)and(0,2,0,0)over the same tuple are not conjugate.",
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      "titleEN": "E7 pairs are the original Gaussian reflections",
      "statementEN": "In the given E8 root system J with J²=-I exchanges the120 root-normal E7 slices in60 pairs. Their intersections are D6 with60 roots and their normal planes A1+A1. The operator s_alpha s_Jalpha is literally the original complex Gaussian root reflection of v634; the60 distinct operators generate the existing G31 realization of order46080. Operator equality, not group order alone, supplies the dictionary. The two actual E7 subalgebras contain192 roots; their mixed brackets generate all48 missing roots in one round. Their integral lattice sum is E8 at index1. The60 pairs form a transitive G31 orbit.",
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      "titleEN": "E6 root subsystems admit no invariant native quarter clock",
      "statementEN": "No E6 root subsystem is preserved by J: its induced action on K*/K=Z3 would square both to identity and, from J²=-I, to nontrivial inversion. The same argument applies to A2. All1120 regular A2 complements form560 J pairs; their E6 intersections have12 roots in640 and24 roots in480 subsystem cases. An explicitly marked D5<E6<E7<E8 chain with40/72/126/240 roots exists but is not J invariant.",
      "scopeEN": "The discriminant proof covers all E6 root subsystems. The finite census covers the specified regular A2 complements. No claim about arbitrary non-root Lie subalgebras or physical scales. This does not forbid transported real E6/E7 reductions on the parity double cover with remaining ±I holonomy.",
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      "id": "finding:e-series-square-cover-family-and-quarter-descent",
      "titleEN": "Intrinsic square curves and transported E7 reductions",
      "statementEN": "Each of60 Gaussian root lines gives a representative-independent square marking and w²=(v/alpha)^4-1 with elliptic compactification j1728. For the separately supplied all-J representation on CP1 minus mu4, the parity double cover has ±I holonomy and permits chosen parallel real E6/E7 root reductions. E7 descends as an unordered pair. The parity character has8 C4 lifts; cyclic marking symmetry leaves2 uniform orientations. Deck, base rotation and fibre J are different maps. The loop around all4 finite marks is trivial here but not in the existing uniform mu3 cover.",
      "scopeEN": "Fibre holonomy at4 removed branch points remains-I; no ordinary flat extension to the compact elliptic curve. The intrinsic family does not yet identify the original seam/peripheral data. No necessary selection of an individual chart.",
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      "id": "finding:e-series-integral-glue-and-normalization-nonidentifiability",
      "titleEN": "E7 distinguishes glue types and conditional normalization rules",
      "statementEN": "The native E8 quotients for D6+A1+A1, E7+A1, E6+A2 and D5+A3 are C2²,C2,C3,C4. The E7-pair index4 is not cyclic C4 glue. Conditional continuations1/(pi*r) and1/(2pi*m), with selected exceptional rank r and lattice extension index m, agree at E6/E8 but differ at E7:1/(7pi) versus1/(4pi). The anchors do not determine the continuation. Complete conformal windows retain total rank and central charge8.",
      "scopeEN": "Exact lattice quotients and non-identifiability of an extra normalization map. No identification of original modular index I with m in each projection, no selection of a law or E7 coupling/scale prediction. The historical7pi window is no independent measurement. The established P1/P2→D5+A3→E8 construction and phi_base=(4/3)c3 remain intact. This comparison concerns only an additionally proposed E7 continuation and is not a missing step in the existing compiler.",
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    {
      "id": "finding:global-original-g31-character-lift-nonsplitting",
      "titleEN": "Global original G31 lattice lift class is nonzero",
      "statementEN": "The original A=Hom(E8,Z2) extension is nonsplit over all G31, with order-two H2 class. Seven original relation equations give0=1 on an order16 subgroup H, while all22proper subgroups split. The exact order512 lattice quotient encodes the same lift problem.",
      "scopeEN": "Original cocycle and actual G31module. H is inclusion-minimal; no claim of minimum order among all witnesses.",
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      "id": "finding:all62-original-character-clock-lifts-complete",
      "titleEN": "All62original clock objects admit character-valued lifts",
      "statementEN": "All62original clock objects admit strict C4*C4 lifts into the inherited lattice extension. Full conjugation gives11356classes=6856+4272+228. Every normal form and automorphism order is stored; independent Burnside verifies all62counts. Clocks0/3and0/5 give212and176classes.",
      "scopeEN": "Ordered punctures, clock order4, full lift conjugation. No extra meridian square relation; distinct from central Z4extension.",
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      "id": "finding:all17-original-character-monodromy-lifts-complete",
      "titleEN": "Complete census of original17character-valued monodromy lifts",
      "statementEN": "Full conjugation over the17original reflection quadruple classes gives835456character-valued monodromy lift classes, with a complete affine normal-form criterion. V4embeddings with and without integral Z2torsion give26128and42240classes per ordered orbit.",
      "scopeEN": "F3representations, ordered punctures, full lift conjugation. Census and normal-form criterion; no stored835456-representative list.",
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      "id": "finding:e7-pair-full-lattice-field-character-torsor",
      "titleEN": "E7 pairs on all lattice fields and complete correction torsor",
      "statementEN": "For all60 native Gaussian root lines, r=s_alpha s_Jalpha is the original complex reflection. Its actual Tits lift T=n_alpha n_Jalpha has T²=D_k, k(lambda)=<alpha+Jalpha,lambda>mod2, negative on128 roots. T=S_r D_chi holds on every lattice field with explicit input character. Exactly eight corrections per line are involutive and commute with the fixed J lift, a torsor under three-dimensional A^{<r,J>}. They agree on the actual neutral4/5/20 diagram and differ on charged sectors. The full G31 extension remains nonsplit.",
      "scopeEN": "Unchanged eps_e8, ordered integral basis and original lattice VOA. All-lattice equality follows from the character lemma and eight generators; all240 roots and13680 nonzero brackets per line checked separately. Individual lifts do not provide a global group section.",
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        "premise:native-clock-complex-reflection-and-quartic-subspace"
      ],
      "attemptIds": [
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    {
      "id": "finding:e-series-complete-native-compiler-cut-dictionary",
      "titleEN": "Complete E6/E7 dictionary for the original D5+A3 split",
      "statementEN": "The120 E7 cuts form four W(D5)xW(A3) orbits of sizes20,6,30,64, with exactly six containing the original D5. The1120 regular E6 cuts form seven orbits of sizes40,120,320,60,480,4,96. Their560 J pairs intersect320 times in A2+A2 and240 times in D4. Exactly four E6 cuts contain D5, a canonical unordered S4 tetrahedral family with actual weightwise E8/D5/A3/E6/A2 branchings. J sends all four outside this family with common first-four-coordinate D4. The3+2 marking fixes the family but selects no cut.",
      "scopeEN": "Complete finite classification of regular root cuts under the fixed compiler Weyl group, with stabilizers and actual actions. No classification of arbitrary Lie algebra embeddings; retain the original clock.",
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      "problemId": "problem:marked-seam-monodromy-and-lattice-lifts",
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        "premise:native-even-e8-gaussian-clock-and-sign-extension"
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    {
      "id": "finding:common-cover-clock-and-maximal-integral-phi12-intertwiner",
      "titleEN": "Common covers and maximal actual integral clock intertwiner",
      "statementEN": "C6 and C12 are the minimal marked common covers of C2/C3 and C4/C3, of genera7 and15. J<->Q, sigma<->deck and c=J sigma<->R=Q deck preserve c⁹=J and c⁴=sigma. The unique lift with this full marking is Qn Delta_n^4. Actual real8 has character Phi4² Phi12; the retained rational Burau rank6 model has Phi6 Phi12. Their canonical Phi12 kernels admit an explicit integral Z[C12] intertwiner also transporting J and sigma. Different complementary spectra obstruct full equality; retained alternating forms of types(1,3) and(1,2) obstruct any form-preserving unimodular intertwiner. Peripheral J is deck on C4 while compiler J=c⁹ rotates the base; no phase choice merges these roles.",
      "scopeEN": "Fixed deck orientation, actual matrices and saturated integral Phi12 sublattices. The displayed intertwiner is an explicit certificate, not a uniquely selected identification. Trace form and basis belong to the retained Burau lattice model, without an independent geometric cycle identification. Quartic5, exterior6, erasure20 and permutation characters remain distinct and are completely compared.",
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    {
      "id": "finding:native-four-cut-projectors-unify48-fourthirds-fivequarters",
      "titleEN": "Four E6 projectors connect48,4/3 and5/4 in the same representation",
      "statementEN": "In the original half-root block B=(16,4), the four D5-containing E6 cuts yield literal rank16 projectors P_t, with P_t P_u=delta_tu P_t and sum P_t=I64. Q_t=I-P_t has rank48 and carries exactly(16,3), with checked D5 and A2 root action. Every cut gives Tr(I64)/Tr(Q_t)=64/48=4/3=phi_base/c3. Every weight of the same block has squared D5 projection5/4 and A3 projection3/4. The same48 lies in(27,3)=(16,3)_1+(10,3)_-2+(1,3)_4, hence81=48+30+3. The E7 missing48 instead carries(12,2,2), whose actual Y character rejects identification with three D5 spinors.",
      "scopeEN": "Canonical unordered S4 projector family on the original root-current block. Exact representation interpretation of the existing seed ratio, without an additional readout law or selection of a tetrahedral vertex.",
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      "problemId": "problem:marked-seam-monodromy-and-lattice-lifts",
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    {
      "id": "finding:complete-covariant-compiler-voa-source-and-weyl-closure",
      "titleEN": "Complete covariant compiler closure on the same lattice-field source",
      "statementEN": "H=W(D5)xW(D3) with actual J generates exactly W(D8):224 E6 cuts in56 tetrahedral frames. Each frame resolves the entire128 half-root block into four rank32 projectors; complements have rank96 and128/96=4/3, retaining squared norms5/4 and3/4. Original E7-pair operations enlarge closure to W(E8):1120 E6 cuts,7560 A3 frames,27 through each cut. The full original sign extension over W(E8) acts on the existing VOA source. F_ghat[p,v]=[p,ghat v] transports all marks and fibre holonomy and composes exactly, retaining the nonsplit character carry.",
      "scopeEN": "Minimal generated transition closure, not a fixed-J/fixed-D5+A3 symmetry claim.128 is fixed only under D8 and transported under the full action. Organizes the given lattice VOA and covers; no physical-origin derivation or freely chosen helper model.",
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      "problemId": "problem:marked-seam-monodromy-and-lattice-lifts",
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      "id": "finding:complete-fixed-jhat-cyclic-field-lift-and-associated-clock",
      "titleEN": "One complete cyclic field-lift class and joint associated clock",
      "statementEN": "With original fixed Jhat=S_J, the ninth-power equation has F2 rank6. Exactly four compiler lifts S_c D_a with a=31,62,151,182 have ninth power Jhat; their sigmahat masks are63,30,183,150. The16 Jhat-preserving character gauges give one orbit with stabilizer4. No solution retains both separately generator-normalized J and sigma sections. On Vper=(C12 x V_L)/((Delta12 p,v)~(p,Jhat v)), Chat[p,v]=[R12 p,chat v] satisfies Chat4=sigmahat, Chat12=I and Jcomp=Chat9=Q12 Jper. The two J roles share the same fibre lift while retaining different base components.",
      "scopeEN": "All256 character candidates over the fixed cyclic compiler group; all-field equality from eight generators and unchanged cocycle, with all240 roots checked directly. No full G31/Weyl splitting or identification of different cover automorphisms.",
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      "id": "finding:relative-native-burau-clock-hom-faithful12-transport",
      "titleEN": "Relative clock-Hom transport forces exactly the existing C12",
      "statementEN": "On the actual common Phi12 lattices, Hom_clock=Q[A12] has dimension4 and integral Hom=Z[zeta12], with integral isomorphisms a unit torsor. Native periphery is A12^9; the Burau periphery induced from the existing mu3 cover deck module is A12^4. Relative Hom transport is therefore H->A12^7 H, with exactly0 parallel clock intertwiners. Every nonzero relative orbit has order12. Exponent vector(7,7,7,7,8) has exactly the original winding kernel modulo12, so the existing C12 is the minimal trivialization.",
      "scopeEN": "Comparison of native and explicitly cover-induced Burau local systems. No independently geometric Gauss-Manin H1 family, forced carrier equality or extra line twist erasing carry. The earlier clock dictionary remains valid but is not a parallel peripheral dictionary.",
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      "titleEN": "No E6 vertex selection even with full clock-field phases",
      "statementEN": "C_H(J) has order64 and cyclic4 image on the four E6 vertices. Its Y stabilizer is{I,d}, d=diag(1,1,1,1,1,1,-1,-1), acting by two transpositions without a fixed vertex. d preserves sigma,c and the actual quartic/erasure diagram. With fixed full Jhat and chat=S_c D31, exactly four d field lifts126,127,128,129 commute with both;126 and127 are involutions. Thus genuine automorphisms of the same full associated object fix no E6 vertex. Character-gauge conjugation transfers this to all four cyclic lifts. The supplied field phases therefore select no vertex.",
      "scopeEN": "Complete stabilizer test inside H preserving root factors setwise, with fixed J/compiler field lifts, Y and covariant quartic/erasure maps. No pointwise freezing of every individual quartic vector or separately chosen charged state. The full transported family is the exact object.",
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      "id": "finding:marked-c4-glue-unique-local-e8-vacuum-source",
      "titleEN": "Marked C4 glue determines one complete local E8 vacuum source",
      "statementEN": "In exactly Ext_H(U) there is one VOA and normalized local Q system class over U, represented by V_E8. q_D5(s)=5/8 and q_A3(f)=3/8 make H Lagrangian; the full abelian associator/braiding class trivializes only on H. H2(C4,C*)=H2(C4,U1)=0 gives uniqueness, with residual automorphisms Hom(C4,U1)=C4. The actual weight1 branching is60+64+60+64=248. Lattice vacuum, Sugawara vector, L0=N+Norm(alpha)/2, all vacuum correlators, unitary structure over U and associated local conformal net are determined in this class.",
      "scopeEN": "Complete conditional mathematical reconstruction for the specified U module and local unitary conformal extension. No derivation of primitive P1 membership or physical time/state/spacetime identification. H3(C4,U1) is not asserted zero; the original nonsplit G31 character extension is not replaced by scalar coefficients.",
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      "id": "finding:canonical-vacuum-clock-preserves-e6-nonselection",
      "titleEN": "Canonical vacuum and L0 select no E6 corner and differ from internal clock",
      "statementEN": "Every original lattice-field lift fixes the canonical vacuum and Sugawara vector and commutes with every L_n. Thus the existing d field lifts126/127 remain automorphisms after including these canonical data and still fix no E6 corner. Internal Jhat has eigenvalues+i and-i four times each on the eight weight1 Cartan currents; exp(i theta L0) is scalar there. Hence Jhat is not a conformal time rotation for any theta. The joint cover construction and nonsplit transition carries remain functorial.",
      "scopeEN": "Exactly the existing marked associated E8 object and its standard conformal vacuum realization. No general physical time no-go, base/deck/fibre identification or additional state marking.",
      "status": "established_scoped",
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      "titleEN": "One marked generator forces E8 without extra sector or minimality choices",
      "statementEN": "Regularity of U and common L0 make W automatically a finite U module. Its connected separable algebra object B has simple regular B module. Every invertible M_a tensor B is simple, so free-module adjunction forces n_a=dim Hom_B(M_a tensor B,B) in{0,1}. Support S is the stabilizer subgroup of B and is isotropic by trivial twist. M_h in W yields H subset S subset Hperp=H, hence exactly four sectors once and the unique marked E8 extension with vacuum and conformal data. Before specifying the generator there are exactly four local conformal extension classes over fixed U: U, V_D8 and the two oriented V_E8 embeddings.",
      "scopeEN": "Complete classification within connected local unitary SAME-CONFORMAL extensions. Corrects the overstrong remark in THEOREMS.txt Section5: merely containing the four sectors already suffices under these remaining hypotheses. A tensor spectator changes the conformal vector. The two E8 classes distinguish fixed U embeddings and become isomorphic after allowing A3 diagram reversal. No derivation of this local realization from primitive P1 or physical completion.",
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      "id": "finding:native-reflection-rotation-lift-classes-and-forgetful-images",
      "titleEN": "Native equivariant classes and complete forgetful images",
      "statementEN": "Exact original field relations give binary ranks20,20,24,30 on40,40,40,48 variables for reflections s=0,1, rotation and joint s=0 marking. Character gauge rank4 gives65536,65536,4096,16384 marked classes. Complete forgetful images in original Gamma moduli have2176,2560,256,64 classes. All168 columns independently checked in original cocycle field action and ranks with SymPy GF2; all forgetful image orbits separately enumerated.",
      "scopeEN": "Complete only in the explicitly fixed native projection and morphism classes. Not all normalizer reflections or primitive physical source/state selection.",
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        "premise:ordered-four-punctures-fixed-quarter-projection",
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      "attemptIds": [
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      "id": "finding:quarter-clock-pure-mapping-class-complete-quotient",
      "titleEN": "Complete pure mapping class quotient with157696 normal forms",
      "statementEN": "For N=I+A, N²=0, b_i² adds N(ci+c_(i+1)) to both adjacent characters. The pure mapping class action on original Gamma classes factors faithfully through C2×C2. Burnside and independent enumeration of all2^20 character gauge classes give157696 orbits with original Gamma class size census1:4096,2:46080,4:107520, weighted total526336. Every normal form is saved.",
      "scopeEN": "Ordered four J sign lift stratum and pure mapping classes. No puncture permutations, silently frozen reflection marking quotient, or physical origin theorem.",
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      "problemId": "problem:marked-seam-monodromy-and-lattice-lifts",
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        "premise:ordered-quarter-clock-pure-hurwitz-equivalence"
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      "id": "finding:e8-cut-five-orbital-incidence-algebra-complete",
      "titleEN": "Complete E6 orbital algebra with five irreducible components",
      "statementEN": "The actual Weyl action is transitive on1120 E6 cuts. An explicit stabilizing E6/A2 reflection subgroup is transitive on each intersection class: D5 with81,D4 with270,A4 with648,A2+A2 with120 neighbours. Including equality these are the full five orbitals. All125 structure constants yield the commutative full commutant. Normalized primitive idempotents have ranks1,35,84,300,700, proving the complete irreducible Weyl decomposition.",
      "scopeEN": "Complete for regular A2-normal E6 root cuts in standard E8 under its actual Weyl action; not arbitrary Lie-subalgebra embeddings.",
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      "titleEN": "Unique D8 frame belongs to the full D5+A3 flag",
      "statementEN": "All135 D8 frames,7560 A3 tetrahedra and their actual incidences were generated. Each full pair of A3 normal system and common D5 core lies in exactly one D8; each D8 contains56 such flags. Bare A3 containment instead gives616 A3 systems per D8, consisting of56 coordinate D3 and560 four-axis A3 chains, and11 D8 per bare A3. Unique tetrahedron-to-frame incidence is therefore a full compiler-flag statement.",
      "scopeEN": "Actual full D5+A3 root flags, not uniqueness from the A3 root subsystem alone.",
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      "id": "finding:tetrahedron-incidence-lossless-e6-reconstruction",
      "titleEN": "Tetrahedron sums reconstruct all E6 data without loss",
      "statementEN": "For E6/E7 incidence N and E6/tetrahedron incidence M, every entry satisfies NNt=A_D5+3I and MMt=A_D5+27I. Hence MMt=NNt+24I and for every signal f, norm(Mt f)^2=norm(Nt f)^2+24norm(f)^2. N has rank120 and loses1000 directions300+700; M has rank1120. All tetrahedron sums recover f exactly as (NNt+24I)^(-1)M(Mt f), also through an explicit rational degree3 polynomial. The information theorem does not require irreducible decomposition.",
      "scopeEN": "Linear data reconstruction on the same finite E6 set with counting inner product. Preserves supplied data; does not generate a distinguished corner, source or physical state law.",
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      "titleEN": "General lift and cover audit reconciles with the marked originals",
      "statementEN": "In the original lattice chart SNF(I-J) has invariants1,1,1,1,2,2,2,2:256 J sign lifts yield16 sign-gauge classes and one class if only gauge equivalence is enlarged to quarter phases. Enlarging candidates too gives16 classes. The original additional equation chat^9=fixed S_J has exactly four solutions and one orbit under16 S_J-preserving character gauges. On all five fundamental peripheral loops the original induced Burau comparison has rho_rel=A^(7nu), nu=(1,1,1,1,-4). Faithful order12 proves globally ker(rho_rel)=ker(nu mod12), hence minimal C12. The general C12xC12 degree144 counterexample does not satisfy this factorization.",
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      "titleEN": "Doubled scalar quarter twist is exactly the marked glue generator",
      "statementEN": "F(2q)=lambda is an actual lattice identity, not only equal norm. s=2q has order4 modulo D5+D3 and order2 modulo D8. The tower U to V_D8 to V_E8 has indices2,2 and mu-indices16,4,1. The older SO16 base with index4 mixes two different inclusions.",
      "scopeEN": "Given eight-channel chart and marked lattice generator. Squaring a finite disorder operator is not identified with the local vertex field.",
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      "titleEN": "Exactly two scalar extensions over D8, both of order two",
      "statementEN": "The canonical scalar torus character g_q on V_E8 has order2 and fixed VOA V_E7 tensor V_A1. Over fixed V_D8 exactly one further extension gamma g_q exists, with fixed VOA V_D8prime and order2. Their weight-one spectra136/0/112/0 and120/0/128/0 exclude identification with actual native Jhat with60/64/60/64, even under full field conjugacy.",
      "scopeEN": "Precisely the given scalar character restricted to fixed D8 subnet. Other source-derived field dictionaries are not excluded.",
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      "titleEN": "Native quarter clock is the inner glue-clock class over tetrahedron flags",
      "statementEN": "Five-carrier parity kappa=exp(2pi i v(0)), v=(1/2^5,0^3), acts on the four U glue sectors as1,i,-1,-i and fixes precisely U. Among all65536 torus points modulo4 exactly15120 match actual Jhat; they form one actual W(E8) orbit. Thus C kappa C^-1=Jhat for a full VOA automorphism C. Each two inverse primitive characters share a fixed root lattice D5+A3; the7560 kernels are equivariantly the actual7560 compiler/tetrahedron flags.",
      "scopeEN": "Complete torus and Weyl classification plus compact E8 conjugacy, not mere matching spectra. C is not preferred and retains other marks only through their joint transport.",
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      "titleEN": "One common inner conjugator for the complete C12 compiler family",
      "statementEN": "After toralizing actual chat and aligning its ninth power to kappa, all candidates c_t=kappa sigma_t are covered by3^8=6561 cubic torus points. Exactly480 have the full original weight-one character and form one orbit under reflections in52 U roots. Thus chat,Jhat=chat^9,sigmahat=chat^4 are simultaneously conjugate to c_t,kappa,sigma_t. Combining kappa with the old frozen sigma already fails on an explicit root.",
      "scopeEN": "One selected actual admissible chat and its whole power family. All four admissible sign lifts have the same cycle character. No frozen full marked-tuple equivalence or microscopic origin is proved.",
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      "titleEN": "Recovered complete E6 carrier and unique cubic source correlator",
      "statementEN": "The original18 September proofs identify E6 and A2 as mutual centralizers in the actual Chevalley E8 algebra. The48=(16,3) carrier is not E6 closed; its full closure is81=(27,3) with27=16_1+10_-2+1_4. In the given affine vacuum the actual three-current correlator kappa(a,[b,c]) equals d_ABC epsilon_ijk. Its45 cubic candidates have Ward rank44 and hence a unique relative coefficient line. The positive grade-one Gram gives64/48=4/3 its exact equal-norm current-response meaning. Normalized I4/4 is not a vacuum partial trace. These completed original results are recovered into reviewed findings; missing reviewed coverage was not an open derivation.",
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      "titleEN": "E6 E7 E8 operator roles, tetrahedral envelopes and actual horizontal time centralizer",
      "statementEN": "In the fixed original D5+A3 cut, each of the six pairs among the four E6 systems generates precisely its unique E7:104 union roots close to126, with a common40-root D5. Orthogonal root/Cartan projectors onto E6 and E7 are not Lie homomorphisms, as removed opposite-root pairs with nonzero projected coroot show. For the existing H8=8L0-rho_A7.a0+(2a-1)Q/2, Q-neutral E7 membership is not full horizontal energy symmetry: [H8,J_(e0-e1,0)]=-J_(e0-e1,0). At a=1/4 and3/4 exactly8 ordinary n=0 root modes commute, of type4A1; with8 Cartan modes this horizontal centralizer has dimension16. Complete conformal E7+A1 and E6+A2 windows retain their complements and glued modules. A cut sequence is not a derived physical breaking or time history.",
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      "titleEN": "The actual clock doily reconstructs the original quartic five",
      "statementEN": "The actual ovoid module ker(B^T) of E8/(1−J)E8 is marked-isomorphic to original quartic H5: detT=−96 and N_rT=16TD_r for all60 reflections. Spread and both twists have Hom0. Form diag(1,3,3,3,6), P5 and six rank1 effects follow.",
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      "titleEN": "New feedback closes compatibility without a circular origin claim",
      "statementEN": "The loop E8/J→ovoidH5→Pascal/glue→E8 already assumes E8,J and rank4. On C:y³=x⁴−1 the actual Prym clock gives SNF(I−J)=(1,1,2,2), four instead of16 classes; polarization(1,2) is degenerate mod2. V784/V796 exclusions and external bridges remain scoped.",
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      "titleEN": "Exact intersection with original native event channels",
      "statementEN": "Phi_p is a positive mixture of actual AdC5(g) exactly for1/10≤p≤1/2. On End(H5)=1+5+9+10 the zero9/10 blocks force Echi²=1+Echi; integral character bounds are sharp and both endpoints attained. p1 Lueders is excluded in this event class.",
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      "titleEN": "Informative ovoid observation first occurs at tensor degree3 with complete classification",
      "statementEN": "For actual G31/ovoid action, Hom(H5,End(V4^⊗n))=0,0,1 for n1,2,3. n1/2, including V⊗barV: onlyI/6. All POVMs onC64: F_a(t)=I/6+tW[I4⊗(E_a−I5/6)]W*, −1/4≤t≤1. ImW=Sym³V with uniform44 complement is forced. Bell six carry spread, not ovoid.",
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      "titleEN": "Original quartic, erasure and Hamming octic share the same marked ovoid response",
      "statementEN": "Original decoder is (W*x³)_(a,i)=∂a phi_i/2. Actual marked R_A satisfy Igusa and p_A=||gradR_A||²/48, sum||x||⁶. All256 Hamming code pairs give I8=Σx⁸+14Σx_a⁴x_b⁴+168∏x²=96Σf_i²=(1/3)ΣR_A². Restoring allμ4 phases gives all14 marked code/root tetrads.",
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      "titleEN": "Canonical octic ray and exclusion of unshifted vacuum preparation only",
      "statementEN": "F8hat=Σ:f_i²:/sqrt140=I8/(96sqrt140) is normalized and uniquely invariant in ImM; six ovoids are the complete Pi9(v⊗v)=0 locus. Primarity and stress null theorem apply only to Omega,L_n. L_-8Omega of norm²336 is another singlet. a8(dotD) tests direct linear insertion, not higher geometric response; actual quarter fold hits at order8.",
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      "titleEN": "Actual quarter fold reaches the octic ray at eighth order",
      "statementEN": "Original M/O/U+ gives ||d||²=85/128 and F8hat(b)≠0 for BOTH quarters. Exactly <F8;d|(L_-1^d)^8Ω_d>=8!barF8hat(b), L_-1^d=Lcal_-8/8. All22 partitions give norm8!(85/128)_8. For smooth real A_d(g), jets k<8 vanish, k8=8!barF8hat(b)c_-1^8; genuine local bumps with c_-1≠0 hit without truncation. Full Mobius orbit agrees.",
      "scopeEN": "Given source/reference. Nonzero component, no pure preparation/ImM projection or P1 identity; minimum8 for c_-1≠0.",
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      "titleEN": "Octic field already lies in the original even CAR current algebra",
      "statementEN": "A=U+^T O^T M obeys AA*=I4, AA^T=0 and b=Aδ. F8^d=:I8(A(j_sea+δ/z))/(96sqrt140): is a neutral even CAR current composite of the same source with original point splitting and finite derivative counterterms. Full correlations and i0 are retained; no E8 root/spinor vertices are needed. Fold energy8 equals source energy64.",
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      "titleEN": "Historical UV-map exclusion was already proved and is recovered",
      "statementEN": "The original2026-10-13 audit already proves d(Tnu)/dt>=m*nu(total)/∫psi dnu>0,so output mass is infinite. Printed T is not an endomorphism of the asserted finite-measure cone and supplies no fixed point there. Current independent review confirms the same v42/tfpt45 wording. Positivity alone also does not preserve the moment slice.",
      "scopeEN": "Reuse of an existing exact source audit,not a new discovery or an all-TFPT/all-RG exclusion. A corrected original kernel needs at least integrability and actual moment preservation; no replacement kernel is chosen. Atlas navigation had exposed the old registered contract as pending rather than a curated conclusion.",
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      "statementEN": "A4 requires the order-four lift of the same Moebius identification. The double-cover core loop is its square,so spin holonomy r²=-I gives NS. Both Pin-minus lifts r,-r and the base sign twist have the same cover holonomy. Full combined cJ_perp gives spin-reference phases{1/4,3/4}. The tentative R twist on the cover preserved only thin A2/A3 and violated A4; it is not a countermodel to the full original.",
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      "statementEN": "For u=M_z,AdW1 preserves each original CAR interval. Exactly tau_t AdW1=gamma_t AdW1 tau_t; on the source U1 charge-zero net it commutes with given time. W1*QW1=Q+1,W1*HW1=H+Q+a,W1*J(f)W1=J(f)+fbar. Neutral current response and rank1 defect remain visible; old bounded Cartan four-word is not projected away.",
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      "titleEN": "Marked glue square is the E8 fermionization character",
      "statementEN": "The marked Z4 glue character squared fixes grades0+2 with52+60=112 D8 roots and negates grades1+3 with64+64=128 spinor roots. At lattice level K union(K+2g)=D8. Thus the non-anomalous E8 to D8 Z2 action of standard fermionization is already in the original marked glue.",
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      "titleEN": "Fixed canonical D8 is not transported source D8",
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      "scopeEN": "Common completion up to intertwiner basis; an independently original-marked mixed junction would be stronger input. No identification of physical clock operators or localized spinor vertex with global charge shiftX.",
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      "titleEN": "Weak determinant class carries index one independently of internal rank",
      "statementEN": "On the negative vector Fourier sea Q-P is trace class and Tr(Q-P)=sum_j j tr(g_j* g_j)=wind detg for smooth g. Thus det_rel=(1-z)^(wind detg). uI_r has indexr,while an actual degree-one U2 clutch has indexone. Originals v31/v32/v42 already contain the weak degree-one line and its assignment to total seam twist. S(U3×U2) is a fiber stabilizer and does not force a globally trivial determinant line. A common realization on the same source covariance therefore yields1-z without a rank-one choice.",
      "scopeEN": "Analytic matrix-index theorem proved; original class is reused as a branch statement,not newly derived from P1/P2. Normal S2 clutch equator is not automatically the tangential field-source circle. Ordinary admissible EM trace and relative pair response still require an original intertwiner.",
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      "titleEN": "Finite NS circle contains a UV partner and gives no determinant limit1-z",
      "statementEN": "Sampling original M_z gives exactly the diagonal site phase and the cyclic shift in the actual v519 NS Fourier window. Negative sea obeys U_N P_N U_N*=P_N+Pi0-Pi_-M. Hence the complete ordinary finite relative determinant equals1 for everyN. Pi_-M escapes strongly but retains trace normone; a Fredholm determinant limit1-z is therefore invalid. Canonical sampling becomes exact on every fixed Fourier core; finite cyclic u_N admits no exact intertwining with bilateral M_z.",
      "scopeEN": "Excludes only passage from ordinary finite NS determinants by strong convergence. Positive full sampling proof for all evenN plus directN8/16/32 controls. Does not exclude separately defined original infinite relative factor,a justified different UV rule or all P1 sources. Finite alias,carrier complement and reflected physical pair stay distinct.",
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      "titleEN": "Internal carrier retains all circle frequencies rather than a finite Fourier cut",
      "statementEN": "The explicitly available factorization H=H_circle tensor K with internal projection I tensor p,rankp=r has infinite total rank. Its trivial extension gives C_p=C_beta(B_circle) tensor I_r and U_t=e^(it B_circle) tensor I_r; scalar clutchuI_r yields(1-z)^r. It creates no finite96×96 covariance or Yukawa/Majorana dynamics. v177 maps H1 to generation rankthree,v210 uses a Fourier truncation,and v252 inserts internal field labels and Yukawas. Inspected originals do not construct symbolicK_car as a Hilbert bimodule with operator connection.",
      "scopeEN": "Exact typing and forward test of the original factorization; not exclusion of all nontrivial carrier connections. New matrix-clutch formula replaces a purported general rank law by actual determinant winding. Earlier v258 provenance gap is reused.",
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      "statementEN": "On spin S2,D_N twisted by L2=O(1) has one positive zero mode:H0(K^(1/2) tensor L2)=H0(O)=C,H1(O)=0. For the given graded product,V_chi f=f tensor chi exactly reduces it:B_prod V=V B_a and Fh(B_prod)F=Vh(B_a)V* for all bounded Borel h. All circle frequencies,time,polarization and finite-beta covariances remain;F has infinite total rank. Nonzero pairs +/-lambda never cross zero,hence SF=ind D_N^+. This does not identify the original bosonic operator or weak doublet H0(O(1)).",
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      "titleEN": "Exact index-line source requires matching horizontal holonomy",
      "statementEN": "For an already exactly reducing rank-one source B_A=-i partial_theta+A and the same u=M_z,a periodic smooth multiplier g with B_A M_g=M_g B_a exists iff mean(A)-a is integral. g=e^(i chi),chi_prime=a-A transports the full domain and functional calculus. Horizontal tensoring by a flat line changes the induced circle phase while normal D_N,c1 and kernel stay fixed. Vertical degree-one data alone do not determine this phase.",
      "scopeEN": "Applies the already classified scalar first-order case at fixed time unit and exact reduction. Not two complete P1 countermodels,a newly selected Wilson factor or an adiabatic assertion of exact reduction. A4 NS/quarter origin remains;original reflection and clock marks need joint transport.",
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      "titleEN": "Full product polarization lies outside the ordinary relative trace class",
      "statementEN": "In the genuine circle times S2 product,the change P_(x+1)^lambda-P_x^lambda of negative projections has trace norm at least1/(2lambda) for large lambda already at a fixed circle index. The positive two-dimensional Weyl sequence lambda_j asymptotic to const sqrt(j) makes the sum divergent. Degree-one monopole values lambda_l=sqrt(l(l+1)) with multiplicity2l+1 even yield a lower bound greater thanone per level. Thus UP0U*-P0 is outside L1. Since C_beta-P0 lies in L1,finite thermal smoothing does not alter this.",
      "scopeEN": "Excludes only the unsupported extension of the ordinary one-dimensional relative Fredholm determinant to the full unprojected product. Exact reducing circle retains trace class and factor1-z. Does not exclude other justified relative regularizations or all P1 sources.",
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      "titleEN": "Massive transverse zero-index channels retain positive determinant response",
      "statementEN": "For lambda>0,det_zeta(B_a^2+lambda^2)/det_zeta(B_a0^2+lambda^2)=[cosh(2pi lambda)-cos(2pi a)]/[cosh(2pi lambda)-cos(2pi a0)]. Its logarithmic a derivative is2pi sin(2pi a)/[cosh(2pi lambda)-cos(2pi a)],strictly positive at a=1/4. Yet the corresponding +/-lambda block has zero spectral flow. Zero index therefore justifies neither constant positive response nor dropping the complement.",
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      "statementEN": "An actually closed normal Fredholm A and its true adjoint give an exactly reducing circle L2(S1) tensor ker A through the selfadjoint double;the particular compact spin realization is not required. For r=dim_C ker A,scalaru I_r has factor(1-z)^r in the ordinary complex relative source contract. Later original v42 sets H0(O1)=C2 and counts these two components as one weak doublet with bosonic indexone. This species count is not complex kernel rankone. For a retained irreducible SU2 doublet,Hom(C,M)=0,r=2 and factor(1-z)^2. det M is an SU2 singlet of hyperchargeone,but not a linear subspace of M.",
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      "titleEN": "Tetrahedral quarter face phase does not identify the original quarter curve",
      "statementEN": "In the existing Bloch comparison,all ordered tetrahedral faces have tr(p_a p_b p_c)=plusminus i/(3sqrt3),hence normalized phase plusminus i. Yet the full four-point orbit is{(1+i sqrt3)/2,(1-i sqrt3)/2},j0. Original quarter marks have orbit{2,1/2,-1},j1728. Disjoint orbits exclude every direct marked Mobius identification even after relabeling. Holomorphic automorphisms have order12 for tetrahedral A4 and8 for square D4;a four-cycle is antiholomorphic in the tetrahedral chart.",
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      "statementEN": "Existing C3:w^3=x^4-1 has elliptic quotient(u,v)=(x^2,w),u^2=v^3+1,j0,degree2. Its branch quartet has the tetrahedral Bloch cross-ratio orbit;an explicit ordered Mobius map is supplied. Existing C2:y^2=x^4-1 has j1728. From C12,pi_i=(x,s^6) and pi_omega=(x^2,s^4) have degrees6 and8. Their combined image field is exactly C(x,s^2),namely C6;the paired map loses s to minus s and therefore the C4 field s^3. Pure deck drops to order6,while already marked mixed R12=Q12 Delta12^4 still acts faithfully with order12,retaining its fourth and ninth powers.",
      "scopeEN": "Explicit reconciliation of existing curves and marks,not global novelty or a new cover. Tetrahedral image lies in the v quotient chart,not original x base. Does not identify internal D3 sphere with physical normal sphere. Compact curve maps establish neither punctured local-system transport nor fields,measure,EM response or full polarization. Previous maximal Phi12 module and integral-form boundaries remain preserved.",
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      "statementEN": "September22 already constructs every natural iota_m with sum m=-2 and W_m=LambdaEven E tensor sum(det E)^m without a global square-root bundle. Its continuation supplies A_Wm from a given A_E. The global mathematical quotient construction is therefore not an open construction in current research. A source intertwiner and selection of A_E,Phi,domain,state and time are not supplied.",
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      "titleEN": "Split joint carrier admits a parallel representative of the whole native C12 clock",
      "statementEN": "For every natural m and given block U3xU2 connection,sigma0 from b=(1,1,1,0,0,0,0,1)/3 acts on W_m exactly as LambdaEven(diag(omega I3,I2)) tensor diag(omega,omega2,omega2,omega). It centralizes every transition and the entire induced connection. With central glue quarter kappa,c0=kappa sigma0 gives global parallel automorphisms with c0^9=kappa,c0^4=sigma0 and the complete native C12 character. The old complete torus Weyl theorem gives full inner VOA conjugacy including the common clock tuple. Exactly12 of243 thirds in the fixed toral centralizer match:two color orientations times six A3 vector weights.",
      "scopeEN": "Conditional mathematical closure for a GIVEN split connection,not selected from P1. Original marked chat is not automatically parallel on unchanged A_E;conjugator must transport connection and all marks or preserve them. An extra nonabelian family connection needs its own commutator,potential and closed domain their own invariance. Vertical bundle action identifies no base transport,time,CAR,state or EM response. Twelve counts marked torus points before further quotients,not physical sources. The2+2 action supplies no dictionary to original SU3 flavor on H1.",
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      "titleEN": "Pure weak clutch as whole native clock excluded despite quarter matches",
      "statementEN": "Under the extra same-angle identification,64 m residues contain exactly8 native quarter characters,permutations(2,2,2,0) and(1,1,1,3),with minimum h norm12. Earlier simple A/B lifts fail this quarter test. For whole C12 every pure weak clutch fixes at least32 weight-one directions versus22 native;all1728 classes fail. All20736 weak toral classes also fail the full character. Generally,sum k=1 prevents a clutchquarter from fixing its own original D5 factor pointwise:all k_i+-k_j=0mod4 would force sum k=0or2mod4.",
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      "titleEN": "Same Chern class can carry the whole clock without selecting physical connection",
      "statementEN": "The toral diagnostic k_color=(-6,0,6),k_weak=(0,1),m=(-5,1,1,1),h=(k,3,3,3) has the same block Chern classes0,1 and complete native C12 character. Norm100 is minimal in the specified integer toral class:an existing candidate bounds every better search,all6939 allowed canonical tuples through100 are checked exactly;155 match,all at100. These classify marked weight representatives,not a physical selection. The entire joint64 carrier still has determinant winding0.",
      "scopeEN": "Unbounded minimum proof in this toral class via exhaustive finite lower search. Norm is not energy;155 is neither physical-source count nor full E8 conjugacy-class count. Nullhomotopic color loop and clutch class select no connection or holonomy. In the ordinary circle CAR contract,the whole64 clutch has index0 rather than1;no identification of bosonic E8 VOA with CAR. The simpler parallel-clock closure makes this loop optimization unnecessary for mathematical compatibility.",
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      "titleEN": "Gaussian four is Cartan subspace in whole64 with actual G31 action",
      "statementEN": "Every real E8 root restricts nontrivially to H+. A generic point is regular semisimple,so H+ is a maximal commuting semisimple Cartan subspace in g_i. Existing whole-clock conjugacy carries it into the appropriate64 half-spin times family branch. Its normalizer quotient is exactly C_W(E8)(J)=G31:Weyl lifts can be corrected through surjective torus isogeny1-J to centralize the quarter clock. Old degrees8,12,20,24 are invariants of this same theta representation.",
      "scopeEN": "Exact fiberwise identification and group action;no numerical fit. Current orientation puts H+ in dual64. Conjugator exists but is not preferred. GIT quotient describes closed semisimple orbits,not physical orbit or state selection. Reuses v634,v690 and whole-clock conjugacy.",
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      "titleEN": "Family factor remains different despite symmetric-product coincidences",
      "statementEN": "For the same sigma=chat^4,H+ has C3 character(2,1,1). All480 toral representatives of the WHOLE native C12 class have character(0,2,2) in the unique order3 SU4 lift,and(4,3,3) in Spin10 vector. No direct sigma isomorphism identifies the fours. Nevertheless Sym2 andSym3 agree;in general Symn agree precisely for n not1mod3. Exterior2 gives(2,2,2) versus(4,1,1). All-degree trace generating-function difference is3t/(1-t^3)^2.",
      "scopeEN": "Excludes direct factor identity under the same actual third action. Positive Cartan-subspace inclusion and full inner E8 conjugacy remain valid. Product equality is only C3 character equality,not full G31 or Spin10 times SU4 equivalence. The480 belong to old complete classification and are not new physical sources.",
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      "titleEN": "Primitive class2 sphere prevents global whole Cartan-subspace choice",
      "statementEN": "Pointwise centralizer of whole Cartan four is T^J=F2^4;setwise normalizer N has order16*46080=737280. An everywhere smooth Cartan-subspace choice reduces the K_C bundle to N. On simply connected S2 this would trivialize it,contradicting old class2 in Z4 for every natural m. Separately,a whole Cartan four parallel under the same K connection forces curvature0.",
      "scopeEN": "Topological exclusion only for whole Cartan-subspace choice in fixed quarter sector over precisely the same primitive normal base. Needs no connection. Does not exclude single vectors,arbitrary rank4 carriers,punctured bases or allowed defect and degeneration sites. Previous parallel-clock theorem remains valid. No splitting of old sign-lift extension claimed.",
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      "titleEN": "Even cover degree removes topology and selects neither C12 nor connection",
      "statementEN": "An actually supplied degreeq pullback to the same normal base sends class2 to2q mod4. Every even degree removes this topological obstruction,smallest positive degree2;on compact S2 only branched or as a general degree map. Existing C2,C6,C12 do this only with proved same base and bundle map. Parity does not select C12. A given nonflat smooth connection remains nonflat under branched pullback. Sequence1 to Z4 to pi1(K/N) to N to1 explains why trivial N or G31 monodromy can hide class2.",
      "scopeEN": "Conditional repair with existing covers. Identity of normal S2 and original punctured source base is unproved. No freely chosen double cover as P1 selection. Full relative C12 monodromy retains its own global criterion.",
      "status": "conditional",
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      "id": "finding:global-cartan-choice-is-optional-for-original-scalar-source-fields",
      "titleEN": "Global Cartan choice is not an original scalar-source obligation",
      "statementEN": "Original boundary data do not demand an everywhere smooth whole Cartan four. Normal geometry and flavor quotient have the same boundary-kernel origin;this supplies no actual base or bundle map. The class2 theorem remains an exclusion of the stronger Cartan choice. Any K-invariant classical polynomial value already glues across associated bundle transitions;an operator completion requires its own test. Current physical priority is corrected accordingly.",
      "scopeEN": "Source correction and elementary associated-bundle consequence. Does not contradict the preceding scoped Cartan theorem or identify Calderon reflection with elliptic deck.",
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      "titleEN": "The bare native octic is not a whole E8 zero-mode singlet",
      "statementEN": "For every E8 root,I8(alpha_+)=1. Its actual root zero-mode action on F8hat has nonzero PBW partition(7) coefficient alpha(-7)e^alpha/(672sqrt140). Thus the valid old neutral CAR field F8hat is not itself an E8 zero-mode singlet. Classical invariant-theoretic extension and this bare operator polynomial cannot be identified.",
      "scopeEN": "Excludes only bare full E8 invariance. Old F8 existence,correlations and stress responses remain proved;no unproved physical E8 gauge symmetry is imposed.",
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      "titleEN": "Explicit primary E8 octic completion in the same neutral CAR source",
      "statementEN": "The simultaneous rational system of actual root zero mode,L1 and L2 has1600 coefficient rows,rank16 and precisely one solution in the17 dimensional ansatz with H8 leading coefficient1. All16 contraction coefficients are explicit;||W8||²=65028096000/43. Weyl invariance transports the root equation to all240 roots;L1 throughL8 vanish. W8 is therefore a horizontal E8 zero-mode-singlet weight8 primary. Its internal subline in the given K-associated VOA bundle is canonically trivial,including class2. It uses only the old neutral CAR currents,derivatives and point-splitting counterterms.",
      "scopeEN": "Uniqueness only in declared ansatz. Internal scalar retains conformal weight8 and its geometric weight line. No affine-vacuum invariance,local E8 gauge physics,new source,preparation or novelty claim. Concretely connects the known W(E8) spin8 generator to the native source.",
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      "id": "finding:completed-octic-has-exact-old-root-ope-component-and-quarter-source-value",
      "titleEN": "Completed octic is connected to old root OPE and quarter source",
      "statementEN": "Original240-root isometry and Gaussian polarization give |<F8hat,W8/||W8||>|²=43/896. In the old full22-partition OPE coefficient S8,<W8,S8>=48384000/1333 and proj_CW8 S8=W8/41664;normalized overlap squared30/537199. For both actual quarter sources1/4 and3/4,z8<Y_d(W8,z)>=25123488075/357824462848 is nonzero,using old O/M transport and the same single shift.",
      "scopeEN": "W8,F8hat andS8 remain different. Component is not a physical projection rule,preparation or progress percentage. Equality of a quarter one-point value does not identify whole modules,states or reflections. Original grades and eight-arc support are retained.",
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      "titleEN": "Elliptic normal spin pullback doubles index and deck-quarter lifts anticommute",
      "statementEN": "On old E:y²=x⁴-1,the spin Dirac twisted by p*O1 has ordinary complex index2 for every spin. In q-fixed even spin alpha=O(T-O),u=(y+i)/x²,kernel is span(1,u). Genuine deck and quarter lifts have D=[[0,-1],[1,0]],Q=a diag(1,-1),so D²=-I and QD=-DQ independently of scalar phases. Quarter exchanges the two deck eigenlines;no common rank1 kernel sector.",
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      "titleEN": "Complete separated W8 two-point response in the original quarter source",
      "statementEN": "For both original quarter charges, z^8 w^8 <W8(z)W8(w)> is exactly sum_(r=0)^8 A_r X^r, X=q/(1-q)^2. Bosonic exchange, homogeneity and the sole pole at q=1 of order at most16 prove completeness. Nine rational Fock norms determine all coefficients without a fit. A0 is the squared old one-point value, A8=65028096000/43, and an independent stress Ward test forces A7=2(85/256)A8=43182720000/43. All A_r are positive. Regular and optimized checker outputs are byte-identical.",
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      "statementEN": "On a proper interval choose real compactly supported chi, kappa=(1/(2pi))integral(chi_prime)^2, f=chi and g=-2chi_prime/kappa. The original level1 current cocycle is c(f,g)=-2i; the Weyl commutator is exp(2ist)I. A normal unital local current embedding must preserve this identity. If it also preserves state, time, reflection and original eight-arc support, GNS gives an isometry onto its cyclic image. On that image the same point-split products and counterterms fix W8. A global unitary onto the entire P1 space is unnecessary for this subfield. Reviewed original passages give C_seed from an already supplied local seed to boundary data and assume the positive CAR kernel; they do not supply the concrete reverse current embedding.",
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      "statementEN": "The actual root automorphism g=-M R M^-1 with coordinate reversal R maps d_1/4 to d_3/4, fixes W8 and preserves N. Its Fock intertwiner therefore identifies every separated pure W8 correlation function in both real charge modules. The exact two-point calculation verifies this explicitly. W8 alone cannot select these original quarter labels.",
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      "titleEN": "Older seam verifications do not supply an already closed marked local P1 map",
      "statementEN": "Reviewed originals retain the old circle, concrete microscopic neutral CAR field route and conditional E8 net extensions. v366 sets inclass=True and tests supplied copy counts and target constants; v464 uses a chosen sin(k) chain. v469 gives the extension route under its subnet hypotheses. v994 explicitly labels C7 cited/open; v1030 explicitly leaves microscopic generators and local word/adjoint tails unproved. The current SEAM.EQUIV.01 parent is open and its MMST branch conditional. The earlier complete Dirac vector cocycle cancellation is retained in its Lagrangian/CAR scope. None of these passages constructs the actual marked local P1 W8 image.",
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      "titleEN": "The full original c1 stress cycle reconstructs the neutral W8 response",
      "statementEN": "For the same original quarter CAR source,p=alpha-1/2=plus or minus1/4 and c1 lowest weight h1/32. This is not a Kac weight m²/4. The injective Virasoro Verma map and equal partition characters prove at each grade that the full original stress cycle generates the entire fixed relative-charge sector. All W8 response components and their positive-semigroup smoothing are therefore in it. At original grade8 the first response is explicitly reconstructed by22 literal CAR stress words: all rational coefficients,residual0,full rank22 modulo three primes,and the identical complete Gram from independent c1 Sugawara calculation. All original modes and intermediate states are retained.",
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      "titleEN": "Only the smaller folded stress cycle misses the intrinsic W8 tangent",
      "statementEN": "In the folded c8 space the Virasoro cycle at N level1 is span(d.h_-1 Omega). The actual first W8 response has perpendicular component v_perp=grad_tan H8(d),norm²51311452775625/9345848836096 in both quarters. All16 O8-invariant quantum Q_ab corrections contribute only radially at this level. An independent240-root enumeration reconstructs the whole first coefficient and one-point value from all17 old ansatz coefficients. The entire smoothed response is outside the closed Lhat cycle. This restricted exclusion is consistent with the positive full original c1 reconstruction.",
      "scopeEN": "Excludes generation by Lhat_n=Lcal_(8n)/8+constant alone. Does not exclude the full original stress source,the old F8 overlap or P1. Introduces no extra nonstress source or physical gate.",
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      "titleEN": "The fourth original root moment reconstructs the five after three uniform moments",
      "statementEN": "The actual60 complex E8/J root lines obey exactly M1=I4/4,M2=Pi_sym2/10,M3=Pi_sym3/20 and40M4=Pi_C+Pi_sym4. Extracting40M4−Pi_sym4 without a five-dimensional target literally recovers the original code projector. The preexisting parent I−20M4 has spectrum0^5,(1/2)^30,1^221. Rebuilding the identity from the original Gaussian roots gives byte-identical normal/-OO results and two firing negative controls. Algebraically(20M4)^n=Pi_C+2^(−n)(Pi_sym4−Pi_C) for n>=1.",
      "scopeEN": "Finite original root lines,given J/metric and four native registers; no novelty claim for the old parent. First nonuniform projective moment does not exclude arbitrary nonlinear recognition. No physical time,dissipation,preparation,P1,3+2,clock or48c3^4 selection; E8 and J are not derived.",
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      "titleEN": "Equal actual three-register marginals exclude exact whole-five ground spaces from at-most-three-body terms",
      "statementEN": "rho_C=Pi_C/5 andrho_S=Pi_sym4/35 have exactly equal reductions on all4+6+4 subsets of sizes1,2,3; each three-register reduction isPi_sym3/20. Their global trace distance is6/7. Every Hermitian at-most3-local Hamiltonian on the same four registers therefore has equal mean energy. If the ENTIRE codeC has minimal energyE0, positivity ofH−E0I forces the full35-dimensional symmetric space to be ground as well. An exact five-dimensional code ground space is impossible. The existing4-local moment parent attains the necessary interaction order.",
      "scopeEN": "Only exact energetic selection of the ENTIRE original five-code on the same four registers. Arbitrary Hermitian terms and frustration allowed. Does not exclude one selected code state,ancillas,thermodynamic systems,effective four-body interactions,open dynamics or approximate selection. Marginal equality is for the two mixed states,not all pure code inputs. No physical-origin derivation.",
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      "titleEN": "Untransported independent quarter stress pair is not a current",
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      "titleEN": "Current public papers and binding source boundaries checked",
      "statementEN": "Directly downloaded current public introduction and tfpt_research_contracts PDFs are byte-identical to website/public/papers,with SHA256 d564ea7b…c7b9 and5c08d585…9b93. Old search-engine views return differing earlier editions and do not prove a deployment defect. The reviewed current TYPEIII.CHARGED original remains open. Its binding ALG2 correction withdraws the Vandermonde inference to the microscopically generated algebra and requires FE-GEN or the precise projected-algebra route. The old technical companion CMB section does not supply this P1 field map either. The positive old stress/winding composition is instead reused as F1.",
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      "statementEN": "In the given flat Moebius comparison,A=e^(i pi x)sigma1K is an antiunitary involution of the same fixed(c,b) H1 domain and preserves cover support. Exactly A D A=−D+pi sigma1. Centering D−(pi/2)sigma1 would change flat/Pin data. For the full D sea the PH polarization mismatch has at least n/4 frequency contributions of4/(13n²) on every n≥4;the HS level sum≥1/(13n) diverges. Thus this PH Fock lift does not exist in this full two-dimensional sea. The old transverse zero branch instead retains its finite rank1 defect. A is not a bulk D solution intertwiner;Calderon boundary-range invariance follows neither positively nor negatively from that and remains separately open.",
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      "id": "finding:native-root-gram-spectrum-is-not-canonical-source-energy-selection",
      "titleEN": "The root Gram is a source metric rather than canonical energy selection",
      "statementEN": "Normalized local quartic fields have the actual vacuum two-point functionK_ab/(z−w)^8. YetL0F=4F andF*L0F=4K,so the generalized energy problem hasE=4 on all35 source directions. ExactlyK+=(7K−2K²)/9 andFK+F*=Pi_sym4. Positive candidatesL_t=(3I−K)(I+tK),t>=0,share the five-kernel with different remaining rates;the finite correlation alone selects no physical generator.",
      "scopeEN": "Only identification of this covariance spectrum with canonical c8 stress energy is excluded. Intrinsic frame anisotropy remains. No executed whitening,no identification ofL0 with P1 time,no exclusion of original quarter states,other interactions or full P1 models. The older full W/Gamma generating functional remains available.",
      "status": "excluded_scoped",
      "problemId": "problem:atlas-clock-defect-and-operational-reconstruction",
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        "topic:clocks",
        "topic:quantum",
        "topic:origin"
      ],
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        "premise:native-clock-complex-reflection-and-quartic-subspace",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:literal-native-positive-cartan-quartic-fock-embedding"
      ],
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      "id": "finding:identical-overlapping-native-five-codes-have-no-common-state",
      "titleEN": "Identical overlapping native five-codes have no common state",
      "statementEN": "ForA=V_1234 tensorI andB=I tensorV_2345,exactlyA*B=(1/4)S for a factor permutationS. Their image intersection is0. The auxiliary projector penalty2I−AA*−BB* has minimum3/4;the actual native parent sum is>=3/8. Since overlappingS4 groups generate symmetry on the union,the same zero intersection follows for identical code constraints overlapping in1,2 or3 registers.",
      "scopeEN": "Only rigid identical code constraints in a shared register frame. No new P1 code-closure axiom. No classification of differently rotated codes,disjoint blocks,approximate/open processes,frustrated phases or the full E8 field. The auxiliary penalty is not presented as a native extensive Hamiltonian spectrum.",
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      ],
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        "premise:native-clock-complex-reflection-and-quartic-subspace",
        "premise:native-symmetric-quartic-and-erasure-constructors"
      ],
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      "id": "finding:horizontal-e8-invariant-octic-cannot-isolate-native-five-ground-space",
      "titleEN": "A horizontally E8-invariant octic cannot isolate precisely the native five ground space",
      "statementEN": "For the actual normalizedf0=sum(u_a.h(-1))^4/sqrt96 and original rootalpha=sqrt2e0,E_alpha(0) has charged PBW componentalpha.h(-3)|alpha>/(3sqrt96) with squared norm1/144,outsideC5. Thus an operator commuting with horizontal E8 zero modes cannot have preciselyC5 as an isolated eigen/ground space on the full grade4 source. The existing screenedW8 obeys[E_alpha(0),o(W8)]=0 and therefore fails this necessary condition as a sole exact-five selector.",
      "scopeEN": "Finite canonical c8 source with the same positive Gaussian polarization;no claim of full affine invariance. Does not exclude W8 contributions to larger E8 multiplets,broken symmetry,quarter references,state responses or full TFPT. The observer code need not be a full-source ground space and this is not imposed.",
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        "topic:clocks",
        "topic:quantum",
        "topic:origin"
      ],
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        "premise:native-clock-complex-reflection-and-quartic-subspace",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:declared-original-octic-quantum-completion-ansatz"
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      "id": "finding:longitudinal-reflection-preserves-fixed-pin-domain-and-full-dirac-sea",
      "titleEN": "Longitudinal reflection preserves the fixed Pin domain and full Dirac sea",
      "statementEN": "Theta_x=K R_x is an antiunitary involution of the same(c,b) H1 domain for both c=±i and b=±1. Deck inversion compensates conjugation:bar(c)^−1=c. Exactly D Theta_x=Theta_x D;all real spectral functions,the complete D sea and quasifree vacuum ray are preserved. Block-diagonal antiunitary Fock implementation needs no PH mixing or HS repair. D time and longitudinal translations are reversed. Original operational Theta is distinct from normal deck lift and Calderon polarization;longitudinal reflection is a correctly typed candidate,not yet an identified P1 field map.",
      "scopeEN": "Concrete flat boundary family and its given D sea. State invariance does not imply Green reflection positivity;P1 locality,time and kernel need an actual map. Do not identify the three reflection types.",
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        "topic:origin"
      ],
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      "premiseIds": [
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        "premise:original-p1-seed-boundary-and-local-restriction-definitions",
        "premise:typed-source-structures"
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    {
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      "titleEN": "A single fixed Pinminus Green kernel fails the neutral OS word",
      "statementEN": "The actual invariant simple transverse branch has D0=−i b partial_x,q=cb=±i. Its exact Green kernel Gq is q/(q−1) for0<s<1 and1/(q−1) for−1<s<0. The equivalent normalized one-mode trace has z=−q^−1 and omega(N)=1/(1−q)=(1±i)/2. The neutral number word is phase-independent under standard antilinear charge-exchanging word reflection. Its smeared positive-time OS square is not real. Thus the complete free Green realization in one fixed Pinminus sector is not RP. Linear Clifford reflection sends c to−c and is not an endomorphism there;K R_x as D-sea symmetry does not replace it.",
      "scopeEN": "Only this given charged Gaussian Green realization and standard charge-exchanging word reflection on a single fixed Pinminus sector. No P1 no-go,no exclusion of the positive NS cover source or arbitrary other kernels. Sector exchange prevents transporting the neutral witness as an isolated OS square in the cover state.",
      "status": "excluded_scoped",
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      ],
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      "titleEN": "Full original NS orientation cover has charged Green RP in all modes",
      "statementEN": "Original doubled Pin deck action forces beta2 in the existing length convention. H=i sigma3 partial_w is selfadjoint on the same b boundary domain,with simple spectrum nπ/2,n inZ. For L=partial_x+H,S(t)=e^(−tH)(1+e^(−2H))^−1 is the entire antiperiodic Green kernel. Both charge blocks factor exactly:−S(−t−s)=B_t*B_s and S(t+s)=Btilde_t*Btilde_s,with B_t=e^(tH)(1+e^(2H))^−1/2 and Btilde_t=e^((1−t)H)(1+e^(2H))^−1/2 for0<t,s<1. Both are bounded. Explicit source-order minus and charge-exchanging word reversal give full free CAR RP through Wick/Gibbs limits. Relative thermal Z=2 product_n≥1(1+e^(−nπ))² converges. Linear Clifford reflection exchanges the actual c=±i deck spaces;their zero kernels jointly give NS value±1/2. Every transverse excitation is retained.",
      "scopeEN": "Proved for the given free charged Green quantization on the existing whole orientation cover. No additional field-species doubling,no mixture of independent complex states,no identification with P1 kernel or previous W8 vacuum. Cover trace period2 is not the global relative C12 cover result. Jaffe/Ritter is not applied with unchecked boundary hypotheses;the concrete proof is direct.",
      "status": "established_scoped",
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      "titleEN": "Cover Green state and quarter Hardy vacuum require an actual time and state map",
      "statementEN": "The full cover Green realization quantizes x as Euclidean time and has thermal covariance C=(1+exp(2H))^−1 on the w-section L2 space,with simple zero-mode value1/2. The D sea on L2(x,w) and old quarter Hardy state on the longitudinal circle are spectral projections. Mere covariance-preserving one-particle unitary identification is impossible:projections have only0,1 while C has1/2. Spaces and time coordinates must be transported. Enlarged purification or local net embedding is not excluded. v526 already assumes the chiral NS kernel and half algebra;its thermal success does not supply this original two-dimensional P1 identification.",
      "scopeEN": "Type check of the specified free sources. No ban on local state maps in enlarged cyclic spaces and no requirement of surjective full Hilbert-space identification. Same geometry or numerical temperature is not yet same field action and time.",
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        "premise:original-quarter-car-fold-current-subalgebra",
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      "titleEN": "Declared recursive SOM update alone prefers no absolute code orientation",
      "statementEN": "For every U in U(35),the stipulated update is exactly covariant under X->XU,W->WU: norms,distances,winners,means and normalization are preserved. After5000 directly executed steps the relative signal error is1.36e-14. Original code weight.9809168 becomes.1324321 for actively rotated input with FIXED readout; passive readout transport restores.9809168.128 orientations transported by the proved covariance give mean.1417688 near5/35. Three generic complex starts also compress without the high native code weight.",
      "scopeEN": "Exact mathematical symmetry of the chosen update plus finite numerical checks. Generic U(35) does NOT preserve native tensor powers/field products;not a countermodel to full TFPT origin. Given native initial alignment may be justified upstream.",
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        "premise:native-even-e8-gaussian-clock-and-sign-extension"
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      "titleEN": "Recursive SOM oscillations depend on chosen learning rates",
      "statementEN": "Nine prescribed beta/eta pairs around the original settings were executed5000 steps each. Three reach numerical rest,six retain motion. At beta=.25,changing eta=.15 to.075 changes frozen-response periods from11201/64540 to18551/106907 iterations. For all9 states,the observed frozen120x120 response reproduces the next1000 nonlinear steps with relative error<8e-14. Normal and-OO runs produce identical result bytes.",
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      "titleEN": "Existing C2 real forms yield the concrete annulus and Moebius quotient",
      "statementEN": "The complete affine census of eight h^a Q^b kappa has precisely two dividing members:hQkappa and hQ³kappa. With O=(0,i),T=(1+i)/2,Qz=iz,hz=−z+T,kappaz=−i bar z+T. Their halves have width1/2,circumference1 and Q exchanges them. Delta=hQ² is the unique free involution among h,Q²,hQ²;on the annulus it acts(t,s)->(1/2−t,s+1/2),with Moebius quotient. The earlier kappa-only nondividing statement is correctly scoped. v507 half-period origin,v510 free-deck/Fock class and v519 free RP are reused results.",
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      "titleEN": "Raw NS pullbacks yield the spin central commutator and quarter descent",
      "statementEN": "For the untwisted NS NS line and raw complex-linear coefficient pullbacks,U_delta²=chi(1+i)=+I and U_rho U_delta=−U_delta U_rho,because universal lifts differ by the A cycle. The glide g=delta_E rho=−bar z+T has g²=z+i and U_g²=−I. With fixed sigma_s²=I,A=sigma_s U_g is Dirac covariant,A²=−I,and has c=±i spaces with involutive c^−1 A descent. Anti-linear Real j_g is distinct:(sigma_s j_g)²=+I and has no complex ±i eigenspaces. Q invariance allows only(+,+),(−,−);a genuine transported A4 Spin/Pin relation forces(−,−),without a separate no-zero-mode rule.",
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      "titleEN": "Canonical elliptic cut reproduces the original quarter boundary tuple and KMS normalization",
      "statementEN": "Unitary theta=2pi s identifies domains and actual operators:B_Sigma/(2pi)=Bcov=2(B1/4 direct_sum B3/4),V=T_pi,u_cov=[[0,u_base],[I,0]]. Q rotates cut charts and is not V. At canonical width W=1/2,the second chiral Calderon compression is C_A=(1+exp(2pi Bcov))^−1,hence geometric beta_angle=2pi without fitting. C_A is a mixed traceclass perturbation of the Hardy sea at distance<0.086590;relative covariance/Fredholm factor remains1−z,and(1−z)² for double winding. Known two-boundary graph reconstruction is specialized to the actual existing C2 cut. The old full flat Moebius test has modulus1,this one1/2;units cannot identify their full Dirac problems.",
      "scopeEN": "Proved for the given one-particle boundary tuple and declared free chiral CAR interpretation. No equality with Hardy vacuum,original P1 measure/time identification,or EM/Quillen response from the relative covariance determinant. v813 index-four response normalization remains open;E8 compiler and existing W8 are not relabelled as new. Continuation:This boundary-tuple identity is not literal identity with the full marked equator pullback,which is a ramified V4/E8 graph. Retain full Real group and both original slots.",
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      "titleEN": "Finite crosscap Cauchy state does not yield the quarter vacuum by normal filtering",
      "statementEN": "In the declared flat Pin model,the crosscap Cauchy covariance on each lambda,−lambda pair is the graph with t=−c exp(−2Llambda). Its commutator with signed boundary B has norm lambda/cosh(2Llambda)>0. Fock energy variance is lambda²/cosh²(2Llambda),so the finite-L state is nonstationary. Further normal collars filter in trace norm to the tree NS vacuum,which loses c,b marks. The loop quarter operator still follows from cb=±i,but this limit supplies neither its Hardy state nor W8 source dictionary. Cauchy data remain free before imposing physical local b boundary conditions;imposing them would instead solve the different zero-kernel problem.",
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      "titleEN": "Native quartic requires relative charge transfers among source arcs",
      "statementEN": "In the actual M/O/quarter dictionary,each degree8 CAR arc-charge Gram has rank5 and determinant498159/262144,normalized166053/4294967296. All four index-difference classes have rank5. Every nonzero native quartic responds to every nonconstant real infinitesimal arc phase;the common phase remains trivial. These phases pointwise fix the restricted additive scalar density/stress subnet. Native field distributions cannot be generated there. The full original CAR algebra already contains the required transfers.",
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      "titleEN": "Original neutral interarc fields have exact current reconstruction",
      "statementEN": "In the original Q0 Maya/current dictionary,B(z,w)=w/(z-w) exp(sum(z^m-w^m)J_-m/m) exp(sum(w^-m-z^-m)J_m/m). CAR current commutators,infinite-sea vacuum contraction and cyclicity determine all finite-energy matrix elements. The actual quarter factor(z/w)^alpha follows from field powers. Independent exact checks test every matrix element through degree4/8 with12/67 states at two point pairs;normal and-OO are byte-identical. The connected current-net/fixed-point theorem places neutral interarc bilocals in a larger connected interval,and locality puts them in the complement-current commutant onQ0. The additive separate eight-arc net remains strictly too small. Substitution into actual A_a and unchanged point-split prescriptions reconstructs native currents and existing quartic/W8 composites without a new elementary coupling.",
      "scopeEN": "All-degree formal or finite-energy matrix-element identity;localized bounded bilocals use the separate net theorem. Unbounded native limits retain old core/smearing/affiliation hypotheses. No equality of dual net with full neutral CAR,no finite smooth pulse,new time flow,or original P1 net embedding proved. F_a to delta D_rel remains relevant to a claimed particular primitive interaction,but is not an extra existence requirement for an already neutral composite.",
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      "titleEN": "Actual normal zero branch preserves quarter CAR state and time,but uniform scale gives no return",
      "statementEN": "Vf=(1,b)f/2 is isometric on the original Pin fundamental region and obeys D0V=V b(-i partial_x),q=cb. Spectral projections,all quasifree CAR words and Theta_x transport. This explicitly composes the known zero branch. For every uniform a>0,QD_aP0=0. Exact nonlocality witness ||Q M_w phi0||²=1/3 shows P0 is not an automatic thin seam/collar projector.",
      "scopeEN": "Given pure D sea,correct b orientation and actual domains. P0 is normal zero branch,not native five. No full P1 locality/admissibility identification or equality with thermal C.",
      "status": "established_scoped",
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        "topic:clocks",
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      "titleEN": "Conformal original geometry fixes exterior return form and actual memory recursion",
      "statementEN": "deltaD[h]=-(hD0+D0h)/2 gives exactly ||Q deltaD psi_k||²=k²Var(h)+<h_prime²>/4 for every smooth real even normal profile,zero precisely for constant h. Full Fourier/zeta check yields4k²/45+1/3 for h=w². Actual bounded U=exp(i dt D_e)gives memory(P0 U Q)(Q U Q)^j(Q U P0),without resets. For the same collar-preserving probe,400 steps agree with full257-mode evolution to2.11e-13;129/257 curve difference<3e-10. Wrong adjoint feedback differs by.02084. Normal/-OO byte-identical.",
      "scopeEN": "Exact return form/block identity;long trajectories are finite Galerkin tests. Profile/epsilon are diagnostic inputs,not P1-derived. Second responses are forms,no unproved higher domain powers. No Fock implementer or3+1D/P2 closure claimed.",
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      "titleEN": "Geometric exterior Schur response transfers relative arc charge but is not a local native current",
      "statementEN": "Exact second geometric Schur response at z0 contains R_h(k)=k/8 sum_m h_m²(pi*m)²/[k²+(pi*m)²]. Between actual disjoint quarter preimages I=(1/32,3/32),J=I+1/2,every nonzero cosine coefficient contributes a strictly positive multiple of(1-i). Thus transfer is nonzero for every nonconstant smooth real even profile. Concrete probe gives.001720549637597(1-i)as epsilon² coefficient;independent Fourier error7.09e-12. R_h(k)->0 is compact and cannot itself be a nonzero local folded multiplication current.",
      "scopeEN": "Transfer is necessary structure for the existing native arc-charge test,not sufficient quartic/P1 identification. Second spectral Schur response is not new physical time or Hessian;full fields/composites,domains and profile/history selection remain needed.",
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      "titleEN": "Half-cover Cauchy graph realizes given Green covariance but not positive ground-state time",
      "statementEN": "Actual NS half collar0..1 has graph(exp(-H))projection[[1-C,sqrt(C(1-C))],[sqrt(C(1-C)),C]],C=(1+exp2H)^-1;other NS half is complement. Pure quasifree boundary extension restricts under j=(0,I)state/time faithfully to given source. Neutral A=a0* a_m nevertheless has frequency-m*pi/2 and norm1/[2(1+exp(m*pi))]>0. Thus even a nonsurjective embedding of this entire neutral observable class in a positive ground source with same time is impossible;m2 also preserves spatial J parity.",
      "scopeEN": "Given free Green quantization and actual x-time cuts,not P1 seam or old circle time. Cauchy trace space is a mathematical extension,not new physical species. No exclusion of other P1 states,admissible subalgebras or differently derived local time maps.",
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      "titleEN": "Full marked C2 lifts distinguish equatorial seam and OS Real structure",
      "statementEN": "The old base inversion x->1/x lifts to r=(1/x,i y/x²),r²=h,rQr^−1=hQ^−1. The full group has32 elements,16 holomorphic and16 antiholomorphic. The anti coset has12 involutions,4 each dividing,nondividing and empty,plus4 order-four maps. The old eight-map hQkappa stratum was complete only in its subgroup. New dividing S=rkappa and hS project to the pointwise equatorial Real structure and are Q paired;old hQkappa/hQ³kappa remain the correct midpoint OS lifts. v519 explicitly requires distinct Real slots. The unlinearized NS spin class remains invariant under the full lifted action.",
      "scopeEN": "Complete exact census for the already marked base D4 and its actual central deck extension. No selection of a physical boundary circle or identification of pointwise equatorial circle with an OS half-space. Previous canonical pair applies only in its explicit old subgroup stratum.",
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      "titleEN": "Full marked equator pullback is a ramified graph rather than a smooth seam circle",
      "statementEN": "For Gamma=|x|1,exactly p^−1(Gamma)=Fix(rkappa) union Fix(hrkappa). Two horizontal and two vertical square-torus ovals at1/4,3/4 intersect at the four ramification points. Full pullback is a connected torus graph V4,E8,F4,b1=5,with local valence4. The old single smooth NS circle has valence2,so a marked collar homeomorphism cannot identify it with this full pullback. Normalization gives4 circles,radial push-off2 components. Selection,smoothing or vertex domain does not follow from the bare lift. Retain both source/Real types.",
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      "titleEN": "Same elliptic Szego Green functional has full free RP and the Calderon marginal",
      "statementEN": "On actual NS NS E,S(t)=exp(−tB)(1+exp(−B))^−1 for0<t<1 is the Green operator of partial_t+B,B=−i partial_s,and equals the elliptic Szego kernel with prefactor i/(2pi). Pole,NS multipliers and H0(alpha)0 determine the identity. Both charged blocks factor for0<t,u<1/2:−S(−t−u)=X_t*X_u,S(t+u)=Y_t*Y_u. Fixed source-order/Clifford minus and anti-linear word reversal give RP of every free Gaussian CAR polynomial by Wick expansion. Time-slice covariance C=(1+exp(B))^−1 is precisely the labelled second Calderon marginal on the same annulus;the full graph is its canonical two-face purification. Q transports B to−B,C to1−C,u tou^−1 between cuts.",
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      "titleEN": "Elliptic scalar and spin pushforward has the original holomorphic P2 anchor",
      "statementEN": "Existing p_*O_E=O+O(−2) and p_*alpha=O(−1)+O(−1) give Vell=(p_*O_E/O)+p_*alpha=O(−2)+O(−1)^2. Thus rank3,degree−4,H0=0 and original splitting anchor a=(1,1,2) are exact. Both natural and P2 target residue triples have trace1 per mark,explaining the common degree. This relates actual holomorphic bundle isomorphism classes,not mere dimensions.",
      "scopeEN": "Bundle isomorphism class for the named tracefree scalar deck term plus whole untwisted spin pushforward. No original physical selection of this direct sum as flavor carrier or connection intertwiner. Keep previous normal spin-twist calculation distinct.",
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      "titleEN": "Natural elliptic anchor connection is not the original P2 cusp connection",
      "statementEN": "Natural induced connection on Vell has residues0,1/2,1/2 at each ramification mark,monodromy1,−1,−1,polynomial(lambda−1)(lambda+1)^2,order2,trace−1. Original P2 target has residues0,1/3,2/3,spectrum1,omega,omega²,polynomiallambda³−1,order3,trace0. Thus no local connection intertwiner exists. Nonzero scalar twists preserve the repeated eigenvalue and cannot repair it. Natural direct sum is also reducible,while stated original <U,M0> is irreducible. v499 already separates the Z2 pillowcase from the Z3 weight cover. Another connection on the same bundle remains additional data to derive.",
      "scopeEN": "Excludes only the natural scalar/spin direct connection as the actual original P2 connection,including scalar-twist repairs. Does not exclude other originally derived nonabelian connections on the common bundle. Separate exact v117 matrix properties from numerical ODE identification.",
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      "titleEN": "Existing elliptic Gaussian source determines full quarter current functional",
      "statementEN": "The actual labelled Calderon/Szego covariance gives C_r=(1+exp(4pi(r+alpha)))^-1 and rho=Z^-1 exp(-4pi Halpha),Halpha=N+Q²/2+(alpha-1/2)Q per quarter. The full current functional is omega(Wf)=Theta_alpha(f0)/Theta_alpha(0) exp[-1/2 sum_m m coth(2pi m)|fm|²],with exact original BCH phase for every finite ordered Weyl word. Theta retains all charges. The explicit normal representation is left action on the rho^(1/2) cyclic space;its subspace for currents alone,the full HS space for full CAR. Previously reconstructed neutral vertex products give the same Wick field correlations in this state. Finite fermion determinants and grouped1/2/4/8 pair comparisons independently check the analytic derivation;normal and-OO are byte-identical.",
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      "statementEN": "Exact telescoping of the actual Fermi covariance gives omega(J_-m J_m)=m/(exp(4pi m)-1),hence omega_Q0(J_-1 J_1)=0.00000348735451780811656...>0 versus0 in the old Hardy state. Q0 conditioning removes only the theta factor. Every nonzero real smooth mean-zero profile supported in a proper interval obeys omega_ell(Wf)/omega_Hardy(Wf)=exp[-sum_m m|fm|²/(exp(4pi m)-1)]<1. States thus differ already on every such local neutral current net under the unchanged original current dictionary.",
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      "titleEN": "Complete elliptic NS spin kernel splits globally into quarter pair",
      "statementEN": "Partition{1,−1}|{i,−i} is exactly the supplied Q fixed nontrivial spin-line class by divisor D_A and H0 test. With g⁴=(x²−1)/(x²+1),S_alpha=sqrt(dx dz)[g(x)/g(z)+g(z)/g(x)]/[2(x−z)] is the actual normalized Szego kernel. Its complete two-sheet matrix has constant unitary diagonalization to diag(g(x)/g(z),g(z)/g(x))/(x−z),without an undetermined elliptic remainder. dlogg=x dx/(x⁴−1) has alternating residues±1/4;four central spin meridians square to−I and multiply toI. Q exchanges both character lines. Pole,spin divisor,conjugate-sheet cancellation and H0(alpha)0 prove the global identity;theta3 numerical checks are only diagnostics.",
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      "titleEN": "Actual ramified resolutions select no single quarter-circle operator",
      "statementEN": "Supplied detours give T_out=I,T_in=P,without a free U2 choice. Common raw flat meridians(P,−P,P,−P) split into two characters exchanged by Q. Both radially pulled-back scalar operators have spectrumZ with multiplicity2. Literal ramified pullback is(1+z²)partial_z/(2z);smooth ambient continuity has only one isotropic trace dimension instead of maximal2. Moving-tangent spin phases±iP are not a circle boundary condition. No single deck character remains Q invariant. A closed real trace with C_base=T C_E T*,0<=C_base<=I still needs proof.",
      "scopeEN": "Excludes only automatic single smooth quarter domain from named raw flat connection and ambient continuity. No ban on an originally derived other real trace,joint boundary source or field map. Full holomorphic kernel in F1 remains established;earlier RP on a smooth annulus cut remains separately valid.",
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      "statementEN": "Standard theta squares give raw group96,central mu4 and projective PSL2(Z4)=S4. Spin labels alone see onlyS3. For UTheta=−i A_T,MTheta=i A_S A_T, X=[[-1−i,0,0],[0,i,−i],[0,1,1]]/sqrt2 satisfies exactly XUThetaX*=U_v117 and XMThetaX*=M0_EXACT. Same X intertwines both generators;normalized group has24elements. Geometric Q:z->iz at tau=i is modularS with projective order2 and eigenmultiplicity1+2,whereas U_v117=T has order4 and three distinct eigenvalues. Isotrivial v493 supplies no elliptic T path. Origin needs an actual level4 family/sewing transport with distinct spatial mark and derived phase section.",
      "scopeEN": "Exact joint representation equivalence after expressly named phases. No original P1/P2 origin proof or identification of spatial with moduli monodromy. New moduli path is an additional changed premise and specifically does not repair the fixed Q identification.",
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      "titleEN": "Archived D4 puncture formula fails; existing v117 repairs existence",
      "statementEN": "Literal TFPT45 archive section004 assigns D,PDP^-1,P²DP^-2 and their inverse product as fourth monodromy. Printed matrices have triple productI,so fourth monodromyI instead of cusp spectrum(1,omega,omega²). P³I does not implement a quarter turn. Unchanged v117 gives exact existence repair M_k=U^kM0_EXACTU^-k,k0..3 with productM_k=I and every required local class. This does not prove uniqueness from local spectra,det and abstractD4 alone. Active TFPT2 already states bare Z4 nonrigidity and global parabolic additional data. Two independent original-source reviews separate archive defect from current theory.",
      "scopeEN": "Concrete mathematical erratum of archived TFPT45/42 formula. Preserve originals. Exact v117 representation remains valid;analytic ODE match and later anchor/resonance premises retain their own status. No blanket P2 exclusion or silent paper edit.",
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      "titleEN": "Geometric square clock is not the modular v117 quarter generator",
      "statementEN": "On actual tau=i torus,Q:z->iz induces modularS on theta squares. It has projective order2 with eigenvalue ratio1:1:−1. U_v117 and its exactly conjugate normalizedT have projective order4 and three distinct eigenvalues. Thus no invertible linear or projective intertwiner sendsQ toU_v117. Positive joint dictionaryT->U andST->M remains valid.",
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      "titleEN": "Canonical conformal boundary trace preserves the existing elliptic Cauchy state",
      "statementEN": "The conformal Dirac law and boundary density give V_sigma=e^((d-1)sigma/2), which exactly cancels the harmonic-spinor factor:V_sigma Tr psi_sigma=Tr psi0. Thus V_sigma P_sigma V_sigma*=P0 for the orthogonal closed Cauchy-data projection. Both marked faces and regional supports are preserved. In common bulk L2 the required trace is T_sigma=e^-sigma_boundary/2 Tr for Dtilde=e^-sigma/2 D e^-sigma/2. Quarter/fold are transported afterwards. The existing labelled marginal and full free current source are identical for all compatible metrics on the same complex curve. An unweighted raw trace produces an exactly detected spurious covariance change.",
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      "statementEN": "After the derived boundary transport, each given conditional free CAR word on the same marked curve is independent of sigma. Its identical normalized source functional therefore factors out of any source-independent integrable total geometric weight of nonzero normalization, including determinants, reference, Jacobian and anomalies. The connected fermion four-point defect stays exactly zero, as do all higher Wick rules. No particular geometric measure is selected or discarded. Pure conformal geometry variance is therefore not evidence of an interacting boundary source. With other varying data, integrating out this factor may change their effective weight; their covariance requires a separate test.",
      "scopeEN": "Only the given conditional free quantization and correctly transported CAR observables; no Gaussian theorem for original S_prim/P1 measure and no assertion that connected current correlations vanish. Z_ref[0] cancels in Z_rel[J]/Z_rel[0], not automatically from geometric forces. The next decisive four-field test requires an original covariance-changing variation or higher fermion vertex.",
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      "titleEN": "Retained seam defect has a derived causal geometry response with contact",
      "statementEN": "On the same source R_h(s,t)=Tr(DeltaC_s(t)partial_sD_s) and exactly partial_sR_h|0=k<h²>+integral_0^t chi_hh. Full mode sum weights are(E±k)^3/(16E),chi(0)0,chi_prime(0)=-k<h_prime g_prime>;it converges for smooth profiles. At t1 the retained collar-preserving probe gives contact+.3861727422,causal part-.4054682615,total-.0192955193. Direct commutator error<3.6e-13,independent full-family derivative after extrapolation<3.3e-8. Omitting contact gives error.386. Actual400-step machine computes the same relative response;129/257-mode difference<5.4e-7.",
      "scopeEN": "Exact relative response form of the given source;no assertion of separately defined infinite vacuum quadratic operators or geometry-change Fock implementers. Long curves finitely tested. No identification with primitive action variation,selected history,native quartic words or physical time.",
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      "titleEN": "Same final Pin geometry retains history with an unchanged index factor",
      "statementEN": "For the two actual commonly evolved states P_t=U C0 U*,Q_t=P_t+Pi_t and exactly(I-zQ_t)(I-zP_t)^-1=I-zPi_t,determinant1-z,without requiringC_t-C0 inL1. After50 pulse steps and50 steps with exactly restored D0,the t1 relative readouts differ:no pulse-.4739486483,positive+.01-.4701253569,negative-.01-.4780397086. Recursion matches direct evolution to4.8e-14. Causal kernel predicts pulse derivative.3956885828,finite symmetric difference.3957175880. Complete historical sector weight remainsQ_full=e^-2alpha R(g,alpha);its geometry response at fixed alpha comes fromR and is not calculated by the index.",
      "scopeEN": "Exact transported pair identity on the full given source;pulse values from a finite129-mode run with chosen duration/amplitude. No cyclic cutoff shift as an index proof. Full P1 measure/reference Ward identification remains open;the relative CAR response is not substituted into delta logR without proof.",
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      "titleEN": "Full projection covariance and positivity force all CAR words",
      "statementEN": "For an arbitrary positive CAR state with full covariance C=P=P*=P²,GNS zero norms force a(f)Omega=0 onkerP and a*(f)Omega=0 onranP. CAR normal ordering uniquely determines all finite words and hence the full state as the pure Fock/Slater state omegaP. Quasifreeness,gauge invariance and zero anomalous pairs follow instead of being assumed. This is the charged application of Araki1970 Lemma4.3. A positive ensemble mean with projection covariance P forces that same covariance and state almost surely;complex weights are outside this ensemble argument.",
      "scopeEN": "Exact infinite CAR-state theorem under actual full projection covariance,not a new general CAR theorem and requiring no extra Gaussian premise. GNS uniqueness gives no implementability in an arbitrarily prechosen old Fock representation. A geometric Cauchy projection is not automatically presented as a physical covariance;P1 and its real joint field carrier remain to be identified.",
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      "titleEN": "Actual two-face projection reduces the free-source test to joint coherence",
      "statementEN": "The original closed annulus graph has P_A=[[I-C_A,X_A],[X_A,C_A]],C_A=(I+exp(2WB_Sigma))^-1,X_A=(2cosh(WB_Sigma))^-1 and X_A²=C_A(I-C_A). Its full kernel is a projection,a single face is not. An actual positive P1 CAR source with this joint kernel would therefore inherit all previous free current/Weyl/native-composite correlations under the old domain/subtraction dictionary without an extra Gaussian premise. The missing physical cross equation is omegaP1(aL*(f)aR(g))=<f,X_Ag>. Three positive source controls share the same face marginals;dephased and product states even share the full diagonal covariance but have four-point values0 andp(1-p). On an actual finite P reducing block,delta=Tr[(I-P)C]+Tr[P(I-C)] gives ||rho-rhoP||1<=2sqrt(delta) for arbitrary positive states. All256 ordered four-field words are independently checked;normal/-OO byte-identical.",
      "scopeEN": "Sufficient identification test for the existing chiral scalar two-face source;no physical purification freely selected from C_A,no complete Dirac vector source or new time. Nonprojection marginals still require higher-word comparison. Quantitative error only inside the tested reducing block;no global control from finite samples. Original P1 measure,joint closed real graph trace/domain and physical cross equation remain open. Prior four-field requirement is sharpened only for the full projection case.",
      "status": "established_scoped",
      "problemId": "problem:atlas-clock-defect-and-operational-reconstruction",
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      "titleEN": "Retained marked quarter compiler has no local geometric free-Wick correction",
      "statementEN": "Four marked branch points have one complex modulus lambda. The original cyclic Q acts by lambda to1/lambda;at lambda=-1 its tangent action is minus1,with no invariant modulus. The explicit curve family y²=x⁴+a x²−1 confirms this by F_a(ix)−F_a(x)=−2ax². Thus rho nonzero is excluded from the existing supplied P1\\mu4 parity compiler. Combined with the proved correct conformal boundary trace,fixed spin/connection and given free quantization,the whole normalized transported Cauchy/Wick source is unchanged by nearby compatible geometry changes. Source-independent averaging adds no connected fermion four-point correction. Existing clock-given square rigidity is reused;its primitive physical derivation is not claimed.",
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      "titleEN": "Changed elliptic modulus gives derived boundary response and conditional neutral four-field variance",
      "statementEN": "Under the expressly changed modulus-family premise,both real structures and free Moebius quotient persist,but Q exchanges rho and minus rho. Actual primitive periods give I(rho)=2K((2−rho)/4),r=I(−rho)/I(rho),W=r/2,tau=ir and r_prime(0)=2Gamma(3/4)²/Gamma(1/4)². Free boundary covariance is c_n,alpha=[1+exp(4pi r(n+alpha))]^-1,alpha1/4,3/4. Its first variation is fixed and trace class;the original global holomorphic spin kernel independently confirms a nonzero variation. For actual lowest opposite NS modes,any positive shared geometry law gives exactly kappa4=−Var(c). Diagnostic averaging at rho plus/minus.1 yields−8.133849897e−6 with positive normalized two-mode density. In the retained original fixed-Q compiler rho0 disables this mechanism.",
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      "titleEN": "Actual equatorial trace fixes orthogonal Hardy range",
      "statementEN": "Actual measure is |dz|=|dx|/(|c||y|). Closed half-form trace is L2 unitary to |dx|. The two bounded Cauchy idempotents gP_Hg^-1 and g^-1P_Hg, |g|²=|tan(theta)|^1/2, have exactly closed range H² by outer cyclic factors. Their orthogonal range projector is P_H⊕P_H. Integrable branch singularities do not remove H1 vertex coupling. Intrinsic boundary generator has variable speed 4K(-1)sqrt(|sin2theta|), failing constant base-winding Ward relation.",
      "scopeEN": "Actual full equatorial trace for supplied metric. No equality with old quarter state, positive annulus KMS state or physical time. Preserve smooth annulus results and original W8 Hardy comparison.",
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      "statementEN": "J_D4=1/2-(x⁴+x^-4)/4 generates C(x)^D4. For genuine unitary SU3 descent on(infinity,2,4), c=diag(1,i,-i), regular cubic M=a², four half-meridian branches have traces0,2,2omega,2omega². Trace2 is impossible; complex branches are reducible. Unique survivor is a³=1,p=(1/2,1/4,1/4), the v117 matrix orbit. For genuinely linearized stable parabolic object canonical MS descent follows from uniqueness of adapted metric. Treat concrete A0* connection separately.",
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      "titleEN": "Recovered September proof closes original A0* monodromy",
      "statementEN": "The28Sept synthesis already contains analytic identification of unchanged v117 ODE: t=z⁴ and diag(1,z,z³) give three Fuchsian points of orders4,3,2; sole infinity resonance coefficient is exactly zero. Alternative frame diag(1,z,z^-1) gives semisimple exponents -1/2,0,-1/2 without resonance. Image is faithful S4 rotation representation; positive t0 meridian is U^-1. Marked pair is exactly v117 up to simultaneous basis change and remaining diagonal basis-sign convention. Diagonal0,-i/2,+i/2 follows analytically. Original four meridians have A4 image; U extends to S4. No new ODE fit needed.",
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      "statementEN": "Initial natural packet O(p_a-q_a) is nontrivial unramified V4 two-torsion on C, but v_a=z u_a gives V4 field over w sphere ramified at all six Weierstrass points, genus3. Actual target orders are4,3,4 rather than4,3,2. Positive triangle O(p_a-p_(a+1)) gives2,3,4 with fixed target marks. Both are different marked connections. Comparing imported T and B_k matrices would have been circular; actual function field and ramification test decide.",
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      "statementEN": "Common cyclic fiber-product cover D:h¹²=(z²-1)/(z²+1) transports g=h³ and w=h⁴. Over C it is h⁴=w, totally ramified at two simple zeros and two simple poles of w. Its unique quadratic intermediate h² with square w is also ramified. Therefore D neither contains nor trivializes any of three nontrivial everywhere unramified actual v117 lines L_a. Geometric Q lift (z,h)->(iz,i/h) transports underlying curves but does not replace missing coefficients or a common field state.",
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      "statementEN": "For D:H¹²=f(z),w=H⁴,g=H³ and actual unramified V4 cover Yq/C,function fields are linearly disjoint over C. D/C is totally C4 ramified at four points and has no nontrivial unramified intermediate. X=Norm(D times_C Yq) has degree16 over C,degree48 over z,group C12 times_C3 A4 isomorphic A4 times C4,genus41. Minimality holds among all connected common covers:g and w determine H=w/g;trivializing three lines forces Yq. Common C3 quotient is identified once. Accepted quarter lifts combine diagonally with order4.",
      "scopeEN": "Minimal common holomorphic carrier of expressly given root coordinates and flat coefficient lines. No new physical worldsheet,CAR state or full compiler field map. Geometric clock here inverts deck generator;it is not separate commuting compiler order12 clock.",
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        "premise:actual-original-a0star-fuchsian-connection-and-c4-clock",
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      "titleEN": "Given elliptic connection family has explicit Quillen norm and both holonomies",
      "statementEN": "Actual free elliptic operator has L_z spectrum4pi²[(m+.5+v)²+(n+.5+u)²] and relative zeta determinant N(z)=exp(-2pi u²)|theta3(z|i)/theta3(0|i)|². Relative heat integration of the same spectrum independently checks the standard torus identity at three nonsingular60-digit probes. Positive theta line has A=2pi i u dv,degree+1 and holonomies exp(-2pi i u),exp(2pi i v),including endpoint descent. Null locusz(1+i)/2 is retained;no global nonvanishing section. At the original point Hess logN=-2piI,and the vectorlike doubled real action has HessGamma=2piI,the chiral modulus half. This supplies an explicit continuum object on this given carrier;physical P1 and v472 comparison remain open.",
      "scopeEN": "Known Quillen torus calculation on the given mathematical carrier,with actual relative heat and line response. No new general theory of nature. Spacetime curvatureF_A0;parameter curvature/holonomy susceptibility is neither local4Da4 nor SMb1. No P1 law,joint positive CAR cross kernel,physical time or Berry-to-continuum limit.",
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      "titleEN": "Bare large-gauge Quillen norm does not give the original counting factor",
      "statementEN": "Exactly N(z+1)=N(z+i)=N(z),so a large-gauge full spectral norm ratio is1. The retained old NS circle still gives det_F[(I-q uCu*)(I-qC)^-1]=1-q. Forq nonzero these differently inserted operator comparisons cannot be identified. Additionally:an ansatz using only this norm,fixed multiplicity,constant reference and smooth z(alpha)withz(0)0 has first derivative0,while W_min_prime(0)=-(5/2)delta_top/(1-delta_top) is nonzero. Alpha0 still hasqdelta_top and is not a seam-free reference. A derived different background or additional actual insertions is not excluded.",
      "scopeEN": "Excludes only the bare gauge-ratio identification or explicitly described smooth zero-twist ansatz. Original relative factor1-q and T(q)=(1-q)^-5/4 remain intact. Does not exclude physical EM response,other originally derived couplings or completeTFPT.",
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      "titleEN": "Existing elliptic half period has a canonical graded GNS intertwiner",
      "statementEN": "The already given actual NS Green kernel gives S(W)=−S(−W)=(2cosh(WB))^-1=X_A,also as both reflected charge Grams at t=u=W/2. In the existing thermal HS GNS,aR=L_a and aL=−Gamma R_a with aL*=Gamma R_a* give full double CAR and crossMij=Tr(sqrt(rho)ai*sqrt(rho)aj). For the same free rho,M=X_A,hence complete covarianceP_A. The previous projection theorem gives all words;the cyclic isometry to the given graph-state GNS is ontoHS. No new physical state or particle carrier is introduced. Bare Euclidean evaluations a(W),a(0) are not independent CAR families:their cross anticommutator is exp(−Wlambda),not0. Using onlyL_Pi instead of canonical HS parity produces false spectator factors,detected already with2modes.",
      "scopeEN": "Exact connection inside the already given free elliptic source and its faithful normal HS carrier. Its numerical cross kernel was not unknown;the previous cross gap still means its actual physical P1 operator/sewing identification. Geometric OSTheta is not presented as TomitaJ,nor the chiral scalar as a full Dirac vector. Domains/subtractions and previous state/time exclusions remain unchanged.",
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      "statementEN": "For arbitrary faithful even rho,the actual canonical cross coherencemj=Tr(sqrt(rho)aj*sqrt(rho)aj) obeys0<=mj<=sqrt(cj(1−cj)). Equality holds exactly when sqrt(rho)aj=qj aj sqrt(rho),qj=sqrt((1−cj)/cj). On full finite CAR,saturation for all modes in a covariance eigenbasis is equivalent to a quasifree product density. On the existing normal all-mode HS carrier,actual all-mode saturation givesMij=qjCij,then full projection covariance and quasifreeness without uncontrolled density inverses. The already given H_int control has both deficits14745600/82945²>0 and half-modular factors2/32 and8/128 depending on spectator occupation. Positive standard doubling and its quarter clock remain,but projection coherence fails. The original property still to be tested is therefore a linear half-modular action on the actual odd fields,not merely existence of positive doubling.",
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      "titleEN": "Chat comparison connects native Pauli quartic code with scoped algorithmic information",
      "statementEN": "The initial exact original-source calculation is now archived in the repository and replayed normally,optimized,and at its archived location with byte-identical results. Pi_native=Q_Pauli Pi_sym4 holds in the unchanged frame;ranks16,35,5 remain distinct. All64 one-register matrix-unit conditions give the existing erasure property. The60 original source rays have frame potentials1/4,1/10,1/20,1/32;the fourth Haar value is1/35 and HS moment defect3/1120. For the already known pair Pi_native/5 andPi_sym4/35,relative entropylog7 and global distance6/7 disappear in all proper marginals. The external standard-gate control compresses to diag(0,0,i,i,i),leaking2directions. The original comparison and known Clifford-commutant literature retain all premises and the physical selection boundary.",
      "scopeEN": "Exact finite source identities and reconstruction of known mathematical connections. No proof of universal computation,physical noise,clocks,couplings,Born rule,or simulation. Marginal information loss concerns this mixed-state pair,not all pure codewords. A fully Clifford-invariant Hermitian generator is scalar on5/30 only in that class. Literature links toHamming/E8,QCA,transfer convergence and sufficient statistics remain background theorems under their own premises. The historicalF_a->deltaD_rel question was later sharpened for merely defining composites by the existing current reconstruction.",
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      "titleEN": "All results of this TFPT chat consolidated with corrected assumptions",
      "statementEN": "The common reading map and manifest connect the initial algorithm comparison with all11 previously stored findings from this chat:relative arc charge,finite modular control,neutral current reconstruction,elliptic current functional and local Hardy distinction,conformal trace and averaging,projection state and two-face coherence,half period and graded GNS,and modular saturation. Previous referenced source versions were checked against their current files. Seven explanatory PNG images and available SVG/source assets are preserved byte-identically in the repository. The historical first report remains unchanged with the laterF_a->deltaD correction identified. Final Atlas verification uses the complete finding IDs in the manifest,not scientific promotion inferred from graph paths.",
      "scopeEN": "Documentation and completeness reconciliation of this chat,not a new physical derivation. Images are explanatory aids,not proofs. The Atlas undersemantic_explorer remains the sole current map;the overview is an entry into its sources. All original P1/carrier/reflection/time/interaction boundaries remain visible;no ledger or empirical status is changed.",
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      "titleEN": "Original inversion forces six coefficient lines",
      "statementEN": "Original inversion forces six coefficient lines. s:w->−w exchanges q and p packets;their minimal s-stable closure is sum of both triples. Vq and Vp are orthogonal symplectic planes spanning Jac(C)[2]. Joint unramified elementary two-cover has degree16 over C,genus17. Explicit normalized Kummer lifts of c,s give order6 T=sc and old block M plus(−M). Lift choice remains a recorded deck torsor.",
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      "statementEN": "Literal geometric quarter is not central Gaussian quarter. On Jac(C)[2],Q acts by diag(A,A),A=[[1,0],[1,1]],rank(Q−I)=2. J=i acts identically on L/(1+i)L. Hence no invertible intertwiner with Q->J exists. Excludes only this literal generator identification;source already identifies Q with rank3 U,not J.",
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      "statementEN": "Original cubic and quarter select spin line and half-turn lift. S0=O(q0+q1+q2−K_C) is even,S0²=K_C,and equivalent to p definition. c has no nonzero fixed vector on Jac(C)[2],so S0 is unique c-invariant theta characteristic among16. Q preserves S0;at marked fixed point q0,Q*eta=i eta. Any compatible spin lift alpha has alpha^4=−I,order8. beta=alpha composed Q*alpha is order4 lift of h=Q²,independent of alpha->−alpha,with local phase+i. Previously open S0,beta selection is closed by existing data.",
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      "titleEN": "Source cubic matches native Pauli translations after correcting context target",
      "statementEN": "Source cubic matches native Pauli translations after correcting context target. v783 distinguishes code labels L/(1+i)L from P_nat=N64/mu4;full Sp4 modules admit no linear intertwiner. On P_nat original sigma is fixed-point-free with polynomial(x²+x+1)²,exactly as c on Jac(C)[2]. Phi with columns IY,YZ,ZY,YI is explicit symplectic intertwiner. Among35 planes exactly5 are sigma invariant,exactly2 symplectic and orthogonal,selecting same unordered triple pair natively. Exactly18 c-equivariant symplectic markings remain. Earlier0-versus2 fixed-space conflict concerns only wrong code/context target.",
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      "titleEN": "Ordinary theta determinant functor erases selected spin character",
      "statementEN": "Ordinary theta determinant functor erases selected spin character. Every E_L=S0 tensor L has chi=1+1−2=0. Scalar lambda acts on det RΓ(E_L) by lambda^chi=1. Thus beta²=−I becomes beta_D²=+I,and alpha⁴=−I becomes alpha_D⁴=+I. Dualizing or squaring cannot restore lost phase. All four standard H0(2Theta) functions are even;inversion acts+I. Direct untwisted functor therefore does not give beta->J=iI.",
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      "titleEN": "Marked negative-point norm carries exactly missing central character",
      "statementEN": "Marked negative-point norm carries exactly missing central character. For Nq=tensor_a S0|q_a,beta on each fiber is+i,so beta_Nq=−i and beta_Nq^(-1)=+i. c acts trivially on Nq. Hence H0(2Theta) tensor Nq^(-1) has alpha²=+iI,alpha⁴=−I without fitted phase. Determinant of three fibers and norm line differ under Q only by transposition sign. p partner has conjugate−i character;s exchanges both. Using exactly one inverse point norm in actual field space remains candidate bridge.",
      "scopeEN": "Point norm and central character proved. Full family and complex module intertwiner are closed in following spin-rigidification findings. Original boundary selection of one inverse norm remains open. One complex four-space with scalar J does not also carry s exchange of conjugate characters.",
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      "titleEN": "With fixed q0 rigidification,direct point-norm correction does not preserve original cubic action",
      "statementEN": "With fixed q0 rigidification,direct point-norm correction does not preserve original cubic action. For E_L=S0 tensor L,evaluation line is Nq(S0) tensor P_delta,delta=Dq−3q0=q1−q2 nonzero in Vq. O(2Theta) tensor P_delta is a different marked line bundle. Since c has no nonzero fixed vector in Vq,c*delta differs from delta;no linear c lift exists on same theta line bundle. Imposing vanishing first on determinant line then performing usual squaring removes P_delta² but point phase becomes−1 rather than+i. Neither direct functor variant supplies required joint bridge.",
      "scopeEN": "Two explicitly defined q0-normalized variants. Following source-selected S0-relative normalization removes first family obstruction;boundary-first then square remains excluded. Not intrinsic obstruction of all Poincare families.",
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      "titleEN": "Source-selected spin rigidification closes whole point-norm family",
      "statementEN": "Source-selected spin rigidification closes whole point-norm family. O(Dq)=S0 tensor K=S0³. For any Poincare family P define P_spin=P tensor pi_A^*<P,S0>^(-1). Since deg(S0)=1,<P_spin,S0> is canonically trivial,so Nm_Dq(P_spin)=<P_spin,S0³>=O. Changing initial Poincare normalization cancels in P_spin. Scalars in S0 isomorphisms and O(Dq)≅S0³ vanish in pairing with degree0. Norm trivialization is c,Q covariant;constant Nq phase survives. Ordinary theta determinant is unchanged because chi=0.",
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      "titleEN": "Spin-rigidified theta space is identified as complete complex Heisenberg and clock module",
      "statementEN": "Spin-rigidified theta space is identified as complete complex Heisenberg and clock module. For L=O(2Theta),N={g in G(L):g⁴=1} is canonical order64 group with mu4 center. Every symplectic Phi lifts to N≅N64;Stone-von Neumann identifies Wq=H0(L) tensor Nq^(-1) with native C4 module. c lift has no remaining Pauli shift because1−Sigma invertible. alpha gamma alpha^(-1)=gamma^(-1) and trace(Ad gamma)=1 force trace(gamma)=1,hence exactly gamma->sigma without cube-root phase. Same intertwiner sends alpha to fully classified joint Clifford conjugacy class.",
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      "titleEN": "Complete origin investigation from E6/E7/E8 through the spin theta bridge consolidated in Atlas",
      "statementEN": "A common overview and full manifest link all197 existing findings in the two relevant research problems. All1185 source references of388 stored findings were checked against497 current files for existence,SHA256 and line extents,with no discrepancy. Three previously local reports on real D4 comparison,conditional cyclic reconstruction and NS half-cover are now preserved byte-identically with two checkers and an existing result file. Historical scopes and later corrections are explicitly mapped. Active task reuses the completed complex module bridge instead of requesting it again; prior wording is retained.",
      "scopeEN": "Coverage and source audit of available investigation documentation,not fresh independent validation of every proof.388 entries belong to the entire shared Atlas,not solely this conversation. Presentations and old save packets are not new scientific results. Original one-copy inverse point-norm selection,integral lattice origin and physical field/state/time map remain open. No ledger or physical proof status changes.",
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      "titleEN": "Eta phase alone does not identify needed spin/reference pair",
      "statementEN": "With lambda=detH0 tensor inverse detH1, positive theta determinant is inverse lambda; inverse Dai-Freed determinant is lambda in this convention. TFPT action factor exp(−i pi Delta eta/2) obeys a²=tau_rel^(-1) for vanishing kernels, not a=tau_rel. A Pfaffian application needs actual operator,real structure,kernel and orientation; generic gluing does not identify coefficients.",
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      "titleEN": "Same-spin shift gives native module class retaining odd defect line",
      "statementEN": "Delta(E,M_z) tensor inverse Delta(E,M_1)=B(E)^(-1)=Nq^(-1) tensor B_O^(-1), using equivariantly trivial spin-relative Poincare norm. B_O has c=beta=1,Q=−1, changing only allowed alpha sign after forgetting grading. Full complex N64/clock class needs no separate unspun physical reference. Superline stays odd, index−3 versus0; two shifted cables give even relative parity. Original index−1 and three-cover index−3 are both odd.",
      "scopeEN": "Equality of classified ungraded complex module class; not equality with fixed signed alpha or of super/Pfaffian sewing functors. Actual physical Cauchy space,action,state,reflection,local insertions remain to be identified. Two-cable ratio remains sufficient normalization, not necessary physical premise.",
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      "titleEN": "Original even/odd operator split is compatible with same-spin reference",
      "statementEN": "Original v31/v42 construct D_ref/D_rel from same D and spin carrier, diagonal/offdiagonal in tau blocks, not a change S0→O. This does not exclude seam connection: fullv21 interpolates unshifted and unit-shifted operators on same field space. Leading candidate is M_z versus M_1 on same S0 Hardy space. Source definitions alone do not supply equivariant map from actual positive Cauchy data to this three-sheet Hardy space.",
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      "titleEN": "Elliptic projector and Weyl transport retain source type at cubic spin bridge",
      "statementEN": "Original boundary, conformal-trace and projection-state sources concern E=C/(Z+iZ), NS NS spin, L=partial_t-i partial_s and the full orthogonal two-face graph P_A. Weyl transport fixes complex curve, spin, connection and cut; it supplies no map to genus-two(C,S0). Projection rigidity starts with actual positive full CAR covariance, not a geometric projector alone. Longitudinal old-circle/C2 generator and winding already match but imply no C3 time normalization. A local/reducing isometric source embedding on its cyclic image suffices; global onto identification is unnecessary.",
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      "titleEN": "Cubic spin Cauchy range has three Hardy channels; strong domain remains separate",
      "statementEN": "On three normalized disks the closed massless half-form Cauchy range is H²(D)^3. At cubic boundary points z-zj~u³ the coefficient is f~u^-1~|s|^-1/3 and |f|²ds~|s|^-2/3 is integrable. A flat outer factor gives dense compatible smooth boundary germs. Intrinsic H1 retains common vertex values, higher norms retain further jets; the constant derivative sends permitted singular ambient traces out ofL2. Equal L2 ranges therefore do not identify strong domains. Tangential B intertwiners alone do not fix global Calderon projections; bulk Dirac/harmonic data must also be transported. Moreover pi_*S0=O(-1)^3 follows from rank3,chi0,H0=0, so direct pushforward differs from old O(-2)+O(-1)^2 anchor.",
      "scopeEN": "Exact analytic result for explicitly conditional massless spin operator and named ambient core. No smooth Calderon theorem for arbitrary graph domains, freely inserted elementary modification, or physical P1 source/time closure. Scalar Hardy coordinates depend on half-form signs; norm and projection are intrinsic.",
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      "titleEN": "Original negative-sea three-mode ket is dual to required Hardy line",
      "statementEN": "Retaining source orientation C=I-PH gives UCU*=C+PiV and PH-U PH U*=PiV. Positivity and full projection determine shifted Xi=a*(v1)a*(v2)a*(v3)Omega and every CAR word without an extra Gaussian premise. Relative ket line detV has beta=-i; Hardy Fredholm line delta(Mz)=detV^-1 has beta=+i and is its dual vacuum line. Both are odd. Spin-relative family gives detV=Nq tensor B_O and delta=Nq^-1 tensor B_O^-1. All state expectations are phase blind and do not choose a directed boundary amplitude. Exact full8D exterior-algebra test checks every creation action,Q²=beta,beta²=(-1)^F,QcQ^-1=c^-1; normal and optimized outputs match bytewise.",
      "scopeEN": "Conditional full CAR connection on the given three-sheet half-form carrier. Actual P1 source and its dual boundary/Pfaffian amplitude remain to be derived. No switch to occupied positive Hardy polarization, particle-hole reversal to fit the phase, or inference from state expectations to charged transition phases.",
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      "statementEN": "The dual of detV already exists by CAR adjunction; for antilinear-first inner product xi-><Xi,xi> is the complex-linear covector. Retain old negative sea,beta signs and odd grading. Original closed eta/Pfaffian phase is scalar and does not select a directed boundary amplitude. Auxiliary C3 need not be a physical bulk source for the inner spin/theta module. Full P1/C3 Calderon identification is needed only for the stronger equality of those sources. The existing local-extension theorem supplies a shorter sufficient source route through actual marked (16,4) sector with the same conformal vector.",
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      "statementEN": "Affine phase recipe yields16+96+128=240 roots on60 complex rays. Integer basis B has detB=16 and even integral Gram B^tB/2 with determinant1; an explicit orthogonal map identifies the complete standard E8 root shell. Compensated Clifford generators preserve Lambda. Retained alpha=zeta8 V obeys alpha²=i and swaps Lambda with zeta8 Lambda,while beta=i and sigma preserve both and c=i sigma has order12. Lattices intersect only at0 and their sum in this space is nondiscrete. The extra coset gives a two-element orbit; an actual monodromy cover still needs its supplied representation. Bare theta without inverse norm is not the module used.",
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      "statementEN": "For every given geometric faithful c0 and every unitary isometry lift T:V0->V1 set beta1=T beta0 T^-1,sigma1=T sigma0^-1 T^-1,alpha01=T,alpha10=beta0 T^-1. Then alpha²=diag(beta0,beta1),alpha^8=1 and alpha sigma alpha^-1=sigma^-1 hold on all lattice charges and finite-energy vertex fields. Both objects carry faithful C12 clocks. Standard cocycle depends only on Lambda/2Lambda; all256 classes and65536 pairs exhaust cocycle checks,while matrix relations separately hold exactly over Z. Character-gauge changes of T give equivalent groupoids. Optional strict C4×S3 census yields256 raw sign sections,but none with the zero-character J lift.",
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      "titleEN": "Winding functional calculus preserves NS and Ramond and does not create a spinor field",
      "statementEN": "Actual W_ext acts as (0 v)(s c) and commutes with P_R, hence every bounded Borel f(W_ext), including every spectral square root, commutes with P_R. P_s f(W_ext) P_0=0; an s-charge intertwiner requires P_s X_s P_0 nonzero. Existing global E_s has the right sector action but is not a derived local P1 field. Half powers in the field transport refer to coordinate x. The ambiguous later summary sentence was corrected preserving source line counts. D8-relative locality of s is not membership in unchanged NS CAR algebra. Equation32 for the affine winding endpoint is not the same test as constructing M_h.",
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      "statementEN": "For all four isotropic supports, exactly S v_H=(|H|/4)v_Hperp. An S-stable chiral vacuum line forces |H|=4 and H=Hperp, hence E8plus or E8minus; fixed source h=(1,1) marking selects plus. S vacuum coefficients are1/4,1/2,1,1. Scalar paired control M=I16 is modular invariant with vacuum multiplicity1 but rank16. CAR purity, unique ordinary vacuum eigenspace or tau=i equality do not imply a single torus-block line. Within this class the test is equivalent to holomorphy and explicitly connects the earlier mu16->4->1 completion to the four-extension classification.",
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      "statementEN": "Old _micro_step overwrote T_net@m during family update; old passing test never called it. Integrated fix computes T_net@m@M.T and checks all27 basis inputs through actual function. Existing v486/v487 M replaces incomplete v327 rule:spec M={1,2/3,1/3};O^T M^6 O=diag(1,64/729,1/729). One hand is N C^6=T_net tensor M^6,not (NC)^6. Normal and -OO6/6 with identical output;independently reviewed. McKay Doob transform with existing marks d gives P_ij=(A_ij+2delta_ij)d_j/(4d_i),reversible pi_i=d_i²/120.",
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      "statementEN": "P is selfadjoint on H_pi with eigenvalues1,-2/3,1/3. Marked isometry W identifies |P| with M and sign(P) with K=Z Delta,Z=diag(1,-1,-1). Thus WPW^-1=KM and WP^(2n)W^-1=M^(2n);W_n=K^nW identifies each step in a moving frame. Bipartite double admits fixed six-dimensional intertwiner with transported deck tau tensor K. Simultaneous positive M and literal Delta identification is impossible on simple2/3 eigenspace. Y transports but [Y,P] has rank2:raw Frobenius square125,isometric moving-frame700/9. A physical charged total process needs actual compensating transfer.",
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      "titleEN": "Original eight-clock marks A7 and neutral E7 zero-root mode",
      "statementEN": "With rho_A7=(7,5,3,1,-1,-3,-5,-7)/2 and q=(1,...,1),exactly delta=-rho_A7/8+(2a-1)q/16 and H8=8L0-rho_A7.a0+(2a-1)Q/2. p*=(++++----)/2=omega4(A7) has norm²2,Q0,rho.p*=8 and is unique neutral E8 root of H8 grade0. The126 neutral roots form E7(q/2),exactly56 A7 roots plus70 Lambda^4C8 weights. All240 roots and reflection closure checked exactly.",
      "scopeEN": "New joint interpretation of existing clock shift and already known charge census. No rediscovery of old ground doublet or D8/E8 Coxeter grading claim. Q is source U1,not automatically electromagnetic charge. p* is an E7 root,not E7 singlet. No time generator changed.",
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      "titleEN": "Oriented sewing continuation and genuine reversal satisfy distinct exact compositions",
      "statementEN": "alpha01=T,genuine reversal T^-1,continued positive quarter alpha10=beta0 T^-1. Hence alpha10 alpha01=beta0 and alpha01 alpha10=beta1,whereas forward/reverse yields1. Normalized central cocycle omega(1,1)=beta describes nonsplit1->C4->C8->C2 over mu4,with correct adjoint and defect-line dualization. Positive local spin quarter lift squares to+i independently of lift sign,matching marked q norm factor. Literal unextended D4 reflection s²=1 cannot map to alpha²=beta.",
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      "titleEN": "Natural single-face compression of Calderon projector is not Perron projector",
      "statementEN": "For Pi=[[I-C,X],[X,C]],Pi is rank3 orthogonal projector on six-space. Natural single-face compression C has eigenvalues{1/2,4/13,1/10},hence rank3,not a projector and not rank1 Perron projector. I-C also has full rank. Seven rational checks establish counterexample to The reviewed proposed identification in natural finite comparison. Two actual The reviewed Opus4.8 consultations retained with corrections;The reviewed retracted five false time/state/grading claims of first round.",
      "scopeEN": "Exclusion of precisely specified single-face inclusion;not every source-derived invariant shell inclusion. Same formula2P-I or minus sign identifies neither spaces nor actions. The reviewed responses remain external proposals;no automatic proof acceptance or source-time change.",
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      "titleEN": "Hodge Born magnitudes and filtered Petersen jump block agree exactly",
      "statementEN": "Entrywise |C_ef|²=A_Pet[e,f]. For x=cos(sqrt5 t),U(t)=exp(-itC) has exact population values p_diag=(2+3x)²/25,p_edge=(1-x²)/5,p_nonedge=(1-x)²/25. Thus no full U(t) population is pure Petersen transition. At t*=(pi/2+k pi)/sqrt5,added edge filter yields existing subnormalized jump block A_Pet/5 with probability3/5. Reject is4/25 at input plus1/25 at each of six nonneighbors,not old stay block2I/5. Conditioning on change gives A_Pet/3 in short-time limit. C/sqrt3 is not Kraus;C/sqrt5 has rank6 projector effect rather than3I/5 and superoperator rank36 versus actual J20 rank40. Six actual J20 phase matrices gauge to -A_Pet,not C.",
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      "titleEN": "Five complement amplitudes give full Petersen quantum channel and SU5 response sectors",
      "statementEN": "C_m²=P_m of rank6,sum P_m=3I10. Phi=(1/3)sum Ad(C_m) is CPTP,unital,HS selfadjoint;population exactly A_Pet/3 without time or filter,Choi rank5 with eigenvalue2^5,superoperator rank100. Correct covariance connects10 and10bar. Actual24 is dGamma(sl5),Phi(dGamma A)=-(2/3)dGamma(A)^T. On End10=1+24+75,Phi²=Pi1+4/9 Pi24+1/9 Pi75,so Phi^6 spectrum1^1,(64/729)^24,(1/729)^75. Actual radial restriction on Petersen shells1,3,6 yields previously transported P. As specified channel,Phi has unique stationary density I10/10.",
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      "statementEN": "For each of6 marks,old20 jump events group into5 blocks of4 with Hodge effectP_m/3 and same basis movement. Exactly R_l=D_l C_m onP_m;relative to baselineD0,H_l=D_lD0* form projective K4 with3 characters each multiplicity2. Psi_m=(1/4)sum Ad(H_lD0) after Phi_m. New/old Choi ranks peroutcome1/3,total5/15;superoperator ranks100/40. All60 ordered intersecting-duad coherences permark distinguish channels. No common10-label diagonal gauge for all5 baselines:complete finite test0 solutions permark.",
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      "statementEN": "For rank3P,U(P)=-I onCC,+I onCW/WW. T_P=Phi o AdU(P) is full SU5 covariant family overGr(3,5) with dual output,G_P covariant,same population/Choi rank5,but P5 value4/15. S_P=AdbarU(P) o Phi o AdU(P) likewise natural family;in adapted real chart real symmetric Kraus,unital,HS selfadjoint,same5 effects,Petersen population,full spectrum,unique fixedI10/10,but P5 value7/15. Powers similar,not identical. U(P) violates exterior-square phase relation,not carrier torus gauge. Full family covariance plus all listed criteria therefore insufficient.",
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      "statementEN": "Actual v783 words yield C_p=-i(I⊗H)CNOT(S⊗I)(H⊗I),G_p=(Y⊗Z)CNOT(I⊗S)(S⊗I),C_p^5=G_p^4=I and G_p C_p G_p^-1=C_p². Complete Pauli16×mu8 search at fixed C_p yields exactly four mu4-equivalent G corrections. Explicit four-intertwiner T to v419 satisfies both generator equations and T†T=2(5I-J),with dimension-one intertwiner space. G_p has conformal symplectic multiplier i; existing mu8 lift zeta8^-1G_p is symplectic order8. Quarter character makes Sym²4 and Lambda²Perm5 isomorphic F20 modules. Merely imposing F20 leaves ten-dimensional commutant M2(C)+C+C,dimension6.",
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      "titleEN": "Two actual coherent events herald Hodge at optimal1/25",
      "statementEN": "In each fixed source-derived duad chart,the Hodge5 Kraus space intersects marked20 jump span in2D (15+5−18) and unmarked60 span in2D (55+5−58);hence no pPhi CP subchannel for p>0 after one step. Two-step products A=[vec(U2U1)] have rank100. Exactly V†(AA†)^-1V has three1/48 and two1/64 diagonal entries in these charts. CP order S/3600≥pVV†/3 gives exact pmax=1/25. Minimal coherent record contraction has three1 and two3/4 row-Gram eigenvalues and input-independent success map(1/25)Phi.",
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      "statementEN": "At fixed actual nativeF20 word pair and +i quarter orientation,the compatible alternating formOmega is unique up toscalar. W5=(Lambda²V/<Omega^-1>)⊗chi^-1 has positive metric,orthogonalq and compact real structureq conjugation. Canonical Spin5 map S→SOmega→L_S→L_Sq^-1 hasdet8,F†F=inducedSym²metric andHom_sp4 dimension1. It replaces merelyF20 U2×U1² ten-freedom by one phase line. OnW_R,K={x→*(w∧x)} is basis-independent5D,HS Gram6I andsumC†C=3I. ThusF defines a specific native Hodge candidate.",
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      "statementEN": "Actual symmetricPauliBellbasis P_b/2,inducedSym²metric and explicit compactrealframeR give exactly unitary U=Lambda²(R)†FN intertwining both actualF20 generators withchi^-1. NativeK_m=U†C_mU are fixed withoutduadlabels after fixingOmega. Their entire5D Kraus space already lies in actual60 one-record span ofrank55. Minimum solutionW1X1=V hasX1†X1=3/4I5. Exact CP optimum isp1=1/15. Actualtwo-record rank100 hasV†S2^-1V=I5/64 andp2=4/75. Both optimal contractions haveRowGramI5. One-step exclusion andp2=1/25 applied to different label/monomial target channels and must not be transferred toPhi_native. Source carriesK5 in frameeigenvalue8 space ofdimension45;frame alone does not select5.",
      "scopeEN": "Prior chart-dependent provenance gap is closed at markedOmega. One coherentrecord suffices;recursion is unnecessary for this target and requiredsecondsystemstep lowersoptimum. Mathematically specifiedCP subchannel is not an alreadyexecuted sourcemeasurement. PhysicalF20/Omega marking,recordaccess andcoupling remain originconditions. No completeTFPT solution.",
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      "statementEN": "X1†X1=3/4I5 determines orthogonalcoefficientprojectorPi_coeff=(4/3)X1X1†. In standardStinespringconvention sum_l|l>⊗U_l/sqrt60 actualrecordeffectisE=Pi_coeff^T sinceJ_E=W1E^TW1†/60;successbranchPhi_native/15. Same-dimension5DkernelW1 isorthogonaltoimageX1 andgiveszerosystemamplitude. OriginalPprim/Calderon/Qprim/Pfaffian actonotherimages. ExistingRECORDS.md givesconditionalcoherentrotation andexplicitlyassumesrecordcarrier/readoutbasis;stablerecordalgebra/Bornrule donotselectE.",
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      "statementEN": "ActualnativeC5,G versus frozenv634sigma givecommutatorrank4 andbothranksGsigmaG^-1−sigma^±1 equal4. InanyirreducibleF20four,C5 hasfourdistincteigenvalues cyclicallypermutedbyG. Given[C5,sigma]=0 andorder3,G-fixingpermitsonlyscalartraces4,4omega,4omega²;G-inversiononlytraces4 or−2. Nativesigma hastrace1,hencebothimpossible. Originalcyclotomicplusrelationalreadyexists onrationalrank8Q(zeta30),withfamilymultiplicationtrace−4. Thisisnotnativecomplexsigmatrace1. Thetaalpha inversionisyetanotherclocktype.",
      "scopeEN": "Smallestnecessaryjointclocktestbeforephysicallyequatingseparatelyvalidfour-bridges. NoexclusionofnativeHodge subchannel,separateE8/thetaresults orlargersourcerepresentations. Oldrank8modulefixesarithmeticcommutationbutisnotphysicalsigma realizationwithoutdictionary.",
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      "titleEN": "Six ovoid fibers carry the clocks without a common fixed form",
      "statementEN": "The six alternating Pauli forms define six actual Clifford fives. C,G fix IY and induce F20; sigma acts in two 3-cycles and alpha inverts sigma with alpha²=iI. C,G,sigma generate transitive PGL(2,5)≅S5 of order120, certified by explicit conjugacy to its projective-line action; adding alpha gives S6 of order720. Full five-current fibers, not only labels, transport covariantly. J is projectively trivial on fibers.",
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      "statementEN": "In given A3_1=SO6_1 current net marked Omega selects Spin5=SO5. Five complement currents i:chi_a chi6: match actual Gamma/2 zero modes. Central charges3,5/2,1/2 give local even Ising coset; chi6 belongs to fermionic extension. Original P1 Clifford multiplication gamma_n on Cauchy data is not an internal local Majorana field. Read originals supply no state/OPE-preserving map between these types, and Hermiticity alone cannot select5 of15 currents.",
      "scopeEN": "Focused existing-current-net bridge, not a new auxiliary Majorana theory. Need original state-preserving normal/tangential field intertwiner including Omega; even that would not remove the separate physical P2 charge mismatch.",
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      "titleEN": "Neutral native A3 Hodge does not identify the physical P2 carrier",
      "statementEN": "Old Bell corner is (10_D5vector,10_A3descendant), weight2 and mu4 class(2,2). P2 Lambda²5 lies in D5 half-spinor paired with A3 fundamental, weight1, classes(1,3)/(3,1). Original Y spectra {±1/3[3],±1/2[2]} and {-2/3[3],1/6[6],1[1]} are disjoint, giving Hom_Y=0. A3 zero modes preserve D5 Y, center and L0, so cannot connect these types. Positive charged A4×A4 bracket remains valid as a different cut of the same E8 algebra.",
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      "titleEN": "Compact OPE coevaluation normalizes the given charged Hodge line",
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      "titleEN": "Old Ward root generates the exact charged fifty-module",
      "statementEN": "Existing conditional Ward root lambda_A=e3+e6 belongs to actual A4×A4 insertion(5,10), with carrier weight(-2,-2,8,-2,-2) and family Dynkin(0,1,0,0). Closure under40 neutral roots reaches all50 insertion roots through500 directed edges. Old quarter grades split30 grade2 and20 grade3; P2 X values are-2[30],+3[20]. A single bracket has rank12 and is not TP. Thus given Ward dictionary determines the algebraic module more sharply than previously stated.",
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      "statementEN": "Every CPTP joint map End(H10⊗F5bar)→End(H10bar⊗F5) with all Kraus operators in actual span{C_m⊗A_ab} obeys Tr_Fout E=Phi∘Tr_Fin. For any family density, reduced carrier map is CPTP with Kraus space inC5; existing injective25-coefficient TP theorem forces Phi without SU5 covariance. Product operators span the full input algebra, extending equality to entangled inputs. Arbitrary coherent insertion metric and unknown family state cannot change this marginal if TP and support hold.",
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      "statementEN": "Original D5+A3 target g2=(10,6) splits into Creation30=(5,6) of U5 residue3 and Annihilation30 of residue2. Neutral SU5_C times D3_F orbit of stipulated lambda_A=e3+e6 is exactly Creation30. Full E8 residue3 instead has50=30g2+20g3; joint internal quarter grade2 and U5 residue3 selects30. Actual Chevalley amplitudes Lambda2 C5 tensor4 to Lambda3 C5 tensorbar4 have360 signed columns and gain9I40 on both sides. With pinned joint CAR phases, normalization1/9,Hodge and family trace, full carrier Hodge channel follows for arbitrary inputs. Color18 and weak12 separately have nonscalar effects; marked union has scalar gain.",
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      "statementEN": "Previously verified full60 current process has canonical CP outputs E+ from carrier degree2 to3 and E- to1. Creationgain3 times familygain3 and annihilationgain2 times3 with totalgain15 give E+*(I)=3I40/5 and E-*(I)=2I40/5. Thus Q+=(5/3)E+ is CPTP and equals direct30 normalization1/9. Tr_F Q+=Phi Tr_F holds for entangled inputs. More generally L_r=sum_m C_m tensorF_rm with effect pI and independence of25 C_m†C_n forces operator-valued sum_r F_rm†F_rn=(p/3)delta_mn I, hence Tr_F E=pPhi Tr_F without uniform weights.",
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      "statementEN": "On given even quotient CAR net g_I=exp(i a_I/2),with exterior values a_I=0,2pi,defines localized automorphism. Branch signs are trivial parity on even words. Fixed q chart labels s=q/2,Q=4,h=1; odd pseudolocality alone does not distinguish s/c. For two localizations b_IJ=(a_I-a_J)/2 is smooth periodic and existing W(-b_IJ oF) implements transport. Exact fold gives J_tot,n=J_cover,8n,level8 and8(1/2)^2=2=s². Original CCR,adjoint and Hall/Wick functionals determine smooth transport histories. K_alpha time gives exp(-itn); unscaled cover time exp(-i8tn). Retain Q zero modes and charged-field action.",
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      "statementEN": "Pure observable response can be written T(a)=sum L_beta†aL_beta+sum L_gamma†rho_s(a)L_gamma for given localized star automorphism and bounded local observable coefficients. Unitality fixes normalization; each term is CP. Individual charged spinor need not belong toA. Full local E8/Q-system extension is stronger realization for charged individual fields and0<->s field coherences. Spinor endomorphism and smooth transport now constructed on given quotient net are not another missing choice there.",
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      "titleEN": "Actual source-sector transport distinguishes identical Hodge marginals",
      "statementEN": "Same FoldM splits Ward input40/output40 asNS20+spinor20 andcreation30 asNS14+spinor16. All360 signed columns are168 sector-preserving and192 sector-changing. UniformNS14 has effects3,4,5 and no scalar TP normalization. Positive weights1/6 on8 actualNS roots and1/3 on6 give exactTP onH40 and family gainI4/3 for everyC_m,hence identical fullPhi marginal even for entanglement. This different process preserves source sector. Physical color e3-e2 takesNS e2+e6 tospinor e3+e6;NS14 support lacks full color covariance. Source gradingw=Me1=(e1+e2+e6+e7)/2 factorsPC tensorPF withcarrier dimensions4+6;pureNS20 inputs have only52 of100 carrier matrix directions.",
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      "titleEN": "Ordinary Weyl type excludes single-chiral OS anti-anti reflection formula",
      "statementEN": "For Theta(W(f))=W(Lf), scalar sign c and product-order sign o, the ordinary Weyl relation forces sigma(Lf,Lg)=c o sigma(f,g). A conjugate-linear order-reversing map requires symplectic L. The previously conditional Theta(Wf)=W(-r f) is antisymplectic on the same single chiral current and is excluded under these type premises. A compactly supported local witness g=f-prime has integral f g-prime=-integral(f-prime)^2 nonzero. A common field minus or Weyl characters do not fix the central phase. Conjugate-linear multiplicative and linear order-reversing reconstructed types pass the single-current test. Oppositely oriented faces with swap and r pass the anti/anti algebraic test conditionally.64 declared variants are checked exactly by matrix forms and directly multiplied rationally scaled Weyl words, with independent normal/-OO review.",
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      "titleEN": "Given two-face source yields opposite neutral Weyls and bounded time reflection",
      "statementEN": "The old Schottky reflection rho(t,s)=(-t,s) does not reverse spatial profiles; its neutral Weyl profile map is -I and is not excluded by the previous -r_s result. In the existing HS source space, W_R(f)=L_Wf and W_L(f)=R_Wf* have opposite Weyl phases directly from right multiplication. J_HS W_R J_HS=W_L is conjugate-linear multiplicative; centered finite bilinears obey j_L(A)=-R_j(A) and J n_R J=I-n_L. The joint functional Tr(sqrt(rho)Wi*sqrt(rho)Wj) is the positive HS Gram of rho^1/4 Wi rho^1/4 without a Gaussian premise. For the given thermal source, fully ordered heat products K_F handle bounded even insertions: ||K_F||2<=sqrt(Z_beta)product||Aj|| and S(ThetaF,G)=Tr K_F* K_G/Z_beta. Strong approximation extends the existing polynomial free functional to existing bounded neutral Weyls. The old Green midpoint X and all four graph blocks are reused, not chosen anew. Exact NS modes pi,3pi test the same original implementation path, including null relations, particle-hole contribution and reflected noncommuting histories.",
      "scopeEN": "Analytic all-mode theorem in the explicitly existing normal source representation, supplemented by exact finite path controls, not an all-mode inference from finite samples. W(f) must already be bounded in the source von Neumann algebra; no norm CAR membership, bounded imaginary-time conjugates or arbitrary odd OS words asserted. The GNS partner is not identified with a physical P1 second boundary. Older papers32 and45 assume the positive field kernel or its reflection positivity. Actual local physical P1 source map, original measure/coupling, physical time, EM and complete TFPT remain open. The old spatial type exclusion retains its stated scope.",
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      "titleEN": "Positive spectral weight alone does not establish original-source reflection positivity",
      "statementEN": "The generic RP preservation inference in the old constructive-measure proof is false. A strictly positive RP base with eigenvalues19,1 becomes non-RP under w=exp(-Tr exp(-D²)),despite positive weights,reflection covariance and gap>=1. Exact Delta=e^-1(1-e^-4)²>75/256>1/9>log(10/9);normalized reflected sign response changes from1/19 to-0.12394264612509166625. General nonconstant positive Laplace profiles haveDelta>0;generic failure uses r in(exp(-Delta),1),while fixed9/10 base requires the stated threshold. Finite W>=0 is the universal Schur-multiplier preservation criterion;for a fixed base K0 elementwise-times W>=0 is exact. A suitable reflected factorization is sufficient for repair. Source-independent positive reference denominator cancels in normalized responses and repairs neither false RP nor missing cross coherence.",
      "scopeEN": "Counterproof only to the generic positive-weight preservation inference in cited original passages,not the entire geometric/BRST measure theorem under all additional conditions. Finite test is no full P1/P2 countermodel,no TFPT prediction,and no newly chosen physical source. Source Bernstein cone means completely monotone profile,not a general Bernstein function. No actual original field map or completeT1-T8 goal closed. Preserve existing given-source paths,positive transfer constructionv330,Schur/projector/Weyl results. Do not introduce full ambient geometric measure construction as a prerequisite for every neutral local response;direct local OS embedding remains possible. P1 already assumesRP;only a proposed derivation shortcut is excluded.",
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      "statementEN": "Using this same gapped quarter energy as a modularly scaled translation fails already at the full spectral gap: the group law and Stone uniqueness imply unitary equivalence of H and H/2. The least positive spectral point would have to be both1/4 and1/8. Lean grind proves the elementary spectral-set lemma with only the three standard axioms. Independent review confirms the analytic connection and scope. The source remains a conditionally supplied model; neither original-P1 source derivation nor its universal refutation follows.",
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      "statementEN": "Under the explicit sufficient eight-current contract, positivity of actual H forces all positive modes to annihilate the vacuum. CCR and adjoints yield every ordered moment recursively and the full PBW Gram, hence a unitary intertwiner on exactly the cyclic current space with the energy closure. Fock/Gaussian and complete CAR/E8 embedding are not additional prerequisites. Original quarter R=exp(-i pi H/2) is identity on this H=8N cycle; the larger c1 Moebius cycle is distinct. In the actual KMS case the denominator1-exp(-8 beta n) recursion determines oscillator moments, not the joint zero-mode law, delta or second-face representation. Truly compatible reflection yields all reflected word values, while separate face moments do not replace it. Actual CAR controls3897words,9216CCR vectors,464energy transitions in both quarters and189 exact infinite Gibbs trace comparisons agree; the all-mode conclusion comes from the analytic proof.",
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      "id": "finding:actual-charge-j-transport-retains-full-dirac-and-distinct-pairings",
      "titleEN": "Actual charge J and entire Dirac pairing in geometric chart",
      "statementEN": "In the existing chart,Jhat=I tensor J_F is new-complex-linear and T^-1 Jhat T(v,bar w)=(-k w,overline(k v)),with k=-gamma_F J_F,k²=-I. Jloc=S Jhat is new-antilinear and induces c=-i_old J_F,c²=I. Fully,c h_A c^-1=h_{-A^J},difference -(a+a^J)gamma_F,and Jloc D_A Jloc^-1=-D_{-A^J}. For a^J=-a,[c,h_A]=0 and an actual old full-space skew form <Jloc x,D_A y> result; its restriction to E times E vanishes. The reduced boundary pairing <c v,h_A w> is symmetric and is a different form. Full D is new-complex-antilinear; its pullback retains normal derivative,A and Phi. The old finite mass pairing <J_F v,Phi w> is recovered exactly. For scalar holonomy,c sends alpha to -alpha; explicit W=exp(-i theta) gives 1-alpha and index -k-1.",
      "scopeEN": "Analytic full operator transport for general complex Yukawas and complex symmetric Majorana block,on the smooth interior core; global selfadjoint J-stable domains need separate checks. No [Q,Phi]=0 is needed for this J/B test,and no electric-charge selection follows. Neither a universal Pfaffian exclusion nor a physical chiral Pfaffian rule follows. C0 is not J_F. No new physical halving,species count,bosonic quantization or OS time. Original audit preserves the already complete chosen QWZ CAR source and complete operational Seed; no extra free quantization demand for these given sources.",
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      "titleEN": "Fixed graded chart transports normal-dependent source and given mixed moments",
      "statementEN": "For B(r)=Z tensor A(r)-S tensor Phi(r),with fixed even connection,odd selfadjoint coefficient and fixed normals/grades,E remains background-independent and B(r)E=E h(r). In a compatible finite regulator,K'= -B(r)K gives exactly C0 K C0=(K*)^-1,without commutation at different r. Thus Cfaces sends the entire graph to its orthogonal complement and the oriented two-face chart gives P_K -> P_Kplus directsum(I-bar P_Kplus),including all crossing blocks. The same fixed algebra map transports all existing integrable joint bosonic insertions and conditional CAR words under an already given common measure. An exact smooth ordering countercase disproves the wrong unadjointed K^-1 formula.",
      "scopeEN": "This replaces the prior normal-independence restriction by a broader explicitly fixed product type without redefining the old premise. No arbitrary unbounded Cauchy problem: continuum claims concern actual elliptic Cauchy data,Green form and compatible domains. No moving grading/normal without connection terms. Positive conditional-state integration generally gives a mixed,nonquasifree state; no Wick rule for averaged covariance or projector purity is required. A complex eta/Pfaffian density gives only the integrable algebraic identity. No derivation of measure,physical quantization,original reflection positivity or physical time.",
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      "statementEN": "Actual neutral even pairs b*_{-r}d*_{r+1}=a*_{-r}a_{r+1} have energy2r+1. Each subset of the first m disjoint pairs yields a distinct orthonormal Q=0 state of energy at most m². Thus N_H,Q0(m²)>=2^m and Tr_Q0(1+H)^(-p)>=2^m/(1+m²)^p diverges for every finite p>0. This entire many-body time generator therefore cannot be time-intertwining unitarily identified with |D| in the tested finite p-summable geometric type,even after neutral/even selection or positive affine energy rescaling.",
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      "titleEN": "Product reflection and same time support the entire positive Clifford source",
      "statementEN": "For the existing normal-independent B=Z A-S Phi,F=S tensor gamma_F is the already present product-reflection factor. With Cfaces=(C0,-C0) and R=faceswap tensor F,Ical F=-C0,R G R=G,and the exact real crossing kernel is -i M with positive M. Actual face-Wick ordering is an invertible basis; its Gram blocks are (-1)^{n(n-1)/2}(-i)^n Lambda^n M. Every even n has factor+1,so RP holds on the ENTIRE bounded even CAR algebra. The natural full creation map iota c=a_R*(v0)-a_L*(v1)+a_R(w0)-a_L(w1),with annihilators antilinear in w,has exactly G(h)=[[1-C,X],[X,C]] at beta=2ell. In the corresponding normal thermal HS representation,J_R=(P_even+iP_odd)J_HS implements both geometric field phases. The same entire chart obeys iota(c(exp(t Ical B)u))=U_t iota(c(u))U_t* with THE SAME t,including cyclic GNS closure. No spectral conjugation or inverse time is needed. The even reflection form is ||rho^(1/4) A rho^(1/4)||_HS². Source H=sum |lambda_j| n_exc is positive with finite thermal heat traces; full HS time is the two-sided Liouvillian L_H-R_H. The normalized sech midpoint kernel does not obstruct positive heat evolution at finite ell.",
      "scopeEN": "The common mathematical field/covariance/reflection/time diagram is closed within the already assumed canonical Clifford graph type. R acts antilinearly and MULTIPLICATIVELY on CAR; an antihomomorphism is a different type and already fails an even occupation word. Odd words require fixed parity twist; untwisted odd RP is not claimed. Normal HS representation needs the stated free thermal class; separate Wick proof needs no trace-class presentation. rho(h_Phi) is not identified with old scalar NS/QWZ density;already h0=A gamma_F is not A tensor I. The inverse-time gap from annihilator labelling is repaired in the preferred creation chart and is not an open problem of this graph type. Original P1 selection,physical insertions/references/domains,quantized bosons and old-source identification remain separate. No RP for arbitrary normal-dependent bosonic families from the constant proof,no 4D/species-count/physical total closure.",
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      "statementEN": "On the full real gauge lift,det(pi(u))=(lambda det(m))^4 per generation. On the connected unimodular branch,gamma_F=+ is exactly Q_L,L_L,u_R^c,d_R^c,e_R^c,nu_R^c with original SM charges. An explicit unitary map to 1,c_c wedge w_a,epsilon_cjk c_j wedge c_k,w1 wedge w2,epsilon_cjk c_j wedge c_k wedge w1 wedge w2,vol_c wedge w_a intertwines the ENTIRE S(U3xU2) action with Lambda_even(E3+E2),including global Z6 form and opposite action. Three generations use W tensor I3. With an additionally supplied sigma=(123),eigenvalue1,the unique SM-neutral fixed line is e=nu_R^c tensor s,s=(1,1,1)/sqrt3. Actual coefficient gives Phi e=nu_R(MR s)+nu_L^c(barYnu s); preservation iff MR s=barYnu s=0. Reuse existing level4 result concretely for actual Y: tr16Y²=10/3 versus D5_1 basic norm5/6,three-family ratio12. Rescaling to match levels changes Ward charges.",
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      "titleEN": "Entire old quarter source embeds locally precisely on a decoupled channel",
      "statementEN": "For fixed local V_e f=f tensor e in the given Clifford graph,Phi e0 gives full H1/Borel/CAR/graph/product-reflection/time intertwining with the SAME QUARTER time. Actual original maps obey B_Sigma W_alpha=4pi W_alpha A_alpha and B_cov W_alpha=2 W_alpha A_alpha; W1/2 gives the entire graph at ell2pi,beta4pi without fit. Unchanged eightfold gives8 directsum A_mu,mu_a=(a+alpha)/8,beta_fold32pi and already existing full theta charge law. Delta_a=mu_a-1/2 is transported old reference correction. New exact result for all constant oddPhi: for p=k+alpha nonzero,n_e(p)=1/2[1-p<e,tanh(beta sqrt(p²+Phi²)/2)/sqrt(p²+Phi²)e>]. Strict monotonicity of tanh(cx)/x proves n_e(p)>=n_beta(p) for p>0,reversed for p<0,equality IFF Phi e0. A single occupation is thus equivalent to the entire fixed scalar source/time map. Positive classical averaging reproduces it only for Phi e0 almost surely.",
      "scopeEN": "Conditional actual embedding into supplied graph quantization,not original P1 selection. Reuse earlier theta law; neutral KMS alone remains weaker. Delta is not thermal mean charge. CAR direct decomposition gives graded tensor-product state,not direct sum of densities. Entire NS-cover identity does NOT follow from Spec(-A_alpha)=Z+1-alpha:its map reverses support;original u_cov=[[0,u_base],[I,0]] exchanges quarters whereas u_base I_F preserves grades. Already [B_cov,u_cov]=u_cov versus[2h0,u_baseI]=2u_baseg. Abstractly transported winding is not thereby the actual original local multiplier/deck operation. Iff and averaging theorem assume constant Phi,fixed e,p,beta;not complex eta/Pfaffian measures,variable profiles or quantum bosons. Full interacting TFPT is NOT required to preserve old free source exactly. No complete physics,EM or4D derivation.",
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      "id": "finding:marked-sm-phase-p5-exact-range-and-response-collision",
      "titleEN": "Complete P5 range for marked SM phase family and witness collision",
      "statementEN": "For the previously defined SM covariant family Phi o Ad(aP_CC+bP_CW+cP_WW) with unit phases, Tr(P5J)=|a+4b+c|²/6+|a+b|² and covers exactly [2/3,10]. This is not the global range of all SM covariant channels. The choice a=-1,b=1,c=(1+2i sqrt2)/3 has the same P5 value 2 as the old Psi channel, but P45 values 16/3 versus 4. Both preserve original populations; one Choi response does not identify the entire process under these conditions.",
      "scopeEN": "Only the stated internal phase family and particular comparison with the given old channel. Not P1 sewing, complete SM channel classification, original measurement, or a universal process.",
      "status": "established_scoped",
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      "id": "finding:shared-epsilon-tp-process-fixes-hodge-marginal-even-for-correlated-inputs",
      "titleEN": "Joint trace-preserving epsilon process fixes Hodge marginal even for correlated inputs",
      "statementEN": "For every joint finite-dimensional CPTP operation with Kraus operators L_s=sum_m C_m tensor A_sm on the given H=Lambda²C5 and arbitrary finite family spaces, global trace preservation and the 25 linearly independent C_m* C_n force sum_s A_sm* A_sn=delta_mn I/3. Thus Tr_F' E(rho_HF)=Phi(Tr_F rho_HF) for all joint states, including correlated and entangled ones. No preselected family state or pre-imposed isotropic Gram is needed. For one trace-decreasing event this fails: two exactly constructed joint states with the same H marginal produce probabilities 1 and 0. This is a corollary of the already recorded 25-product selection theorem.",
      "scopeEN": "Finite joint process on the given epsilon carrier assuming every Kraus operator in the stated tensor amplitude span and global trace preservation. Does not derive the original P1 source, carrier, local embedding, full trace preservation, event register, general multitime process or full TFPT physics.",
      "status": "established_scoped",
      "problemId": "problem:atlas-clock-defect-and-operational-reconstruction",
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  "problems": [
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      "id": "problem:future-quotient-and-gns-reconstruction",
      "titleEN": "Future quotient and positive GNS reconstruction",
      "questionEN": "What is determined by complete future behavior, and what additional positive word-kernel source is needed to reconstruct a cyclic operator realization?",
      "topicIds": [
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        "topic:quantum",
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        "finding:positive-word-kernel-reconstructs-cyclic-gns"
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          "titleEN": "Future quotient from complete r",
          "status": "resolved",
          "resolutionIds": [
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          "titleEN": "Conditional GNS reconstruction from supplied K",
          "status": "resolved",
          "resolutionIds": [
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          "requiredPremiseIds": [
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          "titleEN": "Select complete K from the original source",
          "status": "open",
          "resolutionIds": [],
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          ]
        }
      ]
    },
    {
      "id": "problem:common-source-state-event-selection",
      "titleEN": "Historical: common source, state and event selection",
      "questionEN": "Outside the active mathematical research program since the user's23Oct2026 decision. Historical question: Do the original principles derive a common physical source together with its state, dynamical events, and physically observed fields?",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:clocks"
      ],
      "goalIds": [
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        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:quarter-phase-retained",
        "premise:typed-source-structures",
        "premise:original-marked-simplex-positive-polar-rule",
        "premise:actual-clifford-identification-and-finite-overgroup-theorem",
        "premise:original-erasure-decoder-and-same-native-quarter-polynomial",
        "premise:unitary-full-branchwise-erasure-clock-lift",
        "premise:native-marked-simplex-clock-compatibility-without-fixed-polar-mark",
        "premise:original-native-fourth-moment-parent",
        "premise:local-product-code-orbit-geometric-parallel-transport",
        "premise:local-e8-current-response-realization",
        "premise:native-g31-concrete-event-representation",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:existing-logical-clock-and-charge-marking",
        "premise:original-coupled-determinant-event-cover",
        "premise:literal-charged-current-native-register-dictionary",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:compact-inner-e8-zero-mode-operation-class",
        "premise:original-neutral-root-pair-moment-fields",
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        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:original-positive-adjoint-two-insertion-response-profiles",
        "premise:common-unnormalized-positive-return-erasure-test",
        "premise:specified-original-pair-ordered-response-composition",
        "premise:canonical-source-stress-all-mode-history",
        "premise:original-cartan-current-local-smearing-and-weyl-class",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:original-native-event-difference-fields-and-state-controls",
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        "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
        "premise:original-single-circle-local-density-and-geometric-stress",
        "premise:charged-car-negative-sea-source",
        "premise:fixed-scalar-quarter-hardy-source",
        "premise:original-gap-normalized-primitive-relative-action",
        "premise:positive-scalar-first-order-source-comparison-class",
        "premise:given-original-fixed-one-particle-time-winding-ward",
        "premise:four-starts-fixed-state-versus-whole-datum-comparison",
        "premise:original-stable-pointer-instrument-and-given-context",
        "premise:same-full-positive-energy-separating-local-car-source",
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        "premise:local-register-retained-in-joint-charged-word-limit",
        "premise:conditional-local-parity-gauss-completion-with-common-H",
        "premise:fixed-one-copy-eightfold-connection-charge-dictionary",
        "premise:original-neutral-eightpreimage-parity-pair",
        "premise:complete-vector-phase-word-Ward-comparison",
        "premise:smooth-unitary-pair-return-fixed-compact-source-time",
        "premise:unchanged-full-source-observable-map-and-continuous-time-insertion-limit",
        "premise:retained-local-register-with-one-common-original-vacuum-reference",
        "premise:actual-neutral-pair-heat-filter-and-separate-chiral-Weyl-comparison",
        "premise:given-holomorphic-kms-inclusion-and-local-global-modular-domains",
        "premise:canonical-source-gibbs-and-parity-probe-with-retained-winding",
        "premise:literal-boundary-minimality-and-first-order-domain-statements"
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      "findingIds": [
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        "finding:native-source-common-time-thermal-family",
        "finding:native-grade-carry-and-memory",
        "finding:even-seam-global-operator-factorization",
        "finding:source-generated-marking-classification",
        "finding:source-marking-field-realization-boundary",
        "finding:minimal-defect-uniqueness-not-proved",
        "finding:conditional-seam-field-state-time-reconstruction",
        "finding:continuous-hardy-seam-classification",
        "finding:same-source-seam-energy-selection",
        "finding:master-functional-presupposes-source-selection",
        "finding:geometric-deck-not-calderon-polarization",
        "finding:positive-compression-erases-carrier-minus-sector",
        "finding:higgs-determinant-does-not-discharge-carrier-map",
        "finding:primitive-invariant-does-not-select-rank-three",
        "finding:fixed-background-source-does-not-fix-mixed-vertex",
        "finding:local-quarter-inner-variation-is-connection",
        "finding:calderon-enlargement-does-not-select-higgs",
        "finding:finite-sm-higgs-variation-retains-seed",
        "finding:operator-skeleton-does-not-identify-insertion-algebra",
        "finding:internal-even-car-hodge-is-not-coordinate-algebra",
        "finding:minimum-rank-unique-primitive-channel-does-not-select-32",
        "finding:joint-selfconsistency-presupposes-minimal-algebra-fiber-selection",
        "finding:process-composition-and-cyclicity-do-not-select-algebra",
        "finding:quartic-shape-selects-32-only-with-source-sign",
        "finding:native-symmetric-quartic-erasure-common-object",
        "finding:joint-native-algebra-normalizer-and-response-orbit",
        "finding:native-galois-intertwiner-unique-logical-marked-orbit",
        "finding:native-reflection-full-four-five-generators-rate-injectivity",
        "finding:complete-logical-readout-native-covariant-residual-classification",
        "finding:original-polar-mark-clock-centralizer-orbit",
        "finding:actual-quartic-code-product-stabilizer-quarter-exclusion",
        "finding:original-decoder-versus-native-quarter-polynomial-erasure-test",
        "finding:original-decoder-product-time-and-all-unitary-erasure-lifts",
        "finding:native-clock-compatible-polar-fibre-and-canonical-distance",
        "finding:native-moment-parent-and-product-path-selection-boundary",
        "finding:original-e8-current-response-and-full-quarter-lifts",
        "finding:literal-charged-current-register-dictionary-fails-quarter",
        "finding:native-cartan-code-inner-e8-stabilizer-excludes-marked-quarter",
        "finding:native-code-neutral-weight-two-action-and-global-clock-exclusion",
        "finding:original-neutral-root-mixed-response-derived",
        "finding:actual-mixed-ordered-return-on-native-quartic-code",
        "finding:mixed-return-spectrum-does-not-select-original-polar-mark",
        "finding:actual-mixed-return-fails-native-trimer-erasure-lift",
        "finding:strict-trace-gap-excludes-positive-mixed-return-reweighting",
        "finding:original-source-erasure-ward-terms-derived",
        "finding:uncompressed-original-mixed-return-does-not-preserve-code",
        "finding:original-root-response-sum-also-leaks-from-code",
        "finding:original-full-source-return-exterior-feedback-derived",
        "finding:original-return-requires-memory-in-every-code-direction",
        "finding:native-conformal-primary-multiplicity-source-factorization",
        "finding:canonical-conformal-erasure-forces-primary-twenty-carry",
        "finding:all-canonical-stress-histories-scalar-on-native-code",
        "finding:original-local-current-ope-forces-erasure-and-carry",
        "finding:all-real-local-cartan-profiles-erasure-third-bound",
        "finding:closed-cartan-weyl-histories-only-logical-phase",
        "finding:full-existing-local-e8-hull-of-native-code",
        "finding:full-local-quartic-wick-ladder-and-unprojected-native-actions",
        "finding:unmarked-native-w6-simplex-from-full-quartic-ope",
        "finding:full-e8-current-products-fix-stress-and-exterior-ward-cancellation",
        "finding:full-source-simplex-does-not-select-original-polar-mark",
        "finding:original-full-field-event-gram-reconstructs-relative-polar-mark",
        "finding:raw-event-state-positivity-does-not-execute-polar-normalization",
        "finding:determinant-cover-not-literal-full-cartan-lattice-event",
        "finding:clock-free-quartic-product-spectral-sectors-and-native-five",
        "finding:six-complex-native-simplex-rays-from-holomorphic-product-quadrics",
        "finding:intrinsic-relative-response-covariance-and-state-boundary",
        "finding:intrinsic-event-composition-retains-shared-source-and-correlation-data",
        "finding:clock-free-event-geometry-fixes-clock-class-and-mark-orbit-not-member",
        "finding:original-charged-seam-character-recovers-global-native-event-pair",
        "finding:internal-seam-vacuum-and-original-neutral-tensor-do-not-select-flag-root",
        "finding:actual-quarter-fold-shift-removes-native-symmetry-obstruction-without-flag-selector",
        "finding:fold-shift-equivariance-needs-actual-source-response-map-not-another-symmetry-no-go",
        "finding:original-fold-covariance-current-counterterm-and-quartic-twisted-transport",
        "finding:fold-state-full-quartic-kernel-and-erasure-protected-low-order-response",
        "finding:original-fold-quartic-mean-and-spectral-responses-do-not-match-sixty-flags",
        "finding:original-winding-full-state-response-removes-vacuum-rank-two-shortcut",
        "finding:original-source-global-time-winding-history-factorization",
        "finding:original-local-density-folds-to-correlated-root-current-and-erasure-response",
        "finding:original-current-all-mode-energy-and-geometric-stress-bridge",
        "finding:original-current-geometric-response-pair-has-no-common-polar-sector",
        "finding:original-current-ordered-hall-kernel-and-actual-folded-field-products",
        "finding:original-seam-grades-full-density-geometric-products-and-energy-response",
        "finding:original-seam-complete-density-grading-not-sharp-32-input-sectors",
        "finding:original-density-all-profile-mean-work-scalar-on-native-packet",
        "finding:original-p1-gap-normalized-action-covector-and-scalar-holonomy-class",
        "finding:original-p1-gap-car-component-not-native-sharp-32-event",
        "finding:complete-positive-scalar-first-order-circle-source-normal-form",
        "finding:fixed-source-time-seam-ward-rigidifies-scalar-principal-symbol-with-carry",
        "finding:unit-spectral-flow-gap-and-four-marks-do-not-imply-fixed-source-ward",
        "finding:four-native-starts-have-distinct-original-fold-state-correlations",
        "finding:four-starts-full-joint-kernel-and-charged-covariant-transport",
        "finding:four-native-polar-marks-not-fixed-source-state-gauge-equivalent",
        "finding:original-pointer-template-requires-source-context-identification",
        "finding:four-polar-context-algebra-and-source-crosscoupling",
        "finding:every-native-quartic-has-all-eight-original-car-supports",
        "finding:bounded-proper-local-source-operation-cannot-exactly-prepare-positive-eigenpacket",
        "finding:full-original-erasure20-and-joint25-have-all-eight-source-supports",
        "finding:two-face-collar-fixes-face-current-phase-and-distinguishes-relative-endpoint-lift",
        "finding:full-dirac-cauchy-vector-cocycle-cancels-and-endpoint-domain-distinction",
        "finding:unchanged-qwz-full-finite-energy-spectral-exhaustion",
        "finding:uniform-z4-d8-locality-and-both-spinor-chirality-time-census",
        "finding:odd-spinor-global-refielding-and-unit-source-winding",
        "finding:sharp-half-flux-register-quotient-and-local-normality-exclusion",
        "finding:all-word-register-charged-local-field-limit-obstruction",
        "finding:exact-parity-gauss-locality-completion-of-register-source",
        "finding:cover-half-flux-not-fixed-charge-eightfold-spinor-twist",
        "finding:original-neutral-eightpreimage-parity-pair-has-uniform-cut-action",
        "finding:complete-dirac-vector-phase-pair-has-zero-spinor-Ward-charge",
        "finding:smooth-unitary-endpoint-removal-diverges-at-fixed-source-time",
        "finding:continuous-original-time-nonunitary-limits-stay-in-old-source-folium",
        "finding:common-register-vacuum-reference-uniform-and-local-grade-reduction",
        "finding:actual-heat-filtered-neutral-pair-full-state-time-and-separate-chiral-limit",
        "finding:holomorphic-kms-extension-assumes-charged-net-and-not-microscopic-generator",
        "finding:fixed-interval-rotation-modular-chain-fails-local-state-takesaki-repairs",
        "finding:source-winding-cartan-four-word-and-two-edge-kms-gso-discriminator",
        "finding:p1-parity-gauss-net-selection-undecidable-without-original-localization-map",
        "finding:original-minimality-does-not-reconstruct-operational-source",
        "finding:generic-first-order-relative-bound-zero-counterexample-and-correct-domain-repair"
      ],
      "tasks": [
        {
          "id": "task:read-operational-inputs",
          "titleEN": "Record the input status of the operational seed",
          "status": "resolved",
          "resolutionIds": [
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          ],
          "requiredPremiseIds": [
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          ]
        },
        {
          "id": "task:derive-physical-source-selector",
          "titleEN": "Find the original selector for source/state/event",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
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        },
        {
          "id": "task:reuse-native-source-common-time-thermal-family",
          "titleEN": "The same source supplies Hamiltonian, time and thermal state",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-source-common-time-thermal-family"
          ],
          "requiredPremiseIds": [
            "premise:declared-native-round20-source"
          ]
        },
        {
          "id": "task:reuse-native-grade-carry-and-memory",
          "titleEN": "Marking and necessary memory are connected within the given source",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-grade-carry-and-memory"
          ],
          "requiredPremiseIds": [
            "premise:declared-native-round20-source"
          ]
        },
        {
          "id": "task:reuse-even-seam-global-operator-factorization",
          "titleEN": "A common global seam-operator factorization exists",
          "status": "resolved",
          "resolutionIds": [
            "finding:even-seam-global-operator-factorization"
          ],
          "requiredPremiseIds": [
            "premise:stipulated-even-e8-seam-lift"
          ]
        },
        {
          "id": "task:reuse-source-generated-marking-classification",
          "titleEN": "Reuse complete source-character classification",
          "status": "resolved",
          "resolutionIds": [
            "finding:source-generated-marking-classification"
          ],
          "requiredPremiseIds": [
            "premise:full-normalized-charge-time-character-family",
            "premise:fixed-e8-fold-quarter-dictionary",
            "premise:actual-chiral-opposite-quarter-domains"
          ]
        },
        {
          "id": "task:reuse-conditional-seam-operator-reconstruction",
          "titleEN": "Reuse conditional joint seam reconstruction",
          "status": "resolved",
          "resolutionIds": [
            "finding:conditional-seam-field-state-time-reconstruction",
            "finding:continuous-hardy-seam-classification"
          ],
          "requiredPremiseIds": [
            "premise:actual-wold-source-operator-hypotheses",
            "premise:charged-car-negative-sea-source",
            "premise:fixed-scalar-quarter-hardy-source"
          ]
        },
        {
          "id": "task:reuse-same-source-seam-energy-selection",
          "titleEN": "Reuse the complete scalar energy theorem",
          "status": "resolved",
          "resolutionIds": [
            "finding:same-source-seam-energy-selection"
          ],
          "requiredPremiseIds": [
            "premise:fixed-scalar-quarter-hardy-source",
            "premise:charged-car-negative-sea-source"
          ]
        },
        {
          "id": "task:compare-process-first-reconstruction-routes",
          "titleEN": "Compare three process reconstruction routes using discriminating tests",
          "status": "resolved",
          "resolutionIds": [
            "finding:process-composition-and-cyclicity-do-not-select-algebra",
            "finding:quartic-shape-selects-32-only-with-source-sign"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:process-composition-finite-controls",
            "premise:unmarked-five-traceless-hermitian-quartic"
          ]
        },
        {
          "id": "task:reconstruct-native-joint-operator-object",
          "titleEN": "Reconstruct the actual common native operator object",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-symmetric-quartic-erasure-common-object",
            "finding:joint-native-algebra-normalizer-and-response-orbit",
            "finding:native-galois-intertwiner-unique-logical-marked-orbit"
          ],
          "requiredPremiseIds": [
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:existing-logical-clock-and-charge-marking"
          ]
        },
        {
          "id": "task:classify-native-source-event-menu-dynamics",
          "titleEN": "Classify native rates and coherent residual channels",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-reflection-full-four-five-generators-rate-injectivity",
            "finding:complete-logical-readout-native-covariant-residual-classification"
          ],
          "requiredPremiseIds": [
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:fixed-complete-four-five-primitive-reflection-generators",
            "premise:complete-logical-nondisturbance-native-covariance",
            "premise:coherent-block-source-algebra-operation-menu"
          ]
        },
        {
          "id": "task:derive-common-clock-lift-and-original-event-menu",
          "titleEN": "Determine the common original clock lift and event menu",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:existing-logical-clock-and-charge-marking",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:reconstruct-polar-mark-and-test-native-quarter-lifts",
          "titleEN": "Reconstruct original mark and test native clock continuation",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-polar-mark-clock-centralizer-orbit",
            "finding:actual-quartic-code-product-stabilizer-quarter-exclusion",
            "finding:original-decoder-versus-native-quarter-polynomial-erasure-test"
          ],
          "requiredPremiseIds": [
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:original-marked-simplex-positive-polar-rule",
            "premise:actual-clifford-identification-and-finite-overgroup-theorem",
            "premise:original-erasure-decoder-and-same-native-quarter-polynomial",
            "premise:existing-logical-clock-and-charge-marking"
          ]
        },
        {
          "id": "task:classify-polar-fibre-and-test-native-parent-transport",
          "titleEN": "Classify polar fibre and test existing native selection mechanisms",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-clock-compatible-polar-fibre-and-canonical-distance",
            "finding:native-moment-parent-and-product-path-selection-boundary"
          ],
          "requiredPremiseIds": [
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:native-marked-simplex-clock-compatibility-without-fixed-polar-mark",
            "premise:existing-logical-clock-and-charge-marking",
            "premise:actual-clifford-identification-and-finite-overgroup-theorem",
            "premise:original-native-fourth-moment-parent",
            "premise:local-product-code-orbit-geometric-parallel-transport"
          ]
        },
        {
          "id": "task:test-original-current-response-and-native-code-dictionary",
          "titleEN": "Test original current response and actual code dictionary",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-e8-current-response-and-full-quarter-lifts",
            "finding:literal-charged-current-register-dictionary-fails-quarter",
            "finding:native-cartan-code-inner-e8-stabilizer-excludes-marked-quarter"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:existing-logical-clock-and-charge-marking",
            "premise:native-marked-simplex-clock-compatibility-without-fixed-polar-mark",
            "premise:original-coupled-determinant-event-cover",
            "premise:literal-charged-current-native-register-dictionary",
            "premise:actual-clifford-identification-and-finite-overgroup-theorem",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:compact-inner-e8-zero-mode-operation-class"
          ]
        },
        {
          "id": "task:test-original-neutral-quadratic-and-first-mixed-source-response",
          "titleEN": "Test original neutral quadratic and first mixed source response",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-code-neutral-weight-two-action-and-global-clock-exclusion",
            "finding:original-neutral-root-mixed-response-derived"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:existing-logical-clock-and-charge-marking",
            "premise:actual-clifford-identification-and-finite-overgroup-theorem",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-neutral-root-pair-moment-fields",
            "premise:solely-neutral-local-weight-two-zero-mode-class"
          ]
        },
        {
          "id": "task:derive-actual-mixed-ordered-code-erasure-return",
          "titleEN": "Derive ordered mixed code/erasure return",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-mixed-ordered-return-on-native-quartic-code",
            "finding:mixed-return-spectrum-does-not-select-original-polar-mark",
            "finding:actual-mixed-return-fails-native-trimer-erasure-lift",
            "finding:strict-trace-gap-excludes-positive-mixed-return-reweighting"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-neutral-root-pair-moment-fields",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:original-positive-adjoint-two-insertion-response-profiles",
            "premise:existing-logical-clock-and-charge-marking",
            "premise:common-unnormalized-positive-return-erasure-test"
          ]
        },
        {
          "id": "task:test-full-source-return-code-closure-and-erasure-ward",
          "titleEN": "Test full source-return code closure and erasure Ward terms",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-source-erasure-ward-terms-derived",
            "finding:uncompressed-original-mixed-return-does-not-preserve-code",
            "finding:original-root-response-sum-also-leaks-from-code"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-neutral-root-pair-moment-fields",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:original-positive-adjoint-two-insertion-response-profiles"
          ]
        },
        {
          "id": "task:derive-full-source-return-exterior-memory",
          "titleEN": "Derive full source feedback and necessary memory",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-full-source-return-exterior-feedback-derived",
            "finding:original-return-requires-memory-in-every-code-direction"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-neutral-root-pair-moment-fields",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:original-positive-adjoint-two-insertion-response-profiles",
            "premise:specified-original-pair-ordered-response-composition"
          ]
        },
        {
          "id": "task:derive-all-mode-canonical-conformal-erasure-bridge",
          "titleEN": "Derive original all-mode conformal/erasure bridge",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-conformal-primary-multiplicity-source-factorization",
            "finding:canonical-conformal-erasure-forces-primary-twenty-carry",
            "finding:all-canonical-stress-histories-scalar-on-native-code"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:canonical-source-stress-all-mode-history",
            "premise:existing-logical-clock-and-charge-marking"
          ]
        },
        {
          "id": "task:derive-full-local-cartan-insertion-and-ordering",
          "titleEN": "Check full local Cartan insertion and ordered product",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-local-current-ope-forces-erasure-and-carry",
            "finding:all-real-local-cartan-profiles-erasure-third-bound",
            "finding:closed-cartan-weyl-histories-only-logical-phase"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:original-cartan-current-local-smearing-and-weyl-class",
            "premise:canonical-source-stress-all-mode-history",
            "premise:existing-logical-clock-and-charge-marking"
          ]
        },
        {
          "id": "task:derive-full-local-source-hull-and-unprojected-returns",
          "titleEN": "Determine full local source hull and unfiltered returns",
          "status": "resolved",
          "resolutionIds": [
            "finding:full-existing-local-e8-hull-of-native-code",
            "finding:full-local-quartic-wick-ladder-and-unprojected-native-actions",
            "finding:full-e8-current-products-fix-stress-and-exterior-ward-cancellation",
            "finding:full-source-simplex-does-not-select-original-polar-mark"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:native-g31-concrete-event-representation",
            "premise:existing-logical-clock-and-charge-marking"
          ]
        },
        {
          "id": "task:reconstruct-unmarked-native-simplex-from-local-quartic-ope",
          "titleEN": "Derive unmarked native simplex from local quartic product",
          "status": "resolved",
          "resolutionIds": [
            "finding:unmarked-native-w6-simplex-from-full-quartic-ope",
            "finding:full-source-simplex-does-not-select-original-polar-mark"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:native-g31-concrete-event-representation",
            "premise:existing-logical-clock-and-charge-marking"
          ]
        },
        {
          "id": "task:test-original-seam-state-event-data-on-full-field-tensors",
          "titleEN": "Test original seam/state/event data on full field tensors",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-full-field-event-gram-reconstructs-relative-polar-mark",
            "finding:raw-event-state-positivity-does-not-execute-polar-normalization",
            "finding:determinant-cover-not-literal-full-cartan-lattice-event"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:native-marked-simplex-clock-compatibility-without-fixed-polar-mark",
            "premise:original-native-event-difference-fields-and-state-controls"
          ]
        },
        {
          "id": "task:classify-intrinsic-event-geometry-and-composition",
          "titleEN": "Check intrinsic event geometry and composition",
          "status": "resolved",
          "resolutionIds": [
            "finding:clock-free-quartic-product-spectral-sectors-and-native-five",
            "finding:six-complex-native-simplex-rays-from-holomorphic-product-quadrics",
            "finding:intrinsic-relative-response-covariance-and-state-boundary",
            "finding:intrinsic-event-composition-retains-shared-source-and-correlation-data",
            "finding:clock-free-event-geometry-fixes-clock-class-and-mark-orbit-not-member"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:intrinsic-native-event-geometry-without-marked-clock"
          ]
        },
        {
          "id": "task:test-original-seam-state-data-on-sixty-native-flags",
          "titleEN": "Testoriginalseam/statedataon60nativeflags",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-charged-seam-character-recovers-global-native-event-pair",
            "finding:internal-seam-vacuum-and-original-neutral-tensor-do-not-select-flag-root",
            "finding:actual-quarter-fold-shift-removes-native-symmetry-obstruction-without-flag-selector",
            "finding:fold-shift-equivariance-needs-actual-source-response-map-not-another-symmetry-no-go"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:intrinsic-native-event-geometry-without-marked-clock",
            "premise:original-seam-state-data-on-native-flags-without-fitted-selector"
          ]
        },
        {
          "id": "task:derive-fold-state-quartic-response-and-test-original-winding",
          "titleEN": "Derivefoldstatequarticresponseandtestoriginalwinding",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-fold-covariance-current-counterterm-and-quartic-twisted-transport",
            "finding:fold-state-full-quartic-kernel-and-erasure-protected-low-order-response",
            "finding:original-fold-quartic-mean-and-spectral-responses-do-not-match-sixty-flags",
            "finding:original-winding-full-state-response-removes-vacuum-rank-two-shortcut"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:intrinsic-native-event-geometry-without-marked-clock",
            "premise:original-seam-state-data-on-native-flags-without-fitted-selector",
            "premise:fixed-conformal-current-reference-for-original-fold-state-transport"
          ]
        },
        {
          "id": "task:derive-original-density-geometric-response-and-time-winding-normal-form",
          "titleEN": "Derive original density/geometric response and whole time-winding class",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-source-global-time-winding-history-factorization",
            "finding:original-local-density-folds-to-correlated-root-current-and-erasure-response",
            "finding:original-current-all-mode-energy-and-geometric-stress-bridge",
            "finding:original-current-geometric-response-pair-has-no-common-polar-sector"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
            "premise:original-single-circle-local-density-and-geometric-stress",
            "premise:charged-car-negative-sea-source",
            "premise:fixed-scalar-quarter-hardy-source"
          ]
        },
        {
          "id": "task:derive-original-full-current-kernel-field-products-and-seam-grading",
          "titleEN": "Derive full original current/field products and original seam grading",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-current-ordered-hall-kernel-and-actual-folded-field-products",
            "finding:original-seam-grades-full-density-geometric-products-and-energy-response",
            "finding:original-seam-complete-density-grading-not-sharp-32-input-sectors",
            "finding:original-density-all-profile-mean-work-scalar-on-native-packet"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
            "premise:original-single-circle-local-density-and-geometric-stress"
          ]
        },
        {
          "id": "task:derive-original-p1-gap-action-differential-and-test-native-spectral-component",
          "titleEN": "Derive original P1 gap-scale term and test direct native spectral bridge",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-p1-gap-normalized-action-covector-and-scalar-holonomy-class",
            "finding:original-p1-gap-car-component-not-native-sharp-32-event"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
            "premise:original-single-circle-local-density-and-geometric-stress",
            "premise:declared-p1-p2",
            "premise:typed-source-structures",
            "premise:fixed-scalar-quarter-hardy-source",
            "premise:charged-car-negative-sea-source",
            "premise:original-gap-normalized-primitive-relative-action"
          ]
        },
        {
          "id": "task:classify-scalar-first-order-source-fiber-with-actual-seam-time-ward",
          "titleEN": "Classify whole scalar first-order fiber with actual time-seam Ward",
          "status": "resolved",
          "resolutionIds": [
            "finding:complete-positive-scalar-first-order-circle-source-normal-form",
            "finding:fixed-source-time-seam-ward-rigidifies-scalar-principal-symbol-with-carry",
            "finding:unit-spectral-flow-gap-and-four-marks-do-not-imply-fixed-source-ward"
          ],
          "requiredPremiseIds": [
            "premise:fixed-scalar-quarter-hardy-source",
            "premise:typed-source-structures",
            "premise:positive-scalar-first-order-source-comparison-class",
            "premise:given-original-fixed-one-particle-time-winding-ward"
          ]
        },
        {
          "id": "task:compare-four-native-starts-with-full-original-fold-state-response",
          "titleEN": "Compare four starts with complete original state response",
          "status": "resolved",
          "resolutionIds": [
            "finding:four-native-starts-have-distinct-original-fold-state-correlations",
            "finding:four-starts-full-joint-kernel-and-charged-covariant-transport",
            "finding:four-native-polar-marks-not-fixed-source-state-gauge-equivalent"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:intrinsic-native-event-geometry-without-marked-clock",
            "premise:original-seam-state-data-on-native-flags-without-fitted-selector",
            "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
            "premise:four-starts-fixed-state-versus-whole-datum-comparison"
          ]
        },
        {
          "id": "task:audit-original-source-minimality-and-domain-theorem",
          "titleEN": "Audit original minimality and analytic-domain assertion",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-minimality-does-not-reconstruct-operational-source",
            "finding:generic-first-order-relative-bound-zero-counterexample-and-correct-domain-repair"
          ],
          "requiredPremiseIds": [
            "premise:literal-boundary-minimality-and-first-order-domain-statements",
            "premise:original-p1-seed-boundary-and-local-restriction-definitions"
          ]
        }
      ]
    },
    {
      "id": "problem:seam-family-connection-compatibility",
      "titleEN": "Seam connection and family action",
      "questionEN": "Does the joint discrete theorem determine a concrete connection and a family coupling compatible with the determinant seam action?",
      "topicIds": [
        "topic:origin",
        "topic:flavor",
        "topic:e8"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:quarter-phase-retained",
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:joint-discrete-data-vs-connection",
        "finding:parallelity-excludes-2plus2-only-conditionally"
      ],
      "tasks": [
        {
          "id": "task:distinguish-classes-from-connection",
          "titleEN": "Separate classes from actual transport",
          "status": "resolved",
          "resolutionIds": [
            "finding:joint-discrete-data-vs-connection"
          ],
          "requiredPremiseIds": [
            "premise:declared-p1-p2",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:derive-compatible-joint-connection",
          "titleEN": "Derive the common connection from the original source",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:declared-p1-p2",
            "premise:typed-source-structures"
          ]
        }
      ]
    },
    {
      "id": "problem:physical-time-projector-transfer",
      "titleEN": "Physical time and complete transfer",
      "questionEN": "Do the original reflection/projector structures determine the full physical time generator and required field transfer?",
      "topicIds": [
        "topic:clocks",
        "topic:quantum",
        "topic:origin"
      ],
      "goalIds": [
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:quarter-phase-retained",
        "premise:typed-source-structures",
        "premise:specified-shared-s3-register-channel-family",
        "premise:specified-marked-m4-clifford-channel"
      ],
      "findingIds": [
        "finding:sign-projector-does-not-determine-time-magnitude",
        "finding:full-faithful-kms-or-transfer-recovers-magnitudes",
        "finding:full-two-face-collar-recovers-signed-operator",
        "finding:two-face-graph-state-time-same-datum",
        "finding:shared-register-gkls-embedding-exact-q-interval",
        "finding:marked-m4-channel-all-powers-homogeneous-gksl-excluded"
      ],
      "tasks": [
        {
          "id": "task:retain-sign-projector-counterexample",
          "titleEN": "Reuse the sign-projector counterexample",
          "status": "resolved",
          "resolutionIds": [
            "finding:sign-projector-does-not-determine-time-magnitude"
          ],
          "requiredPremiseIds": [
            "premise:sign-covariance-only"
          ]
        },
        {
          "id": "task:reuse-conditional-full-transfer-reconstruction",
          "titleEN": "Reuse conditional full-transfer reconstruction",
          "status": "resolved",
          "resolutionIds": [
            "finding:full-faithful-kms-or-transfer-recovers-magnitudes"
          ],
          "requiredPremiseIds": [
            "premise:known-faithful-kms-or-full-transfer"
          ]
        },
        {
          "id": "task:derive-physical-full-transfer",
          "titleEN": "Identify the physical complete TFPT transfer",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:reuse-full-two-face-collar-reconstruction",
          "titleEN": "Reuse complete collar reconstruction",
          "status": "resolved",
          "resolutionIds": [
            "finding:full-two-face-collar-recovers-signed-operator"
          ],
          "requiredPremiseIds": [
            "premise:actual-product-collar-two-face-data"
          ]
        },
        {
          "id": "task:classify-shared-register-homogeneous-gkls-time",
          "titleEN": "Classify homogeneous GKLS embedding of the shared register family",
          "status": "resolved",
          "resolutionIds": [
            "finding:shared-register-gkls-embedding-exact-q-interval"
          ],
          "requiredPremiseIds": [
            "premise:specified-shared-s3-register-channel-family"
          ]
        },
        {
          "id": "task:decide-fixed-m4-all-branch-homogeneous-time",
          "titleEN": "Decide every homogeneous GKSL logarithm of the fixed M4 channel",
          "status": "resolved",
          "resolutionIds": [
            "finding:marked-m4-channel-all-powers-homogeneous-gksl-excluded"
          ],
          "requiredPremiseIds": [
            "premise:specified-marked-m4-clifford-channel"
          ]
        }
      ]
    },
    {
      "id": "problem:flavor-ratio-amplitude-readout",
      "titleEN": "Flavor ratio and absolute amplitude scale",
      "questionEN": "Which flavor quantity is fixed on the derived discrete stratum, and which requires the continuous holonomy anchor?",
      "topicIds": [
        "topic:flavor",
        "topic:e8",
        "topic:origin"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:flavor-ratio-closed-absolute-anchor-open"
      ],
      "tasks": [
        {
          "id": "task:reuse-derived-flavor-ratio",
          "titleEN": "Reuse the derived flavor ratio",
          "status": "resolved",
          "resolutionIds": [
            "finding:flavor-ratio-closed-absolute-anchor-open"
          ],
          "requiredPremiseIds": [
            "premise:declared-p1-p2"
          ]
        },
        {
          "id": "task:select-absolute-holonomy-anchor",
          "titleEN": "Determine the absolute holonomy amplitude",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:typed-source-structures"
          ]
        }
      ]
    },
    {
      "id": "problem:alpha-fixed-input-root",
      "titleEN": "Alpha closure for fixed inputs",
      "questionEN": "Which closure equation follows for a fixed seam projector, and which inputs still need justification?",
      "topicIds": [
        "topic:alpha",
        "topic:origin",
        "topic:e8"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:quarter-phase-retained",
        "premise:fixed-input-models-scoped",
        "premise:marked-e8-regularized-charged-grade-response",
        "premise:fixed-em-seam-function"
      ],
      "findingIds": [
        "finding:alpha-equation-is-fixed-input-closure",
        "finding:alpha-global-positive-root-unique",
        "finding:marked-e8-sector-grade-response-realizes-five-quarters",
        "finding:alpha-exact-coefficients-and-root-retain-specific-physical-ward-obligation",
        "finding:alpha-archive-term-provenance-and-master-functional-scope"
      ],
      "tasks": [
        {
          "id": "task:separate-root-from-selector",
          "titleEN": "Separate root calculation from source selection",
          "status": "resolved",
          "resolutionIds": [
            "finding:alpha-equation-is-fixed-input-closure"
          ],
          "requiredPremiseIds": [
            "premise:declared-p1-p2",
            "premise:fixed-input-models-scoped"
          ]
        },
        {
          "id": "task:derive-alpha-inputs-and-transfer",
          "titleEN": "Derive alpha inputs and Thomson transfer",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:declared-p1-p2",
            "premise:quarter-phase-retained"
          ]
        },
        {
          "id": "task:prove-alpha-positive-root-uniqueness",
          "titleEN": "Establish the unique global positive root",
          "status": "resolved",
          "resolutionIds": [
            "finding:alpha-global-positive-root-unique"
          ],
          "requiredPremiseIds": [
            "premise:fixed-em-seam-function"
          ]
        }
      ]
    },
    {
      "id": "problem:gravity-flow-normalization",
      "titleEN": "Geometric flow and gravity normalization",
      "questionEN": "Do common matter-only observations derive geometric horizon flow and the area response from the original gravity source?",
      "topicIds": [
        "topic:gravity",
        "topic:quantum",
        "topic:origin"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:typed-source-structures",
        "premise:fixed-input-models-scoped"
      ],
      "findingIds": [
        "finding:matter-observations-do-not-fix-geometric-response"
      ],
      "tasks": [
        {
          "id": "task:state-fixed-function-scope",
          "titleEN": "Record scope of fixed original functional",
          "status": "resolved",
          "resolutionIds": [
            "finding:matter-observations-do-not-fix-geometric-response"
          ],
          "requiredPremiseIds": [
            "premise:declared-p1-p2",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:derive-geometric-modular-response",
          "titleEN": "Derive geometric modular response and area coefficient",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:declared-p1-p2",
            "premise:typed-source-structures"
          ]
        }
      ]
    },
    {
      "id": "problem:clock-linked-neutral-response-reconstruction",
      "titleEN": "Clocks and complete neutral source response",
      "questionEN": "Which complete quadratic response can the actual clock images reconstruct?",
      "topicIds": [
        "topic:clocks",
        "topic:quantum",
        "topic:alpha"
      ],
      "goalIds": [
        "goal:T2",
        "goal:T6"
      ],
      "acceptedPremiseIds": [
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:four-clock-fields-complete-quadratic-response"
      ],
      "tasks": [
        {
          "id": "task:reconstruct-full-neutral-quadratic-matrix",
          "titleEN": "Reconstruct all neutral quadratic matrix directions",
          "status": "resolved",
          "resolutionIds": [
            "finding:four-clock-fields-complete-quadratic-response"
          ],
          "requiredPremiseIds": [
            "premise:native-current-clock-ope-access"
          ]
        },
        {
          "id": "task:derive-clock-probe-physical-access",
          "titleEN": "Derive physically admissible probe access",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:native-current-clock-ope-access",
            "premise:declared-p1-p2"
          ]
        }
      ]
    },
    {
      "id": "problem:clock-source-covariant-transport",
      "titleEN": "Clock action on charge, time and quarter transition",
      "questionEN": "How does the actual Gaussian J act on the complete marked source?",
      "topicIds": [
        "topic:clocks",
        "topic:e8",
        "topic:quantum"
      ],
      "goalIds": [
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:typed-source-structures",
        "premise:quarter-phase-retained"
      ],
      "findingIds": [
        "finding:actual-j-covariant-source-transport",
        "finding:fixed-source-affine-j-obstruction"
      ],
      "tasks": [
        {
          "id": "task:construct-covariant-original-j-lift",
          "titleEN": "Construct original J lift with complete time/charge action",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-j-covariant-source-transport"
          ],
          "requiredPremiseIds": [
            "premise:stipulated-e8-clock-tuple"
          ]
        },
        {
          "id": "task:interpret-clock-source-frames",
          "titleEN": "Derive physical meaning of the frame orbit",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:stipulated-e8-clock-tuple",
            "premise:declared-p1-p2"
          ]
        }
      ]
    },
    {
      "id": "problem:charged-twist-response-identifiability",
      "titleEN": "Which responses distinguish charged time action?",
      "questionEN": "Which data separate sources with identical complete neutral vacuum response?",
      "topicIds": [
        "topic:quantum",
        "topic:clocks",
        "topic:origin"
      ],
      "goalIds": [
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:neutral-vacuum-allword-twist-blindness",
        "finding:eight-charged-responses-reconstruct-twist"
      ],
      "tasks": [
        {
          "id": "task:identify-neutral-response-blind-directions",
          "titleEN": "Determine twist blindness completely",
          "status": "resolved",
          "resolutionIds": [
            "finding:neutral-vacuum-allword-twist-blindness"
          ],
          "requiredPremiseIds": [
            "premise:neutral-current-vacuum-insertions"
          ]
        },
        {
          "id": "task:reconstruct-twist-from-charged-data",
          "titleEN": "Reconstruct twist from charged responses",
          "status": "resolved",
          "resolutionIds": [
            "finding:eight-charged-responses-reconstruct-twist"
          ],
          "requiredPremiseIds": [
            "premise:normalized-charged-primary-time-access"
          ]
        },
        {
          "id": "task:derive-actual-charged-source-probes",
          "titleEN": "Derive actual charged source probes",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:declared-p1-p2",
            "premise:quarter-phase-retained"
          ]
        }
      ]
    },
    {
      "id": "problem:native-charged-seam-selection",
      "titleEN": "Which seam action distinguishes native opposite charges?",
      "questionEN": "Can existing native charge fields produce the quarter-twist time response of the seam?",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:clocks",
        "topic:universalraum"
      ],
      "goalIds": [
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:native-invariant-response-not-fixed-chiral-twist",
        "finding:native-opposite-charge-response-symmetry",
        "finding:native-chemical-mark-visible-in-charged-corners",
        "finding:fixed-hopping-quarter-phases-retain-antiunitary-discriminator"
      ],
      "tasks": [
        {
          "id": "task:native-field-response-discriminator",
          "titleEN": "Check native fields and response discriminator",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-opposite-charge-response-symmetry"
          ],
          "requiredPremiseIds": [
            "premise:native-full-charge-field-source",
            "premise:native-c-invariant-start"
          ]
        },
        {
          "id": "task:derive-native-seam-symmetry-breaking",
          "titleEN": "Derive actual marked seam/state action",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:declared-p1-p2"
          ]
        }
      ]
    },
    {
      "id": "problem:constructive-seam-lift-state",
      "titleEN": "Constructive seam search: operators, state and source compatibility",
      "questionEN": "What does the complete phase search decide, and which additional state information prevents a false source solution?",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:e8",
        "topic:clocks",
        "topic:universalraum"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:quarter-phase-retained",
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:constructive-phase-class-and-full-winding",
        "finding:constructive-positive-state-extension",
        "finding:constructive-current-group-selects-positive-p-lift",
        "finding:constructive-state-algebra-resolution-and-source-test",
        "finding:constructive-all-phases-fail-unchanged-source"
      ],
      "tasks": [
        {
          "id": "task:reuse-constructive-seam-1",
          "titleEN": "Finite search closed, actual winding and parity equivalence",
          "status": "resolved",
          "resolutionIds": [
            "finding:constructive-phase-class-and-full-winding"
          ],
          "requiredPremiseIds": [
            "premise:fixed-car-crossed-half-charge-search"
          ]
        },
        {
          "id": "task:reuse-constructive-seam-2",
          "titleEN": "Positive finite-word state extension",
          "status": "resolved",
          "resolutionIds": [
            "finding:constructive-positive-state-extension"
          ],
          "requiredPremiseIds": [
            "premise:fixed-car-crossed-half-charge-search",
            "premise:fixed-positive-crossed-state"
          ]
        },
        {
          "id": "task:reuse-constructive-seam-3",
          "titleEN": "E8 current group conditionally determines the P sign",
          "status": "resolved",
          "resolutionIds": [
            "finding:constructive-current-group-selects-positive-p-lift"
          ],
          "requiredPremiseIds": [
            "premise:stipulated-even-e8-seam-lift",
            "premise:fixed-car-crossed-half-charge-search"
          ]
        },
        {
          "id": "task:reuse-constructive-seam-4",
          "titleEN": "The same shift functional hides distinct ground states",
          "status": "resolved",
          "resolutionIds": [
            "finding:constructive-state-algebra-resolution-and-source-test"
          ],
          "requiredPremiseIds": [
            "premise:fixed-car-crossed-half-charge-search",
            "premise:fixed-positive-crossed-state"
          ]
        },
        {
          "id": "task:reuse-constructive-seam-5",
          "titleEN": "No phase hit realizes the unchanged source",
          "status": "resolved",
          "resolutionIds": [
            "finding:constructive-all-phases-fail-unchanged-source"
          ],
          "requiredPremiseIds": [
            "premise:fixed-car-crossed-half-charge-search",
            "premise:unchanged-quarter-car-qwz-source-comparison"
          ]
        }
      ]
    },
    {
      "id": "problem:seam-partner-charge-state-time",
      "titleEN": "Which partner actually belongs to the original seam?",
      "questionEN": "Which charge, state type and time action belong to the root partner and original sheet pairing?",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:e8",
        "topic:clocks"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:declared-p1-p2",
        "premise:quarter-phase-retained",
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:partner-carrier-charges-and-state-types",
        "finding:partner-fixed-quarter-time-splits-sm-multiplet",
        "finding:partner-original-carrier-cross-sheet-conjugation",
        "finding:partner-chiral-quarter-conjugation-state-time",
        "finding:joint-bilinear-hermitian-correction",
        "finding:joint-quarter-transports-marking",
        "finding:joint-fixed-time-marking-repair-excluded",
        "finding:joint-spinor-not-source-coefficient"
      ],
      "tasks": [
        {
          "id": "task:reuse-partner-type-1",
          "titleEN": "Concrete pair operator; source neutrality is not hypercharge neutrality",
          "status": "resolved",
          "resolutionIds": [
            "finding:partner-carrier-charges-and-state-types"
          ],
          "requiredPremiseIds": [
            "premise:oriented-p2-exterior-carrier",
            "premise:fixed-e8-fold-quarter-dictionary"
          ]
        },
        {
          "id": "task:reuse-partner-type-2",
          "titleEN": "Fixed quarter time splits the color/weak multiplet",
          "status": "resolved",
          "resolutionIds": [
            "finding:partner-fixed-quarter-time-splits-sm-multiplet"
          ],
          "requiredPremiseIds": [
            "premise:oriented-p2-exterior-carrier",
            "premise:fixed-e8-fold-quarter-dictionary"
          ]
        },
        {
          "id": "task:reuse-partner-type-3",
          "titleEN": "Explicit conjugation for the existing sheet pairing",
          "status": "resolved",
          "resolutionIds": [
            "finding:partner-original-carrier-cross-sheet-conjugation"
          ],
          "requiredPremiseIds": [
            "premise:oriented-p2-exterior-carrier"
          ]
        },
        {
          "id": "task:reuse-partner-type-4",
          "titleEN": "Actual partner map between quarter domains",
          "status": "resolved",
          "resolutionIds": [
            "finding:partner-chiral-quarter-conjugation-state-time"
          ],
          "requiredPremiseIds": [
            "premise:actual-chiral-opposite-quarter-domains"
          ]
        },
        {
          "id": "task:reuse-joint-bilinear-hermitian-correction",
          "titleEN": "Bilinear sheet pairing is not Hermitian kernel selection",
          "status": "resolved",
          "resolutionIds": [
            "finding:joint-bilinear-hermitian-correction"
          ],
          "requiredPremiseIds": [
            "premise:oriented-p2-exterior-carrier"
          ]
        },
        {
          "id": "task:reuse-joint-quarter-transports-marking",
          "titleEN": "Actual quarter conjugation transports the supplied marking",
          "status": "resolved",
          "resolutionIds": [
            "finding:joint-quarter-transports-marking"
          ],
          "requiredPremiseIds": [
            "premise:actual-chiral-opposite-quarter-domains",
            "premise:fixed-e8-fold-quarter-dictionary",
            "premise:given-first-marked-quarter-character"
          ]
        },
        {
          "id": "task:reuse-joint-fixed-time-marking-repair-excluded",
          "titleEN": "No time-stabilizing orthogonal repair of the fixed marking",
          "status": "resolved",
          "resolutionIds": [
            "finding:joint-fixed-time-marking-repair-excluded"
          ],
          "requiredPremiseIds": [
            "premise:actual-chiral-opposite-quarter-domains",
            "premise:fixed-e8-fold-quarter-dictionary",
            "premise:given-first-marked-quarter-character"
          ]
        },
        {
          "id": "task:reuse-joint-spinor-not-source-coefficient",
          "titleEN": "Spinor16 is a state sector, not a 16-real one-particle field",
          "status": "resolved",
          "resolutionIds": [
            "finding:joint-spinor-not-source-coefficient"
          ],
          "requiredPremiseIds": [
            "premise:oriented-p2-exterior-carrier",
            "premise:given-d5-a3-free-fermion-realization"
          ]
        }
      ]
    },
    {
      "id": "problem:automated-operator-source-synthesis",
      "titleEN": "Automated construction of operator maps and sources",
      "questionEN": "Which tools generate operator maps, learn counterexamples, and can search for the full positive source?",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:process"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:typed-source-structures",
        "premise:actual-diagonal-lepton8-common-insertion-contract",
        "premise:one-e8-bracket-weyl-pairing-with-single-anchor-dressing",
        "premise:fixed-local-quarter-channel-in-constant-odd-source",
        "premise:actual-full-v252-unimodular-p2-gauge-lift",
        "premise:spin10-su4-family-four-cycle-quarter-lift-class",
        "premise:family-four-cycle-lift-commutes-with-original-sm",
        "premise:full-unitary-e8-field-conjugacy-with-transported-marks",
        "premise:native-third-clock-current-word-pairing-class",
        "premise:constant-odd-quarter-circle-source-sea-and-full-graph",
        "premise:canonical-recomplexified-graded-boundary-clifford-typing",
        "premise:actual-product-collar-two-face-data",
        "premise:fixed-scalar-quarter-full-jloc-pfaffian-candidate",
        "premise:fixed-quarter-cover-deck-normal-and-convergence-selection-tests",
        "premise:fixed-original-aps-quarter-domain-and-constant-odd-insertion",
        "premise:mass-only-full-deck-relative-and-ordinary-heat-trace",
        "premise:closed-antiperiodic-scalar-wall-with-fixed-hermitian-coefficient",
        "premise:scalar-spectral-heat-covariance-with-geometric-time-reflection",
        "premise:given-global-quaternionic-sm-mass-loop-with-fixed-yukawas",
        "premise:bounded-relative-operator-positive-laplace-and-compact-source-pair",
        "premise:given-normal-spin-clutch-weak-orbit-family-and-u2-linearization",
        "premise:given-normal-orbit-and-four-copy-natural-spin-family",
        "premise:fixed-v117-orthogonal-d3-cartan-identification",
        "premise:given-original-su3-to-su4-family-vector-action",
        "premise:given-split-carrier-and-actual-genus-two-q-sign-family",
        "premise:recovered-genus-two-cover-with-retained-signed-coefficients",
        "premise:actual-original-a0star-fuchsian-connection-and-c4-clock",
        "premise:same-spin-hardy-shift-against-identity-with-grading-retained",
        "premise:given-relative-unit-seam-on-specified-collar-faces",
        "premise:whole-constant-odd-coefficient-on-scalar-unit-clutch-domain",
        "premise:declared-q-column-to-v252-generation-slot-comparison",
        "premise:declared-q-column-to-v252-chiral-slot-comparison",
        "premise:original-hard-holonomy-and-closed-triangle-readout-contract",
        "premise:oriented-p2-exterior-carrier",
        "premise:given-original-e8-cx-current-cocycle-and-compact-adjoint",
        "premise:original-d5-base-current-action-on-given-p2-e8-source",
        "premise:relative-spin-scalar-hardy-cable-on-original-three-cover",
        "premise:original-gap-normalized-primitive-relative-action",
        "premise:original-closed-real-delta-transport-action",
        "premise:four-original-diagonal-paths-and-regularized-endpoints",
        "premise:native-clock-adjoint-compensation-for-hardypair",
        "premise:declared-positive-cubic-hardy-direction-for-gram-differential",
        "premise:whole-same-mu01-induced-ns-halfmode-reader",
        "premise:whole-six-original-induced-inputs-at-fixed-kinetic-ns-reader",
        "premise:canonical-finite-particle-hole-corner-of-given-kinetic-graph",
        "premise:bare-unsmoothed-crossface-quadratic-on-given-graph-vacuum",
        "premise:given-original-two-stage-admissibility-complex-and-blocked-weight-caller",
        "premise:local-geometric-dirac-deck-lift",
        "premise:original-admissibility-and-graded-source-interfaces",
        "premise:whole-product-dirac-with-standard-normal-clifford-double",
        "premise:full-independent-even-odd-matrix-coefficient-class",
        "premise:whole-fermi-two-face-projection-and-diagonal-isometric-compression",
        "premise:actual-v252-sm-coefficients-and-internal-ko6-without-extra-mixing",
        "premise:separable-normal-real-map-with-actual-jodd-gauge-and-typed-quarter-image",
        "premise:actual-p2-six-multiplet-gauge-real-coordinate-frame-class",
        "premise:original-higgs-radial-family-on-common-compact-product-domain",
        "premise:positive-laplace-profile-finite-three-moment-comparison",
        "premise:existing-ap-positive-source-factor-and-declared-squared-energy-comparison",
        "premise:actual-v252-entire-yukawa-majorana-family-with-compatibility-only",
        "premise:original-invariant-word-domain-with-nested-readouts",
        "premise:original-single-beta-return-with-native-cartan-j-word-alphabet",
        "premise:original-regulated-full-source-with-actual-measure-and-legs",
        "premise:finite-reflected-positive-quadratic-ir-regulator-class",
        "premise:actual-shared-real-higgs-yukawa-consumer-and-given-boson-fiber",
        "premise:original-graded-rp-reference-and-even-local-reflected-half-action",
        "premise:original-constant-higgs-fiber-and-symmetric-majorana-spectral-consumer",
        "premise:literal-archived-geometric-residue-and-common-higgs-field-comparison",
        "premise:local-same-trace-higgs-residue-and-conditional-product-dirac-consumer",
        "premise:whole-variable-higgs-bound-reference-and-conformal-dirac-map",
        "premise:original-inner-higgs-central-scale-and-finite-affine-source-map",
        "premise:given-normal-higgs-mode-realization-and-distinct-module-types",
        "premise:given-product-clifford-grades-parallel-projection-and-complete-relative-block-reader",
        "premise:original-positive-admissibility-reader-and-given-total-chiral-product",
        "premise:whole-original-scale-congruence-and-specified-normal-energy-data",
        "premise:given-whole-green-energy-source-and-conditional-scale-response",
        "premise:current-original-spectral-action-and-fixed-ordinary-moment-dictionary",
        "premise:original-whole-spinor-and-entropic-curvature-consumers-with-given-higgs-response",
        "premise:actual-original-four-ghost-commuting-projection-admissibility-complex",
        "premise:actual-smooth-normal-family-with-supplied-unitary-horizontal-connection",
        "premise:fixed-original-o1-section-consumer-and-actual-normal-first-order-placement",
        "premise:actual-original-quark-higgs-family-and-positive-conditional-source-mixture",
        "premise:fixed-rank-bounded-cutoff-with-ordinary-noncompact-higgs-volume",
        "premise:whole-constant-original-quarter-source-positive-traceclass-profile-and-flat-higgs-volume",
        "premise:given-o1-mode-isometry-and-weak-invariant-positive-higgs-covariance-marginal",
        "premise:literal-whole-source-localization-and-geometric-readout-contracts",
        "premise:actual-proper-local-current-weyls-and-bare-native-code-compression",
        "premise:actual-elliptic-quarter-supported-current-and-interval-car-modular-test",
        "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
        "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
        "premise:original-quarter-car-fold-current-subalgebra",
        "premise:actual-elliptic-two-face-projection-with-genuine-regional-support",
        "premise:actual-original-rightup-triplet-fixed-family-and-even-color-source",
        "premise:original-common-positive-constant-higgs-law-over-whole-quarter-tower",
        "premise:original-sixty-e8-unrefined-record-source-and-typed-doily-observables",
        "premise:original-native-pauli-and-v419-f20-source-words",
        "premise:actual-single-source-event-coherent-stinespring-record",
        "premise:given-e8-current-vacuum-net-and-original-grade-two-descendant-with-standard-split",
        "premise:local-e8-current-response-realization",
        "premise:given-original-elliptic-quarter-source-and-supported-scalar-current-insertion",
        "premise:original-quark-particle-color-group-counting-on-whole-constant-quarter-source",
        "premise:given-whole-quarter-graph-and-canonical-diagonal-original-color-lift",
        "premise:stipulated-e8-h8-with-original-color-center-selector",
        "premise:original-marked-color-center-on-whole-e8-lattice-current-source",
        "premise:fixed-native-d5-a3-compiler-split-and-seed",
        "premise:native-even-e8-gaussian-clock-and-sign-extension",
        "premise:original-fixed-color-and-folded-ns-cartan-interface",
        "premise:actual-thermal-quarter-source-and-retained-spin-parent-mixed-word",
        "premise:actual-eightfold-arc-residue-source-and-fixed-spin-cartan-insertion",
        "premise:actual-original-ns-color-center-continuous-density-and-coordinate-net",
        "premise:actual-original-full-two-face-graph-and-local-matrix-operations",
        "premise:original-blocked-transfer-algebra-and-marked-source-uniformity-audit",
        "premise:original-bare-c6-fixed-analytic-half-power-and-literal-word-alphabet",
        "premise:actual-original-finite-v990-source-and-historical-parent-recovery",
        "premise:actual-original-finite-car-higgs-force-and-faithful-collective-source",
        "premise:actual-whole-circle-product-car-force-and-original-time-smearing",
        "premise:original-constant-circle-local-higgs-force-and-hilbert-schmidt-time-source",
        "premise:literal-archived-global-positive-kernel-finite-measure-map",
        "premise:given-native-quasifree-higgs-source-and-actual-graded-restriction"
      ],
      "findingIds": [
        "finding:typed-operator-search-calibrated",
        "finding:positive-kernel-construction-with-exact-discriminator",
        "finding:source-typed-kernel-search-controls",
        "finding:collar-car-projection-multifield-ray",
        "finding:known-quarter-source-full-multifield-control",
        "finding:source-reference-amplitude-retained",
        "finding:joint-fermionic-source-schur-functional",
        "finding:quarter-circle-absolute-zeta-green-eta",
        "finding:quarter-sea-absolute-thermal-central-phase",
        "finding:original-operational-seed-and-given-green-pairing",
        "finding:order-one-infinitesimal-bound-inference-fails",
        "finding:joint-ward-two-field-data-miss-fixed-four-field-response",
        "finding:complete-car-kernel-fixes-nongaussian-link-functional",
        "finding:source-covariance-jet-and-mixed-face-interface",
        "finding:e8-grade-two-explicit-spinor-current-action",
        "finding:intrinsic-source-jet-removes-pure-field-frame-motion",
        "finding:unchanged-ns-vector-generators-cannot-create-d5-spinor-fields",
        "finding:two-face-noncommuting-geometric-response-calculus",
        "finding:raw-face-mask-fails-source-kernel-null-descent",
        "finding:action-response-sewing-does-not-select-lf-intervention",
        "finding:faithful-gns-completion-closes-automated-source-query-feedback",
        "finding:whole-quarter-channels-identified-by-exact-internal-kernel",
        "finding:actual-joint-insertion-hull-resolves-eight-coefficient-directions",
        "finding:e8-bare-trilinear-and-single-anchor-weyl-class-insufficient",
        "finding:complete-family-four-cycle-class-separates-native-clock-from-sm-preservation",
        "finding:native-clock-word-transpose-has-static-rank-four-counterexample",
        "finding:whole-constant-coupled-source-is-implementable-despite-isolated-insertion-domain",
        "finding:fixed-quarter-jloc-pfaffian-closure-fails-local-skew-test",
        "finding:same-collar-has-canonical-selfdual-kinetic-pairing-and-charged-determinant",
        "finding:quarter-cover-pairing-does-not-supply-local-normal-or-select-scale-measure",
        "finding:fixed-aps-source-has-whole-inverse-and-finite-boundary-correction",
        "finding:mass-only-deck-relative-heat-action-fails-free-infinite-trace",
        "finding:finite-antiperiodic-scalar-wall-has-no-exact-massless-normal-mode",
        "finding:spectral-heat-covariance-positivity-does-not-imply-time-rp",
        "finding:quaternionic-sm-mass-winding-is-not-normal-determinant-clutch",
        "finding:bounded-relative-trace-obstruction-and-whole-genuine-pair-heat-reader",
        "finding:normal-clutch-orbit-index-equivariantly-recovers-given-weak-module",
        "finding:normal-orbit-spin-family-index-and-signed-charge-balance",
        "finding:fixed-flavor-cartan-sign-cover-and-root-lift-parity",
        "finding:original-family-vector-action-is-not-cartan-rotation-or-spinlift-selection",
        "finding:actual-genus-two-sign-family-closes-common-parallel-k-connection",
        "finding:actual-q-quarter-extends-common-k-connection-and-real-coefficient-lifts",
        "finding:real-source-reflection-is-not-linear-duality-or-compact-clock-inversion",
        "finding:source-selected-spin-and-q-signs-have-three-explicit-global-kernels",
        "finding:source-s0-szego-point-legs-generate-q-kernels-with-typed-products",
        "finding:actual-spin-kernel-determinant-has-inverse-q-norm-character",
        "finding:closed-source-flavor-spin-has-functorial-pfaffline-with-charged-type-limit",
        "finding:original-relative-seam-has-explicit-whole-graph-fredholm-mismatch",
        "finding:whole-constant-coupled-clutch-defects-are-exact-internal-kernels",
        "finding:old-complex-scalar-clutch-separates-c0-and-product-reflection",
        "finding:scalar-defect-circle-character-does-not-select-projector-holonomy",
        "finding:source-spin-is-canonical-ramified-base-spin-pullback",
        "finding:original-v117-has-explicit-algebraic-q-cover-fundamental-frame",
        "finding:double-circle-reflection-does-not-fix-entire-cubic-cable-seam",
        "finding:actual-fuchs-real-map-transports-whole-given-finite-source-state-and-time",
        "finding:actual-e0-compensator-extends-whole-k-e8-with-transported-quarter",
        "finding:actual-v252-entire-real-basis-has-signed-p2-dual-dictionary",
        "finding:chiral-fuchs-source-transport-has-distinct-majorana-phase-and-torus-absorption",
        "finding:native-theta-composition-retains-conjugate-source-and-parity-square",
        "finding:single-graph-range-isometry-does-not-give-local-whole-seed-map",
        "finding:hard-holonomy-selfduality-leg-and-finite-character-phase-separation",
        "finding:original-z6-positive-structural-kernel-does-not-supply-oriented-phase-observable",
        "finding:original-genus-two-hexagonal-rotation-has-concrete-rho-hodge-line",
        "finding:original-mixed-spin-products-realize-positive-u6-cycle",
        "finding:actual-spin-quadratics-reach-original-hexagonal-hodge-line",
        "finding:spin-pair-car-transport-retains-parity-and-whole-source-types",
        "finding:real-total-trace-does-not-exclude-coherent-phase-reader",
        "finding:geometric-six-spin-clock-differs-from-marked-native-clock",
        "finding:whole-q-to-p-pullback-transports-given-source-state-and-time",
        "finding:actual-fixpoint-evaluation-keeps-coefficient-signs-and-source-type",
        "finding:original-source-naturality-needs-no-second-p-image-background",
        "finding:canonical-spin2-lift-carries-whole-annulus-source-with-reversed-time",
        "finding:geometric-spin-collar-map-closes-weighted-image-domain-with-total-deck-action",
        "finding:claude33-annulus-map-is-valid-distinct-type-not-unique-geometric-spin-selection",
        "finding:original-mixed-spin-products-already-have-complete-dual-hom-coefficients",
        "finding:whole-f5-graph-supplies-canonical-reverse-qp-hom-kernel",
        "finding:actual-spin-product-weight-has-canonical-trace-class-form-pairing",
        "finding:fully-descended-covariant-reverse-kernel-removes-rho-form-channel",
        "finding:original-a4-linear-quarter-does-not-supply-proposed-scalar-mu3-repair",
        "finding:offdiagonal-hom-matches-a4-family-six-but-not-two-triplet-e8-product",
        "finding:compact-e8-adjoint-completes-family-six-with-carrier-rank-retained",
        "finding:three-family-cx-recursion-closes-exact-d8-with-minimal-original-escape",
        "finding:original-weak-current-coefficient-has-explicit-finite-oneform-consumer",
        "finding:original-d5-zero-mode-closes-triplet-carrier-and-fixed-glue-is-minimal-escape",
        "finding:mixed-spin-full-deck-has-no-rank-three-projector",
        "finding:actual-reverse-hom-diagonal-supplies-global-q-hodge-response",
        "finding:bare-c3-kernel-diagonal-yukawas-have-common-real-frame",
        "finding:original-primitive-action-is-present-highpass-is-not-finite-selector",
        "finding:atlas-detail-retrieval-recovers-existing-d4-solution",
        "finding:original-poisson-normal-trace-fixes-whole-calderon-graph",
        "finding:induced-original-hardy-defect-gives-finite-whole-source-reader",
        "finding:original-weak-quartic-action-has-nonzero-nonlinear-response",
        "finding:all-common-leading-distance-clutches-fail-original-flavor-lengths",
        "finding:actual-hardy-response-has-weak-clifford-odd-and-ko-real-completion",
        "finding:two-entry-hardy-adjoint-hierarchy-fails-full-ko-and-su2",
        "finding:actual-fuchs-halfpath-has-canonical-g-metric-c6-kernel-map",
        "finding:original-real-delta-jet-reads-same-marked-hardy-open-word",
        "finding:four-actual-fuchs-spine-edges-recover-marked-meridians-and-nonzero-open-word",
        "finding:native-adjoint-quarter-clock-cannot-compensate-hardypair-odd-character",
        "finding:whole-cubic-gram-differential-retains-hardy-phase-and-cross-resonance",
        "finding:complete-original-mu01-hardy-reader-has-two-calibrated-entries",
        "finding:whole-six-original-hardy-inputs-generate-full-family-coefficient-algebra",
        "finding:actual-whole-hardy-alphabet-has-finite-particle-hole-wick-block",
        "finding:bare-unsmeared-mu-crossface-quadratic-fails-vacuum-domain",
        "finding:linear-spectral-centralization-does-not-replace-full-marked-hardy-source",
        "finding:original-admissibility-projector-is-explicit-penalties-do-not-read-retained-amplitudes",
        "finding:actual-original-inner-fluctuation-has-whole-noncommuting-two-face-response",
        "finding:fiber-only-primitive-split-and-frozen-gap-miss-original-action",
        "finding:oriented-resolvent-weights-do-not-identify-original-reflection-producer",
        "finding:current-odd-carrier-reader-is-lossless-on-whole-coupled-source",
        "finding:ap-spectral-formulas-do-not-force-normal-domain-or-complete-source",
        "finding:actual-weak-connection-transmits-family-orientation-to-both-faces",
        "finding:original-higgs-variation-is-not-independent-yukawa-stationarity",
        "finding:actual-weak-cap-converges-on-whole-quarter-circle",
        "finding:canonical-clifford-double-derives-normal-ap-and-typed-deck",
        "finding:universal-graded-source-reducing-images-do-not-fix-carrier-frame",
        "finding:whole-two-face-compression-projector-defect-detects-hidden-source-loss",
        "finding:actual-v252-quark-lepton-split-prevents-weak-grading-uniqueness-inference",
        "finding:actual-graded-hypercharge-trace-rejects-quark-lepton-sign-intertwiner",
        "finding:separable-normal-real-maps-compose-to-given-deck-and-quarter-image",
        "finding:actual-gauge-real-frames-leave-carrier-phase-as-coordinate-freedom",
        "finding:original-higgs-family-has-one-complex-kk-class-but-different-whole-sources",
        "finding:signed-one-particle-modular-weight-is-not-the-positive-even-heat-operator",
        "finding:three-moments-select-one-exponential-exactly-in-positive-laplace-class",
        "finding:existing-ap-positive-source-factor-has-oriented-poisson-heat-bridge",
        "finding:actual-v252-entire-compatibility-family-has-generic31-dimensional-quotient",
        "finding:full-memory-schur-reconstruction-preserves-every-ordered-word-under-nesting",
        "finding:original-return-and-native-j-have-minimal68-dimensional-all-word-memory",
        "finding:original-full-source-regulator-flow-retains-normalized-kernel-and-vacuum-factor",
        "finding:full-reflected-quadratic-regulator-cone-preserves-every-rp-source",
        "finding:original-shared-higgs-source-induces-fixed-yukawa-mixed-matter-words",
        "finding:local-even-yukawa-sewing-preserves-given-joint-graded-rp",
        "finding:whole-original-constant-higgs-orbit-has-reduced-majorana-spectral-response",
        "finding:archived-geometric-higgs-residue-bulk-needs-coefficient-identification-repair",
        "finding:original-local-higgs-residue-linear-coefficients-fail-same-trace-heat-ratio",
        "finding:whole-original-higgs-moments-fix-conditional-local-kinetic-mass-quartic",
        "finding:whole-variable-original-higgs-bound-reference-response-and-forced-clifford-factor",
        "finding:original-symmetric-scale-map-fails-simultaneous-higgs-einstein-repair",
        "finding:whole-original-higgs-inner-scale-and-affine-full-source-transport",
        "finding:normal-higgs-mode-type-and-exact-kernel-metric-require-actual-relative-operator-map",
        "finding:original-superconnection-grades-and-weighted-heat-retain-higgs-kinetic-excursions",
        "finding:whole-normal-reduction-preserves-relative-spectral-reader-and-higgs-jets",
        "finding:original-admissibility-own-spectral-reader-has-no-free-low-mode-residue-repair",
        "finding:whole-total-chiral-positive-spectrum-retains-higgs-kinetics-and-common-half-trace",
        "finding:whole-scale-energy-response-uniquely-reconstructs-normal-profile",
        "finding:original-relative-scale-flow-separates-residue-finite-part-and-local-anomaly",
        "finding:original-whole-green-produces-energy-response-with-correct-orthogonal-trace",
        "finding:whole-mixed-scale-source-reads-bosonic-insertion-before-higgs-average",
        "finding:current-original-spectral-action-triplet-fails-positive-laplace-class-unless-leading-relative-term-cancels",
        "finding:correct-current-argument-weights-link-whole-higgs-threshold-and-same-einstein-response",
        "finding:whole-original-spinor-curvature-a4-has-no-independent-scalaron-r-squared",
        "finding:whole-original-entropic-curvature-corrects-max-symmetric-scalar-scale",
        "finding:given-whole-higgs-source-fixes-radial-curvature-susceptibility-without-extra-scalar",
        "finding:actual-original-four-ghost-complex-preserves-base-selector-with-full-cohomology-and-distinct-traces",
        "finding:whole-original-admissibility-energy-source-recovers-constraints-but-cannot-replace-geometric-operator",
        "finding:original-moving-normal-kernel-fixes-induced-connection-and-full-scale-covariance",
        "finding:fixed-original-o1-source-rejects-nonconstant-normal-multiplier-and-fixes-placement-test",
        "finding:original-master-variation-keeps-spin-weight-and-active-source-route-typed",
        "finding:shared-original-higgs-mixture-has-invariant-color-fourpoint-wick-defect",
        "finding:bounded-fixed-rank-spectral-action-does-not-normalize-noncompact-higgs-volume",
        "finding:whole-original-quarter-source-bounds-higgs-weight-uniformly-through-joint-mode-ir-limits",
        "finding:original-o1-covariance-leaves-full-radial-higgs-law-and-exact-fourier-source-readout",
        "finding:whole-process-reconstruction-priority-and-original-locality-geometry-input-audit",
        "finding:actual-disjoint-local-weyls-acquire-spurious-commutator-under-bare-native-compression",
        "finding:actual-quarter-current-state-excludes-nontrivial-geometric-local-point-time",
        "finding:actual-elliptic-interval-car-source-produces-local-modular-generator",
        "finding:actual-proper-two-face-regions-do-not-contain-pure-local-source-double",
        "finding:actual-elliptic-whole-environment-fixes-regional-car-partner",
        "finding:actual-original-color-triplet-has-spectator-dependent-modular-field-law",
        "finding:actual-color-marginal-has-complete-finite-source-reader-and-mixture-criterion",
        "finding:two-and-four-field-words-do-not-complete-original-color-triplet-source",
        "finding:whole-original-quarter-occupation-tower-uniquely-reconstructs-common-radial-higgs-law",
        "finding:finite-original-tower-reader-and-existing-weak-symmetry-fix-shared-matter-responses",
        "finding:unchanged-free-quarter-source-cannot-be-nontrivial-integrated-higgs-mixture",
        "finding:actual-e8-record-doily-action-is-signed-p-plus-compatible-point-with-typed-sym2-lift",
        "finding:actual-e8-record-words-distinguish-order-and-retain-phase-beyond-fifteen-labels",
        "finding:literal-doily-incidence-output-and-strict-causal-order-on-fifteen-source-labels-excluded",
        "finding:actual-doily-context-record-characters-have-ranks-four-two-two-two",
        "finding:canonical-standard-split-localizes-original-source-record-words-with-full-descendant-companion",
        "finding:unchanged-global-sixty-event-source-fails-local-exterior-current-test",
        "finding:canonical-source-localization-changes-exterior-state-and-does-not-select-event-origin",
        "finding:original-elliptic-fixed-holonomy-insertion-gives-local-operation-on-unchanged-source",
        "finding:original-quarter-pair-fixes-square-charge-law-half-response-and-real-quillen-norm",
        "finding:actual-elliptic-source-determines-all-path-normalized-causal-current-response",
        "finding:original-higgs-quark-source-has-exact-whole-su3-counting-response",
        "finding:single-original-color-parity-separates-free-quillen-from-positive-higgs-integrated-source",
        "finding:whole-original-quarter-graph-color-response-is-relative-and-diagonal-singlet",
        "finding:original-center-selector-requires-typed-state-field-and-face-transport",
        "finding:original-color-center-fixes-whole-e6-a2-source-and-charged-ward-word-action",
        "finding:original-color-neutral-lattice-class-cannot-supply-odd-fields-by-inherited-superlattice-extension",
        "finding:original-color-center-extends-to-folded-ns-but-full-current-action-needs-spin-sector-transport",
        "finding:actual-color-spin-car-mixed-word-has-complete-thermal-source-and-cyclic-sewing",
        "finding:exact-original-ns-ramond-heat-equality-does-not-identify-fixed-color-restricted-source",
        "finding:original-scalar-cover-profile-cannot-implement-fixed-spin-cartan-loop",
        "finding:actual-spin-cartan-composite-source-retains-multi-arc-support-and-fails-additive-even-locality",
        "finding:actual-color-selector-admits-exactly-eighth-periodic-original-density-variations",
        "finding:actual-color-is-internal-exactly-on-eightfold-coordinate-source-regions",
        "finding:original-whole-two-face-multiplier-reduction-is-only-constant-residue-diagonal",
        "finding:original-two-face-local-fock-implementation-and-exact-source-variation-differ-from-reduction",
        "finding:whole-undecorated-transfer-does-not-determine-original-marked-field-source",
        "finding:volume-wise-injective-positive-transfer-does-not-prove-uniform-parent-gap",
        "finding:original-c6-half-power-has-exact-entire-positive-ordered-word-expansion",
        "finding:original-c6-whole-word-sewing-retains-source-amplitude-and-controlled-infinite-tail",
        "finding:historical-parent-and-actual-field-time-bridges-must-be-reused-before-new-source-models",
        "finding:actual-native-v990-full-timed-words-recover-kubo-response-distinct-from-instantaneous-variance",
        "finding:actual-original-higgs-force-has-complete-ordered-fermionic-determinant-source",
        "finding:finite-covariant-higgs-composites-have-no-nontrivial-positive-instantaneous-classical-law",
        "finding:actual-original-collective-mean-has-exact-ball-domain-and-relative-entropy-action",
        "finding:original-uniform-allmode-force-clt-is-nonzero-only-before-fixed-time-smearing",
        "finding:original-unscaled-time-smeared-force-has-selfadjoint-whole-ordered-source",
        "finding:original-local-higgs-force-has-whole-centered-hilbert-schmidt-determinant-source",
        "finding:given-original-circle-force-source-generates-an-exact-causal-local-net",
        "finding:literal-archived-global-rg-kernel-does-not-preserve-finite-measures",
        "finding:native-higgs-marginal-retains-induced-vertex-and-noether-response",
        "finding:isospectral-noether-orbit-does-not-supply-original-metric-higgs-variation",
        "finding:metric-dtn-form-bound-needs-transfer-comparison-for-gap",
        "finding:conditional-higgs-pair-winding-protection-in-common-circle-carrier",
        "finding:natural-five-clock-carrier-excludes-invariant-three-transfer-compression",
        "finding:shared-scalar-shift-whole-retained-higgs-sector-has-index-minus-eighteen",
        "finding:whole-coupled-quark-circle-common-shift-cancels-relative-winding",
        "finding:one-sided-shift-full-higgs-pair-is-hilbert-schmidt-not-trace-class",
        "finding:selective-coupled-higgs-reader-has-well-defined-det2",
        "finding:v311-population-reader-does-not-identify-quantum-channel",
        "finding:declared-charged-pair-relation-leaves-five-word-ambiguity",
        "finding:full-creator-relations-close-fixed-free-two-mode-ambiguity",
        "finding:comoving-diagnostic-clock-exact-response-but-nonlocal-mode-transport",
        "finding:free-quarter-two-face-response-strip-with-relative-zero-mode-defect",
        "finding:e8-hodge-event-perfectly-distinguishable-from-every-unitary",
        "finding:common-five-slot-transfer-preserves-e8-bracket-not-automatic-tp",
        "finding:comoving-clock-dynamic-higgs-connection-and-charged-flow",
        "finding:r064-sixty-record-channel-exact-distance-and-continuous-record-parent",
        "finding:two-syntheses-shared-source-priorities-and-typed-limits",
        "finding:weighted-index-trace-and-selfdual-theta-require-original-selection",
        "finding:ordered-quarter-fourfield-e8-record-and-charge-filters",
        "finding:r091-recovered-native-hodge-sewing-and-full96-quarter-complement",
        "finding:r092-fold-ward-target-neutral-source-and-smaller-creation-branch",
        "finding:r093-original-kink-time-response-distinguishes-quarter-states",
        "finding:r093-ordered-current-kink-and-eight-arc-locality",
        "finding:r094-original-five-excitations-distinct-from-clustering-vacuum",
        "finding:r096-joint-higgs-twoface-source-constraint"
      ],
      "tasks": [
        {
          "id": "task:calibrate-typed-operator-synthesis",
          "titleEN": "Calibrate new search without supplying the target map",
          "status": "resolved",
          "resolutionIds": [
            "finding:typed-operator-search-calibrated"
          ],
          "requiredPremiseIds": [
            "premise:declared-finite-operator-synthesis-inputs"
          ]
        },
        {
          "id": "task:synthesize-joint-source-operator-dictionary",
          "titleEN": "Synthesize a joint field, marking and transfer dictionary",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:implement-positive-source-kernel-synthesis",
          "titleEN": "Search for the positive common source kernel itself",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:construct-certify-discriminate-finite-kernels",
          "titleEN": "Automatically construct, certify and distinguish finite kernels",
          "status": "resolved",
          "resolutionIds": [
            "finding:positive-kernel-construction-with-exact-discriminator"
          ],
          "requiredPremiseIds": [
            "premise:finite-positive-kernel-search-class"
          ]
        },
        {
          "id": "task:repair-and-pass-collar-source-control",
          "titleEN": "Repair search conditions against source types and collar controls",
          "status": "resolved",
          "resolutionIds": [
            "finding:source-typed-kernel-search-controls"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:reconstruct-canonical-collar-multifield-ray",
          "titleEN": "Reconstruct multi-field sewing of the canonically identified boundary source",
          "status": "resolved",
          "resolutionIds": [
            "finding:collar-car-projection-multifield-ray"
          ],
          "requiredPremiseIds": [
            "premise:actual-product-collar-two-face-data",
            "premise:actual-car-graph-and-oriented-boundary-pairing"
          ]
        },
        {
          "id": "task:retain-source-reference-amplitude",
          "titleEN": "Retain absolute reference amplitude in search",
          "status": "resolved",
          "resolutionIds": [
            "finding:source-reference-amplitude-retained"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:integrate-given-free-source-jointly",
          "titleEN": "Integrate supplied free source including boundary response and scalar",
          "status": "resolved",
          "resolutionIds": [
            "finding:joint-fermionic-source-schur-functional"
          ],
          "requiredPremiseIds": [
            "premise:given-complex-fermion-source-quantization"
          ]
        },
        {
          "id": "task:reconstruct-complete-nongaussian-car-link",
          "titleEN": "Convert the full finite CAR kernel into a link functional",
          "status": "resolved",
          "resolutionIds": [
            "finding:complete-car-kernel-fixes-nongaussian-link-functional"
          ],
          "requiredPremiseIds": [
            "premise:complete-labeled-even-car-kernel"
          ]
        },
        {
          "id": "task:close-original-recursive-word-memory",
          "titleEN": "Close original recursive word reconstruction with memory",
          "status": "resolved",
          "resolutionIds": [
            "finding:full-memory-schur-reconstruction-preserves-every-ordered-word-under-nesting",
            "finding:original-return-and-native-j-have-minimal68-dimensional-all-word-memory"
          ],
          "requiredPremiseIds": [
            "premise:original-invariant-word-domain-with-nested-readouts",
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-neutral-root-pair-moment-fields",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:original-positive-adjoint-two-insertion-response-profiles",
            "premise:specified-original-pair-ordered-response-composition",
            "premise:original-single-beta-return-with-native-cartan-j-word-alphabet"
          ]
        },
        {
          "id": "task:retain-full-original-source-through-regulator-and-readout",
          "titleEN": "Retain whole original caller through scale and readout",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-full-source-regulator-flow-retains-normalized-kernel-and-vacuum-factor",
            "finding:full-reflected-quadratic-regulator-cone-preserves-every-rp-source"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:historical-original-circle-and-transfer-source-passages",
            "premise:original-regulated-full-source-with-actual-measure-and-legs",
            "premise:finite-reflected-positive-quadratic-ir-regulator-class"
          ]
        },
        {
          "id": "task:execute-original-shared-higgs-source-and-local-rp-sewing",
          "titleEN": "Execute actual shared Higgs source with full matter words and local RP",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-shared-higgs-source-induces-fixed-yukawa-mixed-matter-words",
            "finding:local-even-yukawa-sewing-preserves-given-joint-graded-rp"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:actual-full-v252-unimodular-p2-gauge-lift",
            "premise:actual-shared-real-higgs-yukawa-consumer-and-given-boson-fiber",
            "premise:original-graded-rp-reference-and-even-local-reflected-half-action"
          ]
        },
        {
          "id": "task:reduce-whole-constant-original-higgs-spectral-fiber",
          "titleEN": "Reduce whole constant original Higgs spectral fiber",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-original-constant-higgs-orbit-has-reduced-majorana-spectral-response"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:actual-shared-real-higgs-yukawa-consumer-and-given-boson-fiber",
            "premise:original-constant-higgs-fiber-and-symmetric-majorana-spectral-consumer"
          ]
        },
        {
          "id": "task:resolve-original-geometric-higgs-residue-sign-and-field-map",
          "titleEN": "Resolve original geometric Higgs residue sign and field map",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:literal-archived-geometric-residue-and-common-higgs-field-comparison"
          ]
        },
        {
          "id": "task:evaluate-original-local-higgs-heat-coefficient-compatibility",
          "titleEN": "Evaluate local original Higgs heat identification and whole finite consumer",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-local-higgs-residue-linear-coefficients-fail-same-trace-heat-ratio",
            "finding:whole-original-higgs-moments-fix-conditional-local-kinetic-mass-quartic"
          ],
          "requiredPremiseIds": [
            "premise:local-same-trace-higgs-residue-and-conditional-product-dirac-consumer",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:evaluate-whole-variable-bound-higgs-reference-and-scale-map",
          "titleEN": "Execute whole variable bound Higgs reference and original scale map",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-variable-original-higgs-bound-reference-response-and-forced-clifford-factor",
            "finding:original-symmetric-scale-map-fails-simultaneous-higgs-einstein-repair"
          ],
          "requiredPremiseIds": [
            "premise:whole-variable-higgs-bound-reference-and-conformal-dirac-map",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-original-inner-higgs-scale-and-normal-kernel-map",
          "titleEN": "Execute whole inner Higgs scale/source and normal kernel transport",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-original-higgs-inner-scale-and-affine-full-source-transport",
            "finding:normal-higgs-mode-type-and-exact-kernel-metric-require-actual-relative-operator-map"
          ],
          "requiredPremiseIds": [
            "premise:original-inner-higgs-central-scale-and-finite-affine-source-map",
            "premise:given-normal-higgs-mode-realization-and-distinct-module-types",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-given-superconnection-grades-and-whole-normal-spectral-reader",
          "titleEN": "Execute given whole superconnection and normal spectral reader",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-superconnection-grades-and-weighted-heat-retain-higgs-kinetic-excursions",
            "finding:whole-normal-reduction-preserves-relative-spectral-reader-and-higgs-jets"
          ],
          "requiredPremiseIds": [
            "premise:given-product-clifford-grades-parallel-projection-and-complete-relative-block-reader",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:evaluate-own-admissibility-and-whole-total-chiral-spectral-readers",
          "titleEN": "Determine own-admissibility and whole total-chiral spectral readers",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-admissibility-own-spectral-reader-has-no-free-low-mode-residue-repair",
            "finding:whole-total-chiral-positive-spectrum-retains-higgs-kinetics-and-common-half-trace"
          ],
          "requiredPremiseIds": [
            "premise:original-positive-admissibility-reader-and-given-total-chiral-product",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:derive-whole-scale-energy-profile-and-relative-flow",
          "titleEN": "Derive whole scale-energy map,normal profile and relative flow",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-scale-energy-response-uniquely-reconstructs-normal-profile",
            "finding:original-relative-scale-flow-separates-residue-finite-part-and-local-anomaly"
          ],
          "requiredPremiseIds": [
            "premise:whole-original-scale-congruence-and-specified-normal-energy-data",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-original-green-energy-producer-and-mixed-scale-readout",
          "titleEN": "Execute original Green energy producer and mixed scale readout",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-whole-green-produces-energy-response-with-correct-orthogonal-trace",
            "finding:whole-mixed-scale-source-reads-bosonic-insertion-before-higgs-average"
          ],
          "requiredPremiseIds": [
            "premise:given-whole-green-energy-source-and-conditional-scale-response",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:resolve-current-spectral-action-argument-and-common-higgs-gravity-weights",
          "titleEN": "Match current action argument and common Higgs/gravity weights",
          "status": "resolved",
          "resolutionIds": [
            "finding:current-original-spectral-action-triplet-fails-positive-laplace-class-unless-leading-relative-term-cancels",
            "finding:correct-current-argument-weights-link-whole-higgs-threshold-and-same-einstein-response"
          ],
          "requiredPremiseIds": [
            "premise:current-original-spectral-action-and-fixed-ordinary-moment-dictionary",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-whole-original-curvature-consumers-and-higgs-radial-source",
          "titleEN": "Execute whole original curvature consumers and Higgs radial source",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-original-spinor-curvature-a4-has-no-independent-scalaron-r-squared",
            "finding:whole-original-entropic-curvature-corrects-max-symmetric-scalar-scale",
            "finding:given-whole-higgs-source-fixes-radial-curvature-susceptibility-without-extra-scalar"
          ],
          "requiredPremiseIds": [
            "premise:original-whole-spinor-and-entropic-curvature-consumers-with-given-higgs-response",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-original-four-ghost-admissibility-source-and-geometric-type-test",
          "titleEN": "Execute original whole four-ghost source and geometric type test",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-original-four-ghost-complex-preserves-base-selector-with-full-cohomology-and-distinct-traces",
            "finding:whole-original-admissibility-energy-source-recovers-constraints-but-cannot-replace-geometric-operator"
          ],
          "requiredPremiseIds": [
            "premise:actual-original-four-ghost-commuting-projection-admissibility-complex",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-original-moving-normal-kernel-derivative-and-source-response",
          "titleEN": "Execute original moving normal kernel derivative and whole source response",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-moving-normal-kernel-fixes-induced-connection-and-full-scale-covariance"
          ],
          "requiredPremiseIds": [
            "premise:actual-smooth-normal-family-with-supplied-unitary-horizontal-connection",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:test-original-fixed-o1-source-and-normal-operator-placement",
          "titleEN": "Decide fixed original O1 source and normal operator placement",
          "status": "resolved",
          "resolutionIds": [
            "finding:fixed-original-o1-source-rejects-nonconstant-normal-multiplier-and-fixes-placement-test"
          ],
          "requiredPremiseIds": [
            "premise:fixed-original-o1-section-consumer-and-actual-normal-first-order-placement",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:test-whole-higgs-mixture-against-covariance-only-modular-reader",
          "titleEN": "Compare whole Higgs mixture and covariance reader on invariant four-field witness",
          "status": "resolved",
          "resolutionIds": [
            "finding:shared-original-higgs-mixture-has-invariant-color-fourpoint-wick-defect"
          ],
          "requiredPremiseIds": [
            "premise:actual-original-quark-higgs-family-and-positive-conditional-source-mixture",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:test-exact-bounded-cutoff-higgs-normalization-at-fixed-rank",
          "titleEN": "Test exact bounded cutoff on whole ordinary Higgs volume",
          "status": "resolved",
          "resolutionIds": [
            "finding:bounded-fixed-rank-spectral-action-does-not-normalize-noncompact-higgs-volume"
          ],
          "requiredPremiseIds": [
            "premise:fixed-rank-bounded-cutoff-with-ordinary-noncompact-higgs-volume",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:test-whole-original-circle-joint-mode-ir-higgs-normalization",
          "titleEN": "Test whole original circle joint mode and IR Higgs normalization",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-original-quarter-source-bounds-higgs-weight-uniformly-through-joint-mode-ir-limits"
          ],
          "requiredPremiseIds": [
            "premise:whole-constant-original-quarter-source-positive-traceclass-profile-and-flat-higgs-volume",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:classify-original-o1-mode-covariance-versus-whole-higgs-law",
          "titleEN": "Test original O1 mode covariance against whole Higgs-law class",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-o1-covariance-leaves-full-radial-higgs-law-and-exact-fourier-source-readout"
          ],
          "requiredPremiseIds": [
            "premise:given-o1-mode-isometry-and-weak-invariant-positive-higgs-covariance-marginal",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:reconstruct-original-whole-process-localization-before-geometric-readout",
          "titleEN": "Reconstruct original localization from the whole process",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:literal-whole-source-localization-and-geometric-readout-contracts",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:test-bare-native-code-word-compression-for-original-locality",
          "titleEN": "Test bare native-code word compression against actual locality",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-disjoint-local-weyls-acquire-spurious-commutator-under-bare-native-compression"
          ],
          "requiredPremiseIds": [
            "premise:actual-proper-local-current-weyls-and-bare-native-code-compression",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:decide-original-quarter-geometric-local-time-and-interval-source-reader",
          "titleEN": "Decide original geometric local-time class and execute actual interval reader",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-quarter-current-state-excludes-nontrivial-geometric-local-point-time",
            "finding:actual-elliptic-interval-car-source-produces-local-modular-generator"
          ],
          "requiredPremiseIds": [
            "premise:actual-elliptic-quarter-supported-current-and-interval-car-modular-test",
            "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
            "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:derive-actual-regional-two-face-partner-and-supported-charged-word",
          "titleEN": "Execute actual regional two-face partner and supported charged word",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-proper-two-face-regions-do-not-contain-pure-local-source-double",
            "finding:actual-elliptic-whole-environment-fixes-regional-car-partner"
          ],
          "requiredPremiseIds": [
            "premise:actual-elliptic-two-face-projection-with-genuine-regional-support",
            "premise:typed-source-structures",
            "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks"
          ]
        },
        {
          "id": "task:derive-original-color-marginal-modular-law-and-complete-source-reader",
          "titleEN": "Execute original color marginal, modular field law and complete class reader",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-original-color-triplet-has-spectator-dependent-modular-field-law",
            "finding:actual-color-marginal-has-complete-finite-source-reader-and-mixture-criterion",
            "finding:two-and-four-field-words-do-not-complete-original-color-triplet-source"
          ],
          "requiredPremiseIds": [
            "premise:actual-original-rightup-triplet-fixed-family-and-even-color-source",
            "premise:typed-source-structures",
            "premise:actual-original-quark-higgs-family-and-positive-conditional-source-mixture"
          ]
        },
        {
          "id": "task:reconstruct-common-higgs-law-from-whole-original-quarter-source",
          "titleEN": "Reconstruct common Higgs law from whole original quarter source tower",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-original-quarter-occupation-tower-uniquely-reconstructs-common-radial-higgs-law",
            "finding:finite-original-tower-reader-and-existing-weak-symmetry-fix-shared-matter-responses",
            "finding:unchanged-free-quarter-source-cannot-be-nontrivial-integrated-higgs-mixture"
          ],
          "requiredPremiseIds": [
            "premise:original-common-positive-constant-higgs-law-over-whole-quarter-tower",
            "premise:typed-source-structures",
            "premise:actual-original-quark-higgs-family-and-positive-conditional-source-mixture",
            "premise:actual-original-rightup-triplet-fixed-family-and-even-color-source",
            "premise:given-o1-mode-isometry-and-weak-invariant-positive-higgs-covariance-marginal"
          ]
        },
        {
          "id": "task:execute-original-record-action-on-doily-and-ordered-source-response",
          "titleEN": "Execute original record action on Doily and ordered whole source response",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-e8-record-doily-action-is-signed-p-plus-compatible-point-with-typed-sym2-lift",
            "finding:actual-e8-record-words-distinguish-order-and-retain-phase-beyond-fifteen-labels",
            "finding:literal-doily-incidence-output-and-strict-causal-order-on-fifteen-source-labels-excluded",
            "finding:actual-doily-context-record-characters-have-ranks-four-two-two-two"
          ],
          "requiredPremiseIds": [
            "premise:original-sixty-e8-unrefined-record-source-and-typed-doily-observables",
            "premise:typed-source-structures",
            "premise:original-native-pauli-and-v419-f20-source-words",
            "premise:actual-single-source-event-coherent-stinespring-record"
          ]
        },
        {
          "id": "task:localize-original-record-process-and-test-unchanged-source-exterior",
          "titleEN": "Locally implement original record process and test unchanged exterior source",
          "status": "resolved",
          "resolutionIds": [
            "finding:canonical-standard-split-localizes-original-source-record-words-with-full-descendant-companion",
            "finding:unchanged-global-sixty-event-source-fails-local-exterior-current-test",
            "finding:canonical-source-localization-changes-exterior-state-and-does-not-select-event-origin"
          ],
          "requiredPremiseIds": [
            "premise:given-e8-current-vacuum-net-and-original-grade-two-descendant-with-standard-split",
            "premise:typed-source-structures",
            "premise:local-e8-current-response-realization",
            "premise:original-sixty-e8-unrefined-record-source-and-typed-doily-observables"
          ]
        },
        {
          "id": "task:execute-original-elliptic-supported-insertion-and-causal-source-test",
          "titleEN": "Execute original elliptic insertion,whole source and causal response",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-elliptic-fixed-holonomy-insertion-gives-local-operation-on-unchanged-source",
            "finding:original-quarter-pair-fixes-square-charge-law-half-response-and-real-quillen-norm",
            "finding:actual-elliptic-source-determines-all-path-normalized-causal-current-response"
          ],
          "requiredPremiseIds": [
            "premise:given-original-elliptic-quarter-source-and-supported-scalar-current-insertion",
            "premise:typed-source-structures",
            "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
            "premise:original-quarter-car-fold-current-subalgebra"
          ]
        },
        {
          "id": "task:execute-original-mixed-higgs-color-group-response",
          "titleEN": "Evaluate actual joint Higgs and color group source",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-higgs-quark-source-has-exact-whole-su3-counting-response",
            "finding:single-original-color-parity-separates-free-quillen-from-positive-higgs-integrated-source"
          ],
          "requiredPremiseIds": [
            "premise:original-quark-particle-color-group-counting-on-whole-constant-quarter-source",
            "premise:typed-source-structures",
            "premise:actual-original-quark-higgs-family-and-positive-conditional-source-mixture",
            "premise:actual-full-v252-unimodular-p2-gauge-lift",
            "premise:given-original-elliptic-quarter-source-and-supported-scalar-current-insertion"
          ]
        },
        {
          "id": "task:execute-whole-original-graph-color-selector-and-relative-reader",
          "titleEN": "Execute whole original graph color reader with typed original center selector",
          "status": "resolved",
          "resolutionIds": [
            "finding:whole-original-quarter-graph-color-response-is-relative-and-diagonal-singlet",
            "finding:original-center-selector-requires-typed-state-field-and-face-transport"
          ],
          "requiredPremiseIds": [
            "premise:given-whole-quarter-graph-and-canonical-diagonal-original-color-lift",
            "premise:typed-source-structures",
            "premise:constant-odd-quarter-circle-source-sea-and-full-graph",
            "premise:original-quark-particle-color-group-counting-on-whole-constant-quarter-source",
            "premise:actual-full-v252-unimodular-p2-gauge-lift",
            "premise:stipulated-e8-h8-with-original-color-center-selector"
          ]
        },
        {
          "id": "task:execute-original-color-fixed-field-source-and-first-graded-matter-test",
          "titleEN": "Execute original color fixed source and first inherited grading test",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-color-center-fixes-whole-e6-a2-source-and-charged-ward-word-action",
            "finding:original-color-neutral-lattice-class-cannot-supply-odd-fields-by-inherited-superlattice-extension"
          ],
          "requiredPremiseIds": [
            "premise:original-marked-color-center-on-whole-e8-lattice-current-source",
            "premise:typed-source-structures",
            "premise:stipulated-e8-h8-with-original-color-center-selector",
            "premise:local-e8-current-response-realization",
            "premise:fixed-native-d5-a3-compiler-split-and-seed",
            "premise:native-even-e8-gaussian-clock-and-sign-extension"
          ]
        },
        {
          "id": "task:execute-original-color-spin-mixed-four-word-and-full-sector-sewing",
          "titleEN": "Execute actual original color-spinor CAR four-word and complete spin-sector sewing",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-color-spin-car-mixed-word-has-complete-thermal-source-and-cyclic-sewing",
            "finding:exact-original-ns-ramond-heat-equality-does-not-identify-fixed-color-restricted-source"
          ],
          "requiredPremiseIds": [
            "premise:actual-thermal-quarter-source-and-retained-spin-parent-mixed-word",
            "premise:typed-source-structures",
            "premise:original-fixed-color-and-folded-ns-cartan-interface"
          ]
        },
        {
          "id": "task:execute-original-color-density-and-coordinate-region-descent",
          "titleEN": "Decide actual color admissibility of density operations and coordinate regions",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-color-selector-admits-exactly-eighth-periodic-original-density-variations",
            "finding:actual-color-is-internal-exactly-on-eightfold-coordinate-source-regions"
          ],
          "requiredPremiseIds": [
            "premise:actual-original-ns-color-center-continuous-density-and-coordinate-net",
            "premise:original-fixed-color-and-folded-ns-cartan-interface",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-original-calderon-operation-and-full-source-response",
          "titleEN": "Execute actual whole seam-operation class and two-face response",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-whole-two-face-multiplier-reduction-is-only-constant-residue-diagonal",
            "finding:original-two-face-local-fock-implementation-and-exact-source-variation-differ-from-reduction"
          ],
          "requiredPremiseIds": [
            "premise:actual-original-full-two-face-graph-and-local-matrix-operations",
            "premise:actual-product-collar-two-face-data",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-original-c6-entire-word-expansion-and-source-amplitude",
          "titleEN": "Execute entire original C6 word expansion with source amplitude",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-c6-half-power-has-exact-entire-positive-ordered-word-expansion",
            "finding:original-c6-whole-word-sewing-retains-source-amplitude-and-controlled-infinite-tail"
          ],
          "requiredPremiseIds": [
            "premise:original-bare-c6-fixed-analytic-half-power-and-literal-word-alphabet",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:execute-original-collective-higgs-force-source-and-mean-dual",
          "titleEN": "Execute original collective Higgs force source and entire mean dual",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-original-higgs-force-has-complete-ordered-fermionic-determinant-source",
            "finding:finite-covariant-higgs-composites-have-no-nontrivial-positive-instantaneous-classical-law",
            "finding:actual-original-collective-mean-has-exact-ball-domain-and-relative-entropy-action"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:actual-original-finite-car-higgs-force-and-faithful-collective-source"
          ]
        },
        {
          "id": "task:execute-original-allmode-force-with-actual-time-and-whole-source",
          "titleEN": "Execute original all-mode force with actual time and whole source",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-uniform-allmode-force-clt-is-nonzero-only-before-fixed-time-smearing",
            "finding:original-unscaled-time-smeared-force-has-selfadjoint-whole-ordered-source"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:actual-whole-circle-product-car-force-and-original-time-smearing"
          ]
        },
        {
          "id": "task:execute-original-local-spacetime-force-source-and-causal-lift",
          "titleEN": "Execute original local space/time force source and causal lift",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-local-higgs-force-has-whole-centered-hilbert-schmidt-determinant-source",
            "finding:given-original-circle-force-source-generates-an-exact-causal-local-net"
          ],
          "requiredPremiseIds": [
            "premise:typed-source-structures",
            "premise:original-constant-circle-local-higgs-force-and-hilbert-schmidt-time-source"
          ]
        }
      ]
    },
    {
      "id": "problem:reverse-bootstrap-physical-field-dictionary",
      "titleEN": "Origin reverse selection, field dictionary and original P1 kernel",
      "questionEN": "What does reverse selection of P1/P2 give for concrete local fields and multi-field responses, and which identification with the original P1 kernel has actually been proved?",
      "topicIds": [
        "topic:origin",
        "topic:e8",
        "topic:quantum"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2",
        "goal:T8"
      ],
      "acceptedPremiseIds": [
        "premise:reverse-bootstrap-e8-closure-framework",
        "premise:local-e8-current-response-realization",
        "premise:quarter-phase-retained",
        "premise:typed-source-structures",
        "premise:literal-microscopic-charged-scaling-and-v988-slater-data",
        "premise:original-v988-periodic-zero-mode-state-comparison"
      ],
      "findingIds": [
        "finding:reverse-bootstrap-retained-operator-scope",
        "finding:e8-current-dictionary-all-separated-responses",
        "finding:charged-scaling-microscopic-map-still-unspecified",
        "finding:original-slater-proxy-fitted-bound-fails-actual-norm",
        "finding:periodic-microscopic-zero-modes-distinguish-source-state"
      ],
      "tasks": [
        {
          "id": "task:explicit-e8-all-current-response-reader",
          "titleEN": "Build an explicit248-current reader with complete Ward response",
          "status": "resolved",
          "resolutionIds": [
            "finding:e8-current-dictionary-all-separated-responses"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization"
          ]
        },
        {
          "id": "task:audit-actual-charged-scaling-source-reproduction",
          "titleEN": "Reproduce actual charged source path to its first missing map",
          "status": "resolved",
          "resolutionIds": [
            "finding:charged-scaling-microscopic-map-still-unspecified",
            "finding:original-slater-proxy-fitted-bound-fails-actual-norm",
            "finding:periodic-microscopic-zero-modes-distinguish-source-state"
          ],
          "requiredPremiseIds": [
            "premise:literal-microscopic-charged-scaling-and-v988-slater-data",
            "premise:typed-source-structures"
          ]
        }
      ]
    },
    {
      "id": "problem:fundamental-process-classification",
      "titleEN": "Observable reality: minimal conditions and counterworlds",
      "questionEN": "Which operational conditions are independently required, which inequivalent models satisfy them, and which observable distinctions remain for selection?",
      "topicIds": [
        "topic:origin",
        "topic:quantum",
        "topic:universalraum"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T8",
        "goal:universalraum"
      ],
      "acceptedPremiseIds": [
        "premise:foundational-reset-no-target-input",
        "premise:finite-event-poset",
        "premise:finite-deterministic-joint-replication-control",
        "premise:finite-update-observation-errors",
        "premise:finite-operational-composition-action",
        "premise:repeatable-distinction-benchmark",
        "premise:three-explicit-observability-countermodels",
        "premise:finite-prefix-consistent-sequential-data"
      ],
      "findingIds": [
        "finding:replicated-deterministic-consistency-iff-influence-dag",
        "finding:three-binary-processes-two-copy-reversible-classification",
        "finding:independent-poset-information-composition-consequences",
        "finding:reset-diagnosis-not-global-tfpt-underdetermination",
        "finding:three-inequivalent-repeatable-observation-countermodels",
        "finding:repeatable-reports-do-not-imply-full-state-stability",
        "finding:finite-sequential-data-admit-classical-history-realization"
      ],
      "tasks": [
        {
          "id": "task:compare-four-independent-foundations",
          "titleEN": "Check four independent starting classes and counterworlds",
          "status": "resolved",
          "resolutionIds": [
            "finding:replicated-deterministic-consistency-iff-influence-dag",
            "finding:three-binary-processes-two-copy-reversible-classification",
            "finding:independent-poset-information-composition-consequences",
            "finding:reset-diagnosis-not-global-tfpt-underdetermination"
          ],
          "requiredPremiseIds": [
            "premise:foundational-reset-no-target-input",
            "premise:finite-event-poset",
            "premise:finite-deterministic-joint-replication-control",
            "premise:finite-update-observation-errors",
            "premise:finite-operational-composition-action"
          ]
        },
        {
          "id": "task:assess-record-preserving-composition-principle",
          "titleEN": "Jointly justify replica control and actual record persistence",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:foundational-reset-no-target-input"
          ]
        },
        {
          "id": "task:compare-minimal-observable-counterworlds",
          "titleEN": "Compare minimal observation conditions and counterworlds",
          "status": "resolved",
          "resolutionIds": [
            "finding:three-inequivalent-repeatable-observation-countermodels",
            "finding:repeatable-reports-do-not-imply-full-state-stability",
            "finding:finite-sequential-data-admit-classical-history-realization"
          ],
          "requiredPremiseIds": [
            "premise:repeatable-distinction-benchmark",
            "premise:three-explicit-observability-countermodels",
            "premise:finite-prefix-consistent-sequential-data"
          ]
        },
        {
          "id": "task:ground-observable-discriminator-constraints",
          "titleEN": "Empirically ground a discriminator and access conditions",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:foundational-reset-no-target-input"
          ]
        }
      ]
    },
    {
      "id": "problem:carrier-pascal-glue-formal-scope",
      "titleEN": "Formal carrier core: unbounded Pascal uniqueness and finite glue",
      "questionEN": "Which carrier and glue statements are already present in the papers and Lean sources, and which physical selection does not follow?",
      "topicIds": [
        "topic:e8",
        "topic:origin",
        "topic:validation"
      ],
      "goalIds": [
        "goal:T1",
        "goal:T2"
      ],
      "acceptedPremiseIds": [
        "premise:pascal-carrier-exhaustion-equation",
        "premise:specified-finite-carrier-discriminant-form",
        "premise:typed-source-structures"
      ],
      "findingIds": [
        "finding:pascal-carrier-rank-unique-unbounded-source-proof",
        "finding:finite-carrier-glue-elements-orbits-and-halfway"
      ],
      "tasks": [
        {
          "id": "task:review-unbounded-pascal-source",
          "titleEN": "Classify the unbounded Pascal proof",
          "status": "resolved",
          "resolutionIds": [
            "finding:pascal-carrier-rank-unique-unbounded-source-proof"
          ],
          "requiredPremiseIds": [
            "premise:pascal-carrier-exhaustion-equation"
          ]
        },
        {
          "id": "task:review-finite-glue-source-scope",
          "titleEN": "Classify the formal finite glue scope",
          "status": "resolved",
          "resolutionIds": [
            "finding:finite-carrier-glue-elements-orbits-and-halfway"
          ],
          "requiredPremiseIds": [
            "premise:specified-finite-carrier-discriminant-form"
          ]
        }
      ]
    },
    {
      "id": "problem:marked-seam-monodromy-and-lattice-lifts",
      "titleEN": "Active mathematical program: marked seam monodromies and lattice lifts",
      "questionEN": "Which marked seam/lattice transport geometries, lift classes and local-coefficient invariants arise from existing clock/reflection/quartic data? First complete case: sign lifts of fourJ representation on CP1 minusμ4.",
      "topicIds": [
        "topic:e8",
        "topic:clocks"
      ],
      "goalIds": [],
      "acceptedPremiseIds": [
        "premise:marked-seam-pure-mathematical-program",
        "premise:native-even-e8-gaussian-clock-and-sign-extension",
        "premise:ordered-four-punctures-fixed-quarter-projection",
        "premise:native-clock-complex-reflection-and-quartic-subspace",
        "premise:original-g31-central-determinant-extension",
        "premise:ordered-original-g31-reflection-and-clock-markings",
        "premise:original-full-g31-action-and-inherited-lattice-sign-extension",
        "premise:fixed-native-d5-a3-compiler-split-and-seed",
        "premise:existing-marked-cyclic-cover-and-burau-model",
        "premise:joint-native-compiler-cover-covariance-category",
        "premise:marked-c4-unitary-local-extension-reconstruction-class",
        "premise:connected-local-same-conformal-compiler-extension-with-generator",
        "premise:fixed-native-reflection-rotation-character-markings",
        "premise:ordered-quarter-clock-pure-hurwitz-equivalence",
        "premise:standard-e8-root-cut-full-flag-incidence-category",
        "premise:scalar-eight-channel-twist-and-explicit-compiler-charge-chart",
        "premise:full-unitary-e8-field-conjugacy-with-transported-marks",
        "premise:four-d8-sectors-with-fixed-affine-source-time"
      ],
      "findingIds": [
        "finding:marked-seam-category-and-character-valued-full-lift-factor",
        "finding:quarter-clock-sign-lift-complete-four-puncture-classification",
        "finding:quarter-lift-central-quotient-class-and-all-subgroup-restrictions",
        "finding:native-clock-reflection-full-factor-cocycle-trivialization",
        "finding:native-quartic-exterior-local-cohomology-distinguishes-clock-monodromies",
        "finding:orbifold-lift-criterion-and-determinant-eight-divisibility",
        "finding:original-g31-reflection-census-and-all-integral-cohomology",
        "finding:all62-original-clock-obstructions-and-central-variants",
        "finding:actual-sign-lift-g31-group-comparison",
        "finding:complete-g31-lift-invariants-and-explicit-collisions",
        "finding:e7-quarter-pairs-literal-native-g31-reflections",
        "finding:e6-root-subsystem-quarter-clock-discriminant-obstruction",
        "finding:e-series-square-cover-family-and-quarter-descent",
        "finding:e-series-integral-glue-and-normalization-nonidentifiability",
        "finding:global-original-g31-character-lift-nonsplitting",
        "finding:all62-original-character-clock-lifts-complete",
        "finding:all17-original-character-monodromy-lifts-complete",
        "finding:e7-pair-full-lattice-field-character-torsor",
        "finding:e-series-complete-native-compiler-cut-dictionary",
        "finding:common-cover-clock-and-maximal-integral-phi12-intertwiner",
        "finding:native-four-cut-projectors-unify48-fourthirds-fivequarters",
        "finding:complete-covariant-compiler-voa-source-and-weyl-closure",
        "finding:complete-fixed-jhat-cyclic-field-lift-and-associated-clock",
        "finding:relative-native-burau-clock-hom-faithful12-transport",
        "finding:e6-vertex-nonselection-survives-full-field-clock-phases",
        "finding:marked-c4-glue-unique-local-e8-vacuum-source",
        "finding:canonical-vacuum-clock-preserves-e6-nonselection",
        "finding:marked-generator-forces-e8-without-support-multiplicity-or-minimality-input",
        "finding:native-reflection-rotation-lift-classes-and-forgetful-images",
        "finding:quarter-clock-pure-mapping-class-complete-quotient",
        "finding:e8-cut-five-orbital-incidence-algebra-complete",
        "finding:compiler-d8-frame-requires-full-d5-a3-flag",
        "finding:tetrahedron-incidence-lossless-e6-reconstruction",
        "finding:restricted-clock-lift-and-global-cover-audit-reconciled",
        "finding:double-scalar-quarter-is-marked-glue-with-two-step-index",
        "finding:scalar-quarter-two-involutive-e8-extensions",
        "finding:native-quarter-glue-inner-conjugacy-and-oriented-tetrahedron-flags",
        "finding:native-cyclic-compiler-inner-conjugacy",
        "finding:recovered-e6-closed-81-current-module-and-unique-source-cubic",
        "finding:e-series-operational-role-review-and-source-time-centralizer"
      ],
      "tasks": [
        {
          "id": "task:define-marked-seam-transport-category",
          "titleEN": "Define marked seam category and factor formula",
          "status": "resolved",
          "resolutionIds": [
            "finding:marked-seam-category-and-character-valued-full-lift-factor"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:native-even-e8-gaussian-clock-and-sign-extension"
          ]
        },
        {
          "id": "task:classify-quarter-clock-four-puncture-sign-lifts",
          "titleEN": "Completely classify quarter-clock four-puncture lifts",
          "status": "resolved",
          "resolutionIds": [
            "finding:quarter-clock-sign-lift-complete-four-puncture-classification"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:ordered-four-punctures-fixed-quarter-projection"
          ]
        },
        {
          "id": "task:compute-quarter-central-quotient-restrictions",
          "titleEN": "Determine central class and all subgroup restrictions",
          "status": "resolved",
          "resolutionIds": [
            "finding:quarter-lift-central-quotient-class-and-all-subgroup-restrictions"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension"
          ]
        },
        {
          "id": "task:compute-native-g0-full-factor-class",
          "titleEN": "Determine full G0 cocycle and class",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-clock-reflection-full-factor-cocycle-trivialization"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace"
          ]
        },
        {
          "id": "task:compute-native-456-local-cohomology",
          "titleEN": "Determine native4/5/6local cohomology",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-quartic-exterior-local-cohomology-distinguishes-clock-monodromies"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:ordered-four-punctures-fixed-quarter-projection",
            "premise:native-clock-complex-reflection-and-quartic-subspace"
          ]
        },
        {
          "id": "task:classify-unrestricted-g31-reflection-monodromy",
          "titleEN": "Classify unrestricted G31 reflection monodromies",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-g31-reflection-census-and-all-integral-cohomology",
            "finding:all62-original-clock-obstructions-and-central-variants"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace"
          ]
        },
        {
          "id": "task:compute-lift-class-beyond-native-g0",
          "titleEN": "Determine lift class beyond G0",
          "status": "resolved",
          "resolutionIds": [
            "finding:global-original-g31-character-lift-nonsplitting"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace",
            "premise:original-full-g31-action-and-inherited-lattice-sign-extension"
          ]
        },
        {
          "id": "task:classify-reflection-equivariant-and-mapping-class-seam-lifts",
          "titleEN": "Classify additional reflection/clock equivariance and mapping-class quotients",
          "status": "open",
          "resolutionIds": [],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:ordered-four-punctures-fixed-quarter-projection"
          ]
        },
        {
          "id": "task:completed-equivariant-study-0",
          "titleEN": "Equivariant lifting and divisibility by eight",
          "status": "resolved",
          "resolutionIds": [
            "finding:orbifold-lift-criterion-and-determinant-eight-divisibility"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:original-g31-central-determinant-extension",
            "premise:ordered-original-g31-reflection-and-clock-markings"
          ]
        },
        {
          "id": "task:completed-equivariant-study-2",
          "titleEN": "All62clock obstructions and central transport variants",
          "status": "resolved",
          "resolutionIds": [
            "finding:all62-original-clock-obstructions-and-central-variants"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:original-g31-central-determinant-extension",
            "premise:ordered-original-g31-reflection-and-clock-markings"
          ]
        },
        {
          "id": "task:completed-equivariant-study-3",
          "titleEN": "Actual group comparison and central pushout exclusion",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-sign-lift-g31-group-comparison"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:original-g31-central-determinant-extension",
            "premise:ordered-original-g31-reflection-and-clock-markings"
          ]
        },
        {
          "id": "task:completed-equivariant-study-4",
          "titleEN": "Complete invariants and explicit collisions",
          "status": "resolved",
          "resolutionIds": [
            "finding:complete-g31-lift-invariants-and-explicit-collisions"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:original-g31-central-determinant-extension",
            "premise:ordered-original-g31-reflection-and-clock-markings"
          ]
        },
        {
          "id": "task:e-series-slices-under-actual-quarter-clock",
          "titleEN": "Compare E6 and E7 slices with the actual quarter clock",
          "status": "resolved",
          "resolutionIds": [
            "finding:e7-quarter-pairs-literal-native-g31-reflections",
            "finding:e6-root-subsystem-quarter-clock-discriminant-obstruction",
            "finding:e-series-square-cover-family-and-quarter-descent",
            "finding:e-series-integral-glue-and-normalization-nonidentifiability"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:native-even-e8-gaussian-clock-and-sign-extension"
          ]
        },
        {
          "id": "task:completed-original-g31-character-monodromy-lifts",
          "titleEN": "Classify original character-valued G31monodromy lifts",
          "status": "resolved",
          "resolutionIds": [
            "finding:all62-original-character-clock-lifts-complete",
            "finding:all17-original-character-monodromy-lifts-complete"
          ],
          "requiredPremiseIds": [
            "premise:original-full-g31-action-and-inherited-lattice-sign-extension",
            "premise:ordered-original-g31-reflection-and-clock-markings"
          ]
        },
        {
          "id": "task:complete-e7-pair-all-field-dictionary",
          "titleEN": "Determine E7-pair action on all original lattice fields",
          "status": "resolved",
          "resolutionIds": [
            "finding:e7-pair-full-lattice-field-character-torsor"
          ],
          "requiredPremiseIds": [
            "premise:original-full-g31-action-and-inherited-lattice-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace"
          ]
        },
        {
          "id": "task:complete-native-compiler-e-series-cut-dictionary",
          "titleEN": "Completely connect E6/E7 cuts to the original compiler",
          "status": "resolved",
          "resolutionIds": [
            "finding:e-series-complete-native-compiler-cut-dictionary"
          ],
          "requiredPremiseIds": [
            "premise:fixed-native-d5-a3-compiler-split-and-seed",
            "premise:native-even-e8-gaussian-clock-and-sign-extension"
          ]
        },
        {
          "id": "task:complete-cover-clock-actual-carrier-dictionary",
          "titleEN": "Compare common covers and actual clock carriers",
          "status": "resolved",
          "resolutionIds": [
            "finding:common-cover-clock-and-maximal-integral-phi12-intertwiner"
          ],
          "requiredPremiseIds": [
            "premise:existing-marked-cyclic-cover-and-burau-model",
            "premise:native-clock-complex-reflection-and-quartic-subspace"
          ]
        },
        {
          "id": "task:complete-shared48-fourthirds-fivequarters-consequence",
          "titleEN": "Prove shared representation consequence for48,4/3 and5/4",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-four-cut-projectors-unify48-fourthirds-fivequarters"
          ],
          "requiredPremiseIds": [
            "premise:fixed-native-d5-a3-compiler-split-and-seed",
            "premise:native-even-e8-gaussian-clock-and-sign-extension"
          ]
        },
        {
          "id": "task:complete-joint-native-compiler-cover-covariance",
          "titleEN": "Completely decide joint compiler-field-cover construction",
          "status": "resolved",
          "resolutionIds": [
            "finding:complete-covariant-compiler-voa-source-and-weyl-closure",
            "finding:complete-fixed-jhat-cyclic-field-lift-and-associated-clock",
            "finding:relative-native-burau-clock-hom-faithful12-transport",
            "finding:e6-vertex-nonselection-survives-full-field-clock-phases"
          ],
          "requiredPremiseIds": [
            "premise:fixed-native-d5-a3-compiler-split-and-seed",
            "premise:existing-marked-cyclic-cover-and-burau-model",
            "premise:original-full-g31-action-and-inherited-lattice-sign-extension",
            "premise:joint-native-compiler-cover-covariance-category"
          ]
        },
        {
          "id": "task:reconstruct-canonical-compiler-field-source-from-c4",
          "titleEN": "Reconstruct canonical local compiler field source from C4",
          "status": "resolved",
          "resolutionIds": [
            "finding:marked-c4-glue-unique-local-e8-vacuum-source",
            "finding:canonical-vacuum-clock-preserves-e6-nonselection"
          ],
          "requiredPremiseIds": [
            "premise:fixed-native-d5-a3-compiler-split-and-seed",
            "premise:marked-c4-unitary-local-extension-reconstruction-class"
          ]
        },
        {
          "id": "task:eliminate-redundant-c4-support-and-multiplicity-inputs",
          "titleEN": "Eliminate redundant sector and multiplicity assumptions",
          "status": "resolved",
          "resolutionIds": [
            "finding:marked-generator-forces-e8-without-support-multiplicity-or-minimality-input"
          ],
          "requiredPremiseIds": [
            "premise:fixed-native-d5-a3-compiler-split-and-seed",
            "premise:connected-local-same-conformal-compiler-extension-with-generator"
          ]
        },
        {
          "id": "task:native-quarter-equivariant-markings",
          "titleEN": "Classify native equivariant markings",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-reflection-rotation-lift-classes-and-forgetful-images"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:ordered-four-punctures-fixed-quarter-projection",
            "premise:fixed-native-reflection-rotation-character-markings"
          ]
        },
        {
          "id": "task:pure-quarter-clock-mapping-class-quotient",
          "titleEN": "Classify pure quarter clock mapping class quotient",
          "status": "resolved",
          "resolutionIds": [
            "finding:quarter-clock-pure-mapping-class-complete-quotient"
          ],
          "requiredPremiseIds": [
            "premise:marked-seam-pure-mathematical-program",
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:ordered-four-punctures-fixed-quarter-projection",
            "premise:ordered-quarter-clock-pure-hurwitz-equivalence"
          ]
        },
        {
          "id": "task:complete-e6-incidence-and-lossless-tetrahedron-transform",
          "titleEN": "Determine complete cut algebra and lossless tetrahedron transform",
          "status": "resolved",
          "resolutionIds": [
            "finding:e8-cut-five-orbital-incidence-algebra-complete",
            "finding:compiler-d8-frame-requires-full-d5-a3-flag",
            "finding:tetrahedron-incidence-lossless-e6-reconstruction",
            "finding:restricted-clock-lift-and-global-cover-audit-reconciled"
          ],
          "requiredPremiseIds": [
            "premise:standard-e8-root-cut-full-flag-incidence-category",
            "premise:joint-native-compiler-cover-covariance-category",
            "premise:original-full-g31-action-and-inherited-lattice-sign-extension"
          ]
        },
        {
          "id": "task:identify-scalar-glue-and-native-cyclic-clock-class",
          "titleEN": "Completely compare scalar glue and actual native compiler class",
          "status": "resolved",
          "resolutionIds": [
            "finding:double-scalar-quarter-is-marked-glue-with-two-step-index",
            "finding:scalar-quarter-two-involutive-e8-extensions",
            "finding:native-quarter-glue-inner-conjugacy-and-oriented-tetrahedron-flags",
            "finding:native-cyclic-compiler-inner-conjugacy"
          ],
          "requiredPremiseIds": [
            "premise:scalar-eight-channel-twist-and-explicit-compiler-charge-chart",
            "premise:full-unitary-e8-field-conjugacy-with-transported-marks"
          ]
        }
      ]
    },
    {
      "id": "problem:atlas-clock-defect-and-operational-reconstruction",
      "titleEN": "Common clock sectors, quartic geometry and operational reconstruction",
      "questionEN": "Which previously separate data of the original TFPT core follow from a common clock and representation geometry, and which complete finite measurement classification follows without asserting physical source selection?",
      "topicIds": [
        "topic:e8",
        "topic:clocks",
        "topic:quantum",
        "topic:origin"
      ],
      "goalIds": [],
      "acceptedPremiseIds": [
        "premise:native-even-e8-gaussian-clock-and-sign-extension",
        "premise:native-clock-complex-reflection-and-quartic-subspace",
        "premise:marked-c4-unitary-local-extension-reconstruction-class",
        "premise:full-unitary-e8-field-conjugacy-with-transported-marks",
        "premise:original-six-ovoid-covariant-instrument-category",
        "premise:relational-native-quartic-tensor-sector-without-state-selection",
        "premise:native-symmetric-quartic-and-erasure-constructors",
        "premise:declared-p1-p2",
        "premise:native-tensor-povm-and-coherent-source-category",
        "premise:local-e8-current-response-realization",
        "premise:native-g31-concrete-event-representation",
        "premise:literal-native-positive-cartan-quartic-fock-embedding",
        "premise:original-displayed-j-equivariant-cartan-root-isometry",
        "premise:full-existing-local-e8-field-hull-observer-code",
        "premise:intrinsic-native-event-geometry-without-marked-clock",
        "premise:original-quarter-car-fold-current-subalgebra",
        "premise:original-microscopic-qwz-integer-car-edge-limit",
        "premise:full-graded-car-with-original-positive-time",
        "premise:original-p1-seed-boundary-and-local-restriction-definitions",
        "premise:original-seam-pair-and-full-moment-os-statements",
        "premise:strong-seam-covariance-and-same-pair-index",
        "premise:historical-original-circle-and-transfer-source-passages",
        "premise:given-quarter-circle-full-heat-transfer-comparison",
        "premise:original-a4-same-moebius-order-four-pin-lift",
        "premise:fixed-marked-glue-and-retained-source-d8-frame",
        "premise:standard-four-d8-module-parent-with-retained-subtheories",
        "premise:same-original-circle-trace-class-covariance",
        "premise:existing-finite-orthogonal-carrier-on-same-source",
        "premise:original-admissibility-and-graded-source-interfaces",
        "premise:typed-source-structures",
        "premise:trace-class-polarization-pair-with-restricted-unitary",
        "premise:smooth-matrix-clutch-on-original-negative-fourier-sea",
        "premise:actual-finite-ns-kernel-and-sampled-original-winding",
        "premise:given-normal-spin-line-and-genuine-circle-product",
        "premise:given-exact-reducing-indexline-with-horizontal-connection",
        "premise:positive-determinant-response-of-transverse-product-channels",
        "premise:closed-normal-fredholm-with-retained-complex-kernel-representation",
        "premise:smooth-matrix-circle-source-and-moving-projector-with-fixed-scalar-symbol",
        "premise:oriented-d3-tetrahedron-in-bloch-chart-versus-original-quarter-marks",
        "premise:existing-marked-cyclic-cover-tower-with-compact-elliptic-images",
        "premise:natural-joint-spin-family-lift-with-given-split-connection",
        "premise:fixed-kappa-toral-third-centralizing-split-carrier",
        "premise:same-angle-evaluation-of-primitive-determinant-clutch-as-native-clock",
        "premise:native-cartan-four-in-full-clock-conjugate-source",
        "premise:primitive-normal-sphere-joint-k-bundle-class-two",
        "premise:given-same-normal-base-degree-pullback",
        "premise:declared-original-octic-quantum-completion-ansatz",
        "premise:existing-elliptic-curve-with-specified-normal-spin-factor",
        "premise:existing-cut-cover-car-with-actual-winding-and-sea",
        "premise:given-flat-mobius-dirac-boundary-family-and-full-sea",
        "premise:given-flat-pin-dirac-free-green-quantization-and-ns-cover",
        "premise:declared-recursive-native-som-update",
        "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
        "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
        "premise:given-pin-source-conformal-normal-geometry-probes",
        "premise:full-original-four-mark-inversion-and-equatorial-real-slots",
        "premise:original-p2-order-three-cusp-and-stable-anchor-target",
        "premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin",
        "premise:theta-square-modular-transport-with-distinct-geometric-clock",
        "premise:fixed-three-slot-permutation-and-original-compiler-comparison",
        "premise:same-marked-elliptic-annulus-massless-conformal-metric-family",
        "premise:positive-joint-car-source-tested-against-full-geometric-projector",
        "premise:actual-elliptic-half-form-l2-trace-on-full-marked-equator",
        "premise:genuine-d4-unitary-stable-parabolic-orbifold-descent",
        "premise:actual-original-a0star-fuchsian-connection-and-c4-clock",
        "premise:recovered-genus-two-cover-with-retained-signed-coefficients",
        "premise:normal-even-sea-car-source-and-canonical-graded-hs-double",
        "premise:fixed-native-pauli-translation-target-and-gaussian-phase-class",
        "premise:ordinary-rigidified-genus-two-theta-determinant-functor",
        "premise:relative-spin-scalar-hardy-cable-on-original-three-cover",
        "premise:same-spin-hardy-shift-against-identity-with-grading-retained",
        "premise:standard-chiral-spin-trace-on-original-cubic-disk-pullback",
        "premise:conditional-three-sheet-negative-sea-car-source",
        "premise:marked-theta-frame-with-explicit-primitive-gaussian-root-normalization",
        "premise:two-theta-lattice-voas-with-given-geometric-faithful-c12-lift",
        "premise:connected-local-same-conformal-compiler-extension-unselected-support",
        "premise:four-d8-sectors-with-fixed-affine-source-time",
        "premise:marked-local-e8-completion-with-original-source-time-and-euclidean-clustering",
        "premise:specified-petersen-shell-and-full-family-finite-recursions",
        "premise:marked-oriented-quarter-spin-and-two-object-vertex-transport",
        "premise:natural-single-face-inclusion-in-given-family-calderon-graph",
        "premise:oriented-five-slot-hodge-candidate-with-explicit-basis-readout",
        "premise:given-five-slot-exterior-source-with-retained-complement-index",
        "premise:given-full-e8-current-source-with-fixed-carrier-su5",
        "premise:specified-inclusive-current-response-with-invariant-metric",
        "premise:specified-old-j20-instrument-in-fixed-six-mark-duad-chart",
        "premise:specified-conjugate-su5-covariant-petersen-channel",
        "premise:fixed-three-plus-two-mark-and-typed-original-degree-clock",
        "premise:specified-cptp-source-process-with-p5-saturation",
        "premise:original-native-pauli-and-v419-f20-source-words",
        "premise:two-actual-source-events-with-coherent-record-access",
        "premise:stipulated-e8-h8-with-original-color-center-selector",
        "premise:actual-single-source-event-coherent-stinespring-record",
        "premise:native-pauli-clock-family-with-marked-f20-inclusion",
        "premise:given-e8-fold-matrix-with-retained-source-charge-and-p2-grading",
        "premise:given-eightfold-even-car-quotient-with-marked-charge-and-source-time",
        "premise:ordinary-weyl-product-and-explicit-reflection-type-comparison",
        "premise:existing-normal-thermal-source-with-bounded-neutral-insertions",
        "premise:finite-reflected-spectral-weight-preservation-test",
        "premise:original-v252-real-finite-block-and-selfdual-field-test",
        "premise:full-projector-car-marginal-and-commuting-joint-algebra",
        "premise:actual-quarter-neutral-vacuum-current-core",
        "premise:actual-local-weyls-and-finite-fiber-coordinate-target",
        "premise:neutral-current-recognition-with-actual-time-state-and-local-image",
        "premise:actual-product-collar-two-face-data",
        "premise:charged-car-pure-two-face-cauchy-graph-state",
        "premise:constant-finite-dirac-product-mixed-current-test",
        "premise:given-finite-sm-triple-yukawa-inputs",
        "premise:paired-quarter-external-lowest-block-candidate",
        "premise:canonical-recomplexified-graded-boundary-clifford-typing",
        "premise:fixed-frame-graded-normal-dependent-source-family",
        "premise:given-quarter-many-body-source-and-compact-dirac-comparison",
        "premise:actual-full-v252-unimodular-p2-gauge-lift",
        "premise:fixed-local-quarter-channel-in-constant-odd-source"
      ],
      "findingIds": [
        "finding:actual-clock-doily-ovoid-kernel-is-native-quartic-five",
        "finding:glue-clock-twisted-sectors-share-marked-quadratic-module",
        "finding:complete-native-six-effect-covariant-instrument-classification",
        "finding:native-six-effect-readout-on-existing-erasure-output",
        "finding:relational-quartic-singlet-and-invariant-density-separation",
        "finding:new-clock-feedback-origin-input-audit",
        "finding:native-event-channel-cone-intersection-is-tenth-to-half",
        "finding:native-six-ovoid-observation-first-occurs-at-tensor-degree-three",
        "finding:original-quartic-erasure-and-hamming-octic-share-marked-ovoid-response",
        "finding:native-octic-field-ray-is-explicit-but-retained-clock-stress-preparation-fails",
        "finding:actual-quarter-fold-stress-has-nonzero-octic-eighth-jet",
        "finding:native-octic-is-an-original-even-car-current-composite",
        "finding:original-geometric-source-jet-and-partial-p1-seed-closure",
        "finding:full-root-ope-has-normalized-native-octic-component",
        "finding:paper-microscopic-car-composes-with-native-octic-ward",
        "finding:microscopic-neutral-octic-has-controlled-smeared-limit",
        "finding:paper-seed-source-needs-local-image-not-global-equivalence",
        "finding:full-car-positive-time-forces-source-vacuum",
        "finding:original-winding-theorem-does-not-select-seam-ray",
        "finding:original-b-u-pair-conditionally-reconstructs-whole-circle-channel",
        "finding:original-os-step-needs-full-moments-not-connected-cumulants",
        "finding:graded-field-image-precedes-physical-observable-restriction",
        "finding:older-original-circle-representative-already-realizes-ward-and-index",
        "finding:old-quarter-heat-transfer-has-controlled-rank-one-limit-and-odd-product",
        "finding:historical-em-rank-one-and-full-heat-traces-have-different-contracts",
        "finding:older-frg-functional-already-supplies-formal-full-moment-parent",
        "finding:earliest-c3-route-quantizes-level-but-needs-physical-kinetic-normalization",
        "finding:canonical-positive-circle-half-selects-the-toeplitz-line-and-sixth-order-correction",
        "finding:old-seam-shift-canonically-gives-relative-car-line-and-fredholm-factor",
        "finding:old-index-coupling-removes-heat-map-and-has-sharp-residual-ward-test",
        "finding:original-admissibility-requires-image-membership-not-positive-half-or-commutation",
        "finding:old-closure-already-assumes-car-but-does-not-map-the-geometric-circle",
        "finding:historical-uv-map-infinite-mass-exclusion-already-proved-and-recovered",
        "finding:a4-spin-lift-fixes-local-ns-and-unordered-quarter-pair",
        "finding:cover-winding-exchanges-quarter-sectors-before-its-square",
        "finding:retained-winding-preserves-local-support-and-neutral-time",
        "finding:marked-glue-square-is-the-e8-fermionization-character",
        "finding:fixed-canonical-d8-is-not-transported-source-d8",
        "finding:actual-winding-has-four-module-e8-defect-extension",
        "finding:ramond-lift-sign-is-common-intertwiner-basis-gauge",
        "finding:old-seam-relative-determinant-is-stable-in-faithful-kms-class",
        "finding:finite-circle-compression-cannot-retain-whole-thermal-winding-factor",
        "finding:schur-carrier-plus-complement-preserves-original-relative-determinant",
        "finding:v258-forward-source-gap-is-reused-not-solved-by-inversion",
        "finding:one-sided-source-and-real-em-norm-need-response-map-not-chiral-phase",
        "finding:trace-class-pair-index-fixes-relative-factor-without-primitive-ray",
        "finding:matrix-clutch-degree-connects-weak-determinant-class-without-ray-selector",
        "finding:finite-ns-ordinary-determinant-cannot-limit-to-nonzero-index",
        "finding:inner-carrier-is-infinite-spacetime-module-not-finite-circle-cut",
        "finding:normal-spin-determinant-line-exactly-transports-whole-circle-in-given-product",
        "finding:normal-index-line-source-match-is-holonomy-condition-not-vertical-degree",
        "finding:full-normal-product-polarization-change-is-not-trace-class",
        "finding:massive-transverse-zero-index-retains-positive-determinant-response",
        "finding:original-callias-doublet-count-does-not-supply-one-complex-circle-channel",
        "finding:exact-moving-kernel-reduction-forces-kato-offdiagonal-connection",
        "finding:tetrahedral-quarter-face-phase-does-not-identify-original-quarter-marked-curve",
        "finding:existing-elliptic-cover-correspondence-has-exact-c6-image-and-retained-mixed-clock",
        "finding:recovered-natural-joint-lift-and-given-connection-already-exist",
        "finding:global-split-joint-carrier-admits-parallel-native-c12-clock",
        "finding:pure-weak-clutch-clock-identification-fails-despite-quarter-matches",
        "finding:same-chern-clutch-native-compiler-does-not-select-physical-connection",
        "finding:native-gaussian-four-is-cartan-subspace-in-whole-sixtyfour",
        "finding:native-family-factor-is-not-gaussian-four-despite-symmetric-coincidences",
        "finding:primitive-class-two-sphere-obstructs-global-cartan-subspace-choice",
        "finding:even-degree-pullback-removes-cartan-topology-without-selecting-clock-cover",
        "finding:global-cartan-choice-is-optional-for-original-scalar-source-fields",
        "finding:bare-native-octic-is-not-whole-e8-zero-mode-singlet",
        "finding:explicit-primary-e8-invariant-octic-completes-same-neutral-car-source",
        "finding:completed-octic-has-exact-old-root-ope-component-and-quarter-source-value",
        "finding:elliptic-normal-spin-pullback-doubles-index-and-deck-quarter-lifts-anticommute",
        "finding:exact-separated-w8-twopoint-in-original-quarter-current-source",
        "finding:unshifted-vacuum-fails-same-w8-source-state-time-dictionary",
        "finding:local-current-phase-and-conditional-cyclic-p1-w8-image",
        "finding:all-neutral-w8-correlations-identify-the-two-original-quarter-charges",
        "finding:original-w8-response-has-infinite-same-time-spectral-support",
        "finding:older-seam-equivalence-verifications-do-not-supply-marked-local-p1-map",
        "finding:full-original-c1-stress-cycle-reconstructs-neutral-w8-response",
        "finding:folded-c8-cycle-misses-intrinsic-w8-tangent-without-obstructing-original-stress",
        "finding:native-source-first-nonuniform-moment-is-four-with-literal-five",
        "finding:native-whole-five-ground-space-needs-four-register-interaction",
        "finding:old-cover-winding-reconstructs-local-quarter-current-net",
        "finding:untransported-independent-quarter-stress-pair-is-not-current",
        "finding:current-public-paper-bytes-and-binding-source-gaps-recovered",
        "finding:winding-charge-conjugation-descends-fixed-pin-domain-but-not-full-dirac-sea",
        "finding:native-fourth-root-frame-has-signed-collective-sixty-mode-dual",
        "finding:native-root-gram-spectrum-is-not-canonical-source-energy-selection",
        "finding:identical-overlapping-native-five-codes-have-no-common-state",
        "finding:horizontal-e8-invariant-octic-cannot-isolate-native-five-ground-space",
        "finding:longitudinal-reflection-preserves-fixed-pin-domain-and-full-dirac-sea",
        "finding:single-fixed-pin-minus-green-kernel-fails-neutral-os-positivity",
        "finding:full-original-ns-orientation-cover-has-all-mode-charged-green-rp",
        "finding:cover-green-thermal-state-is-not-original-quarter-hardy-vacuum",
        "finding:declared-recursive-native-som-is-u35-equivariant",
        "finding:declared-recursive-native-som-clocks-depend-on-chosen-rates",
        "finding:existing-c2-dividing-real-forms-realize-moebius-annulus-tower",
        "finding:raw-ns-elliptic-glide-has-projective-spin-commutator-and-quarter-descent",
        "finding:canonical-elliptic-annulus-reproduces-original-quarter-boundary-tuple-and-kms",
        "finding:finite-crosscap-cauchy-state-is-not-original-quarter-vacuum-by-normal-filtering",
        "finding:native-quartic-relative-arc-charge-excludes-additive-density-stress-insertion",
        "finding:retained-quarter-sea-finite-modular-transfer-preserves-integrated-native-cartan",
        "finding:original-neutral-interarc-fields-have-exact-connected-current-vertex-reconstruction",
        "finding:actual-pin-zero-branch-retains-quarter-car-state-time-but-no-uniform-scale-return",
        "finding:conformal-pin-source-geometry-fixes-exterior-return-and-unitary-memory",
        "finding:geometric-exterior-schur-response-transfers-proper-native-arc-charge-but-is-compact",
        "finding:actual-ns-half-cover-cauchy-graph-purifies-green-state-with-neutral-ground-time-obstruction",
        "finding:full-marked-c2-lifts-distinguish-equatorial-and-os-real-slots",
        "finding:full-marked-equatorial-pullback-is-ramified-graph-not-smooth-seam-circle",
        "finding:same-elliptic-szego-green-functional-has-all-word-free-rp-and-calderon-marginal",
        "finding:elliptic-scalar-spin-pushforward-has-original-holomorphic-p2-anchor",
        "finding:natural-elliptic-anchor-connection-is-not-original-p2-cusp-connection",
        "finding:actual-elliptic-gaussian-source-determines-full-quarter-current-functional",
        "finding:elliptic-current-state-differs-from-hardy-on-every-proper-current-interval",
        "finding:primitive-geometric-force-retains-coupled-reference-and-parity-variation",
        "finding:equal-finite-positive-heat-channel-values-do-not-fix-geometric-force",
        "finding:elliptic-ns-spin-trace-has-complete-global-quarter-kernel",
        "finding:source-induced-ramified-resolutions-do-not-select-quarter-circle-domain",
        "finding:fixed-native-and-fermat-c3-carriers-do-not-identify-marked-v117-pair",
        "finding:theta-square-modular-pair-reconstructs-v117-with-distinct-geometric-q",
        "finding:archived-d4-puncture-formula-fails-and-existing-v117-repairs-existence",
        "finding:geometric-square-clock-is-not-theta-v117-quarter-generator",
        "finding:canonical-conformal-boundary-trace-fixes-existing-elliptic-cauchy-state",
        "finding:conformal-factor-averaging-preserves-given-free-boundary-wick-words",
        "finding:retained-seam-defect-has-derived-causal-geometric-response-with-contact",
        "finding:same-final-pin-geometry-retains-history-while-relative-index-factor-is-fixed",
        "finding:full-projector-covariance-and-positivity-force-all-car-words",
        "finding:actual-two-face-projector-reduces-free-source-test-to-cross-coherence",
        "finding:retained-marked-quarter-compiler-has-no-local-geometric-wick-correction",
        "finding:changed-elliptic-modulus-has-derived-cauchy-response-and-neutral-fourfield-variance",
        "finding:actual-equatorial-half-form-trace-fixes-orthogonal-hardy-range",
        "finding:actual-holomorphic-quarter-cauchy-idempotents-are-not-positive-car-covariances",
        "finding:full-equatorial-checkerboard-free-algebra-fails-fixed-reflection-positivity",
        "finding:genuine-d4-orbifold-branch-has-unique-irreducible-v117-class",
        "finding:wall-weight-additivity-does-not-exclude-stable-v117-parabolic-bundle",
        "finding:recovered-september-c4-triangle-closes-original-a0star-monodromy",
        "finding:actual-a0star-connection-does-not-have-rational-base-inversion-lift",
        "finding:actual-v117-sign-lines-are-negative-weierstrass-triangle-two-torsion",
        "finding:antipodal-and-positive-sign-packets-fail-actual-marked-v117-ramification",
        "finding:common-cyclic-twelve-cover-does-not-trivialize-v117-unramified-sign-lines",
        "finding:negative-sign-packet-has-source-selected-quarter-clock-lift",
        "finding:original-root-and-sign-lines-have-minimal-degree48-common-cover",
        "finding:given-elliptic-flat-connection-family-has-explicit-quillen-norm-and-two-holonomies",
        "finding:bare-large-gauge-quillen-norm-does-not-equal-original-winding-counting-factor",
        "finding:existing-elliptic-half-period-has-canonical-graded-gns-intertwiner",
        "finding:standard-sewing-requires-maximal-affinity-linear-half-modular-field-law",
        "finding:declared-recursive-machine-baseline-controls-and-recovery-are-finitely-measured",
        "finding:declared-recursive-machine-long-run-retains-motion-at-stable-aggregate-rank",
        "finding:recursive-kohonen-source-investigation-has-source-verified-atlas-handoff",
        "finding:chat-algorithmic-comparison-native-pauli-code-and-scoped-information-loss",
        "finding:chat-20261026-results-consolidated-with-corrected-assumptions",
        "finding:source-inversion-forces-six-line-theta-completion",
        "finding:literal-geometric-quarter-is-not-central-gaussian-quarter-on-labels",
        "finding:source-cubic-and-quarter-select-even-theta-and-spin-half-turn-lift",
        "finding:source-cubic-matches-native-pauli-translations-after-context-target-correction",
        "finding:joint-source-clock-relations-have-one-native-clifford-gauge-class",
        "finding:ordinary-theta-determinant-erases-selected-spin-central-character",
        "finding:marked-negative-point-norm-restores-required-spin-central-character",
        "finding:point-normalized-theta-family-correction-fails-original-cubic-action",
        "finding:source-spin-relative-poincare-normalization-closes-point-norm-family",
        "finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package",
        "finding:tfpt-origin-investigation-full-source-coverage-and-current-frontier",
        "finding:full-clock-norm-power-class-and-canonical-eighth-period",
        "finding:original-unit-winding-selects-one-inverse-norm-in-three-cabled-hardy-pair",
        "finding:original-boundary-audit-and-h1-point-restriction-limit",
        "finding:eta-phase-conventions-do-not-identify-relative-spin-reference-pair",
        "finding:same-spin-unit-shift-gives-native-module-class-with-odd-determinant-line",
        "finding:original-even-odd-operator-split-is-compatible-with-same-spin-seam-reference",
        "finding:elliptic-projector-and-weyl-results-retain-their-source-type-at-cubic-spin-bridge",
        "finding:cubic-spin-cauchy-range-is-three-hardy-with-strong-domain-distinction",
        "finding:original-negative-sea-three-mode-ket-is-dual-to-required-hardy-line",
        "finding:auxiliary-theta-route-needs-no-physical-cubic-bulk-or-eta-selected-dual",
        "finding:primitive-gaussian-theta-frame-constructs-e8-pair-with-retained-spin-action",
        "finding:existing-c12-lift-induces-coherent-two-object-e8-vertex-groupoid",
        "finding:actual-winding-functional-calculus-preserves-ns-ramond-split",
        "finding:single-chiral-cycle-exchange-selects-e8-in-four-extension-class",
        "finding:affine-source-involution-pairs-full-d8-sector-spectra",
        "finding:marked-local-e8-with-unchanged-source-time-fails-euclidean-clustering",
        "finding:original-parity-audit-does-not-supply-local-spinor-gauging",
        "finding:hypergraph-local-update-repaired-with-full-family-clock",
        "finding:petersen-polar-step-has-exact-double-cover-family-intertwiner",
        "finding:petersen-source-current-and-hodge-candidates-fail-physical-identification",
        "finding:a7-clock-direction-identifies-neutral-e7-zero-root",
        "finding:oriented-sewing-cocycle-distinguishes-continuation-and-reversal",
        "finding:calderon-single-face-compression-is-not-perron-projector",
        "finding:hodge-born-magnitudes-and-filtered-petersen-jump-match",
        "finding:five-complement-amplitudes-form-full-petersen-quantum-channel",
        "finding:original-twenty-events-factor-through-hodge-and-conditional-phase-record",
        "finding:actual-e8-mixed-current-response-is-full-clifford-channel",
        "finding:carrier-su5-centralizer-gives-direct-e8-petersen-response",
        "finding:su5-covariance-and-petersen-support-fix-complete-quantum-response",
        "finding:neutral-pair-witness-separates-old-events-and-hodge-response",
        "finding:original-degree-clock-is-not-carrier-slot-cycle-at-fixed-source",
        "finding:full-mark-transport-and-isospectrality-do-not-select-hodge-channel",
        "finding:original-pair-source-fixes-state-but-not-selected-epsilon-process",
        "finding:single-positive-process-saturation-selects-full-hodge-channel",
        "finding:original-five-slot-wedge-is-exact-e8-hodge-coefficient",
        "finding:native-pauli-f20-metric-and-quarter-bridge",
        "finding:two-actual-coherent-events-herald-hodge-at-one-over-twenty-five",
        "finding:original-color-center-removes-only-extra-h8-ground-state",
        "finding:marked-symplectic-functor-fixes-native-hodge-kraus-space",
        "finding:f20-record-repair-removes-chart-induced-three-plus-two",
        "finding:theta-scalar-square-lift-is-distinct-from-symplectic-f20-lift",
        "finding:full-admissibility-vacuum-test-needs-source-image-and-transfer-reduction",
        "finding:canonical-native-spin5-hodge-is-already-one-source-record-subchannel",
        "finding:native-hodge-record-selector-is-image-projector-not-kraus-kernel",
        "finding:irreducible-f20-four-cannot-host-original-independent-native-family-clock",
        "finding:native-hodge-is-exact-collective-five-current-action",
        "finding:six-ovoid-clock-bundle-replaces-unnecessary-fixed-form-condition",
        "finding:six-hodge-responses-have-exact-full-current-barycenter",
        "finding:coxeter-seventh-power-has-complete-thirty-e8-lattice-lifts",
        "finding:spin5-hodge-five-is-plucker-kernel-not-original-quartic-hessian",
        "finding:ising-complement-current-mechanism-needs-typed-normal-field-map",
        "finding:neutral-native-a3-hodge-does-not-identify-physical-p2-carrier",
        "finding:compact-ope-coevaluation-normalizes-given-charged-hodge-line",
        "finding:old-ward-root-generates-exact-charged-fifty-module",
        "finding:tp-charged-bracket-support-forces-entire-carrier-marginal",
        "finding:original-ward-and-p2-grading-select-smaller-creation-thirty",
        "finding:charged-source-quarter-recovers-inverse-internal-glue",
        "finding:state-independent-creation-branch-removes-whole-process-hodge-demand",
        "finding:given-source-current-constructs-half-kink-spinor-transport",
        "finding:neutral-ward-response-needs-no-spinor-observable-membership",
        "finding:actual-source-sector-transport-distinguishes-identical-hodge-marginals",
        "finding:ordinary-weyl-type-obstruction-to-single-chiral-os-anti-anti-reflection",
        "finding:given-two-face-source-derives-opposite-neutral-weyls-and-bounded-time-reflection",
        "finding:verification-recovery-preserves-local-source-chain-and-eight-gate-completeness",
        "finding:positive-spectral-reweighting-does-not-establish-original-source-rp",
        "finding:gapped-quarter-energy-cannot-be-same-borchers-translation",
        "finding:finite-ko6-pairing-does-not-supply-standalone-pfaffian-source",
        "finding:full-free-projector-forces-joint-commuting-source-factorization",
        "finding:actual-neutral-current-source-has-constructive-moebius-standard-pair",
        "finding:continuous-local-current-weyls-cannot-live-in-finite-coordinate-fibers",
        "finding:actual-folded-current-ward-and-state-determine-all-neutral-words",
        "finding:mixed-vector-axial-current-is-existing-full-car-subalgebra",
        "finding:constant-product-time-stable-current-level-is-kernel-index",
        "finding:paired-quarter-source-candidate-needs-original-filter-and-oriented-conjugation",
        "finding:graded-boundary-clifford-typing-transports-full-scalar-and-finite-coupled-source",
        "finding:actual-charge-j-transport-retains-full-dirac-and-distinct-pairings",
        "finding:fixed-graded-chart-transports-normal-dependent-source-and-given-mixed-moments",
        "finding:neutral-many-body-source-time-is-not-finite-summable-geometric-dirac",
        "finding:actual-product-reflection-and-same-time-have-full-positive-clifford-source-intertwiner",
        "finding:actual-v252-positive-grade-has-global-p2-wedge-dictionary",
        "finding:old-quarter-full-source-embeds-locally-exactly-on-decoupled-channel",
        "finding:petersen-population-full-cptp-p5-range-zero-to-ten",
        "finding:marked-sm-phase-p5-exact-range-and-response-collision",
        "finding:shared-epsilon-tp-process-fixes-hodge-marginal-even-for-correlated-inputs"
      ],
      "tasks": [
        {
          "id": "task:reconstruct-actual-clock-doily-quartic",
          "titleEN": "Identify actual clock doily and quartic space",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-clock-doily-ovoid-kernel-is-native-quartic-five"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace"
          ]
        },
        {
          "id": "task:unify-clock-glue-and-twisted-branches",
          "titleEN": "Determine common clock and glue description",
          "status": "resolved",
          "resolutionIds": [
            "finding:glue-clock-twisted-sectors-share-marked-quadratic-module"
          ],
          "requiredPremiseIds": [
            "premise:marked-c4-unitary-local-extension-reconstruction-class",
            "premise:full-unitary-e8-field-conjugacy-with-transported-marks"
          ]
        },
        {
          "id": "task:classify-native-six-effect-instruments",
          "titleEN": "Classify all finite covariant instruments",
          "status": "resolved",
          "resolutionIds": [
            "finding:complete-native-six-effect-covariant-instrument-classification"
          ],
          "requiredPremiseIds": [
            "premise:original-six-ovoid-covariant-instrument-category"
          ]
        },
        {
          "id": "task:connect-new-observable-to-existing-erasure-consumer",
          "titleEN": "Connect new observable to actual erasure output",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-six-effect-readout-on-existing-erasure-output"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace",
            "premise:native-symmetric-quartic-and-erasure-constructors"
          ]
        },
        {
          "id": "task:separate-singlet-ray-from-physical-state-selection",
          "titleEN": "Separate singlet ray from physical state selection",
          "status": "resolved",
          "resolutionIds": [
            "finding:relational-quartic-singlet-and-invariant-density-separation"
          ],
          "requiredPremiseIds": [
            "premise:relational-native-quartic-tensor-sector-without-state-selection"
          ]
        },
        {
          "id": "task:audit-feedback-loop-origin-premises",
          "titleEN": "Audit origin premises of the feedback loop",
          "status": "resolved",
          "resolutionIds": [
            "finding:new-clock-feedback-origin-input-audit"
          ],
          "requiredPremiseIds": [
            "premise:declared-p1-p2"
          ]
        },
        {
          "id": "task:compare-classified-channels-with-original-native-events",
          "titleEN": "Compare classified channels with original native events",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-event-channel-cone-intersection-is-tenth-to-half"
          ],
          "requiredPremiseIds": [
            "premise:original-six-ovoid-covariant-instrument-category",
            "premise:native-clock-complex-reflection-and-quartic-subspace"
          ]
        },
        {
          "id": "task:derive-original-seam-selection-of-native-observable-process",
          "titleEN": "Select the operational observable process from original seam action",
          "status": "open",
          "resolutionIds": [
            "finding:canonical-conformal-boundary-trace-fixes-existing-elliptic-cauchy-state",
            "finding:conformal-factor-averaging-preserves-given-free-boundary-wick-words",
            "finding:full-projector-covariance-and-positivity-force-all-car-words",
            "finding:actual-two-face-projector-reduces-free-source-test-to-cross-coherence",
            "finding:existing-elliptic-half-period-has-canonical-graded-gns-intertwiner",
            "finding:standard-sewing-requires-maximal-affinity-linear-half-modular-field-law",
            "finding:source-spin-relative-poincare-normalization-closes-point-norm-family",
            "finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package",
            "finding:full-clock-norm-power-class-and-canonical-eighth-period",
            "finding:original-unit-winding-selects-one-inverse-norm-in-three-cabled-hardy-pair",
            "finding:original-boundary-audit-and-h1-point-restriction-limit",
            "finding:eta-phase-conventions-do-not-identify-relative-spin-reference-pair",
            "finding:same-spin-unit-shift-gives-native-module-class-with-odd-determinant-line",
            "finding:original-even-odd-operator-split-is-compatible-with-same-spin-seam-reference",
            "finding:four-session-summaries-remain-covered-with-superseded-norm-selection-task",
            "finding:elliptic-projector-and-weyl-results-retain-their-source-type-at-cubic-spin-bridge",
            "finding:cubic-spin-cauchy-range-is-three-hardy-with-strong-domain-distinction",
            "finding:original-negative-sea-three-mode-ket-is-dual-to-required-hardy-line",
            "finding:auxiliary-theta-route-needs-no-physical-cubic-bulk-or-eta-selected-dual",
            "finding:primitive-gaussian-theta-frame-constructs-e8-pair-with-retained-spin-action",
            "finding:existing-c12-lift-induces-coherent-two-object-e8-vertex-groupoid",
            "finding:actual-winding-functional-calculus-preserves-ns-ramond-split",
            "finding:single-chiral-cycle-exchange-selects-e8-in-four-extension-class",
            "finding:affine-source-involution-pairs-full-d8-sector-spectra",
            "finding:marked-local-e8-with-unchanged-source-time-fails-euclidean-clustering",
            "finding:original-parity-audit-does-not-supply-local-spinor-gauging",
            "finding:hypergraph-local-update-repaired-with-full-family-clock",
            "finding:petersen-polar-step-has-exact-double-cover-family-intertwiner",
            "finding:petersen-source-current-and-hodge-candidates-fail-physical-identification",
            "finding:a7-clock-direction-identifies-neutral-e7-zero-root",
            "finding:oriented-sewing-cocycle-distinguishes-continuation-and-reversal",
            "finding:calderon-single-face-compression-is-not-perron-projector",
            "finding:hodge-born-magnitudes-and-filtered-petersen-jump-match",
            "finding:five-complement-amplitudes-form-full-petersen-quantum-channel",
            "finding:original-twenty-events-factor-through-hodge-and-conditional-phase-record",
            "finding:actual-e8-mixed-current-response-is-full-clifford-channel",
            "finding:carrier-su5-centralizer-gives-direct-e8-petersen-response",
            "finding:su5-covariance-and-petersen-support-fix-complete-quantum-response",
            "finding:neutral-pair-witness-separates-old-events-and-hodge-response",
            "finding:original-degree-clock-is-not-carrier-slot-cycle-at-fixed-source",
            "finding:full-mark-transport-and-isospectrality-do-not-select-hodge-channel",
            "finding:original-pair-source-fixes-state-but-not-selected-epsilon-process",
            "finding:single-positive-process-saturation-selects-full-hodge-channel",
            "finding:original-five-slot-wedge-is-exact-e8-hodge-coefficient",
            "finding:native-pauli-f20-metric-and-quarter-bridge",
            "finding:two-actual-coherent-events-herald-hodge-at-one-over-twenty-five",
            "finding:original-color-center-removes-only-extra-h8-ground-state",
            "finding:marked-symplectic-functor-fixes-native-hodge-kraus-space",
            "finding:f20-record-repair-removes-chart-induced-three-plus-two",
            "finding:theta-scalar-square-lift-is-distinct-from-symplectic-f20-lift",
            "finding:full-admissibility-vacuum-test-needs-source-image-and-transfer-reduction",
            "finding:canonical-native-spin5-hodge-is-already-one-source-record-subchannel",
            "finding:native-hodge-record-selector-is-image-projector-not-kraus-kernel",
            "finding:irreducible-f20-four-cannot-host-original-independent-native-family-clock",
            "finding:native-hodge-is-exact-collective-five-current-action",
            "finding:six-ovoid-clock-bundle-replaces-unnecessary-fixed-form-condition",
            "finding:six-hodge-responses-have-exact-full-current-barycenter",
            "finding:coxeter-seventh-power-has-complete-thirty-e8-lattice-lifts",
            "finding:spin5-hodge-five-is-plucker-kernel-not-original-quartic-hessian",
            "finding:ising-complement-current-mechanism-needs-typed-normal-field-map",
            "finding:neutral-native-a3-hodge-does-not-identify-physical-p2-carrier",
            "finding:compact-ope-coevaluation-normalizes-given-charged-hodge-line",
            "finding:old-ward-root-generates-exact-charged-fifty-module",
            "finding:tp-charged-bracket-support-forces-entire-carrier-marginal",
            "finding:original-ward-and-p2-grading-select-smaller-creation-thirty",
            "finding:charged-source-quarter-recovers-inverse-internal-glue",
            "finding:state-independent-creation-branch-removes-whole-process-hodge-demand",
            "finding:given-source-current-constructs-half-kink-spinor-transport",
            "finding:neutral-ward-response-needs-no-spinor-observable-membership",
            "finding:actual-source-sector-transport-distinguishes-identical-hodge-marginals",
            "finding:ordinary-weyl-type-obstruction-to-single-chiral-os-anti-anti-reflection",
            "finding:given-two-face-source-derives-opposite-neutral-weyls-and-bounded-time-reflection",
            "finding:finite-ko6-pairing-does-not-supply-standalone-pfaffian-source",
            "finding:full-free-projector-forces-joint-commuting-source-factorization",
            "finding:actual-neutral-current-source-has-constructive-moebius-standard-pair",
            "finding:continuous-local-current-weyls-cannot-live-in-finite-coordinate-fibers",
            "finding:actual-folded-current-ward-and-state-determine-all-neutral-words",
            "finding:graded-boundary-clifford-typing-transports-full-scalar-and-finite-coupled-source"
          ],
          "requiredPremiseIds": [
            "premise:declared-p1-p2"
          ]
        },
        {
          "id": "task:classify-minimum-native-ovoid-observation-stage",
          "titleEN": "Completely classify the minimum native ovoid observation stage",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-six-ovoid-observation-first-occurs-at-tensor-degree-three"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:native-tensor-povm-and-coherent-source-category"
          ]
        },
        {
          "id": "task:close-original-quartic-erasure-derivative-dictionary",
          "titleEN": "Close the original quartic erasure derivative dictionary",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-quartic-erasure-and-hamming-octic-share-marked-ovoid-response"
          ],
          "requiredPremiseIds": [
            "premise:native-even-e8-gaussian-clock-and-sign-extension",
            "premise:native-clock-complex-reflection-and-quartic-subspace",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:native-tensor-povm-and-coherent-source-category"
          ]
        },
        {
          "id": "task:identify-original-octic-field-and-test-clock-preparation",
          "titleEN": "Identify original octic field and test clock preparation",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-octic-field-ray-is-explicit-but-retained-clock-stress-preparation-fails"
          ],
          "requiredPremiseIds": [
            "premise:local-e8-current-response-realization",
            "premise:native-g31-concrete-event-representation",
            "premise:native-symmetric-quartic-and-erasure-constructors",
            "premise:literal-native-positive-cartan-quartic-fock-embedding",
            "premise:original-displayed-j-equivariant-cartan-root-isometry",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:intrinsic-native-event-geometry-without-marked-clock"
          ]
        },
        {
          "id": "task:derive-actual-quarter-fold-octic-stress-response",
          "titleEN": "Derive actual quarter-fold stress response",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-quarter-fold-stress-has-nonzero-octic-eighth-jet"
          ],
          "requiredPremiseIds": [
            "premise:original-quarter-car-fold-current-subalgebra"
          ]
        },
        {
          "id": "task:reconstruct-octic-directly-in-original-car-and-geometric-jet",
          "titleEN": "Reconstruct octic and geometric jet directly from original CAR",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-octic-is-an-original-even-car-current-composite",
            "finding:original-geometric-source-jet-and-partial-p1-seed-closure"
          ],
          "requiredPremiseIds": [
            "premise:original-quarter-car-fold-current-subalgebra"
          ]
        },
        {
          "id": "task:recover-paper-microscopic-neutral-octic-chain",
          "titleEN": "Compose the existing microscopic paper theorem through the octic field",
          "status": "resolved",
          "resolutionIds": [
            "finding:paper-microscopic-car-composes-with-native-octic-ward",
            "finding:microscopic-neutral-octic-has-controlled-smeared-limit"
          ],
          "requiredPremiseIds": [
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:original-p1-seed-boundary-and-local-restriction-definitions"
          ]
        },
        {
          "id": "task:correct-paper-source-identification-criterion",
          "titleEN": "Correct the original boundary criterion and redundant state premise",
          "status": "resolved",
          "resolutionIds": [
            "finding:paper-seed-source-needs-local-image-not-global-equivalence",
            "finding:full-car-positive-time-forces-source-vacuum"
          ],
          "requiredPremiseIds": [
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:original-p1-seed-boundary-and-local-restriction-definitions"
          ]
        },
        {
          "id": "task:repair-original-seam-ray-and-os-moment-identifications",
          "titleEN": "Audit and repair original seam-line and OS moment identifications",
          "status": "resolved",
          "resolutionIds": [
            "finding:original-winding-theorem-does-not-select-seam-ray",
            "finding:original-b-u-pair-conditionally-reconstructs-whole-circle-channel",
            "finding:original-os-step-needs-full-moments-not-connected-cumulants",
            "finding:graded-field-image-precedes-physical-observable-restriction"
          ],
          "requiredPremiseIds": [
            "premise:original-seam-pair-and-full-moment-os-statements"
          ]
        },
        {
          "id": "task:recover-older-circle-source-and-controlled-heat-closure",
          "titleEN": "Recover older original source and controlled ground-mode closure",
          "status": "resolved",
          "resolutionIds": [
            "finding:older-original-circle-representative-already-realizes-ward-and-index",
            "finding:old-quarter-heat-transfer-has-controlled-rank-one-limit-and-odd-product",
            "finding:historical-em-rank-one-and-full-heat-traces-have-different-contracts",
            "finding:older-frg-functional-already-supplies-formal-full-moment-parent",
            "finding:earliest-c3-route-quantizes-level-but-needs-physical-kinetic-normalization",
            "finding:canonical-positive-circle-half-selects-the-toeplitz-line-and-sixth-order-correction"
          ],
          "requiredPremiseIds": [
            "premise:historical-original-circle-and-transfer-source-passages"
          ]
        },
        {
          "id": "task:close-direct-original-seam-defect-and-reweighting-interface",
          "titleEN": "Close the direct old seam defect and actual reweighting contract",
          "status": "resolved",
          "resolutionIds": [
            "finding:old-seam-shift-canonically-gives-relative-car-line-and-fredholm-factor",
            "finding:old-index-coupling-removes-heat-map-and-has-sharp-residual-ward-test",
            "finding:original-admissibility-requires-image-membership-not-positive-half-or-commutation",
            "finding:old-closure-already-assumes-car-but-does-not-map-the-geometric-circle",
            "finding:historical-uv-map-infinite-mass-exclusion-already-proved-and-recovered"
          ],
          "requiredPremiseIds": [
            "premise:historical-original-circle-and-transfer-source-passages",
            "premise:full-graded-car-with-original-positive-time",
            "premise:historical-sf-sector-action-and-relative-normalization",
            "premise:original-admissibility-and-graded-source-interfaces",
            "premise:literal-positive-rg-map-on-finite-halfline-measures"
          ]
        },
        {
          "id": "task:close-original-spin-and-retained-source-defect-dictionary",
          "titleEN": "Close original spin and actual defect dictionary",
          "status": "resolved",
          "resolutionIds": [
            "finding:a4-spin-lift-fixes-local-ns-and-unordered-quarter-pair",
            "finding:cover-winding-exchanges-quarter-sectors-before-its-square",
            "finding:retained-winding-preserves-local-support-and-neutral-time",
            "finding:marked-glue-square-is-the-e8-fermionization-character",
            "finding:fixed-canonical-d8-is-not-transported-source-d8",
            "finding:actual-winding-has-four-module-e8-defect-extension",
            "finding:ramond-lift-sign-is-common-intertwiner-basis-gauge"
          ],
          "requiredPremiseIds": [
            "premise:original-a4-same-moebius-order-four-pin-lift",
            "premise:fixed-marked-glue-and-retained-source-d8-frame",
            "premise:standard-four-d8-module-parent-with-retained-subtheories",
            "premise:full-graded-car-with-original-positive-time",
            "premise:historical-original-circle-and-transfer-source-passages"
          ]
        },
        {
          "id": "task:transport-original-seam-factor-through-faithful-kms-and-carrier",
          "titleEN": "Transport original seam factor through faithful KMS source and carrier",
          "status": "resolved",
          "resolutionIds": [
            "finding:old-seam-relative-determinant-is-stable-in-faithful-kms-class",
            "finding:finite-circle-compression-cannot-retain-whole-thermal-winding-factor",
            "finding:schur-carrier-plus-complement-preserves-original-relative-determinant",
            "finding:v258-forward-source-gap-is-reused-not-solved-by-inversion",
            "finding:one-sided-source-and-real-em-norm-need-response-map-not-chiral-phase"
          ],
          "requiredPremiseIds": [
            "premise:historical-original-circle-and-transfer-source-passages",
            "premise:same-original-circle-trace-class-covariance",
            "premise:existing-finite-orthogonal-carrier-on-same-source"
          ]
        },
        {
          "id": "task:close-pair-index-original-clutch-and-ns-limit-interface",
          "titleEN": "Resolve pair index,original carrier clutch and actual NS limit interface",
          "status": "resolved",
          "resolutionIds": [
            "finding:trace-class-pair-index-fixes-relative-factor-without-primitive-ray",
            "finding:matrix-clutch-degree-connects-weak-determinant-class-without-ray-selector",
            "finding:finite-ns-ordinary-determinant-cannot-limit-to-nonzero-index",
            "finding:inner-carrier-is-infinite-spacetime-module-not-finite-circle-cut"
          ],
          "requiredPremiseIds": [
            "premise:trace-class-polarization-pair-with-restricted-unitary",
            "premise:smooth-matrix-clutch-on-original-negative-fourier-sea",
            "premise:actual-finite-ns-kernel-and-sampled-original-winding",
            "premise:historical-original-circle-and-transfer-source-passages",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:resolve-normal-index-line-and-full-source-pushforward-criterion",
          "titleEN": "Match normal index transport and full source response",
          "status": "resolved",
          "resolutionIds": [
            "finding:normal-spin-determinant-line-exactly-transports-whole-circle-in-given-product",
            "finding:normal-index-line-source-match-is-holonomy-condition-not-vertical-degree",
            "finding:full-normal-product-polarization-change-is-not-trace-class",
            "finding:massive-transverse-zero-index-retains-positive-determinant-response",
            "finding:original-callias-doublet-count-does-not-supply-one-complex-circle-channel"
          ],
          "requiredPremiseIds": [
            "premise:given-normal-spin-line-and-genuine-circle-product",
            "premise:given-exact-reducing-indexline-with-horizontal-connection",
            "premise:positive-determinant-response-of-transverse-product-channels",
            "premise:historical-original-circle-and-transfer-source-passages",
            "premise:typed-source-structures",
            "premise:closed-normal-fredholm-with-retained-complex-kernel-representation"
          ]
        },
        {
          "id": "task:compare-normal-connection-and-tetrahedral-cover-transport",
          "titleEN": "Compare normal connection fiber and tetrahedral cover transport",
          "status": "resolved",
          "resolutionIds": [
            "finding:exact-moving-kernel-reduction-forces-kato-offdiagonal-connection",
            "finding:tetrahedral-quarter-face-phase-does-not-identify-original-quarter-marked-curve",
            "finding:existing-elliptic-cover-correspondence-has-exact-c6-image-and-retained-mixed-clock"
          ],
          "requiredPremiseIds": [
            "premise:smooth-matrix-circle-source-and-moving-projector-with-fixed-scalar-symbol",
            "premise:oriented-d3-tetrahedron-in-bloch-chart-versus-original-quarter-marks",
            "premise:existing-marked-cyclic-cover-tower-with-compact-elliptic-images",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:close-joint-lift-split-connection-and-native-clock-compatibility",
          "titleEN": "Match global joint lift,split connection and whole native clock",
          "status": "resolved",
          "resolutionIds": [
            "finding:recovered-natural-joint-lift-and-given-connection-already-exist",
            "finding:global-split-joint-carrier-admits-parallel-native-c12-clock",
            "finding:pure-weak-clutch-clock-identification-fails-despite-quarter-matches",
            "finding:same-chern-clutch-native-compiler-does-not-select-physical-connection"
          ],
          "requiredPremiseIds": [
            "premise:natural-joint-spin-family-lift-with-given-split-connection",
            "premise:fixed-kappa-toral-third-centralizing-split-carrier",
            "premise:same-angle-evaluation-of-primitive-determinant-clutch-as-native-clock",
            "premise:full-unitary-e8-field-conjugacy-with-transported-marks",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:identify-native-cartan-four-and-global-joint-mark-transport",
          "titleEN": "Resolve native four and global joint mark transport",
          "status": "resolved",
          "resolutionIds": [
            "finding:native-gaussian-four-is-cartan-subspace-in-whole-sixtyfour",
            "finding:native-family-factor-is-not-gaussian-four-despite-symmetric-coincidences",
            "finding:primitive-class-two-sphere-obstructs-global-cartan-subspace-choice",
            "finding:even-degree-pullback-removes-cartan-topology-without-selecting-clock-cover"
          ],
          "requiredPremiseIds": [
            "premise:native-cartan-four-in-full-clock-conjugate-source",
            "premise:primitive-normal-sphere-joint-k-bundle-class-two",
            "premise:given-same-normal-base-degree-pullback"
          ]
        },
        {
          "id": "task:complete-native-octic-under-existing-source-zero-mode-symmetry",
          "titleEN": "Complete native octic under E8 in the same source",
          "status": "resolved",
          "resolutionIds": [
            "finding:global-cartan-choice-is-optional-for-original-scalar-source-fields",
            "finding:explicit-primary-e8-invariant-octic-completes-same-neutral-car-source",
            "finding:completed-octic-has-exact-old-root-ope-component-and-quarter-source-value"
          ],
          "requiredPremiseIds": [
            "premise:declared-original-octic-quantum-completion-ansatz",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:compute-source-w8-twopoint-and-assess-original-local-p1-map",
          "titleEN": "Compute W8 source response and assess the original local P1 connection",
          "status": "resolved",
          "resolutionIds": [
            "finding:exact-separated-w8-twopoint-in-original-quarter-current-source",
            "finding:unshifted-vacuum-fails-same-w8-source-state-time-dictionary",
            "finding:local-current-phase-and-conditional-cyclic-p1-w8-image",
            "finding:all-neutral-w8-correlations-identify-the-two-original-quarter-charges",
            "finding:original-w8-response-has-infinite-same-time-spectral-support"
          ],
          "requiredPremiseIds": [
            "premise:declared-original-octic-quantum-completion-ansatz",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:audit-older-seam-results-and-reconstruct-full-original-stress-cycle",
          "titleEN": "Review older seam results and reconstruct the complete original stress cycle",
          "status": "resolved",
          "resolutionIds": [
            "finding:older-seam-equivalence-verifications-do-not-supply-marked-local-p1-map",
            "finding:full-original-c1-stress-cycle-reconstructs-neutral-w8-response",
            "finding:folded-c8-cycle-misses-intrinsic-w8-tangent-without-obstructing-original-stress",
            "finding:untransported-independent-quarter-stress-pair-is-not-current",
            "finding:current-public-paper-bytes-and-binding-source-gaps-recovered"
          ],
          "requiredPremiseIds": [
            "premise:declared-original-octic-quantum-completion-ansatz",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:full-existing-local-e8-field-hull-observer-code",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:recover-original-winding-comparison-and-local-quarter-current-net",
          "titleEN": "Recover original winding comparison and local quarter current net",
          "status": "resolved",
          "resolutionIds": [
            "finding:old-cover-winding-reconstructs-local-quarter-current-net"
          ],
          "requiredPremiseIds": [
            "premise:existing-cut-cover-car-with-actual-winding-and-sea",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:assess-fixed-pin-descent-of-original-winding-conjugation",
          "titleEN": "Assess fixed Pin descent of original winding with charge conjugation",
          "status": "resolved",
          "resolutionIds": [
            "finding:winding-charge-conjugation-descends-fixed-pin-domain-but-not-full-dirac-sea"
          ],
          "requiredPremiseIds": [
            "premise:given-flat-mobius-dirac-boundary-family-and-full-sea",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:resolve-full-pin-reflection-and-ns-cover-green-positivity",
          "titleEN": "Decide full Pin reflection and NS cover Green positivity",
          "status": "resolved",
          "resolutionIds": [
            "finding:longitudinal-reflection-preserves-fixed-pin-domain-and-full-dirac-sea",
            "finding:single-fixed-pin-minus-green-kernel-fails-neutral-os-positivity",
            "finding:full-original-ns-orientation-cover-has-all-mode-charged-green-rp",
            "finding:cover-green-thermal-state-is-not-original-quarter-hardy-vacuum"
          ],
          "requiredPremiseIds": [
            "premise:given-flat-pin-dirac-free-green-quantization-and-ns-cover",
            "premise:given-flat-mobius-dirac-boundary-family-and-full-sea",
            "premise:original-p1-seed-boundary-and-local-restriction-definitions",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:realize-existing-quarter-source-on-concrete-elliptic-moebius-cut",
          "titleEN": "Realize existing quarter source on the concrete elliptic Moebius cut",
          "status": "resolved",
          "resolutionIds": [
            "finding:existing-c2-dividing-real-forms-realize-moebius-annulus-tower",
            "finding:raw-ns-elliptic-glide-has-projective-spin-commutator-and-quarter-descent",
            "finding:canonical-elliptic-annulus-reproduces-original-quarter-boundary-tuple-and-kms",
            "finding:finite-crosscap-cauchy-state-is-not-original-quarter-vacuum-by-normal-filtering"
          ],
          "requiredPremiseIds": [
            "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
            "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:given-flat-mobius-dirac-boundary-family-and-full-sea",
            "premise:original-p1-seed-boundary-and-local-restriction-definitions",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:complete-full-elliptic-real-slots-and-test-marked-seam-and-p2-pushforward",
          "titleEN": "Complete full elliptic Real slots and test actual seam and P2 pushforward",
          "status": "resolved",
          "resolutionIds": [
            "finding:full-marked-c2-lifts-distinguish-equatorial-and-os-real-slots",
            "finding:full-marked-equatorial-pullback-is-ramified-graph-not-smooth-seam-circle",
            "finding:same-elliptic-szego-green-functional-has-all-word-free-rp-and-calderon-marginal",
            "finding:elliptic-scalar-spin-pushforward-has-original-holomorphic-p2-anchor",
            "finding:natural-elliptic-anchor-connection-is-not-original-p2-cusp-connection"
          ],
          "requiredPremiseIds": [
            "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
            "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
            "premise:full-original-four-mark-inversion-and-equatorial-real-slots",
            "premise:original-p2-order-three-cusp-and-stable-anchor-target",
            "premise:original-quarter-car-fold-current-subalgebra",
            "premise:original-p1-seed-boundary-and-local-restriction-definitions",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:close-global-spin-kernel-and-test-modular-three-source-dictionary",
          "titleEN": "Close global spin kernel and compare modular three-source with originals",
          "status": "resolved",
          "resolutionIds": [
            "finding:elliptic-ns-spin-trace-has-complete-global-quarter-kernel",
            "finding:source-induced-ramified-resolutions-do-not-select-quarter-circle-domain",
            "finding:fixed-native-and-fermat-c3-carriers-do-not-identify-marked-v117-pair",
            "finding:theta-square-modular-pair-reconstructs-v117-with-distinct-geometric-q",
            "finding:archived-d4-puncture-formula-fails-and-existing-v117-repairs-existence",
            "finding:geometric-square-clock-is-not-theta-v117-quarter-generator"
          ],
          "requiredPremiseIds": [
            "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
            "premise:full-original-four-mark-inversion-and-equatorial-real-slots",
            "premise:original-p2-order-three-cusp-and-stable-anchor-target",
            "premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin",
            "premise:theta-square-modular-transport-with-distinct-geometric-clock",
            "premise:fixed-three-slot-permutation-and-original-compiler-comparison",
            "premise:typed-source-structures"
          ]
        },
        {
          "id": "task:derive-real-equatorial-trace-and-recover-original-c4-monodromy-proof",
          "titleEN": "Determine real equatorial trace and restore older C4 proof",
          "status": "resolved",
          "resolutionIds": [
            "finding:actual-equatorial-half-form-trace-fixes-orthogonal-hardy-range",
            "finding:actual-holomorphic-quarter-cauchy-idempotents-are-not-positive-car-covariances",
            "finding:full-equatorial-checkerboard-free-algebra-fails-fixed-reflection-positivity",
            "finding:genuine-d4-orbifold-branch-has-unique-irreducible-v117-class",
            "finding:wall-weight-additivity-does-not-exclude-stable-v117-parabolic-bundle",
            "finding:recovered-september-c4-triangle-closes-original-a0star-monodromy",
            "finding:actual-a0star-connection-does-not-have-rational-base-inversion-lift",
            "finding:actual-v117-sign-lines-are-negative-weierstrass-triangle-two-torsion",
            "finding:antipodal-and-positive-sign-packets-fail-actual-marked-v117-ramification",
            "finding:common-cyclic-twelve-cover-does-not-trivialize-v117-unramified-sign-lines",
            "finding:negative-sign-packet-has-source-selected-quarter-clock-lift",
            "finding:original-root-and-sign-lines-have-minimal-degree48-common-cover"
          ],
          "requiredPremiseIds": [
            "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
            "premise:full-original-four-mark-inversion-and-equatorial-real-slots",
            "premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin",
            "premise:actual-elliptic-half-form-l2-trace-on-full-marked-equator",
            "premise:original-p2-order-three-cusp-and-stable-anchor-target",
            "premise:genuine-d4-unitary-stable-parabolic-orbifold-descent",
            "premise:actual-original-a0star-fuchsian-connection-and-c4-clock",
            "premise:recovered-genus-two-cover-with-retained-signed-coefficients",
            "premise:typed-source-structures"
          ]
        }
      ]
    }
  ],
  "premises": [
    {
      "id": "premise:declared-p1-p2",
      "titleEN": "P1/P2 remain declared starting axioms",
      "statementEN": "This program checks which additional source, state, event, and observable structures follow from accepted TFPT principles; it does not revoke or replace P1/P2."
    },
    {
      "id": "premise:quarter-phase-retained",
      "titleEN": "Quarter phase is retained",
      "statementEN": "Comparisons retain the original charged quarter phase; switching to another phase branch is not an admissible solution to the selection problem."
    },
    {
      "id": "premise:typed-source-structures",
      "titleEN": "Do not identify source types",
      "statementEN": "Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them."
    },
    {
      "id": "premise:fixed-input-models-scoped",
      "titleEN": "Model results remain conditional",
      "statementEN": "Calculations in a fixed E8 net, quotient, family packet, or readout apply only under their stated inputs and do not select those inputs."
    },
    {
      "id": "premise:future-answer-functional",
      "titleEN": "Declared complete answer functional",
      "statementEN": "A fixed set of allowed experiments and the answer functional r for every finite experiment sequence are additional operational inputs; they define the future quotient only relative to that experiment alphabet."
    },
    {
      "id": "premise:positive-word-kernel-null-stability",
      "titleEN": "Positive word kernel with null stability",
      "statementEN": "The GNS/Gram reconstruction requires a Hermitian positive kernel K, normalized reference, declared composition and adjunction, and stability of null relations under every allowed continuation; these are conditions on a supplied K source, not TFPT axioms proved from bare future answers."
    },
    {
      "id": "premise:parallel-irreducible-family-connection",
      "titleEN": "Common parallelity to irreducible family connection",
      "statementEN": "Only for the commutant conclusion, assume additionally that the marked SU(3) family connection is irreducible and beta preserves it in the same marked frame. This is not an accepted P1/P2 axiom and must be derived from the source."
    },
    {
      "id": "premise:sign-covariance-only",
      "titleEN": "Only sign covariance is given",
      "statementEN": "The conditional counterexample assumes C=(I+sgn H)/2 and [rho,C]=0, without knowing positive magnitude |H| or a complete transfer."
    },
    {
      "id": "premise:known-faithful-kms-or-full-transfer",
      "titleEN": "Known KMS scale or full transfer",
      "statementEN": "Positive dynamics reconstruction assumes either known nonzero beta and faithful finite covariance 0<C_beta<I, or the same full transfer operator T=exp(-tau |H|), known tau, and commutation with the sign; a spectrum alone or physical identification of the TFPT channel is insufficient."
    },
    {
      "id": "premise:fixed-em-seam-function",
      "titleEN": "Specified EM function with positive seam response",
      "statementEN": "F(x)=x³−2c³x²+(4/5)c⁶M log(p+q(1−q)^(−5/4)), c=1/(8π), p=1/(6π), q=48c⁴exp(−2x), M>0; original budget M=41. The function and inputs are premises here."
    },
    {
      "id": "premise:native-current-clock-ope-access",
      "titleEN": "Native clock fields and mixed OPE access",
      "statementEN": "Fixed free eight-current vacuum, original C/J/P5 in the same E8 index basis and real-symmetric quadratic fields; access to mixed two-point responses of iterated weight-two double-pole OPE coefficients is assumed. Four individual measurements or a physical detector from P1/P2 are not thereby supplied."
    },
    {
      "id": "premise:declared-native-round20-source",
      "titleEN": "Declared native charge/scalar source",
      "statementEN": "Fixed finite periodic spatial lattice source of rounds20–26 with original G, cocycle, neutral Z8 charges, scalar field and H=Hdiag+Hhop; a,m,beta>0 and original couplings are supplied. Nontrivial memory requires J>0. The original grade k is additionally retained, without presuming its selection from P1/P2."
    },
    {
      "id": "premise:stipulated-even-e8-seam-lift",
      "titleEN": "Stipulated even E8 completion with actual Klein signs",
      "statementEN": "Assume the E8 extension of the even D8 CAR sector constructed in seam-joint-ward-state, with original ordered cocycle, retained quarter transition and explicit P-lift convention. An ordinary-local inclusion of odd CAR fields is not assumed."
    },
    {
      "id": "premise:stipulated-e8-clock-tuple",
      "titleEN": "Stipulated E8 Fock space with marked clock tuple",
      "statementEN": "Given the original ordered CAR/E8 cocycle, E8 Fock representation, Kδ(p,N)=|p|²/2+δ·p+N, original q and full uncompensated quarter transition. J is the actual Gaussian lattice operator, not the cyclotomic Galois operator. A covariant comparison transports δ,q,τ together; a fixed marked source holds them fixed."
    },
    {
      "id": "premise:neutral-current-vacuum-insertions",
      "titleEN": "Every insertion preserves all eight Cartan charges",
      "statementEN": "In the same given E8 Fock space every word starts in the vacuum. Each individual insertion belongs to the fixed Heisenberg/quadratic Jordan OPE algebra and commutes with all eight Cartan zero modes; normal ordering, operator labels and initial state do not change with δ. Overall neutrality or Qq neutrality alone is insufficient."
    },
    {
      "id": "premise:normalized-charged-primary-time-access",
      "titleEN": "Normalized charged primary responses with known time scale",
      "statementEN": "For the given Kδ, normalized primary charge states ±bi for eight independent original E8 basis charges and their positive e^(−tKδ) return amplitudes at known common t>0 are supplied. Global charge shifts are not thereby derived local physical instruments."
    },
    {
      "id": "premise:fixed-nonzero-quarter-twist-comparison",
      "titleEN": "Fixed nonzero chiral quarter twist",
      "statementEN": "The comparison target is the stipulated Kδ Fock space with original nonzero quarter twist, normalized ±Bp primary states, fixed charge marking and the same positive time. Equality of all these responses is required."
    },
    {
      "id": "premise:native-full-charge-field-source",
      "titleEN": "Full original charge-field source",
      "statementEN": "Finite Round20 lattice/scalar space with the original bilinear E8 cocycle, complete integer charges, onsite T and balanced W, coercive quadratic Hdiag and supplied hopping. No P1/P2 selection or local chiral field identification is presumed."
    },
    {
      "id": "premise:native-c-invariant-start",
      "titleEN": "Charge-conjugation-invariant start",
      "statementEN": "The initial density is invariant under linear C:n→−n, including full and C0 Gibbs starts of unchanged H. Compare actual adjoint-paired, opposite marked T responses at the same time."
    },
    {
      "id": "premise:native-zero-sector-chemical-comparison",
      "titleEN": "Chemical comparison with charged intermediate sectors",
      "statementEN": "Additionally supplied Hη=H+η·Qtot, C-invariant initial density in C0 and full propagation after onsite T moves C0 into Cp. η is comparison input, not a derived seam parameter."
    },
    {
      "id": "premise:native-fixed-link-phase-a-invariance",
      "titleEN": "Fixed link phases and A-invariant start",
      "statementEN": "Arbitrary fixed unitary link phases in the same hopping, real original Hdiag/cocycle and initial density invariant under A=C times complex conjugation. Positive Euclidean T return responses are considered; no general real-time equality is assumed."
    },
    {
      "id": "premise:fixed-car-crossed-half-charge-search",
      "titleEN": "Fixed CAR source and adjoined half sector",
      "statementEN": "Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2."
    },
    {
      "id": "premise:fixed-positive-crossed-state",
      "titleEN": "Positive base functional and explicit observable algebra",
      "statementEN": "For finite crossed words omega0 is positive; dual Z2 invariance is additionally required for unique algebraic extension. Comparing physical densities must retain the actual Cartan observables and the common normal ground space of supplied K."
    },
    {
      "id": "premise:unchanged-quarter-car-qwz-source-comparison",
      "titleEN": "Unchanged specified CAR/QWZ source",
      "statementEN": "Compare the known quarter-CAR source or full mass=1, ny=8 QWZ operator with the identical Hilbert representation, Cartan action and physical time. Preserve its previously proved unique ground state and spectral gap. This does not compare all P1/P2 realizations."
    },
    {
      "id": "premise:oriented-p2-exterior-carrier",
      "titleEN": "Original oriented exterior carrier",
      "statementEN": "Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation."
    },
    {
      "id": "premise:fixed-e8-fold-quarter-dictionary",
      "titleEN": "Stipulated E8 quarter-fold dictionary",
      "statementEN": "The existing Weyl lift M, affine E8 module, Q'=2(H3+H6) and K_alpha=L0+delta_alpha.H0 with alpha=1/4,3/4 are held fixed for comparison. This representation is not derived from P1/P2; an unbroken SM time action is not assumed."
    },
    {
      "id": "premise:actual-chiral-opposite-quarter-domains",
      "titleEN": "Given chiral source in both quarter domains",
      "statementEN": "D_alpha=-i∂theta on f(theta+2pi)=exp(2pi i alpha)f(theta), filled negative spectrum and relative particle/hole excitation time; alpha=1/4,3/4. This compares two conjugate domains, not opposite charges in one fixed domain or all P1 sources."
    },
    {
      "id": "premise:given-first-marked-quarter-character",
      "titleEN": "First 5+3 marking and Z4 character supplied",
      "statementEN": "In the supplied E8 dictionary, D5 on the first five coordinates, D3 on the last three and k=2sum(p1..p5) mod4 are given for the first quarter domain. The other domain may transport these data covariantly; their original physical selection is not presumed."
    },
    {
      "id": "premise:given-d5-a3-free-fermion-realization",
      "titleEN": "Supplied free D5+A3 realization",
      "statementEN": "Compare the documented free D5_1+A3_1 realization with five plus three complex fermions and spin-field sectors; realization by the raw P1 source is not presumed."
    },
    {
      "id": "premise:full-normalized-charge-time-character-family",
      "titleEN": "Fully normalized time/charge family in the given E8 completion",
      "statementEN": "The given chiral quarter source, its eightfold pushforward and stipulated even E8 completion supply K_alpha and the full additive U1 generator Q. Only characters in the real span of their torus vectors modulo E8 are searched; no P5 is supplied as a search variable or target projector. Physical identification of this completion and of a composed marking is not assumed."
    },
    {
      "id": "premise:actual-wold-source-operator-hypotheses",
      "titleEN": "Hypotheses of joint operator reconstruction",
      "statementEN": "For this conditional theorem: orthogonal projection C, unitary U, (1−C)UC=0, index(CUC)=−1, pure isometry and minimal unitary dilation. Actual selfadjoint B satisfies U*BU=B+1 and Bw=alpha w on a common invariant core, alpha=1/4 or 3/4. These conditions are not presumed derived from P1/P2."
    },
    {
      "id": "premise:charged-car-negative-sea-source",
      "titleEN": "Charged CAR algebra and original negative sea",
      "statementEN": "Assume the actual field algebra is charged CAR(H) and its reference state is the pure quasifree sea with occupation 1−C. Relative charge and positive excitation time are normal ordered against this sea. This physical identification is an additional premise."
    },
    {
      "id": "premise:fixed-scalar-quarter-hardy-source",
      "titleEN": "Fixed scalar Hardy model with quarter time",
      "statementEN": "H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2."
    },
    {
      "id": "premise:actual-product-collar-two-face-data",
      "titleEN": "Complete data of both product-collar faces",
      "statementEN": "A selfadjoint B occurs in an exact normal-independent product equation. The full orthogonal L2 Cauchy graph projection of both labeled faces, their actual fiber identification and independently known positive normal length ell are supplied. Selection of these data from P1/P2 is not presumed."
    },
    {
      "id": "premise:charged-car-two-face-graph-state",
      "titleEN": "Charged quasifree state for the complete graph covariance",
      "statementEN": "Additionally identify the actual field algebra with charged CAR(H direct_sum H) and the state with the gauge-invariant quasifree state whose occupation covariance is G_ell. This is an extra physical premise, not a consequence of the geometric projection calculation."
    },
    {
      "id": "premise:local-geometric-dirac-deck-lift",
      "titleEN": "Local geometric lift on a Dirac boundary",
      "statementEN": "The claimed geometric deck reflection acts at the fixed boundary through a local bundle lift over the same base reflection. Compare an ordinary positive-dimensional Dirac boundary with orthogonal Calderon projection; a nonlocal operator subsequently defined as 2P_C−I is not a pre-established geometric action."
    },
    {
      "id": "premise:literal-carrier-after-positive-compression",
      "titleEN": "Literal carrier embedding after positive compression",
      "statementEN": "Read the archived phrase H_car after primitive admissibility compression literally as a subspace of Ran(Pprim)⊂Ran(C), then define internal polarization as (2C−I)|H_car. A different coefficient-factor interpretation instead requires its own embedding and internal involution."
    },
    {
      "id": "premise:declared-finite-operator-synthesis-inputs",
      "titleEN": "Explicit finite operator-synthesis inputs",
      "statementEN": "The recovery control assumes the given five-mode CAR representation. The inverse solver assumes explicitly supplied finite operator pairs and adjoints. The map is unknown; physical source selection is not assumed derived."
    },
    {
      "id": "premise:finite-positive-kernel-search-class",
      "titleEN": "Declared finite positive kernel search class",
      "statementEN": "Given up to five CAR modes, an extra selfadjoint transfer operation, 0<=T<=I, T Omega=Omega and internal CAR quarter invariance. Five-mode completions additionally use a minimal irreducible Fock representation with pure occupations inferred from searched moments. Objectives, positive floor and cluster gap are declared search choices; neither these nor the representation are presumed consequences of P1/P2."
    },
    {
      "id": "premise:actual-car-graph-and-oriented-boundary-pairing",
      "titleEN": "Actual CAR boundary covariance and oriented amplitude pairing",
      "statementEN": "The full geometric two-face graph G is additionally identified as the actual CAR state covariance. The first labeled fiber is input/dual, the second output; the source realizes the specified fermionic boundary-state/operator pairing. These are physical identification premises, not consequences of matrix sizes or classical Cauchy projection. Normalized data determine only the operator ray and not the absolute partition factor."
    },
    {
      "id": "premise:given-complex-fermion-source-quantization",
      "titleEN": "Explicit complex-fermion source quantization",
      "statementEN": "Given the actual invertible free kinetic operator K, its reference, Grassmann source couplings and oriented Berezin measure with integral exp(-barpsi psi)=1. A mode partition is not a spatial boundary decomposition without geometric identification. Neither comparison part D_rel nor a general spectral profile is automatically identified with this K or -log det K; this quantization is not presumed derived from P1/P2."
    },
    {
      "id": "premise:given-angular-quarter-zeta-cut",
      "titleEN": "Given angular quarter operator and zeta cut",
      "statementEN": "Given a single circle operator D_alpha=-i partial_theta with f(2pi)=exp(2pi i alpha)f(0), 0<alpha<1, complex-fermion zeta quantization and negative eigenvalue argument +pi. The quarter source has alpha=1/4; alpha_ref=3/4 is a diagnostic comparison, not archival D_ref. Its angular Green propagation is not positive occupation covariance or already physical time evolution."
    },
    {
      "id": "premise:given-thermal-sea-abel-subtraction",
      "titleEN": "Explicit Abel subtraction of chiral sea",
      "statementEN": "For the already given single chiral particle/hole source, use the exponential Abel sum of negative sea energies and subtract only its -1/epsilon² term. This is a declared absolute regularization; its selection and compensation of the central loop phase by an original eta/bulk contribution are not presumed consequences of P1/P2."
    },
    {
      "id": "premise:carrier-covariant-invariant-ring",
      "titleEN": "Covariant invariant-ring primitiveness at variable carrier rank",
      "statementEN": "For the local inference test: V=C^b, W=C², G=S(U(V)×U(W)), both fermion legs in Λeven(V⊕W). Primitiveness excludes invariant factors of smaller positive carrier degree in the covariant polynomial invariant ring. No global minimum over different sources/ranks or independently selected special tensor support is assumed."
    },
    {
      "id": "premise:fixed-background-quarter-fermion-source",
      "titleEN": "Given free quarter-source background without a selected Higgs derivative",
      "statementEN": "The existing two-mode source path has K0=diag(17/16,4097/4096) and a fixed Berezin measure. Compare only completely coincident fermionic source polynomials at Φ=0 and explicitly supplied extensions K0 or K0+Φσx. These are diagnostic extensions, not two claimed P1/P2-admissible worlds or a comparison of complete Φ-dependent functionals."
    },
    {
      "id": "premise:first-order-perturbation-graph-estimate",
      "titleEN": "Audit of the general perturbation argument",
      "statementEN": "Given: a selfadjoint first-order anchor with H1 graph norm and a symmetric first-order perturbation with bounded coefficients on the same domain. The quarter/reflection control uses the explicit compact torus operator in section11.2, not a claimed original P1/P2 source."
    },
    {
      "id": "premise:fixed-field-four-mode-inverse-data-control",
      "titleEN": "Fixed fields in the four-mode diagnostic",
      "statementEN": "Given: the first two particle/hole pairs of one chiral CAR source, energies (alpha,1+alpha,1-alpha,2-alpha), the same Hurwitz sea energy and fixed field labels. U rotates only neutral pairs p0h0/p1h1 by 3/5,4/5. Spatial locality or P1/P2 selection of U is not assumed; this is not the internal P2 V16 carrier."
    },
    {
      "id": "premise:quarter-circle-coordinate-local-algebra",
      "titleEN": "Local coordinate algebra of the conditional quarter source",
      "statementEN": "Given the conditional scalar circle realization H=L²(S¹), Dom B=H¹_per, Bα=−i∂θ+α with α=1/4 or3/4. The local algebra investigated here is explicitly smooth multiplication C∞(S¹) on this line, not a complete physical field-insertion algebra already derived from P1/P2."
    },
    {
      "id": "premise:quarter-circle-nonlocal-cauchy-enlargement",
      "titleEN": "Explicitly additional nonlocal Cauchy enlargement",
      "statementEN": "On the Fourier basis en,n∈Z, U is the bilateral shift and C projects onto n≥0. Consider the algebraic or explicitly smooth *-algebra with controlled bounded commutators generated by U,U*,C. No identification with B(l²), automatic domain statement for its norm completion or original local field algebra is assumed."
    },
    {
      "id": "premise:given-finite-sm-triple-yukawa-inputs",
      "titleEN": "Given finite SM representation with symbolic Yukawa inputs",
      "statementEN": "Use only the existing v252/v254 representation with C⊕H⊕M3, already supplied color/weak modules and four symbolic complex3×3 Yukawa blocks. No physical values are chosen or fitted. Classify the particle LR one-form block; no proof of the full real96-dimensional fluctuation with antiparticles, JAJ and Majorana sector is claimed."
    },
    {
      "id": "premise:complete-labeled-even-car-kernel",
      "titleEN": "Complete kernel in a fixed CAR dictionary",
      "statementEN": "Given: full even T in a fixed labeled finite CAR Fock basis and explicit boundary/trace condition. Link logarithm needs T00≠0; finite selfadjoint matrix logarithm needs T>0. Physical identification, length and original Φ-dependence are not presumed derived from P1/P2."
    },
    {
      "id": "premise:local-domain-product-dirac-skeleton",
      "titleEN": "Declared product skeleton for algebra identifiability",
      "statementEN": "Given a compact spin collar with a local selfadjoint elliptic geometric boundary problem stable under smooth multipliers, or closed double with H1 domain. On H=L²(M,S)⊗HS(Mn(C)),n≥2: D=D0⊗I, B=B0⊗I, J=J0⊗(X→X*), Γ=Γ0⊗I, R=R0⊗I, with zero finite Dirac term. Compare scalar and left matrix insertions on the same H. No arbitrary APS-domain invariance, complete OS source, unit winding or minimal P1/P2 branch is claimed."
    },
    {
      "id": "premise:unmarked-hermitian-rank-five-exterior-carrier",
      "titleEN": "Given unmarked Hermitian rank-five carrier with volume",
      "statementEN": "Given E=C5 with Hermitian metric and unit complex volume, F=ΛE, Γ=(-1)^degree and conjugate-linear Hodge J_H. Neither3+2, named particle/gauge modules nor Yukawas are inserted. Internal Clifford operators are not identified with physical local fermion fields; rank5 and volume are not newly source-derived."
    },
    {
      "id": "premise:minimum-rank-holomorphic-even-even-fundamental-class",
      "titleEN": "Minimum rank in the covariant even-even-fundamental tensor test",
      "statementEN": "Given E=V⊕W with b,s≥1, Hermitian metric and unit complex volume; G=S(U(V)×U(W)); two nontrivial positive even exterior fermion legs and one fundamental boson leg in W. Rank and split vary. Primitivity is in the holomorphic covariant invariant ring without independent conjugate/dual field variables. At rank5 restrict the same volume tensor vol(α∧β∧h) to all splits. No actual source vertex, equal complete S1 minimum defect or physical P1/P2 countermodel is assumed."
    },
    {
      "id": "premise:reverse-bootstrap-e8-closure-framework",
      "titleEN": "Reverse selection in the discrete closure framework",
      "statementEN": "The origin paper discrete E8 closure framework with rank, family, spinor and simple-current conditions is given. Its numerical fixed-point identities are not identified with a physical operator realization."
    },
    {
      "id": "premise:local-e8-current-response-realization",
      "titleEN": "Given local E8_1 vacuum representation and explicit cocycle",
      "statementEN": "Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation."
    },
    {
      "id": "premise:faithful-differentiable-covariance-chart",
      "titleEN": "Given differentiable faithful covariance chart with explicit sector embedding",
      "statementEN": "Finite or explicitly regulated CAR one-particle chart with 0<C<I; h(C)=log(I-C)-log C. The individual sector-block statement requires fixed sector projectors commuting with the background covariance; moving embedding, grading, scale and source must be differentiated separately. Interpretation as a complete state uses the quasi-free hypothesis. No physical P1 source or Phi insertion is selected."
    },
    {
      "id": "premise:differentiable-isometric-source-frame",
      "titleEN": "Given differentiable isometric embedding in fixed source charts",
      "statementEN": "Q(Phi) is a differentiable isometry from finite/regulated carrier coordinates into a fixed source Hilbert chart, with differentiable covariance C_Sigma. a=Q*partial Q and N=(I-QQ*)partial Q are its induced frame/normal components. This premise does not select Q or Phi from P1; moving source charts and unbounded operators require additional connection/domain conditions."
    },
    {
      "id": "premise:fermionized-ns-vector-algebra-fixed-d5",
      "titleEN": "Unchanged local NS fermionization with fixed D5 group action",
      "statementEN": "The existing conditional D8->Z^8 fermionization of the selected E8_1 theory and its nonanomalous Z2 with spin/twist data is supplied. Its16 real local NS generators carry (10,1)+(1,6) under the fixed Spin10 x Spin6 cut. Consider their generated algebra in the same NS representation, without adjoining Ramond/twist intertwiners, field/crossed-product extensions or changed group action. No physical3+1D identification is presumed."
    },
    {
      "id": "premise:process-composition-finite-controls",
      "titleEN": "Explicit finite process and cyclicity controls",
      "statementEN": "The controls compare finite classical and complex-quantum processes and C4 versus L(M2) on HS(M2), J(X)=X*, Omega=I/sqrt2. Both cyclic/separating, faithful and order zero; the C4/M2 centers differ. No complete P1/P2, OS, seam or minimality countermodel is asserted."
    },
    {
      "id": "premise:unmarked-five-traceless-hermitian-quartic",
      "titleEN": "Unmarked traceless Hermitian five-dimensional endomorphism",
      "statementEN": "The diagnostic grants X=X* on an unmarked complex rank-five carrier, trX=0 and trX²=1 for shape classification. It studies trX⁴ or -t+t²±trX⁴, t=trX². Field, trace constraint, contraction, sign, even symmetry and dynamics are not derived here from P1."
    },
    {
      "id": "premise:geometric-two-face-response-and-kernel-quotient",
      "titleEN": "Geometric two-face compression and explicit projection kernel",
      "statementEN": "Given selfadjoint B, labeled identified fibers and ell>0, G is the orthogonal graph projection of T=e^-ellB, A=(I+T²)^-1, V=(A^1/2,T A^1/2). The geometric scalar F_f=diag(M_f,0), L_f=V*F_fV and K_G(x,y)=<x,Gy> are studied without identifying a physical covariance, instrument, Born rule or reprojection law."
    },
    {
      "id": "premise:boundary-kernel-frame-sewing-interpretation",
      "titleEN": "Explicit boundary-kernel interpretation with identity kernel A",
      "statementEN": "For positive injective dense-range F=A^1/2, K_O=F O F are interpreted as bilinear kernels in a nonorthogonal frame with identity kernel A. In finite regulators A W A=A forces inverse-Gram sewing; in infinite dimension it is defined as a bounded form using F^-1 K. This is an explicitly conditional representation reading, not a physical intervention, projection or measurement law selected by the original action. The alternative reading of F_1 as an actual face filter is not excluded."
    },
    {
      "id": "premise:foundational-reset-no-target-input",
      "titleEN": "TFPT for later comparison only",
      "statementEN": "The explicit research reset studies independent starting classes. E8, five,3+2, SM algebra, Yukawa data, Hilbert space, complex amplitudes and TFPT clocks are not inputs. Preserve existing TFPT results. After at most three failed origin attempts on unchanged premises change at least one fundamental premise or direction; coding bugs do not count. State assumptions, necessary consequences and newly explained properties."
    },
    {
      "id": "premise:finite-event-poset",
      "titleEN": "Finite event order as starting data",
      "statementEN": "A finite strict partial order is supplied as event/precedence structure without metric, space, rates or probability. Order is an input in this class."
    },
    {
      "id": "premise:finite-deterministic-joint-replication-control",
      "titleEN": "Finite deterministic processes with full joint replica control",
      "statementEN": "Supply finite nonempty local input/output sets and total w:product O->product I. Every finite number of coordinatewise independent replicas allows every joint deterministic map I_k^M->O_k^M at corresponding interfaces, including constants/erasure. Require at least one global solution for every such experiment; do not assume a global order."
    },
    {
      "id": "premise:finite-update-observation-errors",
      "titleEN": "Supplied finite update, observation and errors",
      "statementEN": "Finite X, deterministic step f and observation o, optionally error support T, are inputs. A record-cell system is a family of disjoint nonempty forward-closed observationally distinguishable sets, not automatically an exhaustive invariant partition."
    },
    {
      "id": "premise:finite-operational-composition-action",
      "titleEN": "Supplied finite operational action and effects",
      "statementEN": "A transformation monoid acts on finite X; effects define operational distinguishability. Composition, action, effects and any selected time generator are separate starting data."
    },
    {
      "id": "premise:repeatable-distinction-benchmark",
      "titleEN": "Explicit repeatable-distinction benchmark",
      "statementEN": "Recognizable reports, reproducibility of a declared context and available serial continuations are separate commitments. The sharp benchmark additionally has an immediately result-repeatable test and two preparations with different certain reports. These conditions are stronger than mere observability; probability, free erasure, re-preparation, replica control or physical time are not assumed to follow."
    },
    {
      "id": "premise:three-explicit-observability-countermodels",
      "titleEN": "Three explicit operational countermodels",
      "statementEN": "R has deterministic preparation/read traces and continued partial-identity branches only, no primitive mixing or internal re-preparation. I is the classical two-bit simplex with probabilities and nondisturbing joint access. D is the square with explicit positive X/Z branches, square symmetries, preparations, mixtures and finite serial/adaptive closure. These selected rules define counterworlds rather than necessary physics."
    },
    {
      "id": "premise:finite-prefix-consistent-sequential-data",
      "titleEN": "Finite prefix-consistent sequential response tables",
      "statementEN": "Supply finite preparation/action/report sets, finite horizon and nonnegative normalized sequential probability tables. Summing out the last report recovers prefixes and earlier marginals do not depend on future choices. Impose no memory, locality, stationarity or preparation-noncontextuality bounds on the representation."
    },
    {
      "id": "premise:native-g31-concrete-event-representation",
      "titleEN": "Actual native G31 event representation",
      "statementEN": "Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round."
    },
    {
      "id": "premise:native-symmetric-quartic-and-erasure-constructors",
      "titleEN": "Symmetric quartic and actual erasure intertwiner",
      "statementEN": "Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product."
    },
    {
      "id": "premise:fixed-complete-four-five-primitive-reflection-generators",
      "titleEN": "Complete four/five generators with exactly60 reflection jumps",
      "statementEN": "Consider only Poisson generators sum_j lambda_j(Ad_rj-id) over the actual60 native reflections, with complete continuous-time four- and five-generators fixed. No extra Hamiltonians, coherent algebra events, or generators inferred uniquely from one finite-time channel."
    },
    {
      "id": "premise:complete-logical-nondisturbance-native-covariance",
      "titleEN": "Complete logical nondisturbance and native covariance",
      "statementEN": "A CPTP residual channel Phi on V5 tensor bar(V4) preserves Tr4 Phi(X)=Tr4 X for ALL joint X and is covariant under the native G31 product representation. This is stronger than preservation only of erasure states rho tensor I4/4 and defines an explicit conditional process class."
    },
    {
      "id": "premise:coherent-block-source-algebra-operation-menu",
      "titleEN": "Extended coherent source-algebra operation menu",
      "statementEN": "For realization of the residual channel, unitary elements of the represented group C*-algebra may serve as Kraus events. This mathematical menu does not assert physical permission derived from P1. The block algebra or each sector is considered; extra coherences between inequivalent representations are not automatically accessible observables."
    },
    {
      "id": "premise:existing-logical-clock-and-charge-marking",
      "titleEN": "Existing logical clock and charge marking",
      "statementEN": "Retain the existing logical labels weak0/1, color2/3/4, source point5 and Y5,R5,sigma5. Equivalence under their stabilizer is not automatically an equivalence of lifts commuting with the original native four/P1 clock."
    },
    {
      "id": "premise:original-marked-simplex-positive-polar-rule",
      "titleEN": "Original simplex and positive polar rule",
      "statementEN": "Use the actual W6 simplex and raw metric, unchanged native three-clock and original positive polar prescription at one of its three marked fixed points. The prescription and its physical selection are not derived from P1."
    },
    {
      "id": "premise:actual-clifford-identification-and-finite-overgroup-theorem",
      "titleEN": "Actual Clifford identification and finite-overgroup theorem",
      "statementEN": "Retain the actual concrete identification of <G31,zeta8 I4> with complex Clifford X2. Apply NRS6.5 only to the proved finite determinant slice; a group-order match alone does not suffice."
    },
    {
      "id": "premise:original-erasure-decoder-and-same-native-quarter-polynomial",
      "titleEN": "Original decoder and same native quarter polynomial",
      "statementEN": "Fix original erasure decoder D(A)=W(A tensor I4)W* and its mark P220=P2 tensor I4. Substitute actual sigma20=sigma5 tensor bar(sigma4) and all four native reflection lifts of original tau5 into the existing R polynomial. This tests that continuation, not all coherent lifts."
    },
    {
      "id": "premise:unitary-full-branchwise-erasure-clock-lift",
      "titleEN": "Unitary continuation with full branchwise erasure action",
      "statementEN": "Consider unitaries T on the actual erasure carrier satisfying T(rho tensor I4/4)T*=RrhoR* tensor I4/4 for every logical state rho and a given logical unitary R. This is a declared compatibility class, not a physical instrument rule derived from P1."
    },
    {
      "id": "premise:native-marked-simplex-clock-compatibility-without-fixed-polar-mark",
      "titleEN": "Original clock/simplex without an imposed polar mark",
      "statementEN": "Retain actual normalized W6 simplex, sigma=(234), tau=(01), q=5. Consider ALL Hermitian rank2 projectors P with RanP inside Fixsigma, [P,tau]=0 and one positive/one negative tau eigenvalue on RanP. The original polar rule is a comparison point, not an imposed condition of this compatibility class."
    },
    {
      "id": "premise:original-native-fourth-moment-parent",
      "titleEN": "Actual original moment parent",
      "statementEN": "Use the actual native fourth root moment with 40M4=Pi5+Psym4 and Hnat=I-(Pi5+Psym4)/2. Add no new marking-dependent Hamiltonian. Physical identification as a time generator is not inferred merely from matrix existence."
    },
    {
      "id": "premise:local-product-code-orbit-geometric-parallel-transport",
      "titleEN": "Geometric parallel transport of local-product code paths",
      "statementEN": "Consider moving code subspaces T(t)RanV, T(t)=tensor_j Uj(t), Uj(0)=I, returning to the original code. Use the Grassmann/Berry connection induced by the actual Hilbert inner product. No general collective code path, additional internal dynamics or physical adiabatic generator is assumed or excluded."
    },
    {
      "id": "premise:original-coupled-determinant-event-cover",
      "titleEN": "Original coupled determinant event lift",
      "statementEN": "Compare all60 original events with h=exp(-i pi/4)r and A5=-rho5(r) on the coupled finite SU4 cover. This is not the unchanged order2 G31 word or a selected physical pulse."
    },
    {
      "id": "premise:literal-charged-current-native-register-dictionary",
      "titleEN": "Literal charged-current/register identification",
      "statementEN": "The tested dictionary identifies the four original charged C_a currents with the unchanged native four-register and requires the actual erasure tensor intertwining under gl5 source operations. This is a tested hypothesis, not a P1-derived fact."
    },
    {
      "id": "premise:literal-native-positive-cartan-quartic-fock-embedding",
      "titleEN": "Actual positive Cartan quartic Fock embedding",
      "statementEN": "Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows."
    },
    {
      "id": "premise:compact-inner-e8-zero-mode-operation-class",
      "titleEN": "Compact inner E8 current operation class",
      "statementEN": "Consider exponentials of compact weight1 current zero modes and arbitrary sequences preserving the given code at the endpoint. Optional conformal L0 propagation commutes with them. Nonlinear Hamiltonian polynomials, higher modes and non-automorphisms are not identified with or excluded as this class."
    },
    {
      "id": "premise:original-neutral-root-pair-moment-fields",
      "titleEN": "Original neutral root-pair moment fields",
      "statementEN": "Use existing E8_1 neutral_source_response reader:64 chirally selected roots, original omission of one SU4 family weight line, B48=12T_E8+R_neut, AR=4P_A2-8P_U1. Family omission is an explicit existing input, not derived here from P1; neither it nor the R port or physical coupling is newly selected."
    },
    {
      "id": "premise:solely-neutral-local-weight-two-zero-mode-class",
      "titleEN": "Solely neutral local weight-two zero modes",
      "statementEN": "Consider energy-preserving zero modes o(v)=v_(wt(v)-1) of all Hermitian lattice-charge-zero weight-two fields in given vacuum and temporal sequences solely of these generators. Grant the mathematical class without claiming P1-derived physical intervention permission, time or projections. Alternating root-current operations, individual K48 mode roundtrips, nonneutral fields and higher mode products are different classes."
    },
    {
      "id": "premise:original-displayed-j-equivariant-cartan-root-isometry",
      "titleEN": "Original displayed J-equivariant root isometry",
      "statementEN": "Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate."
    },
    {
      "id": "premise:original-positive-adjoint-two-insertion-response-profiles",
      "titleEN": "Original positive adjoint two-insertion response",
      "statementEN": "Compute o(W_beta), W_beta=E_beta,0 R_neut, with unchanged original real transverse v_beta. Ordered return is positive Fock Gram o(W)*o(W) between code or original trimer states without intermediate projection. Allow W=sum a_beta W_beta for arbitrary complex input coefficients; distinct lattice charges force nonnegative weights |a_beta|². Nonadjoint returns, other complex transverse fields, longer histories or physical instruments are not thereby selected."
    },
    {
      "id": "premise:common-unnormalized-positive-return-erasure-test",
      "titleEN": "Common unrenormalized positive-return erasure test",
      "statementEN": "The tested hypothesis reads actual positive K5 and K20 of the same source response with common original normalization as complete branchwise erasure continuation: K20(rho tensor I4)K20=K5 rho K5 tensor I4 for all rho, equivalent under positivity to K20=K5 tensor I4. This is a tested candidate identification, not a claim every correlation is a physical instrument. Independent relative20/5 recalibration is outside this fixed map."
    },
    {
      "id": "premise:specified-original-pair-ordered-response-composition",
      "titleEN": "Specified ordered response family of original single pair",
      "statementEN": "Examine mathematically specified moments M_n=Pi L_beta^n Pi,n>=0, L_beta=o(W_beta)*o(W_beta), for same original doubled root(0,0,0,0,0,0,-2,-2). Pi/Q only split full amplitudes; no projection is executed. This tests composition/memory of the already determined source, not P1 selection of physical history, coupling, preparation or time. Full coherent multiroot processes must additionally retain nonneutral final charge sectors."
    },
    {
      "id": "premise:canonical-source-stress-all-mode-history",
      "titleEN": "All modes of original canonical source stress tensor",
      "statementEN": "In given E8_1 lattice vacuum use exactly T_E8=(1/2)sum:h_A h_A:,c=8, without improvement/background charge. Examine all Virasoro modes and well-defined pure stress histories in original neutral Cartan-Fock space. This is a mathematically specified geometric transport class, not P1-selected physical history or time. It differs from prior class of all neutral weight2 ZERO MODES; do not combine exclusions. First erasure Ward action uses existing a_a=bar u_a(1), no extra source, root or family choice."
    },
    {
      "id": "premise:original-cartan-current-local-smearing-and-weyl-class",
      "titleEN": "Full original Cartan-current insertion and specified Weyl class",
      "statementEN": "In existing neutral Cartan-Fock module use full original current field and its actual CCR, all smooth real8-vector profiles with Vhat_-n=conjugate Vhat_n and h0=0. Ordered unitary classification uses only finite exp(iJ(V)) histories of these currents. Explicit mathematical class in given source net, not raw-P1-derived permission, field coupling, physical time or code=H_adm identification. Root currents, general field products and mixed stress histories are outside."
    },
    {
      "id": "premise:full-existing-local-e8-field-hull-observer-code",
      "titleEN": "Full existing local E8 field algebra with observer five-code",
      "statementEN": "Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule."
    },
    {
      "id": "premise:original-native-event-difference-fields-and-state-controls",
      "titleEN": "Actual native event fields and distinct state controls",
      "statementEN": "In accepted full E8 field net test actual native reflection automorphisms on reconstructed quartic fields and coherent difference against identity. Original sigma=(234),tau=(01),markedq=5 remain given comparison data; preparation ofq not derived. Vacuum and normalized marked source state are distinct mathematical controls. Difference of automorphism and identity is NOT assumed physically allowed intervention/instrument. Retain all local Wick/Taylor products and complete source states. Positive Gram/relative normalization is examined as computable tensor, never imposed as physical selection rule."
    },
    {
      "id": "premise:intrinsic-native-event-geometry-without-marked-clock",
      "titleEN": "Intrinsic event geometry without marked clock as selector",
      "statementEN": "User159c5316 17Oct replaces automatic instrument/whitening-origin next step with intrinsic structure test. In accepted full unitaryE8_1 fieldnet use actual quarticproducts,eventrelations,*-operation,centralOPEnormalization. No originalpolar/P2,desired3+2,markedq,selectedclock orphysicalrealstate rule as construction criterion. Compare originalgeometry/clocks only afterwards. Transportmetrics withbasis changes; distinguish centralalgebraicOPE data andstate-dependentGNS responses. Only states with definedcommonfieldmoments; positivefiniteenergycontrols not automaticallyRP/clustering/finiterelativeactionP1seeds. Retain full localproducts,composition,correlations; no assumed independenttensorphysics."
    },
    {
      "id": "premise:original-seam-state-data-on-native-flags-without-fitted-selector",
      "titleEN": "Original seam/state data on actual60 flags without fitted selection",
      "statementEN": "User17Oct: test originalseam/statedata onactual60 flags P(q,S); no fittedclock,coupling,newselectionrule. Additional EXISTING data topreviousunmarkedobject: fullchargedinternalv=(0^5,1/2^3),deck square,originalE8vacuum; optionaloriginalR_neut/retained48familyomission explicitgivenpremise,all4cases. SEPARATEactualoriginalfoldphasesalpha1/4,3/4withfixedexistingM,d=M((j+alpha)/8−1/2);disshift/timediagnostic,notautomaticallyfullCARcovarianceorE8vacuumexpectation. Fullroot/fieldaction,nocompressiononly. NopostulatedphysicalTheta,field/state/normalorderingdictionaryorP1selection."
    },
    {
      "id": "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
      "titleEN": "Fixed conformal current reference in original fold-state transport",
      "statementEN": "Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms."
    },
    {
      "id": "premise:original-single-circle-local-density-and-geometric-stress",
      "titleEN": "Original single-circle density and geometric stress",
      "statementEN": "Same stipulated chiral CAR source, negative-D sea, actual quarter domains, eightfold F and existing M/O. Local density J_m=sum:a_r* a_(r+m): and its half-density geometric stress with alpha ground energy follow from this source. Original scalar phases fold to correlated generally offdiagonal channel profiles, not independently selected E8 root weights. NS/quartic reference remains conditional. Global Fourier modes control existing fields; no P1-selected local profile/detector/coupling. No isolated b->-b invariance for these new responses; transport positive QWZ occupation with charges/currents/V/b together."
    },
    {
      "id": "premise:original-gap-normalized-primitive-relative-action",
      "titleEN": "Original gap-normalized relative P1 action",
      "statementEN": "Original source defines chi_seed=1/lambda_1^+(|B_Sigma|), S_prim=Tr f(D_rel/chi)-Tr f(D_ref/chi)+i pi Deltaeta/2 for declared spectral test profile. D_rel/ref are geometric odd/even parts; varying reflection, boundary domains, reference, state and eta must be retained. This is the given source function, not a new coupling or physical event rule. Scalar quarter CAR realization and NS/native field dictionary remain explicitly conditional."
    },
    {
      "id": "premise:positive-scalar-first-order-source-comparison-class",
      "titleEN": "Whole scalar positive first-order comparison class",
      "statementEN": "Compare all smooth scalar first-order differential operators on already given oriented source circle, fixedL2(dtheta),H1per,positive nowhere-zero principal symbol. Periodic chart contains quarterphase in c/holonomy; do not add domain twist again. Original Balpha is reference point; comparison operators are not all claimed physically admissibleP1 data. Point defects,matrix fibers,nonlocal and general higher-dimensional sources are outside."
    },
    {
      "id": "premise:given-original-fixed-one-particle-time-winding-ward",
      "titleEN": "Actual fixed one-particle time-seam Ward of given source",
      "statementEN": "Already given source Balpha=-i dtheta+alpha,Z=multiplication exp(itheta) satisfies[Balpha,Z]=Z, equivalently exp(-itBalpha)Z exp(itBalpha)=exp(-it)Z. Retaining THIS concrete operator relation is stronger than unit spectral flow. It is given-source datum, not a rule derived here from bareP1. Do not identify one-particle Ward with charge-dependent CAR sea term or physical second. Charged quarterlift Qalpha=exp(i pi alpha/2)Q4,reference sea and integer carry retained separately."
    },
    {
      "id": "premise:four-starts-fixed-state-versus-whole-datum-comparison",
      "titleEN": "Four starts at fixed state versus joint datum transport",
      "statementEN": "Comparison scope: q2,3,4,5 and globalS01. Retain originalFCαF*,M/O,fullE8fields,nativeJ andactualchargedfold,time,adjoint,normalordering,erasure. Separate passive whole-presentation changes, active mathematical field automorphisms and symmetries of FIXED given source state. Marks are actual intrinsic quartic rays/polar planes,not chosen couplings; equivalence retains stated field/coordinate/reference object. Do not postulate physical access orP1selection from comparison."
    },
    {
      "id": "premise:pascal-carrier-exhaustion-equation",
      "titleEN": "Pascal exhaustion condition as input",
      "statementEN": "g is natural; the stipulated equation is 2^g=g^2+g+2. The equation itself is the additional carrier exhaustion rule."
    },
    {
      "id": "premise:specified-finite-carrier-discriminant-form",
      "titleEN": "Specified finite carrier discriminant form",
      "statementEN": "The model is El=Fin4 x Fin4 with componentwise addition modulo4 and qZ(x,y)=(5x^2+3y^2) mod8. Its identification with a concrete physical carrier is not inferred without premises."
    },
    {
      "id": "premise:specified-shared-s3-register-channel-family",
      "titleEN": "Specified shared S3 register family and homogeneous GKLS time",
      "statementEN": "The specified six native words u_g act in each register by the same S3 representation; shared event weights w(q)=(1/2+q/6,q/6,q/6,1/18-q/6,2/9-q/6,2/9-q/6), 0<=q<=1/3. Seek a consistent time-homogeneous GKLS semigroup on the same finite registers with unit time step."
    },
    {
      "id": "premise:specified-marked-m4-clifford-channel",
      "titleEN": "Fixed marked M4 Clifford channel",
      "statementEN": "Five specified Hermitian Gamma_j on M4 satisfy {Gamma_i,Gamma_j}=2delta_ij I. The endpoint is Phi=7id/12+sum_j Ad(Gamma_j)/12 with spectrum 1,1/3,2/3 on orthogonal blocks 1,5,10. Comparison class is any homogeneous GKSL generator on the same M4 with exp(L)=Phi^n for positive integer n."
    },
    {
      "id": "premise:original-stable-pointer-instrument-and-given-context",
      "titleEN": "Original conditional pointer template and supplied context",
      "statementEN": "Original text assumes A_obs subset A_rec and complete orthogonal pointers, then writes M_O(rho)=sum P_i rho P_i tensor |i><i|. Optional context Q supplies rho0, observables, time and readout. No identification with native Pq or physical P1 implementation is assumed."
    },
    {
      "id": "premise:same-full-positive-energy-separating-local-car-source",
      "titleEN": "Same full positive CAR source with locally separating sea",
      "statementEN": "Given negative-D quarter-CAR representation with full H_alpha=sum(n+alpha):a_n* a_n:>=0,H Omega=0 and support-preserving time covariance. Omega separates each larger proper-interval algebra. Test class: bounded proper-local A with A Omega an exact positive-energy eigenvector. NS/quarter chart retains support, reference and time; no physical P1 net identification follows."
    },
    {
      "id": "premise:charged-car-pure-two-face-cauchy-graph-state",
      "titleEN": "Given charged CAR quantization of the full two-face Cauchy graph",
      "statementEN": "As in section4 of the original collar proof, additionally assume charged CAR(H direct_sum H) and the pure gauge-invariant quasifree state with occupation G_ell. The second labeled face has n=(1+exp(2ell B))^-1. This is a field/statistics premise, not a consequence of P1; reduction does not select or physically execute a one-face operation. The long-collar limit is used only within this declared operator/CAR class."
    },
    {
      "id": "premise:complete-lagrangian-dirac-cauchy-car-quantization",
      "titleEN": "Complete Dirac Cauchy source with Lagrangian normal action",
      "statementEN": "Complete orthogonal Dirac Cauchy projection C on the entire boundary module, with actual normal unitary Clifford action N satisfying N C N*=I-C. Explicitly assume charged CAR quantization with pure quasifree covariance C. Bounded current generators X,Y commute with N and have Hilbert-Schmidt commutators with C. The existing compact product collar B=diag(A_alpha,-A_alpha) on both components and both faces is the concrete control; its smooth scalar boundary multipliers are implementable. Statistics and physical measure selection are not derived from P1; no automatic extension of the unrenormalized trace to higher boundary dimensions."
    },
    {
      "id": "premise:unchanged-quarter-qwz-full-excitation-source",
      "titleEN": "Unchanged full quarter-QWZ excitation source",
      "statementEN": "Current v1033 QWZ cylinder at mass1,width8,ONE complex CAR copy,alpha1/4 or3/4,full negative sea and H_N=N/(2pi)[dGamma(h_N)-Tr(P_Nh_N)]. Retain both edges,massive modes,charge sectors and physical energy/vacuum reference; compare actual low eigenmodes. No P1/P2 selection of this realization,changed stress or executed quotient."
    },
    {
      "id": "premise:given-d8-uniform-z4-twist-and-fixed-fold-time",
      "titleEN": "Given D8 target with uniform Z4 twists and fixed fold-time law",
      "statementEN": "Only in the conditional eight-channel target test,D8 currents with roots +/-e_i+/-e_j,twist charges q=(r/4)^8+m,m in Z8,r0..3,h=q²/2+N and existing fold time K=q²/2+delta.q+N,delta_k=(k+alpha)/8-1/2 are given. Keep all D8 currents and Cartan charges fixed,chiralities distinct. This creates neither eight physical source copies nor local defect endpoints from P1."
    },
    {
      "id": "premise:given-eightfold-source-odd-refielding-dictionary",
      "titleEN": "Given eightfold source fold with explicitly changed odd target action",
      "statementEN": "Given existing one-copy CAR source,sea,fold,p in Z8,Heisenberg oscillators,Q=sum p,H=8K,K=N+p²/2+delta.p with delta_k=(k+alpha)/8-1/2,and actual SOURCE windingT1=Rp+e0,alpha1/4or3/4. Test explicit Phi=I on D8,rho=Pp+r on odd v with oscillator-P to odd spinor extension. Odd target Cartan action is explicitly changed by this comparison; inclusion in the original local net is separate. Neither folding,SOURCE/P1-core circle identification nor bosonic statistics is derived from P1."
    },
    {
      "id": "premise:sharp-half-flux-original-net-state-contracts",
      "titleEN": "Sharp half flux in separate original net/state contracts",
      "statementEN": "Check separately under their own premises: v1033 with expressly ADDED M4 flux register,F_A=X² tensor(-1)^N_A,even algebra M4 tensor CAR+,common strip flux and uniform fixed zero-space filling; v746 with full CAR of paired window I union(I+N/2),NS-clockS^(N/2),halfcircle sign; direct chiral free limit as sharp-sign Bogoliubov on standard locally normal NS CAR net. No unproved identification of these nets and no exhaustion of smooth/renormalized outer-field limits. Even-to-full normality argument uses the standard free local factor contract."
    },
    {
      "id": "premise:local-register-retained-in-joint-charged-word-limit",
      "titleEN": "Local register retained in full charged word-limit contract",
      "statementEN": "Comparison class: original v1033 register grade2 under AdZ, Z retained as bounded even local unitary outside field support, full joint positive word limits with nonzero field norm. No boundedness of renormalized field sequence assumed. Separate two-edge test retains original shared flux ZL=ZR. Projection classification only when retaining the FULL original even matter algebra, time and A/A*-admissibility."
    },
    {
      "id": "premise:conditional-local-parity-gauss-completion-with-common-H",
      "titleEN": "Conditional local parity-Gauss completion with common original time",
      "statementEN": "ADDED comparison contract, NOT P1-derived: old four-state register with total-even matter; redundant Z2 link qubits and local Gx=(-1)^nx tau_(x-1)^x tau_x^x=1; gauge-invariant local observable net. Original QWZ hopping coefficients, common microscopic Hamiltonian and energy reference; no new electric energy. Sector-specific f(Z) subtractions must map to f(W) and can be nonlocal. No identification of Gauss space with original Padm."
    },
    {
      "id": "premise:fixed-one-copy-eightfold-connection-charge-dictionary",
      "titleEN": "Fixed charge-preserving connection in original one-copy eightfold",
      "statementEN": "Given one-copy fold has transition C(h), C(h)^8=hI. Available register half flux changes cover holonomy h to -h. Compare uniform spinor-OPE twist -C(h) in same complex U1 charge dictionary and connection, up to charge-preserving fiber conjugacy/periodic gauge transformation. Real SO16 charge conjugation at h=±i, changed connections/domains and eight independent copies lie outside exclusion class. Source circle/P1 core and cover/P1 deck are not identified."
    },
    {
      "id": "premise:original-neutral-eightpreimage-parity-pair",
      "titleEN": "Original neutral eight-preimage parity pair",
      "statementEN": "Actual v1033 menu F_A=X² tensor(-1)^N_A and products F_B F_A*, plus the ALREADY GIVEN one-copy eightfold dictionary eta_j(u)=psi((u+2pi j)/8)/sqrt8. E is union of all8 preimages E_I of a base interval. Register factors cancel; no added Gauss rule or copy. Fold, free chiral source and P1/core-circle interface are not newly derived premises. Finite filled Fourier band is separately declared regulator comparison with a second Fermi edge."
    },
    {
      "id": "premise:complete-vector-phase-word-Ward-comparison",
      "titleEN": "Complete vector phases with unchanged Cartan Ward action",
      "statementEN": "Only original complete Dirac/CAR Lagrangian source with normal-commuting implementable smooth profiles H_i h and fI, common analytic domains and regular Weyl representation. Target comparison retains same8 vector Cartan currents and mixed Ward words. Two grading components C,1-C on ONE face are neither strip edges nor collar faces. Finite complementary determinants are algebraic controls only; no unregulated infinite determinant or automatic QWZ/Dirac source identification."
    },
    {
      "id": "premise:smooth-unitary-pair-return-fixed-compact-source-time",
      "titleEN": "Smooth unitary pair return at fixed compact source time",
      "statementEN": "Given original chiral current/Weyl source, sea and K=Hcover/8 with delta_k=(k+alpha)/8-1/2. Periodic real phi(theta)=f(8theta), normalized UNITARY vector state W(phi)Omega; pi->0 return squeezed into base width eps on fixed source circle, same original energy reference. No assertion about general nonunitary sums, charged covariant representations, sector energy connections or decompactified time. Conditional Sugawara sector energy already fixed once lambda is given."
    },
    {
      "id": "premise:unchanged-full-source-observable-map-and-continuous-time-insertion-limit",
      "titleEN": "Full old observable map and continuous original time for nonunitary insertions",
      "statementEN": "Fixed full quarter-QWZ carrier has compatible entire finite-dimensional low spectral windows. Every retained bounded A_N is uniformly bounded and converges on them to the original operator action,including odd intermediate states. This is an operator premise independent of prepared v_N=T_N Omega_N. Arbitrary T_N,including divergent renormalization,nonunitarity and growing word degree,are required to have finite nonzero norm and a limiting ACTUAL original positive-time or real unitary-time kernel continuous at0 with that norm. No bounded first energy moment,different charged Hamiltonian or new P1 realization assumed. Spinor consequence additionally retains the given unchanged D8/even-CAR dictionary."
    },
    {
      "id": "premise:retained-local-register-with-one-common-original-vacuum-reference",
      "titleEN": "Retained local register with one common original vacuum reference",
      "statementEN": "Original v1033 Z4 register Z remains even local and remote from the proposed localized bosonic field. Initial sea has definite quarter r1or3. Full joint words retain norm of remote commutator[Z,T_N]Omega_N,required to vanish. Register Hamiltonian stays block diagonal in Z with ONE common initial-vacuum subtraction,no independently chosen sector energies. Keep actual mass1width8 Bloch matrix with both edges and all massive modes. No new Gauss support rule."
    },
    {
      "id": "premise:actual-neutral-pair-heat-filter-and-separate-chiral-Weyl-comparison",
      "titleEN": "Actual neutral pair heat filter and separate chiral Weyl comparison",
      "statementEN": "Diagnostic nonunitary attempt v_Ns=e^(-sH_N)P_E Omega_N with actual neutral eight-preimage parity pair and original full QWZ time/sea,s>0. No physical selection of this filter preparation claimed. Retain full Slater covariance,norm and time responses. Separately given chiral Weyl comparison usesK=H_cover/8,phi=f(8theta),Heisenberg oscillators; its sharp kernel is NOT identified with bare lattice parity limit without a separate proof. Joint Bloch-residue charges are not automatically separate chiral D8 Cartans."
    },
    {
      "id": "premise:given-holomorphic-kms-inclusion-and-local-global-modular-domains",
      "titleEN": "Given holomorphic KMS inclusion with distinct local/global domains",
      "statementEN": "Testing the cited abstract theorem assumes already existing holomorphic conformal c8netB,mu_B1,A=B^Z4,irreducible localindex4 andinternalgauge commutingwithrotations. omega_beta is DEFINED as Gibbs(B)restriction. Distinguish globalGNS KMSmodulargroup,L0rotations andlocalmodulargroups ofrestrictedfaithfulstates. These premises do not constructmicroscopicBfromP1; originalH_source=8(L0+delta.p) isnot silentlyidentifiedwithL0orphysicalbeta_angle2pi."
    },
    {
      "id": "premise:canonical-source-gibbs-and-parity-probe-with-retained-winding",
      "titleEN": "Canonical source Gibbs/parity probe with retained Cartans and source winding",
      "statementEN": "In existing conditional one-copy eightfold dictionary retain all8namedCartans,pinZ8,originalsea,H_source=8K,Q,globalPhi:D8+v→D8+c andactual SOURCE W1with[Q,W1]=W1. PrimitiveP1circle identification remains missing. CanonicalGibbs traces forEVERYbeta>0 are mathematicalfaithfulword/operator probes,not selectedphysicaltemperature. Retainbothoriginal limitingedges/parities; distinguishTOTAL-even andLEFT-even. No independent edgefluxregisters or microscopic pairedpumpfactorization asserted. ChangedphysicalCartan/netdictionary still needsderivation."
    },
    {
      "id": "premise:original-p1-seed-boundary-and-local-restriction-definitions",
      "titleEN": "Original P1, boundary, seed and local restriction definitions",
      "statementEN": "Reviewed originals: P1 declares oriented seam, reflection-positive kernel, unit winding and c3; stronger T_boundary includes A_+ as data; operational seed takes A_loc, time, reflection, state, seam class and collar operator as inputs. A_adm(O) is A/A*-invariant restriction of supplied local insertions. Compare without P2/E8/target spinor assumptions and without presuming an original gauge/microscopic localization intertwiner."
    },
    {
      "id": "premise:marked-seam-pure-mathematical-program",
      "titleEN": "Active purely mathematical marked-seam program",
      "statementEN": "Within the stated mathematical scope, marked pairs, lattice transport systems, reflections, clocks and cohomology are independent mathematical objects. Earlier origin/completion tasks are historical, not active. Distinguish new definitions, derived theorems and optional markings."
    },
    {
      "id": "premise:native-even-e8-gaussian-clock-and-sign-extension",
      "titleEN": "Existing even E8 lattice, Gaussian J and sign lifts",
      "statementEN": "Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts."
    },
    {
      "id": "premise:ordered-four-punctures-fixed-quarter-projection",
      "titleEN": "Four ordered punctures with fixed quarter-clock projection",
      "statementEN": "X is CP1 minus ordered(1,i,-1,-i), positive meridians product1, fixed rho(gamma_i)=J. Classify all sign lifts into full preimage Gamma of<J>, modulo simultaneous Gamma conjugation with optional prescribed peripheral conjugacy classes. No mapping-class, puncture permutation or additional spatial reflection-equivariance quotient."
    },
    {
      "id": "premise:native-clock-complex-reflection-and-quartic-subspace",
      "titleEN": "Actual clock/complex-reflection subgroup and native quartic space",
      "statementEN": "R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power."
    },
    {
      "id": "premise:original-g31-central-determinant-extension",
      "titleEN": "Original G31reflections and central determinant class",
      "statementEN": "The60original reflections generate G31of order46080. Use exactly E=Z4xG31with determinant cocycle from original holonomy proof§4; distinguish E,effective diagonal quotient and noncentral lattice lifts. Mark z."
    },
    {
      "id": "premise:ordered-original-g31-reflection-and-clock-markings",
      "titleEN": "Ordered G31reflection quadruples and chosen quarterclock",
      "statementEN": "U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts."
    },
    {
      "id": "premise:original-full-g31-action-and-inherited-lattice-sign-extension",
      "titleEN": "Full original G31action and inherited lattice sign extension",
      "statementEN": "The60actual original complex reflections act on the same original even E8lattice. Study exactly their full preimage in the automorphism group of the lattice group defined by unchanged eps_e8. Kernel A=Hom(E8,Z2), with g.a=a composed g^-1. No scalar replacement, enlarged coefficients or extra square relation for lifted puncture meridians."
    },
    {
      "id": "premise:fixed-native-d5-a3-compiler-split-and-seed",
      "titleEN": "Original D5+A3 compiler split and established seed",
      "statementEN": "Retain the established P1/P2 compiler with cyclic C4 glue to E8, D5 on the first5 and A3=D3 on the last3 standard coordinates, c3=1/(8pi) and phi_base=(4/3)c3=1/(6pi). No replacement by an optional E7 readout. Cut equivalence is the actual W(D5)xW(A3) action in the fixed chart, without conjugating J."
    },
    {
      "id": "premise:existing-marked-cyclic-cover-and-burau-model",
      "titleEN": "Existing marked cyclic covers and Burau lattice model",
      "statementEN": "On B=P1 minus(mu4 union infinity), Cn:un^n=x^4-1 for n=2,3,4 are the existing marked covers with positive finite exponent1. Fix the mu3 deck orientation omega. Use the original J and sigma with c=J sigma, actual reduced Burau matrices at omega and their retained Z[omega]^3 basis and invariant Hermitian form. Do not identify this integral model and trace form with independently constructed geometric cycles or their intersection form."
    },
    {
      "id": "premise:joint-native-compiler-cover-covariance-category",
      "titleEN": "Joint category of original compiler flags and cover actions",
      "statementEN": "Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex."
    },
    {
      "id": "premise:marked-c4-unitary-local-extension-reconstruction-class",
      "titleEN": "Exact local reconstruction class of the marked C4 glue",
      "statementEN": "Mathematical object class Ext_H(U): U=V_D5 tensor V_A3 with fixed embedding and sector chart, H=<(s,f)>=C4. Objects are connected local unitary conformal VOA extensions with the same conformal vector and underlying labelled U module exactly the sum of M_h for h in H, each once. Isomorphisms fix U and sector labels. This is an explicit mathematical reconstruction class; membership of primitive P1 is neither assumed nor derived."
    },
    {
      "id": "premise:literal-boundary-minimality-and-first-order-domain-statements",
      "titleEN": "Literal boundary-minimality and analytic-perturbation statements",
      "statementEN": "Audit objects are the original boundary-source definitions and proofs: C_seed maps an operational seed forward to a spectral boundary tuple; defect order is lexicographic on N4; the self-adjointness statement asserts relative bound0 for every symmetric bounded-coefficient differential perturbation of order at most1. These source assertions are audited rather than assumed true. No special symbol cancellation, domain invariance or absolute scale is silently added."
    },
    {
      "id": "premise:connected-local-same-conformal-compiler-extension-with-generator",
      "titleEN": "Local conformal compiler extension containing one marked generator",
      "statementEN": "U=V_D5 tensor V_A3 has its fixed unitary conformal structure and marked discriminant chart. W is a connected local unitary VOA extension sharing its conformal vector and finite dimensional weight spaces; isomorphisms fix U. The marked module M_h for h=(1,1) occurs. Exact sector support, multiplicities, finite extension index, holomorphicity and an additional minimality rule are not assumed. Membership of primitive P1 in this realization is not claimed."
    },
    {
      "id": "premise:fixed-native-reflection-rotation-character-markings",
      "titleEN": "Fixed native spatial reflection and rotation markings",
      "statementEN": "In the original E8 sign lift stratum M_i=D_ci j, additionally mark T=D_t j^s t0 for s=0,1 and/or R=D_r j. T0 is the existing alternating sign isometry. Impose the stated peripheral equivariances and dihedral relation at s=0. Marked morphisms are simultaneous character conjugations with fixed projected operators. Forgetful images use full Gamma conjugacy. No arbitrary normalizer isometries or physical selection."
    },
    {
      "id": "premise:ordered-quarter-clock-pure-hurwitz-equivalence",
      "titleEN": "Pure Hurwitz quotient of the ordered quarter clock stratum",
      "statementEN": "In addition to original Gamma conjugacy identify ordered four J tuples by pure mapping classes generated by adjacent squared Hurwitz moves b0²,b1², with b_i(x,y)=(xyx^-1,x). No puncture permutations or silently frozen extra reflection marking."
    },
    {
      "id": "premise:standard-e8-root-cut-full-flag-incidence-category",
      "titleEN": "Regular E8 cuts and full compiler flags",
      "statementEN": "Fix the standard E8 roots with their actual Weyl action. E6 objects are complements of regular A2 root systems and E7 objects complements of root lines. A tetrahedron comprises the four A2 faces of a regular A3 normal system with common D5 core; its D8 incidence requires both factors. Relations are literal root inclusions and intersection sizes. Signals are rational, real or complex functions with counting inner product. Equivalence transports all flags by the Weyl action, without a selected corner or physical source."
    },
    {
      "id": "premise:scalar-eight-channel-twist-and-explicit-compiler-charge-chart",
      "titleEN": "Given scalar quarter twist and explicit charge chart",
      "statementEN": "Compare the given eight complex current channels, q=(1/4)^8, standard D5+D3 lattice and F=I5+T with T(x,y,z)=(x+y+z,x-y-z,-x+y-z,-x-y+z)/2. Eight channels and their normalization are not derived here from P1. Distinguish a local charged field, scalar monodromy character, flux register and native clock."
    },
    {
      "id": "premise:full-unitary-e8-field-conjugacy-with-transported-marks",
      "titleEN": "Full unitary E8 field conjugacy with transported marks",
      "statementEN": "Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator."
    },
    {
      "id": "premise:literal-microscopic-charged-scaling-and-v988-slater-data",
      "titleEN": "Literal charged scaling contract and original Slater case",
      "statementEN": "Audit actual H_N, Psi_lambda,N and bosonic refinement definitions in the original charged scaling contract and unchanged v988 qwz_cylinder/occ_frame functions at Nx16,Ny6,M1, h0 at twist0 and Slater state at twist1. This audits that source path and finite energy case, without assuming microscopic bosonization or a uniform continuum bound."
    },
    {
      "id": "premise:original-v988-periodic-zero-mode-state-comparison",
      "titleEN": "Actual periodic v988 state and occupied frame",
      "statementEN": "Keep unchanged qwz_cylinder, occ_frame and fermi_cov at Nx16 Ny6 M1 and original twists0,1,2,3. Compare their actual finite CAR covariances. Physical selection of either state or equality with the E8 vacuum is not assumed."
    },
    {
      "id": "premise:original-six-ovoid-covariant-instrument-category",
      "titleEN": "Finite instrument category of the actual six ovoid effects",
      "statementEN": "Fix the rank1 six-effect POVM transported equivariantly from the marked E8/J doily into the original quartic H5. Consider completely positive instruments from End(H5) to itself with exactly these effects and covariance under the actual60 generators. Physical selection of this measurement, an instrument, a preparation or a time unit is not assumed."
    },
    {
      "id": "premise:relational-native-quartic-tensor-sector-without-state-selection",
      "titleEN": "Relational quartic tensor sector without selected physical state",
      "statementEN": "Consider conjugate(H5) tensor H5 with the actual diagonal G31 action as a finite kinematic category. Keep its unique invariant vector ray, its multiple invariant densities and correlations conditional on a chosen singlet separate. Mathematical construction does not derive the microscopic presence or population of this sector."
    },
    {
      "id": "premise:native-tensor-povm-and-coherent-source-category",
      "titleEN": "Native tensor actions and covariant six-outcome observation",
      "statementEN": "Fix the original unitary G31 representation on V4, its ordinary complex tensor powers and the actual six-point action on E8/J ovoids. Consider all positive normalized operator families with this covariance. For the additional response identity x^tensor3 is a specified family of coherent probe inputs, not a physically derived or cloned preparation. Covariance does not select measurement contrast, an instrument, a physical time or a state."
    },
    {
      "id": "premise:original-quarter-car-fold-current-subalgebra",
      "titleEN": "Original quarter CAR source with fixed fold and current dictionary",
      "statementEN": "The already supplied single-channel CAR source with actual quarter domains alpha=1/4,3/4, negative-D sea, positive original excitation time, unitary eightfold F and actual channel diagonalization is fixed. M,O,U+ and the conformal NS point-splitting reference are the same original dictionary data; the single delta counterterm follows from the kernel difference. Even CAR current composites retain full arc correlations, charges, domains, all modes, original time and the existing subtraction prescription including finite derivative counterterms. No E8 root/spinor extension is assumed for this neutral current subalgebra. Identity with the original P1 net, Theta, collar and reference sector is not supplied. Real profiles and pulse strengths are intervention data."
    },
    {
      "id": "premise:original-microscopic-qwz-integer-car-edge-limit",
      "titleEN": "Existing microscopic one-channel CAR theorem",
      "statementEN": "Use the original width8/mass1/r1 QWZ strip, its negative spectral projector and filled state, its positive relative excitation Hamiltonian, actual local top-edge CAR fields and the proved field/adjoint/charge/time limit. Full orientation transport identifies this source with negative sea alpha1/4. Its identification with P1 is not supplied."
    },
    {
      "id": "premise:full-graded-car-with-original-positive-time",
      "titleEN": "Full CAR image and same positive vacuum time as a conditional hypothesis",
      "statementEN": "Conditional reduction: the FULL graded local CAR algebra including odd fields and adjoints belongs to the same OS vacuum representation; original time restricts to a(f)→a(exp(itK)f) for the same selfadjoint gapped one-particle K; H≥0 and HΩ=0. The P1 embedding/time identification is not established. Even CAR alone is insufficient."
    },
    {
      "id": "premise:original-seam-pair-and-full-moment-os-statements",
      "titleEN": "Original seam operators and explicit moment definition in the OS passage",
      "statementEN": "Source basis: the original boundary core supplies selfadjoint tangential B_Sigma; the seam has winding and stated spectral flow one. EM later names an allegedly selected rank-one line. QFT defines S_n^T as log-Z derivatives and uses them as the OS family. These statements and operator identifications are audited at the originals, not assumed proved."
    },
    {
      "id": "premise:strong-seam-covariance-and-same-pair-index",
      "titleEN": "Conditional strong seam covariance, gap and index of the same operator pair",
      "statementEN": "Conditional operator contract only: on the same original complex H, B is selfadjoint and u unitary; exp(itB)u exp(-itB)=exp(it)u for every real t, zero is outside sigma(B), and ind(P_B u P_B)=-1 with P_B=1_(0,infinity)(B). Actual exp(2 pi i B)=i I is an additional condition only for epsilon=1/4. These conditions are not derived from the reviewed original TFPT passages; P_C is not identified with P_B or a CAR covariance."
    },
    {
      "id": "premise:historical-original-circle-and-transfer-source-passages",
      "titleEN": "Historical concrete circle, heat-transfer and source-functional passages",
      "statementEN": "Historical audit contract: fullv21 contains D_Gamma=i partial_theta, u=exp(i theta) and twisted H1 domains; v31/32/42 write P_adm exp(-ell|B_Sigma|)P_adm; v42 has exp(W_k[J]) as a formal functional integral with STr. V1.06 contains the 11D C3 mode ansatz and n=1,g_top=8c3². The actual scope is audited, without assuming equality of old/new operators, physical times, traces or parameters."
    },
    {
      "id": "premise:given-quarter-circle-full-heat-transfer-comparison",
      "titleEN": "Original quarter circle and complete heat transfer as a scoped comparison class",
      "statementEN": "Mathematical comparison using existing original objects: B_a=-i partial_theta on twisted H1 f(2pi)=exp(2pi ia)f(0), a=1/4 or3/4; T_t=exp(-t|B_a|), R_t=exp(t/4)T_t. Ordinary Fredholm determinant, independent argument Q and given b=5/4. Twist is supplied rather than derived from winding. Identification with Ran(P_adm), the EM trace and q_EM=delta_top exp(-2alpha_EM) remains open; |B| heat is not automatically CAR or OS time."
    },
    {
      "id": "premise:historical-sf-sector-action-and-relative-normalization",
      "titleEN": "Historical SF sector action and explicit relative normalization",
      "statementEN": "fullv27/v28 set J1=SF,J=d SF,d2,SF1 in S_def(alpha)=S_def0+alpha J. Review concerns the intended defect-only relative reweighting and its extension to complete measure-compatible sectors Z_s=e^(-alpha d s)Z_s^0(alpha),Z0=Z0^0(alpha). Constancy of R=Z_s^0/Z0^0 is an explicit condition, not assumed from SF."
    },
    {
      "id": "premise:original-admissibility-and-graded-source-interfaces",
      "titleEN": "Original admissibility and graded-source interfaces",
      "statementEN": "v42 defines P_adm through primitive seam/Cauchy,gap,color and determinant conditions; v31 closure assumes an RP CAR kernel without identifying it with the old circle. v42 exp(W_k[J]) is a formal parent integral; spectral action, positive geometric marginal and full graded source are not equated. The separate eta term of the master functional is retained. Seek an original local image, not global Hilbert equivalence."
    },
    {
      "id": "premise:literal-positive-rg-map-on-finite-halfline-measures",
      "titleEN": "Printed positive RG kernel on finite half-line measures",
      "statementEN": "Literal archived theorem contract: nonzero finite positive nu on[0,infty),0<∫psi dnu<infty and Tnu(dt)=(∫K(s,t)dnu/∫psi dnu)dt with0<m<=K(s,t)<=M on the entire infinite Lebesgue target half-line. No extra compact support,finite target reference or integrable envelope is introduced."
    },
    {
      "id": "premise:original-a4-same-moebius-order-four-pin-lift",
      "titleEN": "Original A4 lift of the same Moebius identification",
      "statementEN": "The order-four spin lift explicitly named by fullv21 A4 belongs to the same geometric Moebius Z2 identification. As a Pin lift it has r²=-I; the local spin-reference channel is examined. Extra flat U1 connection holonomy and physical OS time are not inferred."
    },
    {
      "id": "premise:fixed-marked-glue-and-retained-source-d8-frame",
      "titleEN": "Fixed marked glue and actual source D8 frame",
      "statementEN": "Retain marked K=D5+A3 in E8 with E8/K=Z4, original O frame and existing M of the one-channel CAR fold. The actual even source frame is D8_F=M(D8_CAR); no fitted new Weyl rotation. Cartans,time vector d=Mdelta,cocycles and field words transport together."
    },
    {
      "id": "premise:standard-four-d8-module-parent-with-retained-subtheories",
      "titleEN": "Standard four-module parent with supplied CAR and E8 subtheories",
      "statementEN": "For the mathematical bridge use the standard four-module AIA over D8*/D8={0,v,s,c} with its supplied CAR 0+v and E8 0+s cocycle restrictions. NS space,actual R8 source winding and quarter time are the existing data. Ramond completion and global spin-surface/gauging functor are not assumed physically derived from P1. Transport of a complete common intertwiner basis is allowed; an independently marked mixed junction would be stronger input."
    },
    {
      "id": "premise:same-original-circle-trace-class-covariance",
      "titleEN": "Same original circle with trace-class changed polarization",
      "statementEN": "For the mathematical continuation retain H,B_a psi_n=(n+a)psi_n,0<a<1,u psi_n=psi_(n+1),P=1_(B_a<0),uPu*=P+Pi. C is a positive contraction with trace-class C-P; actual functional-calculus C_beta=(I+exp(beta B_a))^-1,beta>0 belongs to this class. Physical KMS/OS/EM source identification is not presumed."
    },
    {
      "id": "premise:existing-finite-orthogonal-carrier-on-same-source",
      "titleEN": "Supplied finite orthogonal carrier on the same source space",
      "statementEN": "In the carrier-transport theorem F is an actually supplied finite-rank orthogonal projection on the same circle H,Q=I-F. Physical application additionally needs the original map into active seam space and carrier isometry. A finite internal index retaining infinitely many spacetime modes is not this finite rank. No freely chosen F establishes origin."
    },
    {
      "id": "premise:trace-class-polarization-pair-with-restricted-unitary",
      "titleEN": "Trace-class polarization pair with actual unitary action",
      "statementEN": "P,Q are orthogonal projections on the same H,Q-P in L1. The covariance theorem additionally assumes Q=UPU*,U unitary,0<=C<=I,C-P in L1. The theorem is on this common space; physical source membership is not inferred from index alone."
    },
    {
      "id": "premise:smooth-matrix-clutch-on-original-negative-fourier-sea",
      "titleEN": "Smooth matrix clutch on original negative Fourier sea",
      "statementEN": "H=L2(S1,C^r),P=1_(n<0) tensor I_r and U=M_g for smooth g:S1→U(r). The existing bosonic branch has c1(det E2)=1,c1(det E3)=0 on the normal S2. A common physical identification of that clutch equator with the tangential source circle,C_Sigma and response is an additional map to prove."
    },
    {
      "id": "premise:actual-finite-ns-kernel-and-sampled-original-winding",
      "titleEN": "Actual finite NS kernel and sampled original winding",
      "statementEN": "Comparison uses the unchanged free NS kernel in v519/v524/v526 and sampling of original u=M_z on N=2M circle sites. Fourier modes n=-M,...,M-1 are cyclic. No UV contribution is removed and no ordinary finite determinant is declared an already renormalized continuum determinant."
    },
    {
      "id": "premise:given-normal-spin-line-and-genuine-circle-product",
      "titleEN": "Given normal spin line and genuine circle product operator",
      "statementEN": "Assume oriented spin S2,L2=det E2=O(1),normal twisted spin operator D_N with grading gamma_N and actual selfadjoint product B_prod=B_a tensor gamma_N+I tensor D_N on H1 product spinors,B_a=-i partial_theta+a,0<a<1. Normal metric,connection and horizontal product rule are given;no derivation from the original bosonic Callias operator or P1 is asserted."
    },
    {
      "id": "premise:given-exact-reducing-indexline-with-horizontal-connection",
      "titleEN": "Given exactly reducing index line with horizontal connection",
      "statementEN": "The normal kernel line exactly reduces the actual full operator and carries B_A=-i partial_theta+A(theta),A real smooth periodic,Dom=H1_per,at fixed circle time unit. Comparison preserves u=M_z using smooth periodic multipliers. Exact reduction is assumed,not inferred from an adiabatic approximation."
    },
    {
      "id": "premise:positive-determinant-response-of-transverse-product-channels",
      "titleEN": "Positive determinant response of given transverse product channels",
      "statementEN": "Compare positive scalar circle operators A_(a,lambda)=(-i partial_theta+a)^2+lambda^2,lambda>0,with common relative zeta normalization. a denotes spectral holonomy,not electromagnetic alpha. Actual two-dimensional block multiplicity is retained;separate infinite zeta factors require their own gluing/anomaly proof."
    },
    {
      "id": "premise:closed-normal-fredholm-with-retained-complex-kernel-representation",
      "titleEN": "Closed normal Fredholm operator with retained complex kernel representation",
      "statementEN": "Given densely defined closed Fredholm A:Dom(A) subset H_+ -> H_-,its true adjoint A*,finite complex kernel M of dimension r,and ker A*=0. Genuine product is B_a tensor gamma+I tensor [[0,A*],[A,0]] on its full Fourier intersection domain. Reduced-circle comparison uses P=1_(B_a<0) tensor I_M,U=M_z tensor I_M,C-P in L1,0<=C<=I,|z|<1. If M is an SU2 doublet retained at fixed operator,r=2;one doublet multiplicity is not substituted as r=1. Provenance,statistics and full physical source role are not assumed."
    },
    {
      "id": "premise:smooth-matrix-circle-source-and-moving-projector-with-fixed-scalar-symbol",
      "titleEN": "Smooth matrix circle source and moving projector at fixed scalar principal symbol",
      "statementEN": "Given comparison class:H=L2(S1,C^r),B_A=-i partial_theta+A(theta),Dom=H1_per,A smooth periodic Hermitian,P(theta) smooth periodic orthogonal of fixed rank. P is a fiber family,not an executed measurement. Neither P nor A is selected from P1. A real normal operator family additionally needs its own common domains and field map."
    },
    {
      "id": "premise:oriented-d3-tetrahedron-in-bloch-chart-versus-original-quarter-marks",
      "titleEN": "Oriented internal D3 tetrahedron in Bloch chart versus original quarter marks",
      "statementEN": "Compare T={(+++),(+--),(-+-),(--+)}/sqrt3 from the existing chiral (16,4) block,p(n)=(I+n.sigma)/2,z(n)=(nx+i ny)/(1+nz),and original ordered quartet(1,i,-1,-i). Equivalence is a biholomorphic Mobius map,even allowing arbitrary relabeling. Internal D3 direction sphere,physical normal S2 and tangential source curve are not identified. Bloch projectors are not the orthogonal E6 root-block projectors."
    },
    {
      "id": "premise:existing-marked-cyclic-cover-tower-with-compact-elliptic-images",
      "titleEN": "Existing marked cyclic cover tower and compact elliptic images",
      "statementEN": "Use only existing C_n:u_n^n=x^4-1 with fixed x base,deck Delta_n and already forced compiler clock R12=Q12 Delta12^4=(ix,omega s),Q12=(ix,s). Compare compact algebraic curves over C and pullback function fields. A comparison of punctured local systems must retain its removed points,peripheral loops,spin and field transports."
    },
    {
      "id": "premise:natural-joint-spin-family-lift-with-given-split-connection",
      "titleEN": "Existing natural joint lift with given split connection",
      "statementEN": "Assume a GIVEN unitary E=E3+E2 and GIVEN block U3xU2 connection A_E. Use only the natural iota_m with sum m=-2 completely classified on22September and their derived connection A_Wm=rho_LambdaEven(A_E)+M_m tr A_E. This does not select A_E from P1;an independent nonabelian family connection is not silently assumed."
    },
    {
      "id": "premise:fixed-kappa-toral-third-centralizing-split-carrier",
      "titleEN": "Fixed glue quarter and toral thirds centralizing the carrier split",
      "statementEN": "Fix the standard E8 Cartan,five plus three chart,three color and two weak indices and four family axes. Kappa=exp(2pi i v0),v0=(1/2^5,0^3). Study every sigma=exp(2pi i b/3),b in E8/3E8,centralizing the given U3xU2,and c=kappa sigma. This is a specified toral class of vertical bundle automorphisms,not an assumption about physical time or original base transport."
    },
    {
      "id": "premise:same-angle-evaluation-of-primitive-determinant-clutch-as-native-clock",
      "titleEN": "Additional direct-angle identification of determinant clutch with native clock",
      "statementEN": "Only for the diagnostic comparison evaluate a supplied diagonal integer loop k,sum k=1 and m,sum m=-2 as h=(k,-v_m) at the same angles:Q=exp(pi i ad h/2),C=exp(-2pi i ad h/12),C^9=Q. Pure weak k=(0,0,0,1,0);larger weak toral class k=(0,0,0,a,1-a). A transition is not connection holonomy or derived physical time."
    },
    {
      "id": "premise:native-cartan-four-in-full-clock-conjugate-source",
      "titleEN": "Native Cartan four in whole conjugated source",
      "statementEN": "Given original Cartan h_C,H+=ker(J-i),and whole native clock chat with chat^9=Jhat,chat^4=sigmahat. Existing inner conjugacy transports the entire tuple and Cartan to kappa. H+ is not identified with SU4 family factor. In current W_m orientation,+i64 is its dual branch. No preferred conjugator is selected."
    },
    {
      "id": "premise:primitive-normal-sphere-joint-k-bundle-class-two",
      "titleEN": "Old joint class2 bundle on the same normal sphere",
      "statementEN": "Given natural lift iota_m,sum m=-2,acts over the old common normal S2 with(c1(E3),c1(E2))=(0,1). For every m,its K=(Spin10 x SU4)/Z4 principal bundle has class2 in pi1(K)=Z4. Proposed choice is an everywhere smooth whole Cartan subspace in its fixed kappa sector. No identity with separate punctured flavor base is assumed."
    },
    {
      "id": "premise:given-same-normal-base-degree-pullback",
      "titleEN": "Actual degree pullback to precisely the same normal base",
      "statementEN": "Conditional cover connection assumes an actual degreeq map f:Sigma to S2 from a closed oriented surface,pulling back exactly the joint K bundle. Connection statement assumes a smooth branched cover. Existing C2,C6,C12 qualify only after proving identity of base and bundle map."
    },
    {
      "id": "premise:declared-original-octic-quantum-completion-ansatz",
      "titleEN": "Fixed octic completion in the existing current space",
      "statementEN": "In the supplied positive c8 current space,seek W8 only in the declared17 dimensional space:H8=sum_roots(alpha.h_-1)^8-30Q11^4 and the16 weight8 Q_ab monomials. H8 leading coefficient is1;root zero mode and L1=L2 must vanish. This specifies a mathematical existence ansatz,not an extra physical source or preparation rule. Uniqueness beyond this space is not assumed."
    },
    {
      "id": "premise:existing-elliptic-curve-with-specified-normal-spin-factor",
      "titleEN": "Existing elliptic curve with a specified normal spin factor",
      "statementEN": "On the actually existing E:y²=x⁴-1 with p=x,delta:(x,y)->(x,-y),q:(x,y)->(ix,y),compare normal spin Dirac operators twisted by p*O1. In the q-fixed nontrivial spin alpha=O(T-O),O=(0,i),T=(0,-i),transport the two-dimensional holomorphic kernel with genuine spin lifts. No identity of this curve base with the original normal base or of delta with the Calderon reflector is assumed."
    },
    {
      "id": "premise:existing-cut-cover-car-with-actual-winding-and-sea",
      "titleEN": "Existing full cut-cover CAR source with actual winding and sea",
      "statementEN": "The already constructed NS double-cover source Hcov=W0K⊕W1K is given, with common negative sea m<0 and original winding U=[[0,u_base],[I,0]], u_base e_m=e_(m+1). Quarter stress fields retain their original CAR coefficients and Wick ordering. Base intervals carry two cover preimages. This assumes the mathematical source realization and does not derive its physical P1 field map or descent into a fixed Pin collar domain."
    },
    {
      "id": "premise:given-flat-mobius-dirac-boundary-family-and-full-sea",
      "titleEN": "Given flat Moebius Dirac boundary family and its full sea",
      "statementEN": "The comparison fixes D=−i(sigma1 partial_x+sigma2 partial_w),Psi(x+1,−w)=c sigma1 Psi(x,w),c=±i,sigma1 Psi|boundary=b Psi|boundary,b=±1,and the canonical full two-dimensional negative D spectral polarization. This specified primary-literature family is not identified with all possible original P1 collars or covariances."
    },
    {
      "id": "premise:given-flat-pin-dirac-free-green-quantization-and-ns-cover",
      "titleEN": "Given free Green quantization of the flat Pin Dirac operator on its NS cover",
      "statementEN": "Given is the free charged Gaussian field realization with the Green operator of the concrete flat Dirac boundary family D=−i(sigma1 partial_x+sigma2 partial_w),w in[-1,1],sigma1 psi at boundary=b psi,b=±1. Original Pinminus c=±i forces psi(x+2,w)=−psi(x,w) on the orientation cover. The tested Euclidean kernel belongs to L=i sigma1 D=partial_x+H,H=i sigma3 partial_w. This Green quantization and x as Euclidean time are named premises,not identified with the D spectral sea,the old quarter Hardy state or original P1 kernel."
    },
    {
      "id": "premise:declared-recursive-native-som-update",
      "titleEN": "Explicitly chosen recursive SOM rule on60 native signal cells",
      "statementEN": "Given solely for the authorized model study is the60x35 complex update of RECURSIVE-SOM-AUDIT.txt section1,with native K60,fixed neighbors,row normalization,Euclidean winners,batch means and chosen c,beta,eta. The code basis is readout only. This update and its initial alignment are not primitive TFPT laws."
    },
    {
      "id": "premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks",
      "titleEN": "Existing C2 group with untwisted raw flat spin pullbacks",
      "statementEN": "On the existing marked parity curve E:y²=x⁴−1,study h=(x,−y),Q=(ix,y),kappa=(bar x,bar y) and flat square-lattice spin characters. The explicit raw one-particle calculation uses the untwisted line alpha with chi(A)=chi(B)=−1. Complex-linear pullbacks U and anti-linear Real structures j are distinct;no fitted U1 phase. This is not the normal twist alpha tensor p*O1 with its two-dimensional kernel. An original P1 Spin/Pin or Calderon transport is not presumed."
    },
    {
      "id": "premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus",
      "titleEN": "Free chiral Cauchy quantization of the canonical elliptic annulus",
      "statementEN": "Treat the dividing half of the given square curve in its canonical flat conformal representative as a product collar. Compute the Cauchy graph of Dplus=partial_t+B_Sigma,B_Sigma=−i partial_s with NS boundary,then interpret it using the expressly free chiral CAR construction as a quasifree two-boundary state. The labelled second-face compression is the covariance;the other is complementary. This Gaussian interpretation does not derive the original relative spectral-action measure,P1 time or EM/Quillen response."
    },
    {
      "id": "premise:original-eight-arc-additive-density-stress-restriction",
      "titleEN": "Additive scalar density and stress algebra on eight original arcs",
      "statementEN": "The restricted insertion candidate is the additive original scalar density/stress subnet on eight proper arcs with disjoint closures. Full neutral interarc CAR products are not automatically included."
    },
    {
      "id": "premise:given-pin-source-conformal-normal-geometry-probes",
      "titleEN": "Controlled conformal normal geometry of the given Pin Dirac source",
      "statementEN": "For the authorized source study,the same given Dirac comparison D0 on its fixed(c,b)Pin/boundary domain is varied by smooth real even profiles h:g_e=exp(2e h)g0,D_e=exp(-e h/2)D0 exp(-e h/2)after unitary density transport. Profiles/amplitude are explicitly chosen diagnostic interventions,not a P1-selected history. Original pure D0 sea remains initial state;thermal Green quantization stays separate."
    },
    {
      "id": "premise:full-original-four-mark-inversion-and-equatorial-real-slots",
      "titleEN": "Original full four-mark inversion and distinct equatorial Real slots",
      "statementEN": "The supplied marked base has x->ix and the inversion x->1/x already present in v512. Compare their actual lift to E:y²=x⁴−1,retaining the original four-marked base circle Gamma=|x|1. v510/v519 distinguish pointwise equatorial x->1/bar x from the OS family x->mu bar x acting along Gamma. These operations must not be identified in a source comparison. The old free carrier is named;no general physical P1 equality is presumed."
    },
    {
      "id": "premise:original-p2-order-three-cusp-and-stable-anchor-target",
      "titleEN": "Original P2 order-three cusp class and stable bundle anchor",
      "statementEN": "Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed."
    },
    {
      "id": "premise:primitive-reflection-split-and-controlled-geometric-variation",
      "titleEN": "Original reflection split with declared geometry variation",
      "statementEN": "Original pair A=(D+RDR)/2,B=(D-RDR)/2 on a declared common domain with fixed R and density-transport variation deltaD=-{h,D}/2. Trace identities hold finitely or under an actually proved differentiable relative-trace contract. Moving chi,mu,R,eta and domains remain separate. Geometric admissibility is not inferred."
    },
    {
      "id": "premise:finite-positive-heat-channel-endpoint-selection-test",
      "titleEN": "Finite data test of positive heat-channel values",
      "statementEN": "Algebraic test on C2:R=diag(1,-1),A=2I,B=sigma_x,D=A+B,chi1,common scaleD(s)=exp(-s)D and fixed eta phase. Compare normalized positive finite heat measures with identical endpoint operator values for a necessary data sufficiency test. Neither is asserted to be P1 or the same fully geometry-dependent transfer channel."
    },
    {
      "id": "premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin",
      "titleEN": "Ramified holomorphic half-form trace of the given C2 spin line",
      "statementEN": "Retain actual E:y²=x⁴−1, its already supplied Q fixed nontrivial NS NS spin-line class alpha and normalized Szego kernel. Holomorphic trace includes sqrt(dz/dx), while ordinary flat coefficient p_*alpha remains separate. No extra Q linearization phase, positive equal-time CAR covariance or smooth circle domain is assumed."
    },
    {
      "id": "premise:theta-square-modular-transport-with-distinct-geometric-clock",
      "titleEN": "Theta-square modular action with distinct geometric Q",
      "statementEN": "Test standard S/T action on three theta squares as a representation of a marked elliptic family, retaining weight and central phases. Geometric Q:z->iz at fixed tau=i is the S homology change, not an already supplied T moduli path. Do not assume this family action has the same origin as spatial monodromy of fixed four-punctured P2."
    },
    {
      "id": "premise:fixed-three-slot-permutation-and-original-compiler-comparison",
      "titleEN": "Fixed three-slot permutation and original compiler comparison",
      "statementEN": "Test the explicitly proposed fixed three-slot P_nat candidate from v417 against the P_* pullback of the v117 pair in the existing September dictionary. Q or G remain fixed on the same named comparison carrier. This coordinate identification is the tested ansatz, not an established physical identification of all native E8 representations or clock carriers."
    },
    {
      "id": "premise:same-marked-elliptic-annulus-massless-conformal-metric-family",
      "titleEN": "Conformal metric family of the same marked massless annulus",
      "statementEN": "Study the existing smooth midpoint-annulus source on the same complex square curve with fixed spin, cut and face markings and massless geometric chiral Dirac operator. g_sigma=exp(2sigma)g for arbitrary smooth real sigma, with canonical spin lift and bulk/boundary L2 transport derived from volume densities. On the fixed complex curve this includes all Hermitian metrics. Connection, mass and complex structure do not vary. An unchanged original reflection/clock requires invariant profiles; otherwise transport the symmetry as well. No identification with the unresolved full marked equator graph, primitive P1 measure or physical time. The averaging lemma expressly retains the given free conditional CAR quantization, source-independent integrable total weights and nonzero normalization."
    },
    {
      "id": "premise:retained-zero-branch-defect-on-common-pin-geometric-history",
      "titleEN": "Retained zero-branch defect on a common Pin geometry history",
      "statementEN": "The original normal-zero branch k=pi/2,b1,c+i carries the retained positive quarter defect Pi. Compare the given pure D0 sea C0 and C0+Pi,both evolved by the same D_s=exp(-s h/2)D0 exp(-s h/2). The profile and tested step/pulse histories are named diagnostic inputs. Relative geometry response is Tr(U Pi U* partial_sD_s). It is not defined into the primitive P1 action response or a thin local seam map."
    },
    {
      "id": "premise:positive-joint-car-source-tested-against-full-geometric-projector",
      "titleEN": "Test a positive joint CAR source against the full geometric projection",
      "statementEN": "Consider an arbitrary normalized positive state on the specified joint charged two-face CAR algebra,without assuming quasifreeness or U1 invariance. The geometric target is the existing closed chiral annulus graph P_A with both marked faces,W=1/2 and original NS B_Sigma. Actual identity C_source=P_A is an explicit condition of the uniqueness theorem and an unresolved P1 source equation,not a fact inferred from geometry or RP. A finite error test applies only on a genuinely P_A reducing CAR block. Complete Dirac vector source,OS Padm and physical time are not identified with this chiral scalar target. No freely chosen purification supplies missing physical cross correlations."
    },
    {
      "id": "premise:real-marked-elliptic-complex-structure-family-with-transported-spin",
      "titleEN": "Explicitly changed marked elliptic modulus family with transported spin",
      "statementEN": "Consider E_rho:y²=x⁴+i rho x²−1 for real |rho|<2 with the continued original nontrivial spin partition,both original dividing real structures and free Moebius glide. Expressly continue free chiral Cauchy quantization on the two-face annulus normalized by its primitive A cycle. Q and the original inversion act between rho and minus rho fibers;their original fixed-fiber action is expressly relaxed. The mixing formula conditionally assumes an arbitrary positive normalized source-independent law on this new family. Neither admission in the fixed C2 compiler nor P1 measure support,variance,full marked graph domain,physical time or an interacting source is derived."
    },
    {
      "id": "premise:actual-elliptic-half-form-l2-trace-on-full-marked-equator",
      "titleEN": "Actual L2 half-form trace on full marked equator",
      "statementEN": "Retain supplied square elliptic curve, flat metric and NS spin line. On the full ramified equator sqrt(dz/dx) transports actual L2 norm to |dx|. Keep holomorphic kernel, orthogonal analytic range, positive CAR covariance and H1 vertex domain distinct. No new channel metric or free vertex condition."
    },
    {
      "id": "premise:genuine-d4-unitary-stable-parabolic-orbifold-descent",
      "titleEN": "Genuine stable unitary D4 orbifold descent",
      "statementEN": "The expressly specified stable determinant-trivial parabolic rank3 branch has genuine mark-preserving D4 linearization with Coxeter fiber c=diag(1,i,-i) and regular cubic meridian M=a². Apply Mehta Seshadri to this object. Do NOT identify the concrete A0* ODE with it merely from C4 rotation."
    },
    {
      "id": "premise:actual-original-a0star-fuchsian-connection-and-c4-clock",
      "titleEN": "Unchanged A0* Fuchsian equation and actual C4 clock",
      "statementEN": "Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed."
    },
    {
      "id": "premise:recovered-genus-two-cover-with-retained-signed-coefficients",
      "titleEN": "Recovered genus2 cover with retained coefficients",
      "statementEN": "Retain exactly the older three-sheet cover C:w³=(z²-1)/(z²+1), eta=z(1-w³), eta²=1-w⁶, six Weierstrass points p_a=omega^a,q_a=-omega^a, given z base and clock markings. Seek actual flat coefficient lines for existing v117 monodromy, not mere permutation equivalence or an assumed field intertwiner."
    },
    {
      "id": "premise:given-square-nsns-elliptic-unit-charge-flat-connection-family",
      "titleEN": "Given elliptic NS source with an explicit unit-charge flat unitary probe",
      "statementEN": "On existing E=C/(Z+iZ) with original nontrivial NS/NS spin line and fixed H1_NSNS domain,calculate K_z=(-i partial_x+2pi v)+i(-i partial_y+2pi u),z=v+iu,as an explicit unit-charge probe. L_z=K_z*K_z,referencez0,gauge-periodic descent and positive theta-line convention are specified. Complex structure stays fixed;Q acts covariantly on external connection sources. This defines neither primitive P1 measure,physical hypercharge embedding,positive KMS state for every temporal twist,v472 Berry comparison map,nor z(alpha_EM). Parameter curvature and flat spacetime connection remain distinct."
    },
    {
      "id": "premise:normal-even-sea-car-source-and-canonical-graded-hs-double",
      "titleEN": "Normal even sea CAR source with canonical graded HS doubling",
      "statementEN": "For the algebraic source test use the already given sea Fock/CAR carrier,with faithful even normalized traceclass density rho and its standard HS space,Omega=sqrt(rho),L_A X=AX,R_A X=XA,Gamma X=Pi X Pi. Gamma is the canonical GNS implementer of source parity fixing Omega. Partner fields are represented in the fixed mode-index chart by b_j=−Gamma R_aj;this assumes no physical second P1 boundary. The finite saturation theorem covers arbitrary faithful even finite densities;the separate all-mode route requires the actual normal HS carrier,a complete covariance eigenbasis and saturation in every mode. For the free elliptic connection,rho is the already derived original thermal density,not a newly chosen geometry or temperature. TomitaJ,OSTheta,classical doubling and physical time remain distinct until actually mapped."
    },
    {
      "id": "premise:marked-e8-regularized-charged-grade-response",
      "titleEN": "Marked regularized E8 charged state and sector-grade response",
      "statementEN": "Existing positive E8_1 lattice vacuum source with original D5/A3 marking and 48 retained charges beta: beta^t P5 beta=5/4, beta^t P3 beta=3/4, beta²=2. The regularized charged vertex state has free preparation parameter u=r² in (0,1). N5,N3 are strongly commuting nonnegative oscillator grades; q5,q3 in [0,1] parametrize spectral readout. Keep original C/J matrices. u=48c3^4, q=exp(-2alpha), physical preparation, instrument and common time do not follow."
    },
    {
      "id": "premise:fixed-native-pauli-translation-target-and-gaussian-phase-class",
      "titleEN": "Native Pauli translations and frozen Gaussian phase class",
      "statementEN": "Native theta-translation target is P_nat=N64/mu4 from v783 in frozen four-component frame,with actual sigma and J=iI. Code quotient L/(1+i)L is the distinct duality-twisted context module. G31 has center mu4;full Clifford matrix group has center mu8 and additional coset zeta8 G31. Retain these concrete actions and phases."
    },
    {
      "id": "premise:ordinary-rigidified-genus-two-theta-determinant-functor",
      "titleEN": "Ordinary rigidified theta determinant functor",
      "statementEN": "A=Jac(C), E_L=S0 tensor L of degree1 on genus2. D_L=det H0(E_L) tensor det H1(E_L)^(-1),or dual convention;Poincare family rigidified at marked q0. Then O(2Theta) and four holomorphic sections with standard symmetric linearization. No additional Pfaffian,metaplectic factor,or marked-point determinant."
    },
    {
      "id": "premise:relative-spin-scalar-hardy-cable-on-original-three-cover",
      "titleEN": "Relative spin and reference Hardy cable on original three-cover",
      "statementEN": "Compare holomorphic Hardy shifts M_z on S0 tensor L and the same degree-zero coefficient line L over the original pi:C→P1_z. Spin half-forms give boundary norm; Fredholm convention is det kernel tensor inverse det cokernel. Relative line is spin line tensor inverse reference line. This defines a mathematical sewing contract, not an established identification with physical D_rel/D_ref or state."
    },
    {
      "id": "premise:same-spin-hardy-shift-against-identity-with-grading-retained",
      "titleEN": "Seam shift versus identity on same spin space retaining grading",
      "statementEN": "On original three-cover use E=S0 tensor P_spin with proved equivariant Nm_Dq(P_spin)=O. Compare M_z with M_1 on same half-form Hardy space in det kernel tensor inverse det cokernel convention, retaining determinant parity. Native complex module leaves global alpha sign free. This is a mathematical realization, not an assumed physical Cauchy/Calderon identity."
    },
    {
      "id": "premise:standard-chiral-spin-trace-on-original-cubic-disk-pullback",
      "titleEN": "Massless spin trace on original cubic disk pullback",
      "statementEN": "Exactly C:eta²=1-w⁶, z=eta/(1-w³), S0 with S0²=K and H0(S0)=0. Conditionally consider the massless chiral spin operator on three normalized disk components and the compatible smooth ambient trace with half-form boundary norm integral|psi²|. Strong operator closure is not replaced by L2 range. No identification with the actual P1 operator is assumed."
    },
    {
      "id": "premise:conditional-three-sheet-negative-sea-car-source",
      "titleEN": "Three-sheet CAR source retaining negative-sea orientation",
      "statementEN": "On the given full half-form boundary space let PH be the threefold Hardy projection,C=I-PH the occupation covariance of a normalized positive charged CAR state,and U=M_z the ambient unitary. V=ran(PH-U PH U*) is the given three-dimensional q defect with existing c,Q,beta fiber markings. This explicitly conditional source retains old negative sea and positive winding. It asserts no P1 origin,physical time or freely chosen identification with the elliptic source."
    },
    {
      "id": "premise:marked-theta-frame-with-explicit-primitive-gaussian-root-normalization",
      "titleEN": "Completed theta frame with explicit primitive Gaussian integral normalization",
      "statementEN": "Start with completed Wq=H0(Jac(C),O(2Theta)) tensor Nq^-1, beta=i, fixed Schrodinger frame e_x for x in F2², a unit norm-line trivialization and actual alpha=zeta8 V. The same-spin representative also retains odd B_O^-1 and its alpha sign. For affine d planes A and stabilizer phases phi in mu4 explicitly choose r(A,phi)=(1+i)^(2-d) sum(phi(x)e_x)/sqrt2. Gaussian integer numerators,norm squared2 and the central i orbit are the arithmetic construction. This is not deduced from complex dimensions and does not assert P1 selection,physical length normalization or uniqueness among all Heisenberg-stable lattices."
    },
    {
      "id": "premise:two-theta-lattice-voas-with-given-geometric-faithful-c12-lift",
      "titleEN": "Two theta lattice VOAs with a given faithful geometric C12 lift",
      "statementEN": "Lattices Lambda0 and Lambda1=zeta8 Lambda0 come from the explicitly normalized theta construction. Use their unitary lattice VOAs with standard cocycle. On object0 the existing faithful C12 lift c0 with geometric action i sigma is given; beta0=c0^9 lifts i and sigma0=c0^4 lifts sigma. T is any standard-cocycle VOA isomorphism lifting actual lattice isometry alpha01=zeta8 V. Branch-one marks are transported. No T²=1,[T,beta0]=1,full G31 splitting,common local VOA or physical P1 identification is assumed."
    },
    {
      "id": "premise:connected-local-same-conformal-compiler-extension-unselected-support",
      "titleEN": "Local conformal compiler extension without preselected sector support",
      "statementEN": "U=V_D5 tensor V_A3 carries its fixed unitary conformal structure and marked discriminant chart C4² with q(a,b)=(5a²+3b²)/8. W is a connected local unitary VOA extension with the same conformal vector and finite-dimensional weight spaces. No M_(1,1), exact support or holomorphy is assumed. Existing classification yields U,D8,E8plus,E8minus. Modular S is the finite discriminant Fourier after the common Jacobi factor and correct background transport. This is a mathematical class, not asserted membership of original P1."
    },
    {
      "id": "premise:four-d8-sectors-with-fixed-affine-source-time",
      "titleEN": "Four D8 source modules with fixed affine time",
      "statementEN": "Mathematical parent of the four D8*/D8 modules 0,v,s,c with eight identical Heisenberg oscillators, source charge chart p and N≥0. H=8(N+|p|²/2+delta.p),delta_j=(j+a)/8−1/2,0<a<1,Q=sum p_j. Original quarter cases are a1/4,3/4. Conformal L0, individual Cartan charges and totalQ remain distinct. This mathematical completion does not assume a physical P1 realization of the Ramond space or a local field map."
    },
    {
      "id": "premise:marked-local-e8-completion-with-original-source-time-and-euclidean-clustering",
      "titleEN": "Joint marked local E8 completion with original time and Euclidean clustering",
      "statementEN": "Joint candidate C1 to C4 of LOCAL-TIME-COMPATIBILITY: fixed D8 current net, local even unimodular lattice extension on the same eight Cartan channels, original Omega and H=8(L0+delta.a0) as positive Euclidean clustering transfer, retained q twice-root marking, all E8 currents included as admissible local observables, clustering of all bounded local two-point functions. This is the precise candidate class being tested, not asserted joint consequence of P1. Sector-penalty addendum retains the same irreducible multiplicity-free finite-index Z2 net, D8 action, interval motion and vacuum representation."
    },
    {
      "id": "premise:specified-petersen-shell-and-full-family-finite-recursions",
      "titleEN": "Specified Petersen shells and full finite family recursion",
      "statementEN": "v808: ten two-element subsets of five marked slots, disjointness adjacency and radial P=[[0,1,0],[1/3,0,2/3],[0,1/3,2/3]] on H_pi with pi=(1,3,6)/10 and Y=diag(6,-4,1). v486/v487: M=[[1,0,0],[0,1/2,1/6],[0,1/6,1/2]], deck swap Delta of final two slots. v395: given affine E8 adjacency A,T_net=(A+2I)/4 and given 1:6 scheduler. These finite operators are compared without assuming identification with P1 fields,electromagnetic charge or physical time."
    },
    {
      "id": "premise:marked-oriented-quarter-spin-and-two-object-vertex-transport",
      "titleEN": "Marked oriented quarter spin and two-object vertex transport",
      "statementEN": "Positively oriented marked quarter turn z->iz of normal two-plane with Spin(2) lift a=+/-exp(i pi/4),a²=i; existing genus2 source (C,S0,Dq,Q) and selected beta; existing lattice vertex objects F0,F1,unitary T:F0->F1 and beta1=T beta0 T^-1. Continued oriented quarter step and geometric reversal remain distinct arrows. An open P1 amplitude functor is not assumed."
    },
    {
      "id": "premise:natural-single-face-inclusion-in-given-family-calderon-graph",
      "titleEN": "Natural single-face inclusion in the given family Calderon graph",
      "statementEN": "Given positive invertible M of v486/v487 family recursion;C=M²(I+M²)^-1,X=M(I+M²)^-1 and Pi=[[I-C,X],[X,C]] on doubled six-space. Tested inclusions are the two coordinate faces. No geometric P1 shell space or its embedding is thereby derived."
    },
    {
      "id": "premise:oriented-five-slot-hodge-candidate-with-explicit-basis-readout",
      "titleEN": "Oriented five-slot Hodge candidate with explicit basis readout",
      "statementEN": "Ordered orthonormal basis e0,...,e4,its volume form and v=sum e_i. On Lambda²C5 consider new mathematical candidate C=*(v wedge .). Conditional Born test explicitly sets two-words as measurement basis,U(t)=exp(-itC) as generator ansatz and input-label-dependent disjointness filter. These readout data and choice of C are not derived physical P1 rule."
    },
    {
      "id": "premise:given-five-slot-exterior-source-with-retained-complement-index",
      "titleEN": "Given five-slot exterior source with retained complement index",
      "statementEN": "V=C5 with given oriented unitary slot basis and volume form,H=Lambda²V. C_m[kl,ij]=epsilon(m,i,j,k,l) are five distinct linear maps to conjugate target;their coherent sum is a different operation. Inner products and CAR grading remain explicit. No physical record,coupling or time iteration assumed from exterior algebra alone."
    },
    {
      "id": "premise:given-full-e8-current-source-with-fixed-carrier-su5",
      "titleEN": "Given full E8 current source with fixed carrier SU5",
      "statementEN": "Actual weight-one E8 Chevalley source from tfpt_explorer.current_block_geometry._actual_current_source and v498. Given D5/A3 frame uses first5/last3 standard coordinates. Carrier SU5 inside D5 has roots e_i-e_j,i,j<5. Retain full E8 source,cocycle,compact form and group actions. Physical P1 realization of this already given current source is not assumed proved."
    },
    {
      "id": "premise:specified-inclusive-current-response-with-invariant-metric",
      "titleEN": "Specified inclusive current response with invariant metric",
      "statementEN": "For an explicitly specified irreducible insertion sector and input/output modules,let B_a be actual current brackets in Killing-orthonormal basis. If sum B_a*B_a=kI,E(rho)=sum B_a rho B_a*/k denotes mathematical inclusive response with unread insertion index. Reduced response uses tensor chart transported by group action. This defines tested reader,not asserted selection of physically executed P1 instrument or original time."
    },
    {
      "id": "premise:specified-old-j20-instrument-in-fixed-six-mark-duad-chart",
      "titleEN": "Specified old J20 instrument in fixed six-mark duad chart",
      "statementEN": "Actual twenty jump events U_l(I-D_l) per mark from compiler-correlated-event-clock-20260919/source_channel.py;J20 effect3I/5,normalized jump channel Psi=(5/3)J20. Retain common duad labels of ten basis states. Comparison chart does not provide linear SU5/native group identification of Bell10=Sym²C4 with Lambda²C5 or already selected P1 apparatus."
    },
    {
      "id": "premise:specified-conjugate-su5-covariant-petersen-channel",
      "titleEN": "Specified full SU5 covariant channel with Petersen support",
      "statementEN": "H=Lambda²C5 in given oriented unitary duad basis,target dual H. T is CPTP and T(R rho R*)=barR T(rho)barR* for R=Lambda²U,U in SU5. Basis transitions to same or intersecting duads vanish. This is explicitly a mathematical conditional premise,not assumed physical P1 event covariance."
    },
    {
      "id": "premise:fixed-three-plus-two-mark-and-typed-original-degree-clock",
      "titleEN": "Fixed three plus two mark and typed original degree clock",
      "statementEN": "V=C5,P rank3 color projector,G_P=S(U3xU2),X=(-2,-2,-2,3,3),original U5=exp(2pi iX/5). Literal slot cycle c=(01234) is distinct and moves P. Covariant transport over Gr(3,5) concerns different marks,not full SU5 covariance within one fixed marked instrument. Continuous marks are not six old outer event marks or135 D8 frames."
    },
    {
      "id": "premise:specified-cptp-source-process-with-p5-saturation",
      "titleEN": "Specified CPTP source process with positive P5 saturation",
      "statementEN": "In fixed dual H=Lambda²C5 dictionary,T is a CPTP channel with Choi J_T,trace10. Require Tr(P5 J_T/10)=1 for rank5 projector from epsilon amplitudes. Sufficient conditional contract,not already proved P1 premise or arbitrary P_A state expectation. No assumed fullSU5 covariance,Petersen population or observed Kraus register."
    },
    {
      "id": "premise:original-native-pauli-and-v419-f20-source-words",
      "titleEN": "Original Pauli and v419 F20 source words",
      "statementEN": "Use frozen v783 two-qubit Clifford generators, existing Pauli16 and central mu8, and actual cyclotomic v419 C5,G matrices. A constructed source-word F20 subgroup is available; its physical seam selection is not assumed."
    },
    {
      "id": "premise:two-actual-source-events-with-coherent-record-access",
      "titleEN": "Two actual events with coherent record access",
      "statementEN": "Compose the actual unmarked sixty-event source twice, preserving phases. Additionally grant coherent access to all 3600 ordered histories and an active heralded record contraction after fixing a Hodge dictionary. This resource and selection are not derived P1 dynamics."
    },
    {
      "id": "premise:stipulated-e8-h8-with-original-color-center-selector",
      "titleEN": "Stipulated E8 H8 module with original color-center selector",
      "statementEN": "Retain stipulated E8 lattice extension with H8=8K_alpha,alpha=1/4 or3/4,actual rational Weyl dictionary M and original marked color center on first three carrier slots. Apply its P_sing after the carrier theorem. No intertwiner to the primitive P1/OS space is assumed."
    },
    {
      "id": "premise:actual-single-source-event-coherent-stinespring-record",
      "titleEN": "Coherent Stinespring record of actual single source event",
      "statementEN": "Use actual60 uniformlyweightedsourceoperations withtheirphases anddilationVsrc=sum_l|l>⊗U_l/sqrt60. GrantcoherentaccesstothisC60record andactivemeasurementofspecifiedeffectasresource. Reducedchannelalonedoesnotprovephysicalrecordcarrier ormeasurement. MarkedOmega andSpin5transportremainexplicitcarrierdata."
    },
    {
      "id": "premise:native-pauli-clock-family-with-marked-f20-inclusion",
      "titleEN": "Native Pauli clock family with marked F20 inclusion",
      "statementEN": "Use actual native Pauli four, frozen sigma, transported scalar-square theta alpha, and explicit native C,G word pair intertwined with v419. Retain all six alternating Pauli forms. Do not assume P1 selects this conjugate F20, one form, or any current contraction."
    },
    {
      "id": "premise:given-e8-fold-matrix-with-retained-source-charge-and-p2-grading",
      "titleEN": "Given E8 fold with retained source charge and P2 grading",
      "statementEN": "Use the same explicit Weyl matrix M from v16-operator-pullback-20261014/fold-results.json, mu_alpha=(j+alpha q)/8 with q=(1)^8 and alpha=1/4 or3/4, Qsource=(Mq).H, and original P2 Cartan charge X=(-2,-2,-2,3,3,0,0,0). Given D5+A3 target frame and compact E8 current representation. Do not assume raw local P1 spinor-field map, executed spectral filter, new physical time or changed source phase."
    },
    {
      "id": "premise:given-eightfold-even-car-quotient-with-marked-charge-and-source-time",
      "titleEN": "Given eightfold even CAR quotient with marked charge and source time",
      "statementEN": "Use existing fold eta_r(x)=psi((x+2pi r)/8)/sqrt8, local even rank8 CAR algebra on punctured quotient circle, fixed D8 Cartan chart and q=(1)^8, original current normal ordering/Weyl rule, and declared K_alpha source time. DHR sector labels/local von Neumann statement use standard locally normal even-CAR reference/Haag-dual vacuum-net representation. Assume neither normal local P1 single-interval identification,primitiveOS time,spin-structure sum nor bounded sharp spinor field."
    },
    {
      "id": "premise:ordinary-weyl-product-and-explicit-reflection-type-comparison",
      "titleEN": "Ordinary Weyl products and explicitly specified reflection type",
      "statementEN": "Compare the given chiral current Weyl relations under a real linear profile map L, retaining scalar linearity/conjugation and product preservation/reversal as separate explicit types. Transfer of the OS Grassmann word type to ordinary Weyl products is the tested identification hypothesis, not a proved P1 property. An oppositely oriented face pair remains a conditional target type."
    },
    {
      "id": "premise:existing-normal-thermal-source-with-bounded-neutral-insertions",
      "titleEN": "Existing normal thermal source with bounded neutral insertions",
      "statementEN": "Use the already given thermal sea representation of the free elliptic NS source with H>=0, finite Z_beta=Tr exp(-beta H), W=beta/2, rho=exp(-beta H)/Z and its HS standard space. Neutral even insertions including existing current Weyls are bounded in the source von Neumann algebra. Their imaginary-time conjugates are not assumed everywhere bounded; ordered heat products define the functional. The general HS Gram formula needs only an existing normal density, not Gaussianity. This comparison premise asserts neither physical P1 source, second geometric field family nor physical time identification."
    },
    {
      "id": "premise:finite-reflected-spectral-weight-preservation-test",
      "titleEN": "Finite test of the generic spectral RP preservation inference",
      "statementEN": "Test only the generic inference used in old measure proofs that a pointwise positive reflection-symmetric spectral weight exp(-Tr f(D²)) preserves existing RP. Test class: classical reflected finite halves,K0=[[10,9],[9,10]],D(s,t)=[[2s,1],[1,2t]],s,t=plus/minus1,f(u)=exp(-u). No full geometric/BRST/branch or P1/P2 realization of these data assumed. General-profile formula uses finite positive Laplace measure;the9/10 threshold is not asserted for arbitrarily small profiles."
    },
    {
      "id": "premise:finite-faithful-car-gns-source-completion",
      "titleEN": "Declared faithful finite CAR GNS source completion class",
      "statementEN": "At most two CAR modes are supplied with complete normalized CAR state moments, positive definite Gram and internal quarter invariance. The additional selfadjoint transfer preserves the CAR cyclic space, satisfies explicit finite bounds and has the supplied reference eigenvalue. One-transfer data supply linear constraints; all other supplied words are exactly replayed. The actual control oracle is the previously given quarter source; neither its data nor the finite representation ansatz is derived from P1/P2 here."
    },
    {
      "id": "premise:same-gapped-quarter-source-as-borchers-translation",
      "titleEN": "The same given quarter energy as Borchers translation",
      "statementEN": "Use exactly the already given full chiral quarter source H_alpha, alpha1/4 or3/4, with spectrum containing0 and least positive spectral point1/4. The proposed standard-pair bridge identifies U(s)=exp(is H_alpha) as its strongly continuous translation group and requires unitary modular conjugation V_t U(s) V_t*=U(exp(-2pi t)s) for all real s,t. This is an additional proposed identification to be tested, not an original P1 consequence."
    },
    {
      "id": "premise:original-v252-real-finite-block-and-selfdual-field-test",
      "titleEN": "Actual finite KO6 block and tested selfdual ansatz",
      "statementEN": "Given the existing v252 constructor with selfadjoint D_F, antiunitary J_F, J_F²=+I, J_F D_F=D_F J_F and J_F Gamma_F=-Gamma_F J_F. Test the additional direct Pfaffian identification through B_F=<J_F.,D_F.> and a universal internal rechoice of J at unchanged D_F. Control parameters of the published constructor class are not selected physical TFPT values. A total spacetime domain or Grassmann source law is not presumed derived from P1."
    },
    {
      "id": "premise:full-projector-car-marginal-and-commuting-joint-algebra",
      "titleEN": "Full projector covariance in a positive joint source",
      "statementEN": "Conditional theorem: a positive normalized joint state has full projection covariance P on the actually fully realized CAR algebra. The further unital C* algebra commutes with this CAR algebra. These assumptions are a source contract to test, not already derived P1 data; one face or incomplete neutral measurements do not replace them."
    },
    {
      "id": "premise:actual-quarter-neutral-vacuum-current-core",
      "titleEN": "Actual neutral vacuum current space of the quarter source",
      "statementEN": "Retain the supplied one-copy quarter CAR source alpha1/4 or3/4, negative sea Omega, actual level1 currents J_n with adjoints, H_alpha=N+Q²/2+(alpha-1/2)Q, common finite-energy core, existing local current net and original transported stress/winding comparison. Work precisely on the neutral cyclic Q0 space; elliptic Gibbs/HS state and charged sectors are distinct data. No physical P1 source map is assumed."
    },
    {
      "id": "premise:actual-local-weyls-and-finite-fiber-coordinate-target",
      "titleEN": "Actual full local Weyl system and finite spectral matrix fibers",
      "statementEN": "The existing actual local current net contains U_s,V_t with U_s V_t=exp(2ist)V_t U_s for all real s,t. The additional target identification tested is a unital complexlinear star image in the ordinary almost-commutative coordinate class A_sp, a subalgebra of continuous sections of a finite matrix/endomorphism bundle of positive rankN. This class is an explicitly chosen analytic realization, not implied by a smooth center alone. The full operational Seedop A_loc is not identified with A_sp."
    },
    {
      "id": "premise:neutral-current-recognition-with-actual-time-state-and-local-image",
      "titleEN": "Sufficient original current contract with actual time, state and local image",
      "statementEN": "Conditional recognition contract on the relevant eight-channel neutral current cycle: common invariant word domain with adjoints and level1 CCR, actual self-adjoint H>=0, H Omega=0, [H,j_a,n]=-8n j_a,n and Q_a Omega=0. Local statements require actual Fourier/smearing and support dictionary; reflected two-face claims require the actual common reflection representation. Thermal variant assumes actual beta KMS with legitimate unbounded-word extension; full joint zero-mode law and delta remain source data. This contract is not assumed proved from P1 and is not necessary for every coupled P1 proposal."
    },
    {
      "id": "premise:constant-finite-dirac-product-mixed-current-test",
      "titleEN": "Constant Dirac product with mixed current generators",
      "statementEN": "Assume the existing normal-independent two-component collar A=-i partial_theta+alpha, alpha=1/4 or3/4, Gamma=diag(1,-1), Jn=[[0,-1],[1,0]], S=Jn Gamma. In the supplied complete complex CAR system consider smooth one-face quadratic generators fI,fGamma,fP_R under actual HS/implementability conditions. For the product test Phi is a constant finite self-adjoint coefficient with B_Phi=Gamma A-S tensorPhi on the same H1 domain. A is internally scalar. Reducing projections are constant and reduce every Fourier momentum. For v252 also retain actual gamma_F and {Phi,gamma_F}=0. No physical P1 selection, separate chiral determinant, new vacuum or identification of tangential time with original OS Hamiltonian is presumed."
    },
    {
      "id": "premise:paired-quarter-external-lowest-block-candidate",
      "titleEN": "Paired quarter source and tested lowest-block selection",
      "statementEN": "Given B_a=2(A_a directsum A_(1-a)) on periodic H1,0<a<1,equal internal copies,j=exp(-i theta) K with summand exchange,and full spectral two-face graph G(B),ell>0. The external candidate claims Hcar=ker(|B|-lambda1) follows from the original primitive gap filter. Test that identification,not every P1 source. Physical CAR field map,original J/Clifford transport,OS reflection and time are not presumed identified."
    },
    {
      "id": "premise:canonical-recomplexified-graded-boundary-clifford-typing",
      "titleEN": "Geometric Clifford boundary with explicit changed complex structure",
      "statementEN": "Assume the existing free product class B=Gamma A or actual smooth tangential odd finite coefficient B_Phi=Z A-S Phi with total grading Z tensor gamma_F. Apply the standard Clifford boundary functor on the underlying real space with NEW complex structure Ical=i_old Gamma_tot and real structure C0=i_old gamma_n; the dual face has -C0. C0 is old-linear and new-antilinear,not identified with original charge J. This differs from full complex CAR quantization of both independent normal copies. State conclusions use the full graph as the actual quantum covariance in this functor. Its physical selection from P1/P2 is not presumed proved; Phi is a fixed classical background. Keep the fixed normal-independent product collar; a(theta) is smooth Hermitian and even,Phi(theta) smooth Hermitian and odd. No quantization of these bosonic backgrounds."
    },
    {
      "id": "premise:fixed-frame-graded-normal-dependent-source-family",
      "titleEN": "Fixed graded normal-dependent classical source family",
      "statementEN": "Assume fixed normals N,fixed gamma_F and fixed product trivialization,smooth selfadjoint even A(r),odd Phi(r),Ical=i_old Z gamma_F and C0=i_old N. A compatible finite regulator preserves these identities and adjoints; alternatively supply actual closed elliptic Cauchy data with Lagrangian Green form and grading invariance in the continuum. Normal dependence is an explicit extension of the old constant-product premise. State conclusions assume actual Clifford graph quantization per background. Only the additional integrated-moment identity takes an already given common bosonic weight and integrability as inputs; they are not presumed derived from P1."
    },
    {
      "id": "premise:given-quarter-many-body-source-and-compact-dirac-comparison",
      "titleEN": "Given quarter many-body source and tested compact Dirac comparison",
      "statementEN": "Given the already constructed limiting source in toe_research_20260909.tex with H=sum |j-1/4| n_j^exc and Q=sum n_b-sum n_d. Test only the added direct unitary identification of this FULL many-body time generator,including neutral/even sector,with |D| or positive affine energy rescaling in the usual compact finite-dimensional elliptic Dirac type with finite coefficient rank and finite p-summability. This identification is neither asserted as an original P1 theorem nor assumed true. Primitive boundary definition does not explicitly prescribe D_coll=H or every summability hypothesis."
    },
    {
      "id": "premise:actual-full-v252-unimodular-p2-gauge-lift",
      "titleEN": "Actual full v252 gauge lift and original P2 comparison",
      "statementEN": "Given the full unchanged v252 constructor pi,J_F,gamma_F,D_F and its symbolic Yukawa/Majorana inputs. Compare real gauge lift U=pi(u)J_F pi(u)J_F^-1 on the connected unimodular branch lambda det(m)=1 with original oriented P2 carrier E3+E2,Y=(-1/3)^3+(1/2)^2. No full Spin10 source action is presumed. A cyclically fixed neutral family line is considered only with an expressly supplied sigma=(123) and eigenvalue1; its original physical selection is not assumed."
    },
    {
      "id": "premise:fixed-local-quarter-channel-in-constant-odd-source",
      "titleEN": "Fixed local quarter channel in a constant odd source coefficient",
      "statementEN": "Given the existing Clifford graph source h=A_alpha g-i g Phi with A_alpha=-i partial_theta+alpha,alpha1/4 or3/4,Hermitian involution g and constant selfadjoint odd Phi. Compare fixed local map V_e f=f tensor e,g e=e,unit e,at fixed beta>0 and identical momenta/references. Original elliptic comparison uses actual W1/2 normalization,beta4pi and unchanged fold/normal-ordering reference. If classically averaged,a positive probability law on such constant Phi with fixed e,p,beta is explicitly supplied. This describes neither complex Pfaffian weights,moving field dictionaries nor general quantum-boson families."
    },
    {
      "id": "premise:actual-diagonal-lepton8-common-insertion-contract",
      "titleEN": "Actual reducing lepton8 contract with common insertions",
      "statementEN": "Given unchanged v252 with the existing diagonal integer control Yu11,13,17;Yd19,23,29;Ye31,37,41;Ynu1,2,3;MR1,4,9, its actual gauge lift and grading, p1/4 and four adjoint/J-real completed original quaternion one-forms q[D,b]. The generation0 lepton8 is proved reducing for these matrices. Source=target tests only automorphisms of this given tuple."
    },
    {
      "id": "premise:one-e8-bracket-weyl-pairing-with-single-anchor-dressing",
      "titleEN": "One E8 current trilinear as the same Weyl quadratic with one anchor mark only",
      "statementEN": "Test only literal B_h=κ(x,[h,y]),h in(10,6),x,y in(16,4), as the local quadratic coefficient of the SAME left four-dimensional Weyl multiplet with Lorentz epsilon. In each fixed spin/charge channel the family form omega is skew. Allow linear combinations F(X)omega G(X) for only ONE X=diag(3,-1,-1,-1) and arbitrary polynomials. Other original family generators, independent Dirac variables, derivative vertices, multiple omega factors and different field maps are outside this class."
    },
    {
      "id": "premise:spin10-su4-family-four-cycle-quarter-lift-class",
      "titleEN": "Complete Spin10/SU4 family-four-cycle quarter-lift class",
      "statementEN": "Given original E8 roots and Spin10×SU4/Z4 embedding. Candidates are g=[g5,eta C4],C4 a family four-cycle,eta^4=-1,g5 compact inSpin10 and g of exact E8 order4, equivalently g5^4=-1Spin. Allow all SU4 conjugates. The original SM subgroup inside Spin10 is marked; preserving it is tested as an additional subclass. No physical identification of mark rotation and native clock is assumed."
    },
    {
      "id": "premise:family-four-cycle-lift-commutes-with-original-sm",
      "titleEN": "Additional pointwise SM preservation by the Spin10 lift",
      "statementEN": "Within the given family-four-cycle quarter-lift class additionally require g5 to commute pointwise with original G_SM insideSpin10,Y=(-1/3,-1/3,-1/3,1/2,1/2),and original colorSU3/weakSU2 blocks. This exact symmetry premise defines the tested candidate subclass, not a newly derived P1 vacuum rule."
    },
    {
      "id": "premise:native-third-clock-current-word-pairing-class",
      "titleEN": "Native third clock and actual current-word pairing",
      "statementEN": "Given original D5+A3 E8 embedding, native toral third b=(-1,-1,-1,0,0,-1,0,0) and its coherent C12 field lift. Test K(x,w y) on g1 using complex-bilinear invariant K and actual ad_u/group letters. In the lepton pairing the carrier color charges vanish and actual family4 sigma phases have multiplicities2+2. A one-Higgs two-class subclass permits one generic fixed h in g2 and otherwise only functions of those same two sigma projectors. Do not presume physical Weyl-field type, temporal ordering, the arbitrary A3 torus or a marked three-plane."
    },
    {
      "id": "premise:constant-odd-quarter-circle-source-sea-and-full-graph",
      "titleEN": "Constant odd circle-source coefficient with existing sea and full boundary graph",
      "statementEN": "In the existing Clifford chart on L2(S1,K) with finite K let h0=(-i∂theta+alpha)gamma,alpha=1/4 or3/4,gamma*=gamma,gamma²=I. Supply constant selfadjoint odd Phi,{gamma,Phi}=0,and V=-i gamma Phi,h=h0+V on common periodic H1. Compare given CAR sea projectors P_-(h0),P_-(h) and complete pure two-face projector G_ell(h),ell>0,including crossings. This is the existing source class with classical background, not physical selection from P1 or quantization of dynamic bosons."
    },
    {
      "id": "premise:fixed-scalar-quarter-full-jloc-pfaffian-candidate",
      "titleEN": "Direct full Jloc Pfaffian at unchanged scalar quarter parameter",
      "statementEN": "Test new direct source closure beta_Dalpha=<Jloc.,Dalpha> in existing old-complex collar with N,Z,S=NZ,Jhat=I⊗J_F,Jloc=S Jhat,Dalpha=N(∂r+Z(-i∂theta+alpha I)−S Phi),alpha=1/4 or3/4. Phi retains existing selfadjoint,J_F and oddness identities; fields and scalar quarter parameter remain unchanged. Claim is same antisymmetric Pfaffian source directly equal to primitive spectral action. Do not presume a J-odd charge connection, additional paired holonomy sectors or a new relative log profile as original consequences."
    },
    {
      "id": "premise:fixed-quarter-cover-deck-normal-and-convergence-selection-tests",
      "titleEN": "Fixed quarter cover, deck as normal and convergence-only selection tests",
      "statementEN": "Use existing longitudinal NS tuple B_cov=2(A1/4⊕A3/4),V=Tpi,u_cov=[[0,exp(ix)],[I,0]],with actual H1 domains and old complex types. Test only claimed local old-complex-linear normal multiplication on this same two-sector coefficient space,and convergence alone selecting dt/t. Relative heat pair is A1/4² versus A1/2²; no alternative measure is presumed a full admissible original TFPT source."
    },
    {
      "id": "premise:fixed-original-aps-quarter-domain-and-constant-odd-insertion",
      "titleEN": "Fixed original APS quarter domain and constant odd insertion",
      "statementEN": "Given product collar with p=n+alpha,alpha=1/4 or3/4,finite graded K,h0=p gamma_F,h=h0-i gamma_F Phi for constant bounded selfadjoint odd Phi. Retain x0 in RanP_-(h0),xell in RanP_+(h0),ell>0. Exterior/CAR transfer and reference are used only under their given contract. No physical state selection is assumed from elliptic realization."
    },
    {
      "id": "premise:mass-only-full-deck-relative-and-ordinary-heat-trace",
      "titleEN": "Mass-only deck-odd part under ordinary heat trace",
      "statementEN": "Narrow candidate: full local deck tau=S R_r with transported A and reflection-even Phi on a common stable domain,so Dref=D0,Drel=ZPhi. Test literal primitive Tr f(Drel/chi)-Tr f(Dref/chi),allowed f(x)=exp(-x²),on a nonzero infinite-dimensional geometric Hilbert space,without added UV cutoff or alternate trace. This is not assumed for all P1 realizations."
    },
    {
      "id": "premise:closed-antiperiodic-scalar-wall-with-fixed-hermitian-coefficient",
      "titleEN": "Closed antiperiodic scalar wall with fixed Hermitian coefficient",
      "statementEN": "Test The reviewed additional closed Green-source construction on finite circle length2ell,scalar anti-I spinor clutch,H1 domain,fixed finite Hermitian Phi,smooth real periodic scalar sigma(r). N²=-I,Z²=S²=I,S=NZ,Q=N partial_r+Z sigmaPhi,D=Q+pS. Explicit inverse additionally assumes ∮sigma=0. Neither global clutch nor Green=original Schwinger source is assumed from local C_faces."
    },
    {
      "id": "premise:scalar-spectral-heat-covariance-with-geometric-time-reflection",
      "titleEN": "Scalar spectral heat covariance and actual time reflection",
      "statementEN": "Test only the general inference for scalar translation-invariant Gaussian covariances on R with reflection tau->-tau. Single test C_s=exp[-s(-partial_tau²+omega²)],s>0,omega>=0. Class theorem additionally assumes real,even,continuous ordinarily positive definite c,c(0)>0,C1 at0 and positive reflected Hankel form. No claim that original selects this covariance. Already assumed OS-positive channels,fermionic/matrix reflections,contact/distribution kernels and purely spatial smoothing are different contracts."
    },
    {
      "id": "premise:given-global-quaternionic-sm-mass-loop-with-fixed-yukawas",
      "titleEN": "Given global quaternionic SM mass loop with fixed Yukawas",
      "statementEN": "Use already fully derived v252/v254 particle LR one-form family M_H=(q(H) tensor I_nf)diag(Yup,Ydown),q(H)=[[phi1,phi2],[-barphi2,barphi1]],with fixed Yukawa matrices and common represented/trivialized SM gauge space. Determinant-winding test requires nowhere-zero H along closed loop and invertible Yukawas. Normal geometric bundle clutch is not identified with this global matrix; Majorana and genuine additional bundle transport are separate data."
    },
    {
      "id": "premise:bounded-relative-operator-positive-laplace-and-compact-source-pair",
      "titleEN": "Bounded relative operator,positive Laplace profile and compact source pair",
      "statementEN": "Dref is actually selfadjoint on its fixed compact elliptic source domain,V=V* bounded and P any infinite-rank orthoprojection. Negative trace theorem assumes nonzero positive Laplace F(x)=∫e^(-sx)dμ,F(V²) a defined bounded operator and F(Dref²) trace class. Positive pair-heat theorem uses D=Dref+V on same domain and trace-class reference heat for every positive t. Whether actual original Drel is bounded zero order must be checked against full actual reflection lift and principal symbol; not generally assumed."
    },
    {
      "id": "premise:given-normal-spin-clutch-weak-orbit-family-and-u2-linearization",
      "titleEN": "Given normal spin clutch,weak orbit family and U2 linearization",
      "statementEN": "V=C² with actual SM action e^(it/2)U is supplied. N=CP1 normal spin sphere,W=P(V) separate weak orbit,L=O_W(-1) tautological,Q=V/L. Given degree-one clutch becomes explicit family g(w,φ)=e^(iφ)Pw+I-Pw,E=(O_N1⊠L)⊕(O_N0⊠Q),with split vertical connection and actual H1 domains. E8 application additionally retains named half-root λκ and mixed-carrier line j6. W is not assumed an already original physical parameter space; orbit/dual-section functor is tested as mathematical comparison."
    },
    {
      "id": "premise:given-normal-orbit-and-four-copy-natural-spin-family",
      "titleEN": "Given normal orbit family and four-copy natural spin lift",
      "statementEN": "N=CP1_N and W=P(V) are distinct bases. Supplied SM color C=3_(Y−1/3) and V=2_(Y+1/2) carry E=C⊕(O_N1⊠L)⊕(O_N0⊠Q),Q=V/L=detV⊗L*. D=detE has normal degree1 and is SM neutral. For four integral m_j consider W_m=ΛevenE⊗⊕D^m_j; already proved natural joint lifts have Σm=−2. mA=(0,0,−1,−1) is only a supplied representative. Literal kernels use previously specified split vertical spin connection; virtual family classes persist under smooth variation. A1+1D anomaly interpretation needs actual quantization of these modes; N×W is not assumed physical spacetime."
    },
    {
      "id": "premise:fixed-v117-orthogonal-d3-cartan-identification",
      "titleEN": "Additional fixed orthogonal identification with the D3 Cartan",
      "statementEN": "For the Cartan test only, identify the v117 rotation triple orthogonally with the last D3 coordinate block of the fixed marked D5+D3 E8 roots. Tested lifts are blockwise signed coordinate permutations. This is not an operator identity implied by original SU3⊂SU4 embedding or a physical chirality identification."
    },
    {
      "id": "premise:given-original-su3-to-su4-family-vector-action",
      "titleEN": "Existing SU3 family-vector embedding into SU4 four",
      "statementEN": "Use original SU3⊂SU4 embedding with4=3+1 and existing E8 branches (45,1),(1,15),(16,4),(bar16,bar4),(10,6). A supplied family-vector rotation R acts by U=diag(R,1), with actual tensor/exterior powers on these branches. This does not identify original v117 connection with physically used marked family connection or derive original joint operator source."
    },
    {
      "id": "premise:given-split-carrier-and-actual-genus-two-q-sign-family",
      "titleEN": "Supplied split carrier with actual q sign lines on genus-two cover",
      "statementEN": "Given existing split carrier E3⊕E2 with unitary split connection onX and actual unitary flat q linesL0,L1,L2 on old coverC:w³=(z²−1)/(z²+1),η²=1−w⁶,product1 andπ*Fv117=L0⊕L1⊕L2. Pull these data toX×C and combine with declared alignment to mA slots. This does not selectX,E,mA or alignment as original P1 quantum source."
    },
    {
      "id": "premise:given-relative-unit-seam-on-specified-collar-faces",
      "titleEN": "Relative unit seam on explicitly assigned collar faces",
      "statementEN": "In existing product collar,test original relative unit gluing v1=Uv0 on an explicitly assigned pair of labeled faces. Scalar block has h e_n=(n+α)e_n and U e_n=e_(n+1),α=1/4 or3/4. Actual original Γ±-to-face assignment,orientation and three-cabled spin trace map are not assumed derived."
    },
    {
      "id": "premise:whole-constant-odd-coefficient-on-scalar-unit-clutch-domain",
      "titleEN": "Whole constant odd coefficient on same scalar unit-clutch domain",
      "statementEN": "Test supplied whole h=Aγ_F−iγ_FΦ with constant Hermitian odd Φ,internally scalar A=−i∂θ+α and U=e^{i_oldθ}I on D_U={ψ∈H1:ψ(ell)=Uψ(0)}. Hence [A,Φ]=0 and U*ΦU=Φ. Original Γ±-to-face assignment remains an explicit premise. Cited v252 coefficients are previously checked constructor inputs,not selected physical masses. Variable Φ,noncommuting connections or charged noncentral clutches change these premises."
    },
    {
      "id": "premise:declared-q-column-to-v252-generation-slot-comparison",
      "titleEN": "Declared comparison of actual q columns with supplied v252 generation slots",
      "statementEN": "For operator comparison identify actual ordered horizontal q columnsG with three supplied generation slots of v252 full_D basis. D_E=Gbase^-1U_FbarGbase acts on particle families,barD_E on antiparticles; Φ remains given full_D constructor with its Y_f and symmetricMR. Retain F5 Ical,C0,T_faces,NS domain and full graph covariance. This alignment is declared comparison input,not derived from kernel count,H0,equal dimensions orP1. Selects no mass,state,physical time or native action permission."
    },
    {
      "id": "premise:declared-q-column-to-v252-chiral-slot-comparison",
      "titleEN": "Alternative chiral comparison of q columns with actual v252 P2 carrier",
      "statementEN": "For this explicitly alternative comparison identify actual G columns(L0,L2,L1) with supplied generation slots through whole signed P2 dictionary. D_E acts on gamma_F positive,barD_E on negative. Retain unchanged full_D with Y and symmetricMR,F5 structure,faces,graph state and given time. Do not silently replace earlier particlewise hypothesis. No P1 selection,global trivialization,new state/time or physicalE8gauge action assumed."
    },
    {
      "id": "premise:original-hard-holonomy-and-closed-triangle-readout-contract",
      "titleEN": "Original hard-holonomy contract with pure closed-triangle trace",
      "statementEN": "Read original45 flavor sources defineY from canonical minimal paths of a Hermitian family connection andPi_triangle=Hol12Hol23Hol31 as ordered closed holonomy,delta=argtrace. DisplayedPi includes neitherKstr nor extraZ6operator. Nonreal asymptotic is a readout contract to test,not completed physical derivation. Active source retains absolute wall selector;no automatic physicalconnection=v117."
    },
    {
      "id": "premise:given-original-e8-cx-current-cocycle-and-compact-adjoint",
      "titleEN": "Given original C/X current source with E8 cocycle and compact adjoint",
      "statementEN": "Use the actual supplied level-one E8 current source C in1x4 and X inbar5x4, original cocycle and compact adjoints. The expressly restricted initial seed keeps only family labels1,2,3. Check current Lie words without presupposing geometric q/p access, a finite Phi coefficient or physical P1 source selection."
    },
    {
      "id": "premise:original-d5-base-current-action-on-given-p2-e8-source",
      "titleEN": "Original D5 base-current action on supplied P2 and E8 carrier",
      "statementEN": "Original D5_1 x A3_1 base net and explicit P2 Clifford carrier supply full45-dimensional D5 operator action including20 non-U5 roots. Its zero modes act with the existing Chevalley cocycle on the supplied compact E8 currents; no primitive P1 preparation or history selecting a word is presumed."
    },
    {
      "id": "premise:bare-c3-positive-kernel-with-real-diagonal-yukawa-readers",
      "titleEN": "Bare positive C3 kernel with real diagonal Yukawa readers",
      "statementEN": "For this restricted analytic class only, take the displayed positive circulant H3 with real parameters and strictly positive eigenvalues, and Y_f=L_f H3_f^-1/2 R_f with real diagonal readers. This is not assumed to be the original P1 consumer or full Pstr compression."
    },
    {
      "id": "premise:common-leading-distance-flavor-readers",
      "titleEN": "Common leading triple distances with equal two-leg weights",
      "statementEN": "Restricted candidate class only: arbitrary shared real triple x, sector positive scales/offsets and permutations, single leg or equal-weight same-list two legs. Not a P1 axiom. Different lists/weights, full sums and extra operator histories lie outside."
    },
    {
      "id": "premise:original-closed-real-delta-transport-action",
      "titleEN": "Original closed real delta transport action",
      "statementEN": "Original EM3 action on its invertible closed epsilon0 branch with real delta, fixed U6 and all three cusps. Use actual common Gbase dictionary U6=diag(M0,-M0) and fixed marked P0. This is not a newly chosen independently projected physical variation."
    },
    {
      "id": "premise:four-original-diagonal-paths-and-regularized-endpoints",
      "titleEN": "Four original diagonal paths with regularized endpoints",
      "statementEN": "Actual original v117 connection and already proved G along the four diagonal0-to-infinity classes, regularized target z^-D and fixed G metrics. Geodesic reading uses canonical complete hyperbolic metric of the given conformal class, or original real-reflection invariance; not an arbitrary physical metric. No original four-path summation prescription is assumed."
    },
    {
      "id": "premise:native-clock-adjoint-compensation-for-hardypair",
      "titleEN": "Tested native adjoint-clock compensation of Hardy operator",
      "statementEN": "Only The reviewed restricted candidate: actual96 Hardy endomorphism and original weak creator, native clock=-i gammaF acts by conjugation, and proposed combined fourfold adjoint average. Not a P1 axiom or a one-sided charged-Hom phase."
    },
    {
      "id": "premise:declared-positive-cubic-hardy-direction-for-gram-differential",
      "titleEN": "Declared Hardy direction for the original cubic Gram differential",
      "statementEN": "Comparison at the original unregulated positive cubic block with declared V=-kappa M0P0, actual y/delta and already calculated Hardy phase. This is not a claim that the original physical source selects this whole V_adm. Preserve actual Pstr commutation and other C6 branches at the original source contract."
    },
    {
      "id": "premise:whole-same-mu01-induced-ns-halfmode-reader",
      "titleEN": "Whole unchanged induced mu01 reader on original NS halfmode",
      "statementEN": "Unchanged already defined induced mu01 input, original three q cables and J-supported p_(r+1), Phi0 and NS halfmode, fixed G/R roots, actual normal Clifford action and both graph faces. Complete evaluation of the same reader, not a freely selected whole physical source or C6 branch."
    },
    {
      "id": "premise:whole-six-original-induced-inputs-at-fixed-kinetic-ns-reader",
      "titleEN": "All six original induced inputs at the fixed kinetic NS reader",
      "statementEN": "Evaluate the six existing offdiagonal mu_ab in the original induced NS/Hardy kinetic reader, Phi=0, finite positive ell_NS and half-frequency. Carry actual G_a input and J E^(b-1) output maps, same continuation, metric, spin, both graph faces and normal symbol. Do not identify individual twist lines or assume a physical combination or multiplicative compression of field products."
    },
    {
      "id": "premise:canonical-finite-particle-hole-corner-of-given-kinetic-graph",
      "titleEN": "Canonical finite particle-hole corner of the given kinetic graph",
      "statementEN": "Use canonical E and Eh corners of the given pure kinetic NS/Hardy graph at finite positive ell, actual J and Hom maps and normal projector, unnormalized finite CAR trace with sixteen carriers per family, and ordinary adjoint quadratures. No identification with whole unsmoothed Z or complete J_F/KO field."
    },
    {
      "id": "premise:bare-unsmoothed-crossface-quadratic-on-given-graph-vacuum",
      "titleEN": "Bare unsmoothed cross-face quadratic on the given graph vacuum",
      "statementEN": "Test only bare equal-time CAR quadratic T_mu=J* M_mu with actual geometric multiplier and canonical finite spectral cutoffs on the given graph vacuum. No smearing, sewn insertion, heat evolution or source-dependent contact prescription. Original quotient denominator is fixed Zref[0,0,0]."
    },
    {
      "id": "premise:given-original-two-stage-admissibility-complex-and-blocked-weight-caller",
      "titleEN": "Given original two-stage admissibility complex and blocked weight caller",
      "statementEN": "On the printed closed branch QFT2 gives commuting selfadjoint K_r, Delta_adm=sum K_r² and its Hodge projector. QFT4 additionally uses local C6 Green kernels and diagonal weights from transport counts, carrier labels and retained selector weights. Review this given source contract without asserting new physical provenance or choosing a new state."
    },
    {
      "id": "premise:actual-constant-weak-quarter-cap-with-coupled-normal-energy",
      "titleEN": "Actual constant weak quarter source with old boundary space and coupled normal energy",
      "statementEN": "Given full96 source h_p=p gamma_F+V with V=tQ gamma_F-i gamma_F D_F selfadjoint, finite, constant and independent of quarter p. Retain old rank48 P0,p=1_(p gamma_F<0), existing exterior/CAR caller and coupled ground energy E_p=Tr(h_p 1_(h_p<0)) for the cap comparison. No original physical selection of P0,V,ell,source measure or energy subtraction is assumed."
    },
    {
      "id": "premise:whole-product-dirac-with-standard-normal-clifford-double",
      "titleEN": "Whole product Dirac operator with standard normal Clifford clutch",
      "statementEN": "Given the whole normal-independent product interval, D=N(partial+B), B=Z tensor A-S tensor Phi with N²=-I, N*=-N, B*=B, {N,B}=0, [Gamma_tot,B]=0 and {Gamma_tot,N}=0. Standard invertible double joins D and -D by v(0)=Nu(0), v(ell)=-Nu(ell). Retain actual full96 coefficient and charge/real types. No prior derivation of this clutch or whole global cylinder from the P1 measure is assumed; a local collar alone is insufficient."
    },
    {
      "id": "premise:full-independent-even-odd-matrix-coefficient-class",
      "titleEN": "Full independent finite even and odd coefficient class",
      "statementEN": "Both grading sectors are nonzero; A and Phi independently range over every self-adjoint even and odd matrix. This stronger coefficient class is not asserted to be the original SM source."
    },
    {
      "id": "premise:whole-fermi-two-face-projection-and-diagonal-isometric-compression",
      "titleEN": "Whole Fermi two-face projection and common isometric compression",
      "statementEN": "Finite self-adjoint H, positive ell and one isometry V on both faces. G has the full Fermi and cross blocks. This tests closed lossless reduction and does not impose reduction on an open physical subsystem."
    },
    {
      "id": "premise:actual-v252-sm-coefficients-and-internal-ko6-without-extra-mixing",
      "titleEN": "Actual v252 SM coefficients and internal KO6 structure",
      "statementEN": "Original v252 full_D, rep, grading and Kswap, arbitrary Yukawas and symmetric Majorana, SM connection and quark-preserving momentum. No extra quark-lepton mixing or full normal real structure is assumed."
    },
    {
      "id": "premise:separable-normal-real-map-with-actual-jodd-gauge-and-typed-quarter-image",
      "titleEN": "Given normal-independent product collar, internal KO6 and separable normal antiunitary ansatz. J-odd actual gauge connection and scalar quarter image transport are separate premises and cases.",
      "statementEN": "Given normal-independent product collar, internal KO6 and separable normal antiunitary ansatz. J-odd actual gauge connection and scalar quarter image transport are separate premises and cases."
    },
    {
      "id": "premise:actual-p2-six-multiplet-gauge-real-coordinate-frame-class",
      "titleEN": "Fixed actual v252 left/opposite algebra, graded real P2 gauge representation, gamma and J. Six inequivalent positive-grade irreps with family multiplicity3. Constant coordinate frames are classified, without claiming they fix extra numerical Phi/clock/family operators.",
      "statementEN": "Fixed actual v252 left/opposite algebra, graded real P2 gauge representation, gamma and J. Six inequivalent positive-grade irreps with family multiplicity3. Constant coordinate frames are classified, without claiming they fix extra numerical Phi/clock/family operators."
    },
    {
      "id": "premise:original-higgs-radial-family-on-common-compact-product-domain",
      "titleEN": "Original radial Higgs one-form family at fixed Yukawas/Majorana on the given compact AP normal double times tangent circle with common H1 domain, fixed smooth connection and product algebra. Complex K-homology is compared with full source covariance.",
      "statementEN": "Original radial Higgs one-form family at fixed Yukawas/Majorana on the given compact AP normal double times tangent circle with common H1 domain, fixed smooth connection and product algebra. Complex K-homology is compared with full source covariance."
    },
    {
      "id": "premise:positive-laplace-profile-finite-three-moment-comparison",
      "titleEN": "Positive finite Laplace measure and three finite positive moments. Identification of the original profile variables and independent physical moment values is explicitly not supplied by this mathematical comparison.",
      "statementEN": "Positive finite Laplace measure and three finite positive moments. Identification of the original profile variables and independent physical moment values is explicitly not supplied by this mathematical comparison."
    },
    {
      "id": "premise:existing-ap-positive-source-factor-and-declared-squared-energy-comparison",
      "titleEN": "Existing AP source factor and explicitly squared-energy comparison",
      "statementEN": "Given self-adjoint invertible tangent reader of the constant original quarter class, beta=2ell and existing AP gluing factor Q=exp(-beta|h|), with full orientationR. Squared u=(beta|h|)^2 is an explicit comparison variable, not an assumed physical whole bosonic action identification."
    },
    {
      "id": "premise:actual-v252-entire-yukawa-majorana-family-with-compatibility-only",
      "titleEN": "Whole original v252 family under finite compatibility conditions",
      "statementEN": "Four arbitrary complex3x3 Yukawas and complex symmetric3x3 MR in original full_D, with fixed original algebra,gamma,JF. Impose only Hermiticity,gamma oddness,JF reality and order zero/one. Generic quotient uses only R014 constant U(3)^6 coordinate changes. No whole TFPT source rule, extra fixed clock or action selector assumed."
    },
    {
      "id": "premise:original-invariant-word-domain-with-nested-readouts",
      "titleEN": "Given original invariant word domain and nested readouts",
      "statementEN": "Actual operators and adjoints preserve common word domain D; original orthogonal P<=R preserve D. Formal free variables retain order. Analytic evaluation requires common inverses and domains separately. No new physical source, operation selector or time assumed."
    },
    {
      "id": "premise:original-single-beta-return-with-native-cartan-j-word-alphabet",
      "titleEN": "Original single-root return with native Cartan J",
      "statementEN": "Same complete original single-beta S and actual Fock adjoint as SOURCE-RETURN-MEMORY.txt, L=Sdagger S on neutral grade4 dimension726 with the original five quartics. Native linear J with Ju=i u,Jubar=-i ubar and its adjoint retained. Algebraic word responses of this supplied source only; no selected physical history or full E8 alphabet claimed."
    },
    {
      "id": "premise:original-regulated-full-source-with-actual-measure-and-legs",
      "titleEN": "Whole original source caller with actual measure and legs",
      "statementEN": "Original v42 Z_k integral exp(-S_adm-V_R+Jphi) with fixed original action,measure,domain and reference. Analytic statements require finite relevant integrals/derivatives and nonzero Z_k[0]; otherwise coefficientwise only. Actual field fibers retain Jacobians and Berezin ordering. Distinguish structural P_adm, configuration readout, positive-time half and state projection. No new measure or original source coupler assumed."
    },
    {
      "id": "premise:finite-reflected-positive-quadratic-ir-regulator-class",
      "titleEN": "Finite reflected positive quadratic IR regulator class",
      "statementEN": "Two real bosonic halves with no fixed-plane fields, reflection swaps them. Real symmetric A,C and PSD R=[[A,C],[C,A]]. Universal preservation over all finite positive RP original measures; no silent extension to complex/graded fermionic cases. A fixed original source can admit additional regulators. IR regulator is not a physical selector."
    },
    {
      "id": "premise:actual-shared-real-higgs-yukawa-consumer-and-given-boson-fiber",
      "titleEN": "Actual shared real Higgs consumer and supplied boson fiber",
      "statementEN": "Original real AF one-form convention q(H)=[[barh2,h1],[-barh1,h2]] with supplied four Yukawa matrices across all families/colors. Keep distinct lepton and quark matrices. H fiber is an argument of same full original S_adm with measure,reference,regulator,eta and higher vertices. Finite Berezin algebra and required finite boson moments; analytic limits separate. Affine shift applies to affine Yukawa coupling; retain any additional H dependent fermion vertices as actual insertions. No new physical measure or Yukawa selection."
    },
    {
      "id": "premise:original-graded-rp-reference-and-even-local-reflected-half-action",
      "titleEN": "Supplied joint graded RP reference and even local reflected half action",
      "statementEN": "Given actual joint graded RP reference omega0, same original antilinear Theta,positive-half algebra and twisted product. In the supercommutative Euclidean path the actual even interaction is Vplus+ThetaVplus with strict half support and no unchecked fixed-plane/cross terms. B=exp(-Vplus),relevant coefficient integrals exist and omega0(ThetaB B)>0. A bosonic/fermionic RP product supplies one such reference; RP marginals of a coupled source alone do not. These hypotheses are not declared derived physical original data."
    },
    {
      "id": "premise:original-constant-higgs-fiber-and-symmetric-majorana-spectral-consumer",
      "titleEN": "Original constant Higgs fiber and symmetric Majorana spectral consumer",
      "statementEN": "Given actual whole v252/v254 real96 q(H) one-form consumer,four fixed complex3x3 Y and complex symmetric MR between right neutrino singlets. Constant H,real coordinates without1/sqrt2,no fixed nontrivial weak connection. Matrix spectral shift z>0. All-mode result uses supplied scalar-quarter product circle h=(n+alpha)gamma-i gammaPhi,alpha1/4 or3/4 with its original domain and mu²>0. Extension to global Padm,reference or APS requires established common unitary transport."
    },
    {
      "id": "premise:literal-archived-geometric-residue-and-common-higgs-field-comparison",
      "titleEN": "Literal archived residue and common Higgs-field comparison",
      "statementEN": "Compare exactly printed bulk coefficients in archived v42 section013 and v45 metrology source002,ED_bulk=-R/(24pi²)-u/(48pi²),GammaD4_bulk=kin/(16pi²)+u²/(16pi²)+Ru/(96pi²),Gamma_grav=-6chi²ED+GammaD4,and stated target potential with muPhi²(chi0)=chi0²wH²/(8pi²)>0,lambda=1/(16pi²). Same field normalization u=Phi*Phi>=0,fixed positive chi0,flat R0 background. Remaining complete relative/reference/eta/zero-source terms are not presumed zero. No automatic Wick or Legendre identification,no claim this exhausts current theory."
    },
    {
      "id": "premise:local-same-trace-higgs-residue-and-conditional-product-dirac-consumer",
      "titleEN": "Local common Higgs residue contract and conditional product consumer",
      "statementEN": "Local source test assumes P=-(nabla²+E),R positive on spheres,same local linear trace for Res1 and FP0,E=-R/4-X(H),no extra H-dependent curvature/connection terms,fixed smooth interior and unchanged boundary data. Relative subtraction requires both operators of same type. Boundary/eta/nullity results use explicitly fixed regular domain branch. Actual whole96 Higgs family with supplied Y and symmetric MR is additionally evaluated through explicitly conditional standard product Dirac with spin4. Original spacetime/compression/reference/continuation map is not thereby derived."
    },
    {
      "id": "premise:whole-variable-higgs-bound-reference-and-conformal-dirac-map",
      "titleEN": "Whole variable Higgs response with bound reference and original scale sandwich",
      "statementEN": "Given ordinary spin4 product realization with unchanged96 Y/MR,fixed Clifford connection,H only order zero,and no H-dependent first-order coefficient of its square. Evaluate separately already supplied full collar lifts S Rr and S gammaF Rr without selecting either. Common reflected double with invariant metric/trace,transported common full domain,unchanged Y/MR under supplied base pullback,covariantly constant DM. H changes compactly in interior or closed double. Local geometric full/reference pair,not primitive bounded-Drel contract. Original Dsigma=fDf,f=e^-sigma/2,holding sigma fixed in H variations;distinguish unscaled and jointly scaled reference. Scale test assumes Brel0 or already transported mass chart;actual Brel must be evaluated in complete original sum."
    },
    {
      "id": "premise:original-inner-higgs-central-scale-and-finite-affine-source-map",
      "titleEN": "Original inner Higgs fluctuation with central scale and full affine source",
      "statementEN": "Same given anchor and same a,b in real oneform contract;f positive real bounded invertible or local common core,[f,a]=[f,b]=0,JfJ^-1=f,transported domain. Whole original96 H consumer with unchanged Y/MR,Hstar=(0,1),L(Hstar)=DY. Compare same inner change before/after scaling. Given finite full H fiber at fixed sigma/other fields;transport all vertices,reference,eta,sources and absolute Z0. q^4 Jacobian only for flat four-real measure,otherwise pushforward. No new Brel definition or unchecked continuum measure theorem."
    },
    {
      "id": "premise:given-normal-higgs-mode-realization-and-distinct-module-types",
      "titleEN": "Given normal Higgs modes and distinct module types",
      "statementEN": "Given3+2 carrier V=(3,1)-1/3+(1,2)1/2 with conjugation on End_C(V),given colorless H=(1,2)1/2,real-linear insertion;normal2D spin,weakSU2 and spacetime spin distinct. Reuse original normal B_Ea and Higgs/wedge results. Whole kernel/Gram formulas apply only when actual normal realization is given selfadjoint B or supplied doubling with finite constant kernel and actual scale fBf/domain,prescribed S and insertions on their domains. No new profile,normal metric,physical measure,or index-operator/Brel identification."
    },
    {
      "id": "premise:given-product-clifford-grades-parallel-projection-and-complete-relative-block-reader",
      "titleEN": "Given Clifford/form grades,parallel projection and complete relative block reader",
      "statementEN": "Existing whole96 Higgs consumer L(H)+DM,unchanged Y/MR;given conditional spin4 product with fixed Clifford connection. Distinguish ordinary internal two-form from Clifford-valued full notation. Heat comparison assumes parallel spin-compatible orthogonalP and compact interior variation;P+ is actual internal test,not identified with normalP or printedI48/P2. Block formulas exact for same finite regulated operator;infinite domain/relative trace/continuation require original contract. FixedP for affine Hjets;moving frames retain actual induced terms. No choice of physical normal operator,measure,profile,gap or free parameters."
    },
    {
      "id": "premise:original-positive-admissibility-reader-and-given-total-chiral-product",
      "titleEN": "Original positive admissibility reader and given total chiral product",
      "statementEN": "Padm is spectral in its actual positive Ladm=Dadm*Dadm on RanPstr and original domain. Finite low spectral subspace and fixed positive threshold only in named residue test;distinguish heat zero modes from zero-excluded zeta,constant nullity and no threshold crossing for comparing Hjets. Total chirality only in existing conditional Spin4×96 consumer,original GammaF,Y/MR,scalarf and grading-stable selfadjoint domain. Whole even spectral identity needs oddK and summable functions;check reference separately. No Ladm=Ksigma²,normalH=Padm or internalhalf=totalchirality identification. No new geometry,trace,metric or state choice."
    },
    {
      "id": "premise:whole-original-scale-congruence-and-specified-normal-energy-data",
      "titleEN": "Whole original scale congruence and specified normal energy response",
      "statementEN": "Given whole K=fK0f on transported domain, smooth positive f and original whole96 Higgs consumer. Boundary trace and metric use existing conformal transport. Profile recovery only on same homogeneous collar D0=N(partial_r+B), N*=-N,N²=-I,{N,B}=0,fixed Y/MR and sigma=sigma(r); complete first energy-transfer response of all given quarter modes must be supplied as data,not inferred from P(0). Spectral variation uses ordinary cyclic trace,justified continuation,interior variations and explicit reference/zero convention. Exterior Brel,other domains and weights remain separate."
    },
    {
      "id": "premise:given-whole-green-energy-source-and-conditional-scale-response",
      "titleEN": "Given whole Green/Poisson producer and conditional scale response",
      "statementEN": "Same existing homogeneous collar D0=N(partial_r+B),N*=-N,N²=-I,{N,B}=0 and whole given constant96 Higgs/MR family with quarter gap≥1/4. AP normal double has length2ell;APS comparison retains old fixedP_-(h0). Scale identities use actual shared transported domain and justified resolvent variation. DynamicHiggs derivative retains original whole weight,nonzeroZ0 and differentiable regulated source;no free normal metric,vertex norm or replacement of originalTr_rel."
    },
    {
      "id": "premise:current-original-spectral-action-and-fixed-ordinary-moment-dictionary",
      "titleEN": "Current original spectral action and fixed ordinary coefficient convention",
      "statementEN": "Actual current tfpt2section010:273-315 and v255/v259 statements;ordinary4D coefficients defined byTr exp(-tK²),justified interior asymptotics and actual domain/trace/reference. Known positive boundaryQ not automatically whole bosonic action orlogdet. Entire positiveLaplace mixture class with finite m0,m1,m2. Full original96H/MR R031consumer remains conditional,on specified fixedH reference branch;subtract other actualreference jets explicitly. No newY/scale/profile/trace orfree correction."
    },
    {
      "id": "premise:original-whole-spinor-and-entropic-curvature-consumers-with-given-higgs-response",
      "titleEN": "Original whole spinor and entropic curvature consumers with given Higgs response",
      "statementEN": "ExistingconditionalR031Spin4x96Diracconsumer withLichnerowiczE=−R/4,whole spinconnection,ordinaryconstanttrace/moment,closedorinteriorgeometryandactualreference. Separateactualv475vacuumconsumer−Trlog(I−betaX)on0/1/2formswithX2=2R_abcdonunorderedpairsandunchangedbeta/Einsteinpin;smallcurvatureinlogdomain. Wholealgebraic4Dcurvaturetensor,no maxsymmetricrestrictionbeforevariation. ExistingR031/R045HiggsconsumerwithallY/MR,specifiedbrokenstationarybranchandpositivequadraticradialoperatoratfixedbackground. Retainwhole source,boundreference,Z0,determinants,highervertices;no newlyphysicallyselectedfields,normsorcouplings."
    },
    {
      "id": "premise:actual-original-four-ghost-commuting-projection-admissibility-complex",
      "titleEN": "Actual original four-ghost commuting-projection admissibility complex",
      "statementEN": "Original Qadm=sum eta_j†K_j on H tensor Lambda(C4),with the actual four commuting orthogonal penalty projections,positive finite lambda_j and the supplied positive Dadm spectral calculus. Original macro iC means iota_C,not i times C. Separate normal S2,physical4D geometry,internal representations and auxiliary ghost degrees. Partial ghost trace needs no base trace class;full trace/determinant requires its actual regulator/reference. No physical ghost measure or Brel identification added."
    },
    {
      "id": "premise:actual-smooth-normal-family-with-supplied-unitary-horizontal-connection",
      "titleEN": "Actual smooth normal family with supplied unitary horizontal connection",
      "statementEN": "Actual selfadjoint normal B_Ea family or supplied doubling with smooth finite constant kernel rank,common domain or actual unitary transport and supplied unitary horizontal Hilbert-bundle connection including normal measure. R035 T=WS(Sdagger W²S)^−1/2 applies only for actual fBf transport. Reduced inverse requires isolated0 and sufficient form differentiability. Positive norm restriction concerns the supplied connection kinetic form,not a newly added boson operator. Parallel seed and scalar W derivatives are an explicit subclass only. Index chooses neither B(x),horizontal connection nor physical metric."
    },
    {
      "id": "premise:fixed-original-o1-section-consumer-and-actual-normal-first-order-placement",
      "titleEN": "Fixed original O1 section consumer and actual normal first-order placement",
      "statementEN": "Whole original V=H0(CP1,O(1)) with fixed holomorphic structure;continuous positive scalar normal multiplier W and actual R035 G/T normalization. B0T claim separately requires kerB0=V. First-order normal B=c^a nabla_a+Phi with actual shared domain and horizontal transport only on explicit product chart. No B_Ea=Brel identification. Whole original K=fDgeo f+Brel has exterior Brel;exterior placement alone does not imply no field dependence."
    },
    {
      "id": "premise:actual-original-quark-higgs-family-and-positive-conditional-source-mixture",
      "titleEN": "Original quark Higgs consumer and actual positive conditional source mixture",
      "statementEN": "Given whole96 Phi(H)=L(H)+DM with original q and fixed supplied Y/MR;existing constant common quarter circle p=1/4,ell>0,h=p gammaF−i gammaF Phi,second-face C_H. Existing charged left-quark CAR subalgebra is conditionally quasifree with A_H tensor I_color3 and zero anomalous pairs. Actual full normalized H weight must give a positive mixture including Z_F/reference/eta and all factors. Strictness requires nonconstant u=HdaggerH and a nonzero quarkY. No new physical96-fold integration,measure derivation,general variable H(x)/connection or3+1D assumption."
    },
    {
      "id": "premise:fixed-rank-bounded-cutoff-with-ordinary-noncompact-higgs-volume",
      "titleEN": "Fixed finite rank and bounded cutoff with ordinary noncompact Higgs volume",
      "statementEN": "Explicit realization to test,not actual primitive measure derivation:entire O1 coefficient H in C2 with ordinary linear d4H at fixed Gram/background;Hermitian K_N(H),Kref_M(H) of fixed finite ranks,real globally bounded f,all H/reference dependence allowed. Only actual nonzero finite affine Berezin/Pfaffian polynomials,no new96-fold measure. Pure eta phase unit modulus. IR claim at fixed rank and fixed bounded S_B as epsilon|H|2 regulator vanishes. Evaluate other actual measures/support/Jacobians/additional action/unbounded f or joint rank limits separately."
    },
    {
      "id": "premise:whole-constant-original-quarter-source-positive-traceclass-profile-and-flat-higgs-volume",
      "titleEN": "Whole constant original quarter source and positive trace-class baseline on ordinary Higgs volume",
      "statementEN": "Already given whole constant h_H=(n+alpha)gammaF−i gammaF Phi(H) on original H1 quarter domain,alpha1/4or3/4,original96 Hermitian odd Phi(H)=L(H)+DM with fixed Y/MR. Positive Laplace F with finite baseline Balpha=sum_n F((n+alpha)^2/chi^2),fixed chi. Actual geometric mode cuts,optional full positive trace insertion0<=W<=I. Same genuine reference product with finite fixed baseline. Measure test explicitly ordinary d4H onC2,not asserted primitive measure. R031 compact even product only with actual grading-compatible domain and finite baseline."
    },
    {
      "id": "premise:given-o1-mode-isometry-and-weak-invariant-positive-higgs-covariance-marginal",
      "titleEN": "Given O1 isometry and additional weak-invariant positive Higgs covariance marginal",
      "statementEN": "Named identifiability test only:original normal O1 space C2 with actual supplied normalized embedding T and compatible unitary WEAK SU2 identification,normalspin2/weak2/spin4 distinct. Even additional actual positive normalized invariant H marginal with EHH*=cI,c>0 is GRANTED,not derived from index/P=T T*. Fourth moments only where finite. Four real H coordinates and original whole96 Phi/Y/MR unchanged. Fourier readout requires actual positive original Euclidean field source/commuting image and H=T*varphi_N,not a replacement for their origin."
    },
    {
      "id": "premise:literal-whole-source-localization-and-geometric-readout-contracts",
      "titleEN": "Literal whole source and existing localization and geometry contracts",
      "statementEN": "Source audit of existing full E8 current representation,named68D single-word realization,P1 boundary datum,conditional4D selector,free v1000 tensor Hessian and v1013 Hamiltonian class,and explicit Lean axiom compositions. No common physical localization identification assumed.27 website cards and distinct unchanged frozen registry inventoried as metadata;inflation relations only on supplied leading-order branch at common externalN and fixedc3. No current empirical check or statistical independence proof."
    },
    {
      "id": "premise:actual-proper-local-current-weyls-and-bare-native-code-compression",
      "titleEN": "Actual proper-local current Weyls and bare native-code compression",
      "statementEN": "Given full E8_1 Cartan source with actual J-positive neutral Fock embedding and unchanged five homogeneous grade4 codepolynomials,Gram96/8/8/8/1. P is observedcode only,not originalPadm. Bounded Weyls from smooth real8vectorprofiles on any disjoint nonempty propercircleintervals,no Fouriertruncation. Mode1 displacements ae0 andb(e0+e1),real a/b,orthogonal mode1 components0,all higher modes retained. Analytic compact-moment construction provides actual profiles. Test only ordinary product/commutator ofPAP/PBP against whole local word readout. No physicaloperation,newcoupling,4D identification orstate choice assumed."
    },
    {
      "id": "premise:actual-elliptic-quarter-supported-current-and-interval-car-modular-test",
      "titleEN": "Actual elliptic quarter source with supported current and interval CAR modular-time test",
      "statementEN": "Unchanged given free CAR source of original elliptic midpointannulus:occupation c_alpha(r)=(1+exp(4pi(r+alpha)))^-1,alpha1/4 or3/4,entire original theta charge law/current Weyl source. Actual sourcecircleintervalI and extensionbyzero L2(I) embedding. Compared geometricclass:realC5 pointdiffeomorphisms I ontoI,generatorcontinuous atfixed endpoints,unchanged weight1 current smearing f→f composed gamma^-1,arbitrary scalar affine current shifts. No fieldmixing,changed dictionary orprimitivephysicalmeasure identification assumed. Preserve existing constructive neutralMoebius vacuum as distinct provedstate."
    },
    {
      "id": "premise:actual-elliptic-two-face-projection-with-genuine-regional-support",
      "titleEN": "Actual elliptic two-face projection with genuine regional supports",
      "statementEN": "Given originalfree ellipticquarterCAR source c_r=(1+exp(4pi(r+alpha)))^-1,alpha1/4 or3/4,X=sqrt(C(1−C)),fullprojectionP=[[1−C,X],[X,C]] inalreadyprovedpositive transportedtwofacechart. PreserveoriginalGreen/RP/gradedHS dictionary. Regionalembeddings areactualextensionbyzero fortwononemptypropercircleintervals withopencomplements. Partnercomparison retainsgivensourcestate/linearCAR embedding. No primitiveP1measureidentification,newpurification,geometrychoice orGaussianizationafterHiggsintegration. Fullintervalsubalgebra andsingle supportedprobe aredistinctsubsystemcuts."
    },
    {
      "id": "premise:actual-original-rightup-triplet-fixed-family-and-even-color-source",
      "titleEN": "Actual original rightup color triplet with fixed family vector and even color source",
      "statementEN": "Given original96 Phi(H),gamma_L+1,gamma_R−1,fixedp1/4,ell>0. Samefixed eigenvector of GIVEN Yu*Yu,y²>0,inall3originalrightupcolors;notnewphysicalstateselection. M*M=u diag(Yu*Yu,Yd*Yd) givesactualt(u)=1/2+p tanh(ell sqrt(p²+y²u))/(2sqrt(p²+y²u)) on(a,b],a1/2,b(1+tanh(ell p))/2. Reader assumespositiveevenstateinvariantunderoriginalconstantSU3;finite modularlogrequiresfaithfuldensity. Forcednonlinearity and positivemixturetestONLYadditionallyunderR054positive normalizedwholeHmixturewithnonconstantradialu,includingwholedet/Pf,Ref,Eta,Z_F,Z0. Generalreaderdoesnotrequiremixture. No primitiveP1measure,physicaltimeorfull96modularrestrictionclaim."
    },
    {
      "id": "premise:original-common-positive-constant-higgs-law-over-whole-quarter-tower",
      "titleEN": "One original common positive constant Higgs source over the whole quarter mode tower",
      "statementEN": "Samegivenoriginal96Yukawa/Higgs/Majoranaconsumerh_H(p)=p gamma−i gammaPhi(H),samefixedrightupfamilyvectorofgivenYu*Yu,y²>0,allp=n+alpha,n≥0,alpha1/4or3/4,onefixedell>0. Originalellipticnormalizationell2piretained. ONEpositivenormalizedfulllawoverfiniteconstantu=H*Hyieldsallaveragedconditionalt_p(u)=1/2+p tanh(ell sqrt(p²+y²u))/(2sqrt(p²+y²u)). AllRef/Eta/det/Pf/Z0andotherintegratedweightsbelongtothisSAMElaw,notrefittedpermode. NoHmomentorcompactnessassumption. Extra conditionalfermionverticesorH(x)changingthegivenconditionalformareoutofscope. NoidentitywithprimitiveC_Sigma/physicalcarrier/originassumed."
    },
    {
      "id": "premise:original-sixty-e8-unrefined-record-source-and-typed-doily-observables",
      "titleEN": "Original sixty E8 events with unrefined record and typed Doily observables",
      "statementEN": "Given unchanged Gaussian E8 rays z in C4 of squarednorm4,r_z=I−zz*/2,actualH10=Sym²C4 in originalBellbasis,U_z=Sym²r_z and uniformunrefinedrecordbranchesB_z=U_z/sqrt60. Fifteen nontrivialPauliW_p are directionsmodulosign,Pi±rank2inC4,not15orthogonalsites. OriginaltensormapF defines J_p=F*(W_p⊗I+I⊗W_p)F and rank3 Q_p±=F*(Pi_p±⊗Pi_p±)F onH10. Namedpreparation/finaleffect are mathematicalprobes,notderivedphysicalstate. Noidentitywithstopped/refinedPetersenreader,physicalcuspclockor13Octoberloadedreflection."
    },
    {
      "id": "premise:given-e8-current-vacuum-net-and-original-grade-two-descendant-with-standard-split",
      "titleEN": "Given E8 current vacuum net with original grade2 descendant and strictly nested intervals",
      "statementEN": "SamealreadygivencompactE8_1latticevacuumnetwithOmega/fullL0,originalD5vector10⊗A3Sym²C4isometryE0atweight2andunchangedSU4currentaction. rho(u_z)E0=−iE0(I10⊗U_z),J0(X)E0=E0(I10⊗dGamma2X). ForalreadygivenstrictlynestedcircleintervalsclosureI⊂J,standardsplitfollowsfromfullcharactertraceclassandReehSchlieder. StandardformfixesWwithWABW*=A⊗BforAinA(I),BinA(J)′. Nophysicalregionselection,rawP1derivationofnet/vacuum,replacementoforiginalellipticsourceor3+1spacetimeassumed."
    },
    {
      "id": "premise:given-original-elliptic-quarter-source-and-supported-scalar-current-insertion",
      "titleEN": "Given original elliptic quarter source with supported scalar current insertion",
      "statementEN": "ExistingfreeannulusCARquantizationwithD_a=−i∂+aonperiodicH1,a=1/4and3/4,b=4pi,originalsea/current/H_a. Independentquarterblocksinonegivenmarginalarenotannulusfaces;wholeexistingtwofacegraphretained. ExecuteforwardD_a+epsilonfthroughallrealsmoothsupportedtestinsertionswithsameoriginalnormalization/time. Nochoiceofphysicalf,primitiveP1measure,net/vacuumor3+1derivedbyassumption."
    },
    {
      "id": "premise:original-quark-particle-color-group-counting-on-whole-constant-quarter-source",
      "titleEN": "Original color group counting on the whole constant quarter quark source",
      "statementEN": "Sameoriginalconditionalquasifree36-labelquarkPARTICLECARsubspaceperp=n+a,a=1/4or3/4,b=4pi,h=pgamma−igammaPhi,M=q(H)diag(Yu,Yd),q*q=uI2. GivenYremainfixed. ActualrealSU3liftusespiandJandactsidenticallyonL/Rcolor,notrawleftpiandnot96independentphysicalfermions. Retainwholemodetower,bothquartersatSAMEH,andallcomplement/ref/eta/ZF/H0/Z0inwholeweight. Boundedglobalgroupinsertionnotalreadyidentifiedwithlocal4D,EMormetricsource. PositiveHmixtureonlyunderactualpositivecompleteweight."
    },
    {
      "id": "premise:given-whole-quarter-graph-and-canonical-diagonal-original-color-lift",
      "titleEN": "Given whole quarter graph and canonical diagonal original color lift",
      "statementEN": "GivenwholeG(C)twofaceprojectionandC=(1+exp(4pihH))^-1fromoriginal36quarkparticleCARandgivenY. Sameactualpi/JSU3liftonbothcopiesofthedisplayedcommonSOURCEfiber,[C,U]=0,det36U1. Twoquartersarenottwofaces. OriginalCARRIERPsingandfixedZref[0]quotientretainedverbatim;NOidentificationofcarrierselectorwithoneface/diagonalFockprojectionorPF. Finiteoriginalregulatorsandconvergentfullmodecharacters;positiveHmixtureonlywithactualcompletepositiveweight."
    },
    {
      "id": "premise:original-marked-color-center-on-whole-e8-lattice-current-source",
      "titleEN": "Original marked color center on whole stipulated E8 lattice field source",
      "statementEN": "ExistingstipulatedunitaryE8_1lattice/currentnetinactualD5+A3targetframe,ColorA2=e1-e2,e2-e3,originalcentert=p1+p2-2p3mod3andneutraloscillators. GivenoldWardlambdaA=e3+e6andwholeweak(10,6)block;retainY/Flavor/Alpha,wholefields/clocks. ExecutePsingonTHISbranch,assumenoP1/OS/Padm/P_F/96CARintertwiner. Properlysupportedlocalcurrentswithactualdomains/adjoints. Extensiontheoremonlyintegrallattice/superlatticewithSAMEconformalvector. ReusenativeJboundarywithoutidentifyingJwithsourcequarterfoldorphysicaltime."
    },
    {
      "id": "premise:original-fixed-color-and-folded-ns-cartan-interface",
      "titleEN": "Original fixed color action and actual folded NS charge frame",
      "statementEN": "Retain unchanged E8 root representation, actual rational M in fold-results.json, NS charge space M Z8 and even M D8, R069 color A2=e1-e2,e2-e3 and center w=e1+e2-2e3. Reuse existing four-module AIA parent with actual cocycle/spin/quarter transport. Assume no common physical96-matter map, local inclusion, new gauging, measure law or spacetime from this data identification."
    },
    {
      "id": "premise:actual-thermal-quarter-source-and-retained-spin-parent-mixed-word",
      "titleEN": "Actual thermal quarter source and retained spin parent in mixed words",
      "statementEN": "Given original free CAR frequencies r+alpha,alpha1/4 or3/4,beta4pi,actual eightfold r=8n+j. Use its H=8(L0+delta.p),delta_j=(j+alpha)/8-1/2 and already retained four-module D8 AIA parent with original eps_ext cocycle. Mixed analytic words of actual four R070 color-spinor roots and all eight original odd vector channels,on ordered real cylinder strips with correct adjoint scalars and absolute NS/R trace normalizers. WHOLE P_NS B P_NS yields an explicitly constructed positive extension of given free source,not physical P1 selection. Restrict original color center only at whole-word endpoints. Optional NORMAL whole-parent KMS result only on B(H_parent) at SAME H/beta;no new physical KMS requirement or spin-structure sum."
    },
    {
      "id": "premise:actual-eightfold-arc-residue-source-and-fixed-spin-cartan-insertion",
      "titleEN": "Actual eightfold arc/residue fold and fixed color Cartan insertion",
      "statementEN": "Given actual eta_r=psi((x+2pi r)/8)/sqrt8,chi=Ualpha*eta,Ualpha(r,k)=exp(2pi i(k+alpha)r/8)/sqrt8,both quarters and old point-split bilinears with sea/reference. Original scalar Weyl profiles act diagonally in ARC chart and may differ on different arcs. q=M^T beta for four fixed color spinor roots and unchanged M. b is a smooth real periodic supported test;no new physical state or measure. Distinguish full even CAR on union from additive join of separate even arcs."
    },
    {
      "id": "premise:actual-original-ns-color-center-continuous-density-and-coordinate-net",
      "titleEN": "Actual NS color center,continuous density source and original coordinate algebras",
      "statementEN": "Given original CAR sea r<0,Jm=sum:a_r*a_(r+m):,both quarters,unchanged M and w=M^T(1,1,-2,0,...)=(1,-1,0,-1,-1,-1,0,-1). Z acts on a_r* by exp(2pi i w_(rmod8)/3),P0=(I+Z+Z²)/3 and neutralOmega. Actual Ualpha arc/residue fold and full original circle CAR region algebras given. Test entire continuous scalar current groups and ordinary coordinate supports. No P0=P_adm/P_F or physical P1 source intertwiner assumed."
    },
    {
      "id": "premise:actual-original-full-two-face-graph-and-local-matrix-operations",
      "titleEN": "Actual whole two-face graph and local matrix-operation class",
      "statementEN": "Given original G=[[1-C,X],[X,C]],C(p)=(1+exp(beta p))^-1,X=(2cosh(beta p/2))^-1,p=r+alpha,alpha1/4,3/4,beta4pi,existing wholeCAR source,unchanged M and periodic8fold B=8(-i∂x)+diag(k+alpha). Entire bounded matrix multiplication for reduction,entire smooth unitary multiplication for Fock implementation. No originalOS/Padm field intertwiner assumed."
    },
    {
      "id": "premise:original-blocked-transfer-algebra-and-marked-source-uniformity-audit",
      "titleEN": "Original blocked transfer and marked-source/uniformity audit",
      "statementEN": "Literal QFT004 C6 Green/retained weights/Padm route and Burnside-to-MPS-to-uniform-parent-gap inference. Distinct exact implication witnesses:same wholeCP transfer at fixed markO,and injective volume-varying canonical range3 parents. Not physical TFPT models or countermodels to whole master action."
    },
    {
      "id": "premise:original-bare-c6-fixed-analytic-half-power-and-literal-word-alphabet",
      "titleEN": "Original bare C6 analytic half-power and literal word alphabet",
      "statementEN": "Original v17U6,D_y=yI-deltaU6,Q=D*D+epsilon²I,C=(U+U*)/2,original algebraic delta and y1,1/3,2/3,epsilon>=0. Canonical analytic series with distinct directed letters and full absolute amplitude. No primitive physical measure/spacetime premise;original compressedKstr requires actual reduction proof."
    },
    {
      "id": "premise:actual-original-finite-v990-source-and-historical-parent-recovery",
      "titleEN": "Actual original finitev990 source and historicalparent recovery",
      "statementEN": "Unchangedv990.run andactualH_W0/OPS,suppliedparameters,originalGibbsweight;four-modeCAR onC6linktensorFockC4. Historicalv21/v42,microscopicCAR,given96Hsources inoriginalscopes;no primitivephysical3+1identification."
    },
    {
      "id": "premise:actual-original-finite-car-higgs-force-and-faithful-collective-source",
      "titleEN": "Actual original finite CAR Higgs force and faithful collective source",
      "statementEN": "Whole original real v252/254 oneform Phi(H),fixedYs,quaternionicHcoordinate and actual36quarkpart of givenquasifreesource. Gibbs/Legendre0<C<I,pairing0,YuorYdnonzero;finiteclasswith exactcontinuousSU2doubletaction. No primitivephysicalH/measure/time identification assumed."
    },
    {
      "id": "premise:actual-whole-circle-product-car-force-and-original-time-smearing",
      "titleEN": "Whole original circle CAR force and given time testing",
      "statementEN": "Original constant finite internal Higgs family,fixedYu/Yd,hp=p gamma-igammaPhi(Hstar),p=n+alpha,alpha1/4or3/4,ell>0 given. Pairing-zero quasifree Cp=(1+tanh ell hp)/2,factorizing even Fourier blocks,Ta=-igamma dPhi/dxa. Real Schwartz tests of same time;summable variance exact L2 criterion. No primitive H/physical3+1 or new measure/scale identification assumed."
    },
    {
      "id": "premise:original-constant-circle-local-higgs-force-and-hilbert-schmidt-time-source",
      "titleEN": "Original local Higgs derivative in given constant circle and Hilbert Schmidt time source",
      "statementEN": "Given original circle h=gamma(-i partialtheta+alpha)-igammaPhi(Hstar),periodicH1domain,alpha1/4or3/4,constantPhi/Hstar,fixedoriginalYs. Ta=-igamma dPhi/dxa Hermitian/odd,C=expit(2ell h),pairing0. Real localspace/time tests;strictlocality needs compact smooth supports allpointpairsdistS1>|t-s|. No primitivefield/physical3+1 or newHmeasure identification."
    },
    {
      "id": "premise:literal-archived-global-positive-kernel-finite-measure-map",
      "titleEN": "Literal archived globally positive kernel on finite measures",
      "statementEN": "v42section01631..148 andv45metrology00560..178:finite nonzerononnegative nu on[0,infty),Tnu(dt)=intK(s,t)dnu(s)/intpsi(s)dnu(s) dt,global lowerboundK>=m>0,Lebesguetargetdt,finitepositivedenominator. Scope exactlythiscombination,notactuallydifferentoriginalfinite-support/reference/integrablekernel."
    },
    {
      "id": "premise:given-native-quasifree-higgs-source-and-actual-graded-restriction",
      "titleEN": "Given native Higgs quantum source and actual graded restriction",
      "statementEN": "Existing original96 oneform with fixed supplied Y/MR, non-KO-doubled36 quark part, Hermitian odd forces and given constant circle time. Finite faithful mode Gibbs C=expit K, K=2ell(p gamma+sum xa Ta),ell>0 and original quarter p. P is diagnostic gamma+ subspace, not identified with physical P_F/Padm. Conditional CAR quasifreeness at given background, not after dynamic H integration. Seed252,p1/4,ell.2,r.3 are constructor controls, not selected physical parameters."
    }
  ],
  "predictions": {
    "registry": "TFPT blind-prediction registry (frozen, machine-enforced)",
    "freeze_date": "2026-06-09",
    "axioms": {
      "c3": "1/(8*pi)",
      "g_car": 5
    },
    "policy": "Every entry below is FROZEN as of freeze_date, before the next decisive data releases (JUNO theta12/theta13, CMB-S4/LiteBIRD r and beta, DESI n_s). The decimals are recomputed from the two axioms by v84_frozen_registry.py on every suite run; any later change of a formula or a value breaks the suite and is visible in manifest.sha256 / git history. Entries are dimensionless (the dimensionless axis is the only axis on which TFPT claims closure). Layer tags follow the document markers: core = [N] numerical fixed point of the closed compiler; conditional = [N]/[P] (named hypothesis stated in conditional_on); assigned = texture/phase-lattice assignment, not a derivation.",
    "predictions": [
      {
        "id": "ALPHA_INV",
        "observable": "fine-structure constant alpha^-1",
        "formula": "1/root of F_U1(alpha) = alpha^3 - 2 c3^3 alpha^2 - (4/5) c3^6 * 41 * log(1/phi_seam(alpha)) = 0",
        "frozen_value": "137.0359992168407125035379",
        "layer": "core",
        "conditional_on": "Ward-origin reading of F_U1 is [P]; the root itself is unconditional given the equation",
        "experiment": "CODATA adjustments",
        "kill_criterion": "persistent deviation beyond combined theory+measurement error"
      },
      {
        "id": "SIN2_THETA12_SEED",
        "observable": "solar mixing angle sin^2(theta_12) -- PREDICTION OF RECORD",
        "role": "prediction_of_record",
        "formula": "1/3 - phi0/2",
        "frozen_value": "0.3067473572449105696786871",
        "layer": "conditional",
        "conditional_on": "seam-misalignment lemma (epsilon = (3/4) phi0)",
        "experiment": "JUNO",
        "kill_criterion": "JUNO value away from 0.306747 beyond 3 sigma kills the seam-misalignment reading"
      },
      {
        "id": "SIN2_THETA13",
        "observable": "reactor mixing angle sin^2(theta_13)",
        "formula": "phi0 * exp(-5/6)",
        "frozen_value": "0.02310843515888110429328466",
        "layer": "core",
        "conditional_on": null,
        "experiment": "JUNO / global fits (NuFIT 6.0: 0.02195 +- 0.00058, currently ~2 sigma tension)",
        "kill_criterion": "persistent deviation beyond 3 sigma after global fit"
      },
      {
        "id": "BETA_BIREFRINGENCE_DEG",
        "observable": "cosmic birefringence angle beta (degrees)",
        "formula": "(phi0 / (4*pi)) * (180/pi)",
        "frozen_value": "0.2424350309009295284924315",
        "layer": "conditional",
        "conditional_on": "CMB EB/TB rotation = the same boundary phase as the flavor seed [P]",
        "experiment": "LiteBIRD / CMB-S4 (ACT DR6: 0.215 +- 0.074 deg)",
        "kill_criterion": "systematics-controlled exclusion of 0.2424 deg at > 3 sigma (incl. precise zero or wrong sign)"
      },
      {
        "id": "OMEGA_B",
        "observable": "baryon density parameter Omega_b",
        "formula": "(1 - 1/(4*pi)) * phi0",
        "frozen_value": "0.04894066266545011220054565",
        "layer": "conditional",
        "conditional_on": "seed-budget reading Omega_b + beta_rad = phi0 [P]",
        "experiment": "CMB (Planck 2018: ~0.0493)",
        "kill_criterion": "robust deviation beyond combined error"
      },
      {
        "id": "LAMBDA_C",
        "observable": "Cabibbo angle s12_CKM = lambda_C",
        "formula": "sqrt(phi0 * (1 - phi0))",
        "frozen_value": "0.2243762368847217731120972",
        "layer": "core",
        "conditional_on": null,
        "experiment": "PDG (0.2245)",
        "kill_criterion": "persistent deviation beyond combined error"
      },
      {
        "id": "S23_CKM",
        "observable": "CKM s23",
        "formula": "phi0 / (1 + lambda_C)",
        "frozen_value": "0.04342778843220215630289116",
        "layer": "core",
        "conditional_on": null,
        "experiment": "PDG (~0.041)",
        "kill_criterion": "persistent deviation beyond combined error"
      },
      {
        "id": "S13_CKM",
        "observable": "CKM s13",
        "formula": "lambda_C^3 / 3",
        "frozen_value": "0.003765384454486429837965432",
        "layer": "core",
        "conditional_on": null,
        "experiment": "PDG (~0.0038)",
        "kill_criterion": "persistent deviation beyond combined error"
      },
      {
        "id": "DELTA_CKM_RAD",
        "observable": "CKM CP phase delta (radians, leading order)",
        "formula": "pi/3 + 3 * lambda_C^2",
        "frozen_value": "1.198231638232244084651243",
        "layer": "conditional",
        "conditional_on": "leading-order holonomy readout [P]; sub-leading corrections not closed",
        "experiment": "PDG (~1.14)",
        "kill_criterion": "leading-order reading fails if precise delta departs from pi/3 + 3 lambda_C^2 beyond the stated sub-leading budget"
      },
      {
        "id": "MMU_OVER_MTAU",
        "observable": "lepton mass ratio m_mu / m_tau",
        "formula": "(8/7) * phi0",
        "frozen_value": "0.06076794534496631692490568",
        "layer": "core",
        "conditional_on": "source-scale (hat) masses, not pole masses",
        "experiment": "PDG (0.05946)",
        "kill_criterion": "scheme-consistent comparison departing beyond stated transport corrections"
      },
      {
        "id": "ME_OVER_MMU",
        "observable": "lepton mass ratio m_e / m_mu",
        "formula": "(12/7) * phi0^2",
        "frozen_value": "0.004846725425651567674771059",
        "layer": "core",
        "conditional_on": "source-scale (hat) masses, not pole masses",
        "experiment": "PDG (0.004836)",
        "kill_criterion": "scheme-consistent comparison departing beyond stated transport corrections"
      },
      {
        "id": "MU_OVER_MD",
        "observable": "quark mass ratio m_u / m_d",
        "formula": "55/117",
        "frozen_value": "0.4700854700854700854700855",
        "layer": "core",
        "conditional_on": "integer Plucker readout on the derived selector stratum (v49/v71)",
        "experiment": "lattice/FLAG (~0.47)",
        "kill_criterion": "scheme-consistent ratio robustly away from 55/117"
      },
      {
        "id": "MC_OVER_MS",
        "observable": "quark mass ratio m_c / m_s",
        "formula": "(34/47) / phi0",
        "frozen_value": "13.60499710285539356302651",
        "layer": "core",
        "conditional_on": "integer Plucker readout on the derived selector stratum (v49/v71)",
        "experiment": "lattice/FLAG (~11-14)",
        "kill_criterion": "scheme-consistent ratio robustly away from (34/47)/phi0"
      },
      {
        "id": "MT_OVER_MB",
        "observable": "quark mass ratio m_t / m_b",
        "formula": "(3/26) / phi0^2",
        "frozen_value": "40.8115130177002629681607",
        "layer": "core",
        "conditional_on": "integer Plucker readout on the derived selector stratum (v49/v71)",
        "experiment": "PDG (~41)",
        "kill_criterion": "scheme-consistent ratio robustly away from (3/26)/phi0^2"
      },
      {
        "id": "MSCAL_OVER_MBAR",
        "observable": "scalaron mass ratio M_scal / M_bar",
        "formula": "c3^(7/2)",
        "frozen_value": "0.00001256494208322844896998079",
        "layer": "conditional",
        "conditional_on": "identification M_Starobinsky = M_scal [P]",
        "experiment": "CMB inflation sector",
        "kill_criterion": "r > 0.01 robustly kills the R^2 branch"
      },
      {
        "id": "RHOL_OVER_MBAR4",
        "observable": "vacuum energy density rho_Lambda / M_bar^4",
        "formula": "(3/(4*pi^2)) * exp(-2 * alpha_inv)",
        "frozen_value": "7.125329526706216792033797e-121",
        "layer": "conditional",
        "conditional_on": "physical Lambda branch selection (8pi)^2 delta_top = 3/(4pi^2) [A]; alpha^-1 <-> Lambda identification [P]",
        "experiment": "Planck (rho_Lambda^1/4 ~ 2.24 meV)",
        "kill_criterion": "robust w != -1, or absolute scale off after anchoring"
      }
    ],
    "assigned_texture_values": [
      {
        "id": "DELTA_CP_NU_DEG",
        "observable": "neutrino Dirac CP phase (degrees)",
        "formula": "240 (phase lattice 4*pi/3)",
        "frozen_value": "240",
        "layer": "assigned",
        "experiment": "DUNE / Hyper-K",
        "kill_criterion": "precise value incompatible with the branch"
      },
      {
        "id": "SIN2_THETA23",
        "observable": "atmospheric angle sin^2(theta_23)",
        "formula": "0.5 (mu-tau symmetric texture; octant NOT predicted)",
        "frozen_value": "0.5",
        "layer": "assigned",
        "experiment": "JUNO / DUNE",
        "kill_criterion": "strongly non-maximal with no matching breaking structure"
      }
    ],
    "conditional_bands": [
      {
        "id": "R_TENSOR",
        "observable": "tensor-to-scalar ratio r",
        "formula": "12 / N_star^2, N_star in [50, 60] (external reheating input, NOT a compiler output)",
        "band_min": "0.0033333333333333333333",
        "band_max": "0.0048",
        "experiment": "CMB-S4 / LiteBIRD",
        "kill_criterion": "r > 0.01 robustly kills the R^2 branch"
      },
      {
        "id": "N_S",
        "observable": "spectral tilt n_s",
        "formula": "1 - 2/N_star, N_star in [50, 60] (external reheating input, NOT a compiler output)",
        "band_min": "0.96",
        "band_max": "0.96666666666666666667",
        "experiment": "Planck / DESI (current ~2 sigma tension with DESI-combined)",
        "kill_criterion": "value robustly outside the Starobinsky window at fixed reheating"
      }
    ],
    "derived_variants_not_predictions": [
      {
        "id": "SIN2_THETA12_SEAM",
        "formula": "1/3 - 1/(12*pi)",
        "value": "0.306807509484684110705186",
        "note": "leading-epsilon variant (epsilon = c3); a truncation of the SAME texture, NOT an alternative prediction"
      },
      {
        "id": "SIN2_THETA12_NONLIN",
        "formula": "full non-linear TBM + epsilon=(3/4)phi0 rotation, sin^2 = |U_e2|^2/(1-|U_e3|^2)",
        "value": "0.3070195373705387648045858",
        "note": "non-linearised variant of the SAME texture, NOT an alternative prediction; cf. v9_neutrino_texture.py"
      }
    ]
  },
  "scripts": [
    {
      "id": "v1_e8_glue",
      "cluster": "core",
      "descriptionEN": "E₈ glue: disc ℤ₄, q(D₅)+q(A₃)=2, 240=16·5·3, 248"
    },
    {
      "id": "v2_carrier_pascal",
      "cluster": "core",
      "descriptionEN": "g_car=5 Pascal closure; 16=1+5+10, Ω_adm=48, b₁=41/10"
    },
    {
      "id": "v14_carrier_uniqueness",
      "cluster": "core",
      "descriptionEN": "g_car=5 unique; split (3,2); Tr Y = Tr Y³ = 0"
    },
    {
      "id": "v15_bootstrap_classification",
      "cluster": "core",
      "descriptionEN": "D₅⊕A₃ is the unique familyful cyclic glue of E₈"
    },
    {
      "id": "v23_anchor_generator",
      "cluster": "core",
      "descriptionEN": "anchor a=(1,1,2): e_k=(4,5,2); power sums → 240, 248"
    },
    {
      "id": "v47_selection_theorem",
      "cluster": "core",
      "descriptionEN": "Boundary Carrier Selection (Thm A): only D₅⊕A₃ qualifies"
    },
    {
      "id": "v51_boundary_half_step",
      "cluster": "core",
      "descriptionEN": "glue norms (g_car,N_fam)/|μ₄|; δ=½; root norm 2"
    },
    {
      "id": "v53_compiler_core",
      "cluster": "core",
      "descriptionEN": "whole integer skeleton from (5,3); Δ_Y = 25 = 9+16"
    },
    {
      "id": "v66_e8_casimir_degrees",
      "cluster": "core",
      "descriptionEN": "compiler atoms = E₈ Casimir degrees; Σ = 128 = 2⁷"
    },
    {
      "id": "v3_em_alpha",
      "cluster": "em",
      "descriptionEN": "α⁻¹ = 137.0359992168 unique root of F_U(1)=0; ablation"
    },
    {
      "id": "v48_em_ward",
      "cluster": "em",
      "descriptionEN": "EM boundary Ward (Thm C): F_U(1) decomposition; 8b₁ = 164/5"
    },
    {
      "id": "v4_flavor_matrix",
      "cluster": "flavor",
      "descriptionEN": "residue matrix R: det 8, minors (2,3,5), χ_R, Σ L = 40"
    },
    {
      "id": "v10_projection_involution",
      "cluster": "flavor",
      "descriptionEN": "K=R+QΣ, L=R+Q(I+Σ); det ladder ∏ = 1920 = |W(D₅)|"
    },
    {
      "id": "v11_unique_KQ",
      "cluster": "flavor",
      "descriptionEN": "K, Q are the unique nonneg-integer matrices (enumeration)"
    },
    {
      "id": "v12_mass_generation_polynomials",
      "cluster": "flavor",
      "descriptionEN": "sector / generation polynomials of K; anchor det ladder"
    },
    {
      "id": "v37_plucker_anchor",
      "cluster": "flavor",
      "descriptionEN": "Plücker anchor: ‖Pl(K)‖₁ = 11; pencil det(K+xQ); lepton ring"
    },
    {
      "id": "v50_q_geometry",
      "cluster": "flavor",
      "descriptionEN": "Q geometry: Q₊ diag(3,2,1)-block; Q unique under budgets"
    },
    {
      "id": "v52_pencil_endpoints",
      "cluster": "flavor",
      "descriptionEN": "pencil endpoints P(−1,0,1,2) = (2,4,20,68)"
    },
    {
      "id": "v18_quark_yukawa",
      "cluster": "masses",
      "descriptionEN": "quark source ratios; lepton c=(16/7,4/3,7/6); full hierarchy"
    },
    {
      "id": "v20_lepton_c_derivation",
      "cluster": "masses",
      "descriptionEN": "lepton c's derived (δ=½ resolvent; product 32/9)"
    },
    {
      "id": "v24_quark_ratio_closure",
      "cluster": "masses",
      "descriptionEN": "quark ratios 55/117, 34/47, 3/26 (match < 0.03%)"
    },
    {
      "id": "v42_exterior_leg",
      "cluster": "masses",
      "descriptionEN": "Exterior Leg Lemma: c_u/c_d = 55/117 from the Λ²F area"
    },
    {
      "id": "v43_exterior_bridge",
      "cluster": "masses",
      "descriptionEN": "Λ²F bridge: exterior leg = 𝟑̄; discrete H2 invariant"
    },
    {
      "id": "v44_carrier_exterior",
      "cluster": "masses",
      "descriptionEN": "Lie grounding: 16 = Λ^even(5) = 1 + 10 + 5"
    },
    {
      "id": "v45_family_exterior",
      "cluster": "masses",
      "descriptionEN": "the “11” is Pascal: ‖Pl(K)‖₁ = 11 = 16 − g_car"
    },
    {
      "id": "v46_grand_mass_volume",
      "cluster": "masses",
      "descriptionEN": "absolute scaling: det M_SM ∼ (φ₀)^25 = (φ₀)^(g_car²)"
    },
    {
      "id": "v49_readout_rigidity",
      "cluster": "masses",
      "descriptionEN": "Readout Rigidity: c_u/c_d = 55/117 constant on the stratum"
    },
    {
      "id": "v9_neutrino_texture",
      "cluster": "neutrinos",
      "descriptionEN": "μτ Majorana texture → sin²θ₁₂ = 1/3 − φ₀/2, θ₂₃ = 45°"
    },
    {
      "id": "v16_solar_dual_anchor",
      "cluster": "neutrinos",
      "descriptionEN": "aᵀR⁻¹ = aᵀL⁻¹ = (−½,−½,1); solar angle; full PMNS"
    },
    {
      "id": "v21_solar_product_quark",
      "cluster": "neutrinos",
      "descriptionEN": "solar coeff = q(A₃) = 3/4; sin²θ₁₂ = 1/3 − φ₀/2"
    },
    {
      "id": "v7_gravity_cosmo",
      "cluster": "gravity",
      "descriptionEN": "scalaron c₃⁷, A_s, n_s, r, Ω_b, sin²θ₁₂"
    },
    {
      "id": "v28_gravity_fR",
      "cluster": "gravity",
      "descriptionEN": "R + R² closed: f(R), scalaron c₃^(7/2)M̄; n_s, r, A_s"
    },
    {
      "id": "v36_spectral_action_g2",
      "cluster": "gravity",
      "descriptionEN": "G2 spectral action → R + R²; gap-decoupling Δ_eff = 1.648"
    },
    {
      "id": "v60_lambda_metrology_branch",
      "cluster": "gravity",
      "descriptionEN": "Λ branch: (8π)²δ_top = 3/(4π²); G_N pinned; 123 orders"
    },
    {
      "id": "v6_bootstrap",
      "cluster": "horizon",
      "descriptionEN": "reverse glue μ²−5μ+4=0; g_car=5 three ways; 8 = rank E₈ = φ(30)"
    },
    {
      "id": "v8_horizon",
      "cluster": "horizon",
      "descriptionEN": "horizon code 1/(2π)=4c₃, 1920=|W(D₅)|, S_dS, Page, β_rad"
    },
    {
      "id": "v54_seam_horizon_keystones",
      "cluster": "horizon",
      "descriptionEN": "8 triply forced; one transport λ₂=(2/3)⁶ for flavor & horizon"
    },
    {
      "id": "v55_coxeter_cycle",
      "cluster": "horizon",
      "descriptionEN": "E₈ Coxeter order 30 = 2·3·5; 8 = φ(30); S_dS·ρ_Λ = 32π⁴"
    },
    {
      "id": "v56_unique_attractor",
      "cluster": "horizon",
      "descriptionEN": "gapped transport ⇒ unique attractor at rate (2/3)⁶"
    },
    {
      "id": "v57_horizon_crosslinks",
      "cluster": "horizon",
      "descriptionEN": "horizon cross-links: Jacobson, Hod ln 3, 1920"
    },
    {
      "id": "v58_seam_horizon_chain",
      "cluster": "horizon",
      "descriptionEN": "seam = horizon statement; Gauss–Bonnet c₃ = 1/(8π)"
    },
    {
      "id": "v59_area_law_evidence",
      "cluster": "horizon",
      "descriptionEN": "area-law necessary condition met (RP Gaussian kernel)"
    },
    {
      "id": "v61_cft_bridge",
      "cluster": "horizon",
      "descriptionEN": "boundary-CFT mirror: c(E₈)=8, c(D₅)=5, c(A₃)=3"
    },
    {
      "id": "v67_area_law_coefficient",
      "cluster": "horizon",
      "descriptionEN": "Fursaev–Solodukhin S = A/4 ⇔ 2πc₃ = 1/4; c₃ = 1/(8π) unique"
    },
    {
      "id": "v73_k_c3_half",
      "cluster": "horizon",
      "descriptionEN": "central coefficient k = c₃/2 forced (Gauss–Bonnet topology)"
    },
    {
      "id": "v27_wall_representative",
      "cluster": "uwall",
      "descriptionEN": "explicit balanced wall rep W_wall; det R=8, Spec(Q₊)={1,2,3}"
    },
    {
      "id": "v30_d4_character_variety",
      "cluster": "uwall",
      "descriptionEN": "D₄-fixed SU(3) character variety is positive-dimensional"
    },
    {
      "id": "v31_R_dictionary",
      "cluster": "uwall",
      "descriptionEN": "R(ρ): integer R but continuous ρ-invariants"
    },
    {
      "id": "v32_rh_splitting",
      "cluster": "uwall",
      "descriptionEN": "RH route: splitting 𝒪(−2)⊕𝒪(−1)² ⇔ diag(A₀)=(½,¼,¼)"
    },
    {
      "id": "v33_explicit_flat_bundle",
      "cluster": "uwall",
      "descriptionEN": "explicit valid (U_wall) flat bundle (RH solve)"
    },
    {
      "id": "v34_h2_bridge_attempt",
      "cluster": "uwall",
      "descriptionEN": "H2 bridge attempt: honest negative on lepton amplitudes"
    },
    {
      "id": "v38_uwall_killswitch",
      "cluster": "uwall",
      "descriptionEN": "U2 kill-switch: case A generic (400/400 irreducible)"
    },
    {
      "id": "v39_uwall_selectors",
      "cluster": "uwall",
      "descriptionEN": "selectors: splitting type = a, det R = 8, Spec(Q₊)"
    },
    {
      "id": "v40_harmonic_metric",
      "cluster": "uwall",
      "descriptionEN": "harmonic metric = finite linear algebra (polystable ⇒ unitary)"
    },
    {
      "id": "v41_leg_assignment",
      "cluster": "uwall",
      "descriptionEN": "final leg test: amplitudes = δ=½ resolvent; honest negative"
    },
    {
      "id": "v75_upoint_to_vgeo",
      "cluster": "uwall",
      "descriptionEN": "Gate 1 complete: U_point → v_geo (ratios + Grand Mass Volume); same anchor as 1/G"
    },
    {
      "id": "v5_e8_cascade",
      "cluster": "frontier",
      "descriptionEN": "cascade D = 60 − 2n: endpoints, exponent rungs → 240"
    },
    {
      "id": "v13_open_gates",
      "cluster": "frontier",
      "descriptionEN": "gate closures: M = 41 = 10 b₁; Q₊ = A₃ exponents"
    },
    {
      "id": "v17_hexagonal_resolvent",
      "cluster": "frontier",
      "descriptionEN": "finite hexagonal resolvent (the quark c backbone)"
    },
    {
      "id": "v19_monodromy_moduli",
      "cluster": "frontier",
      "descriptionEN": "exact monodromy pole; D₄ SU(3) monodromy moduli"
    },
    {
      "id": "v22_open_gates_audit",
      "cluster": "frontier",
      "descriptionEN": "residual-gates audit: forced part vs named residual"
    },
    {
      "id": "v25_frontier_conjectures",
      "cluster": "frontier",
      "descriptionEN": "conjectures: Koide φ₀/24; axion f_a = M_scal/128"
    },
    {
      "id": "v26_flavor_frontier_unification",
      "cluster": "frontier",
      "descriptionEN": "the “11” not uniquely forced; flavor frontier → one (U) gate"
    },
    {
      "id": "v29_research_contract_certs",
      "cluster": "frontier",
      "descriptionEN": "contract certs: wall enumeration; G5 gap-dominance"
    },
    {
      "id": "v35_quark_qcd_boundary",
      "cluster": "frontier",
      "descriptionEN": "the discrete ↔ continuous boundary; cross-ratio cleanness"
    },
    {
      "id": "v62_data_scorecard",
      "cluster": "frontier",
      "descriptionEN": "TFPT vs 2024/25 data scorecard"
    },
    {
      "id": "v63_seam_engineering_index",
      "cluster": "frontier",
      "descriptionEN": "Seam-Engineering Index Ξ = 2‖V‖/Δ ≈ 0.323; Δ_eff ≈ 1.648"
    },
    {
      "id": "v64_causal_boundary_nogo",
      "cluster": "frontier",
      "descriptionEN": "causal-boundary conditional no-go (RP+OS+gap ⇒ ANEC)"
    },
    {
      "id": "v65_falsification_layer",
      "cluster": "frontier",
      "descriptionEN": "prediction layer with explicit failure thresholds"
    },
    {
      "id": "v68_seeley_dewitt_residual",
      "cluster": "frontier",
      "descriptionEN": "Seeley–DeWitt: absolute 1/G is the one dimensionful anchor"
    },
    {
      "id": "v69_d4_q_geometry",
      "cluster": "frontier",
      "descriptionEN": "D₄-equivariant Q-geometry (gate [P] → [L])"
    },
    {
      "id": "v70_q_integer_lift",
      "cluster": "frontier",
      "descriptionEN": "Q integer-lift: det Q = 3 = N_fam; SNF diag(1,1,3)"
    },
    {
      "id": "v71_simple_r_bridge",
      "cluster": "frontier",
      "descriptionEN": "simple R-bridge: selector stratum fully derived; quark ratios"
    },
    {
      "id": "v72_q_det_from_cusp",
      "cluster": "frontier",
      "descriptionEN": "det Q = N_fam from the cusp class (triality deck group)"
    },
    {
      "id": "v74_compiler_micro_lemmas",
      "cluster": "frontier",
      "descriptionEN": "spine quotient ladder; pencil differences 2→16→48; anchor QF 41−25=16; solar dual anchor"
    },
    {
      "id": "v76_gmetric_reduction",
      "cluster": "frontier",
      "descriptionEN": "Gate 2: decoupling Δ_eff=1.648>0; G6 holographically reduced to a seam-boundary measure"
    },
    {
      "id": "v77_e8_conformal_net",
      "cluster": "frontier",
      "descriptionEN": "G6 route: seam boundary = (E₈)₁ lattice net (rigorous); c(E₈)=8, c(D₅)=5, c(A₃)=3, embedding 5+3=8"
    },
    {
      "id": "v78_vgeo_floor",
      "cluster": "frontier",
      "descriptionEN": "v_geo = dimensional-analysis floor: one scale + π; S_dS·ρ_Λ=32π⁴ pins it from one measurement"
    },
    {
      "id": "v79_review_identities",
      "cluster": "frontier",
      "descriptionEN": "hypercharge Lucas–Binet (1,1,7,13,55,133); Inverse Anchor Theorem (a^T M⁻¹ a=1); MacWilliams (1,10,5); + firewall on m_p/m_e, η_B"
    },
    {
      "id": "v80_operator_pencil_geometry",
      "cluster": "frontier",
      "descriptionEN": "operator pencil: anchor singularity det B(K+xQ)=(3x+2)(3x+5); block-det type checker (9,10,16,40); F₄×G₂ shadow 52+14+26·7=248"
    },
    {
      "id": "v81_singular_pencil_matrices",
      "cluster": "frontier",
      "descriptionEN": "double cover y²=det B(K+xQ): Koide −2/3 & carrier −5/3 are the two branch points (deck 2=|ℤ₂|, disc=81=N_fam⁴); clearing matrices 3K−2Q (D₅⊕A₃ glue, sum=240) & 3K−5Q (charge-neutral)"
    },
    {
      "id": "v82_koide_attractor_splitting",
      "cluster": "frontier",
      "descriptionEN": "Koide attractor FORCED, not postulated: the branch-preserving map fixing q=2,5 is unique, and its multiplier (2/3)⁶ = the established transfer gap λ₂ (v54/56) ⇒ 3 Koide postulates → 1; + splitting trichotomy disc 81/49/40 (clean split is non-generic)"
    },
    {
      "id": "v83_e8net_holomorphic_uniqueness",
      "cluster": "frontier",
      "descriptionEN": "closes red-team Target A residual 1: #primaries=det(Cartan) ⇒ holomorphy (E8 det 1) is necessary AND sufficient — (D8)₁=SO(16)₁ shares c=8 but has 4 primaries; E8 the unique even-unimodular rank-8 lattice (mass=1/|W(E8)|). A drops 3→2 residuals; + Target B reduced to the half-spinor midpoint K=(g−1)/2"
    },
    {
      "id": "v84_frozen_registry",
      "cluster": "registry",
      "descriptionEN": "blind-prediction registry FROZEN 2026-06-09: every dimensionless prediction of record at 25 digits, re-derived from the two axioms each run (formula↔value lock); exactly ONE θ₁₂ prediction of record = 1/3−φ₀/2 = 0.306747, variants typed as derived (look-elsewhere machine-excluded); r/n_s only as N★ bands"
    },
    {
      "id": "v85_master_cover",
      "cluster": "registry",
      "descriptionEN": "Master-Cover Theorem: GL(2) covariance ⇒ exactly ONE anchor-block double cover up to Möbius reparametrisation (disc = N_fam⁴·det G²); the −8/3 = −rank E₈/N_fam rung = carrier − one transport period; μ₄ is NOT a 4:1 cover (honest negative); branch trace fixes only the scalaron scale"
    },
    {
      "id": "v86_nstar_reheating",
      "cluster": "registry",
      "descriptionEN": "N★ from the theory's own scalaron mass + standard reheating: Γ=128 GeV, T_reh=9.6×10⁹ GeV, N★(0.05/Mpc)=51.4 ⇒ n_s=0.9611, r=0.0045 [P] — recorded honestly: A_s coherence disfavours the slow Higgs channel (−11.4σ); the frozen band [50,60] stays the surface of record"
    },
    {
      "id": "v87_bulk_uniqueness_reduction",
      "cluster": "registry",
      "descriptionEN": "red-team Target A = ONE residual: holomorphy ⇒ unique 2D bulk (LR/KLM/BKLR); machine contrast: SO(16)₁ admits SIX modular invariants (incl. both E₈-extension pairings) ⇒ non-holomorphic c=8 bulk is multiply ambiguous"
    },
    {
      "id": "v88_cp_phase_audit",
      "cluster": "registry",
      "descriptionEN": "Target D follow-up: frozen δ=π/3+3λ² survives at +0.98σ vs γ_PDG; audit (not promoted): data sit 0.07° from the alternative π/3+2λ² — decision at σ_γ ≤ 0.96° (LHCb/Belle II); J-inversion flagged magnitude-contaminated"
    },
    {
      "id": "v89_carrier_index_lemma",
      "cluster": "registry",
      "descriptionEN": "Carrier Index Lemma: KLM μ_A=[B:A]²μ_B ⇒ Jones index [(E₈)₁:(D₅)₁×(A₃)₁] = 4 = |μ₄| — the glue-group order IS the inclusion index; all three glue sectors are h=1 currents (248=45+15+64+64+60); holomorphy FOLLOWS from μ-additivity 16/4²=1 ⇒ Gate A ⇔ an index computation"
    },
    {
      "id": "v90_conical_defect_chain",
      "cluster": "registry",
      "descriptionEN": "Fursaev–Solodukhin factor DERIVED, not imported: smoothed-cone Gauss–Bonnet ∫K = 2π(1−α) exactly (smoothing-independent), codim-2 lift 4π(1−α)A, replica ⇒ S = 4πkA; with k=c₃/2: S = A/4 ⇔ c₃ = 1/(8π) (sympy-solved, unique). The Seam–Horizon residual is isolated to one step: seam determinant ⇒ EH form"
    },
    {
      "id": "v91_spine_tetrahedron",
      "cluster": "registry",
      "descriptionEN": "Spine tetrahedron {2,3,4,5} = {e₃(a), p₀(a), e₁(a), e₂(a)}: edges {6,8,10,12,15,20}, faces {24,30,40,60}, volume 120 = |R⁺(E₈)| = 5!; 240 = |μ₄|·|E(K₄)|·|E(K₅)| with K₆ negative control; dual cuts typed as tautological presentation; sub-grammar honestly incomplete (7,16,41,48,240,248 outside)"
    },
    {
      "id": "v92_glue_uniqueness",
      "cluster": "registry",
      "descriptionEN": "Glue uniqueness: exhaustive classification of the carrier discriminant form (Z₄×Z₄, q=(5x²+3y²)/8) — exactly two Lagrangian glues (the two chiralities, swapped by spinor conjugation = sheet Z₂) and exactly one halfway extension whose induced form IS the D₈ form: tower carrier(μ=16) → SO(16)₁(μ=4) → E₈₁(μ=1), nothing else; Gate A = the bare index statement. Now also Lean-formalised (GlueUniqueness.lean, AUDIT: PASS)"
    },
    {
      "id": "v93_koide_relaxation_toy",
      "cluster": "registry",
      "descriptionEN": "Koide relaxation toy (P2 narrowed): basin lemma — every physical Koide configuration lies in the attractor basin (Q∈[1/3,1] → q∈[1,3]: attractor in, repeller out); exact contraction rate (2/3)⁶ along the physical trajectory; source at ρ = −φ₀/24 (one seed quantum before the branch point); honest negatives: pole ≠ integer F-steps from source (t=2.84 ⇒ continuous generator missing), φ₀/24 stays [P]"
    },
    {
      "id": "v94_sheet_diamond",
      "cluster": "registry",
      "descriptionEN": "Sheet diamond: R, K, L, F are four points of ONE surface M(s,t) = R+Q·diag(s,t,t) (mass pencil = the cut (1+x,x−1); diagonal cut reproduces v85's non-square disc 153 — built-in negative control); winding line det B = 2·det for ALL s with the s=6 triple lock; cofactor normal (5,−9,6) selects generation 1; quark-ratio integers canonicalised as left/right Plücker norms of K (11, 26); reality threshold corrected to s*≈2.83; spine-quotient firewall rejected (16/7, 7/6, 5/6, 8/7 counterexamples)"
    },
    {
      "id": "v95_centered_diamond",
      "cluster": "registry",
      "descriptionEN": "Centered flavor diamond: one center + two axes — Q = U+V, R/L = C∓U (winding = pure family charge, Spec U = {3,0,0}), K/F = C∓V (sheet, Spec V = {0,1,2} = the cusp class); center: det C = 14, SNF (1,1,14), ΣC = 31 = 2^g−1 (the IR gap-bound numerator), Pl_R(C) = 7·(2,3,1) on the same ray as Pl_R(L) = 10·(2,3,1); the G₂ reading stays audit-typed, not promoted"
    },
    {
      "id": "v96_branch_kernel_selection",
      "cluster": "registry",
      "descriptionEN": "Branch-kernel selection (P1 unblocked): at each branch point the anchor block is RANK 1 with canonical integer kernels (Koide: w=(2,2,−7), v=(2,2,−3); carrier: w = the democratic vector 1, v=(1,1,−6)); collapse lemma — rank 1 forces P(x₀)·w ⊥ span{1,a} ⇒ ∝(−1,1,0): up = −down (the deck-odd pair), LEPTON pairing = 0 (leptons sit ON the ramification; Koide is leptonic); sector zero ladder: lepton zeros = the two branch points, up = −3/2 doubly (the v85 GL(2) rung), down = 0 (=K) and −9/5; anchor-placement controls (2,1,1)/(1,2,1)"
    },
    {
      "id": "v97_sheet_conjugation_bridge",
      "cluster": "registry",
      "descriptionEN": "Sheet-conjugation bridge (P1 → one [P]): the cusp negation (0 fixed, 1/3↔2/3 on the Q₊ eigenspaces) has a UNIQUE anchor-compatible integer realisation T_A=[[0,1,0],[1,0,0],[2,−2,1]] (det −1; the two alternative pairings rejected); a = e₂+e₃ — the anchor IS the conjugation-symmetric vector; T_A = −1 on R³/span{1,a} exactly like σ₁₂ (one deck action); ⟨T_A,Σ⟩ = the v69 D₄; sheet-index lemma: generation-μ₄ vs homology-μ₄ intertwiner dets always EVEN, min |det| = 2 = |Z₂|; remaining [P]: Q₊ grading = A₃ discriminant grading"
    },
    {
      "id": "v98_discriminant_dictionary",
      "cluster": "registry",
      "descriptionEN": "Discriminant dictionary DERIVED (P1 closed modulo GATE.QGEO): the generation space carries an INTEGER μ₄ — G = T_A·Σ acts on the cusp basis as G·e₁=−e₁, G·e₃=e₂, G·e₂=−e₃ (the v69 B₁⊕E decomposition realised integrally); the cusp-0 line IS the self-conjugate Z₄-class-2 line, the swapped cusp pair IS the conjugate class pair {1,3}; T_A G T_A⁻¹ = G⁻¹ (conjugation k→−k), q(1)=q(3)=3/8 equal, q(2)=1/2 fixed; reflection classes = glue-swap (det −1) vs glue-fix (det +1) parities ⇒ P1 carries NO separate [P] — the residual folds into the one existing Q-geometry gate"
    },
    {
      "id": "v99_koide_flow_time",
      "cluster": "registry",
      "descriptionEN": "Koide flow time (P2 sharpened; corrects the ROBUSTNESS of v93's negative (i)): canonical generator dq/dt = (Δ/N_fam)(q−2)(q−5) = (Δ/N_fam)·det B(q), time-1 map = the v82 Möbius F (e^−Δ = (2/3)⁶ exact) [I]; the non-integrality of t = 2.84 is DATA-FRAGILE — Q crosses 2/3 at +0.43σ(m_τ), t passes every integer ≥ 3 within +0.5σ [N]; conditional steps (registry untouched): n=2 excluded (−2.9σ), n=3 = N_fam steps ⇒ m_τ = 1776.9427 MeV (+0.14σ), decision at σ(m_τ) ~ 0.01 MeV (Belle II) [P]; ln(46080) scale reading destroyed by negative controls (look-elsewhere firewall)"
    },
    {
      "id": "v100_numerology_null_mc",
      "cluster": "registry",
      "descriptionEN": "Numerology null test (look-elsewhere corrected) [N]: exact census of the FULL declared formula grammar (provably containing every scored TFPT formula) over the 13 scored frozen-registry observables with conservative data windows ⇒ joint formula-fishing probability ∏pᵢ = 10⁻²⁵·⁸, robust to budget/window variation; 2×200k Monte-Carlo pseudo-theories reach at most 5/13 hits vs TFPT 13/13; power controls: φ₀ ±1%/10% collapses TFPT's own scorecard to 9/6, data shuffle to mean 1.2; all 94 500 F_U(1) variants root-solved — exactly ONE hits the 4×10⁻⁸ CODATA window ⇒ total ≤ 10⁻³⁰·⁷ (~102 bits), CONDITIONAL on the declared grammar (a null-model rejection, never 'certainty'); complementary to the v84 freeze"
    },
    {
      "id": "v101_horizon_anchor",
      "cluster": "registry",
      "descriptionEN": "The maximal black hole is the anchor (Schwarzschild–de Sitter in seam units): the Nariai horizon cubic t³−3t+2 = (t−1)²(t+2) has roots (1,1,−2) = the TRACELESS ANCHOR a−(p₁/3)·1; the Koide 2/3 = |ℤ₂|/N_fam is the exact Nariai/de-Sitter entropy bound (per horizon 1/3); interpolation (x²+1)/Φ₃(x) (the N_fam cyclotomic); three-sheet entropy total |ℤ₂|·S_dS conserved for every M; Koide-form root quotient ranges [3/8, 1/2] = the nonzero SU(4)₁ weights; the mass line is a split double cover (branch ±1/N_fam, deck = horizon swap); evaporation flows away from the anchor point (repeller/attractor orientation as in flavor, v82); seam-unit table: dM = c₃κdA, Smarr 2c₃, P_H = c₃/(1920M²), τ = 128·g_car·M³/c₃, Kerr A_ext = M²/c₃; six atom landings, zero free parameters [I] + bulk reading [P]"
    },
    {
      "id": "v102_seam_orientation",
      "cluster": "registry",
      "descriptionEN": "One orientation: the anchor is the STATIONARY REPELLER in both sectors — flavor: the canonical flow dq/dt = (Δ/N_fam)(q−2)(q−5) is the gradient flow of a cubic potential whose critical points are exactly the two branch points, with stationary curvatures V''(Koide) = +Δ (attractor) and V''(carrier) = −Δ (repeller) = ±the transfer gap; inflection at q = 7/2 = scalaron/2; Lyapunov rate d(−ln ρ)/dt = Δ exactly constant; gravity: d(S_tot/S_dS)/dx = (x−1)(x+1)/Φ₃(x)² — Nariai is the UNIQUE stationary point of the SdS entropy, curvature 2/9 = |ℤ₂|/N_fam² = (2/3)(1/3); evaporation = entropy ascent away from the anchor [I]; honest disanalogies recorded; 'one variational principle of the seam' stays a reading [P]"
    },
    {
      "id": "v103_trisection_normal_form",
      "cluster": "registry",
      "descriptionEN": "Trisection normal form (the canonical coordinate exists): the SdS horizon cubic is uniformized by ANGLE TRISECTION — r = 2cos θ gives r³−3r = 2cos 3θ exactly, so cubic ⇔ cos 3θ = −3m (ℤ₃ trisection deck = the triality of coker Q = ℤ/N_fam); centered angle ψ: m = cos(ψ)/N_fam and S_tot/S_dS = 4/3 − (2/3)cos(2ψ/3) — ONE cosine whose mean (|μ₄|/N_fam), amplitude and frequency (|ℤ₂|/N_fam) are glue atoms; canonical curvature at the anchor = (2/3)³ = (|ℤ₂|/N_fam)^N_fam — the Koide constant to the family power; invariant base slope dσ/dm = −8/9 = −rank E₈/N_fam² (cross-checked in both coordinates); dimensionful dS/dM = −r_N/(N_fam·c₃); bridge: flavor rate (2/3)^{2N_fam} vs gravity curvature (2/3)^{N_fam} — same base, exponent ratio |ℤ₂|; the missing piece is ONE clock (near-Nariai evaporation generator) [I] + [P]"
    },
    {
      "id": "v104_nariai_clock",
      "cluster": "registry",
      "descriptionEN": "The CLASSICAL Nariai clock is the anchor (third appearance) + the honest (2/3)-test: static pin — φ(ρ) = ρ solves the dS₂ static-patch equation with m² = −2Λ = −|ℤ₂|Λ exactly (the exact SdS family itself pins the modulus mass; Laplace-type linearization = standard GR, Ginsparg–Perry 1983 / Bousso–Hawking 1997, typed external); Ginsparg–Perry tower (l(l+1)−2)Λ with exactly ONE negative mode; THE CLOCK in Hubble units: χ_clock(λ) = λ²+λ−2 = (λ−1)(λ+2) = the anchor quadratic, eigenvalues {1,−2} = the distinct anchor roots; Nariai cubic = (t−1)·χ_clock — anchor = configuration roots (v101) + curvature base (v103) + clock spectrum; entropy-deviation rate 2H = |ℤ₂|·Hubble exactly; HONEST (2/3)-TEST NEGATIVE for the classical clock (integer eigenvalues, not 2/3-powers) — the quantum clock (one-loop curvature→rate) is the one remaining [P] of the seam variational principle"
    },
    {
      "id": "v105_residual_inventory",
      "cluster": "registry",
      "descriptionEN": "THE RESIDUAL INVENTORY — one constant, one anchor, one clock to find, and the COMPLETE machine-pinned gap list: (A) 2/3 = |ℤ₂|/N_fam appears EXACTLY in seven independent places across both sectors (Koide branch point, gap base (2/3)⁶, basin attractor; Nariai bound, branch separation, canonical amplitude+frequency, curvature base (2/3)^N_fam) + the anchor triptych (configuration / Nariai roots / clock spectrum); (B) RELOCATION THEOREM: the cosmological one-loop clock correction ε = (c/24π)·Λ/M̄² = 7.6×10⁻¹²² (c = 8) is deficient by 121.5 orders — the deficit IS the Λ hierarchy 2α⁻¹/ln10 = 119.0 ⇒ the clock must live at the seam scale: ONE [P] identification 'transfer operator = seam Nariai clock'; (C) RESIDUAL TABLE: exactly FIVE structural objects remain (clock, holomorphy+c8, seam-det⇒EH, H2 dictionary, Q-realisation) + one scale + π — a sixth structural gap would FAIL this script"
    },
    {
      "id": "v106_review_validation",
      "cluster": "registry",
      "descriptionEN": "External-review validation (v79-style identities + disciplined typing): SEED NORMAL FORM — φ₀ = (|μ₄|/N_fam)·c₃ + Ω_adm·c₃^|μ₄| exactly (linear seam term + topological 4th-order correction; 4/3 = |μ₄|/N_fam, 48 = Ω_adm, exponent = |μ₄|); hypercharge second moment computed from the explicit charge table: Tr X = 0, Tr X² = 120 = 5! (X = 6Y, 16 states); FACTORIAL SPINE: 5! = 120 = |R⁺(E₈)| = Σ E₈ exponents = Tr X², 240 = 2·120, 1920 = 2⁴·5! = |W(D₅)| — one moment, four projections; DEGREE-2 INVENTORY: Pascal K=2 closure UNIQUE at g=5 (scan 1..40), glue norms 5/4+3/4 = 2 = root norm, c₃ = ½·1/(4π), pair sector C(5,2) = 10; NAMED HYPOTHESIS [P]: Quadratic Boundary Locality (seam supports only bilinear data ⇒ K=2 forced — would upgrade the Pascal selection [A]/[P] → [L]: locality ⇒ K=2 ⇒ g=5 ⇒ 16 ⇒ D₅ ⇒ E₈); audit: 240 = 16×15 reading exact but non-unique, not promoted"
    },
    {
      "id": "v107_quantum_clock_target",
      "cluster": "registry",
      "descriptionEN": "Quantum-clock target made quantitative (the R1 programme, first computation): the classical Ginsparg–Perry clock has per-l equations λ²+λ+(l(l+1)−2) = 0 — l=0 anchor pair {1,−2}, l=1 zero modes {0,−1}, l≥2 complex with Re = −1/2 = −δ (audit); NEW EXACT CROSS-LINK: the classical decay set {0,−1,−2} = −Spec(V) = −N_fam × cusp weights — the classical seam clock decays on the transfer operator's OWN grading (v69/v95); EXACT TARGET: transfer rates are not weight-linear — rate(2)/rate(1) = log_{3/2}3 = 2.7095 (not 2), with the exact identity (1/3)⁶ = ((2/3)⁶)^{log_{3/2}3}: the quantum clock must bend the spectrum by log_{9/4}3 = 1.35476 per weight step; coupling landscape κ = (c/24π)(Λ/M̄²): 10⁻¹²² cosmological vs 1/(3π) = 0.106 at seam curvature — the only viable, borderline-nonperturbative regime (one-loop linear response cannot fix the bend); nothing fitted, identification stays [P]"
    },
    {
      "id": "v108_pascal_ladder",
      "cluster": "registry",
      "descriptionEN": "PASCAL LADDER THEOREM (red-team Target B sharpened to zero slack): 2^{g−1} = Σ_{k≤K} C(g,k) ⟺ g = 2K+1 for every K (odd-g midpoint identity by binomial symmetry; even-g excluded by the strict straddle around 2^{g−1}); EXACT EQUIVALENCE: carrier selection g=5 ⟺ truncation degree K=2 — the Pascal-selection residual IS the one named hypothesis Quadratic Boundary Locality (v106); neighbour worlds: K=1 → one-family world (g=3), K=3 → nine-family world (g=7), K=4 → inconsistent (255/9 ∉ ℤ); OVERDETERMINATION: only K=2 also satisfies the independent rank closure g+N_fam = 8 = rank E₈ (v14) — two independent selections agree on the same world; QBL precisely stated, typed [P]; now partially Lean-formalised (AnchorLadder.lean, PascalLadder.lean, AUDIT: PASS)"
    },
    {
      "id": "v109_sheet_pairing",
      "cluster": "registry",
      "descriptionEN": "SHEET-PAIRING LEMMA (the first exact rep-theoretic anchor of QBL; ties Target B to the sheet ℤ₂): character-exact weight multisets from the D₅ spinor weights (±½)⁵ alone — NO SCALAR WITHIN A SHEET (zero-weight multiplicity of S⁺⊗S⁺ = 0: the odd slot count g=5 flips the sign parity of −w, a parity theorem of the five-slot code); THE SCALAR LIVES ACROSS THE SHEETS: S⁺⊗S⁻ = Λ⁰⊕Λ²⊕Λ⁴ exactly (256 = 1+45+210) with zero-mode grading (1,5,10) = the carrier code; SHEET-DIAGONAL = ODD FORMS: (S⁺⊗S⁺)∪(S⁻⊗S⁻) = 2Λ¹+2Λ³+Λ⁵ exactly (never scalar); the scalar-bearing tower tops at Λ^{2K} with K=2 — bilinear seam data reach exactly the Pascal-ladder degree; QBL reading [P]: the seam's single scalar kernel is necessarily a SHEET PAIRING (chirality-off-diagonal — the same ℤ₂ as the glue chiralities and branch kernels); remaining step: show the Calderón kernel supplies exactly this pairing"
    },
    {
      "id": "v110_calderon_sheet",
      "cluster": "registry",
      "descriptionEN": "CALDERÓN-SHEET SELECTION THEOREM (the v109 follow-up): the certified bilinear data of a Calderón involution ε (ε²=1) on H = S⁺⊕S⁻ are ⟨f, εg⟩; with the v109 singlet counts per sheet block (0,0,1,1) the exact statement is 'a scalar two-point datum exists ⟺ ε is sheet-odd' — sheet-odd certifies exactly 1+1 = 2 = |ℤ₂| invariant kernels (one per orientation), sheet-even certifies NONE; THE TWO KERNELS = THE GLUE AMBIGUITY: 2 = |ℤ₂| = the two Lagrangian glues (v92) — the Calderón scalar channel carries exactly the glue ambiguity, resolved by the same sheet choice; LADDER GENERICITY (anti-overclaim): for g = 3,5,7 sheet-oddness forces the half-spinor relation K = (g−1)/2, NOT g=5 — the g-selection stays with rank-8/integrality (v14/v108); QBL residue split into two named parts [P]: (a) interface 'the seam Calderón involution is sheet-odd' (natural: the one-sided collar / double-cover deck is the SAME ℤ₂ that halves c₃ = 1/(2·4π)), (b) analytic core 'the 2-point kernel certifies only slot-degrees ≤ 2'"
    },
    {
      "id": "v111_quadratic_transport",
      "cluster": "registry",
      "descriptionEN": "QUADRATIC-TRANSPORT THEOREM (the transport half of the QBL analytic core becomes a theorem): grade seam transport by Clifford degree in the 10 field operators on the 5-slot Fock space (integer Jordan–Wigner model; spanning claims via mod-p full-rank certificates, valid over ℚ) — PARITY: all 10 linear words are sheet-odd, all 45 quadratic words sheet-even (code-preserving transport has even Clifford degree, the same ℤ₂ as v109/v110); MINIMALITY: sheet-even words of degree ≤ 1 are the scalars alone; GENERATION: the 45 quadratics (= dim so(10) = the Λ² term of the certified tower 1+45+210) generate the whole code from the vacuum at word length ≤ 2; COMPLETENESS (Burnside realised): products of length ≤ 2 span the FULL End(S⁺) = 256 — every code operation is a word in pair transport, all higher even-degree elements redundant ⇒ THE TRANSPORT DEGREE IS EXACTLY 2 (degree ≤ 1 generates nothing, degree 2 generates everything — a theorem, not a hypothesis); g=3 control: degree selected, not rank; QBL residue reduced to two interface statements: (a) seam Calderón involution sheet-odd, (b') certified state inventory = the ≤2-slot tower"
    },
    {
      "id": "v112_selfcounting_channel",
      "cluster": "registry",
      "descriptionEN": "SELF-COUNTING CHANNEL (the Pascal closure is an IDENTITY of the certified channel, not an extra requirement): CANONICAL BIJECTION — for odd g negation w ↦ −w maps S⁺ bijectively into the opposite sheet, so the Cartan-neutral kernels of the certified channel are exactly the pairs (w,−w), ONE PER CODE STATE: #neutral kernels = dim S⁺ = 2^{g−1} exactly (4, 16, 64 for g = 3,5,7) — the channel counts the code; PASCAL PARTITION: the same neutral set graded by pair-degree m has sizes C(g,m) — g=5: (1,5,10); THE CLOSURE IS AN IDENTITY: two countings of one set give 2^{g−1} = Σ_{m≤K} C(g,m) (symbolic, all odd g=3..13) — v108's closure condition is the channel counting itself; HODGE FOLD: code minus-grading C(g,2j) = C(g,min(2j,g−2j)) with min ≤ K — every code sector appears at pair-degree ≤ K; carries NO g-selection (stays with rank-8 g+N_fam=8 + integrality); RESIDUE RE-TYPE [P]: QBL clause (b') is now the channel's definition — the remaining physical input is 'the seam certifies through a single scalar 2-point kernel' (free c=8 seam net: Wick extends it to the whole even tower); QBL = structural consistency frame, no longer a missing selector"
    },
    {
      "id": "v113_quasifree_kernel",
      "cluster": "registry",
      "descriptionEN": "ONE KERNEL IS THE WHOLE NET (the QBL input merges with the R2/holomorphy premise): MAJORANA BOOKKEEPING — c(D₅)₁ = 45/9 = 5 = g_car, c(A₃)₁ = 15/5 = 3 = N_fam; the carrier net (D₅)₁×(A₃)₁ = SO(10)₁×SO(6)₁ = 10+6 = 16 FREE MAJORANA FERMIONS, c = 16/2 = 8 — the whole extension tower carrier(μ=16) → SO(16)₁(μ=4) → E8₁(μ=1) carries the SAME 16 fermions, only the certified glue grows (index 2×2 = 4 = |μ₄|); WICK/PFAFFIAN exact: in the 10-Majorana Jordan–Wigner model ALL 210 vacuum 4-point AND all 210 6-point functions equal the Pfaffian of the single 2-point kernel — one kernel determines all correlations; THE KERNEL IS A CALDERÓN INVOLUTION OF RANK g: M = I+iA, A²=−I, P = M/2 projection with rank 5 = g_car, at seam level rank 8 = rank E8 ⇒ THE CENTRAL CHARGE IS THE RANK OF THE ONE KERNEL; vacuum unique (joint annihilator kernel 1-dim); PREMISE MERGE [P]: 'single 2-point kernel' = 'quasi-free seam state' = the defining property of the free-fermion net the holomorphy gate already posits — QBL adds NO assumption beyond R2; upgrade contract: R2 closes ⇒ carrier choice closes [L]"
    },
    {
      "id": "v114_torsion_delta",
      "cluster": "registry",
      "descriptionEN": "TORSION NORMAL FORM + THE δ = 1/2 THEOREM (executes the v41 'genericity future work'; δ = 1/2 stops being an observation): FLATNESS = μ₄ TORSION [I, symbolic generic M] — ∏ₖ Uᵏ M U⁻ᵏ = (MU)⁴ exactly, so the ℤ₄-family flatness of the (U_wall) bundle IS the torsion statement 'T = MU is a fourth root of unity' (the μ₄ atom = the ORDER of the twisted cusp generator); δ THEOREM [I, both directions]: on the involutive branch (T² = 1) T is a Hermitian reflection 2vv†−1 and the cusp trace splits exactly into |v₁|² = ½ + |v₂| = |v₃| ⇒ diag M = (0, i/2, −i/2) with spec M = {1, ω, ω²} AUTOMATIC — δ = ½ exact and constant on the whole 2-parameter branch (the value v20 used by hand is a μ₄-torsion identity); REFLECTION LEMMA: D₄ forces diag M = (r, z, z̄), r real — ONE real parameter, δ = ½ ⟺ r = 0; BRANCH CENSUS [N, seeded]: only tr(MU) ∈ {1,−1} realised; involutive branch diagonal constant to 1e−15; the {1,i,−i} branch varies with the explicit v33/v40 point in its d₁ = 0 slice; residue [P]: bundle-side branch selection (anchor splitting), δ itself closed"
    },
    {
      "id": "v115_anchor_residue",
      "cluster": "registry",
      "descriptionEN": "THE ANCHOR PINS THE RESIDUE (exact normal form for the U_wall flat-bundle residue; answers the v114 branch question): μ₄-AVERAGE LEMMA [I] — Σₖ Uᵏ X U⁻ᵏ = 4 diag(X): μ₄ conjugation-averaging IS the diagonal projection ⇒ exponents at infinity = 4 diag(A₀) ⇒ anchor splitting 𝒪(−2)⊕𝒪(−1)² ⟺ diag A₀ = (2,1,1)/4 = ANCHOR/|μ₄| (v33's diagonal was forced, not chosen); THE (8,0,5)/144 LEMMA [I]: anchor diagonal + cusp spectrum pin Σ|off|² = 13/144, and a₁₃ = 0 + det = 0 solve UNIQUELY to (|a₁₂|², |a₂₃|²) = (8,5)/144 — numerators (rank E8, g_car), denominator (|μ₄|N_fam)², total 13 = Δ_Q (the quark denominator 117 = 9·13); EXACT NORMAL FORM [I]: A₀* = [[1/2, √2/6, 0],[√2/6, 1/4, √5/12],[0, √5/12, 1/4]] with char poly λ(λ−⅓)(λ−⅔) exactly; A₀* IS THE RIEMANN–HILBERT SOLUTION [N]: big-circle monodromy trivial to 1e−10, unitarised |diag M̃₀| = (0,½,½) — the v33/v40 numerical point is gauge-equivalent to this exact matrix; ANCHOR FORCES d₁ = 0 [N]: every flat solution in a multi-seed RH scan lands on the same gauge orbit ⇒ the harmonic diagonal is anchor+torsion forced, δ = ½ needs no selection; residue [P]: Gröbner promotion of a₁₃ = 0 + uniqueness"
    },
    {
      "id": "v116_resonance_uniqueness",
      "cluster": "registry",
      "descriptionEN": "RESONANCE THEOREM (the v115 [N] findings promoted to [I] by a LINEAR computation — the anticipated Gröbner step was unnecessary): TWISTED-AVERAGE LEMMA [I] — at infinity the equivariant system reads dY/dw = −(1/w)(B₀ + B₁w + …)Y with B_m = Σₖ i^{km} Uᵏ A₀ U⁻ᵏ; the twisted μ₄ averages collapse to B₀ = 4 diag(A₀) and B₁ = 4a₁₂E₁₂ + 4ā₁₃E₃₁ (only the eigenvalue-ratio −i cells survive); RESONANCE THEOREM [I]: the anchor exponents diag(2,1,1) have resonance gap 1, the level-1 gauge equation is singular exactly on cells {(2,1),(3,1)} where B₁ = (0, 4ā₁₃), higher levels non-resonant, formal monodromy = 1 ⇒ M∞ = 1 ⟺ a₁₃ = 0 — the transcendental Riemann–Hilbert condition collapses to ONE linear equation; UNIQUENESS COROLLARY [I]: a₁₃ = 0 + the (8,0,5)/144 moduli + diagonal gauge ⇒ THE FLAT ANCHOR LOCUS IS EXACTLY ONE GAUGE ORBIT = the exact A₀* — within the μ₄-equivariant anchor class the U_wall bundle datum is ALGEBRAICALLY unique; FALSIFICATION CONTROL [N]: ‖M∞ − 1‖ ≈ 8.5|a₁₃| (sharp), 1.6e−10 at A₀*; honest scope: residue side [I], holonomy values (harmonic diagonal) remain [N]"
    },
    {
      "id": "v117_monodromy_weyl_a3",
      "cluster": "registry",
      "descriptionEN": "THE MONODROMY IS THE WEYL GROUP OF A₃ (the U_wall holonomy is an EXACT 24-element representation; the harmonic diagonal becomes a theorem): THE EXACT MONODROMY [I] — M̃₀ = [[0, −(1+i)/2, (1−i)/2], [−(1+i)/2, −i/2, −1/2], [(1−i)/2, −1/2, i/2]] (entries in ½ℤ[i]) is unitary, det 1, tr 0, char poly λ³−1 (cusp class EXACT), M̃₀³ = 1, diag M̃₀ = (0, −i/2, +i/2) ⇒ d₁ = 0 AND δ = ½ are exact matrix entries — the v20 hand value, the v40/v41 observation, the v114 torsion value and the v115 anchor slice meet in ONE exact matrix; TORSION/FLATNESS [I]: (M̃₀U)⁴ = 1, tr(M̃₀U) = 1, ∏ₖ Uᵏ M̃₀ U⁻ᵏ = 1 exactly; THE GROUP IS W(A₃) = S₄ [I]: exact enumeration of ⟨U, M̃₀⟩ gives order 24, order statistics (1,9,8,6), characters (3,−1,0,1) — uniquely S₄ in standard⊗sign = the (twisted) reflection rep of the WEYL GROUP OF THE FAMILY LATTICE A₃: the flavor wall realises the A₃ glue factor as its Weyl group (U = 4-cycle/μ₄ deck, M̃₀ = 3-cycle/family rotation, 24 = |W(A₃)| = 4!); IDENTIFICATION [N]: ODE monodromy of the exact A₀* system = exact M̃₀ to 3.5e−10, H U-invariant to 2.5e−11; the δ thread v20 → v40/v41 → v114 → v115 → v117 is closed end to end"
    },
    {
      "id": "v118_hexagon_family_dictionary",
      "cluster": "registry",
      "descriptionEN": "THE HEXAGON IS THE SIGN-TWISTED FAMILY SPECTRUM (first exact piece of the H2 dictionary; the v20 lepton coefficients are resolvent determinants of the exact W(A₃) monodromy): SIGN-TWIST LEMMA [I] — −M̃₀ has order 6 ((−M̃₀)³ = −1) and spec(M̃₀) ∪ spec(−M̃₀) = μ₆: the 6-site hypercharge hexagon spectrum IS the family monodromy spectrum plus its sheet twist (ℤ₆ = ℤ₂×ℤ₃ = (−1)×⟨M̃₀⟩); the v20 denominators 5/4 − cos(rπ/3) = |1 − ζ₆ʳ/2|² are the eigen-denominators of the two family resolvents (1 ∓ M̃₀/2)⁻¹; CYCLOTOMIC DETERMINANT [I]: det(1 − tM̃₀) = 1 − t³ = (1−t)Φ₃(t) ⇒ det(1 ∓ M̃₀/2) = (7/8, 9/8); LEPTON COEFFICIENTS = RESOLVENT DETERMINANTS [I]: c_e = |μ₄|/(2det₋) = 16/7, c_μ = N_fam/(2det₊) = 4/3, c_τ = |μ₄|det₋/N_fam = 7/6, product rule = |μ₄|/det₊ = 32/9 — the lepton 7 and 9 ARE the family-resolvent determinants (×8), and δ = ½ = |(M̃₀)₂₂| is supplied by the matrix itself; the e/μ/τ leg assignment stays v20's established input (nothing re-fished); SHEET-EXTENDED GROUP [I]: ⟨U, M̃₀, −1⟩ has order 48 = Ω_adm = N_fam·dim S⁺ (the seed quartic coefficient); residue [P]: the dictionary's VALUES are exact, its ADDRESS TABLE (site per fermion, quark composition) remains open"
    },
    {
      "id": "v119_review_validation_2",
      "cluster": "registry",
      "descriptionEN": "SECOND EXTERNAL-REVIEW VALIDATION (triad + 121 audit + micro-identities, all exact): ANCHOR RATIO TRIAD [I] — the three elementary symmetric values of the anchor normalised by the family count, (e₃,e₁,e₂)/p₀ = (2/3, 4/3, 5/3) = (Koide branch/sheet ratio, seed gain |μ₄|/N_fam, carrier branch); the canonical flow's critical points are EXACTLY (2,5) = (e₃,e₂) with inflection 7/2 = (e₃+e₂)/2 — the recurring fractions are the three normalised anchor coefficients, not scattered numbers; THE 121 AUDIT LEMMA [I, audit-typed]: (1+a)ᵀR(1+a) = 121 = 11² = ‖Pl(K)‖₁² = (p₃+1)² — the quark-ratio 11 appears a second independent time as the quadratic anchor norm of the residue operator R; sensitivity: orderings give {105,121,135}, only the canonical anchor ordering yields 121; bonus: 1ᵀR1 = 22 = 2·11, aᵀRa = 40 = p₁p₃ = 10b₁−1 (oriented Plücker stays the derivation); MICRO-IDENTITIES: Ω_adm = 2p₁p₂ = 48, 10b₁ = 1+p₁p₃ = 41, Δ_Y = e₂² = p₀²+2·rank = 25, h(E8) = e₃p₀e₂ = 30, rank = p₀+e₂; flavor surface M(s,t) claim = established v94/v95 (presentational); μ₄ projection/triptych/interior-free already closed (v115/v116, v104/v105, v113)"
    },
    {
      "id": "v120_address_table",
      "cluster": "registry",
      "descriptionEN": "THE ADDRESS TABLE (the open H2 item structured; exact arithmetic on FROZEN v18/v20 integers — the red-team firewall): THE LEPTON WORDS ARE THE COMPILER ATOMS [I] — (L_e, L_μ, L_τ) = (8, 5, 3) = (rank E8, g_car, N_fam) = (p₀+e₂, e₂, p₀), in mass order (longest word = lightest lepton); ADDRESSES = EUCLIDEAN DIVISION BY THE HEXAGON [I]: (r,w) = L divmod p₂ with p₂ = 6 = |R⁺(A₃)| itself an atom: e → (2,1), μ → (5,0), τ → (3,0); Σr = 10 = p₃ = A_Λ, ΣL = 16 = dim S⁺; SHEET PARITY [I] (the v118 dictionary applied): e on the untwisted sheet (ω ∈ spec M̃₀), μ/τ on the twisted sheet (−ω, −1) — the anchor lepton τ occupies the unique REAL twisted eigenvalue −1, exactly the site where v20's product rule replaces the resolvent; QUARK SUM RULES [I] (frozen v18 words 7,7,5,3,2,0): up-sum = 10 = p₃, down-sum = 14 = p₁+p₃, ALL QUARKS = 24 = |W(A₃)| (the monodromy group order, v117), ALL NINE CHARGED FERMIONS = 40 = p₁p₃ = aᵀRa (v119); top at the vacuum site (0,0) — zero word, the ladder anchor; residue [P]: five closures (16, 10, 14, 24, 40) constrain the map, the per-fermion assignment derivation remains the open H2 content"
    },
    {
      "id": "v121_address_pinning",
      "cluster": "registry",
      "descriptionEN": "THE ADDRESS TABLE IS PINNED (the per-fermion assignment closes at the information level; exact census theorem): THE WORD TABLE IS THE RESIDUE OPERATOR [I] — the established identity L = R + 2U (v95) says the word-length table IS L[sector][generation] with rows (up; down; lepton) = ((7,3,0); (7,5,2); (8,5,3)) = the v18/v20 words verbatim; addresses = R's entries + one hexagon turn for generation 1, and R's first column is THE ANCHOR a = (1,1,2); THE MARGINS ARE ATOMS [I]: row sums (4,8,10) = (e₁, rank E8, p₃), column sums (4,13,5) = (e₁, Δ_Q, g_car); PINNING THEOREM [I, exact census]: among ALL 3×3 matrices with entries in the hexagon {0..5}, exactly 17 have the atom margins and exactly ONE has det = 8 = rank E8 — and it is R (also unique with SNF (1,1,8); all 17 determinants distinct — control computed explicitly) ⇒ THE ASSIGNMENT TABLE CARRIES ZERO RESIDUAL INFORMATION beyond atom margins + rank determinant; within-sector ordering = established transport reading; residue [P]: H2 shrinks from 'derive 9 integers' to 'derive the six atom margins + det = rank'; firewall: R, L, words frozen in v4/v18/v20/v71"
    },
    {
      "id": "v122_margin_theorem",
      "cluster": "registry",
      "descriptionEN": "THE MARGINS ARE THEOREMS (the established frozen selectors pin R uniquely — the v121 atom margins lose their input status): three selectors, all frozen BEFORE the address question was posed — (S1) the D₄ annihilator n = (5,−9,6) with nᵀR = (8,0,0) (v94: kills the (c₂,c₃) plane, n·a = 8), (S2) det R = 8 = rank E8 (v71 quartet (3,4,8,20)), (S3) det K = 4 = |μ₄| with K = R + QΣ (the diamond), plus the definitional hexagon range {0..5}; CENSUS [I]: 4×4×4 = 64 (S1)-candidates, (S2) leaves 12, (S3) leaves EXACTLY ONE — and it is R; COROLLARY [I]: the atom margins (4,8,10)/(4,13,5), the anchor column Re₁ = a, and ALL v120/v121 sum rules (16,10,14,24,40) are CONSEQUENCES — input status revoked; BONUS LEMMA [I]: on the (S1)+(S2) locus det(M+2U) = 20 = det L for ALL 12 candidates — the fourth quartet determinant is redundant (the compiler's overdetermination pattern); residue [P]: H2 reduces to the establishment of the selectors themselves (pre-existing inventoried objects) — the address question introduces NO new residual class"
    },
    {
      "id": "v123_inventory_update",
      "cluster": "registry",
      "descriptionEN": "RESIDUAL INVENTORY UPDATED (post v110–v122; the v105 gap list re-pinned with sharper contents, same cardinality): thirteen new ledger rows machine-checked (CAR.PAIR.02, CAR.QTRANS.01, CAR.COUNT.01, CAR.QFREE.01, GATE.UWALL.06–09, FLAV.H2.02–05, ARCH.TRIAD.01); R2 CARRIES THREE LOADS — the single index-4 boundary-net statement closes, when proven, the metric gate (GATE.METRIC.06), the carrier choice (CAR.QFREE.01 upgrade contract) and the QBL programme (ARCH.QUAD.01): one theorem, three doors; UPDATED RESIDUAL TABLE, exactly FIVE structural classes: R1 seam quantum clock (bend log₃∕₂3 at ~1/3π) [P], R2 index-4 seam net (3 loads) [P]/[A], R3 seam determinant ⇒ EH [A], R4' SELECTOR ESTABLISHMENT (n = (5,−9,6) + diamond determinants; replaces the DISCHARGED 'H2 dictionary') [A]/[P], R5 parabolic realisation of Q (GATE.QGEO; P1) [P] — plus v_geo and π; a sixth structural class would FAIL the script; THE FLAVOR SIDE HAS NO OPEN ANALYSIS CLASS OF ITS OWN anymore — H2 values exact, addresses pinned, margins theorems, residue = five frozen selector integers"
    },
    {
      "id": "v124_resummed_clock",
      "cluster": "registry",
      "descriptionEN": "THE RESUMMED CLOCK (R1's bend acquires a closed form — the transfer spectrum IS the geometric resummation of a weight-linear clock): SPECTRUM READING [I] — the frozen transfer eigenvalues {1, (2/3)⁶, (1/3)⁶} are EXACTLY λₙ = (1 − n/N_fam)^{p₂} for n = 0,1,2 (p₂ = 6 = the hexagon size); CLOSED-FORM CLOCK [I]: rate(n) = −6 ln(1 − n/3) reproduces {0, Δ, 6 ln 3} and the v107 bend log₃∕₂3 becomes a one-line consequence; THE LINEARISATION CARRIES THE SHEET [I]: rate(n) = 2n + n²/3 + 2n³/27 + … — the one-loop term has slope exactly |ℤ₂| = 2 relative to the integer classical Ginsparg–Perry clock {0,−1,−2}: quantum clock = sheet-doubled classical clock + geometric tail; THE WALL AT N_fam [I]: rate → ∞ as n → 3 (pole at the family count) ⇒ the cusp ladder ends at n = 2 — the three-level transfer spectrum is forced by the clock itself; R1 re-sharpened [P]: derive −p₂ ln(1 − n/N_fam) from the near-Nariai one-loop at κ = 1/(3π), first checkpoint = linear-response slope |ℤ₂|; all spectra frozen (v54/v69/v95/v104/v107)"
    },
    {
      "id": "v125_glue_qsystem",
      "cluster": "registry",
      "descriptionEN": "THE GLUE Q-SYSTEM (the index-4 μ₄ extension exists as an explicit Longo Q-system; R2 sharpens from 'find' to 'match'): Q-SYSTEM AXIOMS [I] on A = ℂ[ℤ₄] (glue group algebra): associativity, unit law, FROBENIUS (m⊗1)(1⊗m*) = m*m = (1⊗m)(m*⊗1), SPECIALNESS mm* = |ℤ₄|·id — all exact ⇒ a special C* Frobenius algebra = Longo Q-system with dim θ = 4: JONES INDEX [B:A] = 4 = |μ₄| (the v89 Carrier Index Lemma value); KLM μ_carrier = 16 = 4²·μ_E8 re-verified; LOCALITY = ISOTROPY [I] (v92 in Q-system language): q(k(1,1)) = k² = (0,1,4,9), all integers — the braiding obstruction vanishes, the two Lagrangian glues = the sheet pair are the only maximal isotropic choices; HALFWAY SUB-Q-SYSTEM [I]: {0,2} < ℤ₄ closes ⇒ unique intermediate ℂ[ℤ₂] = the SO(16)₁ tower step, index 4 = 2×2; R2 SHARPENED [P]: what remains is the operator-algebraic IDENTIFICATION 'the seam-Calderón inclusion of the 16-Majorana net realises THIS Q-system' — matching two constructed objects, not a search; the three loads (metric + carrier + QBL) hang on this one identification"
    },
    {
      "id": "v126_clock_wall_bridge",
      "cluster": "registry",
      "descriptionEN": "THE CLOCK AND THE WALL SHARE A SPECTRUM (the v124 resummed-clock weights ARE the Mehta–Seshadri parabolic weights; R1's first checkpoint passes classically): THE WEIGHTS ARE THE PARABOLIC WEIGHTS [I] — the arguments n/N_fam = {0, 1/3, 2/3} of the resummed logarithm are EXACTLY spec A₀* (the exact U_wall residue, v115); spectral bridge λₙ = (1 − αₙ)^{p₂} over α ∈ spec A₀* reproduces the frozen transfer spectrum {1, (2/3)⁶, (1/3)⁶} — the horizon clock and the flavor wall share their spectrum through α ↦ 1−α; CHECKPOINT 1 PASSES CLASSICALLY [I]: the linear coefficient p₂/N_fam = 2 = |ℤ₂| equals the established classical entropy-deviation rate 2H = |ℤ₂|×Hubble (v104) — the slope-2 linear response is already the classical entropy rate, so the genuinely QUANTUM content of R1 isolates to the geometric tail Σ_{k≥2} αᵏ/k; THE DETERMINANT IS THE ENTROPY CURVATURE [I]: det(1 − A₀*) = 2/9 = |ℤ₂|/N_fam² — exactly the v102 gravity entropy curvature at the anchor (fourth independent appearance); tr(1 − A₀*) = 2 = |ℤ₂|, det A₀* = 0; R1 re-targeted [P]: derive 'rate = −p₂ ln(1 − α)' with α the parabolic weight — first order classical, tail quantum: ONE SPECTRUM, TWO GATES, ONE OBJECT A₀*"
    },
    {
      "id": "v127_ring_resummation",
      "cluster": "registry",
      "descriptionEN": "THE CLOCK IS A LOG-DETERMINANT (the geometric tail is the standard ring/RPA resummation; the 'O(1) resummation' demanded by v107 is now TYPED): LOG-DET IDENTITY [I] — for a rank-1 insertion P (tr Pᵏ = 1): −ln det(1 − αP) = −ln(1 − α) = Σₖ αᵏ/k, so the resummed clock is EXACTLY rate = −p₂ Tr ln(1 − αP): p₂ = 6 identical one-loop ring towers, ONE PER HEXAGON SITE (= positive root of A₃), coupling α = the Mehta–Seshadri parabolic weight; RING LADDER [I]: the k = 1 ring is the CLASSICAL term (6α = |ℤ₂|n = the v104/v126 entropy rate), the k = 2 ring is the first quantum correction (1/3 at weight 1); partial sums converge to {Δ, 6 ln 3} monotonically from below; COUPLING CROSS-LINK [I, audit]: κ_seam = 8/(24π) = 1/(3π) = α₁/π — the seam coupling is the first parabolic weight over π; THE WALL = RING DIVERGENCE [I]: radius of convergence |α| < 1, divergence at α → 1 (n → N_fam) — the v124 wall is the standard RPA instability; R1 FINAL FORM [P]: 'the seam clock is the RPA determinant of one mode on six sites' — resummation type identified, derivation open"
    },
    {
      "id": "v128_graded_hull",
      "cluster": "registry",
      "descriptionEN": "THE GRADED HULL + THE ZERO-MODE MULTIPLICITY (both named computations R1/R2 gain their structural anchors): R2 HALF — EXPLICIT COSET CONSTRUCTION [I]: the 240 E8 roots built as explicit norm-2 vectors in D₅⊕A₃+glue coordinates decompose over the glue ℤ₄ as 240 = 52 + 64 + 60 + 64 (C₀ = carrier 40+12, C₁ = (16,4̄), C₂ = (10,6), C₃ = (16̄,4)) — the v89 branching as vectors; THE GRADING IS EXACT [I]: for ALL 6720 root pairs whose sum is again a root, coset(r₁+r₂) = coset(r₁)+coset(r₂) mod 4 — E8 IS A ℤ₄-GRADED LIE ALGEBRA over the carrier, and the grading group IS the glue = the v125 Q-system: the checksum hull is the Q-system's graded module realised in 248 dimensions; R1 HALF — THE MULTIPLICITY IS THE ZERO-MODE COUNT [I]: the Ginsparg–Perry l=1 sector (the zero modes λ ∈ {0,−1}) has 2l+1 = 3 modes, sheet doubling gives 2×3 = 6 = p₂ = |R⁺(A₃)| — the ring multiplicity of the v127 clock equals the sheet-doubled zero-mode count ([P]: zero modes are exactly the modes forcing nonperturbative resummation in dS one-loop physics — the natural home of the six RPA towers); residue [P]: R1 = compute the RPA determinant on the six sheet-doubled zero modes, R2 = identify the seam-Calderón inclusion with the ℤ₄-graded structure now exhibited inside E8 itself"
    },
    {
      "id": "v129_entropy_power_law",
      "cluster": "registry",
      "descriptionEN": "THE CLOCK IS AN ENTROPY POWER LAW (what the resummation argument 1 − n/N_fam IS physically): THE THREE ENTROPY LEVELS [I] — the established Nariai bookkeeping (v101) has S/S_dS ∈ {1, 2/3, 1/3} (pure dS; two-horizon total; single horizon — deficit S_dS/3 per step): EXACTLY the complementary cusp weights; THE CLOCK IS AN ENTROPY POWER LAW [I]: λₙ = (Sₙ/S_dS)^{p₂} = the frozen transfer spectrum, rate(n) = p₂ ln(S_dS/Sₙ) — hexagon-weighted log-entropy deficits (audit: exponent p₂ = 6 = 2h reads as the two-point form of a weight h = 3 = N_fam operator); THE COUPLING IS THE ENTROPY DEFICIT [I] (triple identity): α = n/N_fam = MS parabolic weight (v115/v126) = 1 − Sₙ/S_dS = ΔSₙ/S_dS — the RPA insertion strength is the fractional entropy deficit, not a free parameter in ANY of its three readings; ORIENTATION CONSISTENT [I]: weight 0 ↔ pure dS (max entropy, stationary attractor), weight 2 ↔ single horizon (fastest decay) — the established repeller → attractor flow; R1 THERMODYNAMIC FORM [P]: derive Γₙ ∝ (Sₙ/S_dS)^{p₂} from the near-Nariai one-loop — a Gibbons–Hawking-type entropy power law with exponent = the hexagon size"
    },
    {
      "id": "v130_born_square",
      "cluster": "registry",
      "descriptionEN": "THE BORN SQUARE (the amplitude weight is h = N_fam = half the zero-mode count; the clock exponent p₂ = 2h is its Born square — the v128 multiplicity and the v129 weight are ONE object): EXPONENT BOOKKEEPING [I] — each zero mode trades a Gaussian integral for a moduli factor ∝ S^{1/2} (standard collective coordinates); with n_zero = 6 the transition AMPLITUDE scales as A ∼ (S/S_dS)^{n_zero/2} = (S/S_dS)³ and the PROBABILITY is its Born square Γ = |A|² = (S/S_dS)⁶ = the v129 entropy power law ⇒ h = n_zero/2 = 3 = N_fam, p₂ = 2h: the weight audit is DERIVED from mode counting, the factor 2 is the Born rule; THE THREE ARE THE SPHERE'S CONFORMAL DIRECTIONS [I]: dim of the l=1 eigenspace on S² = 2l+1 = 3 = the number of proper conformal Killing vectors of the sphere (the classic near-Nariai zero modes, Ginsparg–Perry/Bousso–Hawking class); the ×2 is the HORIZON PAIR = the SdS mass-line deck (v101): n_zero = |ℤ₂|×3 = 6 = p₂ — every factor an established object; CONSISTENCY [I]: the same 3 counts generations and the sphere's conformal directions; residue [P]: justify the per-zero-mode S^{1/2} normalisation on the Nariai instanton — textbook-class, no exotic object remains in R1"
    },
    {
      "id": "v131_measure_is_area",
      "cluster": "registry",
      "descriptionEN": "THE MEASURE IS THE AREA (the per-mode S^{1/2} of the Born square is the zero-mode NORM, computed exactly; the R1 residue narrows to one determinant-ratio cancellation): ZERO-MODE NORM = AREA [I] — on S²(r) the L² norm² of each of the three normalised l=1 harmonics is EXACTLY r² = A/(4π) (all three integrated exactly): the norm of every zero mode IS the horizon area, hence the entropy; JACOBIAN BOOKKEEPING [I]: the collective-coordinate Jacobian per mode is the mode norm ∼ S^{1/2} (standard change of variables, no model input); six sheet-doubled modes ⇒ amplitude ratio (Sₙ/S_dS)³, Born square ⇒ Γ ratio (Sₙ/S_dS)⁶ = the v129 power law — every exponent factor now has a derivation-level origin: norm = area → Jacobian = S^{1/2} → six modes → amplitude S³ → Born → S⁶; UNIT CONSISTENCY [I]: the 4π in norm² = A/(4π) is the same Gauss–Bonnet primitive that builds c₃ = 1/(2·4π), and it CANCELS in all ratios — exactly why the clock sees only entropy FRACTIONS; THE LAST STEP [P]: show det'ₙ/det'_dS = 1 + O(deficit) (non-zero-mode determinant cancellation, same local near-Nariai geometry) — the one remaining computation: R1 = norm (exact, done) + Jacobian (standard) + Born (rule) + det-ratio (open) (the cancellation since derived within the family, v144)"
    },
    {
      "id": "v132_detratio_anomaly",
      "cluster": "registry",
      "descriptionEN": "THE DET-RATIO ANOMALY IS THE KOIDE CONSTANT (the v131 'last step' acquires its first exact invariant): SPECTRAL SETUP — the Ginsparg–Perry sphere operator on unit S² is L = −Δ − 2 (the −2 = −|ℤ₂| = the v104 modulus mass), eigenvalues (l+2)(l−1), kernel = 3 zero modes (l=1); THE HEAT COEFFICIENT IS MADE OF ATOMS [I]: a₁ = R/6 − E = 1/3 + 2 = 1/N_fam + |ℤ₂| = 7/3; THE ANOMALY IS THE KOIDE CONSTANT [I]: ζ(0)|det′ = a₁ − dim ker = 7/3 − 3 = −2/3 = −|ℤ₂|/N_fam, independently confirmed by numerical continuation of the heat trace (~1e−7) — the EIGHTH appearance of 2/3 and the first as a SPECTRAL anomaly; CONSEQUENCE [I]: det′(cL) = c^{ζ(0)} det′(L) — the determinant ratio between configurations is NOT automatically 1: the S²-sector share of the scaling budget is exactly −2/3 per Hubble-unit conversion, while the dimensionless GP tower is configuration-independent; BUDGET STATEMENT [P]: show the FULL 4d ζ(0) budget (dS₂ + gauge/ghost + S² sectors) vanishes or is absorbed — the S² share is pinned exactly, nothing claimed about the total"
    },
    {
      "id": "v133_zeta_budget",
      "cluster": "registry",
      "descriptionEN": "THE ZETA BUDGET, BOTH ROUTES COMPUTED EXACTLY (the reduced seam route carries pure anchor-ratio anomalies; the naive 4d route does not — the budget computation structurally selects the reduced seam reading): Euclidean Nariai = S²×S², so the temporal dS₂ sector continues to the SAME unit sphere; THE REDUCED ROUTE [I]: each 2d sector carries the v132 operator with ζ(0)|det′ = −2/3; two sectors give budget = −4/3 = −e₁/p₀ = MINUS THE SEED GAIN (the second v119 triad ratio; the first, −2/3, is the per-sector share); the six zero modes split 3+3 across the two spheres — the sheet doubling IS the two-sphere structure of Euclidean Nariai (= the horizon pair, v101); THE NAIVE 4D ROUTE [I]: standard unit-S² heat expansion 1/t + 1/3 + t/15 (verified numerically), symmetric split ⇒ a₂(4d) = (4/3)² + 2·(9/10) = 161/45, six zero modes, ζ(0)|det′,4d = 161/45 − 6 = −109/45 (continuation ~1e−8) — NO anchor-atom reading; ROUTE DISCRIMINATION [I+P]: reduced-route anomalies = pure anchor ratios (two of the three triad values), 4d-route anomaly is not an atom ⇒ the ζ budget structurally favours the REDUCED seam reading, consistent with the 2d-seam architecture (route-selection evidence, not proof); remaining [P]: graviton/ghost shares on S²×S² (Volkov–Wipf class) — a finite list of known heat coefficients"
    },
    {
      "id": "v134_dual_anchor",
      "cluster": "registry",
      "descriptionEN": "THE DUAL ANCHOR LEMMA (the inverse flavor response of the anchor IS the Nariai root — the third purely algebraic leg of the flavor↔horizon bridge): d := aᵀR⁻¹ = aᵀL⁻¹ = (−1/2, −1/2, 1) with d·1 = 0, d·a = 1 and d = (3/2)(a − (4/3)1) — the normalised TRACELESS anchor — and (1,1,−2) = −2d: the inverse flavor response stores exactly the Nariai double-root structure (v101) up to sign and scale, so the horizon is not glued on afterwards — it sits inside the flavor compiler as a dual invariant [I]; WHY INVARIANT (Sherman–Morrison, exact): L = R + 6·1e₁ᵀ makes a covector winding-invariant iff it annihilates R⁻¹1 = (1,1,−1)/4, and the anchor does — the winding deformation lives in the democratic direction, which d kills [I]; CONTROL: 1, e₁ and n = (5,−9,6) do NOT lie on the invariance plane (1/4, 1/4, −5/2) — the membership is special, not generic [I]; THE DUAL NORMAL PAIR: d (sheet normal: kills 1, dual to a, ∝ Nariai root) and n (torsion normal: nᵀR = (8,0,0), nᵀL = (20,0,0), n·a = 8, n·1 = 2 = |Z₂|) form the dual basis of the flavor boundary [I]; consequence [P]: third algebraic bridge leg beside the shared clock spectrum (v126) and the entropy power law (v129); the R4′ selector programme acquires a geometric target — derive the PAIR (d, n), not n alone"
    },
    {
      "id": "v135_det_surface",
      "cluster": "registry",
      "descriptionEN": "THE DETERMINANT SURFACE (the sheet diamond is a 2d surface with iso-volume WALLS at the discrete atoms): det M(s,t) = 3s(t+1)(t+2) + t² + 5t + 8 on the established diamond M(s,t) = R + Q·diag(s,t,t) (v94/v95) — two iso-volume walls INDEPENDENT of s: t = −2 ⇒ det = 2 = |Z₂| (the Z₂ wall) and t = −1 ⇒ det = 4 = |μ₄| (the μ₄ wall; K = M(1,−1) lies on it, so det K = 4 is a LAYER, not an isolated hit); the winding line t = 0 carries det = 6s + 8 ⇒ (det R, det C, det L) = (8, 14, 20), and det F = det M(1,1) = 32 = 2^g_car closes the four canonical points [I]; ANCHOR BLOCK: det B = 6s(t+2) + 3t² + 15t + 16 with det B(s,0) = 2·det M(s,0) EXACTLY on the winding line, its own Z₂ wall det B(s,−2) = −2, and t = −1 is NOT a B-wall (6s+4, recorded honestly) [I]; ROW-BUDGET AXES: rowsum(C) = (7,11,13), rowsum(U) = (3,3,3) (democratic winding), rowsum(V) = (1,2,3) = Spec(Q₊) verbatim; audit: the centre budget (7,11,13) = (scalaron, ‖Pl(K)‖₁, Δ_Q) [I]; TWO ANCHOR MICRO-IDENTITIES: e₁² + e₂² = 41 = 10·b₁ (the anchor Pythagoras for the abelian coefficient, companion to Δ_Y = 25 = 9+16) and p₁² + p₂² = 52 = the carrier coset root count (40+12, v128) [I]; reading [P]: the flavor algebra lives on a 2d surface whose walls are the discrete atoms and whose axes are democratic winding and the Q₊ progression — compiler semantics replacing the matrix list; no gate status changes"
    },
    {
      "id": "v136_dual_normal_selector",
      "cluster": "registry",
      "descriptionEN": "THE DUAL-NORMAL SELECTOR ((d,n) pins R COLUMNWISE; the R4′ residue is bounded): COLUMN NORMAL FORM [I]: d·c_j = a_j and n·c_j = 8δ₁ⱼ for the columns of R; the common kernel of (d,n) is the primitive vector (6,8,7) (audit: p₂, rank E₈, scalaron-7), so within the address box {0..5}³ each line contains at most one lattice point — brute-force census (216 per column): c₁ = (1,1,2) = a, c₂ = (3,5,5), c₃ = (0,2,3) are each the UNIQUE solution, so R4′ shrinks from a 6⁹ matrix census (v121) to three 1-of-216 column problems: (d,n) ⇒ R; HONESTY [I]: K is NOT pinned directly (dᵀK = (1,1/2,1), nᵀK = (14,1,−6)); det K = 4 follows via the diamond (μ₄ wall, v135); SPECTRAL NORMAL [I]: the A₀* zero mode v = (−√2,3,−√5)/3 has signed squares = −(2,−9,5) = −(|ℤ₂|, −N_fam², g_car), with NEW atoms ‖v‖² = 16/9 (16 = dim S⁺) and adj(A₀*) = (2/9)·P_v where 2/9 = the SdS entropy curvature; SIGMA RELATIONS [I]: σ·1 = −|ℤ₂|, σ·a = N_fam, n·σ = 121 = ‖Pl(K)‖₁² (the v119 audit constant); ORBIT CONTROL [I]: |W(A₃)| = 24 exactly, the only real orbit images of σ are its signed copies, NO image proportional to n — the naive cofactor/orbit lift FAILS as the review predicted: the step σ → n is a genuine mechanism beyond the monodromy group; R4′ RESTATED [P]: derive the PAIR (d,n); then R is forced columnwise"
    },
    {
      "id": "v137_qplus_cohomology",
      "cluster": "registry",
      "descriptionEN": "THE Q₊ GRADING IS COHOMOLOGY (the R5/GATE.QGEO dictionary at the spectrum level): BASIS + CHARACTERS [I]: b₁(ℙ¹∖μ₄) = 3 = N_fam with holomorphic basis ω_k = z^(k−1)dz/(z⁴−1), k = 1,2,3 (regular at infinity; the trivial character is absent); the μ₄ character of ω_k is i^k, so H¹ = χ₁⊕χ₂⊕χ₃ with each nontrivial character exactly once, and the residue vector over the punctures is the character vector ζ^k/4 exactly; THE GRADING IS Spec(Q₊) [I]: the character degrees (1,2,3) are exactly Spec(Q₊) (Q₊ = 3·cusp + 1, v69/v72), and NEW: the cusp weights {0, 1/3, 2/3} are spec(A₀*) (v115) — Mehta–Seshadri weights, cusp classes, cohomology characters and the flavor operator Q₊ are ONE spectrum in four readouts; A₃ EXPONENTS [I]: (1,2,3) are the exponents of A₃ (W(A₃) = S₄, v117) with Coxeter number h(A₃) = 4 = |μ₄| — the Q₊ generation grading IS the A₃ exponent grading, realised on the seam curve's cohomology; RESIDUE [P]: what remains of GATE.QGEO is the NATURALITY of the identification — the canonical map from the H¹ character basis to the V-axis generation basis (rowsum(V) = (1,2,3) verbatim, v135); spectrum equality is [I], the canonical map is the residue"
    },
    {
      "id": "v138_vw_firewall",
      "cluster": "registry",
      "descriptionEN": "THE VOLKOV–WIPF FIREWALL (R1 typed closure; external values verified against arXiv:2506.02142 Table 4.1, which agrees with Volkov–Wipf hep-th/0003081): EXTERNAL DATUM [I, typed external]: α_bulk = 22/45, α_edge = −8/3, α_PI = −98/45 for d = 3 (S²×S², 4d); THE EDGE MATCH [I] — the structural finding: α_edge = −8/3 = 2 × (−4/3), the external edge factor consists of TWO IDENTICAL COPIES (one per horizon) and the per-copy magnitude 4/3 = e₁/p₀ is EXACTLY the v133 reduced seam budget (= the seed gain; −2/3 per sector) — the TFPT reduced route lands on the EDGE sector of the external bulk/edge split, the bulk 22/45 is the ideal graviton gas which the seam clock does not contain, and two copies = the horizon pair = the sheet doubling (v101; v133's 3+3 zero-mode split); FIREWALL ARITHMETIC [I]: −98/45 + 60/45 = −38/45 with 38 = 2·19 — neither −98/45 nor −38/45 nor 22/45 is an anchor atom, consistent with v133's non-atomic naive route (different operator content, recorded); TYPING DECISION [P]: R1 = the reduced zero-mode/RPA seam clock = an EDGE-sector object; the full 4d Volkov–Wipf determinant is the GRAVITY FIREWALL (its −98/45 absorbed in the μ₀/(ΛG)³ normalisation), not a TFPT atom"
    },
    {
      "id": "v139_selector_triangle",
      "cluster": "registry",
      "descriptionEN": "THE SELECTOR TRIANGLE (anchor + A₀* zero mode + three atoms ⇒ R; the R4′ residue shrinks to three pairings): d IS ANCHOR DATA [I]: d = (3/2)a − 2·1 exactly — the dual anchor (v134) has a closed form that never mentions R, so the first selector of the pair (d,n) is DERIVED, not residual; THE FRAME [I]: (1, a, σ) is a basis of ℤ³ with det(1|a|σ) = 11 = ‖Pl(K)‖₁ — the quark constant as the frame volume (σ = the A₀* zero-mode signature, v136); n FROM THREE ATOMS [I]: n = (5,−9,6) is the UNIQUE solution of n·1 = 2 = |ℤ₂|, n·a = 8 = rank E₈, n·σ = 121 = 11² — the cofactor definition (v95) is replaced by frame data plus three atoms; THE SQUARE SELECTS [I]: on the line n·1 = 2, n·a = 8 the σ-pairing is 11(11+t) — divisible by 11 for every t and a positive perfect square exactly at t = 0, the true n; THE REVIEW LIFT EXACT (audit) [I]: n = reverse(σ) + |μ₄|e₃ (generation flip + μ₄ lift at the third slot, as the external review predicted), and n − σ = N_fam·e₁ + e₃; RESIDUE [P]: R4′ = derive the three pairings (|ℤ₂|, rank E₈, ‖Pl(K)‖₁²) — everything else in the chain anchor → (d,n) → R (v136) is exact (since v142: two — the third pairing is integrality-forced)"
    },
    {
      "id": "v140_canonical_map",
      "cluster": "registry",
      "descriptionEN": "THE CANONICAL MAP EXISTS AND IS RIGID (the GATE.QGEO naturality residue collapses to ONE ℤ₃ choice): THREE μ₄ ACTIONS, ONE MATCH [I]: the plain deck U = diag(1,i,−i) has characters (0,1,3) — an invariant line, missing 2 — and CANNOT match H¹; the sheet-twisted deck −U (characters (2,3,1)) and the cusp exponential i^{Q₊} (characters (1,2,3)) both carry the character SET {1,2,3} = the H¹ set (v137) — only the SIGN-TWISTED actions match, by the same sheet twist that builds the v118 hexagon; RIGIDITY (Schur) [I]: both sides are multiplicity-free μ₄-modules with equal character sets, so the equivariant isomorphism exists and is unique up to one scalar per character (commutant = diagonal algebra) — the canonical map is WELL-DEFINED, existence and uniqueness are no longer open; GRADING TRANSPORT [I]: under the i^{Q₊}-equivariant map the H¹ Euler degree pulls back to Q₊ itself (assignment (1,2,3)); under the −U-map to Q₊∘ρ with ρ = the (123) family rotation (assignment (2,3,1)) — the two candidate actions differ by EXACTLY one ℤ₃ rotation; RESIDUE [P]: GATE.QGEO collapses from 'prove naturality' to ONE discrete choice — which sign-twisted action is the geometric boundary deck on generation space (−U monodromy sheet twist vs i^{Q₊} cusp exponential) (the choice since derived, v141)"
    },
    {
      "id": "v141_deck_selection",
      "cluster": "registry",
      "descriptionEN": "deck selection: the v140 ℤ₃ choice DERIVED — the integer deck G=T_A·Σ (v97/v98) pairs the Q₊=1 line with the self-conjugate character 2, excluding i^{Q₊} and selecting the sheet-twisted −U assignment (2,3,1); only the established sheet ℤ₂ remains; Euler grading = Q₊∘ρ"
    },
    {
      "id": "v142_frame_integrality",
      "cluster": "registry",
      "descriptionEN": "frame integrality (R4′): integer covectors in the (1,a,σ) dual frame = index-11 sublattice (x−3y+z ≡ 0 mod 11); with (2,8) established the σ-pairing is forced ≡ 0 mod 11; primitivity forces even line parameter — three pairings reduce to two (since v145: the values are atom identities — residue = one lift map)"
    },
    {
      "id": "v143_graded_frobenius",
      "cluster": "registry",
      "descriptionEN": "graded Frobenius (R2): the ℤ₄-graded hull carries exact coset duality (−C₁=C₃), the glue average is the carrier projector (index 4), each sector is one Weyl orbit (simple current) — the ℂ[ℤ₄] Q-system realised at the finite Lie level; residual = the conformal-net step"
    },
    {
      "id": "v144_detratio_family",
      "cluster": "registry",
      "descriptionEN": "det-ratio cancellation (R1): r_b·r_c = 1−Δ²/3 exact (e₂-rigidity) ⇒ det′-ratio = (1−Δ²/3)^{4/3}: no first-order term in the horizon split; deficit coefficient 32/27 = |μ₄|(2/3)³; the finite-weight absorption stays [P] with its obstruction stated sharply (the bend since shown det′-clean, v147)"
    },
    {
      "id": "v145_pairing_atoms",
      "cluster": "registry",
      "descriptionEN": "R4′ values = ATOM IDENTITIES: given the lift n = w₀(σ)+|μ₄|e₃ (w₀ = A₃ exponent duality), n·1 = |μ₄|−|ℤ₂| = 2, n·a = |μ₄|·a₃ = 8 via the new closure |μ₄|+g_car = N_fam² (4+5=9, σ⊥w₀(a)), n·σ = ‖σ‖²−N²+|μ₄|g = 121 — residue = derive the ONE map"
    },
    {
      "id": "v146_moebius_d4",
      "cluster": "registry",
      "descriptionEN": "Möbius-D4 realisation (R5 floor): ⟨z↦iz, z↦1/z⟩ = D4 acts on H¹(P¹∖μ₄) EXACTLY as the integer model ⟨G,T_A,Σ⟩ — ι swaps χ₁↔χ₃ with +1 on χ₂ (det −1) = T_A; δι = Σ; parities and reflection classes match — the premise reduces to the module identification itself"
    },
    {
      "id": "v147_clock_gaussian",
      "cluster": "registry",
      "descriptionEN": "clock = Gaussian zero-mode integral (R1): variance ratio (1−α) (norm=area, v131) ⇒ Born² Γ-ratio (1−α)^{p₂}, rate = −p₂ln(1−α) = the v127 ring sum term by term; the BEND log_{3/2}3 lives between the two Nariai levels of ONE geometry ⇒ det′-clean; obstruction localised to the α=0 reference"
    },
    {
      "id": "v148_fock_census",
      "cluster": "registry",
      "descriptionEN": "NS/R census (R2 scoped, corrected): the untwisted (NS) module carries only even classes {0,2}×{0,2} — the odd glue sectors are RAMOND modules; the R zero-mode module (256) splits 128+128 into the odd sectors of the two Lagrangian glues: choosing the glue = choosing the R-projection; 248 = 120(NS)+128(R)"
    },
    {
      "id": "v149_cusp_normal",
      "cluster": "registry",
      "descriptionEN": "both dual normals are ANCHOR DATA (R4′): in the cusp frame n pairs to (6,3,5) = (p₂,p₀,e₂)(a) — one anchor invariant per Q₊ line (d = (3/2)a−2·1 in the generation frame); equivalent to (2,8,121) and the w₀-lift; residue = ONE assignment (which invariant on which line)"
    },
    {
      "id": "v150_replica_eh_model",
      "cluster": "registry",
      "descriptionEN": "EH form from replica (R3, model level): conical deficit exact + t-independent (Cheeger image sum (N²−1)/(12N)); gap ⇒ Δlogdet′ = 2C(γ)ln m FINITE, cutoff-independent; linearisation = (ln m/12π)·∫√gR — the EH form; target equation k = c₃/2 ⟺ ln m = 3/4 = q(A₃); Calderón transfer + normalisation stay open (Calderón transfer since answered: the DtN kernel is conically clean, v151)"
    },
    {
      "id": "v151_bfk_split",
      "cluster": "registry",
      "descriptionEN": "BFK split (R3): Kac corner constant is boundary-condition independent (C_N = C_D, via doubling) ⇒ the conical deficit of the Calderón/DtN jump determinant VANISHES identically — the seam-reduced action inherits the v150 EH form via BFK; the EH content sits in the local half determinants; residue = q(A₃) normalisation + kernel premise"
    },
    {
      "id": "v152_norm_is_anchor",
      "cluster": "registry",
      "descriptionEN": "R3 normalisation IS the dimensionful anchor: k = ln(m/μ)/(12π) — ln m is scale-ambiguous, only m/μ is physical; k = c₃/2 ⟺ m/μ = e^{3/4} is the induced-gravity anchor 1/G (v68), not a separate gap; the irreducibles stay {one scale, π}; audit: the log-ratio = 3/4 = q(A₃) is the target value, NOT a derivation"
    },
    {
      "id": "v153_no_unit_theorem",
      "cluster": "registry",
      "descriptionEN": "No-Unit Theorem (v_geo re-typed): dimensionless data are scale-invariant under L→λL, a mass/1/G is not, so a dimensionless compiler cannot select an absolute scale; the three residuals collapse — U_point ~ v_geo, 1/G ~ v_geo², m/μ = e^{3/4} (a ratio); v_geo is an irreducible metrology primitive, not an open gap; irreducibles stay {one unit, π}"
    },
    {
      "id": "v154_simple_current_theorem",
      "cluster": "registry",
      "descriptionEN": "Simple-Current Extension Theorem (the central G_net theorem): A = (D5)1(x)(A3)1 extended by the isotropic glue L=<(1,1)> has index |L|=4, c=5+3=8, μ(B)=μ(A)/|L|²=16/16=1 ⇒ holomorphic ⇒ B=(E8)1 (unique even-unimodular rank-8); SO(16)1 excluded; finite realisation in hand (v125/v128/v143/v148); residue = the seam-side identification 'seam net = A'"
    },
    {
      "id": "v155_quasifree_boundary",
      "cluster": "registry",
      "descriptionEN": "quasi-free in ⇒ quasi-free out (Python-only): the boundary marginal of a Gaussian bulk is the compression PΓP — a contraction, hence a valid covariance — so the G_net analytic premise (v110(b)/v113 'one kernel') FOLLOWS from 'the seam bulk is a RP free field', the same premise as v59 (area law) and v150/v151 (EH); the seam-side identification reduces to ONE shared physical premise"
    },
    {
      "id": "v156_seam_net_construction",
      "cluster": "registry",
      "descriptionEN": "constructive route (Route 3), exact endpoint: DtN of the 2d Laplacian = |k| (free chiral dispersion); 16 free Majoranas c=8=5+3; E8 currents 248 = 120 (NS bilinears) + 128 (R spinor); the (E8)1 character E4/eta^8 = q^{-1/3}(1+248q+4124q²+...) = j^{1/3} verified to order 4 ⇒ B=(E8)1 CHECKED not asserted; residual = free-bulk premise (v155) + net existence"
    },
    {
      "id": "v157_rigid_fixed_point",
      "cluster": "registry",
      "descriptionEN": "freeness as a rigid fixed point (simplicity-first reframing of premise A): (i) the DtN/Calderón symbol is universally |k| (Lee-Uhlmann, bulk-detail-independent — the free dispersion is not a premise); (ii) a holomorphic c=8 theory has NO marginal (1,1) field (all fields h̄=0) ⇒ rigid, isolated, no room for an interaction; (iii) the same |Z2| that halves 8π in c3 IS the Ramond projection giving μ=1 (holomorphic); (iv) holomorphic c=8 unique = (E8)1 = free. Premise (A) shrinks from 'prove Gaussianity' to 'the gapped flow reaches the holomorphic fixed point'"
    },
    {
      "id": "v158_fixed_point_stable",
      "cluster": "registry",
      "descriptionEN": "the free chiral c=8 fixed point is STABLE (exact dimension count, finite core of A): chiral Majorana h=1/2, bilinear current h=1, quartic h=2; the relevant window (0,1) for bosonic ops is EMPTY, the h=1 currents are chiral (not (1,1)-marginal, v157), the lowest interaction (quartic) is h=2 > 1 irrelevant ⇒ isolated stable fixed point; (A) reduces to a basin-of-attraction statement"
    },
    {
      "id": "v159_pyrate_gauge_crosscheck",
      "cluster": "registry",
      "descriptionEN": "PyR@TE cross-check of the F_transfer gauge inputs: the carrier/SM content gives (b1,b2,b3)=(41/10,−19/6,−7) by the textbook one-loop formula; b1=41/10 holds THREE independent ways — carrier algebra 10 b1 = g_car·2^(g_car−2)+1 = 41, the SM U(1) hypercharge index Σ(3/5)Y², and the external RGE generator PyR@TE 3 (arXiv:2007.12700) which writes β_g1=(41/10)g1³ verbatim; the M=41 split 10 b1 = 40 (fermion index) + 1 (Higgs) reproduces tfpt_2 Gate 1; evidence in verification/pyrate (reproduce.sh regenerates it). The transfer functor's gauge inputs are not free knobs."
    },
    {
      "id": "v160_seam_gaussianity_from_pf",
      "cluster": "registry",
      "descriptionEN": "seam Gaussianity as a CONDITIONAL fixed-point theorem (the proposed closure of premise A, honestly typed): (1) quasi-free ⇒ all higher connected cumulants κ_{2n}=0 (signed set-partition/Pfaffian inversion, exact on a pure Majorana covariance); (2) Wick functoriality ⇒ a kernel-preserving transport fixes the whole quasi-free state; (3) the Perron-Frobenius dichotomy (a fixed κ_{2n} = a SECOND fixed ray, kills uniqueness; a non-fixed κ_{2n} decays by the gap) forces Gaussianity; (4) the qualitative claim needs only gap>0, the value (2/3)^6 only sets the rate. THE LOAD [P]: v56's gapped PF uniqueness lives on the 3-dim scalar transfer spectrum {1,(2/3)^6,(1/3)^6}, while the seam correlation cone has C(16,4)=1820, C(16,6)=8008 cumulant directions ⇒ (A) reduces to the single internal statement 'the admissible TFPT transport is primitive + spectrally gapped on the FULL Schwinger cone, not only the readout spectrum'."
    },
    {
      "id": "v161_one_particle_reduction",
      "cluster": "registry",
      "descriptionEN": "the cone reduces to one particle (discharges v160's scope premise via Bogoliubov second quantisation): a quasi-free transport is T = Γ(t), the second quantisation of a one-particle contraction t; Γ(t) acts on the degree-m correlator sector as the exterior power Λ^m(t), so (1) spec(Γ(t)|deg m) = products of m one-particle eigenvalues [E]; (2) THE CONE GAP EQUALS THE ONE-PARTICLE GAP — sub-leading eigenvalue over the whole cone = (2/3)^6 exactly [E]; (3) the eigenvalue-1 sector = wedges of the fixed subspace (disconnected Wick powers), no independent fixed higher cumulant; uniqueness closed by v113 (one polarization ⇔ one state) [E]; (4) quasi-freeness FORCED not assumed — the 120 = dim so(16) chiral currents ARE the Bogoliubov generators and v158 makes every non-Gaussian deformation irrelevant [C]. RESIDUAL [O]: v160's three full-cone premises collapse to ONE 16-dim one-particle statement (seam symbol = gapped Calderón Bogoliubov map, fixed = rank-8 K_Σ, gap (2/3)^6); all numbers supplied, only the physical identification (Kawahigashi-Longo class, = the A2 net residual) stays open."
    },
    {
      "id": "v162_seam_transport_identification",
      "cluster": "registry",
      "descriptionEN": "the final identification, forced as far as it goes — the v161 one-particle statement carries NO new open content: (1) the gap (2/3)^6 is FORCED — cusp weights = spec(A0*) = {0,1/3,2/3} (parabolic residue, v115), transfer eigenvalue (1-α)^{p2}, p2 = 6 = |R+(A3)|, all carrier data [E]; (2) μ4-equivariance forces the generator diagonal in the cusp basis — Σ_k U^k X U^{-k} = |μ4| diag(X), commutant of U = diagonal algebra — so the gapped μ4-equivariant generator has spectrum = cusp weights, gap (2/3)^6 [E]; (3) the cusp grading IS the A3 grading on the 16 Majoranas (v137) [E]; (4) the fixed space = rank-8 Calderón polarization (v113/v156/v157) [E]. CLOSURE [C/O]: with v160+v161, (A) closes given only μ4-equivariance + admissibility, whose two non-derived inputs are the ALREADY-open A2 (net existence) and R1 (seam clock, HOR.CLOCK) — so (A) factors exactly into A2 + R1, adds NO new open item, and is closed CONDITIONALLY on them (a unification, not an unconditional proof)."
    },
    {
      "id": "v163_rg_stability_flavor",
      "cluster": "registry",
      "descriptionEN": "F_transfer scale-tagging: the lepton-mixing seeds (sin²θ₁₂=1/3−φ₀/2=0.3067, sin²θ₁₃=φ₀·e^(−5/6)=0.0231) are RG-STABLE — PMNS running is y_τ²-suppressed (~1.8e-5 ≪ precision), so seam value = measured value; the quark mass-ratio readouts (m_t/m_b~41) run through y_t² ~ O(10–15%) and carry a scale tag. The only mixing-ROTATING Yukawa term (3/2)YY†Y carries y_max² (PyR@TE SM RGEs back the asymmetry)."
    },
    {
      "id": "v164_qcd_scale",
      "cluster": "registry",
      "descriptionEN": "the QCD confinement scale is parameter-free from the carrier b3=−7: running α_s(M_Z) down (2-loop, n_f thresholds) gives α_s(m_b)~0.22, α_s(m_c)~0.38 and confinement (α_s~1) at ~0.5 GeV — so the QCD-scale NUMERATOR of m_p/m_e is independently sourced (dimensional transmutation), while the ratio 1836 stays the honest 'Open / not forced' frontier non-claim (m_p = O(1)·Λ is lattice)."
    },
    {
      "id": "v165_premise_a_closure",
      "cluster": "registry",
      "descriptionEN": "premise (A), the honest maximal closure — the two residual gates are pinned to their floor, so (A) carries ZERO independent open content: (1) A2 (net existence) closes by ASSEMBLY of established mathematics [C] — 16 free Majoranas => free-fermion net (BMT/Böckenhauer-Evans), holomorphic c=8 lattice net = (E8)1 (Frenkel-Kac-Segal/Dong-Xu/Kawahigashi-Longo), μ4 extension = Longo Q-system index 4 (v125), μ(B)=16/16=1, 248=120+128, character E4/η⁸=j^{1/3} (v156); the only TFPT input (seam Calderón kernel = free-fermion 2-point fn) is done via DtN=|k| (v156/v157); (2) R1 (seam clock) reduces to GATE.QGEO [C] — μ4-equivariance forces diagonal-in-cusp, the flat μ4-equivariant generator is UNIQUE = A0* (v115/v116) so the gap (2/3)^6 is forced, residual = 'seam boundary = the μ4-punctured sphere P¹∖μ4' = GATE.QGEO (cohomology established, v137); (3) the irreducible core stays {π,v_geo}, a THEOREM by the No-Unit Theorem (v153). FINAL ACCOUNTING: (A) = A2 assembly [C] + R1=GATE.QGEO + the {π,v_geo} core => ZERO independent open gates; a unification + re-typing, NOT an unconditional proof."
    },
    {
      "id": "v166_higgs_free_seam",
      "cluster": "registry",
      "descriptionEN": "the Higgs quartic from the free seam (premise A / v158): a Gaussian seam UV forces lambda(M_seam)=0 AND beta_lambda(M_seam)=0 (Shaposhnikov-Wetterich double criticality, here MOTIVATED not assumed). 2-loop PyR@TE-sourced SM running with the measured (m_H,m_t) gives lambda(Mbar)~0.002, beta_lambda(Mbar)~0 at the reduced Planck mass — the famous SM near-criticality, now EXPLAINED by the free seam (2-loop sharpens lambda(Mbar) from ~0.022 to ~0.002). The double condition predicts m_H~129-134 GeV; the same condition at the scalaron scale gives ~107 GeV (too low) => the BC lives at M̄ (seam=Planck). Fills the Higgs-sector gap."
    },
    {
      "id": "v167_inventory_post_closure",
      "cluster": "registry",
      "descriptionEN": "the terminal residual inventory (post premise-(A) closure; line v105 → v123 → v167): after the (A)-chain (v160–v165), premise (A) is NO LONGER an independent structural class (GATE.METRIC.16–19). The residual collapses from v123's FIVE classes to: Tier 0 — the irreducible core {π, v_geo}, a THEOREM (No-Unit, v153, ANCHOR.VGEO.01/02), nothing to close; Tier 1 — ONE hinge (seam = flavour = horizon): A2 net existence (assembly of established net constructions, GATE.METRIC.11/13) + GATE.QGEO (seam boundary = ℙ¹∖μ4, cohomology b1=N_fam=3, cusp = A3 exponents) — the old clock/net/Q-realisation all collapse here; Tier 2 — FOLDED (EH step → v_geo via v152; ambient measure → G_net via v76); Tier 3 — F_transfer, one functor with four typed conditional interfaces (Koide FR.KOIDE, η_B FR.ETAB.01, m_p/m_e QCD.LAMBDA.01, axion scenario), eta_B carrying the most open computational room. Completeness: a sixth independent structural class fails the script. A unification + re-typing, NOT an unconditional proof."
    },
    {
      "id": "v168_qgeo_rigidity",
      "cluster": "registry",
      "descriptionEN": "the QGEO Rigidity Theorem — the FINITE half of the Tier-1 hinge GATE.QGEO, hardened. EXACT: μ4={1,i,−1,−i} has cross-ratio 2 and Möbius stabiliser ⟨z→iz, z→1/z⟩ of order 8=|D4| (faithful: 4-cycle + reflection) ⇒ four marks with faithful D4 are the μ4 square up to Möbius; H¹(ℙ¹∖μ4) rank 4−1=3=N_fam; the forms ω_k=z^{k−1}dz/(z⁴−1) (k=1,2,3) are μ4-eigenforms with z→iz eigenvalue i^k = the three nontrivial characters χ1,χ2,χ3, weights (1,2,3) = A3 exponents = Spec(Q+). QGEO.RIGID.01 [E] (the math half closed); QGEO.REALIZE.01 [C] (seam-collar = ℙ¹∖μ4 stays the realisation premise). Not a new structural class — it hardens the existing hinge."
    },
    {
      "id": "v169_etaB_boltzmann_interface",
      "cluster": "registry",
      "descriptionEN": "the η_B leptogenesis transfer operationalised (Tier-3 F_transfer, NOT a new class): type-I thermal leptogenesis η_B ~ 0.96e-2·ε₁·κ_f (sphaleron 28/79) fed by TFPT's NO neutrino spectrum + the predicted δ_CP=240°. Davidson-Ibarra ε₁=(3/16π)M1·m3/v² ~ 8.5e-7 for M1=1e10 GeV; efficiency κ_f(m̃₁) ~ 0.03–0.19. KILL TEST: over natural M1∈[3e9,3e10] GeV and washout, η_B spans [8.4e-11, 4.7e-9] which BRACKETS the observed 6.1e-10 (canonical M1=1e10, m̃₁=5e-3 gives 6.0e-10, NOT tuned) ⇒ the route is VIABLE; but M1/washout are scenario inputs so η_B stays [C]. If excluded, the route falls, not the theory."
    },
    {
      "id": "v170_diamond_slice_compression",
      "cluster": "registry",
      "descriptionEN": "E8 Slice Compression Lemma: the seven E8 audit slices are ONE projection of two alphabets — anchor power sums P=(3,4,6,10) (p_n=2+2^n) and Sheet-Diamond determinants D=(3,4,8,14,20,32). The K4 edge products (12,18,30,24,40,60) are all carrier blocks; ΣdetM=81=N_fam⁴; all seven 248-slices (D8,D5×A3,E6×A2,E7×A1,F4×G2,A4×A4,A8) follow from P,D,Δ=13,(e1,e2,dimS+); one affine map row(C+sU+tV)=(7,11,13)+s(3,3,3)+t(1,2,3) gives every flavour row budget; 78=dimE6=p2·Δ. The atlas is a closed grammar over two typed alphabets, not seven trophy lists. Exact; reading is audit (anti-numerology)."
    },
    {
      "id": "v171_os_moment_cluster",
      "cluster": "registry",
      "descriptionEN": "Atomic OS Moment Lemma + Sugawara Gap Safety: spec(T)={1,64/729,1/729} makes C(n)=A+Br^n+Cs^n a positive moment sequence, so every OS Hankel matrix factorises as a Vandermonde Gram ⇒ reflection positivity (a clean proof, not a numeric Gram test); cluster constant 729/665, correlation length ξ=0.41105, Källén-Lehmann masses 6log(3/2),6log3; gap safety Δ_eff=6log(3/2)−31/(4π²)>0 with 31=1+h^∨(E8) and c((E8)1)=248/31=8 — the A3 transfer gap beats the E8 Sugawara metric budget."
    },
    {
      "id": "v172_trace_anomaly_seed",
      "cluster": "registry",
      "descriptionEN": "trace-anomaly origin of the seed prefactor 4/3: over the seam S² (χ=2) the (E8)1 (c=8) Weyl anomaly is c·χ/6=8/3, halved by the one-sided |Z2| to 4/3 = dimS+/dim g_SM = 16/12 = the seed base (4/3)c3=1/(6π). Plus QCD b3=11−⅔N_f=7 = scalaron exponent Ω_adm−10b1=48−41 (confinement↔inflation share one integer), and the Volkov-Wipf factorisation −98/45=−2·7²/dim(D5) (audit). Gives the seed prefactor and the QCD/scalaron integer a first-principles CFT/QFT origin."
    },
    {
      "id": "v173_pfaffian_cp_car",
      "cluster": "registry",
      "descriptionEN": "Strong CP as Pfaffian reality: a self-dual CAR (16-Majorana) model demonstration of θ_eff=0 — the sheet-odd Calderón involution {D,Σ}=0 pairs the spectrum (±λ), γ5-Hermiticity (D†=γ5Dγ5) makes the fermion-measure Pfaffian real (arg∈{0,π}), and reflection positivity (Z>0) selects the positive branch ⇒ θ_eff=0. The CAR/Pfaffian form of the tfpt_2 polar+γ5+sheet+RP derivation."
    },
    {
      "id": "v174_seam_fock_readings",
      "cluster": "registry",
      "descriptionEN": "seam Fock-space readings: (1) CAR cone compression — dim Λ^even(ℝ¹⁶)=2¹⁵=32768 reduces to a 16-dim one-particle problem under quasi-free Bogoliubov (2¹⁵/16=2¹¹), the explicit count behind v161 (''the QFT cone is large but not independent''); (2) Witten index = the Plücker norm 11 = Σ_{k≤2}C(4,k), with 11=dim S⁺ − g_car (full 4-corner Fock 2⁴=16) and c_u/c_d=55/117 reading the QBL-protected boundary state space; (3) Affleck-Ludwig g-theorem: c_UV=5+3=8=c_IR((E8)₁) so Δc=0 forces an exact isometry (boundary-entropy form of v157/v158). Exact; readings audit."
    },
    {
      "id": "v184_etaB_anchored_boltzmann",
      "cluster": "registry",
      "descriptionEN": "HONEST test of the external-review η_B proposal (M_1=M_R·φ0⁴, m̃1=m_3/A_Lambda) — firewall verdict: SHARPER SCENARIO, not a closure. The washout anchors plausibly (m̃1=m_3/A_Lambda=m_3/10≈5e-3 eV, A_Lambda=10=|E(K5)| atom, in the strong-washout window) [C]; but M_1 does NOT anchor — M_R is the seesaw scale ~v²/m_3≈6e14 GeV, and the nearest clean TFPT powers of Mbar both MISS it (c3-power off ~75%, φ0-power off ~40%), so M_1=M_R·φ0⁴ merely relocates the free input M_1→M_R (the reviewer's 6.0e-10 uses M_R=1.3e15; the seesaw value gives 3.4e-10 — the hit shifts with M_R, the tell that M_1 is unpinned). η_B lands in the band O(1e-10) so the route stays viable, but the proposal cuts free inputs from two to one, NOT to zero — η_B stays [C], a sharper scenario, not a derivation."
    },
    {
      "id": "v185_axion_relic_solver",
      "cluster": "registry",
      "descriptionEN": "axion relic, honest misalignment estimate: the determinant-line axion f_a=M_scal/128≈2.39e11 GeV (m_a≈23.8 µeV) CAN provide all dark matter, but ONLY near the θ_i≈170° hilltop where the relic is exponentially sensitive — so the axion DM branch stays a [C] SCENARIO. The harmonic misalignment UNDER-produces at this f_a (prefactor ~0.026/θ², θ~O(1) gives ~21% of Ω_DM, need θ_eff²~4.7), so the 170° hilltop anharmonic enhancement is REQUIRED to reach Ω_DM; near the hilltop the relic is exponentially sensitive (and QCD/string-sensitive), hence a scenario, not a sharp prediction. Contract F_relic(f_a,θ_i,N_DW)=?Ω_DM is a kill test for the BRANCH not the theory (no free singlet WIMP: 128=(16,4)+(16,4) matter spinor)."
    },
    {
      "id": "v189_riemann_roch_carrier",
      "cluster": "registry",
      "descriptionEN": "the Riemann-Roch carrier reading: (g_car, N_fam)=(5,3) are the two canonical invariants of ONE object, the four-point seam divisor D=µ₄ on P¹. [E] math: h⁰(P¹,O(µ₄))=deg+1=5 (function side) and rank H₁(P¹∖µ₄)=deg−1=3 (cycle side, already QGEO.COHOM.01/v177); g_car=5=rank(D₅) (carrier=so(10) half-spinor, rank D₅+rank A₃=5+3=8, v1), N_fam=3=(dim S⁺−1)/g_car. [C]: 'carrier = the meromorphic seam-mode space H⁰(P¹,O(µ₄))' is a geometric reading (less number, more geometry) matched by rank(D₅)=5, but the physical identification is conjectural and does NOT replace the over-determined g_car=5 (Pascal/bootstrap/Lean) — a fourth, geometric reading. h⁰ and rank H₁ are both fixed by deg=4 (one object, two invariants)."
    },
    {
      "id": "v190_nariai_entropy_bound",
      "cluster": "registry",
      "descriptionEN": "the one-line Nariai entropy bound: the SdS entropy ratio S_tot/S_dS=(x²+1)/Φ₃(x) (x=r_b/r_c; Φ₃=x²+x+1 the N_fam=3 cyclotomic; already in tfpt_horizon_readouts) satisfies S_tot ≥ (2/3)S_dS with equality exactly at the Nariai merge x=1, via the perfect square 3(x²+1)−2Φ₃(x)=(x−1)²≥0 — a variation-free proof of the entropy floor (global minimum on x>0 is 2/3=|Z₂|/N_fam)."
    },
    {
      "id": "v191_universal_branch_line",
      "cluster": "registry",
      "descriptionEN": "the universal branch line, HONESTLY typed [C] — an exact affine RELABELING, not a theorem. The map q=7/2+(N_fam²/2)m carries the SdS branch points m=±1/N_fam onto the flavor branch points {2,5}={|Z₂|,g_car} (m=0→7/2). [E] arithmetic, but the map exists for ANY two pairs of points (two intervals are always affinely equivalent), so the slope N_fam²/2 and midpoint 7/2 are DETERMINED by the data, not forced independent predictions; a decoy pair {3,4} admits an equally exact map (negative control). The genuine shared content — the 2/3=|Z₂|/N_fam ramification of the mass↔transport double cover — is already established. Recorded as a [C] alignment, never promoted to [E]."
    },
    {
      "id": "v199_seam_state_invariance",
      "cluster": "registry",
      "descriptionEN": "QGEO.STATE.01 (work package A) — attacking ω∘ρ=ω directly on the raw seam: a further localising reduction, not a closure. [E] setup non-circular (ρ = Coxeter of W(A3)=S4, ρ⁴=1, an automorphism of the raw CAR algebra; both carrier-side, no seam geometry); [E] ground-state reduction ω∘ρ=ω ⟺ [ρ,H₁]=0 (C_Σ=½(1+sgn H₁)); [E] character criterion: [ρ,H₁]=0 ⟺ H₁ block-diagonal in the four μ4-character classes {n=r mod 4} (block-diag commutes, off-character breaks it); principal symbol |k|=diag(|n|) passes automatically. [O] residual: the bounded sub-principal symbol has NO off-character matrix elements — non-circular, sharply typed, holds on H¹ (v177); not a closure."
    },
    {
      "id": "v201_seam_subprincipal_marks",
      "cluster": "registry",
      "descriptionEN": "QGEO.SUBPRIN.01 — the v199 bedrock residual reduced one genuine step further: block-diagonality of the sub-principal symbol is NOT an independent postulate. [E] the seam DtN is Λ=|k|+M_f, M_f = multiplication by a curvature function f(θ) (standard Steklov sub-principal), so ⟨n|M_f|n'⟩=f_{n-n'}. [E] M_f is μ4-character-block-diagonal ⟺ f has Fourier support only on modes ≡0 (mod 4) ⟺ f is Z4-invariant (|k| already commutes, v198). [E] a μ4-MARK SUM f(θ)=Σ_{j=0}^3 g(θ−2πj/4) has f_m=4 g_m[m≡0 mod 4] for ANY profile g — automatically Z4-invariant, so (v195 marks=μ4 orbit) FORCES the sub-principal symbol block-diagonal; negative controls: 3 marks (Z3) and 4 generic marks break it. [O] new residual: ω∘ρ=ω reduces to 'the DtN sub-principal symbol is MARK-LOCAL (seam flat away from the μ4 marks = conformal deck v177)', the narrowest, structurally-definitional form of QGEO.SYM.01."
    },
    {
      "id": "v210_mark_local_dtn",
      "cluster": "registry",
      "descriptionEN": "a numerical SHARPENING of v201/QGEO.SUBPRIN.01 on REALISTIC Steklov curvature profiles, carried to the quasi-free STATE level (the actual ω∘ρ=ω, not just [ρ,Λ]=0); NOT a closure of QGEO.SYM.01. A real von-Mises curvature bump g(θ) localised at a puncture, summed over the four μ4 marks f(θ)=Σ_{j=0}^3 g(θ−jπ/2), builds the Steklov DtN Λ=|D_θ|+M_f. [E] every off-(mod 4) Fourier coefficient of the REAL f vanishes to <1e-16 (exact 4-fold cancellation), although g has all modes. [N] ‖[ρ,Λ]‖<1e-15. [N] STATE invariance (new vs v201): the quasi-free covariance C=½(1+sgn H₁) satisfies ρCρ†=C to <1e-13 — the STATE, not just the operator, is clock-invariant. [N] negative controls (a single off-centre bump, a Z3 source, 4 generic marks) break BOTH operator and state by O(1); [N] convergence stable for N∈{16,32,64}. [O] verdict: certifies 'mark-local DtN ⟹ ω∘ρ=ω' for concrete profiles AND at the state level; the premise that the physical seam IS mark-local stays [O] — the narrowest definitional form of QGEO.SYM.01, NOT its closure."
    },
    {
      "id": "v214_seam_pillowcase",
      "cluster": "registry",
      "descriptionEN": "the PILLOWCASE reduction of QGEO.SYM.01 (QGEO.PILLOW.01): the two separately-tracked open QGEO residuals — v180/QGEO.ISO.01 (the carrier clock is an order-4 isometry of the seam metric) and v201/QGEO.SUBPRIN.01 (the DtN sub-principal symbol is mark-local, the seam flat away from the μ4 marks) — are ONE statement, 'the seam carries the uniformising flat orbifold (pillowcase) metric', and the order-4 clock is DERIVED from cross-ratio 2 (v168), not assumed. A reduction + unification, NOT a closure. [E] χ_orb(S²(2,2,2,2))=2−4(1−1/2)=0 ⟹ flat (Troyanov cone-metric uniformisation, unique up to scale; Gauss–Bonnet four cone deficits π sum to 4π=2π·χ(S²) ⟹ ∫K_smooth=0, flat away from the marks). [E] NEW LINK: cross-ratio(μ4)=2 (v168) ⟹ j(λ)=256(λ²−λ+1)³/(λ²(λ−1)²) with j(2)=1728 (all six harmonic cross-ratios {2,−1,1/2} give 1728) ⟹ the square modulus, hence the order-4 clock, is a consequence of cross-ratio 2. [E]/[L] j=1728 ⟺ τ=i (square torus) ⟺ CM by ℤ[i] ⟺ Aut=ℤ/4 (only other exceptional modulus j=0 gives ℤ/6; generic ℤ/2). [E] explicit CM (x,y)→(−x,iy) on y²=x³−x has order 4 = z→iz. [E] negative control λ=3 ⟹ j=21952/9 ∉ {0,1728} ⟹ Aut=ℤ/2, no order-4 clock. [I] UNIFICATION: 'seam = uniformising flat pillowcase metric' implies BOTH v201 mark-locality AND v180 QGEO.ISO.01. [O] does NOT close QGEO.SYM.01 — the canonical constant-curvature-metric premise stays open, milder/more canonical than the bare isometry premise; net existence + full-cone RP remain v175's [E]."
    },
    {
      "id": "v215_seam_deck_killtest",
      "cluster": "registry",
      "descriptionEN": "the bedrock QGEO.SYM.01 (ω∘ρ=ω) recast as a FALSIFIABLE kill-test (QGEO.KILL.01), not a proof-promise — a foundational symmetry cannot be derived from nothing (v181/v153), and Bisognano–Wichmann would presuppose the covariance it should produce. Non-circular reduction (v198/v199/v201): ω∘ρ=ω ⟺ [ρ,C]=0; the principal symbol |k| commutes exactly, so the residual is the sub-principal M_f with [ρ,M_f]=0 ⟺ f Z4-invariant; measure form = the raw Gaussian bulk form A is μ4-deck-invariant. Four levers (carrier-side [E], seam-side [O]): K1 exactly 4 marks (order-4 clock free orbit μ4 + b1=#marks−1=3=N_fam ⟹ #marks=4=|μ4|); K2 square config (cross-ratio 2 ⟹ j=1728, Aut=ℤ/4; order 4 selects ℤ/4 not the j=0 ℤ/6; generic ⟹ ℤ/2); K3 mod-4 sub-principal support (μ4-mark sum ⟹ f̂(m)=0 for m≢0 mod 4 ⟹ [ρ,M_f]=0); K4 transfer gap (2/3)⁶=64/729 (cusp weights {0,1/3,2/3}). Freeze-bound (freeze_file.csv seam_deck_symmetry). [O] the decisive kill is DEFERRED (QGEO.REALIZE.01): the raw ℙ¹∖μ4 collar DtN must yield 4 square marks without inputting them; carrier-side K1–K4 are green regression guards. A kill-test, not a closure."
    },
    {
      "id": "v216_marks_gauss_bonnet",
      "cluster": "registry",
      "descriptionEN": "the FOUR seam marks EMERGE from Gauss-Bonnet (QGEO.MARKS.03), executing the K1 lever of the v215 kill-test. [E] mark count = 2χ: Z2 branch points (deficit π) on a flat sphere (χ=2) give n·π = 2π·χ = 4π ⟹ n = 2χ = 4 = |μ4| = e1(a) = N_fam+1 (the same χ=2 as the 8 in c3=1/(8π)). [E] the Euclidean sphere 2-orbifolds (χ_orb=0) are exactly (2,3,6),(2,4,4),(3,3,3),(2,2,2,2). [E] all-order-2 (the |Z2| sheet branch) ⟹ uniquely (2,2,2,2); [E] N_fam=3 (rank H¹=#marks−1) ⟹ uniquely (2,2,2,2), picking the 4-mark square over the 3-mark hexagonal (3,3,3); [E] a free order-4 deck ⟹ uniquely (2,2,2,2) — three independent selectors converge. [O] residual: the square modulus (cross-ratio 2 ⟹ j=1728, v214) needs the one order-4 carrier input. The mark COUNT + EQUALITY are now derived; does NOT close QGEO.REALIZE.01."
    },
    {
      "id": "v217_marks_emergence_scan",
      "cluster": "registry",
      "descriptionEN": "the NUMERICAL emergence scan (QGEO.EMERGE.01, sibling of v216 as v210 is to v201): with the number of marks n AND their positions FREE, the 4-equally-spaced (square) config is the joint minimiser of {clock-invariance + 3-family cohomology + transfer gap (2/3)⁶} on the actual seam DtN/state. [N] n=4 equally-spaced ⟹ ‖ρ4 C ρ4†−C‖~1e-14 (clock-invariant state); [N] family count selects n=4 (scan n∈{2..8}, only n−1=3=N_fam; n=3→2 families = hexagonal, excluded; n=5→4); [N] gap: μ4-equivariant ⟹ diagonal in the cusp basis (deck-average off-diagonal ~1e-16), cusp weights {0,1/3,2/3} ⟹ transfer (1−w)⁶={1,(2/3)⁶,(1/3)⁶}, gap (2/3)⁶=64/729; [N] position scan: the square minimises the defect (square ~1e-14 vs off-square ~0.4); [N] negative controls n=3,5,6. [O] confirms the emergence on the real DtN (number scanned, positions free); selector = the 3-family/gap structure (carrier input); executes K1 numerically, does NOT close QGEO.REALIZE.01. Python-only."
    },
    {
      "id": "v211_axion_spine_angle",
      "cluster": "frontier",
      "descriptionEN": "axion DM, the spine-angle branch θ_i = π N_fam/g_car = 3π/5 (108°): an ALTERNATIVE, more robust misalignment angle than the determinant-line hilltop θ_i≈170° (v185, which over-produces Ω_a h²≈0.66). [I] rational motive θ_i=π N_fam/g_car=3π/5=108° exactly (spine quotient 3/5, no fit); [N] the v185 finite-T solver (θ=1→0.03) reaches Ω_DM at θ≈106°, so 108° is essentially on target (harmonic 0.03θ²=0.107); [C] ROBUST: 108° is 62° below the 170° hilltop, in the mild-anharmonic regime — not exponentially sensitive, a far more robust landing than v185; [O] HONEST: an alternative ansatz to θ_i=π(1−φ_seam)≈170° (mutually exclusive, neither forced — the full solver decides, DM.AXION.SPINE.01); [X] kill test: full finite-T solve, Ω_a h² outside ~[0.08,0.16] demotes the branch. A [C] kill-test branch, NOT a closure."
    },
    {
      "id": "v212_leptogenesis_decuple",
      "cluster": "frontier",
      "descriptionEN": "leptogenesis, the scalaron-decuple branch: both Boltzmann inputs share the decuple A_Λ=10=|E(K₅)| — M₁=M_scal φ₀²/A_Λ≈8.65×10⁹ GeV and m̃₁=m₃/A_Λ≈5 meV. A cleaner [C] route than v184's M₁=M_R φ₀⁴ (which only relocated the free input to the un-pinned seesaw M_R). [I] shared decuple A_Λ=10; [N] M₁=M_scal φ₀²/A_Λ=8.65×10⁹ GeV (M_scal=c₃^{7/2} M̄≈3.06×10¹³, φ₀≈0.053172) inside the thermal window [3×10⁹,3×10¹⁰] (v169); [N] m̃₁=m₃/A_Λ≈5 meV (m₃≈0.05 eV), K≈4.6 strong washout; [N] Davidson–Ibarra ε₁≈8.5×10⁻⁷, η_B≈0.96×10⁻²·ε₁·κ_f≈1.2×10⁻⁹ brackets observed 6.1×10⁻¹⁰; [C] HONEST: M₁ uses only {M_scal,φ₀,A_Λ} (no hidden seesaw scale), but the coupling mechanism is posited not derived — η_B stays [C]; [X] kill test: a precise Boltzmann/RGE solve outside the band falls the route, not the theory (Route A/Ω_b independent, FR.ETAB.01)."
    },
    {
      "id": "v213_ftransfer_functor",
      "cluster": "frontier",
      "descriptionEN": "the F_transfer FUNCTOR CONTRACT (CONTRACT.F.01): the four frontier transfers are ONE typed functor F_transfer = F_observable∘F_threshold∘F_RG with four structural axioms, each instance a discrete compiler kernel [E/I] composed with an external continuous solver [C/O]; consolidates v183 (F_pole), v212 (F_Boltzmann), v211 (F_relic), v164/FR.MPME (F_QCD), the structural complement of the v187 typing guard. [I] axiom 1 μ4-deck equivariance (Koide multiplier λ₂=(2/3)⁶=64/729 IS the μ4-deck transfer eigenvalue, v82/v54); [I] axiom 2 Plücker preservation (Koide 53=aᵀ(R+Q)1, F-corner Pl(F)=(1,22,30); quark ratio ‖Pl(K)‖₁=11); [I] axiom 3 positivity/stochasticity (spec T={1,(2/3)⁶,(1/3)⁶} positive, washout κ_f∈(0,1)); [I] axiom 4 external modules explicit (pole transfer, DNN Boltzmann, finite-T χ(T), lattice C_N; b₃=−7 a carrier output, |b₃|=Ω_adm−10b₁=7); [C] verdict: a typed functor, NOT a bag of open topics; discrete kernels [E], external solvers [C]; an architectural consolidation guarded by v187, NOT a closure."
    },
    {
      "id": "v200_seam_variational_scan",
      "cluster": "registry",
      "descriptionEN": "QGEO.VARI.02 (work package B) — the discretised variational test: v196 had E_fail=0 at the μ4 config on the fixed block; here the clock angle is free and E_fail is minimised over it. [N] scanning E_fail(θ)=‖ρ(θ)⁴−1‖²+faithfulness over θ∈(0,π) gives the unique minimum at θ=π/2 (E_fail=0: order-4 AND three distinct characters i,−1,−i = the (1,2,3) grading); decoys (0, π, π/3, 2π/3) all lose. So argmin E_fail = the μ4 normal form. [E] consistent with v196/v199/v198. [O] a numerical sign on the finite H¹ grading, not the full operator-valued minimisation (the v199 residual)."
    },
    {
      "id": "v198_modular_commutator_reduction",
      "cluster": "registry",
      "descriptionEN": "QGEO.MODULAR.01 — cracking the bedrock to a state-invariance. The v194 residual ([ρ,Λ_Σ]=0 on full L², framed as Bisognano–Wichmann with a circularity worry) is (a) made EXACT at the principal-symbol level and (b) reduced via Tomita–Takesaki to a clean state-invariance. [E] the DtN principal symbol |k|=√(−d²) is diag(|n|) and the clock ρ:z↦iz is diag(i^n) in the Fourier basis — both diagonal ⇒ [ρ,|k|]=0 exactly on ALL of L²; [E] Tomita–Takesaki: [ρ,Λ_Σ]=0 follows from ω∘ρ=ω (modular Δ is intrinsic to (M,Ω); a state-preserving symmetry commutes with it), needing NO conformal covariance — removes the BW circularity; [E] the finite H¹ block is already μ4-invariant (v177). [O] residual: the full raw seam state is μ4-invariant (ω∘ρ=ω) ⇐ the Gaussian bulk measure is deck-invariant = operational QGEO.SYM.01. Sharpest non-circular form; not a closure (a foundational symmetry can't be derived from nothing)."
    },
    {
      "id": "v195_marks_lefschetz_character",
      "cluster": "registry",
      "descriptionEN": "QGEO.MARKS.02 — a Lefschetz/character derivation of the four seam marks (review path 2): instead of positing D4 marks, they are FORCED to be a free µ4 orbit. [E]: ρ:z↦iz (order 4) fixes only {0,∞}; H¹=χ1+χ2+χ3 (A3 exponents (1,2,3), v177) has no trivial character ⇒ no puncture at a fixed point; rank H_1=n−1=3 ⇒ n=4 ⇒ exactly one free orbit ⇒ marks = µ4 (cross-ratio 2); Tr(ρ|H¹)=i+i²+i³=−1, L(ρ)=1+1=2=#fixed. [O] residual relocates from 'seam is P¹∖µ4 (geometric)' to 'H¹ carries the A3-exponent rep' — less circular."
    },
    {
      "id": "v196_seam_energy_functional",
      "cluster": "registry",
      "descriptionEN": "QGEO.VARI.01 — the E_fail variational functional (review path 1), a numerically-testable sharpening of the v194/ENERGY.02 target: E_fail = ‖[ρ,Λ_Σ]‖²_HS + ‖ρ⁴−1‖² + ‖ΘρΘ−ρ⁻¹‖², with E_fail=0 ⇔ the QGEO.SYM.01 conditions. [E] on the finite H¹ block (ρ=diag(i,−1,−i), Λ diagonal, Θ=conj) all three vanish ⇒ E_fail=0; [N] a µ4-breaking Λ or non-order-4 ρ gives E_fail>0 (µ4 is a true minimiser); [I] the zero locus = the seam-deck conditions; [O] the full-operator minimisation = the v194 Bisognano–Wichmann residual — a discretisable handle on the open bedrock, a target not a closure."
    },
    {
      "id": "v197_rr_carrier_clifford_d5",
      "cluster": "registry",
      "descriptionEN": "ARCH.RRCAR.02 — hardening the Riemann–Roch carrier from g_car to D5 itself (review point 7): the 5-dim carrier mode space C_car:=H⁰(P¹,O(µ4)) generates the so(10)=D5 half-spinor by its even Clifford algebra. [I] dim Λ^even(C⁵)=C(5,0)+C(5,2)+C(5,4)=1+10+5=16=2^{g_car−1}=dim S⁺; so(10)=D5 half-spinor 2⁴=16=S⁺=Λ^even(C_car); closes the chain µ4→g_car→D5 from one four-point divisor (cycle side rank H_1=3=N_fam(A3); function side h⁰=5=g_car; Clifford spinor=16=D5 half-spinor=carrier). [C] 'carrier = Clifford spinor of the seam mode space' stays physical (as v189), does not change the over-determined g_car=5."
    },
    {
      "id": "v194_raw_seam_dtn_rp",
      "cluster": "registry",
      "descriptionEN": "QGEO.DTN.01 — subclaim 2 of QGEO.ENERGY.02 ('the raw RP-seam DtN is RP-definable') tackled: a REDUCTION, not a closure. (1) [E] finite core: ρ:z↦iz on H¹=⟨w1,w2,w3⟩, wk=z^{k-1}dz/(z⁴−1), acts as ρ*wk=i^k wk (characters i,−1,−i distinct), so [ρ,Λ_Σ]=0 on H¹ (v177). (2) [I] hinge met: the DtN/transfer operator is RP-canonical (Osterwalder–Schrader, v54) on a quasi-free seam state (v155), so Λ_Σ is RP-definable without assuming P¹∖µ4. (3) [O] residual relocates: full-L² commutation ⇔ Bisognano–Wichmann geometric modular flow of the quasi-free seam state (must be intrinsic, else re-circulates). QGEO.SYM.01 reduces from a definitional postulate to intrinsic BW modular covariance — sharper, still [O], not closed to [E]."
    },
    {
      "id": "v193_qgeo_energy_commutator",
      "cluster": "registry",
      "descriptionEN": "QGEO.ENERGY.02: the energy-COMMUTATOR contract (non-circular sharpening of QGEO.ENERGY.01): [ρ,Λ_Σ]=0 on the rank-8 Calderón polarisation, ρ the carrier clock, Λ_Σ the DtN energy form of the RAW RP seam. Three sub-claims: (1) [E] ρ is carrier-defined (Coxeter of W(A3)=S4, order 4=|µ4|, v117), not imported from the seam; (2) [O] the non-circularity hinge — Λ_Σ must be the RAW RP-seam DtN, not the µ4 normal form (circular if it needs P¹∖µ4 assumed); (3) [E] [ρ,Λ_Σ]=0 forces the (1,2,3) grading on H¹ (QGEO.COHOM.01). Exact [I] rider: the induced EH coefficient k=(ln m)/(12π) reproduces c3/2 iff ln m=q(A3)=3/4 (FAMILY), not q(D5)=5/4 (carrier, off by 5/3) — gravity is family-geometry-induced. Proof target [O]; if proven, QGEO.SYM.01 becomes an energy-rigidity theorem."
    },
    {
      "id": "v192_qgeo_energy_clock",
      "cluster": "registry",
      "descriptionEN": "the energy-conserving-clock reformulation, HONESTLY typed: QGEO.ENERGY.01 (ρ*Λ_Σρ=Λ_Σ — the carrier clock preserves the Calderón/DtN energy form) RELOCATES the bedrock QGEO.SYM.01 to a more checkable operator identity, but does NOT eliminate it. The downstream chain (conformality→Möbius→z↦iz→µ₄→QGEO.SYM.01) already exists (v180/v181); the new upstream link (energy invariance ⇒ conformality in 2D) is checkable on the DtN operator Λ=|k|. But 'ρ preserves Λ' is plausibly equivalent to 'ρ is the conformal deck', so it is a sharper restatement, not a theorem, unless ρ is independently defined and the invariance independently proven (open). Bedrock stays [O]."
    },
    {
      "id": "v188_frontier_wording_guard",
      "cluster": "registry",
      "descriptionEN": "the frontier-wording guard (a prose sentinel, no new physics): v187 protects the ledger typing, this guard protects the prose. It forbids stale sentences whose semantics an earlier round superseded — 'leptogenesis interface (mE)' (v184 retyped Route B to [C]), 'closed branch also fixes the leptogenesis inputs' (M_R is the seesaw scale, not a compiler power), 'relic scale f_a,m_a open' (v185: the scales define the branch, the open item is the hilltop-sensitive relic abundance), 'computed Q=0.664 … near, not exact' (v183 gave 53/54 an exact operator origin), 'Suite now 187 modules' (the count is 187, highest ID v188; v186 skipped). Scans the 10 active .tex documents; passes when none of the stale phrasings appear, so a frozen Zenodo release can never re-introduce half-true wording."
    },
    {
      "id": "v187_ftransfer_laws",
      "cluster": "registry",
      "descriptionEN": "the F_transfer discipline guard (CI-style firewall, the machine-readable form of the Frontier rule): the four continuous-transfer frontier observables — Koide (FR.KOIDE.*), η_B (FR.ETAB.*), axion (FR.DM.*), m_p/m_e+Λ_QCD (FR.MPME.*/QCD.LAMBDA.*) — must NEVER be typed as primitive [E]/[N] compiler predictions; each must carry a [C]/[A]/[O] marker. Reads the ledger and enforces it (all active frontier rows pass; exact algebraic sub-parts like Koide 53/54 v183 or b3=−7 v159 may be [I]/[E], the physical prediction never is). Records the four interfaces F_pole/F_Boltzmann/F_relic/F_QCD and that the raster rule + frozen registry + null model bind — preventing any future edit from quietly promoting a pretty frontier number to a closed compiler output."
    },
    {
      "id": "v183_koide_f_corner_transfer",
      "cluster": "registry",
      "descriptionEN": "the Koide source→pole factor 53/54 has an OPERATOR origin (external-review proposal, validated): u_pole/u_source = aᵀ(R+Q)𝟙 / (2·𝟙ᵀRa) = 53/(2·27) = 53/54. (1) 𝟙ᵀRa=27 is the E6×A2 cubic family block (248 = dim E6 78 + det R 8 + 2·27·N_fam); (2) F=R+Q is the MISSING Sheet-Diamond corner M(1,1) with det B_F=52=dim F4 (v80) and anchor response aᵀ(R+Q)𝟙=53=52+1; (3) 53/54 = 1−1/(2N_fam³) exactly (54=2·27); (4) negative control: only F gives 53 (R/Q/K/L give 32/21/35/56). Upgrades FR.KOIDE.03 from a bare guess to an operator identity; [I] factor exact, [C] the leptonic source→pole reading the F corner stays an F_transfer conjecture."
    },
    {
      "id": "v182_reviewer_residual_map",
      "cluster": "registry",
      "descriptionEN": "the four external-review concerns reduce to the two already-named residuals + one data-only item: A (mapping 2D→4D, ''why read the masses as Plücker 11?''), B (UV-gravity abstractness), C (grammar-tuning/meta-look-elsewhere), D (Koide Möbius flow). A [I]/[C]: 11=Σ_{k≤2}C(4,k)=dim S⁺−g_car=16−5 (QBL count, v174), the 2D→4D step is the holographic reduction ⇒ A = QGEO.SYM.01 + F_transfer. B [I]: 'gravity = geometric boundary readout' IS QGEO.SYM.01. C [N]/[C]: frozen (v84) + minimal + over-determined (anchor a=(1,1,2) in ≥6 independent structures) ⇒ look-elsewhere-resistant, residual falls only to data (JUNO/CMB-S4). D [I]/[C]: Möbius forced by Aut(ℙ¹)=PSL(2,ℂ) on the deck, t~2.84 locates the point, the missing RG generator IS F_transfer ⇒ D = QGEO.SYM.01 + F_transfer. NET: three reduce, one waits for data — a unification of the concern structure, NOT a closure (premises stay [O]/[C])."
    },
    {
      "id": "v181_clock_is_conformal_symmetry",
      "cluster": "registry",
      "descriptionEN": "the FINAL honest reduction + the explicit identification of BEDROCK. The isometry premise QGEO.ISO.01 (v180) is weakened one more genuine step — to a CONFORMAL-symmetry premise QGEO.SYM.01 — via equivariant uniformisation (Nielsen realisation, finite cyclic on genus-0): a CONFORMAL automorphism (strictly weaker than an isometry — every isometry is conformal, conformal allows any positive scale) already suffices to put the order-4 clock in the form z↦iz. The clock is the Coxeter element of W(A3)=S4 (v117), order h(A3)=4=|μ4| = the μ4 deck, and a deck transformation preserves the pulled-back conformal structure by definition. So below QGEO.SYM.01 the residual is DEFINITIONAL, not a missing theorem: 'the seam's conformal structure IS the μ4-deck structure of the carrier clock' (the carrier μ4 glue is the conformal monodromy of the seam) — a physical/definitional identification tying P2 to the seam's conformal geometry, NOT reducible further without relabeling. The chain REALIZE(v175)→theorem(v176)→MARKS+KERNEL(v177)→finite cores(v178)→CONF.01(v179)→ISO.01(v180)→SYM.01(v181) has reached bedrock; we stop here honestly."
    },
    {
      "id": "v180_clock_is_mobius",
      "cluster": "registry",
      "descriptionEN": "cracking QGEO.CONF.01 one level further — the conformal-realisation premise is REPLACED, via three classical theorems, by a strictly MILDER, non-circular premise QGEO.ISO.01: 'the carrier clock is an order-4 orientation-preserving ISOMETRY of the RP seam metric'. (1) Uniformisation (Poincaré-Koebe): a genus-0 Riemann surface is conformally ℙ¹, so the genus-0 seam double (P1, v179) IS ℙ¹ — the target is produced, not assumed; (2) Kerékjártó (1919; Constantin-Kolev 2003): a finite-order orientation-preserving homeo of S² is conjugate to a rotation, so the order-4 clock (h(A3)=4) is topologically z↦iz; (3) isometry⇒conformal⇒Möbius: an orientation-preserving isometry of a Riemannian surface is holomorphic, Aut(ℙ¹)=PSL(2,ℂ); (4) order-4 Möbius = z↦iz (verified: projective order 4, fixed {0,∞}, multiplier i, free orbit μ4). So QGEO.CONF.01 is IMPLIED by the milder isometry premise; the new residual QGEO.ISO.01 (the seam transport is a metric-compatible holonomy preserving the RP inner product) does not name ℙ¹/μ4 and is milder, but its surface realisation from the raw seam stays OPEN."
    },
    {
      "id": "v179_conformal_realisation",
      "cluster": "registry",
      "descriptionEN": "the two residual premises (MARKS-residual + KERNEL-residual) collapse to ONE geometric premise QGEO.CONF.01 (honest unification, the conformal realisation stays OPEN). MARKS's residual is NOT three independent assumptions: (1) genus-0 IS P1 — c3=1/(|Z2|·2π·χ(S²))=1/(8π) with χ(S²)=2 ⇒ g=0, the same χ=2 that gives the '8' (v58/v73); (2) the order-4 clock IS the carrier — Coxeter element of W(A3)=S4, order h(A3)=4=|μ4| (v117); (3) marks=one orbit is FORCED — parabolic marks ≠ the 2 elliptic fixed points {0,∞}, a free order-4 orbit has exactly 4=e1(a) points. So MARKS reduces to 'the carrier clock acts conformally on the genus-0 double'. KERNEL then FOLLOWS from that geometry: DtN symbol |k| (Lee-Uhlmann, v156) ⇒ rank-8 c=8 fixed polarisation; no drift (v158); the cusped-surface residual data is H¹=ℂ³=N_fam = the 3-dim gapped transfer block (Schur, v162/v178). Both ⇒ ONE premise QGEO.CONF.01: 'the raw RP seam normal double is conformally (ℙ¹,μ4) with the carrier clock as the order-4 Möbius map'. The two doors of v177/v178 are one geometric realisation — the single structural residual of the whole theory, left honestly OPEN."
    },
    {
      "id": "v178_marks_kernel_attempt",
      "cluster": "registry",
      "descriptionEN": "an honest ATTEMPT at the two open obligations QGEO.MARKS.01 + QGEO.KERNEL.01 — NOT a closure (they are genuine constructive-geometry/AQFT statements, not finite computations), but a deeper reduction: each finite/algebraic CORE is closed [E] and each obligation reduced to ONE sharper irreducible [O] premise. MARKS finite core [E]: a genus-0 double with an order-4 clock ρ (~z↦iz, fixed points {0,∞}) and sheet reflection τ forces the marks to be a generic ρ-orbit {1,i,−1,−i}=μ₄ (NOT the fixed points), distinct iff b₁=4−1=3=N_fam (collision drops b₁, excluded), τρτ=ρ⁻¹ ⇒ faithful D4 order 8 — so MARKS reduces to the single topological/clock premise. KERNEL finite core [E]: on the 3-dim multiplicity-free H¹ the μ₄ characters (1,2,3) are distinct, so by Schur a μ₄-equivariant operator is forced DIAGONAL and completely determined by its eigenvalue per character — two such with the same forced transfer spectrum {1,(2/3)⁶,(1/3)⁶} (v162) and the unique residue-normalised basis (v177) are EQUAL as operators (the 'spectrum vs operator' worry vanishes on the finite block); so KERNEL reduces to the single full-operator-reduction premise (principal symbol |k| Lee-Uhlmann v156, no drift v158). NET: neither obligation closed; both sharpened to milder premises."
    },
    {
      "id": "v177_seam_marking_kernel",
      "cluster": "registry",
      "descriptionEN": "the QGEO proof tree — the v176 'one realisation premise' split into its honest constituents: REALIZE = MARKS · UNIFORM · COHOM · MODULE · KERNEL · NET, with FOUR nodes closed CONSTRUCTIVELY as [E] and the open residual sharpened into exactly TWO named obligations. [E] UNIFORM (Lemma 2, constructive): an order-4 Möbius map conjugates to z↦iz, a non-fixed orbit scales to μ₄, the reflection z↦1/z gives στρτ=ρ⁻¹ (D4 order 8) — a genus-0 4-marked faithful-D4 boundary IS (ℙ¹,μ₄); [E] COHOM: ρ*ωk=i^kωk so the H¹ grading is (1,2,3)=A3 exponents=Spec(Q₊); [E] MODULE: the H¹→generation iso is UNIQUE (multiplicity-free + residue normalisation; reflection ω1↔ω3, ω2 fixed); [E] NET (v154/v175): index-4 μ₄ extension of (D5)₁⊗(A3)₁ = (E8)₁. The TWO genuine open obligations [O]: QGEO.MARKS.01 (raw seam → genus-0 four-marked D4 boundary) and QGEO.KERNEL.01 (raw seam Calderón kernel = the μ₄-equivariant free gapped contraction as an OPERATOR). Removes the near-circular v176 framing (which took the marks as a hypothesis); the whole structural residual is now exactly these two doors."
    },
    {
      "id": "v176_seam_collar_realisation",
      "cluster": "registry",
      "descriptionEN": "the Seam Collar Realisation Theorem assembled as ONE central statement + reduction certificate (the single structural proof obligation of the theory, formulated cleanly — NOT a fabricated proof): the theorem 'the seam collar's conformal boundary is ℙ¹∖μ₄ with H¹ grading (1,2,3), Calderón contraction = rank-8 K_Σ, index-4 extension = (E8)₁' decomposes into six steps, FIVE already [E] — geometry/uniformisation (v168: cross-ratio 2, faithful Möbius D4 order 8, b₁=N_fam=3), Calderón DtN symbol |k| (v156), H¹ character = generation grading (1,2,3)=A3 exponents=Spec(Q₊) (v137/v168), μ₄-equivariant one-particle contraction diagonal in the cusp basis with gap (2/3)⁶ (v162), AQFT closure (D5)₁⊗(A3)₁⋊μ₄=(E8)₁ + full-cone RP all m (v154/v175) — so the WHOLE structural residual reduces to EXACTLY one open premise: that the physical seam collar's conformal boundary IS this ℙ¹∖μ₄ (QGEO.REALIZE.01), a constructive-geometry statement left honestly OPEN."
    },
    {
      "id": "v175_net_existence_full_cone",
      "cluster": "registry",
      "descriptionEN": "the heavy artillery — net existence (A2) and full-cone reflection positivity discharged to [E], isolating QGEO.REALIZE.01 as the single open premise (nothing fabricated): (1) FULL-CONE RP FOR ALL m — the CAR second-quantisation functor Γ(t)=⊕_m Λ^m(t) of the one-particle seam contraction t acts on the complete Fock space Λ^•(ℝ¹⁶) (dim 2¹⁶=65536); the exterior-power spectrum theorem makes every eigenvalue a subset product of spec(t), verified on all 2¹⁶ products to lie in [0,1] (contraction), with eigenvalue-1 multiplicity 2⁸=256 (wedges of the rank-8 K_Σ polarisation) and sub-leading = the one-particle gap (2/3)⁶, so full-cone RP reduces for ALL m to the one-particle contraction (Shale-Stinespring CAR functor), discharging the v160 scope premise by a theorem; (2) A2 net existence assembled + verified — (E8)₁ character E4/η⁸=1+248q+4124q²+34752q³+213126q⁴+1057504q⁵ (level-1=248), conformal embedding 248=120+128, E8 Cartan even unimodular (det 1, the unique rank-8 even unimodular lattice); the net EXISTS by cited theorems (free-fermion net, lattice VOA, index-4 Q-system); (3) both collapse to the SAME seam-kernel identification = QGEO.REALIZE.01, whose finite half is v168 and whose realisation half is left OPEN."
    },
    {
      "id": "v202_rare_kaon",
      "cluster": "flavor",
      "descriptionEN": "the rare kaon sector K → π ν ν̄ from the TFPT CKM point ([C] downstream flavor readout, NOT a compiler power). The kernel fixes the full CKM matrix (s₁₂=λ_C, s₂₃=φ₀/(1+λ_C)=|V_cb|, s₁₃=λ_C³/3, δ=π/3+3λ_C²=1.19823 rad; v18/v88) with no flavor fit; feeding the exact point into the standard Brod–Gorbahn–Stamou short-distance formulas gives BR(K⁺→π⁺νν̄)=9.45×10⁻¹¹ and BR(K_L→π⁰νν̄)=3.33×10⁻¹¹. [I] CKM point + exact apex (ρ̄,η̄)=(0.1374,0.3509), λ_t=V_ts*V_td; [C] the branching ratios (NOT parameter-free — X_t, P_c, κ_± are external EW/QCD input); [N] BR(K⁺)=9.45×10⁻¹¹ lands ON the NA62 2016-2024 combination (9.6 +1.9/-1.8)×10⁻¹¹ (La Thuile 2026), +0.08σ (the 2016-2022 obs (13.0₋₃.₀⁺³·³)×10⁻¹¹ was ~1.2σ above) — a strong consistency hit, but an F_transfer bridge, not a unique TFPT-vs-SM discriminator (SM (8.6±0.4)×10⁻¹¹ also inside the NA62 error); [I] Grossman–Nir ratio 0.352 ≪ 4.3; [X] dated kill test: a stable NA62 BR(K⁺) outside [7,12]×10⁻¹¹, or a KOTO-II BR(K_L) off the GN point, breaks the flavor bridge (core untouched)."
    },
    {
      "id": "v203_eht_achromatic",
      "cluster": "horizon",
      "descriptionEN": "the EHT achromatic polarization-rotation intercept β_BH around a horizon-scale black hole — the surviving core of the old UFE/BH notes (NOT the RN-type metric, superseded by Nariai/seam=horizon v101–v104/v190; only the polarization signature). [E] the coupling is a compiler number: β_BH(r)=Q_e^eff Q_m^eff/(256π⁴r²)=16c₃⁴(Q_eQ_m/r²), 16c₃⁴=1/(256π⁴) exactly. [E] α-kernel cross-link: 16c₃⁴=δ_top/3, δ_top=48c₃⁴=3/(256π⁴) — the SAME top-form coefficient that fixes the α-kernel precision-zone correction (one row from β_rad=φ₀/4π). [C] TFPT fixes the 1/r² profile, achromaticity, and sign-flip under E·B reversal; the amplitude Q_eQ_m is an MHD/GR weight, not a compiler output. [X] dated kill test: the residual intercept must pass three nulls (frequency/achromatic, spatial 1/r², sign-flip); a null-consistent residual after honest GRMHD subtraction falsifies the channel (core untouched)."
    },
    {
      "id": "v204_muon_seam_g2",
      "cluster": "frontier",
      "descriptionEN": "the muon anomalous magnetic moment a_μ=(g_μ−2)/2 as a seam vertex readout ([C] downstream, NOT a compiler power; archive integration of the old muon-g2 note). The carrier fixes the second-order topological defect δ₂=B·γ·δ_top²=(5/4)δ_top² (δ_top=Ω_adm c₃⁴=48c₃⁴=3/256π⁴, v2/v23; B·γ=(3/2)(5/6)=5/4 the carrier compression quotient, v51); the seam-loop projection a_μ^seam=δ₂/(2π)=45/(524288π⁹)≈2.879×10⁻⁹. [E] the VALUE is an exact compiler number (trace reading δ₂=4!·Tr_{S⁺}(X²)·c₃⁸, Tr=120=5!); [C] the IDENTIFICATION of δ₂/(2π) as a magnetic vertex (1/2π=4c₃ seam unit) is a physical bridge; [N] 0.81σ vs dispersive Δa_μ=(2.49±0.48)×10⁻⁹; [C] HONEST: lattice/CMD-3 HVP shrinks Δa_μ (~1.5×10⁻⁹), the FIXED value then ~1.5σ high; [X] dated kill test: a converged Δa_μ outside 2.879×10⁻⁹±0.5×10⁻⁹ excludes the mechanism (core untouched)."
    },
    {
      "id": "v205_xi_threequarter",
      "cluster": "gravity",
      "descriptionEN": "ξ=c₃/φ_tree=3/4: an INDEPENDENT route to the gravitational 3/4 (archive integration of the old Eliminating-K note). The torsion-compression ansatz κ²=ξφ₀/c₃² with the Einstein limit κ²=8πG fixes ξ=8πc₃²/φ₀=c₃/φ₀ (since 8πc₃²=c₃), and at tree level φ_tree=1/(6π) this is EXACTLY 3/4. [E] tree identity ξ=c₃/φ_tree=(1/8π)/(1/6π)=3/4; [E] Einstein-limit reduction 8πc₃²=c₃ ⇒ G is an output, not an input; [E] OVER-DETERMINATION 3/4=q(A₃)=N_fam/|μ₄|=ln(m/μ) (v152 k=c₃/2) — the torsion-compression ξ and the gapped EH replica are two independent appearances of the same gravitational 3/4; [N] with the retained seed ξ=c₃/φ₀=0.748303 (−0.226% from 3/4); [C] the κ²=ξφ₀/c₃² ansatz is the [C] bridge, the modern derivation is v152/v68, 1/G stays the one anchor (v60/v153)."
    },
    {
      "id": "v206_h0_lambda_branch",
      "cluster": "gravity",
      "descriptionEN": "the cosmological-constant Letter branch: the seam number δ_Σ=m₁e^{−2πα⁻¹/3} with the lowest-block m₁=Ω_adm=48, the Planck-convention split, and the quantitative H₀=66.5..67.1 (archive integration of the tfpt-31 CC note; sharpens the qualitative H₀-vs-Λ row). [N] δ_Σ^lead=48 e^{−2π·137.036/3}≈1.085×10⁻¹²³; [E] convention split M_Pl⁴/M̄_Pl⁴=(8π)²=631.65 (unreduced vs reduced Planck density — must be displayed); [N] ρ_Λ=M_Pl⁴δ_Σ≈2.4×10⁻⁴⁷ GeV⁴ (observed vacuum order); [N] 123-orders cross-link |log₁₀δ_Σ|≈122.96, consistent with the v60 split 122.948; [C] under flatness closure H₀^lead=66.51, H₀^disc=67.10 km/s/Mpc — 1.6..0.5σ BELOW Planck, ~6σ below SH0ES: the Hubble tension is NOT relieved by raising the vacuum scale; H₀ uses imported Ω_b, ω_DM, so it is a [C] readout."
    },
    {
      "id": "v207_asymptotic_safety",
      "cluster": "frontier",
      "descriptionEN": "Asymptotic Safety as a SECOND, independent, [P] route to the metric sector (archive integration of the old Five-Problems note). The current theory closes QG via the discrete seam-net/AQFT programme (v175; open premise QGEO.REALIZE.01); this records a COMPLEMENTARY continuum-FRG argument that a non-Gaussian UV fixed point exists from the same axioms — a plausibility cross-check, NOT load-bearing, does NOT close G_metric. [E] the axion coupling is fixed and small y²=16c₃²=1/(4π²)=0.025330 exactly (g_aγγ=−4c₃ from the cubic); [N] existence by IVT: d_t g/g=2+η_R is +2>0 as g→0 and <0 at large g (torsion K² damping) ⇒ a root g_*>0; [N] g_*=2/a₁>0 finite, Reuter-type O(1), no new free input; [P] HONEST: a continuum-FRG ansatz the discrete-compiler theory did NOT adopt, primary stays the seam-net (v175), G_metric open either way."
    },
    {
      "id": "v208_bh_thermodynamics",
      "cluster": "horizon",
      "descriptionEN": "black-hole thermodynamics from the seam: the modular Hawking temperature and the scalaron-corrected Wald entropy (archive integration of the dropped tfpt-42 stationary-horizons/compact-objects section, re-typed through the current Tomita–Takesaki seam work v198/v199). [E] MODULAR T_H: Δ^{it}=e^{−2πtK_H} (Bisognano–Wichmann) ⇒ KMS at β=2π and T_H=κ/(2π); the 2π IS the seam unit 1/(4c₃) (1/2π=4c₃, v58), reproducing T_H=c₃/M (v57); seam↔horizon identification is [P]; [E] WALD ENTROPY: with the induced f(R)=R+R²/(6M_s²) (M_s=c₃^{7/2}M̄, v28), S_W=(f_R A)/(4G)=(A/4G)(1+R_h/3M_s²) — leading A/4G is the c₃ area law (1/4=1/|μ₄|), the correction an EXACT scalaron consequence; [E] SdS T_H=(1/4πr_h)(1−Λr_h²), 1/4π=2c₃; [E] area law S_BH=M²/(2c₃)=4πM²=A/4G."
    },
    {
      "id": "v209_bh_regular_core",
      "cluster": "horizon",
      "descriptionEN": "the black hole as a seam fixed point (topological defect), NOT a singularity: the honest [P] synthesis of the old Five-Problems BH picture, with the SUPERSEDED torsion/RN-vortex metric DROPPED (RN-type metric superseded by Nariai/seam=horizon v101–v104/v190; cf. v203). Built entirely from established suite atoms. [E] NO SINGULARITY = SEAM FIXED POINT: the gapped attractor multiplier λ₂=(2/3)⁶=64/729 with gap Δ=6log(3/2)>0 (v56) — the core is a fixed point (dφ/dt=0), ρ→∞ replaced by φ→φ_* [P]; [E] MAXIMAL BH = ANCHOR: Nariai cubic t³−3t+2=(t−1)²(t+2), roots (1,1,−2) = the traceless projection of a=(1,1,2) (a−(p₁/3)1=−(1/3)(1,1,−2), p₁=4; v101/v190); [E] INFORMATION stays in the boundary field: Page recovery decays at λ₂=(2/3)⁶ per step (v54), no interior loss [P]; [E] holographic remnant S_BH=A/4G, 1/4=1/|μ₄|, a Planck-scale floor [P]; [P] HONEST: a qualitative structural synthesis (reuses v54/v56/v57/v101/v190), NOT load-bearing, NOT a metric."
    },
    {
      "id": "v218_diamond_axis_geometry",
      "cluster": "registry",
      "descriptionEN": "Diamond axis geometry: the Sheet Diamond (v94) is a discrete geometry with two axes around the centered cross (v95: C center, U winding, V sheet), and the missing corner F is the transfer completion. Three exact [E] lemmas + honest audit/heuristic typing, no new numbers. [E] AXIS CURVATURE: det(C+xU)=14+6x linear (winding flat, slope 6=|R⁺(A₃)|), det(C+yV)=14+14y+4y² quadratic with 2nd difference 8=rank E₈; det B(C+xU)=2 det(C+xU), anchor-block sheet 2nd difference 6=|R⁺(A₃)| — curvatures (8,6)=(rank E₈, |R⁺(A₃)|). [E] PLÜCKER TRANSFER LADDER: Pl(K)=(−1,6,4)→Pl(C)=(0,14,14)→Pl(F)=(1,22,30); C−K=(1,8,10)=(N_Φ,rank E₈,A_Λ), F−C=(1,8,16)=(N_Φ,rank E₈,dim S⁺) — decuple 10 then full generation 16 (identity [E], transfer reading [C]). [E] SPECTRAL RAMIFICATION: for Q,K,C,F the cubic discriminant=q(r)²·Disc(q); squares (1,3,4,6)=(N_Φ,N_fam,|μ₄|,|R⁺(A₃)|), kernels (13,48,65,105)=(Δ_Q,Ω_adm,g_car·Δ_Q,N_fam·g_car·7), F carries 105=3·5·7. [I] audit anchor-defect spectrum Δ_a(Q,K,R,C,L,F)=(8,16,18,21,24,26); [I] audit pair-sum atlas {12,24,28,36,40,52} total 64, staircase sum 81=N_fam⁴ — 28=2 dim G₂, 52=dim F₄ audit-only, not spine. [C] heuristic: F the transfer-completion corner, a search principle not a hard filter."
    },
    {
      "id": "v219_icosahedral_mckay",
      "cluster": "core",
      "descriptionEN": "McKay bedrock: WHY the atoms are 2,3,5. E₈ is the exceptional TOP of the McKay tower of finite SU(2) subgroups (2T→Ê₆, 2O→Ê₇, 2I→Ê₈). The McKay graph is BUILT from the group: the 120 icosians close to 2I (orders {1,2,3,4,5,6,10}, containing the 2,3,5 axes), 9 conjugacy classes, Dixon irrep degrees {1,2,2,3,3,4,4,5,6} (sum 30=h(E₈), sum² 120=|R⁺(E₈)|); the natural-rep McKay adjacency IS the affine E₈ graph with marks = degrees. Backward certificate of E₈, not a P2 proof."
    },
    {
      "id": "v222_cm_norm_duality",
      "cluster": "core",
      "descriptionEN": "CM-norm duality: the two exceptional moduli give 41 and 7. SQUARE (Gaussian ℤ[i], j=1728): N(g_car+i|μ₄|)=5²+4²=41=10b₁ (the EM index IS the Gaussian norm of the carrier-glue vector); N(3+2i)=13=Δ_Q. HEX (Eisenstein ℤ[ω], j=0): N(3+2ω)=3²−3·2+2²=7=scalaron. The (3,2)=(colour,weak) split generates (5,6,7,13) by sum/product/Eisenstein/Gauss norm; rings rigid (Eisenstein of (5,4)=21≠41)."
    },
    {
      "id": "v223_coxeter_totative_clock",
      "cluster": "horizon",
      "descriptionEN": "The μ₄ clock is the order-4 character of the E₈ Coxeter cycle. (ℤ/30)ˣ={1,7,11,13,17,19,23,29}=the E₈ exponents=live Coxeter phases; φ(30)=8=rank E₈, 30=2·3·5. (ℤ/30)ˣ≅ℤ/4×ℤ/2 with order-4 generator 7 (7²=19,7⁴=1 mod 30): a literal μ₄ clock ⟨7⟩={1,7,13,19} on the exponents; conjugate pairs m+(30−m)=30 give |μ₄|=4 invariant planes the clock permutes. Only E₈ has φ(h)=rank AND h=2·3·5."
    },
    {
      "id": "v227_degree_exponent_channel_split",
      "cluster": "core",
      "descriptionEN": "248=120+128 as a magnitude/phase channel typing (replaces the over-strong S⁻-dark-matter reading). E₈ exponents {1,7,11,13,17,19,23,29} sum 120=|R⁺(E₈)| (MAGNITUDE channel); E₈ degrees {2,8,12,14,18,20,24,30} sum 128=2^(rank−1)=rank·dim S⁺ (PHASE/GLUE channel); 248=120+128=adj(D₈)+spinor(D₈); deg−exp sum=rank=8. Ratios+products (Target D) live in 120, the un-covered CP phases in 128; the dark-sector reading of 128 stays Frontier."
    },
    {
      "id": "v228_rr_index_gate",
      "cluster": "core",
      "descriptionEN": "P2 as the mode space of a degree-4 seam divisor (a Riemann-Roch INDEX GATE; bedrock language for P2, not a proof from nothing). deg D=4=|μ₄| on P¹ ⇒ h⁰(O(D))=5=g_car (Riemann-Roch, genus 0); rank H₁(P¹ minus 4 marks)=3=N_fam; h⁰+rankH₁=8=rank E₈, h⁰−rankH₁=2=|Z₂|; Λᵉᵛᵉⁿ(C⁵)=1+10+5=16=dim S⁺. Conditional on the raw seam producing the degree-4 divisor (=QGEO.SYM.01)."
    },
    {
      "id": "v220_cp_hexagonal_modulus",
      "cluster": "flavor",
      "descriptionEN": "CP lives in the HEXAGONAL phase fiber, not the square seam deck (a geometric sharpening of red-team Target D). The seam deck is the SQUARE modulus j(i)=1728, Aut Z/4 (the μ₄ clock); its exceptional partner is the HEX modulus j(ρ)=0, ρ=e^{iπ/3}, Aut Z/6, CM by ℤ[ω], arg(ρ)=π/3. The frozen leading CKM reading δ=π/3+3λ_C²=68.654° (v88) has leading term π/3=arg(ρ); the |Z₂| alternative π/3+2λ_C² is the frozen look-elsewhere trap. The deck stays Z/4 (v215); CP is the Z/6 fiber over it."
    },
    {
      "id": "v225_dual_normal_frame",
      "cluster": "flavor",
      "descriptionEN": "The dual normal frame (d,n): one boundary compass for flavor+horizon, with CP as orientation. d=aᵀR⁻¹=aᵀL⁻¹=(−1/2,−1/2,1)=−½(1,1,−2) (the Nariai/horizon normal; d·1=0, d·a=1); n=(5,−9,6) with nᵀR=(det R,0,0)=(8,0,0), nᵀL=(20,0,0) (first-generation/torsion selector). Oriented volume det(1,d,n)=21=N_fam·scalaron. CP as orientation: Im det(1,d,ρn)=21 sin(π/3) for ρ=e^{iπ/3} — CP sits where the magnitude bijection (Target D) fails."
    },
    {
      "id": "v224_diamond_ftransfer_path",
      "cluster": "masses",
      "descriptionEN": "F_transfer as ONE path on the Sheet Diamond, not four external interfaces. Family M(s,t)=R+Q diag(s,t,t): the transfer corners K=M(1,−1),C=M(1,0),F=M(1,1) lie on the SHEET axis (s=1). The sheet axis is curved det M(1,t)=4t²+14t+14 (2nd diff 8=rank E₈), the winding axis det M(s,0)=6s+8 flat (slope 6=|R⁺(A₃)|); Plücker ladder Pl(C)−Pl(K)=(1,8,10), Pl(F)−Pl(C)=(1,8,16). F_transfer = the K→C→F sheet geodesic, Koide source→pole its 1-D Möbius projection."
    },
    {
      "id": "v229_lepton_frobenius_algebra",
      "cluster": "masses",
      "descriptionEN": "The charged leptons as an étale Frobenius algebra over the μ₆ resolvent. Exact coefficients (c_e,c_μ,c_τ)=(16/7,4/3,7/6), ring closure c_e c_τ=8/3=|Z₂|c_μ, product 32/9=2^g_car/N_fam². A=ℚ[t]/(m), m=∏(t−c): the trace-form Gram is nondegenerate (det=disc(m)≠0, A étale) — a commutative Frobenius algebra. The C₆ shift has charpoly t⁶−1 (spectrum μ₆=μ₃ family ×μ₂ sheet). PMNS extension via Aut(A)+the hex CM point is [C]."
    },
    {
      "id": "v221_seam_qecc",
      "cluster": "horizon",
      "descriptionEN": "The seam as a FINITE recoverability code (recovery rate (2/3)⁶=64/729). Code dimension dim S⁺=16=2^(g_car−1); the gapped transport T on the cusp-weight 3-space (deviations (1,−1,0) and the Nariai anchor (1,1,−2)) is symmetric, doubly stochastic (CPTP), spectrum {1,(2/3)⁶,(1/3)⁶}; on the trace-zero subspace ||T δ||≤(2/3)⁶||δ|| so ||Tⁿ δ||≤(2/3)^{6n}||δ|| (Knill-Laflamme-type bound); a free ratio breaks it. Holographic/Petz reading [C], gated on the Seam-Horizon theorem. Python-only (numerical contraction bound)."
    },
    {
      "id": "v230_center_budget_norms",
      "cluster": "flavor",
      "descriptionEN": "The Sheet-Diamond center budget (7,11,13) is the three LOCAL NORMS of the theory. C=R+Q diag(1,0,0), row sums (7,11,13): 7=N_ℤ[ω](3+2ω) (Eisenstein/hex = scalaron), 13=N_ℤ[i](3+2i) (Gauss/square = Δ_Q), 11=C(4,0)+C(4,1)+C(4,2)=1+4+6 (QBL boundary Fock count on 4 μ₄ corners=dim S⁺−g_car=16−5). The two CM rings (v222) and the seam boundary count meet in one operator center; the ring assignment is rigid."
    },
    {
      "id": "v231_cp_mu6_phases",
      "cluster": "flavor",
      "descriptionEN": "Both CP phases are mu6 powers of ONE hexagonal CM unit, split by the sheet -- a structural reduction of red-team Target D. With rho=e^{i pi/3} (the j=0 Eisenstein CM unit): delta_CKM leading = arg(rho^1)=pi/3 (quark), delta_PMNS = arg(rho^4)=4pi/3 (lepton, the phase lattice); rho^4=-rho so delta_PMNS = delta_CKM,lead + pi = (CM unit)x(Z2 sheet, rho^3=-1) -- the two CP phases are ONE hexagonal unit on the two sheets, not two independent inputs. The C6/mu6 monodromy carries rho (charpoly t^6-1); the dual-frame orientations are sheet-flipped Im det(1,d,rho^1 n)=+21 sin(pi/3), Im det(1,d,rho^4 n)=-21 sin(pi/3). NEG: mu3 has no -1 (no sheet). Removes one of Target D's two phase inputs; the quark seam correction 3 lambda_C^2 stays the v88 misalignment, the deck stays Z/4."
    },
    {
      "id": "v232_e8_kleinian_seam",
      "cluster": "horizon",
      "descriptionEN": "The seam as the E8 Kleinian singularity -- a canonical algebraic-geometric MODEL for the open seam-realisation premise QGEO.REALIZE.01 (the du Val side of the McKay correspondence, v219). The McKay graph of 2I is affine E8; dropping the unique trivial-rep node (single Kac mark=1) leaves the finite E8 Dynkin on 8 nodes = the dual intersection graph of the 8 exceptional P^1's of the minimal resolution of C^2/2I (trivalent node, arms 1,2,4). EIGHT exceptional P^1's = rank E8 = g_car+N_fam = #nontrivial 2I irreps -- a fourth reading of the '8'. The negated intersection form is the E8 Cartan (even, det 1, positive definite; each P^1 a (-2)-curve). One degree-1 irrep => 2I perfect => H1(S^3/2I)=0: the link is the Poincare homology 3-sphere, pi1=2I order 120=|R+(E8)|. Seam reading: the raw RP seam (P^1 with marks) is canonically a du Val exceptional curve -- a model for QGEO.REALIZE.01, not a P2 proof (presupposes E8)."
    },
    {
      "id": "v233_cp_triality_phase",
      "cluster": "flavor",
      "descriptionEN": "The CP phase is the UNIVERSAL family/triality phase, only sheet-split -- one step past v231 in mu6 = mu3 (family/triality) x mu2 (sheet). mu6=mu3 x mu2 (gcd(3,2)=1): the Z3 triality centre {1,omega,omega^2} = cusp weights {0,1/3,2/3}x2pi, sheet mu2={1,-1}; rho = omega^2*(-1). SAME FAMILY CLASS: rho^4 = rho^1*rho^3 with rho^3 in mu2, so quark (rho^1) and lepton (rho^4) CP phases have the SAME image in mu6/mu2~mu3 and OPPOSITE sheet -- so the CP phase IS the universal triality phase omega^2 (the 2/3 cusp-weight Z3 centre), NOT a free power choice. omega is the mu3 factor of the order-30 (2,3,5) monodromy that builds E8 (v232). Reduces Target D from 'one hexagonal unit + sheet' (v231) to 'the universal triality phase + sheet'; physical CP derivation stays [C]."
    },
    {
      "id": "v234_seam_holomorphy_selection",
      "cluster": "frontier",
      "descriptionEN": "The Seam-Holomorphy selection certificate: the entire structural residual is ONE condition with THREE equivalent faces, all forcing E8. THE CONDITION: 'the seam carries no nontrivial abelian sector.' (i) AQFT: the boundary net is holomorphic (mu-index 1); (ii) geometry: the seam link is a homology 3-sphere <=> the binary group is perfect <=> 2I; (iii) rep theory: exactly one 1-dim irrep. #(mark-1 affine-Dynkin nodes) = |Gamma^ab| = |H1(S^3/Gamma)| = #(1-dim irreps): A_n->n+1, D_n->4, E6->3, E7->2, E8->1 -- ONLY E8 gives 1 (2I the unique perfect SU(2) subgroup, Poincare the unique homology-sphere space form). Holomorphic c=8=g_car+N_fam => unique even unimodular rank-8 lattice = E8. The three faces (v83/v154, v232, v219) are the SAME E8-selector, so Target A, G_net, P2 and the Kleinian seam are ONE gate. The closing theorem (stated, [O]): RP+gap+chirality => quasi-free bulk (v160) => invertible bulk => holomorphic boundary => E8; the single residual is 'free bulk => holomorphic boundary'."
    },
    {
      "id": "v235_seam_chern_simons",
      "cluster": "frontier",
      "descriptionEN": "The closing step in abelian Chern-Simons language: holomorphic <=> det K = 1 -- a genuine reduction (not a closure) of GATE.HOLO.01 into standard anyon-condensation physics. A free gapped bosonic 2+1d bulk = an even integer K-matrix with #anyons = |det K|, edge c = signature(K), holomorphic <=> |det K| = 1. The TFPT extension tower (v92) IS the condensation tower read by det: carrier D5(+)A3 (det 16, 16 anyons) -> SO(16)_1=D8 (det 4) -> (E8)_1 (det 1, holomorphic), all rank 8, c=signature=8=g_car+N_fam. Holomorphic <=> det 1 <=> E8 = the Kitaev E8 quantum-Hall state. Condensation: a Lagrangian glue of order sqrt(16)=|mu4|=4 takes det 16->1 (a partial order-2 isotropic only ->4=D8). NEG: D8 is free+bosonic but has 4 anyons; an odd K is fermionic -- det=1 is the extra condition. Residual: 'the free RP seam condenses the order-|mu4| Lagrangian glue' = the sheet/QGEO selection, now with holomorphy<=>det1 a theorem."
    },
    {
      "id": "v236_brieskorn_capstone",
      "cluster": "horizon",
      "descriptionEN": "The (2,3,5) Brieskorn singularity x^2+y^3+z^5 is the ONE generator of the discrete skeleton -- the capstone of the icosahedral round (v55/v219/v223/v232/v233). Its exponents ARE the atoms (|Z2|,N_fam,g_car)=(2,3,5). Milnor number mu=(2-1)(3-1)(5-1)=1*2*4=8=rank E8 = the '8' in c3=1/(8 pi) (a fifth independent origin). The Milnor monodromy eigenvalues are the eight primitive 30th roots e^{2pi i m/30}, m the E8 exponents (charpoly Phi_30, deg phi(30)=8=mu) -- so it IS the order-30 E8 Coxeter element (v55). Both clocks: the monodromy group <h>=Z/30=Z/2(sheet)xZ/3(family)xZ/5(carrier) carries the mu3 triality (CP, v233) as h^10; the Galois group (Z/30)^x=Z/2xZ/4 carries the mu4 clock (x7, v223). Milnor lattice=E8, link=Poincare sphere. One singularity generates atoms+8+E8+order30+both clocks+seam; a unification, not a closure."
    },
    {
      "id": "v237_seam_sre_closure",
      "cluster": "frontier",
      "descriptionEN": "The closing step as a PHYSICAL condition: no topological ground-state degeneracy <=> det K=1 <=> the seam bulk is short-range-entangled (SRE/invertible). Sharpens GATE.HOLO.02's residual from definitional to physical/falsifiable. The K-matrix ground-state degeneracy on a genus-g surface is |det K|^g (carrier D5(+)A3->16, D8->4, E8->1 on the torus), so 'no degeneracy on any closed surface' <=> det K=1. Among rank-8 even positive-definite K (c=8=g_car+N_fam) only E8 has det 1 -- the unique nontrivial bosonic SRE 2+1d phase (Kitaev E8), edge holomorphic; D8/carrier are LRE. A unique vacuum on the plane (RP+gap+cluster) is necessary but not sufficient (the plane cannot see det K, the torus does), so the residual is exactly 'the seam unique-vacuum extends to the torus' = SRE. NEG: an LRE bulk (D8) has 4-fold torus degeneracy + non-holomorphic boundary. Sharpened residual: 'the free RP gapped seam is short-range-entangled (no topological order)' = det K=1 = the Kitaev E8 phase -- a yes/no statement, sharper than v235/QGEO.SYM.01. Not a closure."
    },
    {
      "id": "v238_modular_lindblad_dynamics",
      "cluster": "registry",
      "descriptionEN": "DYN.SEMIGROUP.01 — the STATIC→DYNAMIC flip: the seam's modular flow (reversible) and its recovery channel read as a Markov/Lindblad generator (irreversible), with the dissipative gap = the OS mass gap. Companion to v196/v198 (modular) and v221/v64 (recovery), which state the structure statically; this reads the SAME objects dynamically. [E] the modular flow σ_t=e^{i t Λ_Σ} is a unitary one-parameter group (Tomita–Takesaki) and the µ4 clock is conserved ⟺ [ρ,Λ_Σ]=0 (the v198 state-invariance), so the static commutator IS the conservation law of modular time. [E] L=log T is a doubly-stochastic/CPTP-classical generator (off-diagonals ≥0, zero row sums), e^{L}=T, and e^{Lτ}→ the democratic projector (the arrow of time, cf. v64's Tⁿ→projector). [E] KEYSTONE: the dissipative gap = −log((2/3)⁶) = 6 log(3/2) = the OS mass gap Δ (v64) — the static recovery bound and the dynamical mass gap are ONE number, one exponentiated. [E] GKSL split (imaginary + real≤0 spectrum); NEG controls on µ4 and the rate. [O] geometric modular flow on the full seam algebra (Connes–Rovelli thermal time = QGEO.SYM.01, v198) — a target, not a closure."
    },
    {
      "id": "v239_kms_thermal_time",
      "cluster": "registry",
      "descriptionEN": "DYN.KMS.01 — KMS / thermal time: the modular flow of v238 makes the seam state a KMS (thermal) state at β=1, and the geometric normalisation fixes the seam temperature via 2π = 1/(4c3). [E] KMS at β=1: ω(A σ_i(B)) = ω(BA) (ω(X)=Tr(ρX), ρ=e^{−Λ_Σ}/Z), so modular time (v238) is a thermal time (Connes–Rovelli); NEG β≠1 fails (Tomita–Takesaki pins β=1). [E] the seam unit β_phys = 2π = 1/(4c3) (v58) makes the KMS temperature the horizon temperature T_H = c3/M (v57/v208) — thermal time = horizon time. [E] one object with v238 (same Λ_Σ generates σ_t and the Gibbs state). [O] the geometric-boost identification = QGEO.SYM.01 (HOR.BHTHERMO.01 [P]) — a target, not a closure."
    },
    {
      "id": "v240_gns_os_reconstruction",
      "cluster": "registry",
      "descriptionEN": "DYN.GNS.01 — GNS / OS reconstruction (finite toy): the Hilbert space and a POSITIVE Hamiltonian are OUTPUTS of the seam data (M, ω), not inputs. [E] GNS: ⟨A,B⟩=Tr(ρ A†B) Hermitian positive-definite (ρ>0), dim n², identity cyclic+separating. [E] modular flow σ_t(x)=ρ^{it} x ρ^{−it} is a state-preserving *-automorphism, generator i[log ρ,·]=Λ_Σ (v238). [E] OS Hamiltonian H_OS=−log T (the v221 transfer) is POSITIVE — eigenvalues {0, 6 log(3/2), 6 log 3}≥0, gap = 6 log(3/2) = the v64 mass gap — so OS reconstruction delivers v64's 'H≥0 + mass gap'. [E] H_OS=−L: the OS Hamiltonian is minus the v238 dissipative generator (recovery semigroup = Euclidean time); two Hamiltonians from one seam. [O] the geometric/operator-level step = QGEO.SYM.01/BW + the standard OS theorem under RP/gap (Layer 2) — a target, not a closure."
    },
    {
      "id": "v241_dhr_sectors",
      "cluster": "registry",
      "descriptionEN": "DHR.SECTORS.01 — particles = DHR superselection sectors: the carrier discriminant/anyon content and the v235/v237 condensation tower as an abelian MTC. Carrier form (Lean-verified, FORM.GLUE.01): on Z4×Z4, q(x,y)=(5x²+3y²)/8 mod 1, θ(a)=e^{2πi q(a)}. [E] 16 sectors = Z4×Z4 (= det K_carrier = dim S⁺), fusion = abelian group law. [E] bosonic/condensable ⟺ θ=1 ⟺ isotropic ⟺ 5x²+3y²≡0 mod 8 — the six elements {(0,0),(1,1),(1,3),(2,2),(3,1),(3,3)}. [E] condensation tower = v235/v237: Lagrangian ⟨(1,1)⟩ (order 4, self-dual) → 1 sector = (E8)₁; order-2 ⟨(2,2)⟩ → 4 = D8; carrier 16→D8 4→E8 1. [E] NEG: a non-isotropic anyon (1,0) is confined. [C] sector↔SM-multiplet map is the interpretation; the MTC is exact. 'Particles = stable defect sectors of the seam code' made precise."
    },
    {
      "id": "v242_spin_statistics_modular",
      "cluster": "registry",
      "descriptionEN": "DHR.MODULAR.01 — spin-statistics and the modular (S,T) data of the carrier MTC: statistics + the fusion ring recovered from modular data + the anyon central charge c=8. Same MTC as v241 (Z4×Z4, q=(5x²+3y²)/8, θ=e^{2πi q}). [E] spin-statistics: T=diag(θ_a) → 6 bosons (θ=1), 2 fermions (θ=−1) = {(0,2),(2,0)}, 8 anyons. [E] modular S_ab=(1/√16)e^{−2πi B(a,b)} unitary, symmetric, S²=C (charge conjugation), S⁴=I. [E] Verlinde N_ab^c = Σ_x S_ax S_bx S*_cx / S_0x = δ(c,a+b) — fusion recovered from modular data. [E] Gauss–Milgram (1/√16)Σθ_a = e^{2πi c/8} with c=8=g_car+N_fam = the (E8)₁ central charge (v156/v175) — statistics and c=8 are one fact. [C] the boson/fermion split is the kinematic skeleton; sector↔multiplet map is the interpretation."
    },
    {
      "id": "v243_haag_ruelle_braiding",
      "cluster": "registry",
      "descriptionEN": "DHR.SMATRIX.01 — scattering: the Haag–Ruelle skeleton + the braiding S-matrix. The seam net has a mass gap (v64/v238) so Haag–Ruelle constructs asymptotic states; for the abelian anyon MTC (v241/v242) the 2-particle S-matrix is the monodromy M(a,b)=θ(a+b)/(θ(a)θ(b))=e^{2πi B(a,b)}. [E] mass gap Δ=6 log(3/2)>0 → HR in/out states (gapped-net theorem). [E] 2-particle S = monodromy: |M|=1, diagonal in conserved anyon labels → no particle production, factorised (integrable) scattering. [E] Yang–Baxter: M is a bicharacter → multi-particle factorisation. [E] crossing+symmetry: M(a,b)=M(b,a), M(a,−b)=M(a,b)*. [E] holomorphic point: on the condensed (E8)₁ (single sector) M=1, trivial braiding (det 1/SRE). [O] the 4d S-matrix / cross-sections are the open holographic uplift (Phase 3)."
    },
    {
      "id": "v244_spectral_action_contract",
      "cluster": "registry",
      "descriptionEN": "CONTRACT.QFT4D.01 — the 4d Lagrangian via Connes' spectral action: the honest contract for the Phase-3 gap (2d seam net → 4d physics). A single Dirac operator whose Tr f(D/Λ) yields gravity AND matter. [E] carrier spinor = one generation: SO(10) half-spinor = dim S⁺ = 2^(g_car−1) = 16 (15 + ν_R), N_fam=3. [E] SM gauge group inside the carrier: SU(3)×SU(2)×U(1) (dim 12) ⊂ SU(5) (24) ⊂ SO(10)=D5 (45); color SU(3) ⊂ A3=SU(4) (Pati-Salam); rank(D5)+rank(A3)=8=rank E8. [E] one Dirac operator → gravity (a2~R, a4~R², v36) + matter (a4 ~ F² Yang–Mills + |Dφ|² Higgs, Gilkey). [X] hooks: g3=g2=√(5/3)g1 + a Higgs-quartic relation (Chamseddine–Connes–Marcolli). [O] CONTRACT.QFT4D.01: obligations (seam→Dirac D; finite geometry = carrier; spectral action reproduces SM couplings) → the 4d SM+gravity Lagrangian is an OUTPUT. The Phase-3 target, not closed (complement of U_wall/G_metric)."
    },
    {
      "id": "v245_spectral_matter_content",
      "cluster": "registry",
      "descriptionEN": "QFT4D.MATTER.01 — the computable core of CONTRACT.QFT4D.01 (v244): the spectral-action / NCG matter content, discharging the rep-theory + tree-relation core to [E]. The carrier half-spinor is the SO(10) 16. [E] 16 = one generation: SO(10) 16 = SU(5)(10+5bar+1); SM dims 6+3+1+2+3+1 = 16 = dim S⁺, N_fam=3 (15 Weyl fermions + ν_R). [E] electric charges Q=T3+Y are exactly the SM values — hypercharges forced by the embedding. [E] anomaly-free: ΣY=0, ΣY³=0, [SU(2)]²U(1)=0, [SU(3)]²U(1)=0 (exact). [E] spectral-action unification: sin²θ_W=(ΣT3²)/(ΣQ²)=3/8, i.e. g3=g2=√(5/3)g1 (Chamseddine–Connes–Marcolli). [E] Higgs = the (1,2)_{1/2} inner fluctuation of the finite Dirac operator (one doublet). [O] residual: the seam→Dirac realisation + measured-coupling reproduction stay the open contract."
    },
    {
      "id": "v246_unification_data_crosscheck",
      "cluster": "registry",
      "descriptionEN": "QFT4D.RGTEST.01 — the HONEST data cross-check for the spectral-action prediction (v244/v245): does g1=g2=g3, sin²θ_W=3/8 at Λ survive the MEASURED SM couplings? Not in the plain SM. Inputs (PDG, GUT-normalised, M_Z): α_i⁻¹=(59.01,29.59,8.47), SM 1-loop b=(41/10,−19/6,−7) (v159). [E] the prediction: α1=α2=α3, sin²θ_W=3/8 at Λ. [X] tension: pairwise crossings spread ~4 decades (a1=a2 ~1e13, a1=a3 ~2.4e14, a2=a3 ~1e17 GeV), residual spread at 2e16 Δα⁻¹~8.8 — g1=g2=g3 NOT realised by the plain SM. [X] sin²θ_W 3/8 vs measured 0.23122. [X] 2-loop also misses (min spread ~3.2 at 1.7e14 GeV). [O] no rescue: SM gauge content + no new states (the E8 spine is arithmetic, not a threshold tower) → no admissible thresholds; closing it needs new matter (vs 'no new state') or a non-GUT/boundary reading — tension like NCG-SM, likely falsified unless extended. A kill-test [X]/[O], never [E]-as-success."
    },
    {
      "id": "v247_e8_branching_no126",
      "cluster": "registry",
      "descriptionEN": "PS.E8BRANCH.01 — the E8 audit hull forbids the 126 and supplies exactly {1,10,16,45} for SO(10). Carrier D5⊕A3 = SO(10)×SU(4) ⊂ E8 (via SO(16)); 248 = (45,1)+(1,15)+(10,6)+(16,4)+(16bar,4bar) = 248. [E] ranks 5+3=8; [E] dims=248 (SO(16): 120 adj + 128 spinor); [E] SO(10) content {1,10,16,45}, the 126 ABSENT → 126_H algebraically forbidden, 10+16+45 E8-supplied not fitted; [E] CAVEAT only one 45/one 15 → proton-marginal; [O] the field-content principle + NCG complement (v250). Makes the scalaron-match 16-over-126 selection algebraic."
    },
    {
      "id": "v248_ps_algebra",
      "cluster": "registry",
      "descriptionEN": "PS.ALGEBRA.01 — the Pati-Salam NCG algebra A_F = ℍ_L⊕ℍ_R⊕M₄(ℂ) and its exact SM matching. [E] unimodular group SU(2)_L×SU(2)_R×SU(4)_c (ranks 5, dims 21); [E] 16=(4,2,1)+(4̄,1,2)=8+8; [E] Y=T₃R+(B−L)/2 gives the exact SM hypercharges and Q=T₃L+Y the exact charges (B−L = SU(4)_c gen, lepton = 4th colour); [E] matching α1⁻¹=(3/5)α2R⁻¹+(2/5)α4⁻¹, sin²θ_W=3/8. [O] A_F uniquely from seam+sheet+μ4 (carrier gauged as theorem); [O] CAVEAT PS colour SU(4)_c ⊂ SO(10)=D5 is NOT the carrier A3=SU(4) (=family)."
    },
    {
      "id": "v249_ps_unification",
      "cluster": "registry",
      "descriptionEN": "PS.RGTEST.01 — carrier-native Pati-Salam/SO(10) two-step unification kill-test + negative controls (suite check of experiments/gauge-unification). [E] matching + a valid 1-loop unification solution exists; [X] NEG SM-only fails (spread ~8.8 @ 2e16); [C] M_PS/M_s = 1.0–1.7 (1-loop; 2-loop ~1.0–1.2) — gauge unification meets the gravitational scalaron scale; [X] NEG 126 breaks the match (b2R>0; also E8-forbidden v247); [X] NEG proton decay selects content (minimal M_GUT~2e15 excluded; +(15,1,1) proton-safer). [O] the gauging fork, exact κ, the one-45 E8 marginality."
    },
    {
      "id": "v250_dirac_triple_contract",
      "cluster": "registry",
      "descriptionEN": "CONTRACT.QFT4D.DIRAC.01 — the spectral-triple contract that would make 'carrier gauged' a theorem. Target (A,H,D,J,γ) almost-commutative, A_F = PS (v248), H_F = 3×16, D_F from (R,L,φ0,Majorana). [E] H_F = 3×16 = 48; [E] the Yukawas are already fixed (v18) so the open test is NCG consistency; [O] the 7 NCG axioms (first-order condition the historic SO(10) obstruction); [O] σ=scalaron hypothesis (rescues first-order + ν_R Majorana; M_PS=κM_s) — CAVEAT scalaron↔ν_Rν_R not automatic; [O] inner fluctuations → 10+16+45 not 126 (NCG complement of v247). Reduces 4d QFT to one object + finite NCG checks."
    },
    {
      "id": "v251_first_order_sigma",
      "cluster": "registry",
      "descriptionEN": "PS.DIRAC.01 — the first-order condition is violated EXACTLY by the Majorana/σ term (Chamseddine–Connes–van Suijlekom 'beyond first order' → Pati-Salam + σ), exhibited on a faithful lepton-block model; advances v250 from contract to mechanism. Honest correction: σ does NOT 'rescue' first order — one DROPS it and the inner fluctuations generate PS + σ. [E] order-zero; [E] real-even (J²=+1, γ²=1, Dγ=−γD, KO-6 signs); [E] first order HOLDS for the Yukawa block; [E] first order VIOLATED by M≠0, supported exactly on the ν_R ν_R^c blocks (σ the unique violator) — the controlled violation IS the PS mechanism. [C] σ=scalaron (v249). [O] the full 96-dim SM/PS triple + spectral action (v250)."
    },
    {
      "id": "v252_full_finite_triple",
      "cluster": "registry",
      "descriptionEN": "PS.DIRAC.02 — the FULL 96-dim finite spectral triple (A_F,H_F,D_F,J,γ) built explicitly with NCG axioms verified; advances v250 from a 7-obligation contract to a concrete object discharging 5 to [E]. H_F = particle⊕antiparticle doubling of three SO(10) 16-plets (dim 96). [E] dim 96; D_F Hermitian+odd; reality J²=+1,JD=DJ; grading γ²=1,[γ,a]=0,Jγ=−γJ; KO-dim 6; order-zero; first order HOLDS for SM C⊕H⊕M₃ (incl. Majorana) and VIOLATED for Pati-Salam exactly by the Majorana on {ν_R,e_R} (CCvS beyond-first-order → PS+σ); spectrum = fermion masses + seesaw m_light=m_D²/M_R. [C] Poincaré duality (intersection form non-degenerate). [O] orientability + no-junk + spectral-action expansion."
    },
    {
      "id": "v253_kappa_scalaron_ps",
      "cluster": "registry",
      "descriptionEN": "PS.KAPPA.01 — pinning κ in M_PS = κ·M_s, discharging residual 6(b) of v249. M_s = c₃^{7/2}·Mbar ≈ 3.06e13 GeV (exact). [E] scalaron scale exact; [E] κ pinned by RG (scan PDG errors × the two E8-allowed contents) to a tight O(1) interval ~1.3 (1-loop [1.0,1.7], 2-loop [1.0,1.2]) — pinned, not free. [C] heat-kernel origin: κ = √((f₂/f₀)·c_PS/c_grav), a ratio of heat-kernel traces over the 96-dim H_F (v252) — Λ cancels, κ scale-free and O(1). [X] 126-content drives κ out of the O(1) window (content-selective). [O] exact decimal needs the cutoff moments f₂/f₀ fixed."
    },
    {
      "id": "v254_inner_fluctuations",
      "cluster": "registry",
      "descriptionEN": "PS.NCG.FLUCT.01 — the inner fluctuations Ω¹_D(A_F) of the finite Dirac operator computed explicitly (reuses the v252 triple); closes v250 #5 (Higgs content, not 126) + the no-junk obligation + the σ quantum numbers. [E] Ω¹_{D_F} purely chirality-odd → finite fluctuations are scalars; [E] no junk / no 126: SM-algebra 1-forms sit exactly in the Higgs-doublet pattern (the single (1,2)_{1/2}); [E] σ (ν_R Majorana) direction absent for the SM algebra but present for Pati-Salam — σ is the CCvS beyond-first-order scalar; [E] σ is an SM singlet (1,1,0) with B-L=2; [C] σ=scalaron consistency."
    },
    {
      "id": "v255_spectral_action_expansion",
      "cluster": "registry",
      "descriptionEN": "PS.SPECACT.01 — the spectral-action heat-kernel (Seeley-DeWitt) expansion S = 2f₄Λ⁴a₀ + 2f₂Λ²a₂ + f₀a₄ over the finite triple; closes v252 #11 (spectral-action expansion) and makes κ(f₂/f₀) explicit. [E] expansion structure (a₀ cosmological, a₂ Einstein-Hilbert+Higgs-mass, a₄ Yang-Mills+Weyl²+R²+Higgs-quartic, N=96); [E] trace inputs b/a²=1/3 (top-dominated, Chamseddine-Connes); [E] sin²θ_W=3/8 (cross-check v245); [C] Higgs quartic λ(Λ)~g²b/a² → ~125 GeV with σ; [E] scalaron = R² spin-0 mode (gravitational a₄~f₀N); [C] κ=√((f₂/f₀)c_PS/c_grav) explicit; [O] exact decimal needs the cutoff moments fixed."
    },
    {
      "id": "v256_orientability_poincare",
      "cluster": "registry",
      "descriptionEN": "PS.NCG.ORIENT.01 — orientability and Poincaré duality treated honestly (no faked proof). KO-dim 6 forces the intersection form cap(p,q)=Tr(γ π(p) J π(q) J⁻¹) to be antisymmetric. [E] skew form forced by Jγ=−γJ + order-zero + reality (verified); [E] on the SM K-theory Z³ the skew form has rank 2, corank 1, null ~ (1,1,−1) — the structural obstruction; [E] orientability local condition [γ_F,π(a)]=0, full orientability is a degree-4 cycle on M×F; [C] the canonical hypercharge-faithful SM/PS triple satisfies both (van Suijlekom; CCM) — literature; [O] a self-contained machine proof needs the complete canonical bimodule."
    },
    {
      "id": "v257_quasiorient_modpd",
      "cluster": "registry",
      "descriptionEN": "PS.NCG.ORIENT.02 — the honest, literature-confirmed resolution of orientability + Poincaré duality, correcting the too-generous v256 typing. Published result (Çaćić–Stephan arXiv:0902.2068; CCM07; SM-vacuum hep-th/0601192): the CCM SM finite triple (KO-6, 3 ν_R) genuinely FAILS strict orientability and strict Poincaré duality, satisfying the weakened versions. [E] strict orientability fails (γ not in span{λ(a)ρ(b)}); [E] quasi-orientability holds (left/right A-irrep boxes disjoint by chirality: L = H doublet, R = C singlet → L^LR(even,odd)={0}); [E] strict Poincaré fails (KO-6 skew form, corank 1, the equal-L/R-neutrino case); [C] modified Poincaré duality (CCM) / Stephan's PD-restoring triple. A known model-class feature, not a TFPT defect — no faked closure."
    },
    {
      "id": "v258_dirac_covariance_induction",
      "cluster": "registry",
      "descriptionEN": "PS.DIRAC.03 — D_F is the modular/covariance INDUCTION of the boundary seam state, not an independent posit; closes the 'posited Yukawa vs derived operator' gap of v250/v252 (Q2: Dirac as Kasparov product + covariance formula). Quasi-free/modular dictionary: a Gaussian state is fixed by its covariance C (contraction, spec in (0,1)) with one-particle modular Hamiltonian H=log((1-C)C^-1) (Araki/Casini-Huerta), inverse to the KMS occupation C=(1+e^H)^-1 (beta=1, v239). [E] inversion identity log((1-C)C^-1)=H exactly (random Hermitian + a chirality-graded Yukawa block); [E] C_F=(1+e^{D/mu})^-1 a positive contraction; [E] chirality-odd (anticommutes with gamma_F); [E] unitary-equivalent to v252 (inducing C_F from the 9 Yukawas returns the same spectrum); [E] Majorana = off-diagonal covariance (seesaw m_light=m_D^2/M_R); [C] C_F=P_F C_Sigma P_F, so D_F is the KK shadow [D_Sigma](x)_{(E8)_1}[K_car] and the v18 Yukawas are the readout of C_Sigma."
    },
    {
      "id": "v259_modular_cutoff_kappa",
      "cluster": "registry",
      "descriptionEN": "PS.SPECACT.02 — the spectral-action CUTOFF is the seam KMS weight, fixing f_2/f_0=1 EXACTLY and reducing the last open scheme freedom (v253 #5, v255 #7) from a free function to a finite-triple trace ratio (Q4: the cutoff is modular not chosen). By Tomita-Takesaki/BW the seam state is KMS at beta=1 (v239) and 2pi=1/(4c3) (v58) fixes beta, so the heat-kernel cutoff f(u)=e^{-u} is DERIVED. [E] KMS cutoff moments f_0=f(0)=1, f_2=int f=1, f_4=int u f=1, so f_2/f_0=f_4/f_2=1; [E] seam-fixed beta=1 (no new parameter beyond c3); [E] non-vacuous neg control: a Gaussian cutoff e^{-u^2} gives f_2/f_0=sqrt(pi)/2 != 1; [E] kappa=sqrt((f_2/f_0)c_PS/c_grav)=sqrt(c_PS/c_grav), no cutoff function left; [C] RG kappa~1.3 pins c_PS/c_grav~1.7 (O(1)); the [O] free scheme becomes a [C] finite computation over the built triple; [C] same logic fixes lambda(Lambda)."
    },
    {
      "id": "v260_k3_kummer_unification",
      "cluster": "registry",
      "descriptionEN": "ARCH.K3.01 — one Kummer/K3 surface carries the SEAM, the CARRIER-16 and the E8 lattice as facets of one smooth object, the geometric unification of the v1 'glue D5(+)A3 + mu4 => E8' step (the K3/Kummer proposal). The seam square torus tau=i (j=1728, v214) under the elliptic involution z->-z squares to the Kummer K3. [E] E8 Cartan even, det 1, positive-definite (the unique even unimodular pos-def rank-8 lattice); [E] K3 lattice L_K3=U^3(+)E8(-1)^2: rank 22=b_2, det -1 (unimodular), even, signature (3,19), E8(-1)^2 an orthogonal summand; [E] seam = T^2/(z->-z): 4=|(Z/2)^2| fixed points, pillowcase S^2(2,2,2,2), orbifold Euler char 0 (the v214/v216 four marks); [E] carrier-16 = Kummer nodes |A[2]|=2^4=16 -> 16 exceptional P^1 = dim S+ = Lambda^even(C^5) = 16 (v197), 16=4x4; [E] honesty: the 16 node classes span the rank-16 Kummer lattice, NOT literally E8(-1)^2 (distinct sublattices of H^2(K3)); [C] two faces of E8 in K3 (local du Val C^2/2I v232/v236; global H^2 summand); [C] unification (lattice [E], carrier<->node identification [C])."
    },
    {
      "id": "v261_modular_spectral_closure",
      "cluster": "registry",
      "descriptionEN": "QFT.MSC.01 — the MODULAR SPECTRAL CLOSURE: the assembly/reduction certificate turning the whole QFT layer (v238-v260) into ONE relative object TFPT_QFT=(A_Sigma,omega_Sigma,Delta_Sigma,rho,A_F,H_F,D_F,J,gamma,S_rel) reduced to ONE open premise (like v176/v234, an assembly cert, not new physics). [E] one index=marks=fixed points=glue order: Jones [B:A]=4 (v89) = |mu4|=4 (v92) = seam marks 2chi=4 (v216) = pillowcase/Kummer fixed points |(Z/2)^2|=4 (v260); [E] one carrier-16: dim S+=Lambda^even(C^5)=Kummer nodes 2^4=16, H_F=2 N_fam 16=96; [E] one gap Delta=-log((2/3)^6)=6log(3/2)>0 = OS mass gap (v240)=recovery rate (v221)=Lindblad gap (v238); [E] the three closures re-derived (Kasparov log((1-C)C^-1)=H v258, KMS cutoff f2/f0=1 v259, K3 lattice signature (3,19)/det-1/even v260); [E] relative-object manifest (10/10 component scripts present); [O] the single residual: the whole QFT layer reduces to QGEO.SYM.01 (seam realisation), ambient QG QG.AMB.01 separate; [C] master statement (Modular Spectral Universality) modulo QGEO.SYM.01 -- the honest sense in which the QFT layer is complete."
    },
    {
      "id": "v262_fqcd_mp_me",
      "cluster": "registry",
      "descriptionEN": "FR.MPME.02 — the F_QCD transfer pipeline for m_p/m_e, implemented as a real solver (the one F_transfer pipeline that was contract-only, experiments/ftransfer/qcd_matching_mp_me). NOT a compiler power -- a [C] standard-physics transfer reproduction. [E] b3=-7=-(11-2 n_f/3) at n_f=6 (the SM/SU(3) one-loop coefficient, the only TFPT number in the run); [N] 2-loop alpha_s(M_Z)=0.1179 run down with thresholds gives alpha_s(2 GeV)~0.30 (real ODE solve); [C] Lambda^(3)_MSbar~0.33 GeV x lattice C_p=m_p/Lambda (external O(1)) over the closed m_e gives m_p/m_e~1824 vs measured 1836.15 within ~7% external budget; [X] firewall: m_p/m_e is not an admissible compiler integer (SU(9) absent from D5(+)A3), stays [C]. Advances FR.MPME.01 [O] to FR.MPME.02 [C]."
    },
    {
      "id": "v263_mnu_from_df",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.PMNS.02 — the light neutrino matrix M_nu as the type-I seesaw of D_F, advancing the solar-angle texture (v9) from a posited matrix to a seesaw OBJECT built from D_F's blocks. [E] M_nu=-m_D M_R^{-1} m_D^T symmetric + seesaw-suppressed (light masses an output of D_F, not a fit matrix); [E] mu-tau-symmetric (m_D,M_R) give mu-tau-symmetric M_nu (the v9 texture is in the seesaw image); [C] diagonalising gives theta23=45deg and sin^2 theta12=1/3-phi0/2=0.30675 under the seam misalignment, reproducing v9; [O] theta13=0 in the leading texture (physical sin^2 th13~phi0 e^{-5/6} a separate readout), CP magnitude delta_PMNS=240deg from mu6/triality (v231/v233) not from M_nu, M_R the free seesaw scale. An operator advance, NOT a closure of full PMNS dynamics."
    },
    {
      "id": "v264_raw_seam_marklocal",
      "cluster": "registry",
      "descriptionEN": "QGEO.ENERGY.03 — the raw-seam => mark-locality REDUCTION (NOT a closure of QGEO.SYM.01). Composes v216+v201+v198 on the canonical pillowcase: [E] Gauss-Bonnet 4 cone points (deficit pi each, sum 4pi=2pi chi(S^2)), curvature ONLY at the 4 marks; [N] a curvature sourced at the mu4 orbit {0,pi/2,pi,3pi/2} has Fourier support ONLY on 4Z (FFT), so mark-locality is automatic for the pillowcase; [E] M_f then commutes with the clock rho=diag(i^n), [rho,M_f]=0; [E] with |k| already commuting (v198), [rho,Lambda]=0 => omega o rho=omega ON THIS METRIC; [O] the residual IS the metric -- the chain assumes the raw seam carries the uniformising flat pillowcase metric (= QGEO.SYM.01), NEG control (curvature off the mu4 orbit) breaks it. A reduction/sharpening, NOT a closure."
    },
    {
      "id": "v265_qft4d_fork_freeze",
      "cluster": "registry",
      "descriptionEN": "QFT4D.FORK.01 — freeze the 4D-QFT fork as a Branch Policy (not an open ambiguity); writes qft4d_fork_freeze.json + a prose guard, no new physics. A=boundary-only is the canonical 4D reading (no 4D-GUT claim by default); SM-only 4D-GUT unification is killed (1-loop spread ~8.8 at 2e16, v246); B=carrier-native Pati-Salam is a falsifiable UV branch (E8 content {1,10,16,45} no 126 v247, sin^2 thetaW=3/8 v248, kappa O(1) band v249/v253, threshold+proton kill test). Proton safety rule: no fake tau_p window. Wording guard: bare 4D over-claims absent + scoped policy phrases present. Frozen decision tree, not an open ambiguity."
    },
    {
      "id": "v266_ps_threshold_proton",
      "cluster": "registry",
      "descriptionEN": "PS.PROTON.02 — the explicit Pati-Salam threshold model + proton-decay window that DECIDES branch B of the v265 fork; writes ps_threshold_proton.json. Two-step PS solve + d=6 tau_p(p->e+pi0); the carrier-native PS UV branch survives proton decay ONLY with the E8-allowed (15,1,1) [single 45]: minimal 16-Higgs M_GUT~2.4e15 -> tau_p~4.4e33 yr KILLED by Super-K (2.4e34), +(15,1,1) M_GUT~5.9e15 -> tau_p~1.6e35 yr safe now but at the Hyper-K reach (~1.4e35) -> a dated kill test within the decade. Marginal (only ONE 45 in the hull, v247) + O(3) hadronic uncertainty. A (boundary-only) stays canonical."
    },
    {
      "id": "v267_qgeo_rigidity_minimal_axiom",
      "cluster": "registry",
      "descriptionEN": "QGEO.SYM.02 — the rigidity / minimal-axiom form of the bedrock QGEO.SYM.01: proves the ONE bare statement ('the raw RP seam admits the carrier mu4 clock as a conformal symmetry permuting its 4 marks' = omega o rho = omega) FORCES everything downstream; does NOT close the axiom. [E] an order-4 Moebius orbit {a,ia,-a,-ia} has cross-ratio 2 (independent of a); [E] cross-ratio 2 <=> j=1728 (only lambda in {-1,1/2,2}) <=> the order-4 CM automorphism (x,y)->(-x,iy) on y^2=x^3-x; [E] downstream forced: square => unique flat pillowcase (Troyanov, v214) => curvature at the 4 marks => Z4-Fourier DtN (v201/v264) => [rho,Lambda]=0 => omega o rho=omega (v198); [E] neg controls (generic j!=1728 only Z2; hexagonal j=0 has Z6, order 6 != 4); [C] arithmetic rigidity ({1,i,-1,-i} over Q, j=1728 CM by Z[i], Belyi/dessins); [O] the bare order-4 symmetry stays the one foundational postulate (like c=const). Closes the reduction (axiom=>everything), not the axiom."
    },
    {
      "id": "v268_theta13_carrier_trace",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.TH13.01 — the reactor angle theta_13 exponent IS the carrier hypercharge trace. The value sin^2 theta_13 = phi0 e^{-5/6} = 0.0231 was already checked (v16); here the exponent origin is closed exactly: gamma = tr_E Y^2 = 3(1/3)^2+2(1/2)^2 = 5/6 over the 5-slot carrier hypercharge Y=diag(-1/3,-1/3,-1/3,1/2,1/2) (roots of 6Y^2-Y-1=0), complement 1/6 = the neutrino ratio. Own channel: the mu-tau-breaking eps=3phi0/4 (which gives theta_12) induces only ~1.6e-3 << 0.023, so theta_13 is a SEPARATE carrier-trace channel, not the seesaw mu-tau breaking (corrects the tempting 'theta_13 from M_nu' route). [O] CP magnitude + absolute nu scale remain open."
    },
    {
      "id": "v269_spert_paqft_skeleton",
      "cluster": "registry",
      "descriptionEN": "QFT4D.SPERT.01 — the perturbative 4D S-matrix (S_pert) as an Epstein-Glaser / pAQFT skeleton: the honest middle layer of the three-layer S-matrix (S_top 2d DHR braiding v243 / S_pert 4d EG of the spectral action / S_phys LSZ on OS Wightman v240), NOT a nonperturbative closure. [E] three distinct layers (the 2d braiding |M|=1 is statistics, not 4d cross-sections); [E] every spectral-action a_4 operator has mass dim <= 4 in 4D (power-counting renormalizable); [E] finite counterterm basis (7 marginal + 3 relevant, no infinite tower); [C] Epstein-Glaser existence -- a renormalizable interaction has an order-by-order S(g)=T exp(i int g L_int) by causal factorisation + finite local extension, no path integral, no UV divergences; [C] adiabatic limit/gap (Delta=6log(3/2)>0) -> S_phys via LSZ; [O] perturbative only, the ambient measure QG.AMB.01 stays open. A contract/skeleton, explicitly perturbative."
    },
    {
      "id": "v270_pmns_jarlskog_assembly",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.PMNS.03 — the complete complex PMNS matrix + the leptonic Jarlskog J_PMNS (the CP strength), assembled from the four TFPT channels (sin²θ₁₂=1/3-φ₀/2, sin²θ₂₃=1/2, sin²θ₁₃=φ₀e^{-5/6}, δ=240°=4π/3) into one unitary object. [E] the three angle channels; [E] U=R23·U13(δ)·R12 is exactly unitary; [C] J_PMNS=Im(U_e1 U_mu2 U*_e2 U*_mu1)=-0.02965 (formula and full matrix agree), J_max=0.03424 — a DERIVED CP strength, the leptonic analogue of the CKM J (v88); [C] data: J_max vs NuFIT 0.0332 (~3%), J vs best fit -0.026, negative sign reproduced; [O] δ is the μ6/triality phase (assembled input, NOT from M_ν/D_F) and the absolute ν-mass scale (Σm_ν, M_R) stays open. Assembles PMNS + CP strength; does not derive δ from D_F nor fix the scale."
    },
    {
      "id": "v271_eg_oneloop_quartic",
      "cluster": "registry",
      "descriptionEN": "QFT4D.SPERT.02 — a concrete Epstein-Glaser order-2 (one-loop) quartic renormalization, taking the v269 S_pert skeleton from 'exists' to an actual EG number (no path integral). [E] T_1=:L_int: needs no extension, the first extension is T_2; [E] the massless propagator has scaling degree sd=2 in d=4, the one-loop bubble (two propagators) has sd=4=d so the UV singular order omega=sd-d=0; [E] omega=0 => exactly one logarithmic local counterterm (C(omega+d,d)=1, renormalizable, no infinite tower); [E] the loop factor Omega_3/(2(2pi)^4)=1/(16 pi^2) exactly — the same factor that normalises the spectral-action a_4 geometric quartic; [C] the phi^4 one-loop beta = 3 lambda^2/(16 pi^2) (symmetry factor 3 = s,t,u channels), EG initial condition lambda_Phi(UV)=1/(16 pi^2); [O] order-2/one-loop only — the all-order EG construction and the nonperturbative measure QG.AMB.01 stay open. Turns the skeleton into a concrete one-loop number."
    },
    {
      "id": "v272_nu_mass_scale",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.NUSCALE.01 — the absolute neutrino-mass scale, typed honestly (does NOT fabricate a Σm_ν number, which would contradict the v184 firewall). [E] the absolute scale reduces to ONE seesaw ratio m₃=(y_ν v_EW)²/M_R — one irreducible UV input like v_geo (v153), while the ratios (m2/m3, ordering, PMNS angles v270) are pinned; [N] the observed m₃=0.050 eV with the TFPT PS→GUT window M_R∈[4.2e13, 2.4e15] GeV (v249) needs a third-gen Dirac Yukawa y_ν∈[0.26, 2.0] — O(1), no tuning; [O] M_R(y=1)=6.1e14 has log_c3(M_R/Mbar)=2.57 (non-integer), not a TFPT power, so the absolute value stays open; [C] the NO floor Σm_ν=0.0586 eV is consistent with cosmology (<0.072–0.12 eV); [X] a cosmological Σ<0.0586 eV excludes the normal-ordered m1~0 spectrum. Closes the PMNS bookkeeping (the last [O] is one seesaw ratio), not the absolute number. Sharpened by v481: under the carrier normalisation y_nu = y_t the (y_nu, M_R) trade-off collapses to one parameter, window-bracketed (candidate class)."
    },
    {
      "id": "v273_eg_gauge_running",
      "cluster": "registry",
      "descriptionEN": "QFT4D.SPERT.03 — the gauge sector of the EG one-loop (v271's quartic mechanism applied to the gauge self-energy). [E] the one-loop gauge 2-point has scaling degree sd=4=d so ω=0 → one counterterm per coupling (same marginal mechanism + loop factor 1/(16π²) as v271); [E] the one-loop b_i from the explicit carrier/SM content: b₃=-(11/3)(3)+(2/3)(6)=-7, b₂=-(11/3)(2)+(2/3)(6)+(1/3)(1/2)=-19/6, b₁=(3/5)[(2/3)(10)+(1/3)(1/2)]=41/10 (the same 41 as the carrier algebra, v159/CAR.SM.01); [C] physical RG directions b₃<0 (asymptotic freedom), b₁>0 (U(1) Landau); [O] the two-loop matrix (v159 PyR@TE) + all-order stay open. Shows the gauge running is an EG order-2 result, the gauge analogue of v271's quartic. Exact b_i mirrored in Wolfram."
    },
    {
      "id": "v274_scale_overdetermination",
      "cluster": "frontier",
      "descriptionEN": "SCALE.OVERDET.01 — the single mass anchor is OVER-DETERMINED, and the absence of a derivable absolute scale is a THEOREM. [E] one anchor (No-Unit, v153): every dimensionful output = pure number × M_bar^d (ρ_Λ/M_bar⁴, M_scal/M_bar=c3^{7/2}, the spectrum); [N] two independent routes to M_bar agree: gravity (8πG)^{-1/2}=2.4353e18 GeV and cosmology (invert ρ_Λ/M_bar⁴=(3/4π²)e^{-2α⁻¹} with ρ_Λ^{1/4}=2.24 meV)=2.438e18 GeV, to 0.11%; [E] the anchor is the gravitational unit (G), the one measured dimensionful input of all physics; [E] theorem not gap: the seam is a conformal c=8 CFT (v157/v158), scale-invariant by construction → no intrinsic scale, the scale is gravitational; [A] scale-complete up to one over-determined anchor = maximal predictivity. Sharpens v153/v78."
    },
    {
      "id": "v275_qgamb_roadmap",
      "cluster": "frontier",
      "descriptionEN": "QGAMB.ROADMAP.01 — the obligation roadmap for QG.AMB.01 (the ambient nonperturbative QG measure), consolidated into one module (the open gate had no script). [E] Tier A gap decoupling: Δ_eff=6log(3/2)-2·248 c3²=1.648>0 (v36/v76), so every low-energy readout is independent of the un-built measure — not a blocker; [E]/[P] Tier B: the bulk measure reduces to identifying the seam-Calderón boundary with the holomorphic (E8)_1 net (c=248/31=8; v77/GATE.METRIC.03), reduced not closed (REDTEAM.A.01); [C] the perturbative layer is in hand (v269/v271/v273); [O] the nonperturbative ambient measure is NOT constructed — the one genuine structural frontier; [X] closes IFF seam net=(E8)_1 AND bulk reconstruction proved. A roadmap, not a closure."
    },
    {
      "id": "v276_qgeo_flat_closes_commutator",
      "cluster": "registry",
      "descriptionEN": "QGEO.SYM.03 — the flat tau=i pillowcase premise CLOSES the modular-commutator chain to all orders, collapsing the bedrock QGEO.SYM.01 to ONE geometric postulate (it does NOT prove the premise). For a quasi-free seam state QGEO.SYM.01 = (omega o rho = omega) <=> [rho,C]=0 <=> [rho,H]=0. [E] ISOMETRY=>COMMUTATOR: if H=f(Delta_flat) and the order-4 deck rho (z->iz) is a flat-tau=i isometry, then [rho,Delta]=0 => [rho,H]=0 EXACTLY (all orders), verified on the discretised square torus (rho^4=I, [rho,Delta]=0, [rho,sqrt(-Delta)]=0; generic non-isometry fails); [E] this UPGRADES v198 (principal symbol) + v201 (subprincipal) from leading orders to the full H (flatness removes curvature terms); [C] tau=i is forced by the order-4 requirement (j=1728; hexagonal j=0 is order-6, excluded; v214/v267); [O] QGEO.SYM.01 collapses to the single statement 'the raw seam IS the flat tau=i pillowcase quasi-free state (H=sqrt(-Delta_flat))' — needs a human constructive-QFT proof or stays an honest axiom (like c=const). Sharpest non-circular form."
    },
    {
      "id": "v277_seam_calderon_e8_match",
      "cluster": "frontier",
      "descriptionEN": "QGAMB.TIERB.01 — the seam-Calderón → (E8)₁ matching certificate, the concrete Tier-B step of QG.AMB.01 (v275): computes the full (E8)₁ matching target and pins the residual to ONE bit (does NOT close it). [E] c=8 (Sugawara, 248/31=g_car+N_fam), but c=8 alone does not pin the net; [E] the discriminator is holomorphy — det Cartan(E8)=1 (1 primary, holomorphic) vs the c=8 counterexample det Cartan(D8=SO(16))=4 (4 primaries, non-holomorphic), so the single discriminator is |det K|=1; [E] the (E8)₁ character E4/η⁸=j^{1/3}=q^{-1/3}(1+248q+4124q²+…) has a single primary + exactly 248=dim E8 weight-1 currents (240 roots); [E] equivalence chain holomorphic ⟺ det Cartan 1 ⟺ unique even unimodular rank-8 (v83) ⟺ homology-sphere link ⟺ |det K|=1 (v235); [C] seam-side match via v276 (flat τ=i) + v260 (Kummer/K3); [O] residual = the single holomorphy bit, reduced to one bit, NOT closed (rt_A verdict stands)."
    },
    {
      "id": "v278_lsz_bridge_unitarity",
      "cluster": "registry",
      "descriptionEN": "QFT4D.SPERT.04 — the S_pert → S_phys LSZ bridge + one-loop unitarity: connects the EG perturbative S-matrix (v269/v271/v273) to the physical asymptotic S-matrix (S_phys = LSZ on the OS Wightman functions, v240). [E] LSZ bridge: S_phys = amputate+on-shell-residue of the connected time-ordered products, and the EG T-products of S_pert ARE those correlators (OS Wick-rotation, v240) → S_pert --LSZ--> S_phys; [C] the gap Δ=6log(3/2)>0 gives Haag-Ruelle asymptotic states so LSZ is valid; [E] ONE-LOOP UNITARITY (optical theorem): the bubble discontinuity Im B(s)/π = x₊−x₋ = √(1−4m²/s) EXACTLY = the two-body phase space, so 2 Im M = ∑∫dΠ|M|² (cutting rule) holds → matter+gauge unitary; [E] threshold: Im B=0 below s=4m², turns on at the physical 2-particle threshold; [O] gravity caveat — the Stelle ghost makes the gravity perturbative S-matrix non-unitary → QG.AMB.01; [C] finite field-strength Z from the EG scheme. Connects S_pert to S_phys and verifies one-loop unitarity for matter+gauge."
    },
    {
      "id": "v279_qgeo_obligation_lemma",
      "cluster": "registry",
      "descriptionEN": "QGEO.OBLIG.01 — the QGEO.SYM.01 bedrock written as ONE precise constructive-QFT lemma with a proof-tree completeness check (the formal obligation write-up; does NOT close it). Lemma: given the premise 'the raw seam carries the flat τ=i pillowcase metric g_flat (Troyanov-unique, cone angles π), so Λ_Σ=√(−Δ_flat)', then ω_Σ∘ρ=ω_Σ → mark-locality → E8. [E] the reduction DAG from 'free RP seam' has 8 closed nodes and EXACTLY ONE open leaf ('the raw seam state IS the flat state'); [E] the geometric side is fixed (flat DtN diagonal, ρ commutes to all orders, Lean FORM.QGEO.04), so the obligation is STATE-IDENTIFICATION not geometric ambiguity; [C] two proof routes (RP-state uniqueness on the flat orbifold, or Troyanov + OS); [F] the all-orders step is a Lean theorem (SeamDeckClosure.flat_all_orders_clock, standard axioms, no sorry); [O] one constructive-QFT statement closes QGEO.SYM.01."
    },
    {
      "id": "v280_pillowcase_steklov",
      "cluster": "registry",
      "descriptionEN": "QGEO.STEKLOV.01 — a direct numerical experiment on the flat τ=i pillowcase orbifold (the seam geometry): the self-investigable test of the QGEO.SYM.01 obligation (does NOT prove the premise). [E] pillowcase = Z2-even sector of the flat torus, 4 cone points (2-torsion, deficit π each, sum 4π=2π·2 Gauss-Bonnet); [E] the order-4 deck ρ (z→iz) descends ([ρ,Z2]=[ρ,P_even]=0) and permutes the 4 cone points as 2 fixed + 1 swapped pair; [E] the geometric chain is exact: [ρ,Δ]=0 ⇒ [ρ,H]=0 (H=√(−Δ)) ⇒ [ρ,C]=0 (C=(1+e^H)⁻¹) ⇒ ω∘ρ=ω (the v276/v279 all-orders closure on the real orbifold); [C] the Steklov DtN is positive self-adjoint, fixed by the flat metric (route-(i) evidence for RP-state uniqueness); [O] confirms the conclusion given flat geometry, the premise stays open."
    },
    {
      "id": "v281_holomorphy_modular_data",
      "cluster": "frontier",
      "descriptionEN": "QGAMB.MODULAR.01 — the modular-data / anyon-condensation layer of the QG.AMB.01 Tier-B holomorphy bit. [E] #anyons=|det Gram|: E8→1 (holomorphic), D8=SO(16)→4, carrier D5⊕A3→16; [E] the anyon-condensation tower D5⊕A3 (16) → D8 (4) → E8 (1), each step condensing a Lagrangian Z2 boson (|det|/4=2²), the Kitaev E8 endpoint (v92/v235); [E] Gauss-Milgram (D8 spins 0,1/2,1,1) recovers c=8; [E] E8 is the unique holomorphic c=8 (det 1) — SO(16) has the same c=8 but 4 anyons, so holomorphy is THE discriminator (v277); [O] the open bit: does the seam-Calderón net condense to det K=1? Reduced to one bit, not closed."
    },
    {
      "id": "v282_e8_tau_i_unification",
      "cluster": "frontier",
      "descriptionEN": "QGAMB.UNIFY.01 — the E8-at-τ=i unification: the candidate SIMPLIFICATION reducing the two open obligations to ONE object (the raw seam is the (E8)₁ holomorphic CFT at τ=i). χ_E8(τ)=Θ_E8/η⁸=j^{1/3}. [E] τ=i is the order-4 CM point (j=1728, the QGEO.SYM.01 flat geometry); [E] χ_E8=j^{1/3} (χ³=q·j as q-series), so χ_E8(i)=1728^{1/3}=12 (the (E8)₁ character at the SAME τ=i, the QG.AMB.01 holomorphy); [E] one object, two faces — τ=i is both the order-4 CM point and the (E8)₁ partition-function modulus; [C] obligation count 2→1: prove 'the raw seam is (E8)₁ at τ=i' and BOTH QGEO.SYM.01 and QG.AMB.01 follow; [O] the unified premise stays open but is now ONE object. A genuine reduction of the open count, not a closure."
    },
    {
      "id": "v283_live_killtest_scorecard",
      "cluster": "frontier",
      "descriptionEN": "PRED.LIVE.01 — a live kill-test scorecard of the recent round's computable predictions vs current data, each tagged with the decisive experiment. [N] PMNS angles sin²θ₁₂=0.30675 (+0.31σ), sin²θ₁₃=0.02311 (+1.45σ), θ₂₃ maximal; [C] leptonic CP J_PMNS=−0.0297 (J_max ~3% from NuFIT), δ=240° vs ~232(30); [X] Σm_ν NO floor=0.0586 eV (consistent at the edge, DESI/CMB-S4 decisive); [X] proton decay τ_p=1.55e35 yr for +(15,1,1) (safe; minimal excluded; Hyper-K decisive); [N] closed readouts α⁻¹, Ω_b=0.04894, β=0.2424°; [N] all rows consistent < 2σ, 2 sharp near-term kill tests. A consolidation, not new physics."
    },
    {
      "id": "v284_route_i_rp_uniqueness",
      "cluster": "registry",
      "descriptionEN": "QGEO.ROUTEI.01 — Route (i) (RP-state uniqueness) decomposed into a 6-lemma chain with every dischargeable lemma discharged and the one irreducible lemma isolated (does NOT prove the open premise). [E] 5/6 lemmas are existing [E]/[F] results — L1 RP→DtN (v194), L2 the four marks force the pillowcase class (v195/v216/v214), L3 the flat orbifold Steklov operator = √(−Δ_flat) (verified: flat-strip DtN(k)/k→1, principal symbol |ξ| with no curvature term), L4 covariance C=(1+e^H)⁻¹ from the DtN (v258), L5 μ4-invariance (v276/Lean/v280); [C] Troyanov reduction: 4 cone points of angle π ⇒ χ_orb=0 ⇒ the unique constant-curvature representative is flat (τ=i), so L_open reduces to 'the raw seam is the constant-curvature geometric state'; [O] the ONE open lemma — the raw RP seam state is the constant-curvature/geometric state (the state-identification theory-selection step)."
    },
    {
      "id": "v285_route_ii_seam_condensation",
      "cluster": "frontier",
      "descriptionEN": "QGAMB.ROUTEII.01 — Route (ii) (seam-net condensation) decomposed into a 4-lemma chain, with the key result that its open lemma COINCIDES with Route (i)'s (the v282 unification at the lemma level). [E] 3/4 lemmas discharged — M1 no abelian sector ⟺ |det K|=1 ⟺ holomorphic (v234/v235), M2 holomorphic c=8 ⟺ (E8)₁ with the SO(16) det-4 counterexample (v277/v281), M3 the condensation tower 16→4→1 = the unique μ4-Lagrangian glue (v92/v281/v235); [E] the discriminator det 1 vs det 4; [E] Routes (i) and (ii) share ONE open lemma — both say 'the raw seam IS the (E8)₁-at-τ=i object' (v282), so proving either closes both; [O] the unified open lemma (the reviewer's exact demand): construct a canonical equivalence between the raw seam KMS/DtN state and the holomorphic (E8)₁ net at τ=i. M_open = QGEO.SYM.01."
    },
    {
      "id": "v286_seam_equivalence_contract",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.01 — the Seam Equivalence contract: concentrates the whole open structure on ONE named arrow (does NOT prove it). Theorem: the raw RP seam state reconstructs canonically the holomorphic (E8)₁ boundary net at the τ=i pillowcase ('the raw seam is (E8)₁ at τ=i'). [E] four objects (RawRPSeam, FlatTauIPillowcaseDtN, HolomorphicE8Net, SREKitaevE8Phase); [E] six arrows — 2 closed (Route i v276/v280/v284, Route ii v281/v277/v235), 1 conditional (v282), 2 open premises (the same lemma, v284/v285), 1 Gral; [E] IMPORT FIREWALL: no Route-i module imports any Route-ii module or vice versa, so the routes converge only through SEAM.EQUIV.01 (kills the 'E8 into the geometry and back' circularity); [E] exact-label taxonomy (Formal/Numerical/Conditional Exact, Open Selection); [O] the one open arrow 'free Gaussian RP bulk ⇒ holomorphic single-sector boundary'. Writes seam_equivalence.json."
    },
    {
      "id": "v287_free_rp_bulk_to_holomorphic_boundary",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.A01 — Route A (AQFT) theorem-dependency checker for the open Gral arrow 'free Gaussian RP bulk ⇒ holomorphic single-sector boundary net': types the 5-lemma chain and isolates the ONE missing standard theorem. [E] L1 RawRPSeam→OS→CAR/quasi-free boundary (v155/v160/v240/v175); [E] L2 gap+chirality ⇒ gapped chiral bulk (v64/v156); [O] L3 invertible/SRE Gaussian bulk ⇒ single-sector boundary (μ-index 1) — THE ONE MISSING STANDARD THEOREM (the AQFT form of det K=1); [E] L4 μ=1+c=8 ⇒ holomorphic (v234); [E] L5 holomorphic c=8 ⇒ (E8)₁ (v83/v277/v281). Verdict: 4/5 discharged, Route A reduces SEAM.EQUIV.01 to one cited theorem (invertibility ⇒ single-sector)."
    },
    {
      "id": "v288_full_l2_subprincipal_z4",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.B01 — Route B (DtN): the full-L² lift of the sub-principal Z4 block-diagonality, proving (numerically, on the full Toeplitz operator) the 'probably provable' step and lifting v201/v284 from the finite block to L². [E] character orthogonality: a mark-sourced curvature f=Σ_{j<4} g(θ−jπ/2) has Fourier support only on m≡0 mod 4 (nonzero modes {0,±4,±8,±12}; the 4th-root mark sum Σ_{j<4}((−i)^m)^j=4[4|m]); [E] full-L² block-diagonality: Λ=|n|+M_f commutes with ρ=diag(i^n), ‖[ρ,Λ]‖~2e-15 on −50..50; [E] negative control: a generic curvature gives ‖[ρ,Λ]‖=4.44>0; [E] Lean consistency (geom_sum_fourth_root + markLocal_blockDiagonal); [O] the sharper residual — why is the raw seam sub-principal term mark-local? (the curvature-locality half of SEAM.EQUIV.01)."
    },
    {
      "id": "v289_marklocal_raw",
      "cluster": "frontier",
      "descriptionEN": "MARKLOCAL.RAW.01 — decompose the ONE remaining Route-B residual ('why is the raw seam sub-principal term mark-local?') into a 5-lemma chain and isolate the single open analytic lemma. [C] L1 conic parametrix: conic Steklov DtN = |D_θ| + M_f + cone sources (b-calculus/Melrose, imported); [O] L2 FLAT-AWAY — the one open analytic lemma: RP+gap+4 marks ⇒ curvature vanishes away from the marks (the flat-pillowcase realisation); [E] L3 orbit-source: the 4 marks form one μ4 orbit (v195/v216); [E] L4 Fourier-support: μ4-orbit curvature ⇒ support only on 4ℤ (v264/v288, FFT, with a non-orbit negative control); [E] L5 full-operator: ⇒ [ρ,Λ]=0 on full L² (v288). 4/5 discharged; Route B reduces to exactly L2 — if it falls, Route B closes and SEAM.EQUIV.01 needs only the holomorphy half (Route A)."
    },
    {
      "id": "v290_z4_smooth_curvature_adversary",
      "cluster": "frontier",
      "descriptionEN": "SEAM.ADVERSARY.01 — the Route-B red-team that Z4 block-diagonality is NOT mark-locality. Constructs the strongest adversary, a SMOOTH Z4-symmetric off-mark curvature f=ε·cos(4θ), and shows honestly: [E] it is Z4-symmetric (Fourier support {±4}⊂4ℤ) but NOT mark-local (f(π/4)=−ε≠0); [E] it PASSES the v288 commutator ‖[ρ,Λ]‖=0 (same as mark-local; i^n−i^(n±4)=0) — so [ρ,Λ]=0 is necessary but NOT sufficient; [E] BUT it shifts the DtN spectrum off the flat ladder (max|Δλ|=0.155 at ε=0.3) and the heat trace Tr(e^{−tΛ}) (Δ=0.019) — the KMS weight + (E8)₁ character shift; [O] verdict: a genuine candidate modulus the commutator misses but the spectral data excludes — Flat-Away must use spectral/energy/heat input. An honest red-team, no manufactured pass."
    },
    {
      "id": "v291_flataway_contract",
      "cluster": "frontier",
      "descriptionEN": "FLATAWAY.RP.01 — the Flat-Away lemma as its own named mini-theorem with three proof routes opened by v290. LEMMA [O]: given RawRPSeam with gap, chirality and the four μ4 marks, the smooth curvature density in the DtN sub-principal symbol vanishes away from the branch divisor. [E] SPECTRAL-RIGIDITY route: scanning f=ε·cos(4θ), the spectrum deviation from the flat ladder is 0 only at ε=0 and grows with |ε| — flat is the unique spectral match. [E] HEAT-KERNEL route: the heat-trace deviation is 0 only at ε=0 — smooth off-mark curvature adds heat coefficients. [C] RP-ENERGY route: RP+gap = Dirichlet-energy minimisation in the Troyanov conformal class ⇒ constant-curvature minimiser (Troyanov; cf. v284). [O] verdict: 3 independent routes to ONE lemma; closing any one closes Route B."
    },
    {
      "id": "v292_flataway_heat_reduction",
      "cluster": "frontier",
      "descriptionEN": "FLATAWAY.HEAT.01 — the Heat-Kernel route made precise: reduces Flat-Away to ONE named spectral invariant. [E] the heat-trace deviation ΔTr(e^{−tΛ}) = c(t)ε²+O(ε⁴) is a positive-definite quadratic form in the off-mark curvature (c_k(t)>0 for every Z4 mode 4k: 0.066/0.047/0.033; positive on all combinations, no cancellation), so ΔTr=0 ⟺ f=0; leading invariant ‖f‖_L² (Parseval). [O] reduction: Flat-Away ⟸ 'the seam fixes the a₂ heat coefficient (=Tr e^{−tΛ}) to its flat value' ⇒ ∫f²=0 ⇒ f=0 off the marks. One external invariant, far smaller than 'prove Flat-Away'."
    },
    {
      "id": "v293_flataway_spectral_hessian",
      "cluster": "frontier",
      "descriptionEN": "FLATAWAY.SPEC.01 — the Spectral-Rigidity route upgraded from one direction (v291) to the FULL smooth-Z4 space. [E] splitting mechanism: the degenerate flat Steklov level |n| is split by mode 4k at |n|=2k by ±ε/2 (modes 4,8,12 split levels 2,4,6); [E] S=Σ(Δλ)² ~ 0.50ε² (quadratic); [E] the Hessian over modes {4,8,12} has eigenvalues ~{0.497,0.500,0.503}>0, so the flat seam is a STRICT isolated minimum over the whole Z4 space (answers the 'only one direction' objection); [E] clock-filter contrast: a non-Z4 mode (cos2θ) breaks the commutator outright. [O] consequence: pinning the seam's forced Steklov spectrum excludes every smooth Z4 deformation ⇒ Flat-Away — the same external fact as v292, reached independently."
    },
    {
      "id": "v294_flataway_rp_energy",
      "cluster": "frontier",
      "descriptionEN": "FLATAWAY.ENERGY.01 — the RP-energy / Troyanov route: the flat pillowcase is the unique constant-curvature minimiser at the fixed mark divisor. [E] prescribed divisor + Gauss-Bonnet: S² with 4 cone angles π, Σ(2π−θ)=4π=2πχ; [C] Troyanov/Euclidean-cone uniqueness: the prescribed cone divisor carries a UNIQUE constant-curvature (flat) metric (Troyanov 1991), excluding smooth off-mark curvature; [E] strictly convex curvature energy E[f]=∫f², Hessian 2π·I (PD) ⇒ unique minimiser; [E] no zero-cost deformation (E[ε cos4kθ]=π ε²>0). [O] premise: 'RP+gap realises the energy-minimiser', converging with v292 (a₂) and v293 (spectrum) on ONE selection principle for the flat metric; closing any one closes Route B."
    },
    {
      "id": "v295_flataway_a2_exact",
      "cluster": "frontier",
      "descriptionEN": "FLATAWAY.A2.01 — the EXACT analytic a₂ coefficient, upgrading the v292 positive-definiteness from numerical to a proof. [E] first variation vanishes: d/dε Tr e^{−tΛ(ε)}|₀ = −t f̂(0) Σe^{−t|n|} = 0 (smooth off-mark curvature has zero mean, Gauss-Bonnet); [E] convexity ⇒ global minimum: φ(x)=e^{−tx} strictly convex ⇒ Λ↦Tr e^{−tΛ} convex (Klein/Peierls), so ε=0 is the GLOBAL minimum and ΔTr≥0 for ALL ε (verified even in ε over [−1,2]), =0 iff f=0; [E] exact a₂ kernel ΔTr⁽²⁾=(1/4)Σ_n[Φ(|n|,|n+4k|)+Φ(|n|,|n−4k|)]g² with the divided-difference kernel of e^{−tx} (Duhamel), matched to numerics to rel.err<1e−6; [E] degenerate-split closed form 2e^{−2kt}(cosh(tg/2)−1)≥0; [O] reduction now analytic: Flat-Away ⟸ 'the seam fixes a₂'. Lean-mirrored (FORM.FLATAWAY.01)."
    },
    {
      "id": "v296_flataway_a2_closed",
      "cluster": "frontier",
      "descriptionEN": "FLATAWAY.A2.CLOSED.01 — the exact a₂ coefficient in CLOSED FORM (replacing v295's validated-numeric sum). [E] the operator tails telescope to a geometric series (Φ(n,n+4k)=e^{−nt}C_k/(16k²), C_k+D_k e^{−4kt}=4kt(1−e^{−4kt})), giving W_k^tail=t(1−e^{−4kt})/(8k(1−e^{−t})); [E] the middle is finite (4k−1 terms incl. the degenerate diagonal Φ(2k,2k)=(t²/2)e^{−2kt}); [E] closed form W_k(t)=t(1−e^{−4kt})/(8k(1−e^{−t}))+½Σ_{m=1}^{4k−1}Φ(m,4k−m), with W_1(t)=t(2t eᵗ+e³ᵗ+3e²ᵗ+eᵗ−1)e^{−3t}/8, reproducing v295 to machine precision; [E] positivity manifest term by term; [E] small-t W_k=t/2+O(t³) (mode-independent leading 1/2, the t² term cancels)."
    },
    {
      "id": "v297_route_a_literature_stack",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.A.LIT.01 — Route A's 'one standard import' written out as a precise CITABLE theorem stack. [E] LIT-A invertible/SRE bulk ⇒ trivial bulk topological order (Kitaev, Ann.Phys.321 2006; Freed-Hopkins, Geom.Topol.25 2021); [E] LIT-B bulk MTC = Drinfeld center, trivial center ⇒ holomorphic boundary net (Müger, JPAA 180 2003; Kawahigashi-Longo-Müger, CMP 219 2001); [E] LIT-C holomorphic c=8 net ⇒ (E8)₁ (Conway-Sloane lattice uniqueness; Dong-Xu/Kawahigashi-Longo). [E] the three COMPOSE (out=in matches) into 'invertible bulk ⇒ (E8)₁', all legs established; [O] the ONE open hypothesis is the input of LIT-A, 'the raw quasi-free seam bulk is invertible (SRE)' = Route B's Flat-Away. [E] non-circular with Route B (no DtN/Steklov; the v286 firewall holds)."
    },
    {
      "id": "v298_e8_network_attractor",
      "cluster": "frontier",
      "descriptionEN": "E8 as a network attractor — a computational-substrate (Wolfram-program) reading of the (2,3,5)/McKay hull, on the v219 bedrock; an HONEST audit of the 'TFPT as hypergraph' idea, not a derivation. [E] the E8 Kac marks (1,2,3,4,5,6,4,2,3) are the UNIQUE attractor of the minimal lazy graph update m←(A+2I)/4·m on the affine-E8/(2,3,5) McKay graph (converges from any positive start; A·m=2m, top eigenvalue 2). [E] the spherical McKay tower terminates at E8 (2T→E6 h12, 2O→E7 h18, 2I→E8 h30=2·3·5). [E] the E-cascade is nested-subgraph coarse-graining (|roots| 240→126→72→40→20). [C] Wolfram-bridge reading: 'E8 is the universal attractor of a minimal (2,3,5) network' at the Coxeter-skeleton level. [O] the deep question (a minimal hypergraph REWRITE whose attractor is the FULL TFPT structure) stays open; NEG: φ₀=(4/3)c₃=1/(6π) is the analytic seed not an edge fraction, recovery is a 1-D Möbius fixed point not a rewrite."
    },
    {
      "id": "v299_hypergraph_0d_autonomous",
      "cluster": "frontier",
      "descriptionEN": "The hypergraph/Wolfram bridge, hardened: **the 3-arm (2,3,·) branch type** is the irreducible seed on which a LOCAL witness-based spectral growth REACHES E8 as the last finite point (E6→E7→E8, then stops at the affine boundary Ê8; E8 = maximal FINITE point) — NOT 'P2 derived'. [E] HYP.ICO.01 icosahedron = 3-uniform hypergraph (12 V, 20 hyperedges, 30 edges=h(E8)), |Aut(graph)|=120=|I_h|=A5×ℤ₂ (the McKay group is the SU(2) binary lift 2I, a different order-120 group; v219); [E] HYP.REWRITE.01 the 1→4 subdivision rewrite is equivariant (120 symmetries at every scale, 12→42→162); [E] HYP.FUSION.01 the 2I fusion tensor is a nonneg-integer weighted 3-uniform hypergraph, 2-dim slice = affine E8 McKay graph; [E] HYP.SMITH.01 ρ≤2 selects (affine) ADE (Smith); [E] HYP.CW.01 Collatz–Wielandt: ρ≤2 ⟺ positive witness m with A m≤2m (local per-node), gate decided by the diffusion witness (Perron only as audit); [E*] HYP.AUTO.01 self-generated witness (projectively-normalised diffusion) + LOCAL gate A m≤2m runs E6→E7→E8→Ê8 (critical witness = Kac marks) under the (2,3,·) seed + grow-longest-leg policy; [E] HYP.NEG.AN.01 ρ(A_n)=2cos(π/(n+1))<2 for all n, never reaches E8; [E] HYP.NEG.DN.01 ρ(D_n)=2cos(π/(2n−2))<2 for all n, never reaches E8; [O] HYP.SEED.01 the (2,3,·) seed is the single irreducible input = P2. A second network reading of the (2,3,5)/E8 core, honestly bounded; builds on v219/v298."
    },
    {
      "id": "v300_flataway_rigid",
      "cluster": "frontier",
      "descriptionEN": "FLATAWAY.RIGID.01 — the attack on Flat-Away: (I) HARDEN the obstruction from a soft minimum (v291/v293) to a DISCRETE one, (II) DERIVE the external pin from the (E8)₁ data. [E] FLAT FINGERPRINT: spec(|D_θ|) is the integer ladder, every level n≥1 of multiplicity exactly 2 — a discrete fingerprint, not a norm. [E] EXACT DEGENERATE SPLIT: the resonant mode 4k splits level 2k by gap = g_k (sympy 2×2 [[2k,g/2],[g/2,2k]] ⇒ 2k±g/2, + an O(g²) common shift), validated vs the full Toeplitz operator. [E] HARD OBSTRUCTION: any g_k≠0 lifts the resonant degeneracy (mult 2→1+1) and changes the spectral MULTISET ⇒ preserving the flat ladder *with its mult-2 degeneracies* forces every g_k=0 ⇒ f=0 (strictly stronger than the v291/v293 'deviation grows' scan). [E] TWO-METRIC ISOLATION: with the v295 convexity (ΔTr≥0, =0 iff f=0), flat is isolated analytically AND discretely. [C] THE PIN DERIVED: the (E8)₁ vacuum character E4/η⁸=q^{-1/3}(1+248q+4124q²+…) has integer q-coefficients with 248 at level 1 (E8 even unimodular ⇒ integer weights); by 2d Steklov conformal rigidity the integer ladder ⟺ flat seam, so (E8)₁ rationality ⇒ f=0 (conditional on Route A). [O] VERDICT: Flat-Away carries ZERO new open content beyond Route A; convexity unconditional, the discrete obstruction exact; the two SEAM.EQUIV.01 routes converge on one rationality fact."
    },
    {
      "id": "v301_route_a_invertible",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.A.INV.01 — discharge Route A's single open hypothesis (v297 LIT-A: 'the raw quasi-free seam bulk is invertible/SRE') via the free-fermion classification. Since v300 collapsed Route B into Route A, this is the LAST unconditional gap of SEAM.EQUIV.01. [E] carrier = 16 Majoranas, c=8: g_car+N_fam=5+3=8=16/2 (v148). [E] SO(16)₁ ⊃ (E8)₁: dim so(16)=120 currents + 128 spinor = 248 = dim E8 (v154). [C] free-fermion invertibility (Kitaev periodic table): a gapped Gaussian/quasi-free 2+1D fermion phase (class D) is Z-classified by the chiral Majorana number and is ALWAYS invertible (no intrinsic anyons; topological order needs interactions) — so a gapped quasi-free bulk is invertible automatically. [E] invariant |det K|=#anyons: E8 Cartan det=1 (invertible) vs the gauged contrast D8=SO(16) det=4 (anyons), which free fermions cannot produce. [O] reduction: LIT-A's 'invertible bulk' now FOLLOWS from 'gapped 16-Majorana quasi-free bulk' (v148+v160+the gap); the residual changes from a topological-order statement to the spectral input 'is the quasi-free bulk gapped?'. [O] verdict: with v300 the WHOLE open content of SEAM.EQUIV.01 reduces to the single spectral statement 'the quasi-free seam bulk is gapped'."
    },
    {
      "id": "v302_seam_gap",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.GAP.01 — the LAST lever: after v300/v301 the whole open content of SEAM.EQUIV.01 was 'the quasi-free seam bulk is gapped'; this shows that gap is NOT a free input but the established RECOVERY GAP of the seam transfer operator. [E] TRANSFER SPECTRUM spec(T)={1,(2/3)^6,(1/3)^6}={1,64/729,1/729} (v160/v162), simple leading Perron eigenvalue 1, rest <1. [E] THE GAP IS DERIVED: Δ=−ln((2/3)^6)=6 ln(3/2)=2 N_fam ln(3/2)≈2.4328>0 (mult. gap 1−(2/3)^6=665/729); 6=2 N_fam, (2/3)^6 the μ4-deck transfer eigenvalue — forced by the carrier, not assumed. [E] POSITIVITY MARGIN Δ−31/(4π²)=1.6476>0 (the v76 decoupling margin), robust not marginal. [C] OS/QUASI-FREE CLUSTERING: for an RP quasi-free system a transfer gap Δ>0 ⟺ exponential clustering ⟺ a mass gap of the OS-reconstructed Hamiltonian (bulk mass gap = transfer gap). [O] VERDICT: the last TFPT-internal input is realised by the derived Recovery gap 6 ln(3/2)>0, so SEAM.EQUIV.01's residual is a composition of cited theorems (OS/clustering + Kitaev v301 + the v297 AQFT stack) over established TFPT facts — no undischarged TFPT-internal assumption remains; stays [O] (not machine-proved end-to-end)."
    },
    {
      "id": "v303_ftransfer_dynamics",
      "cluster": "frontier",
      "descriptionEN": "FR.DYNAMICS.01 — the discrete->dynamic lens on F_transfer: is the main-branch mechanism (a gapped, positivity-preserving relaxation to a UNIQUE attractor; local-averaging update -> E8 marks, gap 6 log(3/2)) also the shape of the four F_transfer instances? [E] F_pole (Koide) IS the seam dynamics: the source->pole/branch relaxation is the Moebius map with fixed-point multiplier F'=(2/3)^6 = lambda_2 = the seam clock subleading eigenvalue (v82/v54), |F'|<1 => a unique attractor at the SEAM rate. [C] F_Boltzmann (eta_B): a gapped positive contraction (washout kappa_f in (0,1) -> the unique balance, H-theorem), compiler fixes A_Lambda=10, rate thermal. [C] F_relic (axion): the misalignment comoving number freezes to a constant (adiabatic fixed point), seed theta_i=3pi/5, rate cosmological. [I] F_QCD (m_p/m_e): the 1-loop RG flow with the carrier b3=-7 (asymptotic freedom) has the Gaussian UV fixed point as attractor, Lambda_QCD dynamically generated. [O] VERDICT: ONE shape underlies all four = the main-branch principle; F_pole has the seam rate exactly, the other three share only the shape (external rates); F_transfer is the readout end of the one principle, not a bolt-on; simplicity preserved by honest fencing (v187)."
    },
    {
      "id": "v304_idg_nonlocal_ghost",
      "cluster": "frontier",
      "descriptionEN": "QGAMB.IDG.01 — the KMS spectral-action cutoff suggests a NONLOCAL (infinite-derivative) graviton form factor; under entire analyticity the Stelle ghost is a TRUNCATION ARTEFACT — a CONDITIONAL narrowing of the red-team rt_F perturbative-gravity attack, NOT a closure of QG.AMB.01 (no new parameter: the cutoff is the v259 seam KMS weight). [E] STELLE GHOST = TRUNCATION ARTEFACT: the local four-derivative propagator 1/(p^2(p^2+M^2)) has a healthy massless graviton (residue +1/M^2 at p^2=0) and a massive spin-2 mode with NEGATIVE residue -1/M^2 at p^2=-M^2 (reproduces rt_F/v278). [E] ENTIRE FORM FACTOR HAS NO EXTRA POLE: for an IDG operator p^2 a(p^2) with a(z)=e^{z/M^2} entire and nowhere zero, the dressed propagator 1/(p^2 a) keeps its ONLY pole at p^2=0 — no ghost (Tomboulis 1997; Biswas-Mazumdar-Siegel 2006). [E] NEG CONTROL: a polynomial truncation a(z)=1+z/M^2 (the R+R^2 order) has a real zero at z=-M^2 = the ghost pole, so 'entire vs polynomial' is the discriminator. [C] the cutoff is the beta=1 seam KMS weight f(u)=e^{-u} (v259), not a choice; truncating Tr e^{-D^2/Lambda^2} at a_4 is what makes the local R+R^2 and the ghost. [C] CONDITIONAL NARROWING: IF the resummed form factor is entire (Biswas-Mazumdar-Siegel class) THEN the perturbative graviton is ghost-free and rt_F's attack is a truncation artefact. [O] OPEN: (i) the trace regulator f is NOT the kinetic form factor a(Box) — entire analyticity is unproven; (ii) a ghost-free perturbative graviton is not the nonperturbative ambient measure — QGAMB.IDG.01 stays [C]/[O], not a QG.AMB.01 closure."
    },
    {
      "id": "v305_witness_independence",
      "cluster": "registry",
      "descriptionEN": "Witness-independence / generator-economy audit (structural anti-numerology firewall, companion to v100): every headline integer (4,5,2,3,16,8,240,248,120,30,1920,48,6) is a documented image of the single anchor a=(1,1,2) with pi the only transcendental primitive -> 13 atoms / 2 free primitives = 6.5x compression made explicit (the recurring 2/3=|Z2|/N_fam and (2/3)^6 are ONE ratio); E8/g_car/anchor are overdetermined (>=3 structurally independent witnesses), the 5 pure-seed readouts single-witness (correlated, scored jointly by v100); alpha^-1's 137 is foreign to the anchor family = an independent witness, so the map discriminates"
    },
    {
      "id": "v306_seed_crossval",
      "cluster": "registry",
      "descriptionEN": "Leave-one-out cross-validation of the shared seed phi0: back-solving phi0 independently from the 5 seed-grammar observables (sin^2 theta12, sin^2 theta13, beta, Omega_b, lambda_C; NuFIT 6.0 / ACT DR6 / Planck / PDG centrals) and error-weighted-averaging reproduces the axiom seed phi0=1/(6pi)+48 c3^4=0.0531720 to 0.04% (one real seed, not a per-observable fit); leave-one-out predicts every held-out observable within 3 sigma out of sample (most-tensioned = sin^2 theta13, ~2 sigma); a data<->formula shuffle explodes the chi^2 by >100x (power control)"
    },
    {
      "id": "v307_data_watchdog",
      "cluster": "frontier",
      "descriptionEN": "Data watchdog: the operational decision pipeline over the frozen registry (v84). Classifies all 20 entries (16 predictions + 2 assigned textures + 2 bands) PASS/WATCH/TENSION/KILL by live n_sigma vs repo-documented current bests (CODATA 2022, NuFIT 6.0, ACT DR6, Planck 2018, PDG 2024, BK18); no decidable prediction in KILL (theory alive); watch items alpha^-1 (1.9 sigma), s23 (1.8 sigma) and the most-tensioned core sin^2 theta13 (2.0 sigma); the prediction-of-record sin^2 theta12=0.306747 compatible (JUNO, the fastest falsifier); r in [0.0033,0.0048] below BK18 and the kill 0.01; w=-1 flagged the dark-energy watchdog"
    },
    {
      "id": "v308_seam_equiv_chain",
      "cluster": "frontier",
      "descriptionEN": "TOE attack 1 -- the SEAM.EQUIV.01 keystone as a composition certificate: the v297 stack as ONE well-typed chain gap>0 -> invertible (Kitaev) -> holomorphic c=8 (Mueger/KLM) -> (E8)_1 (Conway-Sloane), with exact discriminators |det Cartan(E8)|=1 vs |det Cartan(D8)|=4, carrier tower 4*4=16->4->1, c=g_car+N_fam=8; the input gap = the derived recovery gap 6 ln(3/2)>0. The only undischarged premise is the cited OS step -- an assembly/reduction certificate (Lean target), NOT a closure"
    },
    {
      "id": "v309_modular_bedrock",
      "cluster": "frontier",
      "descriptionEN": "TOE attack 2 -- the bedrock omega o rho = omega via an explicit modular Hamiltonian, non-circular: marks=4 from Gauss-Bonnet (v216) + the order-4 clock from cross-ratio 2 (v214) => mu4 derived not assumed; the quasi-free modular Hamiltonian K=log((1-C)/C) round-trips to the seam operator and [rho,K]=0 <=> omega o rho = omega (a Z2 profile breaks it). Residual = Bisognano-Wichmann intrinsicality of the raw collar; NOT a closure of QGEO.SYM.01"
    },
    {
      "id": "v310_carrier_sm_anomaly",
      "cluster": "registry",
      "descriptionEN": "TOE attack 3 -- the carrier half-spinor IS one anomaly-free SM generation with the SM RG: 16=Lambda^even(C^5)=1+10+5 -> SU(5)<SO(10) as 10+5bar+1 = one generation; gauge-anomaly-free (sum Y=0, sum Y^3=0, [SU2]^2U1=[SU3]^2U1=0, exact) and reproduces (b1,b2,b3)=(41/10,-19/6,-7); a 4th family shifts b1, so N_fam=3 pins the RG. Closes the spectrum+anomaly+beta sub-question; the full EG S-matrix + Yukawa stay open"
    },
    {
      "id": "v311_gap_decoupled_measure",
      "cluster": "frontier",
      "descriptionEN": "TOE attack 4 -- the physical QG measure as a gap-decoupled, exponentially-clustering object: the transfer spectrum {1,(2/3)^6,(1/3)^6} gives geometric clustering at rate (2/3)^6, correlation length 1/(6 ln(3/2)), finite susceptibility 729/665; the v76 margin 6 ln(3/2)-31/(4pi^2)~1.648>0 decouples the physical sector from the un-built ambient. Spectral-action a2=EH/a4=R^2/72 (v36) + KMS beta=1 (v259); QG.AMB.01 stays open (gap-decoupled, inert)"
    },
    {
      "id": "v312_hypergraph_substrate",
      "cluster": "frontier",
      "descriptionEN": "TOE attack 5 -- the hypergraph substrate sharpened: the affine-E8 Kac marks are the Perron eigenvector (A.marks=2 marks) and the subleading eigenvalue is exactly 2cos(pi/5)=golden ratio (the 5-fold/icosahedral signature), but the recovery rate (2/3)^6 is NOT any eigenvalue and phi0 is not a rational edge fraction; so a full-structure rewrite must inject exactly two non-graph-spectral data: the cusp weight 2/3=|Z2|/N_fam (gap 6 ln(3/2)) and the analytic seed phi0. The open question is precisely delimited, not closed"
    },
    {
      "id": "v313_golden_atoms_spectral",
      "cluster": "frontier",
      "descriptionEN": "The golden ratio is the g_car=5 signature -- the (2,3,5) atoms ARE the affine-E8 network spectral angles (v312 follow-up, exact). The charpoly factors x(x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1), the golden minimal polynomial x^2-x-1 divides it (phi=2cos(pi/5) exact eigenvalue); the non-trivial eigenvalues are 2cos(pi/k) for k in {2,3,5}={|Z2|,N_fam,g_car} (0,1,phi,1/phi) so the golden ratio is the g_car=5 (5-fold) signature -- a new spectral witness for g_car (raises v305 multiplicity). The (2,3,5) spherical excess 1/2+1/3+1/5=31/30 ties the v63/v76 margin numerator 31=2^g_car-1=248/8=1+h to the Coxeter 30 (a unification, not new evidence). HONEST NULL: phi is structural, not a readout. DIRECTION: phi is the E8-quasicrystal ratio (candidate substrate)"
    },
    {
      "id": "v314_rate_translation",
      "cluster": "frontier",
      "descriptionEN": "Do the discrete-kernel and dynamic rates translate? An exact number-field split. The affine-E8 adjacency spectrum splits into the rational part {0,+-1,+-2} in Q and the golden part {+-phi,+-1/phi} in Q(sqrt5); the dynamic transfer spectrum {1,(2/3)^6,(1/3)^6} is entirely rational. So the family rates DO translate (the rational discrete angles 2cos(pi/2)=0=|Z2|, 2cos(pi/3)=1=N_fam build the recovery rate (2/3)^6=(|Z2|/N_fam)^(2 N_fam) in Q), but the carrier 5-fold (golden, Q(sqrt5)) has NO rational dynamic image -- a field obstruction, so it is static-only. The shared object is the order-30 Coxeter element (30=g_car*(2 N_fam)=5*6); the phi quasicrystal carries the carrier, not the cusp-weight family dynamics"
    },
    {
      "id": "v315_coxeter_coupling",
      "cluster": "frontier",
      "descriptionEN": "Couple or factorize? The order-30 Coxeter sectors couple as a cyclotomic field (v314 follow-up). GROUP FACTORIZES: gcd(5,6)=1 => Z/30 = Z/5 x Z/6 (direct product, group-decoupled). EXPONENTS FACTOR: E8 exponents=totatives(30)=totatives(5)x totatives(6), phi(30)=4*2=8=rank E8. FIELD COUPLES: [Q(zeta30):Q]=8=rank E8, Q(zeta30) is the compositum of the carrier Q(zeta5) (contains sqrt5) and the family Q(zeta3). GALOIS = mu4 x Z2: (Z/30)^x=(Z/5)^x x (Z/3)^x=Z/4 x Z/2 (order 8, max order 4); mu4=(Z/5)^x is the carrier Galois (the seam clock = the carrier golden-field Galois), Z2=(Z/3)^x the family Galois. sqrt5=phi+1/phi=Gauss sum in Q(zeta5). VERDICT: 30=5*6 does not dynamically entangle -- it couples as the cyclotomic compositum, bridge is Galois not dynamical"
    },
    {
      "id": "v316_galois_readout",
      "cluster": "frontier",
      "descriptionEN": "Does the Galois mu4 x Z2 (v315) organize the physical readouts? Yes -- the CP phases. The hexagonal CP unit rho=zeta_6=zeta_30^5 is the order-6=2 N_fam FAMILY factor c^5 of v315 (the carrier is zeta_5=zeta_30^6); delta_CKM,lead=pi/3=arg(zeta_6), delta_PMNS=4pi/3=arg(zeta_6^4) (power 4=|mu4|), zeta_6^4=-zeta_6 the sheet flip (mu6=mu3 x mu2). The Galois Z2=Gal(Q(zeta6)/Q)={zeta_6->zeta_6^5=conj} IS CP conjugation (delta->-delta). The carrier factor zeta_5 (golden sqrt5) carries no phase (static). The magnitudes (sin^2 theta12=1/3-phi0/2, transcendental phi0) are NOT cyclotomic. So the Galois bridge reaches phenomenology through the family factor (phases), the carrier is static, the magnitudes are the analytic seed"
    },
    {
      "id": "v317_galois_family",
      "cluster": "frontier",
      "descriptionEN": "Are the 3 generations a mu3/Galois orbit? Yes (mu3), refined 1+2 by Galois (v316 follow-up). The 3 cusp weights {0,1/3,2/3} (Spec(Q+)={1,2,3}=A3 exponents) map under w->e^{2pi i w} to the 3rd roots of unity {1,zeta_3,zeta_3^2}; the cyclic family symmetry w->w+1/3 is a transitive mu3 orbit (the family IS the cube-root orbit). The transfer eigenvalues (1-w)^6={1,(2/3)^6,(1/3)^6} attach to the weights (attractor=eigenvalue 1 at w=0). The Galois Z2 (zeta_3->zeta_3^2, = CP conjugation v316) fixes w=0 (rational) and swaps w=1/3<->2/3 -> orbit 1+2; so the Galois-FIXED generation is the recovery ATTRACTOR (the law), the Galois PAIR is the decaying hierarchy. The family structure is arithmetic; the mass magnitudes stay the analytic seed"
    },
    {
      "id": "v319_translation_clock",
      "cluster": "horizon",
      "descriptionEN": "The translation clock: the discrete<->dynamic bridge IS the order-30 Coxeter element, a clock with two coprime hands 5 x 6, read law-inclusive (0..5) or live-only (1..5). The only object holding both the static carrier (Q(sqrt5)) and the dynamic family (rational) data is the order-30 Coxeter element (v314); 30 = h(E8) = g_car*(2 N_fam) = 5*6, gcd(5,6)=1 => Z/30 = Z/5 x Z/6 (v315). DYNAMIC HAND (Z/6 = 2 N_fam, ticks 0..5): the recovery exponent is exactly 6 = 2 N_fam, rate (2/3)^6; order-6 generator c^5 (zeta_6 = zeta_30^5). LAW vs LIVE: the resummed clock rate(n) = -p2 ln(1 - n/N_fam) (v124) has rate(0)=0 (the attractor/law) and live modes from n=1; the E8 live phases (totatives of 30) start at 1, so '0,1,2,3,4,5' is law-inclusive, '1,2,3,4,5' live-only. STATIC HAND (Z/5 = g_car, ticks 1..5): golden Q(sqrt5), field-obstructed (no rational dynamic image, v314) -- static-only. ONE FULL TURN = lcm(5,6) = 30 = the Coxeter period (rank E8 = 8 = phi(30) live phases, 4 invariant planes, v55). The arithmetic is [E]; the 'it is one clock' reading is [C]."
    },
    {
      "id": "v318_arithmetic_capstone",
      "cluster": "core",
      "descriptionEN": "The capstone: the SM structural sector is Q(zeta30) + Galois mu4 x Z2, and the complete input is {a, pi, v_geo} (synthesis of the v313-v317 arc). The structural sector (3 generations, 2 CP phases, orbit/hierarchy) lives in Q(zeta30): [Q(zeta30):Q]=phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=h(E8), Galois (Z/30)^x=mu4 x Z2. Rigidity: forced by the anchor atoms {2,3,5}=e(a). Magnitude reduction: phi0=(4/3)c3+48 c3^4 is a pure function of pi (no free parameters), so the magnitudes are {a,pi}-determined, not free. Free inventory: 0 dimensionless free parameters; pi the unique transcendental; v_geo the one dimensionful unit. Residual: the raw-seam realisation stays open (QGEO.SYM.01/SEAM.EQUIV.01)"
    },
    {
      "id": "v320_galois_cp_relation",
      "cluster": "frontier",
      "descriptionEN": "A NEW falsifiable prediction from the Galois structure: the two CP phases are locked, delta_PMNS = delta_CKM,lead + pi. Both phases are powers of one hexagonal unit rho=zeta_6 (v316): delta_CKM,lead=arg(rho)=pi/3=60deg, delta_PMNS=arg(rho^4)=4pi/3=240deg, rho^4=-rho, so delta_PMNS-delta_CKM,lead=pi exactly (the Z2 sheet flip) -- the quark and lepton leading CP phases are NOT independent. Prediction of record: delta_PMNS,lead=240deg (upgrades the assigned texture to a Galois-forced relation to the measured quark phase); currently within the broad NuFIT 6.0 range; KILL TEST: delta_PMNS robustly away from 240deg (>3 sigma; DUNE/Hyper-K/JUNO) falsifies the Galois-CP organization"
    },
    {
      "id": "v321_killtest_board",
      "cluster": "frontier",
      "descriptionEN": "The forward kill-test board: the decisive upcoming measurements, ranked (forward companion to v307). Each entry is locked to a freeze_file.csv kill row: #1 JUNO sin^2 theta12=0.306747 (fastest), #2 CMB-S4/LiteBIRD r (kill r>0.01), #3 DESI/Euclid w=-1, #4 DUNE/Hyper-K delta_PMNS=240deg (the new v320 Galois-CP relation), #5 DUNE/LEGEND/nEXO ordering+m_betabeta, #6 PSI n2EDM nEDM, #7 JUNO sin^2 theta13 (~2 sigma now), #8 LiteBIRD beta. Coverage locked to freeze_file; >=7 decisive tests; a forward decision ledger complementing v307, no new physics"
    },
    {
      "id": "v322_cp_lock_sharpened",
      "cluster": "frontier",
      "descriptionEN": "The v320 CP-lock prediction sharpened into a quantitative kill test. The leading CP phase sits on the six hexagonal nodes arg(rho^k)=k*60deg; delta_CKM,lead=60deg (node 1), delta_PMNS,lead=240deg (node 4). The node selector IS the deck order: delta_PMNS,lead=|mu4|*delta_CKM,lead=4*(pi/3)=4pi/3=delta_CKM,lead+pi (rho^4=-rho). Sub-leading budget: delta_CKM,full=pi/3+3 lambda^2 (~68.7deg); the lepton sub-leading is a separate open holonomy term bounded by ~3 lambda^2~8.7deg => band 240 +- ~9deg. Current tension: vs NuFIT 6.0 NO (delta_CP=212 +26/-41 deg) the leading 240deg sits at +1.08 sigma, band edge ~231deg at +0.74 sigma -- compatible today. Kill: the nearest wrong node is 60deg away >> budget; DUNE/Hyper-K (~5-15deg) discriminate, >3 sigma outside kills the lock. An anarchic delta_CP hits the band with p~5% and node 4 1-in-6 -- a low-prior pre-registered prediction"
    },
    {
      "id": "v323_bw_geometric_modular",
      "cluster": "frontier",
      "descriptionEN": "The keystone pushed one step: Bisognano-Wichmann makes the seam modular flow GEOMETRIC, so the mu4 deck invariance is not an independent axiom. The clock rho=diag(i^n)=exp(i(pi/2)L) is a geometric rotation (L=diag(n) the rotation generator, rho^4=I, mu4 in U(1)_rot). For a rotation-covariant covariance C=f(L) the modular Hamiltonian K=log((1-C)/C)=g(L), so [K,L]=0 (the modular flow commutes with all rotations) and [rho,K]=[rho,C]=0 -- the clock is a modular symmetry for free. Discriminator vs v309: a period-4 curvature preserves the clock but its modular flow is NOT geometric ([K,L]!=0); BW is strictly stronger. Linked bedrock: GIVEN the seam is the (E8)_1 chiral net (v308), BW/Hislop-Longo make the vacuum modular flow geometric => QGEO.SYM.01 (omega o rho=omega) is downstream of SEAM.EQUIV.01 + a rotation-invariant vacuum. Residual: the raw collar vacuum is rotation-invariant (cleaner than v309, shared with v308)"
    },
    {
      "id": "v324_hypergraph_fiber",
      "cluster": "frontier",
      "descriptionEN": "The minimal hypergraph substrate that works: a (2,3,5)-network FIBERED by a 3-fold cusp channel reproduces both the E8 skeleton and the recovery gap (the constructive v312 follow-up). Network factor (carrier/5-fold): T_net=(A+2I)/4 fixes the Kac marks (eigenvalue 1=E8 skeleton), subleading (phi+2)/4 (golden); (2/3)^6 NOT in its spectrum (v312 negative). Cusp fiber (family/3-fold): T_cusp=diag((1-w)^{2N_fam}), w in {0,1/3,2/3} (N_fam=3, v317), spectrum {1,(2/3)^6,(1/3)^6}. Fibered substrate carries both: T=T_net (x) T_cusp has top eigenvalue 1 (eigenvector marks (x) w=0 = skeleton x democratic cusp) AND (2/3)^6 as a genuine eigenvalue (network attractor x cusp subleading). Honest injection (v312): without the fiber (T_net (x) I_3) there is no (2/3)^6 -- the cusp datum is injected, not graph-spectral; substrate = network x cusp = the v315 split 30=g_car*(2 N_fam)=5*6. Minimality: the smallest fiber is the 3-node cusp (=N_fam) = order-30 (v319 clock). Verdict: a consistent fibered realization, not a derivation of 2/3 from a pure rewrite"
    },
    {
      "id": "v325_pillowcase_keystone",
      "cluster": "frontier",
      "descriptionEN": "The seam keystone as ONE citable theorem: the raw seam state is the flat tau=i pillowcase state, with EXACTLY ONE open lemma. IF (raw RP-collar seam state = flat tau=i pillowcase / rotation-invariant state) THEN 4 square marks + mu4 deck + omega o rho=omega + holomorphic c=8 => (E8)_1, with the carrier recovery gap as input. Pieces (all exact): n=2chi(S^2)=4 (v216); cross-ratio 2 => j=1728 (v214/v267); flat orbifold S^2(2,2,2,2), chi_orb=0 (Troyanov); DtN sub-principal at m=0 mod 4 (v201/v280); det Cartan(E8)=1 vs D8=4 (v83/v143/v308); gap 6 ln(3/2)>0 (v76/v302). The ONE open lemma -- the raw collar state IS the flat tau=i pillowcase / rotation-invariant state -- is SHARED by both bedrock IDs (QGEO.SYM.01, SEAM.EQUIV.01; v323), machine-pinned in Lean (FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). An assembly/reduction certificate (like v176/v261), not a closure."
    },
    {
      "id": "v326_ftransfer_suite",
      "cluster": "frontier",
      "descriptionEN": "The F_transfer solver suite: the four transfer readouts as ONE runnable harness, each a gapped relaxation to a unique attractor, with values + uncertainties + honest typing (productises v303). F_pole (Koide): Moebius contraction to Q*=2/3 with multiplier (2/3)^6 = the seam-clock subleading eigenvalue, unique attractor at the SEAM rate. F_Boltzmann (eta_B): washout contraction to the balance 6.1e-11, monotone (H-theorem), external thermal rate. F_relic (axion): the comoving number N=E/omega is the adiabatic invariant, the relic freezes (seed theta_i=3pi/5), cosmological rate. F_QCD (m_p/m_e): 1-loop RG with carrier b0=7 flows to the Gaussian UV fixed point, Lambda_QCD generated; standard RG. F_RareKaon (bridge): BR(K+)=9.45e-11 on NA62 2016-2024 (v202). Only F_pole has the seam rate (2/3)^6; the other four share the SHAPE with external rates honestly fenced (v187/v303) -- the suite makes the frontier measurable without claiming the external rates."
    },
    {
      "id": "v327_hypergraph_rewrite",
      "cluster": "frontier",
      "descriptionEN": "Deriving the cusp weight 2/3 from a minimal rewrite (sharpening v324/v312). A minimal local rule -- a family channel with N_fam=3 slots, one absorbing attractor (w=0) and |Z2|=2 surviving -- supplies the cusp fiber. M=[[1,0,0],[0,1/3,1/3],[0,1/3,1/3]] has spectrum {1,2/3,0}; 2/3=(N_fam-1)/N_fam=|Z2|/N_fam EMERGES from the rule arity, not injected. Over the order-6=2 N_fam hand the rate is (2/3)^6=64/729 = the recovery gap (v76/v324), derived. NON-GRAPH-SPECTRAL (proof, sharpening v312): 2/3 is NOT a root of the affine-E8 charpoly (p(2/3)=-3520/19683 != 0), so the recovery rate cannot be an adjacency eigenvalue. The absorbing state (eigenvalue 1) is the democratic w=0 law (v317). Honest residual: the arity {2,3} (the anchor atoms) is still the input (P2) -- v324's injected datum reduced to the anchor arity. Substrate = (2,3,5)-adjacency (carrier) x this minimal branching rule (family)."
    },
    {
      "id": "v328_theta13_pressure",
      "cluster": "neutrinos",
      "descriptionEN": "The theta13 pressure point: the honest weak spot, quantified and pre-registered. sin^2 theta13 = phi0 e^{-5/6} = 0.023108 vs NuFIT 6.0 NO (0.02195 +- 0.00058) is +2.0 sigma -- the most-tensioned CORE prediction (v62/v307). theta13 is the carrier hypercharge trace (exponent gamma=tr_E Y^2=5/6 EXACT, v268), NOT the seesaw mu-tau breaking that sets theta12 -- distinct channels (a theta13 pull does not move theta12). No cheap relief: the back-solved seed phi0(theta13)=0.0505 is ~5% below the axiom, PMNS theta13 RG running is <1% (NO) so cannot close it, and 5/6 is exact -- the prediction stays frozen (REG.FREEZE.01). Pre-registered kill: a stable sin^2 theta13 away from 0.023108 at >3 sigma (JUNO/global) flags the carrier-trace/seed-grammar reading. Exactly the v306 seed-hyperplane outlier -- one honest pressure point, not a pattern of failures."
    },
    {
      "id": "v329_os_gap_reduction",
      "cluster": "frontier",
      "descriptionEN": "Closing the OS step: the bulk gap IS the transfer gap (second quantization), and assembling the single sufficient premise of the whole bedrock. [E] OS GAP = TRANSFER GAP: the OS/Euclidean many-body Hamiltonian dGamma(-log T) has spectrum = sums of single-mode energies, so the many-body (bulk) gap = the smallest positive single-mode energy = 6 ln(3/2), INDEPENDENT of system size -- exactly 'OS bulk gap = transfer gap' (the v308 open input), discharged at the many-body level. [E] H => omega o rho = omega: under H the covariance is rotation-covariant C=f(L), K=g(L) commutes with rho=exp(i(pi/2)L) (v201/v309/v323). [E] H => SRE => (E8)_1: det Cartan(E8)=1 vs D8=4 (v237/v308). [E] both bedrock IDs follow from the ONE premise H = 'raw collar = rotation-invariant flat tau=i pillowcase state' (Lean-pinned FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). [O] residual: the NECESSITY of H on the raw collar. Sharpens v240/v308/v323/v325; does NOT close the bedrock."
    },
    {
      "id": "v330_qgamb_admissible_measure",
      "cluster": "frontier",
      "descriptionEN": "Constructing the ambient measure on the gap-decoupled admissible sector as a bona fide Osterwalder-Schrader object (with the honest fence). [E] REFLECTION POSITIVITY: the seam transfer T (v221/v311; spectrum {1,(2/3)^6,(1/3)^6}) is symmetric PSD => the finite-volume measures are RP. [E] EXPONENTIAL CLUSTERING: C(n)/C(n-1) -> lambda_2=(2/3)^6 (xi=1/Delta, Delta=6 ln(3/2)). [E] TIGHTNESS => PROJECTIVE LIMIT: chi=sum_n C(n)=729/665 finite, so Var(S_L)/L -> chi => uniform tightness (Prokhorov) => the projective limit mu=lim mu_Lambda EXISTS on the admissible sector. [E] OS RECONSTRUCTION: H_OS=-log T>=0, unique vacuum, gap 6 ln(3/2) (v240) -- mu is a genuine OS measure on the admissible sector. [E] NEG: an ungapped transfer (lambda_2->1) sends chi->inf, tightness FAILS. [O] THE FULL AMBIENT MEASURE: the non-admissible (metric) sector is NOT built; by the gap margin Delta-31/(4pi^2)~1.648>0 (v76/v275) physical readouts do not need it, but QG.AMB.01 stays open. A genuine PARTIAL construction, NOT a closure."
    },
    {
      "id": "v331_necessity_of_H",
      "cluster": "frontier",
      "descriptionEN": "The NECESSITY of H reduced: rotation-invariance of the seam DtN is EQUIVALENT to the four marks at the square (tau=i), so the one open lemma is exactly the order-4 CM selection (the necessity companion to v329's sufficiency). [E] SQUARE => [rho,Lambda]=0: marks at {0,pi/2,pi,3pi/2} make the curvature Z4-symmetric (Fourier support 0 mod 4), M_f couples only modes differing by 4Z (the v201/v210 sufficiency as a DtN |k|+M_f operator). [E] NECESSITY: a mark off the square breaks Z4 and gives [rho,Lambda]!=0 -- equivalence, not just implication. [E] ORDER-4 CM: cross-ratio(1,i,-1,-i)=2 => j=1728, the unique 4-config with an order-4 automorphism (v214/v267); H <=> square <=> mu4. [E] NEG: a Z2 mark set commutes with rho^2 but not rho. [O] residual: the dynamical SELECTION of the square (the 4 marks forced by v216, the square the unique order-4 CM) is the irreducible postulate. Does NOT close the bedrock."
    },
    {
      "id": "v332_qgamb_metric_sector",
      "cluster": "frontier",
      "descriptionEN": "The metric sector of QG.AMB.01 characterized: the obstruction is exactly the conformal-mode (Gibbons-Hawking-Perry) problem, the physical sector is gap-insulated, and the only route is the conditional IDG taming (makes v330's metric-sector fence precise). [E] CONFORMAL-MODE WRONG SIGN: c_conf(D)=-(D-1)(D-2)/4 is NEGATIVE for D>=3 (c_conf(4)=-3/2) while the TT coefficient is positive -- the conformal-factor problem. [E] MEASURE OBSTRUCTION: the kinetic form is indefinite (one negative eigenvalue), so exp(-S) diverges over the conformal mode; the bare metric-sector measure is not directly definable. [E] PHYSICAL SECTOR INSULATED: the obstruction is in the conformal/gauge direction, not the admissible sector (positive+gapped, v330), margin Delta-31/(4pi^2)~1.648>0 (v76). [C] IDG CONDITIONAL ROUTE: the entire form factor e^{z/M^2} (v259/v304) dresses the conformal propagator with no extra pole. [E] NEG: a polynomial (R+R^2) truncation crosses zero (the Stelle ghost, v304/v278). [O] METRIC SECTOR OPEN: the full QG.AMB.01 needs the conformal mode resolved nonperturbatively (GHP rotation / IDG), gap-decoupled but NOT constructed. A precise characterization, NOT a closure."
    },
    {
      "id": "v333_cm_selection",
      "cluster": "frontier",
      "descriptionEN": "The CM-selection mechanism: tau=i (the square) is the UNIQUE order-4 modulus and the attractor of mark-equilibration, so the last keystone residual is 'the deck acts geometrically', not 'why the square' (the v331 follow-up). [E] ORDER-4 MODULAR ELEMENT: S=[[0,-1],[1,0]] in SL(2,Z) has S^2=-I, S^4=I and fixes tau=i; the unique order-4 elliptic point of PSL(2,Z) (the other, rho, is order 3). [E] DECK ORDER -> CM POINT: order-4 -> cross-ratio 2 -> j=1728 -> tau=i (square); order-3 -> j=0 -> tau=rho (hexagon); |mu4|=4 selects tau=i. [E] MARK-EQUILIBRATION ATTRACTOR: 4 points minimizing the log-energy (Fekete) converge to equal spacing (gaps pi/2 = the square); the square beats random perturbations -- the unique (mod rotation) attractor. [E] NEG: an order-3 deck selects rho not the square. [O] residual: only 'the deck acts GEOMETRICALLY (= QGEO.SYM.01) vs an abstract Z4 on labels' stays open -- the residual sharpens from 'why the square' to 'the deck acts geometrically'. NOT a closure."
    },
    {
      "id": "v334_conformal_resolution",
      "cluster": "frontier",
      "descriptionEN": "The conformal-mode resolution: the Gibbons-Hawking-Perry contour rotation (IR sign) + the infinite-derivative (IDG) dressing (UV pole) give a CONDITIONALLY convergent metric-sector measure; the residual is the nonperturbative validity of the contour (the v332 follow-up). [E] THE OBSTRUCTION: c_conf(4)=-3/2<0, the unrotated conformal Gaussian diverges. [E] GHP CONTOUR ROTATION: phi->i phi flips the sign, int(exp(-|c|phi^2))=sqrt(pi/|c|) FINITE (GHP 1978). [E] IDG UV DRESSING: the entire form factor exp(p^2/M^2)>0 (no ghost pole, v304), dressed per-mode action positive post-GHP and UV-suppressed. [E] NEG: a polynomial truncation crosses zero (the Stelle ghost, v304/v278). [C] CONDITIONAL CONSTRUCTION: GHP+IDG give a convergent metric-sector Gaussian mode-by-mode (conditional on the contour + entire analyticity). [O] residual: whether the GHP-rotated IDG-dressed measure is the CORRECT nonperturbative QG.AMB measure (the contour's nonperturbative validity); gap-decoupled (v76), blocks no test. NOT a closure."
    },
    {
      "id": "v335_seam_equiv_unify",
      "cluster": "core",
      "descriptionEN": "The keystone unification: there is ONE open theorem (SEAM.EQUIV.01), QGEO.SYM.01 is its COROLLARY (a conformal-net axiom), and QG.AMB.01 is a decoupled general QG problem, NOT a TFPT gate. [E] QGEO.SYM.01 = COROLLARY: a chiral conformal net has a Moebius-covariant vacuum (the unique invariant positive-energy vector, a NET AXIOM); rotations U(1) < Moebius => rotation-invariant vacuum; the mu4 clock = rotation by pi/2 fixes it => omega o rho = omega. So SEAM.EQUIV.01 => QGEO.SYM.01 with NO extra premise (the v323 'rotation-invariant vacuum' is a net axiom); the two bedrock items collapse to ONE. [E] THE ONE KEYSTONE THEOREM: SEAM.EQUIV.01 = a construction theorem (c=8, det Cartan E8=1, gap 6 ln(3/2)>0 derived) with EXACTLY 1 open continuum lemma. [E] QG.AMB.01 DECOUPLED: margin Delta-31/(4pi^2)~1.648>0 => no TFPT prediction depends on it; the general Euclidean-QG conformal-factor problem (GHP 1978), inherited not TFPT-specific. [E] COLLAPSED STATUS: structural items 2->1; live residual = v_geo + G_net(=SEAM.EQUIV.01) + F_transfer. [O] the one residual = SEAM.EQUIV.01's continuum OS reconstruction (Lean-pinned). A synthesis/inventory; does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v336_continuum_limit",
      "cluster": "core",
      "descriptionEN": "The ONE open lemma of SEAM.EQUIV.01, stated precisely and mapped to the recent rigorous literature: the continuum-limit and OS-reconstruction legs are now CITABLE, the genuinely-open residual narrows. [C] CONTINUUM LEG: Osborne-Stottmeister (CFT from Lattice Fermions, CMP 398 (2023) 219; legacy internal token MMST, on the MMST/Morinelli-Morsella-Stottmeister-Tanimoto OAR framework, arXiv:2010.11121) give the chiral-CFT scaling limit of free lattice fermions (Koo-Saleur -> Virasoro); WZW range rank<=c<=D. (E8)_1: c=8, rank 8, D=16 Majoranas (SO(16)_1 -> (E8)_1), 8<=8<=16 in range; the collar is gapped quasi-free (CAR). [C] OS LEG: Adamo-Moriwaki-Tanimoto (arXiv:2407.18222) prove the conformal OS axioms for unitary VOAs incl. lattice VOAs; (E8)_1 is the even self-dual rank-8 lattice VOA. [C] CONSTRUCTION + COVARIANCE (Adamo-Giorgetti-Tanimoto): arXiv:2506.01008 builds the 2d conformal net of an even lattice and classifies the 2d Heisenberg-net extensions as lattice nets (the lattice IS a conformal net), and arXiv:2508.07109 shows every DHR rep of such a net is automatically positive-energy + diffeomorphism-covariant (covariance is a consequence, not an input; defuses the v215 BW circularity). [E] TARGET PINNED: c=g_car+N_fam=8, det Cartan E8=1, the D=16 Majorana SO(16)_1 IS the carrier-16 and the mu4 extension IS the seam glue. [O] OPEN RESIDUAL: c=8 alone does NOT pin the net (SO(16)_1 also c=8, det 4), so the open lemma = the holomorphy discriminator det K=1. Does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v337_decoupling_theorem",
      "cluster": "frontier",
      "descriptionEN": "THE DECOUPLING THEOREM (consolidated, citable): every TFPT dimensionless readout factors through the gapped admissible sector and is INDEPENDENT of the ambient QG measure -- the theorem that makes the QG.AMB.01 reclassification (v335) honest. [E] GAP MARGIN: Delta - 2*dim(E8)*c3^2 = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0. [E] FACTORIZATION: every readout (mass ratios, mixings, alpha^-1, R+R^2) is a function of the admissible gapped transfer spectrum {1,(2/3)^6,(1/3)^6} + the discrete kernel alone. [E] THE THEOREM: margin>0 => exponential clustering (lambda_2=(2/3)^6), finite susceptibility chi=729/665, so the ambient contribution is gap-suppressed => no readout depends on the ambient measure => QG.AMB.01 decoupled. [E] NEG CONTROL: gap closed => chi diverges => fails (non-vacuous). [E] COROLLARY: zero frozen kill tests depend on the ambient measure. Consolidates v76/v311/v330/v332; does NOT construct the ambient measure."
    },
    {
      "id": "v338_theta13_budget",
      "cluster": "neutrinos",
      "descriptionEN": "THETA13.BUDGET.01: the theta13 sub-leading budget vs the current global fit -- the +2 sigma pressure point (v328) quantified against BOTH NuFIT 6.0 NO variants and JUNO; neither defuses nor excludes, fences correctly. [N] SK-DEPENDENT TENSION: sin^2 theta13 = phi0 e^{-5/6} = 0.023108 is +2.0 sigma vs the no-SK central (0.02195) and +1.7 sigma vs the with-SK NO best fit (0.02215) -- NOT a clean 2 sigma. [E] WITHIN 3 SIGMA: 0.023108 < the with-SK 3 sigma upper 0.02388 (not falsified). [N] BUDGET: moving to the central needs -4.1%; RG < 1% and the exponent 5/6 is exact => no freeze-respecting relief (v328). [X] JUNO DECIDES: sub-1% precision turns 1.7-2.0 sigma into ~6 sigma -- near-term pre-registered kill/confirm. One honest pressure point (v306/v328), fenced."
    },
    {
      "id": "v339_firstprinciples_boundary",
      "cluster": "frontier",
      "descriptionEN": "The honest first-principles boundary, mapped precisely -- answers (a) can M_Planck/v_geo be computed? and (b) can F_transfer be made first-principles? Both NO, with the exact reason per residual, classified into FOUR kinds. [E] v_geo/M_Planck FORBIDDEN BY THEOREM (No-Unit, v153): a mass is dimension 1 while every TFPT datum is dimension 0, so no dimensionless map outputs a scale; over-determined (gravity = dark energy to 0.11%, v274); every RATIO is derived, only the unit is not. [E] m_p/m_e structure in place (v262): carrier b3=-7 + closed electron; 2 external inputs: alpha_s(M_Z) and the lattice C_p~2.83. [X] alpha_s external BECAUSE OF A TENSION: SM content does NOT unify (spread ~8.8 at 2e16, v246). [N] C_p parameter-free but lattice-only (v35). [C] eta_B/axion = cosmological initial conditions; Koide Q=2/3 structural [E]. [A] FOUR kinds: forbidden-by-theorem / external-parameter-free / external-tensioned / cosmological. A boundary audit, NOT a solution."
    },
    {
      "id": "v340_As_reconcile",
      "cluster": "frontier",
      "descriptionEN": "A_s is predicted (not missing), is DIMENSIONLESS (so it does NOT define an absolute mass), and the apparent -11 sigma tension is a reheating-channel question -- reconciled with the archived old derivation (TFPT v4.5/v4.6). [E] A_s = N_star^2 (M_scal/Mbar)^2/(24 pi^2) = N_star^2 c3^7/(24 pi^2) (v86); M_scal/Mbar = c3^{7/2} = 1.2565e-5 is a PURE RATIO, so A_s carries NO absolute scale -- it pins N_star, NOT M_bar => A_s does NOT define mass (No-Unit v153 stands). [N] the -11 sigma is the slow-vs-fast reheating dichotomy (slow Higgs N_star=51.44 -> A_s=1.764e-9; A_s-preferred N_star=56.2). [P] ARCHIVE: old v4.5/v4.6 used the algebraic N_star=3/phi0-1=55.42 and reported A_s=2.124e-9; at that N_star the current normalisation gives 2.047e-9 (-1.9 sigma). [E] nothing lost: the current theory DERIVES N_star (record band [50,60], v84). [X] decisive test: fast preheating or slow reheating confirmed."
    },
    {
      "id": "v351_continuum_realisation_sharpened",
      "cluster": "core",
      "descriptionEN": "The continuum realisation sharpened -- the 'c=8 ambiguity' flagged open in v277/v344 (E8 vs SO(16) both have c=8) is RESOLVED, non-circularly, by the seam's ORDER-4 mu4 clock. [E] E8 (det 1, holomorphic) vs D8=SO(16) (det 4). [E] the order-4 clock (the four Gauss-Bonnet marks, |mu4|=4=h(A3), v216): index-4 extension D5(+)A3 = E8 (v154), index-2 = SO(16); so order-4 selects E8 (16->4->1) not SO(16) (16->4) -- the order (4 not 2) is the discriminator. [E] bootstrap agrees: h(E8)=30 (max prime 5) vs h(D8)=14 (max prime 7). [E] which-net ambiguity RESOLVED (det K=1 is then a consequence). [O] residual = PURELY the chiral-edge/scaling-limit existence (bulk-edge, v336), net+holomorphy pinned. Does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v364_vgeo_sharpen",
      "cluster": "core",
      "descriptionEN": "Direction 6: v_geo is the SINGLE dimensionful input of the WHOLE theory -- confirmed across matter AND gravity -- and irreducibly primitive. The No-Unit Theorem (v153) showed the dimensionless compiler cannot output a scale (v_geo a forced unit). SHARPENING after the parameter-free gravity (v358/v359/v361): the gravity sector adds NO new scale either -- 1/c3=8pi and the Lambda prefactor (8pi)^2*48c3^4=3/(4pi^2) are dimensionless, so 1/G~v_geo^2 and Lambda~(dimensionless)*v_geo^4 reduce to v_geo times atoms. [E] FINAL TALLY: every dimensionful quantity = (dimensionless ratio)*(power of v_geo); free content = {v_geo, pi}, ZERO dimensionless dials, vs the SM's ~26. [O] v_geo irreducibly primitive (No-Unit; 1/G UV-sensitive Sakharov v68). Clarity, not a new derivation; the last dimensionful anchor named."
    },
    {
      "id": "v361_grav_backreaction",
      "cluster": "core",
      "descriptionEN": "Direction 5: the matter-gravity backreaction with the explicit J3 flux -- two forced facts + an honest decline. [E] the carrier's gravitational anomaly is FORCED: c_-=8 (16 Majoranas=dim S+=2^(g_car-1); chiral central charge 16/2=8=g_car+N_fam, the same c_- as v358). [E] the backreaction is FINITE -> Lambda from alpha: the matter vacuum energy through G_ab=(1/c3)T_ab is gap-suppressed (Decoupling Thm v337) to rho_Lambda/Mbar^4=(3/(4 pi^2))e^{-2 alpha^-1} (v60), 123 orders = 119.03+3.92 -- NOT the M_Pl^4 catastrophe; the loop (v359+v60+carrier vacuum) closes. [E] J3 explicit (v358): delta<K_B>=(8 pi^2 R^4/15) delta<T_00>, carrier T_ab=the SM's (v159). [C] SM trace anomaly reproduced (content=SM). [O] DECLINE: no new atom-forced read-out of individual SM quantum numbers beyond c_-=8 and Lambda -> per v354/v355 declined. Finite & consistent backreaction, no new dial."
    },
    {
      "id": "v360_grav_gap_corrections",
      "cluster": "core",
      "descriptionEN": "Direction 2: the gap-induced higher-curvature correction to GR -- the disciplined result. Does the seam transfer gap give a NEW forced near-term GR-deviation beyond the known R^2 scalaron? With the v354/v355 discriminator: NO. [E] the leading higher-curvature term IS the R^2 scalaron (a4=R^2/72 v36; M_scal^2/Mbar^2=c3^7, exp 7=g+N-1) -- already in TFPT, already falsifiable via n_s=1-2/N*, r=12/N*^2 (CMB-S4). [C] it is the entanglement-equilibrium next order (Wald first law -> f(R)=R+R^2, v28/v359). [E] the gap Delta=6 ln(3/2) is dimensionless, so a distinct gap term sits at xi*v_geo ~ seam/Planck scale, not near-term observable. [O] DECLINE: a new dimensionless coefficient is not atom-forced (needs heat-kernel/Wald data; exp 7 vs 6, c3^7 vs (2/3)^6 unrelated) -> per v354/v355 declined. No new near-term forced prediction; the answer is the scalaron."
    },
    {
      "id": "v359_grav_nonlinear_einstein",
      "cluster": "core",
      "descriptionEN": "The FULL covariant Einstein equation, parameter-free (Direction 1, extends v358). The Jacobson-2015 fixed-VOLUME entanglement equilibrium brings in the EINSTEIN TENSOR G_ab (not just R_ab) -- the unique divergence-free metric tensor (Lovelock) -- so equilibrium yields the full covariant G_ab + Lambda g_ab = (1/c3) T_ab with conservation built in. [C/E] fixed-volume -> Einstein tensor (trace=-R in d=4). [E] Lovelock => nabla^a T_ab=0 is an output. [E] coeff 1: 8pi=1/c3 fixed (v358). [E] coeff 2: Lambda from alpha -- rho_Lambda/Mbar^4=(3/(4pi^2))e^{-2 alpha^-1}, prefactor (8pi)^2*48c3^4=3/(4pi^2) (v60); BOTH coefficients TFPT-fixed. RESULT: the original B6 full covariant field equation is now parameter-free at the LOCAL level. [O] residual: the equation-of-state status + the global ambient measure (C7)."
    },
    {
      "id": "v358_grav_entropy_equilibrium",
      "cluster": "core",
      "descriptionEN": "The entanglement-equilibrium derivation (Jacobson 2015 / Faulkner et al., delta S = delta<K>) with TFPT's atoms -> the LINEARISED Einstein equation falls out PARAMETER-FREE, and the two formerly-open pieces are closed. [E] parameter-free Einstein coeff G_ab=(1/c3)T_ab, 1/c3=8pi fixed (Unruh 1/(2pi)=4c3, EH 1/(16pi)=c3/2, Einstein 8pi=1/c3, Bekenstein 1/4=1/|mu4|). [E] THERMO=GEO coincidence (new): the first law's 2pi/eta = the geometric |Z2|2pi chi iff |mu4|=|Z2|chi=4 -- the thermodynamic and geometric origins of c3 are ONE. [C] J3 solved: the CHM ball modular Hamiltonian (boost via BW v323) gives the matter flux delta<K_B>=8pi^2 R^4/15 delta<T_00>. [E] entropy density atom-fixed: 1/4=1/|mu4| + central charge c=g_car+N_fam=8 (form v59). [O] residual: the equation-of-state status + the global ambient measure (C7) + the v_geo unit (v359 closes the FULL covariant equation); no free dimensionless gravity dial remains."
    },
    {
      "id": "v356_continuum_mmst_applicability",
      "cluster": "core",
      "descriptionEN": "Direction A: the continuum chain of SEAM.EQUIV.01 assembled theorem-by-theorem, residual reduced to ONE lattice-realization input tied to the seam's one-sidedness. Chain: gapped (v302) quasi-free 16-Majorana collar (v155/v160) -> invertible (Kitaev, v301) -> the NONTRIVIAL invertible phase (c_-=8) -> a non-gappable anomalous chiral edge (anomaly inflow: edge EXISTENCE follows, reducing the v336/v351 'does the edge exist?' residual) -> chiral-CFT scaling limit (MMST v336, 8<=8<=16) -> order-4 mu4 clock + AQFT stack pin (E8)_1 (v351/v297). The only non-theorem link is the seam one-sidedness = the c3 origin (c_-!=0 = one-sided = the |Z2| oint K = 8 of c3=1/(8pi), v58). [C]/[E]/[O]; ADVANCES SEAM.EQUIV.01 to one realization input, does not close. (Direction B was found already closed by v40.)"
    },
    {
      "id": "v355_e8_unmapped_bandwidth",
      "cluster": "registry",
      "descriptionEN": "A disciplined bandwidth search of the unmapped E8 region (next step after v354), run with a STRICT discriminator: a candidate counts only if it is a FORCED identity (a theorem about E8), never a numerical coincidence -- because mining a rich object for matches is the numerology temptation. [E] FORCED SET-LEVEL (the real collective content): sum(degrees)=128=dim S+ the spinor half of 248=120+128; sum(exponents)=120=#pos roots; product(degrees)=|W(E8)|; exponents = the phi(30)=8 totatives of 30 (v223); max(degree)=30=h(E8). [C] SUGGESTIVE not forced: 12=h(E6),18=h(E7),30=h(E8) (the McKay ladder v219), but 'deg(E8) contains subalgebra Coxeter numbers' is not a theorem -- flagged. [O] DECLINED: the per-degree atom matches (14=dim G2, 20=det L, 24=|W(A3)|, 12=|R(A3)|, 18=p4(a); v66 flagged 12,18 fittable) each admit multiple readings, and the |W| prime 7='scalaron exponent' is one of many -- declined. [E] RECONCILES v66 (degrees=atoms) and v354 (only 2,8,30 primary): NO new forced physical hit; the disciplined decline IS the anti-numerology result."
    },
    {
      "id": "v354_e8_reverse_audit",
      "cluster": "registry",
      "descriptionEN": "The REVERSE numerology audit + what the golden ratio means. The forward rule (v305) is one-directional (every readout must map onto E8); the reverse question: how much E8 structure maps onto NOTHING? [E] E8 has 8 Casimir invariant degrees {2,8,12,14,18,20,24,30}; THREE feed a readout (2 metric, 8 rank->c3, 30 Coxeter->g_car) and FIVE are UNMAPPED {12,14,18,20,24} -- 3/8 load-bearing, 5/8 not. [E] ECONOMY, NOT PROMISCUITY: a small fixed set (rank 8, h=30, det 1, D5(+)A3, 16-spinor) generates ALL readouts; the higher Casimirs are never reached into to fit -- the anti-numerology signature (economy + a large untouched remainder). [E] the golden ratio is UNMAPPED (the affine-E8 attractor eigenvalue v312, the icosian lattice v348; no FROZEN readout is phi -- the sole [C] exception, cos(theta_i=3pi/5)=-1/(2phi) on the axion spine branch, refined in v429) -> NOT numerology (which needs a number fitted to an observation); phi is the algebraic signature of the 5-fold h=30=2*3*5, the carrier announcing it is icosahedral. [O] the 5/8 unmapped is the HULL OVERHEAD (E8 = minimal unimodular hull, not the world group); where future readouts could live, or over-capacity. The reverse complement of v305."
    },
    {
      "id": "v353_selfloop_capstone",
      "cluster": "core",
      "descriptionEN": "The bird's-eye rethink (refining v352): TFPT is not a LINEAR theory (axioms -> theorems) but a CLOSED SELF-CONSISTENT LOOP whose outputs fix its own inputs, with ZERO free adjustable dimensionless parameters. [E] IT IS A LOOP: outputs fix inputs (g_car=5 = max prime of h(E8)=30; the 8 in c3 = rank E8) and inputs fix outputs; the arrows close into a circle (v6) with a unique fixed point (v56). [E] ZERO FREE DIALS: every load-bearing number is forced/over-determined; free dimensionless parameters = 0. [E] pi is a FIXED geometric constant (the boundary 2-sphere, oint K = 4pi -> c3=1/(8pi)), not a dial. [A] refining v352: pi is not a free input and the framework is a SELECTION PRINCIPLE (the unique self-consistent loop), not a free meta-axiom. [O] the one residual is the CONTINUUM closure (the chiral-edge existence, v336/v351); structural-realism (loop complete) vs constructivism (continuum open) is a philosophical, not numerical, fork."
    },
    {
      "id": "v352_framework_irreducible",
      "cluster": "core",
      "descriptionEN": "The framework reduction -- the deepest answer to 'the inputs are not axioms, they are back-determined'. [E] P2 (g_car=5) FORCED three ways (v6): rank-fill, Coxeter-match (g_car=max prime of h(E8)=30), Pascal. [E] P1 (c3=1/(8pi)) FORCED: c3 is the boundary Gauss-Bonnet coefficient (v216/v342), the 8 over-determined four ways (8=rank E8=h(D5)=phi(30)=det R, v54/v6); only pi is non-forced (the boundary transcendental). [E] the E8 hull is forced (unique even unimodular rank-8, v83/v154). [E] so the ONLY irreducibles are the FRAMEWORK (a discrete reflection-positive boundary with a finite Clifford carrier) + pi -- the precise content of 'the inputs are not axioms'. [O] the framework is the META-AXIOM (not math-reducible, the foundational stance, tested by its consequences not proved). TFPT's true input is ONE framework + pi."
    },
    {
      "id": "v350_bootstrap_inputs_correction",
      "cluster": "core",
      "descriptionEN": "The honest CORRECTION of v349's framing (prompted by the right objection: the inputs are not axioms, they are back-determined by the theory). [E] THE INPUTS ARE BOOTSTRAP-FORCED, NOT FREE AXIOMS (v6): g_car=5 over-determined three ways (rank-fill, Coxeter-match g_car=max prime of h(E8)=30, Pascal). [E] g_car=5 IS THE ICOSAHEDRAL 5-FOLD, not the D5 rank: g_car = max prime of h(E8)=30=2*3*5; the atoms (2,3,5) ARE the primes of the icosahedral Coxeter number 30. [E] THE GOLDEN RATIO IS EMERGENT FROM THE GLUE, not external: D5 (h=8)/A3 (h=4) non-golden alone, but the mu4 extension D5+A3 -> E8 (v154) gives h(E8)=30 (golden, 5|30) -- the seam's own mu4 produces the 5-fold; v349 checked the un-glued pieces, the wrong object. [E] v349's pentagon-vs-count dichotomy was FALSE (the bootstrap forces the pentagon). [O] residual relocates: the golden is bootstrap-forced; the only open thing is the physical continuum realisation (v336), NOT the golden ratio. Caveat: self-consistency within the framework, not from-nothing. Supersedes v349's interpretation."
    },
    {
      "id": "v349_raw_seam_golden_test",
      "cluster": "core",
      "descriptionEN": "The honest test of the v348 question 'does the RAW seam carry the golden ratio (before assuming E8)?'. Answer, computed: NO -- the bare carrier (D5, A3) and the dynamical seam data are NOT golden; the golden ratio genuinely enters WITH the icosahedral/E8 identification. A NEGATIVE result that prevents a false closure. [E] RAW CARRIER NOT GOLDEN: golden needs 5|h; D5 has h=8 (2cos(2pi/8)=sqrt2), A3 has h=4 -- no 5-fold, bare carrier gives sqrt2. [E] DYNAMICAL DATA NOT GOLDEN: cusp {0,1/3,2/3} rational, recovery (2/3)^6 rational, seed phi0 transcendental. [E] GOLDEN IS THE ICOSAHEDRAL/E8 SIDE: 5|h only for A4=SU(5) (h=5), H3 (h=10), E8 (h=30). [E] phi alone insufficient: 2I != C5 x mu4 (20 != 120). [O] RESOLUTION: L2 reduces to 'is g_car=5 a PENTAGON (5-fold => golden => icosahedral) or a COUNT (rank D5 => sqrt2)?' -- the golden ratio is the pentagon-reading of P2, NOT derivable from P2-as-a-count; no hidden phi. Foundational: P2-as-pentagon closes it (mode C) else rigidity must produce the 5-fold (mode B); NOT closed."
    },
    {
      "id": "v348_seam_rigidity_route",
      "cluster": "core",
      "descriptionEN": "Route B (the rigidity route) for the one open arrow L2, carried out -- it lands on a strikingly SIMPLE statement: the whole keystone reduces to ONE question, does the raw seam carry the golden ratio? [E] rigidity closes the crystallographic part: the order-4 clock has a UNIQUE normal form z->iz (Nielsen/Kerckhoff, v180), Troyanov gives the unique flat pillowcase metric (v284). [E] downstream is a LATTICE identity: E8 = the ring of icosians (Conway-Sloane), rank 8, so 2I->E8 is a lattice identity (stronger than McKay v219). [E] the single bridge is phi: crystallographic orders {1,2,3,4,6}, the 5-fold non-crystallographic; mu4 crystallographic, 2I (5-fold) not, phi=2cos(pi/5) extends one to the other (quasicrystal). [O] the keystone reduces to ONE question: 'does the raw seam carry phi?' -- if yes, mu4+phi -> 2I -> E8 closes L2. [O] honest subtlety: TFPT's phi is currently E8-side (v312/v313); the raw-seam data (cusp {0,1/3,2/3}, recovery (2/3)^6, seed phi0) are not manifestly golden, so showing the RAW seam carries phi non-circularly is the residual (= L2). NOT closed."
    },
    {
      "id": "v347_seam_closure_modes",
      "cluster": "core",
      "descriptionEN": "The closure-mode classification of the one open arrow L2 (the seam-orbifold realisation, v346) -- a direct honest answer to 'is there ANY other way to FULLY solve it?'. [E] THE PRECISE LOCUS: the seam gives the pillowcase base S^2(2,2,2,2), chi_orb=0 (EUCLIDEAN, v216), but the 2I/E8 structure is the Seifert base S^2(2,3,5), chi_orb=1/30 (SPHERICAL); L2 must bridge a FLAT 4-marked base to a SPHERICAL 3-marked base -- why L2 needs the carrier (2,3,5) input. [E] FOUR CLOSURE MODES: A construct (the analytic wall, v336); B rigidity (unique realisation => C^2/2I forced, the only theorem-route w/o new input, v284/v300/v331); C axiom P3 (=> TFPT complete rel. 3 axioms); D empirical (test the consequences). [E] no-new-state (tfpt_3/v246) blocks deriving L2 from extra physics => only rigidity (B) closes it as a theorem. [E] VERDICT: no free theorem; only B (theorem) or C (axiom), validated by D; only these four modes. [O] NOT closed."
    },
    {
      "id": "v346_seam_geometric_bridge",
      "cluster": "core",
      "descriptionEN": "The geometric bridge as an explicit six-link chain raw-seam => det K=1, with the ONE open arrow pinned -- the honest capstone of the SEAM.EQUIV.01 investigation. L1 [E/C] raw seam -> mu4 + four marks (v216); L2 [O] mu4 + (2,3,5) atoms -> 2I orbifold C^2/Gamma (THE OPEN ARROW = SEAM.EQUIV.01); L3 [E] 2I -> affine E8 McKay (v219); L4 [E] 2I -> E8 du Val (v232); L5 [E] C^2/2I -> Poincare link H_1=0 (v345); L6 [E] H_1=0 -> det K=1. [E] 5 of 6 links are theorems, only L2 open. [E] the group is FORCED by the seam atoms: axis signatures (2,3,3)/(2,3,4)/(2,3,5) -> 2T/2O/2I, (2,3,5) unique to 2I, and (|Z2|,N_fam,g_car)=(2,3,5) ARE those axes. [E] downstream of L2 forced. [O] L2 (the seam IS an ADE orbifold point) = SEAM.EQUIV.01, NOT closed -- the keystone reduced to one geometric-realisation arrow, group pinned by (2,3,5)."
    },
    {
      "id": "v345_hypergraph_homotopy",
      "cluster": "core",
      "descriptionEN": "EXECUTE the combinatorial core of route R3 (the hypergraph-homotopy route to det K=1, v344): computes explicitly that the (2,3,5)-rewrite attractor graph's plumbing link is the POINCARE homology sphere (H_1=0), only E8 among ADE; R3's combinatorial half is now [E]-complete, the geometric bridge (= SEAM.EQUIV.01) stays [O]. Plumbing topology (Brieskorn): link H_1 = coker(Cartan). [E] SNF(E8 Cartan) = identity => coker(E8)=0 => the E8 plumbing boundary is an INTEGRAL HOMOLOGY SPHERE (the genus-1 meaning of det E8=1). [E] only E8: coker(A_n)=Z/(n+1), D_n->Z/4, E6->Z/3, E7->Z/2 -- all nontrivial, only E8=0. [E] pi_1=2I=SL(2,5) PERFECT (commutator subgroup order 120=|G|, computed) => H_1=0 => THE Poincare sphere; |2I|=120=|mu4|*h(E8). [O] does NOT close det K=1: the graph->geometry half is meanwhile Kronheimer-cited (v479); what stays open is the raw seam supplying the ALE/orbifold datum = SEAM.EQUIV.01."
    },
    {
      "id": "v344_detk_synthesis",
      "cluster": "core",
      "descriptionEN": "The bird's-eye synthesis of the ONE open keystone bit, det K=1 -- a full-spectrum map of SEAM.EQUIV.01's only open residual. [E] det K=1 is ONE statement with SIX equivalent faces (holomorphy / homology-sphere link H_1=0 / one 1-dim irrep / det Cartan=1 / full mu4 condensation / tau=i CM), unified by |det Cartan| = |H_1(link)| = #1-dim irreps across ADE (A_n->n+1, D_n->4, E6->3, E7->2, E8->1) -- ONLY E8 gives 1. [E] THE GENUS-1 OBSTRUCTION: the easy arguments are genus-0, necessary but not sufficient (v87); det K is the torus degeneracy. [E] THREE genus-1 access routes: R1 continuum-modular (v336), R2 anyon-condensation (v281), R3 hypergraph-homotopy. [C] the HYPERGRAPH route is the most promising fresh angle: the (2,3,5) attractor is the affine-E8 graph (v312), the E8 du Val link is the Poincare sphere (v232), so det K=1 becomes COMBINATORIAL. [O] none closes it; R3's combinatorial-to-geometric bridge is meanwhile discharged to Kronheimer 1989 (v479), so the residual transforms to 'the raw seam supplies the ALE/orbifold datum' = SEAM.EQUIV.01."
    },
    {
      "id": "v343_four_routes_analysis",
      "cluster": "frontier",
      "descriptionEN": "The honest investigation of the four black-hole-birth solution routes (A finite causal diamond, B Carlip-Cardy near-horizon CFT, C modular/thermal flow, D self-reproduction attractor) -- direct answer to 'which closes ALL problems and is 100% provable': NONE individually, but they CONVERGE on the same two irreducible residuals as v335 (SEAM.EQUIV.01 + the decoupled QG contour). [E] ROUTE A reframes QG.AMB to a FINITE thermal trace (de Sitter finite, gapped chi=729/665) but c_conf(4)=-3/2<0 still needs the GHP contour (v334 [O]); does NOT touch SEAM.EQUIV.01. [C] ROUTE B: Carlip+Cardy consistent with c=8 but L0 free (ambiguity) and gives c only not holomorphy det K=1 -- inherits SEAM.EQUIV.01. [E] ROUTE C: modular flow KMS beta=1 (v239), geometric identification = SEAM.EQUIV.01 (v335) -- a reduction. [E]/[P] ROUTE D: unique Perron-Frobenius attractor (v56) but cyclic dynamics [P] -- not a closure. [E] VERDICT: independent closures = 0; irreducible residuals = 2 (= v335); BH premise makes both more natural but eliminates neither; focus = SEAM.EQUIV.01."
    },
    {
      "id": "v342_em_ward_heatkernel",
      "cluster": "em",
      "descriptionEN": "The heat-kernel sharpening of EM.WARD.01: DERIVES the ORIGIN and STRUCTURE of the F_{U(1)} determinant-line terms from textbook Seeley-DeWitt/Gilkey coefficients, advancing the EM-Ward origin from [P] to [E] for the STRUCTURE; NO new gate. [E] c3=1/(8pi) the boundary GAUSS-BONNET coefficient (seam = flat pillowcase chi=2, oint K = 4pi, v216). [E] the alpha^2 term is the GILKEY gauge-curvature coeff: a4 has 30 Omega^2/360 = (1/12)Omega^2, Omega=i alpha F => -(alpha^2/12)F^2, so the Calderon -2 c3^3 a^2 is the gauge-curvature piece, not an ansatz. [E] c3-ladder {0,3,6} arithmetic (3 channels; C6 hexagon at 6). [E] multiplicities 2=|Z2|, 8 b1=(rank E8)b1. [E] pi-power test: 2 c3^3 ~ pi^3 => THREE boundary insertions. [C] exact coefficient needs the seam F-normalisation. [O] the cubic alpha^3 + full identification = EM.WARD.01 residual (tied to SEAM.EQUIV.01)."
    },
    {
      "id": "v341_alpha_quillen",
      "cluster": "em",
      "descriptionEN": "The alpha fixed point reformulated as the STATIONARITY of a U(1) determinant line, every ingredient identified as an index / heat-kernel coefficient / discriminant form -- NOT a free knob (the 'derive the FORM' step). F_{U(1)} = a^3 - 2 c3^3 a^2 - (4/5) c3^6 M ln(1/phi_seam) = 0 in Quillen form: [a^3 - 2 c3^3 a^2] (det-line variation) = [8 b1 c3^6] (anomaly counterterm) * [ln(1/phi_seam)] (seam response). [E] M=41 a U(1) INDEX (EM Ward, v48): (4/5)M = 8 b1, b1=41/10 the SM 1-loop U(1)_Y beta. [E] c3=1/(8pi) GAUSS-BONNET boundary coeff (8=rank E8). [E] -5/4 = -q(D5) DISCRIMINANT; phi_base=(4/3)c3, c3/phi_base=3/4=q(A3). [E] delta_top=Omega_adm c3^4. [E] Quillen split holds at the alpha^-1=137.036 root. [E] NO NEW GATE: alpha stays [E]; v341 SHARPENS the pre-existing [P] EM.WARD.01 (no new alpha problem). [O] the one still-open part IS EM.WARD.01's residual: the AQFT proof that this is the EXACT Quillen functional (variational principle derived, not assembled). Sharpens v3/v48/EM.WARD.01 from fixed point to stationarity of a named determinant line."
    },
    {
      "id": "v365_qg_oneloop_saddle",
      "cluster": "core",
      "descriptionEN": "Direction 3: the QG measure (gate C7/QG.AMB.01) as one-loop GAUSSIAN fluctuations around the PARAMETER-FREE saddle (v358/v359). [E] the fluctuation stiffness is 1/c3=8pi (no free dial), so the one-loop problem is well-posed (it was NOT before v358). [E] the fluctuation operator is gapped M>=Delta_eff=6 ln(3/2)-31/(4pi^2)~1.648>0 (v76/v337) => mode-by-mode Gaussian convergence, no zero/negative modes besides the saddle directions. [E] finite model log-det Tr' log M=6 ln(9/2) on the OS spectrum {0,6 ln(3/2),6 ln 3}. [C] the conformal mode converges on the GHP contour (int exp(-|c|phi^2)=sqrt(2pi/3), v334). [E] the admissible projective limit exists (chi=1/(1-(2/3)^6)=729/665, v330). [E] 0 new dimensionless dials (v364). [O] residual: the full non-perturbative projective limit (G6/QG.AMB.01) on the ambient sector. Sharpens C7 from 'diffuse full QG' to a one-loop Gaussian problem around a parameter-free saddle; does NOT close it."
    },
    {
      "id": "v366_mmst_seam_collar",
      "cluster": "core",
      "descriptionEN": "Direction 7: the seam collar is IN MMST's free-lattice-fermion scaling-limit class, so the chiral scaling limit is a chiral CFT of central charge 8 pinned to (E8)_1 -- reducing SEAM.EQUIV.01 to the single S3 lattice-realisation input. Osborne-Stottmeister (arXiv:2107.13834, CMP 398 (2023) 219; legacy internal token MMST, on the MMST/Morinelli-Morsella-Stottmeister-Tanimoto OAR framework): free lattice fermions (CAR quasi-free) -> chiral-CFT massless scaling limit with correct c, range rank<=c<=D. [E] class arithmetic: D=dim S^+=2^(g_car-1)=16 Majoranas, c=g_car+N_fam=8, rank E8=8 => rank<=c<=D is 8<=8<=16 (in MMST range). [E] gapped (Delta=6 ln(3/2)>0, derived v302). [C] quasi-free CAR class (v155/v160). [E] chirality c_-=D/2=8 != 0 (non-gappable anomalous edge, v356 S4). [C] the MMST scaling-limit theorem applies (in-class & gapped & chiral & in-range) => chiral CFT c=8 (v336). [E] target pinned to (E8)_1 by c=8 + the order-4 mu4 clock (det K=1 vs SO(16) det K=4, v351). [O] residual S3: the collar realised as a genuine lattice chiral free-fermion invertible phase (one-sidedness/Flat-Away, v297/v356). Verifies MMST in-class; ADVANCES SEAM.EQUIV.01, does NOT close it."
    },
    {
      "id": "v370_grav_spin2_unitarity",
      "cluster": "frontier",
      "descriptionEN": "Track 2: the Barnes-Rivers spin decomposition of the graviton propagator -- perturbative graviton unitarity SECTOR-BY-SECTOR (extends v304's scalar pole algebra to the tensor spin structure + combines with v334's spin-0 conformal mode). [E] projector completeness: tr P2=(d-2)(d+1)/2, tr P1=d-1, tr P0s=tr P0w=1 sum to d(d+1)/2; the d=4 split is 5+3+1+1=10. [E] EH structure: spin-2 coefficient +1, spin-0 scalar -1/(d-2)=-1/2. [E] the Stelle ghost is a SPIN-2 state (a local 1+p^2/M^2 dressing puts the negative-residue pole at p^2=-M^2). [E] the entire KMS form factor e^{p^2/M^2} (v304/v259) makes spin-2 ghost-free (only the p^2=0 pole). [C] the wrong-sign spin-0 conformal mode (c_conf(4)=-3/2, v332) is cured by the GHP contour (v334) -- a different mechanism. [C] => perturbative graviton unitarity sector-by-sector, conditional on v304's entire-analyticity assumption. [O] perturbative-only + the open analyticity assumption; NOT the non-perturbative measure (QG.AMB.01). [O] decline: tr P2=5 is dimension counting, NOT g_car (v354/v355)."
    },
    {
      "id": "v367_seam_s3_lattice",
      "cluster": "core",
      "descriptionEN": "Track 1: the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1; turns the abstract S3 residual (v356/v366) into a RUNNABLE model with a numerically computed topological invariant. [E] bulk gapped (band gap 2>0 at M=1). [E] non-trivial Chern number (Fukui-Hatsugai-Suzuki: |C|=1 at M=1, C=0 at M=3 trivial control). [E] c_-=8: each chiral Majorana edge carries c_-=C/2=1/2, N_Maj=dim S^+=2^(g_car-1)=16 copies => c_-=16/2=8=g_car+N_fam. [E] edge K-matrix SO(16)_1 (det Cartan D8=4) -> (E8)_1 (det Cartan E8=1) by the order-4 mu4 clock (v351/v369). [C] the mu4 condensation SO(16)_1 -> (E8)_1 (v154/v351). [O] residual: the continuum scaling-limit existence (MMST v336). Reduces S3 to a runnable lattice model; does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v368_seam_s3_inflow",
      "cluster": "core",
      "descriptionEN": "Track 1: the bulk-edge / anomaly-inflow leg (S4 of the v356 chain) made CONCRETE on the v367 lattice model -- a finite STRIP of the gapped p+ip model has a realised, non-gappable chiral edge, so edge EXISTENCE follows from the invertible bulk (anomaly inflow), NOT a separate assumption. [E] topo in-gap edge states (M=1 strip min_kx|E|~0) vs trivial clean gap (M=3, neg control); [E] the near-zero state is edge-localised (weight within 4 sites of a boundary, edge>>bulk); [E] bulk-edge correspondence (edge branch count = bulk Chern |C|=1) => 16-Majorana edge, c_-=8. [C] anomaly inflow: a c_-!=0 invertible bulk cannot have a gappable edge => discharges the v336/v351 'does the edge exist?' residual on the lattice. [O] residual: the continuum scaling limit (MMST v336). Establishes edge EXISTENCE on the lattice; does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v371_ftransfer_pole",
      "cluster": "frontier",
      "descriptionEN": "Track 3 (F_transfer): the Koide source->pole transfer as a typed solver -- promotes the experiments/ftransfer solve into the ledger with a kill test (does NOT re-prove 53/54 v183 or Q*=2/3 v82/v101). [E] Q_pole=Koide(PDG poles)=2/3 to 3e-6, Q_src(phi0-ladder)=0.66446 (0.33% below). [E] QED runs the wrong way: dQ/deps>0 pushes Q ABOVE 2/3; the eps mapping Q_pole->Q_src is NEGATIVE while QED gives eps=+(3/2)(alpha/pi)>0 -- wrong sign for any alpha in [1/137,1/128]. [C] verdict: F_pole is NOT standard QED running; the transfer is the non-QED 53/54 operator / (2/3)^6 Moebius generator (v183/v82). Stays [C], plain running excluded as the kill."
    },
    {
      "id": "v372_ftransfer_boltzmann",
      "cluster": "frontier",
      "descriptionEN": "Track 3 (F_transfer): eta_B from the INTEGRATED Buchmueller-Di Bari-Pluemacher Boltzmann network at the TFPT-frozen heavy scale -- promotes the experiments/ftransfer solve, replacing the v326 toy washout. Frozen M1=M_scal phi0^2/A_Lambda (~8.65e9 GeV, v212), m~1=m3/A_Lambda=5 meV, delta_CP=4pi/3. [E] integrated efficiency kappa_f~0.09 (strong washout K~4.6) agrees with the BDP fit to ~25%. [C] eta_B(ODE)~6.5e-10 at the frozen M1 lands within a factor ~1.1 of the observed 6.1e-10, NO M_R dial; the M1 reproducing eta_B_obs is within a factor ~1.1 of the frozen one. [X] kill: eta_B outside [obs/3,3obs] for all seesaw-consistent M_R would drop the leptogenesis branch. Stays [C] (a transfer target, not a compiler power)."
    },
    {
      "id": "v373_ftransfer_relic",
      "cluster": "frontier",
      "descriptionEN": "Track 3 (F_transfer): the axion relic from the FINITE-T misalignment ODE -- promotes the experiments/ftransfer solve and DECIDES the two TFPT angle branches, replacing the v326 adiabatic toy. f_a=M_scal/128~2.4e11 GeV, m_a~23.8 ueV, lattice chi(T) n=8.16. [I] candidate scales fixed by the determinant-line atoms (128=2 dim S^+ |mu4|). [C] the frozen SPINE angle 3pi/5=108deg=pi N_fam/g_car gives Omega_a h^2~0.12 in [0.08,0.16] (lands on Omega_DM untuned) -- the robust dominant-DM branch. [X] the HILLTOP 170deg over-produces (~5.4x) -- disfavoured/fine-tuned. [X] kill: a finite-T solve robustly outside [0.08,0.16] for the spine across chi(T)/g_* drops the dominant axion-DM branch. Stays [C]/[X]."
    },
    {
      "id": "v374_ftransfer_qcd",
      "cluster": "frontier",
      "descriptionEN": "Track 3 (F_transfer): the m_p/m_e transfer as a typed uncertainty-budget + kill test -- the falsifiability companion to v262 (the precise 2-loop point), replacing the v326 toy (which hardcodes 1836.15). [E] b3=-(11-2 n_f/3)=-7 (n_f=6 carrier). [N] two declared externals (v339): alpha_s(M_Z)=0.1179+-0.0009 and C_p=2.83+-0.15, each ~5.3%. [C] m_p/m_e central ~1824 (v262) with a propagated +-7.5% band [~1690,~1960] CONTAINING the observed 1836.15 -- no free dial. [X] kill: a future lattice C_p+-0.03 + tighter alpha_s shrink the band to ~+-2%; a band then excluding 1836.15 drops the QCD-matching route (b3=-7 + the SU(9)-integer firewall v262 untouched). Distinct from v262's point; stays [C]."
    },
    {
      "id": "v375_observatory_registry",
      "cluster": "frontier",
      "descriptionEN": "Track 4: the prediction OBSERVATORY -- a status-typed CI over the frozen prediction registry (freeze_file.csv) that makes the falsifiability surface machine-checkable (no new physics). [E] falsifiability complete: every registry row carries a non-empty kill criterion (no orphan claims). [E] headline values re-derive from {phi0}: sin^2 theta12=1/3-phi0/2, sin^2 theta13=phi0 e^{-5/6}, beta_rad=phi0/(4pi). [N] live scorecard: theta12 vs JUNO first run (0.3092+-0.0087)=0.28 sigma (compatible); theta13 vs NuFIT 6.0 NO (0.02195+-0.00058)=2.0 sigma (the flagged most-tensioned core prediction); beta_rad vs ACT DR6 (0.215+-0.074)=0.37 sigma (hint); r=12/N*^2~0.004<BK18 (compatible). [E] registry integrity: the single machine-readable falsifiability surface. Tooling for the closed/open pieces."
    },
    {
      "id": "v376_seam_s3_centralcharge",
      "cluster": "core",
      "descriptionEN": "S3 closure stack: the central charge c=8 extracted NUMERICALLY from the lattice via Calabrese-Cardy finite-size entanglement scaling of the critical free-fermion content (the quantitative core of the MMST scaling-limit claim, on OUR model). [E] one critical complex mode gives c=1.0000 (Peschel correlation-matrix EE); [E] the collar is 16 Majoranas = 8 complex modes (v367), c additive => c=8=g_car+N_fam; [E] chiral c_-=8 (Chern |C|=1 per Majorana v367, non-gappable edge v368). [O] the specific net (E8)_1 is v377/v378; the continuum existence is the cited MMST theorem (v336). Numerical c=8 from the lattice; does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v377_seam_s3_e8character",
      "cluster": "core",
      "descriptionEN": "S3 closure stack: the chiral c=8 content is the SPECIFIC net (E8)_1, via the exact affine character q-series + the mu4/GSO promotion. [E] chi = E4/eta^8 = q^{-1/3}(1+248q+4124q^2+...): leading -c/24=-1/3 => c=8; level-1 = 248 = dim E8 (one primary). [E] mu4/GSO promotion SO(16)_1 -> (E8)_1: 248=240+8=(E8 roots)+(Cartan), 240=dim S^+*g_car*N_fam=16*5*3; 248=120+128=dim SO(16)+spinor. [E] discriminator: (E8)_1 has 1 primary/248 currents vs SO(16)_1 4 primaries/120. [O] continuum existence (v336); mu4 projection = seam one-sidedness (S3). Pins the target net; does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v378_seam_s3_modular",
      "cluster": "core",
      "descriptionEN": "S3 closure stack: the GENUS-1 discriminator -- torus ground-state degeneracy = 1 (holomorphic = (E8)_1), the 'hard part' v344 flagged (genus-0 unique vacuum is necessary not sufficient). [E] KLM condensation: mu(carrier)=det Cartan(D5)*det Cartan(A3)=4*4=16; order-4 mu4 glue (|L|=4) => mu(E8)=16/16=1 (torus GSD=1); index-2 => mu(SO16)=16/4=4. [E] |det Cartan(E8)|=1 vs |det Cartan(D8)|=4. [E] the single (E8)_1 character has modular T-eigenvalue e^{-2 pi i/3} (h=0,c=8), 1-dim rep => one character => torus GSD=1. [O] continuum existence (v336). Pins the net at genus-1; does NOT close SEAM.EQUIV.01."
    },
    {
      "id": "v379_seam_s3_rp",
      "cluster": "core",
      "descriptionEN": "C7 / OS reconstruction: reflection positivity (the OS axiom) of the explicit gapped collar measure, verified numerically -- the remaining constructive-QFT ingredient for the OS reconstruction. [E] the gapped p+ip band gives G(tau)=mean_k e^{-eps_k tau} with eps_k>=gap/2~1>0 (v367) -- a positive spectral measure; [E] the Hankel/reflection Gram matrix M_ij=G(tau_i+tau_j) is PSD -- OS reflection positivity holds; [E] neg control: a growing ghost mode makes it indefinite (non-vacuous). [C] RP + gap (v76/v337) + GNS (v240) + c=8 (v376) + (E8)_1 (v377/v378) give the OS inputs; C7/QG.AMB.01 is then DISCHARGED by the redundancy theorem v369, not built. [O] the full constructive measure stays the cited step."
    },
    {
      "id": "v380_grav_kms_hessian",
      "cluster": "frontier",
      "descriptionEN": "The KMS Entire Hessian -- the Stelle ghost is EXACTLY the Seeley-DeWitt truncation, and resummation pushes it to infinity (upgrades v304/v370 from 'assume entire' to a derived statement). [E] the seam KMS cutoff f(u)=e^{-u} (v259) gives the dressed spin-2 propagator e^{-p^2/M^2}/p^2 = 1/(p^2 a) with a(u)=e^u entire & nowhere zero => only the p^2=0 pole (no ghost). [E] each finite truncation carries the ghost: the R+R^2 order T_1=1+u has the Stelle zero u=-1, the Weyl^2 order T_2=1+u+u^2/2 the pair -1+-i; the entire a has none. [E] the ghost DECOUPLES: |nearest zero| of T_n grows monotonically with n (1,1.41,...,3.07 for n=1..8) -> infinity, so 'entire' = 'do not truncate'. [C] => perturbative graviton unitarity. [O] the exact off-shell curved-space Hessian identification is the cited heat-kernel resummation; perturbative-only, not the non-perturbative measure."
    },
    {
      "id": "v369_qgamb_redundancy",
      "cluster": "frontier",
      "descriptionEN": "Track 2 (next-plan v2): the AMBIENT REDUNDANCY statement -- the holographic route to Gravity-complete that SIDESTEPS building the general Euclidean QG measure (C7/QG.AMB.01). Instead of constructing the non-perturbative ambient measure (the conformal-factor swamp, v332/v334), assemble the established discriminators that prove the ambient sector carries NO physical d.o.f. relative to the holomorphic boundary algebra, so QG.AMB.01 is a non-fundamental CERTIFICATION object (like a coordinate choice in GR), not missing dynamics. [E] HOLOMORPHIC => DHR=Vec: #DHR primaries = |det Cartan| = 1 for E8 (ONE primary = vacuum) vs 4 for SO(16)/D8 -- no nontrivial superselection charges => no hidden boundary sector a bulk could carry invisibly. [E] NO TORUS GSD: ground-state degeneracy on T^2 = #anyons = |det K| = 1 for E8 => unique ground state => no topologically protected invisible bulk d.o.f. (the genus-1 obstruction of v344). [E] FINITE PETZ RECOVERY: deviation subspace contracts at (2/3)^6=64/729<1 (v221) => Rec_Sigma bounded. [E] GAP-DECOUPLING MARGIN: Delta_eff=6 ln(3/2)-31/(4 pi^2)~1.648>0 (v76/v337) => ambient sector energetically separated. [C] RECONSTRUCTION MAP: Bisognano-Wichmann gives the modular boost (v258/v309/v323) => Rec_Sigma: (bulk)->A_Sigma^{mu4}. [C] THE REDUNDANCY STATEMENT: under SEAM.EQUIV.01 + BW, O_phys^bulk/Diff ~= A_Sigma^{mu4} => QG.AMB.01 carries no extra physical d.o.f. [O] residual: the two premises -- SEAM.EQUIV.01 (S3 lattice input, v366) + BW intrinsicality (v329); provides the ALTERNATIVE route, does NOT construct the measure. Gravity-complete = Boundary-complete + Ambient-redundancy."
    },
    {
      "id": "v381_qft4d_eg_allorder",
      "cluster": "registry",
      "descriptionEN": "The all-order Epstein-Glaser / BRST contract for the perturbative 4D S-matrix S_pert -- the 'all-order closing statement' for the 4D perturbative QFT leg (review Point 3). Types the standard causal-perturbation-theory closure for the TFPT finite interaction class; does NOT build a path integral and is NOT a new mechanism. [E] POWER-COUNTING: every gauge+matter vertex (Yang-Mills + Dirac-Yukawa + Higgs + ghost) is a mass-dimension-4 marginal operator => the EG singular order omega<=4 => a FINITE counterterm space at every order (power-counting renormalisable). [E] BRST NILPOTENCY s^2=0 verified via the Jacobi identity for su(2) (residual 0) and su(3) (1e-16), the carrier weak+colour gauge content. [E] the seam gap Delta=6 ln(3/2)>0 (v302) gives the IR-safe adiabatic limit g->1. [C] imported: T_n exists to all orders by causal factorisation (Epstein-Glaser 1973 / Brunetti-Fredenhagen). [C] imported: all-order BRST invariance (Piguet-Sorella / Kugo-Ojima) => Slavnov-Taylor to all orders. [E] scope fence: matter+gauge ONLY -- the R^2/Weyl^2 gravity sub-sector is v304/v370/v380, not a local EG vertex; S_pert is not the ambient measure QG.AMB.01 (a [C] redundancy v369). NET [C] (imported EG+BRST on the TFPT dim-4 class + gap), prerequisites [E]; NOT [E], NOT a path integral. Extends v269/v271/v273/v278 from 1-loop to all-order."
    },
    {
      "id": "v382_alpha_quillen_exact",
      "cluster": "em",
      "descriptionEN": "Names the Quillen determinant-line VARIATION delta_tau(log det_zeta Delta_U(1) + 8 b1 c3^6 log phi_seam)=0 as a tracked target (review Point 10), elevating EM.WARD.01's residual to a citable [O] so the one EM obligation is trackable separately from the seam-net identification. [E] the Quillen split holds at the root a^3-2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at alpha^-1=137.0359992 (F_U(1)=0 IS the stationarity of the U(1) determinant line, v341). [E] every coefficient a named atom: 8 b1=(4/5)*41 a U(1) index, c3=1/(8 pi) the Gauss-Bonnet boundary coeff (c3^3,c3^6 heat-kernel powers), q(D5)+q(A3)=2 discriminant forms (v341/v48/v216). [E] the value alpha^-1=137.0359992168 stays closed (unique root+ablation, v3) -- only 'why this functional' is open. [O] THE TARGET: the from-first-principles AQFT/zeta proof that F_U(1) is the EXACT Quillen functional. [C] a FACE of SEAM.EQUIV.01 (v336): the seam U(1) determinant line lives on the same holomorphic seam net, so once the net is (E8)_1 its U(1) sub-determinant is fixed -- not a second independent gate, it names/tracks the pre-existing EM.WARD.01 residual (v341/v342). No new number (reuses v341, Wolfram-mirrored)."
    },
    {
      "id": "v383_dynamics_universal",
      "cluster": "frontier",
      "descriptionEN": "DYNAMICS.UNIVERSAL.01: the UNIVERSAL spectral-gap / Perron-Frobenius principle -- every TFPT sector is the SAME object (a gapped operator with a UNIQUE leading attractor = the physics, a spectral gap = the reason there is no free parameter), extending v303 from F_transfer to the whole theory. [E] FLAVOR (computed): the cusp transfer spectrum {1,(2/3)^6,(1/3)^6} has a simple Perron leading 1 and gap 6 ln(3/2)>0 (v56/v82). [E] DISCRETE COMPILER (computed): the affine-E8 adjacency has Perron eigenvalue 2 (Kac marks, v312/v313) and subleading 2cos(pi/5)=phi, gap 2-phi=1/phi^2>0. [E] DECOUPLING MARGIN: Delta_eff=6 ln(3/2)-31/(4 pi^2)~1.648>0 (v76/v337). [E] TWO NUMBER-FIELD FACETS (v314): golden phi in Q(sqrt5) (carrier-5, static) and (2/3)^6 in Q (family-3, dynamic) are the prime-5 and prime-3 facets of the order-30 Coxeter clock 30=2*3*5=h(E8) -- a factorisation, not an equality. [C] the other sectors are the same structure: gravity (v358/v359), QG saddle (v365), RG fixed point (v157/v158), all-order S-matrix (v381), recovery (v221), F_transfer (v303), tau=i (v333). [C] META-THEOREM: gapped => unique attractor => parameter-free, theory-wide. A synthesis, no new number."
    },
    {
      "id": "v384_residual_certification",
      "cluster": "frontier",
      "descriptionEN": "RESIDUAL.CERTIFICATION.01 (historical compiler scope; corrected 2026-09-05): the then-listed compiler residual is CERTIFICATION, not CONSTRUCTION -- after v369/v381/v382 there is NO open TFPT physics MECHANISM left; every residual is an external CERTIFICATE of already-determined structure, of three kinds: (A) external math proof, (B) theorem-forbidden, (C) external physics. A ledger-CI audit. [E] LEDGER CI: every residual claim id is present (no orphan). [E] NO-UNIT (B): c3=1/(8 pi) and g_car=5 are mass-dim 0, so by No-Unit (v153/v364) no dimensionless compiler outputs an absolute scale => v_geo is forbidden by theorem, not a gap. [E] GAP-DECOUPLED (A): Delta_eff~1.648>0 (v76/v337) => every readout is independent of the ambient measure (QG.AMB.01 a [C] redundancy, v369). [E] CLASSIFICATION COMPLETE: SEAM.EQUIV.01 (continuum existence=cited MMST/Adamo v336)->A, ALPHA.QUILLEN.EXACT.01 (v382)->A, QG.AMB.01 (v369)->A, v_geo->B, F_transfer (v371-v374, firewall v187)->C; ZERO open internal mechanisms within that scoped compiler list. This audit does not classify the physical completeness gates T1-T8; those and a shared 3+1D parent remain open. [C] CONVERGENCE applies only to the historical compiler residual. An audit, no new number."
    },
    {
      "id": "v385_uvbranch_killtest",
      "cluster": "registry",
      "descriptionEN": "UVBRANCH.KILLTEST.01: the optional carrier-Pati-Salam UV branch on the ALL-ORDER footing -- a single citable kill-test surface with the proton-decay SAFETY hierarchy computed (not a fake tau_p window). A consolidation (does NOT re-run the RGE v246 nor re-derive the PS scale v249/v253). [E] carrier content = SM (b=(41/10,-19/6,-7), v159/v246, no new states) and S_pert closes to all orders (v381) => the SM-non-unification is all-order-robust. [X] plain-SM 4D-GUT killed (sin^2 thW=3/8 vs 0.231; v246). [C] optional carrier-PS branch: M_PS=kappa M_scal, M_scal=c3^{7/2}Mbar~3.06e13 GeV, kappa in [1.0,1.2] (v253), content {1,10,16,45} no 126 (v247/v248), not default (v265). [E] proton-decay SAFETY hierarchy: minimal PS gauge bosons are SU(4) leptoquarks (B-L-conserving q<->l, rare LFV K_L->mu e), NOT the p->e+ pi0 dim-6 operator (needs SO(10)/SU(5) diquark, absent); binding bound M_PS>~1e6 GeV cleared by M_PS/M_bound~1e7>>1. [X] frozen kill tests (v265): kappa leaving O(1) / a forbidden rep / a new gauge-charged state / a rare-LFV signal below the bound; no fake tau_p. A consolidation, no new number."
    },
    {
      "id": "v386_grav_amplitude",
      "cluster": "frontier",
      "descriptionEN": "GRAV.AMPLITUDE.01: the entire-form-factor graviton-exchange AMPLITUDE is finite, UV-softened and tree-unitary -- perturbative gravity as an explicit scattering problem, extending v304/v370/v380 from the propagator to the amplitude. The dressed spin-2 propagator P=e^{-p^2/M^2}/p^2=1/(p^2 a), a(u)=e^u, M=M_scal. [E] one pole, ghost-free: a entire & nowhere zero (v380) => only pole at p^2=0, residue e^0=1>0. [E] UV-softened: ratio e^{-u} at u=1,5,10,20=0.37,6.7e-3,4.5e-5,2.1e-9 (faster than any power). [E] low-energy = GR + a calculable -p^2/M^2 correction (M=M_scal=c3^{7/2}Mbar, v253). [C] tree unitarity: a single positive-residue pole, no negative-norm states (v370/v380). [C] the amplitude A=(kappa^2/2)(T.P2.T')e^{-p^2/M^2}/p^2 is GR at low energy, softened in the UV (perturbative-only; QG.AMB.01 a [C] redundancy v369). No new number."
    },
    {
      "id": "v387_corrections_gap",
      "cluster": "frontier",
      "descriptionEN": "CORRECTIONS.GAP.01: the practical harvest of v383 -- the same spectral gap that makes each sector parameter-free ALSO sets the SIZE of its first correction (correction_n ~ (lambda_2/lambda_1)^n). [E] FLAVOR: lambda_2=(2/3)^6=64/729 => first Koide correction ~0.088 (v56/v82). [E] RECOVERY: the same (2/3)^6 Petz rate (v221). [E] QG: capped by chi=1/(1-(2/3)^6)=729/665 (v337). [E] DISCRETE COMPILER: the update (A+2I)/4 fixes the Kac marks (lambda_1=1) and decays at the golden (phi+2)/4~0.905 (carrier-5 facet, v303/v312). [E] the rates split into the (2/3)^6 family (family-3, in Q) and the golden family (carrier-5, in Q(sqrt5)) -- the two facets of v314/v383. [C] the correction calculus: the gap does double duty (WHY no free parameter AND HOW BIG the first correction), so sub-leading magnitudes are organised, not free. No new number."
    },
    {
      "id": "v388_corrections_budget",
      "cluster": "frontier",
      "descriptionEN": "CORRECTIONS.BUDGET.01: the gap correction (v387) is a TYPED budget, NOT a uniform band on every prediction -- the (lambda_2/lambda_1)^n size is a property of the distance from a gapped operator's leading eigenvector, so it is only defined where lambda_2 exists. The prediction surface splits into four correction classes (already implied by v303). [E] SEAM-GAPPED: Koide (F_pole v303), recovery (v221), the QG bound (capped by chi=729/665, v337) carry (2/3)^6; the discrete compiler decays at the golden (phi+2)/4 (v312). [E] KOIDE FIRST CORRECTION (door #2): F_pole is the ONE seam-gapped prediction, rate (2/3)^6~0.0878 (v371), source quotient 0.33% below 2/3 -- the rate is the suppression SCALE, not the residual. [E] EXACT-IDENTITY class (det R=8, N_Phi=1, theta_eff=0, sin^2 theta13): no lambda_2 => correction 0 STRUCTURALLY (a band would contradict [E]). [E] EXTERNAL-RATE class (eta_B/axion/m_p,m_e): the gapped shape with external rates (only F_pole has the seam rate, v303), firewalled (v187). [E] FIXED-POINT class: the texture is already explicit (theta12 eps=c3+36 c3^4). [C] a uniform band on all predictions is WRONG; the size is keyed to the mechanism class. A typing of v387, no new number."
    },
    {
      "id": "v389_grav_loop_finiteness",
      "cluster": "frontier",
      "descriptionEN": "GRAV.LOOP.FINITE.01: the entire-form-factor graviton is UV-finite at the LOOP level by power counting -- the honest next step past v386 (tree). Extends v304/v370/v380/v386 from the propagator to the superficial degree of divergence of loop graphs. HONEST SCOPE: the standard infinite-derivative/nonlocal-gravity finiteness statement (Tomboulis; Biswas-Mazumdar-Siegel; Modesto), NOT a full proof of perturbative QG. [E] super-polynomial falloff p^k e^{-p^2/M^2}->0 for every k. [E] one dressed line turns a quadratic divergence into the finite (M^2/2)e^{-p0^2/M^2}; GR int p^3/p^2 dp diverges quadratically. [E] superficial degree -> -infinity at every loop order (dressed cutoff-ratio ->1 vs GR ->4). [E] the regulator is the SEAM scale M=M_scal=c3^{7/2}Mbar (v253/v259), not a dial. [C]/[O] loop UNITARITY of the nonlocal form factor (Lorentzian continuation/macro-causality) stays the honest open point -- tree unitarity is v386; NOT claimed solved. No new number."
    },
    {
      "id": "v390_prime2_facet",
      "cluster": "frontier",
      "descriptionEN": "COXETER.PRIME2.01: the prime-2 facet of the order-30 Coxeter clock, completing v383. The clock 30=2*3*5=h(E8) had two named number-field facets (golden phi in Q(sqrt5), prime-5/carrier; (2/3)^6 in Q, prime-3/family); this names the prime-2 facet, identified with the Gaussian field Q(i) -- tau=i and the order-4 mu4=Z/4 square seam deck (v282/v288/v333), structure already in the theory. [E] h(E8)=30 factors into exactly three primes. [E] one quadratic facet field per prime, each ramified at its prime: disc Q(i)=-4, disc Q(sqrt-3)=-3, disc Q(sqrt5)=5; ramified primes {2,3,5}, product=30. [E] prime-2 Gaussian facet: min poly x^2+1, tau=i, j(i)=1728=12^3, |mu4|=4=2^2 (the static seam-orientation facet). [E] prime-5 golden (v383): 2cos(pi/5)=phi (pentagon). [E] prime-3 family (v383/v220): (2/3)^6 + Eisenstein zeta_3 (hexagonal CP pi/3). [C] completion: each prime carries a facet (square/hexagon/pentagon = seam/CP/carrier), the prime set forced by h(E8). A synthesis, no new number."
    },
    {
      "id": "v391_alpha_quillen_progress",
      "cluster": "em",
      "descriptionEN": "ALPHA.QUILLEN.PROGRESS.01: an HONEST attempt at the external target ALPHA.QUILLEN.EXACT.01 (v382) -- it REDUCES the open step to a sharper named sub-target via a solvable model but does NOT close it; ALPHA.QUILLEN.EXACT.01 stays [O] (external math, type A, v384). [E] solvable model: det_zeta(-d^2/dtau^2+m^2) on a circle = (2 sinh(beta m/2))^2, m-variation beta*coth(beta m/2) = 2/m + (beta^2/6)m - (beta^4/360)m^3+... a closed heat-kernel/Laurent series (m^3 coeff -1/360 verified). [E] a zeta-det variation is structured heat-kernel data, so v382's 8 b1 c3^6 counterterm (Gauss-Bonnet heat-kernel power) is the right KIND of object. [E] the v382 split re-verified at alpha^-1=137.0359992. [E] reduction: the open step reduces to the named sub-target 'seam Seeley-DeWitt coefficient = 8 b1 c3^6'. [O] NOT CLOSED: the actual seam zeta-det proof stays open, SEAM.EQUIV.01 continuum existence (v336) the other external [O]; alpha^-1 stays [E]. An attempt that reduces, does not close."
    },
    {
      "id": "v433_alpha_quillen_heatkernel",
      "cluster": "em",
      "descriptionEN": "ALPHA.QUILLEN.PROGRESS.02: a SECOND honest step on the external target ALPHA.QUILLEN.EXACT.01 (v382) -- it GROUNDS and CONNECTS, it does NOT close it; ALPHA.QUILLEN.EXACT.01 stays [O] (external math, type A, v384), alpha^-1 stays [E]. [E] a solvable 4D model (flat Delta=-d^2+m^2 on T^4) reaches the a_4 (t^0) conformal-anomaly order: a_4=V m^4/(2(4pi)^2) NON-ZERO -- the order v391's 1D circle toy (a_0,a_1 only) could not reach. [E] it is the order v342 grounds (Gilkey a_4 gauge-curvature 30/360=1/12), unifying v391 (a zeta-det variation IS heat-kernel data) with v342 (the term IS the Gilkey a_4). [E] (4pi)^-2 = 4 c3^2 is c3-built. [E] the v382 split re-verified at alpha^-1=137.0359992. [O] NOT CLOSED: the actual seam a_4 (=8 b1 c3^6) and the cubic a^3 Chern moment stay open (v384); the exact seam F-normalisation stays [C] (v342). Grounds/connects, does not close."
    },
    {
      "id": "v434_alpha_quillen_betafunction",
      "cluster": "em",
      "descriptionEN": "ALPHA.QUILLEN.PROGRESS.03: a THIRD honest step on ALPHA.QUILLEN.EXACT.01 (v382) -- it settles the three residuals named after v433 and shows they are NOT three independent problems; the target stays [O], alpha^-1 stays [E]. [E] b1 IS THE U(1) a_4 HEAT-KERNEL COEFFICIENT (beta=a_4 theorem): from the carrier hypercharge content with spin weights (2/3 Weyl, 1/3 complex scalar), sum (2/3)Y_f^2+(1/3)Y_s^2 = 41/6 (SM), (3/5)*that = 41/10 = b1 (GUT) -- the b1 in 8 b1 c3^6 is the a_4 content coefficient, not a fitted multiplicity (same number as the beta, v159/v246; same KIND as the alpha^2 Calderon a_4, v342). [E] 8 b1 c3^6 = b1*(rank E8)*c3^6 (rank 8 = chi=2 boundary, v216; c3^6 = six insertions, v342). [C] residual (1) 'exact a_4 = 8 b1 c3^6' == residual (3) 'seam F-normalisation' -- ONE obligation. [E] residual (2) located: a^3-2c3^3 a^2 = a^2(a-2c3^3), so a^3 = the first moment of the Chern density (a level, order c3^0); origin stays [O]. [E] net reduction: the EM-Ward residual is 1[C]+1[O], not three. [O] NOT CLOSED; alpha^-1 stays [E]. Python-only."
    },
    {
      "id": "v435_alpha_quillen_chernlevel",
      "cluster": "em",
      "descriptionEN": "ALPHA.QUILLEN.PROGRESS.04: a FOURTH honest step on ALPHA.QUILLEN.EXACT.01 (v382) -- it attacks the SINGLE remaining [O] after v434 (residual (2), the cubic alpha^3 Chern level) and SHARPENS it, it does NOT close it; the target stays [O], alpha^-1 stays [E]. [E] pi-POWER / METRIC-INDEPENDENCE TEST: the three closure coefficients carry pi-powers exactly {0,3,6}=c3-orders (k0=1 -> pi^0; -2c3^3=-1/(256 pi^3) -> pi^3; -8 b1 c3^6=-41/(327680 pi^6) -> pi^6), so the Maxwell a^3 is the UNIQUE pi^0/c3^0 (metric-INDEPENDENT) term -- the signature of a topological level (a heat-kernel boundary coeff always carries the (4pi)^{-d/2} measure; a level carries none). The test partitions the closure into ONE topological level (a^3) + TWO heat-kernel boundary terms, confirming v342/v434's 'a^3 is a level, not a curvature integral' from a test. [C] conditional on the CS/eta reading (EM1, v48), large-gauge invariance forces the level in Z and k0=1 is the unit (minimal) level -- an integer, not a free real. [O] NOT CLOSED: the EM1 CS-boundary derivation stays external (type A, v384); seam F-norm stays the one [C] (v434); alpha^-1 stays [E]."
    },
    {
      "id": "v392_seam_s3_scalinglimit",
      "cluster": "core",
      "descriptionEN": "SEAM.S3.SCALINGLIMIT.01: an HONEST attack on the one open lemma of SEAM.EQUIV.01 (the continuum scaling-limit existence, v336) -- does NOT close it (the abstract existence is the cited MMST theorem); STRENGTHENS the case with the existence-relevant observable v376 did not test: that the finite-size data CONVERGE. [E] the Calabrese-Cardy EE slope c1(L) at L=66..514 is a monotone Cauchy-like sequence approaching 1 (|c1(L)-1| strictly decreasing). [E] a 1/L Richardson fit gives c1(inf)~1.000 so the 16-Majorana collar c=8*c1~8.00 = g_car+N_fam is reached IN the limit. [E] the successive differences are strictly decreasing (numerical Cauchy, necessary for existence). [O] convergent data are evidence, not a proof of existence (MMST v336), and do not pin (E8)_1 vs SO(16)_1 (holomorphy v377/v378). Strengthens the residual, does NOT close it."
    },
    {
      "id": "v393_corrections_numeric",
      "cluster": "frontier",
      "descriptionEN": "CORRECTIONS.NUMERIC.01: the typed correction budget (v388) turned into ACTUAL first-correction magnitudes -- a computed theoretical-uncertainty surface, the short-term testable harvest of v388. [E] FIXED-POINT band (theta12/theta13/lambda_C/beta_rad) = the explicit phi0 puncture term 36 c3^4/c3 = 9/(128 pi^3) ~ 0.227% (exact; = theta12's epsilon first-correction). [E] SEAM-GAPPED band (Koide/recovery) = (2/3)^6 ~ 8.78%; QG bound capped by chi-1 = 64/665 ~ 9.62% (v337). [E] EXACT-IDENTITY band = 0 structurally (det R=8, N_Phi=1, theta_eff=0). [E] EXTERNAL-RATE band = external physics quoted (m_p/m_e +-7.5% v374; n_s/r N* band; eta_B 1.07x v212), fenced v187. [C] each prediction gets 'central value +- computed first correction'. A harvest of v388, no new number."
    },
    {
      "id": "v396_phi0_icosahedral",
      "cluster": "frontier",
      "descriptionEN": "HYP.PHI0.01: the phi0 leading term from icosahedral combinatorics (sharpening v312/v395). [E] ICO COUNTS ANCHOR-FORCED: V=12, E=30=h(E8), F=20=g_car(g_car-1), |Aut|=120=4h=|2I|. [E] LEADING TERM EXACT: phibase=1/(6 pi)=(4/3)c3=F/(4h pi)=F/(|Aut| pi)=(F/(g_car N_fam))c3. [E] F/h=|Z2|/N_fam; phibase=(F/h)/(4 pi)=c3(g_car-1)/N_fam. [E] GAUSS-BONNET: icosahedron deficit sum 4 pi; c3=1/(|Z2| 4 pi) (v216/v342). [E] HONEST NEGATIVE: puncture 48 c3^4 not rational hypergraph fraction. [C] phibase = (hyperedges)/(g_car N_fam) x c3, not independent injection; puncture [O]."
    },
    {
      "id": "v395_hypergraph_coupled",
      "cluster": "frontier",
      "descriptionEN": "HYP.COUPLED.01: finite local carrier/family recursion F(m)=T_net m M^T, checked on all 27 basis inputs; full v486/v487 M has spectrum {1,2/3,1/3}. The joint Perron line is marks tensor e0. One hand is one network step and six family substeps, T_net tensor M^6, not T_micro^6. Exact even/odd conjugacy O^T M^6 O=diag(1,64/729,1/729) supplies the complete v324 readout. v299 growth is unchanged. P2 and phi0 remain inputs. No physical OS-transfer or local charged-field identification is claimed. Correction 2026-10-27: the former routine discarded its network update and its test did not call that routine; the former v327 matrix also erased the odd mode."
    },
    {
      "id": "v394_coxeter_atoms",
      "cluster": "frontier",
      "descriptionEN": "COXETER.ATOMS.01: the clockwork coherence -- the three Coxeter primes ARE the three anchor atoms ARE the three number-field facets, and the prime-2 seam is the common dynamical factor. A bird's-eye [E] re-reading tying v390 (facets) to the anchor a=(1,1,2) and v319 (5x6 clock); also registers the residual-asymmetry question RES.COXETER.SYMMETRY.01 as a tracked [O]. [E] a=(1,1,2): e3=2=|Z2|, p0=3=N_fam, e2=5=g_car, product 30=h(E8). [E] atoms=facet fields (v390): 2->Q(i), 3->Q(sqrt-3), 5->Q(sqrt5), each ramified at its prime. [E] chi=729/665=3^6/(3^6-2^6), (2/3)^6=2^6/3^6, exponent 6=2 N_fam. [E] both dynamic rates carry prime-2: 2/3=|Z2|/N_fam, (phi+2)/4=(phi+2)/|mu4|, |mu4|=4=2^2. [E] 30=5x6=g_car*(2 N_fam). [C] the seam is the universal dynamic engine. [O] RES.COXETER.SYMMETRY.01 registered, not closed."
    },
    {
      "id": "v398_seam_state_rigidity",
      "cluster": "registry",
      "descriptionEN": "Seam State Rigidity Theorem (Paper A / G_net): RP-definability hinge (OS transfer + quasi-free) + band-limited mu4-character core swept N=4..64 + holomorphy=>(E8)_1; [O] full-L^2 / intrinsic BW"
    },
    {
      "id": "v399_gravity_complete",
      "cluster": "frontier",
      "descriptionEN": "Gravity-complete = Boundary-complete + Ambient-redundancy (Paper B): local equation parameter-free (v358/v359) + redundancy discriminators (v369) + O_phys subset A_Sigma over 5 readout classes"
    },
    {
      "id": "v400_grav_nonlocal_action",
      "cluster": "frontier",
      "descriptionEN": "Nonlocal spectral-gravity action; local R+R^2 is its IR projection (Paper C): entire form factor one pole + truncation-zero monotone + IR matching to R+R^2 with scale c3^7"
    },
    {
      "id": "v401_metrology_closure",
      "cluster": "frontier",
      "descriptionEN": "Metrology closure {a, pi, v_geo} (Paper D): 0 dimensionless dials + 1 unit; dimensional bookkeeping (No-Unit) + gravity adds no unit + G-vs-Lambda 0.11%; v_geo theorem-forbidden"
    },
    {
      "id": "v402_ftransfer_suite",
      "cluster": "frontier",
      "descriptionEN": "F_transfer functor suite (Paper E): the four frontier sectors as one F_observable o F_threshold o F_RG with exact atom sources + kill-test coverage + the firewall"
    },
    {
      "id": "v397_external_clock_probe",
      "cluster": "frontier",
      "descriptionEN": "FR.CLOCK.PROBE.01: a FALSIFIABLE probe of how external the 'external physics' really is -- do the frontier scheme-numbers land on the {2,3,5} clock (the anchor atoms, v394) or not? Strict No-Free-Pattern (v100): 'on the clock' = an EXACT match, not a wide-band fit. [E] eta_B decuple A_Lambda=10=|Z2|*g_car ON CLOCK (v212). [E] Koide rate (2/3)^6=(|Z2|/N_fam)^6 ON CLOCK (v303). [E] axion spine angle 3pi/5=pi*N_fam/g_car ON CLOCK (v211). [E] proton C_p~2.83+-0.15 NOT discriminating (2^{3/2},17/6,14/5 all fit the band) => EXTERNAL (firewall v187/v374). [C] 3 of 4 land exactly on the clock; the irreducible external residue shrinks to the proton QCD number C_p -- 'external physics' was overstated. [E] anti-numerology: refuses to assign C_p a clock value."
    },
    {
      "id": "v403_facet_compositum",
      "cluster": "frontier",
      "descriptionEN": "ARITH.HULL.01: the three number-field facets (v390) COMPOSE to K=Q(i,sqrt-3,sqrt5), a degree-8=rank(E8) abelian field -- the arithmetic hull dual to the E8 lattice hull. [E] -1,-3,5 independent in Q*/(Q*)^2 so [K:Q]=2^3=8=rank E8=g_car+N_fam, Galois (Z/2)^3 order 8; 2^omega(h(E8))=2^3=8=rank E8 is E8-special (E7: h=18, omega=2, 2^2=4!=7). [E] exactly 7 quadratic subfields {-1,-3,5,3,-5,-15,15} = 4 imaginary (|mu4|, CM/dynamic) + 3 real (N_fam, RM/static). [E] ramified only at {2,3,5} = the atoms = primes of h(E8)=30. [C] K is the abelian (Z/2)^3 arithmetic hull dual to E8; the 7 subfields a candidate for the 7 E8 audit slices. A synthesis composing v390/v394's facets, no new number."
    },
    {
      "id": "v404_e8_char_cm_duality",
      "cluster": "frontier",
      "descriptionEN": "E8.CMDUAL.01: the (E8)_1 vacuum character chi_E8=E4/eta^8=j^{1/3} at the two CM points -- 12 at the seam (tau=i), 0 at the family (tau=omega), extending v282. [E] chi_E8^3=j (q-series). [E] SEAM tau=i (Q(i), order 4): cross-ratio 2 => j=1728 => chi_E8(i)=12 (E8 present, v214/v282). [E] FAMILY tau=omega (Q(sqrt-3), order 6): equianharmonic lambda (lambda^2-lambda+1=0) => j=0 => chi_E8(omega)=0 (E4 vanishes, v220/v267). [E] tau=i,tau=omega are the only elliptic points of PSL(2,Z) so the values are forced. [C] the seam/flavor rigidity duality: seam rigid where E8 present (chi=12), flavor the soft residual where E8 degenerates (chi=0); dual-keystone candidate. Core values j=1728/j=0 already mirrored (v214/v220/v267/v282)."
    },
    {
      "id": "v405_seam_equiv_omega",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.02: the dual keystone at tau=omega -- the family/flavor sector as the order-3 Eisenstein/A2 face of E8, dual to the seam = (E8)_1 at tau=i (v404). A named contract, NOT a closure. [E] chi_E8(omega)=0 (E8 degenerates at the family CM point) vs chi_E8(i)=12. [E] A2 = the tau=omega Eisenstein lattice (Cartan det 3=N_fam, h=3=N_fam, mu6=mu3 x mu2). [E] R lives in E6 x A2: 248=78+8+2*27*3, ||R||_F^2=78=dim E6, det R=8=dim A2=rank E8. [E] delta_PMNS=4pi/3 = the tau=omega CM phase (arg zeta6=pi/3). [E] cusp weights {0,1/3,2/3}=order-3 deck=N_fam=det Q, Spec(Q+)=(1,2,3). [C] the family sector is the order-3 Eisenstein/A2 face; the two keystones sit at the only two elliptic points of PSL(2,Z). [O] residual = the canonical Z3 map (v140) + the v_geo-class scale (No-Unit v153)."
    },
    {
      "id": "v406_pmns_phase_origin",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.PMNS.CLOSE.01: the PMNS dynamics closed modulo the tau=omega keystone (v405) -- the CP-phase ORIGIN is the family CM point and the absolute scale is one v_geo-class unit, so PMNS has zero free flavor dials beyond v_geo. [E] the three angles (sin^2 th12=1/3-phi0/2, th23=1/2, sin^2 th13=phi0 e^{-5/6}; v9/v268) assemble to a unitary U_PMNS (v270). [C] CP-phase origin: delta_PMNS=4pi/3 is the tau=omega CM phase (v405), upgrading v270's [O] assembled input to a keystone consequence. [C] J_PMNS=-0.0297 derived. [E] absolute scale = one seesaw ratio = v_geo-class UV input (v272/v153), not a flavor gap. [C] net: PMNS = angles+delta[keystone]+J+scale[v_geo], zero free flavor dials. [O] residual = the seesaw operator realisation (M_R not a clean Mbar power, v272/v184; one-parameter carrier-normalised candidate v481) + the NO Sigma m_nu floor kill test."
    },
    {
      "id": "v407_dn_pairings_omega",
      "cluster": "uwall",
      "descriptionEN": "FLAV.SELECTOR.CLOSE.01: the R4' residue folds into the tau=omega keystone (v405) -- the three (d,n) selector pairings ARE the tau=omega family-slice atoms. [E] the frame (1,a,sigma) is a Z^3 basis det=11=||Pl(K)||_1, n=(5,-9,6) the unique covector with n.1=2, n.a=8, n.sigma=121=11^2 -> (d,n)=>R columnwise (v136/v139). [E] n.a=8=rank E8=dim A2=det R = the A2 (tau=omega) block dimension reading R in E6 x A2. [E] the pairings {2,8,121}={|Z2|, dim A2=det R, ||Pl(K)||_1^2} = the tau=omega family-slice atoms. [E] the square selects the point (sigma-pairing 11(11+t), square only at t=0). [C] deriving (d,n) folds into SEAM.EQUIV.02 (d anchor-derived v139, pairings = slice atoms). [O] the lone flagged residue why 11=||Pl(K)||_1."
    },
    {
      "id": "v408_phi0_puncture_heatkernel",
      "cluster": "frontier",
      "descriptionEN": "HYP.PHI0.PUNCTURE.01: name the phi0 puncture term dtop = 48 c3^4 as the order-4 local boundary heat-kernel (Seeley-DeWitt) contact term of the mu4 puncture divisor -- the analytic complement of the icosahedral tree term (v396, HYP.PHI0.01). [E] THE SPLIT IS EXACT: phi0 = phibase + dtop = 1/(6 pi) + 48 c3^4 (v396/ARCH.QUAD.01), a tree term plus a fourth-order boundary correction. [E] PUNCTURE COUNT = Omega_adm = N_fam dim S+ = 48, order = |mu4| = 4: dtop = Omega_adm c3^|mu4|. [E] CLOSED FORM dtop = 48 c3^4 = 3/(256 pi^4); c3 = 1/(|Z2| 4 pi) Gauss-Bonnet (v216/v342) so c3^4 is a fourth-order boundary heat-kernel power (cf. v391). [E] HONEST NEGATIVE: dtop not a rational hypergraph fraction (dtop/phibase < 1%). [O] THE TARGET: prove Omega_adm c3^4 IS the order-4 boundary Seeley-DeWitt contact term of the mu4 puncture summed over the Omega_adm admissible states. [C] two engines, two terms; a FACE of the seam boundary analysis (the c3 datum), not a new gate. No new number."
    },
    {
      "id": "v409_coxeter_prime2_lemma",
      "cluster": "frontier",
      "descriptionEN": "RES.COXETER.SYMMETRY.01 closed as a STRUCTURAL LEMMA: prime-2 is NECESSARY but NOT autonomous -- no prime-2-only attractor exists (resolves the v394/COXETER.ATOMS.01 open question's falsifiable corollary). [E] SHEET INVOLUTION: sigma=diag(1,-1,-1), sigma^2=I, spectrum {+1,-1} (parity). [E] SHEET PROJECTOR P=(I+sigma)/2=diag(1,0,0), P^2=P, spectrum {0,1} (projection). [E] NO CONTRACTION RATE: none of {-1,0,1} lies in (0,1), whereas the genuine rates 2/3 and (1/3)^6 do -- prime-2 alone gives parity/projection, never a nonzero rate. [E] FACTOR NOT RATE: 2/3=|Z2|/N_fam needs prime-3, (phi+2)/4=(phi+2)/|mu4| with |mu4|=2^2 needs the golden phi (prime-5). [E] 30=g_car(2 N_fam), carrier static, 2*3 dynamic. [C] prime-2 is the universal dynamical parity factor, necessary in every sheet-paired transport but not an autonomous attractor; dynamics begins after coupling to family-3 or carrier-5 (mu4=2^2 the Galois bridge). No new number."
    },
    {
      "id": "v410_sheet_generator_binary",
      "cluster": "registry",
      "descriptionEN": "The sheet generator V = Q diag(0,1,1) (the rank-2 sheet axis of the centered cross, v95/v218) is a binary internal compiler: V^n = ((0,2^(n-1),0),(0,2^n,0),(0,2^(n+1)-2,1)) (induction F(n)V=F(n+1)), so V^n.1 = (2^(n-1),2^n,2^(n+1)-1) = (1,2,3),(2,4,7),(4,8,15),(8,16,31); the bilinears 1.V^n.1=7*2^(n-1)-1, 1.V^n.a=7*2^(n-1), a.V^n.a=11*2^(n-1), a.V^n.1=11*2^(n-1)-2 give (6,7,9,11) at n=1 and 13=Delta_Q, 27=1.R.a, 55, 56=dim 56_E7; the binary spine is FORCED by Spec(V)={0,1,2}, the carrier (rank E8=8, dim S+=16, 248/8=31) and theta (sigma3(2)=9=a.V.1, sigma3(3)=28=1.V^3.a=det(I+R), sigma3(5)=126) readings audit-typed."
    },
    {
      "id": "v411_ud_ratio_vpower",
      "cluster": "registry",
      "descriptionEN": "The quark u/d ratio is a pure V-power readout: the established c_u/c_d = g_car ||Pl(K)||_1/(N_fam^2 Delta_Q) = 55/117 (v94) re-expresses EXACTLY via the sheet generator (v410) as c_u/c_d = (1.V^4.1)/((a.V.1)(1.V^2.1)) = 55/(9*13), with 55=1.V^4.1, 9=a.V.1=N_fam^2, 13=1.V^2.1=Delta_Q; the same number as 5*11/(9*13), so the V-power numerator repackages g_car*11=55; an exact RE-ENCODING, the value and u/d identification stay [C] (U_wall readout rigidity)."
    },
    {
      "id": "v412_sheet_source_corner_J",
      "cluster": "registry",
      "descriptionEN": "The sheet source corner J: the determinant surface has two s-independent iso-volume walls (v135) t=-1 (det=4=|mu4|, carries K) and t=-2 (det=2=|Z2|); the operator on the Z2 wall, J=M(1,-2)=C-2V=((4,1,0),(4,1,2),(5,1,1)), was unnamed and completes the sheet axis J,K,C,F=M(1,-2..1). chi_J=lambda^3-6lambda^2+3lambda-2 gives (tr,e2,det)=(6,3,2)=(|R^+(A3)|,N_fam,|Z2|); a.J.a=30=h(E8), det(I+J)=12=dim g_SM=chi_E8(i), det(2I+J)=40=|R(D5)| (Lie readings audit-typed like the G2/F4 shadows of v95/v218)."
    },
    {
      "id": "v413_sheet_characteristic_calculus",
      "cluster": "registry",
      "descriptionEN": "The sheet axis is a discrete characteristic-difference calculator: on M_t=C+tV (v95/v218) the elementary-symmetric coefficients encode atoms as DIFFERENCE ORDERS -- e1(t)=3t+12, e2(t)=2t^2+15t+25, e3(t)=det M_t=4t^2+14t+14 give Delta e1=3=N_fam, Delta^2 e2=4=|mu4|, Delta^2 e3=8=rank E8 (the e3 curvature cross-cited to v218/v224; the full (e1,e2,e3) ladder is new); the anchor energy a.M_t.a=52+11t (step 11=a.V.a, v410) reads 30->41->52->63=(h(E8),10 b1,dim F4,7 N_fam^2); four-row axis J,K,C,F (tr,e2,det)/a.M.a = (6,3,2)/30,(9,12,4)/41,(12,25,14)/52,(15,42,32)/63."
    },
    {
      "id": "v414_center_resolvent_portal",
      "cluster": "registry",
      "descriptionEN": "The center C is a resolvent portal G2 -> F4 -> E8: the center C=M(1,0) (v95) is the t=0 evaluation of V; tr C=12=dim g_SM=chi_E8(i), det C=14=dim G2, sum C=31=2^g_car-1=248/8; the shifted determinants climb the exceptional ladder det(C)=14=dim G2, det(I+C)=52=dim F4, det(2I+C)=120=|R^+(E8)| -- the shell ladder G2->F4->E8^+ as det(C+kI), k=0,1,2 (the resolvent ladder is the new content; reading audit-typed)."
    },
    {
      "id": "v415_gaussian_operator",
      "cluster": "flavor",
      "descriptionEN": "The square CM-norm dictionary is operator-realised and the mu4 deck is intrinsic: the Gaussian arithmetic of v222/v230 lifted from numbers to OPERATORS via J=[U,V]/3, the complex structure born from the commutator of the binary V and ternary U compilers (v410/v95). J=((-1,1,0),(-2,1,0),(-3,1,0)) is integer, Spec(J)={i,-i,0}, J^2=-I on the rank-2 image, J^3+J=0 (a mu4 quarter-turn); 3I+2J has spectrum {3+2i,3-2i,3} and 5I+4J has {5+4i,5-4i,5}, so the Gaussian integers 3+2i (norm 13=Delta_Q) and 5+4i (norm 41=10b1) are LITERAL operator eigenvalues; (3I+2J)(3I-2J) has Spec {13,13,9} and (5I+4J)(5I-4J) has Spec {41,41,25} (kernel slot = real part^2: 9=N_fam^2, 25=g_car^2); the intrinsic deck D=-I+J-J^2 is order 4 (Spec {i,-1,-i}, chi=(x+1)(x^2+1)) with chi2 line ker[U,V]=(0,0,1)=a-1; the geometric-deck identification sharpens (not closes) the v140 Z3 residue."
    },
    {
      "id": "v416_atom_trichotomy",
      "cluster": "core",
      "descriptionEN": "The atom trichotomy: {2,3,5} are ramified/inert/split in the two exceptional CM rings, and each atom is the unique ramified prime of one quadratic facet of K=Q(i,sqrt-3,sqrt5) (v403). Z[i] (square/seam, tau=i): 2=|Z2| ramifies (2=-i(1+i)^2), 3=N_fam inert (norm 9=N_fam^2), 5=g_car splits ((2+i)(2-i)) -- the carrier rank factors on the seam; Z[omega] (hex/flavor, tau=omega): 3=N_fam ramifies (N(1-omega)=3), 2 inert (norm 4), 5 inert (norm 25=g_car^2); the ramified prime of each ring IS its atom (Z[i] at 2 = sheet/seam, Z[omega] at 3 = family/flavor); the three facets Q(i)/Q(sqrt-3)/Q(sqrt5) (disc -4,-3,5) ramify over one atom each, product 2*3*5=30=h(E8); 2 imaginary (CM/dynamic) + 1 real (RM/static)."
    },
    {
      "id": "v417_eisenstein_cp_operator",
      "cluster": "flavor",
      "descriptionEN": "The Eisenstein/CP operator (hex dual of v415): the family rotation P=(1 2 3) and the CP clock W=-P^2 put the tau=omega flavor side on operators, unified with the seam into ONE order-30 clock. P^3=I, P^2+P+I=ONES (Eisenstein omega^2+omega+1=0 on the plane perp to (1,1,1)); (3I+2P)(3I+2P^2)=7I+6 ONES with Spec {7,7,25} -- 7=N_omega(3+2omega)=scalaron on the omega-plane, 25=g_car^2 on (1,1,1) -- the dual of v415's {13,13,9} (NEG: (5I+4P)(5I+4P^2)->{21,21,81}, N_omega(5,4)=21!=41); the CP clock W=-P^2 (order 6, =rho v233) has W^2=P, W^3=-I, Spec {-1,zeta6,zeta6^-1}, so delta_CKM=arg(zeta6)=pi/3 and delta_PMNS=arg(zeta6^4)=4pi/3 (W^4=P^2); all flavor clocks are powers of zeta30 (zeta30^15=-1, ^10=omega, ^6=zeta5, ^5=zeta6 = orders 2,3,5,6) while the seam mu4=i is the Galois side (Z/30)^x of order phi(30)=8=rank E8; the carrier mu5 has no clean 3x3 operator (golden phi an output-side trace, v313/v349)."
    },
    {
      "id": "v418_cyclotomic_norm_triple",
      "cluster": "core",
      "descriptionEN": "The cyclotomic norm triple: the carrier split (3,2) read in the three atom-rings gives (7,13,55)=(scalaron, Delta_Q, quark numerator), and the missing carrier-5 clock of v417 is found as the 4x4 Phi5-companion C5. C5 has chi=x^4+x^3+x^2+x+1, C5^5=I, tr=-1, and the golden ratio is its real-part data (2cos(72)=1/phi, 2cos(144)=-phi; output-side, v313/v349; dim 4, not 3 -- refines v417's gap); N_Z[zeta5](3+2 zeta5)=det(3I+2 C5)=55=5*11=g_car*||Pl(K)||_1=quark numerator (v410/v411); the triple det(3I+2 Comp(Phi_n))=(7,13,55) for n=3,4,5 over Z[omega],Z[i],Z[zeta5] (atoms 3,2,5, v416) is one (3,2)-generated family; NEG control: the anchor (5,4)->(21,41,461), 21=N_omega(5,4) and 41=10b1 named (v222) but the carrier ring 461 prime/un-named, so the carrier ring separates (3,2)->55 from (5,4)->461; HONEST: (3,2) is FORCED (v14) not unique, the clean splits are {(2,1)->(3,5,11),(3,2)->(7,13,55)}; the sharper reading: only (3,2) spans three physics sectors (gravity/flavor/masses), the unique cross-sector seed, while (2,1) is all-core."
    },
    {
      "id": "v419_seam_galois_carrier",
      "cluster": "core",
      "descriptionEN": "The seam mu4 IS the carrier's Galois group: a positive resolution of the v409/RES.COXETER.SYMMETRY.01 cyclic/Galois asymmetry. 30=2*3*5 is squarefree so Z/30 has no order-4 element -- the seam mu4 (order 4) cannot be a cyclic hand and is forced onto the Galois side; (Z/30)^x = (Z/2)^x x (Z/3)^x x (Z/5)^x = mu4 x Z2 (order phi(30)=8=rank E8), where the mu4 (Z/4) factor IS (Z/5)^x (carrier 5) and the Z2 IS (Z/3)^x (family/CP); so the seam mu4 = Gal(Q(zeta5)) = (Z/5)^x, the Frobenius zeta5->zeta5^2, realised on the carrier clock C5 (v418) by the explicit order-4 operator G with G C5 G^-1 = C5^2, G^4=I, G^2 C5 G^-2 = C5^-1; resolves v409 (the 2^2=4 is the order of (Z/5)^x; 'no contraction rate' because the seam is an automorphism, not a hand); Q(zeta30) carries both E8 invariants (cyclic h=30, Galois rank=8)."
    },
    {
      "id": "v420_fpole_qed_running",
      "cluster": "frontier",
      "descriptionEN": "F_pole external: the Koide source->pole transfer is a ONE-LOOP ELECTROMAGNETIC effect, not an algebraic step -- the honest 'do the external physics' follow-up to v93 and the clock No-Go. Koide Q(m)=sum m/(sum sqrt m)^2 is rescaling-invariant (Q(c*m)=Q(m)) so flavor-UNIVERSAL running leaves it fixed -- the source->pole shift is purely flavor-SPLIT; the tree->pole gap 2/3-Q_src=0.002203 is one-loop-EM-sized (= phi0/24 to 0.6%, = alpha/pi to ~5%), and the PDG pole sits AT 2/3 (the IR attractor); a 1-loop QED per-lepton-log pole<->MSbar estimate shifts Koide by EM-loop order toward 2/3 (the v82 IR-attractor direction); the EXACT flow time t~2.84 (v93) is scheme/scale-dependent external QED, NOT derived, only sized. F_pole stays [C], firewalled (v187): the lens forces the RATE (2/3)^6 (v327/v82), this external EM physics supplies the flow. Numerical (mpmath), Python-only."
    },
    {
      "id": "v422_galois_net_bridge",
      "cluster": "core",
      "descriptionEN": "The Galois<->Net bridge: the seam mu4 = Gal(Q(zeta5)) is the SAME cyclic Z/4 as the (E8)_1 simple-current glue -- not a mere order-4 coincidence -- connecting the Galois gearbox of v419 to G_net/SEAM.EQUIV.01. disc(A3)=disc(D5)=Z/4 (one SNF invariant factor 4 = cyclic, det 4); D_n disc is Z/4 for n ODD, Z2xZ2 for n EVEN, so the carrier D5 (rank 5=g_car, odd) is cyclic while the control D8 (rank 8, even) is Klein; Gal(Q(zeta5))=(Z/5)^x=<2> is cyclic order 4 (only the carrier prime 5 among the atoms gives order 4); in disc(D5(+)A3)=Z4xZ4 (16=dim S+) the simple current (1,1) is order 4 (cyclic) with Lagrangian quotient 16/4^2=1 => (E8)_1 (1 primary, holomorphic). NEG control: the Klein Z2xZ2 / order-2 halfway <(2,2)> gives 16/2^2=4=|disc(D8)|=SO(16) (det 4), NOT E8 (det 1) -- cyclic Z/4 -> E8, Klein -> D8, so the carrier-Galois matches E8 not D8. The bridge: one cyclic Z/4 threads the seam deck mu4 (v419), Gal(Q(zeta5)) (v419) and the (E8)_1 glue (v1/v89/v154); cyclicity forced by 5, Klein excluded. This is the four-fold mu4 identity (deck/divisor=Galois=discriminant=simple-current); per v428 a COMPRESSION (one object read four ways, a unification that makes the carrier less axiomatic), NOT four independent witnesses that multiply."
    },
    {
      "id": "v423_no_ambient_dependency",
      "cluster": "frontier",
      "descriptionEN": "The No-Ambient-Dependency certificate: no frozen TFPT readout is COMPUTED from the ambient QG measure QG.AMB.01 -- sharpening the v369/v384 prose redundancy into a machine-checked DAG+script fact, told honestly. Gap-decoupling margin Delta_eff = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (31=2^g_car-1), susceptibility chi=729/665 finite (neg control gap->0 => chi diverges). QG.AMB.01 is a certification SINK: every claim that lists it as a dependency is a gravity-gate/QG row, none a frozen prediction; and it is not a value source -- it has no producing script while every frozen prediction is computed by a compiler script (alpha<-v3, theta12<-v9, theta13<-v268, cosmology<-v7), none an ambient-measure script. The one transitive link is architectural: only theta13 references QG.AMB.01, via the 4D-QFT contract spine (PS.DIRAC->CONTRACT.QFT4D->QG.G2->GATE.METRIC), a coherence edge not a numerical input; alpha/theta12/cosmology have no route at all. So QG.AMB.01 is gap-decoupled certification (v369 as a DAG/script fact); honestly, this does not prove SEAM.EQUIV.01 -- the readouts depend on the seam being (E8)_1; QG.AMB.01 specifically is dead."
    },
    {
      "id": "v424_seam_lto_rp_reduction",
      "cluster": "frontier",
      "descriptionEN": "The (LTO-RP) reduction of the keystone: an honest MMST-style reduction of SEAM.EQUIV.01's one open lemma -- the raw collar's intrinsic Bisognano-Wichmann property (the v308/v309 residual) -- to the cited commuting-projector theorem of Naaijkens-Penneys-Wallick (arXiv:2605.10693), with the central mapping CORRECTED. The mu4 clock is a modular SYMMETRY ([rho,K]=0, v309/v199), not the flow -- 'sigma^psi=rho' is type-wrong (one-parameter group vs finite order-4 element). NPW26's (LTO-RP) means u_Theta=J (reflection = modular conjugation) = the intrinsic BW condition theta K theta=-K, DISTINCT from omega o rho=omega: a positive character-block-diagonal K satisfies [rho,K]=0 but never theta K theta=-K, so clock-invariance does not imply (LTO-RP). Make-or-break: NPW26 prove (LTO-RP) for topologically-ordered models (Toric Code |det K|=4 via a trace; Levin-Wen anyonic), while the seam is invertible (|det Cartan(E8)|=1, no anyons) and a beta=1 KMS state (not a trace) -- in neither proved bucket (neg control D8 |det|=4). So SEAM.EQUIV.01's BW residual reduces to NPW26 Theorem A modulo two open sub-steps (realise the seam as a Z/2-reflection commuting-projector LTO; extend (LTO-RP) to the invertible KMS case) -- a reduction, not a closure; SEAM.EQUIV.01 stays open."
    },
    {
      "id": "v425_dyn_transfer_universal",
      "cluster": "frontier",
      "descriptionEN": "DYN.TRANSFER.UNIVERSAL.01 -- the single-flow reduction of F_transfer: the four frontier transfers (F_pole, F_Boltzmann, F_relic, F_QCD) do not each need a separate external solver; their dynamics is the ONE native flow of the theory (the seam modular/recovery semigroup, v238/v240, gap 6 ln(3/2)) restricted to each sector -- native flow + external anchors, firewall-clean. The recovery spectrum {1,(2/3)^6,(1/3)^6} gives H_OS=-log T with gap 6 ln(3/2) and e^{-Delta}=(2/3)^6 exactly; the v99 Koide generator linearised at q=2 has rate -Delta so the time-1 contraction is the recovery eigenvalue (2/3)^6 (q=5 unstable); the Boltzmann washout is that same contraction with the native CP phase 4pi/3 (v320); the QCD rate is b3=-7, b0(n_f=6)=7. Only the anchors (v_geo, the lattice C_p, the seesaw M_R) stay external; F_relic's cosmological initial condition theta_i is the one genuine wall. Extends v383 (static universal gap) to a dynamic universal flow; a reduction, not internalisation (no new premise)."
    },
    {
      "id": "v426_seam_lto_rp_kms",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.LTORP.KMS.01 -- the invertible beta=1-KMS variant of (LTO-RP), the corner NPW26 leave open (they prove it for topologically-ordered models; the TFPT seam is invertible + beta=1 KMS, not a trace). A constructive Tomita-Takesaki witness advancing v424 sub-step (ii); does NOT close SEAM.EQUIV.01. One-particle dictionary: modular operator Delta=e^{-K}, (LTO-RP)/BW <=> u_Theta=J <=> Theta K Theta=-K, the mu4 clock a modular symmetry [rho,K]=0. Build K reflection-odd/mu4-even/gapped (gap 2k=6 ln(3/2)); C=1/(1+e^K) the beta=1 KMS covariance satisfies C/(1-C)=e^{-K} (genuine KMS, not trace); Theta K Theta=-K and [rho,K]=0 both hold, so (LTO-RP) is satisfied by direct Tomita-Takesaki; |det Cartan(E8)|=1 (invertible, no anyons) vs anyonic |det|=4, and the state is not a trace (Delta!=I) -- neither NPW26 bucket. Neg controls: a side-even reflection gives +K (BW fails), an off-mu4-block K breaks [rho,K]. SEAM.EQUIV.01 stays open (the continuum sub-step (i) remains)."
    },
    {
      "id": "v427_overdet_witness_map",
      "cluster": "registry",
      "descriptionEN": "OVERDET.WITNESS.MAP.01 -- the multiply-vs-compress FRAMEWORK for over-determination (a reviewer correction): witnesses from DISJOINT grammars multiply; projections of the SAME generator compress. The CLASSIFICATION of the seven arithmetic witnesses is REFINED by v428 (they compress). Seven grammar readouts, each from its own syntax landing on the carrier skeleton: Gauss N(3+2i)=13, Eisenstein N(3+2w)=7, cyclotomy N(3+2 zeta5)=55, Galois |(Z/5)^x|=4, root-lattice |det Cartan E8|=1, Pascal C(4,<=2)=11 (2^4=16=dim S+), Coxeter h(E8)=30=2*3*5 with phi(30)=8=rank E8. By v236 these seven are facets of ONE (2,3,5)/E8 object, so syntactic distinctness is NOT statistical independence (v428): they compress, like the anchor a=(1,1,2) emitting (e1,e2,e3)=(4,5,2)+power sums+the E8 data from one generator. The auditable map that replaces the rhetorical claim; the multiplicative strength is the multiply-forced inputs + foreign witnesses (v428), and it does not by itself establish physics (the seam/anchor/transfer bridge problem)."
    },
    {
      "id": "v428_overdet_reclass",
      "cluster": "registry",
      "descriptionEN": "OVERDET.WITNESS.RECLASS.01 -- the honest correction of v427: applying v427's own multiply-vs-compress test to v427 itself. The (2,3,5) Brieskorn singularity (v236) is the ONE generator -- Milnor mu=(2-1)(3-1)(5-1)=8=rank E8, 30=2*3*5=h(E8), phi(30)=8 -- so each of v427's seven witnesses maps to a facet of that same object (primes 2,3,5 of 30, the clock 30, or E8): all seven collapse to one generator (distinct objects=1), which by v427's own definition is COMPRESSION, not seven independent multiplications. What honestly multiplies is the INPUT forced four independent ways -- the '8' in c3=1/(8 pi) from rank(E8), the carrier Coxeter h(D5)=2(5-1), the totient phi(30) and the Milnor number -- plus the FOREIGN witness alpha^-1~137 (prime, not in the (2,3,5)-skeleton; v305). Refines and partly walks back v427; consistent with v236/v305/v313."
    },
    {
      "id": "v430_other_side_reverse_audit",
      "cluster": "registry",
      "descriptionEN": "E8.OTHERSIDE.AUDIT.01: the 'other side of the seam' (double-cover deck / conjugate sheet S⁻) audited against E8's UNMAPPED Casimir degrees -- the sheet/deck complement of v354/v355, answering 'is the leftover where the second sheet lives?' with a disciplined NEGATIVE. [E] DECK=DEGREE-2: the one-sided cover contributes |Z2|=2 (the 1/2 in c3=1/(2·4π)=1/(8π), v58/v216); 2=min(E8 degrees)=the metric, one of the matched primary readouts {2,8,30} (v354), NOT one of the unmapped {12,14,18,20,24}. [E] TWO SHEETS=128-SPINOR: dim S⁺=dim S⁻=16, the spinor block 128=rank·dim S⁺=8·16=2^(rank−1)=Σdeg, the spinor half of 248=120+128; S⁻ is the conjugate (16̄,4̄), 128=2·64, not a spare singlet. [E] FORCED-DISJOINT: the sheet/deck integer set {2,16,32,128} has EMPTY intersection with {12,14,18,20,24}, its only degree-alphabet contact the matched 2. [O] DISCIPLINED DECLINE: each unmapped degree needs an UNFORCED coefficient from sheet atoms and admits ≥2 readings (the v355 discriminator) -- no forced sheet→unmapped-degree identity. [E] 'S⁻=DM' DOWNGRADED + WIMP NO-GO: v227 replaced the over-strong 'S⁻ is dark matter'; 128=(16,4)+(16̄,4̄) is the SAME matter spinor (no spare singlet), DM is the determinant-line axion (a phase). [O] the only live 'other side' question (the 128 phase/glue dark sector, v227) concerns MATCHED structure, not the unmapped degrees. A clean structural NEGATIVE; Wolfram-mirrored, Python (sympy)."
    },
    {
      "id": "v429_axion_pentagon_phi",
      "cluster": "frontier",
      "descriptionEN": "DM.AXION.PENTAGON.01 -- the 'unmapped' icosahedral/golden E8 structure surfaces in the ONE external cosmological input theta_i, answering 'why does idle E8 structure exist?' for the one case where it has a job. Because N_fam=g_car-2, the spine quotient theta_i=pi N_fam/g_car=(g_car-2)pi/g_car is EXACTLY the interior angle of the regular g_car-gon; for g_car=5 the PENTAGON, theta_i=3 pi/5=108 deg -- the carrier polygon, not a bare rational multiple of pi. Its cosine is golden: cos(3 pi/5)=(1-sqrt5)/4=-1/(2 phi), partner 2 cos(2 pi/5)=1/phi=phi-1, phi=2 cos(pi/5)=(1+sqrt5)/2. The golden character is unique to g_car=5: among small n-gons only n=5 has an irrational interior-angle cosine (n=3,4,6 give 1/2,0,-1/2). So the golden ratio v354 called 'unmapped' (and v313 named the g_car=5 signature) IS the angular data of theta_i=3 pi/5 (the spine branch v211/DM.AXION.SPINE.01). A re-reading: it REFINES v354 from 'phi in NO physical readout' to 'phi touches ONLY the [C] spine misalignment angle', and gives a geometric motive for the spine over the hilltop ~170 deg; it does NOT upgrade DM.AXION.SPINE.01 (stays [C]) and closes no gate (the abundance is cosmology, decided by the finite-T solver, v185/v211)."
    },
    {
      "id": "v431_e8_degree_ladder",
      "cluster": "registry",
      "descriptionEN": "E8.DEGREE.LADDER.01: the reverse-audit 'unmapped' E8 Casimir degrees are NOT diffuse overhead but the forced two-family decomposition deg(E8) = 6·spine{2,3,4,5} (+) ({2} U flavor-det-ladder{8,14,20}) -- the structural complement of v354/v355/v430. [E] TWO-FAMILY SPLIT FORCED BY 30=2·3·5: the E8 exponents are the φ(30)=8 totatives of h=30=2·N_fam·g_car (coprime to 6 => ±1 mod 6), so the degrees occupy ONLY the residue classes {0,2} mod 6. [I] 6k family {12,18,24,30}/6 = {2,3,4,5} = the v91 spine {e3,e1,e2,p0}(a=(1,1,2)). [I] 6k+2 family {2,8,14,20}: {8,14,20}=(det R,det C,det L)=the winding line det M(s,0)=6s+8 of the flavor diamond (v135), C=R+Q diag(1,0,0). [E] '18'=6·N_fam is NOT a holdout (spine family, not on the det-ladder). [E] SPECIAL TO E8 among the simply-laced exceptionals (E6/E7/D5 fail the clean {0,2}-mod-6 split). [P] HONEST v355 LINE: the arithmetic decomposition is exact [I], but the FUNCTORIAL claim '6k+2 Casimir stratum = flavor sheet operators' needs a rep-theoretic map ([P]); 12,24 admit two canonical readings each, so this is a STRUCTURE theorem, not a forced functorial flavor map -- reclassifies v354 overhead to zero orphan degrees, closes no gate. Wolfram-mirrored, Python (sympy)."
    },
    {
      "id": "v432_overdet_floor",
      "cluster": "registry",
      "descriptionEN": "OVERDET.FLOOR.01: the UNCONDITIONAL improbability floor + the L1<->L2 bridge -- the defensible number that survives even if a skeptic REJECTS the declared formula grammar, complementing the conditional null model (v100) and the multiply-vs-compress framework (v427/v428). [E] DISJOINT PIECES (v428): compression removed (v427's seven readouts are one (2,3,5)/E8 object, NOT multiplied); what multiplies is the input '8' forced FOUR ways {rank E8, h(D5)=2(g_car-1), φ(30), Milnor(2,3,5)} plus the FOREIGN witness α^-1~137 (prime, outside (2,3,5)). [C] CONSERVATIVE FLOOR: multiplying ONLY across disjoint pieces (input over-determination × the α grammar census 1/94500 (v100) × the φ0-seed cluster counted ONCE × one downstream witness) the floor stays ≤1e-6 across a grid of generous assignments, nominal ~1e-10 -- 'extraordinary' WITHOUT the declared grammar. [E] THE BRIDGE: the floor (~1e-10) sits ~20 orders ABOVE the conditional 10^{-30.7} (v100); that gap = the value of proving the FORMS forced, so L1 (ALPHA.QUILLEN.EXACT.01, v382) and L2 (this floor) are the SAME lever. [C] HONEST SCOPE: bounds ONLY 'is it numerology?', NOT the physics bridge (firewall v187); a conservative FLOOR, not a posterior. Numerical/methodological [C], Python-only (mpmath) -- deliberately NOT Wolfram-mirrored (a bound, not an identity)."
    },
    {
      "id": "v436_overdet_floor2",
      "cluster": "registry",
      "descriptionEN": "OVERDET.FLOOR.02: HARDENS OVERDET.FLOOR.01 (v432) -- the 'is it numerology?' floor does NOT rest on v432's subjective chance assignments. [E] ASSUMPTION-MINIMAL FLOOR: the α cubic-class census (v100 layer D) is PURE COUNTING -- of N=94500 complexity-matched F_U(1) variants only 1 hits the 4e-8 CODATA window (the TFPT instance, α^-1=137.0359992), so p_α ≤ 1/94500 = 1.06e-5 ~ 4.40σ with NO subjective probability, prior or multi-observable grammar. [E] MONOTONE CONCESSION LADDER: -log10(p) full scorecard+α ~30.7 → scorecard-only ~25.8 → α-census-only ~4.98 (=4.40σ); the 'not chance (>4σ)' verdict holds at the MOST adversarial rung. [E] INDEPENDENCE: strip ALL of v432's subjective factors (0.1³×0.012×0.15=1.8e-6) and the floor is still 4.40σ -- they only sharpen it. [C] HONEST 5σ GAP: 5σ (p=5.7e-7) is crossed by the census times exactly ONE more independent factor ≤0.054 (the input-'8' four-fold over-determination or one foreign witness), stated openly. [C] FIREWALL: bounds numerology only, not the physics bridge (v187). Python-only (imports the v100 census live), deliberately NOT Wolfram-mirrored (a counting bound, not an identity)."
    },
    {
      "id": "v437_e8_degree_joint",
      "cluster": "registry",
      "descriptionEN": "E8.DEGREE.JOINT.01: a CONSOLIDATION of v6/v66/v355/v431 deriving more from the E8 Casimir degrees WITHOUT crossing the v354/v355 anti-numerology line -- no new per-degree coincidence, only a joint lock. [E] E8 IS FIXED BY ITS DEGREE MULTISET: from deg(E8)={2,8,12,14,18,20,24,30} alone -- rank=#deg=8 (=2nd deg), h=max=30, #pos roots=Σexp=120, #roots=240, dim=248, |W|=∏deg=696729600, Σdeg=128=2^7=dim S^+. [E] THE JOINT QUADRATIC: the two structural integers (g_car,N_fam) are the UNIQUE root pair of x²-(rank E8)x+(h/2)=x²-8x+15=(x-3)(x-5) -- sum=rank E8=8 (rank-fill, v6), product=h/2=15=g_car·N_fam (2 g_car N_fam=h, v431); {3,5} the unique factor pair of 15 summing to 8, so BOTH are forced together by two degree-invariants. [E] the 3 mapped degrees {2,8,30}=min/rank/max (the canonical invariants); g_car=5=max prime of 30. [E] DISCIPLINE: no new per-degree mining; the 5 non-primary degrees keep their v431 homes. Exact integer/lattice [E], mirrored in wolfram/tfpt_readouts_extension.wl."
    },
    {
      "id": "v438_seam_hsmi_borchers",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.BW.HSMI.01: the half-sided modular inclusion (Borchers/Wiesbrock) as an INTRINSIC Bisognano-Wichmann route for face (ii) of SEAM.EQUIV.01 -- geometric modular flow WITHOUT presupposing covariance. [E] STANDARD-PAIR ALGEBRA (ladder, exact): K=diag(n), P=raising shift give [K,P]=P and the Borchers scaling Δ^{it}PΔ^{-it}=e^{-t}P. [E] BW REFLECTION: J(n→-n) gives JKJ=-K (u_Θ=J) and JPJ=P_- (opposite translation). [E] POSITIVE ENERGY: in the unitary lowest-weight sl(2,R) rep P_line=L0-(L+ +L-)/2 is POSITIVE (min eig>0); the boost direction is the indefinite neg control. [C] SYNTHESIS: P≥0 + [K,P]=P + J is a Longo STANDARD PAIR / +half-sided modular inclusion, so by Borchers the flow is geometric INTRINSICALLY (complementary to Adamo-Giorgetti-Tanimoto); face (ii) becomes a STRUCTURE consequence. [C]/[O] assembling the finite algebra + positive-energy rep into ONE continuum standard pair is the continuum existence; SEAM.EQUIV.01 stays [O]. Python-only (numpy/scipy)."
    },
    {
      "id": "v439_seam_lieb_robinson",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.CONTINUUM.LR.01: the Lieb-Robinson/Nachtergaele-Sims continuum-existence MOTOR for the gapped seam collar -- grounds the BULK half of the cited MMST existence in the explicit v367 p+ip model. [E] UNIFORM GAP: Δ(N)=const>0 across N=20..160 (lattice image of the OS transfer gap 6 ln(3/2)>0, v302/v329). [E] EXPONENTIAL CLUSTERING: |⟨c_i†c_j⟩|~e^{-r/ξ}, ξ finite and STABLE across N (gapped ⇒ Hastings-Koma decay). [E] LIEB-ROBINSON LIGHT CONE: |(e^{-iHt})_ij| confined to |i-j|≲v_LR t with finite v_LR (~2t). [C] NACHTERGAELE-SIMS LIFT: uniform gap + finite LR velocity ⇒ exponential clustering with N-independent ξ AND existence+uniqueness of the thermodynamic-limit ground state with a finite light cone -- derived for OUR collar. [C]/[O] the bulk thermodynamic limit EXISTS; SEAM.EQUIV.01 stays [O], the open step narrowed to the CHIRAL massless EDGE scaling limit. Python-only (numpy/scipy)."
    },
    {
      "id": "v440_seam_lto_rp_beta",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.LTORP.BETA.01: the β-interpolation connecting the NPW26-PROVEN trace case (β=0) to the seam's β=1 KMS case along an obstruction-free homotopy -- upgrading the single-point v426 witness for v424 sub-step (ii) to a PATH argument. [E] β-INDEPENDENT (LTO-RP) DATA: ΘKΘ=-K and [ρ,K]=0 are conditions on K alone. [E] THE KMS PATH: C_β=1/(1+e^{βK}) has β=0→C=½I (NPW26 TRACE), β=1→seam KMS, β→∞→ground state; at every β ΘC_βΘ=1-C, [ρ,C_β]=0, 0≤C_β≤1. [E] NO OBSTRUCTION: β→C_β norm-continuous, stays in the (LTO-RP)-valid set -- a continuous homotopy, no wall. [E] NEG CONTROLS: reflection-EVEN K' breaks ΘC_βΘ=1-C for β>0; off-mu4 K breaks [ρ,C_β]. [C]/[O] a PATH argument for v424 (ii), stronger than v426; SEAM.EQUIV.01 stays [O]. Python-only (numpy/scipy)."
    },
    {
      "id": "v441_seam_applicability_ledger",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.APPLICABILITY.01: the applicability ledger -- a hypothesis-by-hypothesis audit of the THREE external theorems the seam keystone cites (MMST scaling limit, NPW26 (LTO-RP), Adamo-Giorgetti-Tanimoto covariance), isolating the SINGLE imported analytic fact. [E] MMST HYPOTHESES: D=16 Majoranas (=2^{g_car-1}), c=8 (=g_car+N_fam), rank E8=8, range rank≤c≤D (8≤8≤16), gapped, quasi-free CAR -- all internal computed facts. [E] NPW26: (LTO-RP)=u_Θ=J, gap, Z2 reflection, mu4 clock, [ρ,K]=0 internal; the seam invertible |det Cartan E8|=1 + β=1 KMS is OUTSIDE NPW26's bucket (|det|=4) -- a TEMPLATE not a direct import. [E] ADAMO (arXiv:2508.07109): covariance is a CONSEQUENCE (defuses the v215 circularity, downgrades cov. ASSUMPTION→[E]). [E] of 12 hypotheses, 11 internal [E] and exactly 1 external -- the continuum EXISTENCE of the chiral massless scaling limit. [C]/[O] external dependence pinned to ONE analytic fact; SEAM.EQUIV.01 stays [O]. Python-only (dets Wolfram-mirrored via v89/v281/v422)."
    },
    {
      "id": "v442_seam_rigidity_uniform",
      "cluster": "frontier",
      "descriptionEN": "SEAM.RIGIDITY.UNIFORM.01: the uniform-in-N hardening of the Seam State Rigidity Theorem (v398) -- the band-limited rigidity is NOT a small-N artifact. [E] EXACT UNIFORM 4-SECTOR COMMUTATION: ρ=diag(i^n) has EXACTLY 4 distinct eigenvalues for all N, and a genuinely character-block-diagonal K (random Hermitian within each n mod 4 class, not merely diagonal) has [ρ,K]=0 EXACTLY for N=4..256 (extends v398's N≤64). [E] UNIFORM GAP: Δ(N)=const>0 across N=20..160 (Nachtergaele-Sims, shared with v439). [E] N-INDEPENDENT RIGIDITY BOUND: a FIXED local clock-breaking perturbation gives ||[ρ,V]||=2 EXACTLY for all N. [E] NEG CONTROL: a clock-PRESERVING perturbation gives ||[ρ,V']||=0; an order-2 clock has only 2 sectors -- the order (4 marks) is the discriminator. [C]/[O] band-limited rigidity is uniform in N; SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O]: the full-L² forcing is the cited rigidity/MMST step. Python-only (numpy)."
    },
    {
      "id": "v443_seam_e8_discriminator",
      "cluster": "frontier",
      "descriptionEN": "SEAM.E8.DISCRIMINATOR.01: the empirical/structural DISCRIMINATOR bundle (mode D of the v347 closure-mode classification) -- concrete FALSIFIABLE kill tests separating the seam=(E8)_1 from the nearest c=8 rival SO(16)_1=(D8)_1. [E]/[X] PRIMARY/GSD: |det Cartan(E8)|=1 (ONE primary, holomorphic) vs |det Cartan(D8)|=4 (FOUR primaries); torus GSD=#primaries=1 vs 4 -- a seam with GSD>1 falsifies (E8)_1. [E]/[X] CURRENT/THETA: level-1 currents dim E8=248 vs dim SO(16)=120; E8 lattice theta has 240 norm-2 roots (=112 integer + 128 spinor, computed) vs D8's 112. [E] c=8 IS NOT A DISCRIMINATOR (the v277 ambiguity); only the order-4 mu4 condensation + det K=1 (genus-1, v90) selects E8. [E] NON-VACUITY: the discriminators all fire on the rival. [X]/[C] mode D as concrete kill tests; SEAM.EQUIV.01 stays [O]. Python-only (dets Wolfram-mirrored via v89/v281/v422)."
    },
    {
      "id": "v444_seam_edge_scaling",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.CONTINUUM.EDGE.01: the CHIRAL EDGE scaling-limit MOTOR -- the edge half of the one SEAM.EQUIV.01 continuum residual, attacking exactly the step v439 left open. On the SAME explicit v367/v368 p+ip collar it shows the lattice edge carries the kinematic AND conformal data of a chiral c=1/2 Majorana CFT, converging under refinement. [E] CHIRAL LINEAR EDGE DISPERSION: the in-gap branch is linear near its zero-crossing (top v=0.983, R^2=0.9999) and the two boundaries carry OPPOSITE velocities -- one chiral Majorana per edge. [E] ALGEBRAIC EDGE TWO-POINT: the equal-time edge correlator decays as r^{-η} with η=1.00 (the chiral free-fermion CFT exponent) -- a CRITICAL edge. [E] EXPONENTIAL BULK (neg control): the bulk-row correlator decays exponentially (ξ~1.1) -- only the edge is critical. [E] TRIVIAL-PHASE NEG CONTROL: M=3 (no edge) has an exponential edge-row -- the power law is the topological edge, not the geometry. [E] SCALING-COLLAPSE: η is N-stable (1.001±0.007) and the edge profile is Cauchy-convergent under refinement (successive spread shrinks 0.028→0.013, Nx=36..60). [C] EDGE CENTRAL CHARGE: η=1 ⇔ one chiral Majorana (c=1/2)/edge, so the 16-copy carrier has a chiral c_-=8 edge CFT (=(E8)_1, c_-=g_car+N_fam). [C]/[O] grounds the EDGE half of the cited MMST scaling limit (v336) the way v439 grounds the bulk; SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v445_seam_rigidity_forcing",
      "cluster": "frontier",
      "descriptionEN": "SEAM.RIGIDITY.FORCING.01: the rigidity FORCING theorem -- upgrades v442's 'block-diagonal PERMITS commuting' to 'commuting with the order-4 clock FORCES block-diagonal', isolating the single remaining physical premise. [E] FORCING (commute <=> block-diagonal, EXACT): for rho=diag(i^n), [rho,K]=0 iff K_ij=0 whenever i!=j mod 4 -- entrywise, swept N=4..128 (the converse of v442). [E] EXACT COMMUTANT DIMENSION: dim{K:[rho,K]=0}=sum_s n_s^2 (ad_rho nullspace), a PROPER subspace of the N^2 algebra (N=16: 64<256). [E] OFF-BLOCK <=> NON-COMMUTING: every off-block E_ij has ||[rho,E_ij]||>=sqrt(2)>0 -- zero slack. [E] THE ORDER IS THE DISCRIMINATOR: order-4 gives 4 sectors, order-2 a STRICTLY LARGER commutant -- the four marks force strictly more structure than two; only index-4 is (E8)_1 (det 1; v281/v154/v351). [E] NONDEGENERACY: exactly 4 distinct eigenvalues for all N>=4. [C] SYNTHESIS: commuting with the order-4 clock is EQUIVALENT to mu4-block-diagonality, so RP-definability (v54/v398) + clock-invariance FORCE block-diagonal Lambda_Sigma. [C]/[O] the residual is now precisely 'transfer in commutant' (clock-invariance, v323/v335); SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v446_seam_clock_invariance",
      "cluster": "frontier",
      "descriptionEN": "SEAM.RIGIDITY.CLOCKINV.01: DERIVING clock-invariance ('transfer in the commutant') from RP-symmetry, closing v445's single residual one notch further. [E] OS RECONSTRUCTION INHERITS THE SYMMETRY: for a mu4-invariant RP covariance ([rho,C(t)]=0), the OS canonical transfer Lambda=C(1)C(0)^-1=e^{-h} (fixed by RP+reflection, v54) has [rho,Lambda]=0 -- swept N=4..32 over random block-diagonal h. [E] SHARPNESS/NEG CONTROL: a mu4-BREAKING covariance gives [rho,Lambda]!=0 -- iff-sharp. [E] CONTRACTION/OS-VALIDITY: Lambda has spectrum in (0,1]. [E] MODULAR (Tomita-Takesaki): the free modular Hamiltonian K=log((1-C)C^-1) (v258) of the invariant state has [rho,K]=0, broken control [rho,K]!=0. [C] SYNTHESIS: [rho,Lambda_Sigma]=0 is a CONSEQUENCE of mu4-invariance of the seam RP data (the four marks, v216/v323) via OS reconstruction (v54) + the modular fact (v258), not an extra assumption. [C]/[O] the residual reduces from an operator commutation to 'the RP data is mu4-symmetric' (QGEO.SYM.01); SEAM.EQUIV.01 stays [O]. Python-only (numpy/scipy)."
    },
    {
      "id": "v447_seam_edge_chern",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.EDGE.CHERN.01: an INDEPENDENT (topological) reading of the edge c_-=8, corroborating v444's correlator route via the bulk-edge correspondence on the same v367/v368 collar. [E] INTEGER CHERN NUMBER: the M=1 collar lower band has Chern C=+1 (Fukui-Hatsugai-Suzuki lattice link-variable method, an EXACT integer, no fit). [E] TRIVIAL-PHASE NEG CONTROL: M=3 (v368) has C=0; M=-1 has C=-1 (opposite chirality, matching v444). [E] BULK-EDGE CORRESPONDENCE: the open strip has EXACTLY |C|=1 in-gap edge branch per boundary (0 trivial). [E] LI-HALDANE: the half-cut bulk ground state's entanglement spectrum has in-gap modes at 1/2 in the topological phase, none trivial. [C] EDGE CENTRAL CHARGE: c_-=N_Maj*(1/2)*|C|=16*(1/2)*1=8=g_car+N_fam, from the Chern integer -- the SAME c_-=8 as v444 from an independent observable. [C]/[O] corroborates v444's (E8)_1 reading; SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v448_seam_edge_fourpoint",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.EDGE.FOURPOINT.01: pinning the ONE external MMST fact empirically -- the edge FOUR-point is quasi-free and conformally convergent, extending v444's two-point scaling limit. [E] QUASI-FREE/WICK: the ground-state correlation matrix is an exact projector G^2=G (~1e-13), so the state is quasi-free and all n-point are Wick sums -- the MMST quasi-free CAR hypothesis (ledger entry 5) literally holds (structural: the BdG state is free). [E] FOUR-POINT CAUCHY CONVERGENCE: the edge four-point cross-ratio Q=[G01 G23]/[G02 G13] at commensurate positions (1,2,4,7)*(Nx/12) is Cauchy-convergent (spread shrinks, Nx=36..60). [E] CONFORMAL FOUR-POINT: Q converges to the free-fermion conformal cross-ratio sin(5pi/12)/sin(pi/12)=2+sqrt(3)~3.732 (eta=1). [E] BULK NEG CONTROL: the bulk four-point is orders of magnitude off (exponential). [C] the one external MMST fact (continuum existence) is now empirically supported at BOTH 2-point (v444) and 4-point order; SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v449_seam_mmst_uniform",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.MMST.UNIFORM.01: uniform-in-N control of the edge scaling limit -- the empirical content of the cited MMST uniform-convergence theorem, strengthened, on the same v367/v368 collar. [E] CROSS-RATIO A CONVERGES: the edge four-point Q(1,2,4,7)/12 -> sin(5pi/12)/sin(pi/12)=2+sqrt(3), deviation shrinks monotonically (Nx=36,48,60). [E] CROSS-RATIO B (independent positions): Q(1,3,5,9)/12 -> its OWN value 2 -- a second observable with a different limit, so convergence is not a tuned point. [E] UNIFORM 1/N RATE: |Q(N)-Q_inf|*N <= 4 for BOTH cross-ratios and every Nx in {36,48,60} (one N-independent constant across two independent observables). [E] BULK NEG CONTROL: the bulk cross-ratio is off-scale. [C]/[O] the MMST uniform-convergence hypothesis is empirically satisfied (multi-observable uniform rate); SEAM.EQUIV.01 stays [O] -- the abstract existence theorem (v336) is strengthened in evidence, not proved. Python-only (numpy)."
    },
    {
      "id": "v450_seam_edge_entanglement",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.EDGE.ENTANGLEMENT.01: a THIRD independent reading of the edge c_-=8 -- the Calabrese-Cardy entanglement scaling c=1/2 per Majorana (after v444 correlator, v447 topology). [E] CALABRESE-CARDY LAW: the block entanglement entropy of the critical XX (Dirac) chain fits S=(c/3)ln[(L/pi)sin(pi l/L)]+const with c_Dirac=1.000 -- the universal scaling-limit signature. [E] c PER MAJORANA = 1/2: c_Dirac=1=2*c_Majorana (Dirac=two Majoranas), so c_Majorana=1/2. [E] GAPPED NEG CONTROL: a dimerised chain has saturating (c=0) entanglement. [C] c_-=N_Maj*c_Majorana=16*(1/2)=8=g_car+N_fam, the SAME c_-=8 as v444/v447 from a third observable. [C]/[O] three observables agree; SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v451_seam_edge_cardy_tower",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.EDGE.CARDY.01: the Cardy finite-size conformal tower -- the edge spectrum reproduces the EXACT Ising primary dimensions {1/16,1/2}. On the critical Majorana ring (the v367/v368 edge universality class). [E] SPIN DIMENSION h_sigma=1/16: the periodic(R)-antiperiodic(NS) ground-energy split gives (E0_R-E0_NS)*L/(2pi v)=0.06250 (v=2), stable across L=200,400,800. [E] ENERGY DIMENSION h_epsilon=1/2: the lowest NS pair excitation gives 2*eps_min*L/(2pi v)=0.50000. [E] CONFORMAL SCALING: both dimensions are L-independent (gaps scale exactly as 1/L). [C] {c,h_sigma,h_epsilon}={1/2,1/16,1/2} is the chiral Ising (free Majorana) CFT; with N_Maj=16 and bulk-edge (v447), c_-=16*(1/2)=8=g_car+N_fam. [C]/[O] the edge CFT is named by central charge (v450) AND operator content (this); SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v452_seam_e8_modular",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.E8.MODULAR.01: the (E8)_1 torus modular data -- one primary, S-invariant, T-phase e^{-2pi i/3}, the holomorphic c=8 signature (extends v156's q-expansion). [E] S-INVARIANCE: chi(-1/tau)=chi(tau) for chi=E4/eta^8 (weight 0), to ~1e-30 -- ONE primary, S-matrix=[1]. [E] T-PHASE: chi(tau+1)/chi(tau)=e^{-2pi i/3}=e^{-2pi i c/24} (c=8), to ~1e-30. [E] LEADING POWER => c=8: chi*q^{1/3}->1 so the leading power is q^{-c/24}=q^{-1/3} => c=8=g_car+N_fam. [E] HOLOMORPHIC CONSTRAINT: c=8=0 mod 8 (even unimodular rank-8 lattice E8). [C]/[O] the (E8)_1 modular data is the torus signature of the seam net; with the edge readings (v444/v447/v450/v451) (E8)_1 is pinned on torus AND edge; SEAM.EQUIV.01 stays [O]. Wolfram-mirrored."
    },
    {
      "id": "v453_seam_mu4_from_marks",
      "cluster": "frontier",
      "descriptionEN": "SEAM.RIGIDITY.MU4FROMMARKS.01: deriving QGEO.SYM.01 (mu4-symmetry of the seam RP data) from the four marks -- the last structural premise of the rigidity chain folded into the existing realisation premise. [E] THE MARKS ARE mu4: the four Gauss-Bonnet marks are exactly {1,i,-1,-i}=roots of z^4-1 (v168/v216/v323). [E] THE CLOCK PERMUTES THE MARKS: z->iz is a 4-cycle of order 4 (fixes the configuration setwise). [E] MOEBIUS STABILISER: z->iz preserves the cross-ratio 2 (in the D4 stabiliser, v168). [E] THE FORM BASIS IS mu4-GRADED: omega_k=z^{k-1}dz/(z^4-1)->i^k omega_k (k=1,2,3->{i,-1,-i}, weights (1,2,3)=A3 exponents). [C]/[O] QGEO.SYM.01 is a CONSEQUENCE of marks=mu4 + QGEO.REALIZE.01 (the EXISTING premise), NO new premise; the rigidity chain (this->v446->v445) closes modulo realisation, SEAM.EQUIV.01 stays [O]. Wolfram-mirrored."
    },
    {
      "id": "v454_seam_edge_virasoro",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.EDGE.VIRASORO.01: the lattice realises the level-1 current algebra whose Sugawara central charge is EXACTLY 8 (G2) -- testing the cited MMST 'Koo-Saleur -> Virasoro with the correct c' fact on our own free-fermion edge. [E] CASIMIR c: E_0(L)=e_inf*L - pi*v*c/(6L) (Affleck-Cardy) gives c_Dirac=1.000 -> 1/2 per Majorana (parameter-free, L=64..1024). [E] KAC-MOODY LEVEL: the lattice U(1) current has <0|J_n J_-n|0>=n exactly (n=1..8) -- the affine level-1 central extension closes; a gapped control gives a vanishing non-linear term. [E] SUGAWARA EXACT: c=dim G/(1+h^v)=8 for BOTH SO(16)_1 (120/15) and (E8)_1 (248/31). [C] 16 Majoranas => c_-=8=g_car+N_fam. [C]/[O] SEAM.EQUIV.01 stays [O]. Mixed numerical (Casimir, Kac-Moody) + exact (Sugawara, Wolfram-mirrored)."
    },
    {
      "id": "v455_seam_bw_uniform",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.BW.UNIFORM.01: a UNIFORM-IN-N Tomita-Takesaki tower for the intrinsic Bisognano-Wichmann condition u_Theta=J (G1) -- the inductive-limit companion to v426, advancing v424 sub-step (i). [E] UNIFORM BW: ||Theta K Theta + K||=0 exactly at every size (N=8..128) -- u_Theta=J holds uniformly. [E] UNIFORM GAP: the gap of K is N-independent (6 ln(3/2)) -- Delta=e^{-K} uniformly bounded from 1, a genuine KMS (not trace) limit. [E] UNIFORM CLOCK: [rho,K]=0 exactly at every N. [E] N-INDEPENDENT NEG CONTROL: a side-even reflection gives ||Theta K Theta+K||=2||K|| growing with N, an off-mu4 K breaks [rho,K] at every N. [C]/[O] the inductive limit inherits u_Theta=J; SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v456_seam_chirality_from_c3",
      "cluster": "frontier",
      "descriptionEN": "SEAM.S3.FROM-P1.01: the edge chirality (c_-!=0, the S3 phase) is FORCED by the one-sidedness that defines c3=1/(8pi) (P1), so S3 is a CONSEQUENCE of P1 (G6). [E] c3's '8' IS THE ONE-SIDED COUNT: 8pi=|Z2|*intK=2*(2pi*chi(S^2))=2*4pi, integer 8=|Z2|*(intK/pi)=2*4. [E] REFLECTION REVERSES C: the p+ip model has Chern C=+1, its ky->-ky partner C=-1. [E] TWO-SIDED => NON-CHIRAL: a gapped boundary with an orientation-reversing symmetry has C=-C=0 (c_-=0). [C] ONE-SIDED => CHIRAL: the one-sided seam (the same |Z2| of c3) has no such reflection, so C!=0 is forced. [C]/[O] c_-=8=rank E8 is the SAME 8 as c3's 8pi; S3 follows from P1; SEAM.EQUIV.01 existence stays [O]. Mixed exact (arithmetic, Wolfram) + numerical (Chern flip)."
    },
    {
      "id": "v457_seam_character_killtest",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.KILLTEST.01 [X]: an explicit (E8)_1-vs-SO(16)_1 character/sector kill-test for the seam edge -- a FALSIFIABLE test of the (E8)_1 identification (G7). [E] (E8)_1 CHARACTER TOWER: chi=E4/eta^8=q^{-1/3}(1+248q+4124q^2+34752q^3+...) -- exact (E8)_1 tower degeneracies. [E] CURRENT-COUNT DISCRIMINATOR: weight-1 multiplicity=248=dim E8=120(SO16)+128(spinor) -- not the free 120. [E] SECTOR-COUNT DISCRIMINATOR: |det Cartan E8|=1 (invertible) vs |det Cartan D8|=4. [X] KILL CRITERION: 120 (not 248), OR 4 sectors (not 1), OR a wrong tower coeff would FALSIFY (E8)_1; data give 248/1/{1,248,4124}, PASSED. [C]/[O] SEAM.EQUIV.01 existence stays [O]. Exact (Wolfram-mirrored) + det cross-check."
    },
    {
      "id": "v458_seam_mmst_citation_audit",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.MMST.AUDIT.01: the EXACT hypothesis-by-hypothesis citation audit of Osborne-Stottmeister (arXiv:2107.13834, CMP 398 (2023); legacy internal token MMST) against our collar (G3, the Zitat-Abgleich). [C] H1 ALGEBRA CLASS: CAR 16-Majorana collar -- MATCH. [C] H2 MASSLESS POS-ENERGY LIMIT: the edge IS that limit (v439/v444) -- MATCH. [E] H3 PER-COPY c=1/2: 16 chiral Majoranas, each the proven Theorem A case. [C] H4 DECOUPLED ADDITIVITY: 16*(1/2)=8 -> free SO(16)_1 c=8. [C] H5 WZW RANGE 8<=8<=16 + 120 bilinear currents. [O] H6 RESIDUAL = the 128-spinor extension (248=120+128), outside MMST's bilinear method, handed to v459. [C]/[O] the c=8 EXISTENCE is citable; SEAM.EQUIV.01 stays [O], residual narrowed to the holomorphic extension. Python (sympy)."
    },
    {
      "id": "v459_seam_lattice_voa_route",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.LATTICEVOA.01: the SECOND, independent route to (E8)_1 -- the lattice net A_Q(E8) (Adamo-Giorgetti-Tanimoto arXiv:2506.01008 + OS arXiv:2407.18222) supplies the 128-spinor extension MMST (v458) leaves open (G5). [E] E8 EVEN UNIMODULAR: det Cartan=1, all 240 roots norm 2 -- A_{E8} is a holomorphic net (det K=1). [E] LATTICE CURRENTS 248=8+240: weight-1 = 8 Cartan + 240 roots. [E] THE 240 ROOTS SPLIT 112+128=SPINOR: 8+112=120=so(16), the 128 half-integer roots ARE the spinor (248=120+128=8+240). [C] AGT: A_Q of an even lattice IS a 2d conformal net, so the SO(16)_1->(E8)_1 extension is constructed, not assumed. [C]/[O] two complementary routes cover (E8)_1; residual = the one realization input (collar=A_{E8}), tied to P1 by v456; SEAM.EQUIV.01 stays [O]. Exact (Wolfram-mirrored) + structural."
    },
    {
      "id": "v460_seam_s3_lto",
      "cluster": "frontier",
      "descriptionEN": "SEAM.S3.LTO.01: the EXPLICIT flat-band commuting-projector (local-topological-order) realisation of the S3 input -- advancing v424 sub-step (i) ('realise the raw seam as a Z/2-reflection commuting-projector LTO') from abstract existence to an explicit gap-protected quasi-local projector net carrying the Z/2 reflection and the mu4 clock. [E] FLAT-BAND PARENT: the flattened Bloch Q=h/|h| of v367's p+ip collar has Q^2=I (flat +-1 bands) and P=(I-Q)/2 is an exact projector EQUAL to the occupied ground-state projector -- a frustration-free commuting-projector parent with the same gapped ground state. [E] GAP-PROTECTED QUASI-LOCALITY: the gapped projector kernel |P(n,0)| decays rapidly (xi~1.14 at M=1, exp R^2=0.995, |P(20,0)|<1e-6), xi grows as the gap closes (2.7 at gap 0.02); the GAPLESS point decays as a clean power law |R|^{-2.1} (R^2=0.999, tail ~2e-4) -- locality needs the gap. [E] Z/2 REFLECTION + mu4 INHERITED: the flattened modular Hamiltonian K_flat=sign(K) inherits Theta K_flat Theta=-K_flat (BW/u_Theta=J) and [rho,K_flat]=0 (mu4) EXACTLY (sign is odd; rho commutes with any f(K)). [E] NEG CONTROLS: a side-even reflection gives 2||K_flat||!=0 (BW fails), an off-mu4 K mixing opposite-sign marks breaks [rho,sign(K)]!=0. [C]/[O] the chiral c_-=8!=0 (FHS Chern |C|=1) obstructs STRICT finite-range commuting-projector locality (Kapustin-Fidkowski; C!=0 forbids exp-localised Wannier), so the realisation is the quasi-local LTO net of NPW26; advances v424 sub-step (i), SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v461_seam_strict_locality",
      "cluster": "frontier",
      "descriptionEN": "SEAM.S3.LOCALITY.01: the Kapustin-Fidkowski/Thouless strict-locality OBSTRUCTION made explicit -- sharpening v460's verdict from a cited statement to an exhibited topological obstruction. [E] CHERN INTEGER: the FHS Chern of v367's p+ip collar = +1 (chiral M=1), 0 (trivial M=3), -1 (opposite mass). [E] WILSON-LOOP / WANNIER-CENTRE WINDING = CHERN: the hybrid Wannier centre theta(ky) winds by |C|=1 (chiral) and 0 (trivial) over the BZ -- exp-localised Wannier exist IFF winding=0 (Brouder et al.), so the chiral collar has NO exp-localised Wannier basis. [E] OBSTRUCTION INTEGERS TIED: winding=|C|=1!=0 and c_-=16/2=8=g_car+N_fam=rank E8!=0. [E] NEG CONTROL: BOTH phases gapped, yet only the trivial (winding 0) admits exp-localised Wannier and a strict finite-range commuting projector -- it is the CHIRALITY, not the gap. [C]/[O] a strictly finite-range commuting projector would force winding 0, C=0; since C=1!=0 none exists, so v424 sub-step (i)'s realisation is PROVABLY the quasi-local NPW26 LTO net; SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v462_seam_spinor_continuum",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.SPINOR.01: the 128-spinor extension SO(16)_1 -> (E8)_1 exhibited at CHARACTER level and as a finite-L -> continuum convergence -- the constructive companion of the v458 (MMST audit) / v459 (lattice-VOA) residual. [E] (E8)_1 TOWER FROM SO(16)_1 SECTORS: the exact Jacobi/E8 identity theta2^8+theta3^8+theta4^8=2E4 gives chi_{(E8)_1}=E4/eta^8 = chi_o^{SO16}+chi_s^{SO16} -- the holomorphic (E8)_1 character IS the SO(16)_1 vacuum+spinor sector sum. [E] 128 SPINOR WEIGHT-1 STATES FILL 120 -> 248: chi_o gives 120, chi_s gives 128, summing to the exact (E8)_1 tower {1,248,4124,34752}. [E] FINITE 16-MAJORANA FOCK: C(16,2)=120 NS bilinear currents and 2^8=256 Ramond ground states = 128 spinor + 128 cospinor. [E] FINITE-L -> CONTINUUM CONVERGENCE: the critical free-fermion ring Casimir gives c_Dirac -> 1 (c_-=8) and the lowest scaled NS gap x_1 -> 1 with |x_1-1| ~ 1/L^2 (L=64..1024). [C]/[O] the rigorous holomorphic extension is supplied by AGT/AMT (v459); SEAM.EQUIV.01 existence (v336) stays [O]. Mixed exact (theta identity + tower + 120/128/256, Wolfram-mirrored) + numerical."
    },
    {
      "id": "v463_seam_c8_holomorphic_uniqueness",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.UNIQ.01: the c=8 holomorphic-uniqueness pin -- the positive lever that makes the IDENTIFICATION half of SEAM.EQUIV.01 classification-forced (B of the A+B round). [E] c=8 NOT UNIQUE AT LEVEL 1: c=dim/(1+h^v)=8 has THREE simple solutions -- A8 (80), D8=so(16) (120), E8 (248); c=8 is necessary but NOT sufficient. [E] HOLOMORPHY FORCES dim V_1=248: the only weight-0 form with leading q^{-1/3} and integer coeffs is j^{1/3}=E4/eta^8, q^1 coeff 248. [E] dim 248 + level-1 c=8 => E8 uniquely (248/(1+30)=8); A8/D8 excluded. [E] tower {1,248,4124,34752,213126}=E4/eta^8; E4/eta^8=j^{1/3}. [C] Dong-Mason/Schellekens: holomorphic c=8 VOA unique = V_{E8}; so the identification half of SEAM.EQUIV.01 is [C], leaving the realisation R1 (v464). SEAM.EQUIV.01 stays [O]. Exact (sympy q-series + Lie scan; mpmath) Wolfram-mirrored."
    },
    {
      "id": "v464_seam_oneparticle_rigidity",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.ONEPARTICLE.01: the one-particle RIGIDITY + scaling limit attacking R1, the REALISATION input of SEAM.EQUIV.01 (A of the A+B round). The seam is quasi-free (v113/v160/v161), so by Araki self-dual CAR the whole net is the second quantisation of its one-particle symbol P. [C] ARAKI: state <-> symbol P bijectively. [E] RIGIDITY: the gapped ground state is the UNIQUE quasi-free state with P=projection onto K<0 (P^2=P), K fixed by gap+chirality+reflection (v426). [E] ONE-PARTICLE SCALING LIMIT EXISTS: kernel Cauchy on a fixed window (~1/N). [E] LIMIT IS c=1 DIRAC: entanglement S=(c/3)ln L gives c->1, 16 copies -> c_-=8 (ties v450). [C] Shale-Stinespring: implementable Bogoliubov map; R1 upgraded from a bald axiom to an audited [C]; with v463 the SEAM.EQUIV.01 residual is entirely certification, zero open internal mechanism. SEAM.EQUIV.01 stays [O]. Python-only (numpy)."
    },
    {
      "id": "v465_seed_crosssector_joint",
      "cluster": "registry",
      "descriptionEN": "SEED.CROSSSECTOR.01: the cross-sector one-parameter seed test -- the theta13-independent, externally-runnable slice of v306. TFPT says a CMB EB/TB birefringence angle beta and a quark Cabibbo angle lambda_C=|Vus| (unrelated in the SM) are two readouts of the ONE axiom seed phi0. [E] plumbing: axiom phi0 reproduces the frozen lambda_C, sin^2 theta12, beta to 1e-9. [N] CROSS-SECTOR: back-solving phi0 independently from beta (CMB) and lambda_C (quark) gives the SAME seed within 0.01 sigma_combined. [N] one-parameter fit: error-weighted mean of the 3 back-solved phi0 reproduces the axiom to 0.06%; joint chi^2 at the axiom (0 free params, 3 dof)=0.007, chi^2/dof<1, combined pull 0.08 sigma. [N] theta13 EXCLUSION declared: adding sin^2 theta13 raises chi^2 by ~4 (the ~2 sigma pressure, v328/v338). [N] NEG/POWER: data<->formula shuffle explodes the cluster chi^2 by >5e4x. Consistency statistic (subset of v306); phi0 FIXED by axioms, upgrades no prediction. Python-only."
    },
    {
      "id": "v466_seed_leptonmass_channel",
      "cluster": "registry",
      "descriptionEN": "SEED.LEPTONMASS.01: a sixth, new-sector seed channel -- the charged-lepton mass ratio m_e/m_mu=(12/7)phi0^2 points to the SAME axiom seed phi0, extending the empirical seed over-determination (v306/v465) from three measurement sectors to five. [E] plumbing: axiom phi0 reproduces the frozen m_e/m_mu=(12/7)phi0^2=0.00484673 to 1e-9. [N] NEW SECTOR: back-solving phi0 from the measured POLE ratio m_e/m_mu=0.004836 (PDG) gives phi0=0.053113, only -0.11% from the axiom -- a charged-lepton MASS reading, a 4th measurement sector beyond {neutrino mix, CMB, quark mix}. [C] TRANSPORT CAVEAT: the formula is the SOURCE ratio, the datum the POLE ratio; the source->pole shift is the seam-gapped class (bound (2/3)^6=8.78%, v393) but largely cancels for m_e/m_mu, so the 0.11% residual sits far inside the band -- added as a CONSISTENT new sector, not a sharper pin. [N] EXTENDS: the 6-channel cluster (v306's 5 + this) has chi^2/dof<1 and the error-weighted mean stays at the axiom to <0.1%; with the corroborating heavy-quark m_t/m_b (-0.23%) it is a 7-channel, FIVE-sector agreement. [N] NEG/POWER: a data<->formula shuffle explodes the cluster chi^2 by >100x; m_c/m_s=(34/47)/phi0 is scheme-ambiguous (FLAG 11-14), a WEAK channel not counted. [C] HONEST: still the ONE seed (v305 multiplicity 1) -- an EMPIRICAL strengthening, NOT theoretical independence; phi0 FIXED by axioms, upgrades no prediction, closes no gate. Python-only."
    },
    {
      "id": "v467_thirdgen_mixing_pattern",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.THIRDGEN.PATTERN.01: the three ~2σ mixing tensions (θ13, θ23, V_cb) are ONE common −φ₀ suppression (−5.36%±1.67%, 3.2σ non-zero); ladder-base forms λ_C²e^(−5/6) / (1−φ₀)/2 / λ_C²/(1+λ_C) drop the joint χ² 10.4→0.11 with zero parameters; post-hoc candidate [O], record unchanged, JUNO/Belle II decide"
    },
    {
      "id": "v468_dm2_ratio_jarlskog",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.DM2RATIO.01: Δm²₂₁/Δm²₃₁ = |J_PMNS(frozen)| = 0.029653 vs measured 0.029805(792) → −0.19σ — a parameter-free candidate relation for the (previously unpredicted) splitting ratio; the informal m₂/m₃ ~ πφ₀ heuristic is now −2.44σ and retyped; NO floor Σm_ν = 0.0588 eV consistent; post-hoc [O], JUNO decides"
    },
    {
      "id": "v469_seam_crossedproduct_route",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.CROSSEDPRODUCT.01: the net-level crossed-product certification of the 128-spinor extension + the R1 -> R1' invariant reduction. [E] LR LOCALITY INTEGER: h_s(SO(16)_1) = 16/16 = 1 in Z => statistics phase +1 -- the Longo-Rehren criterion HOLDS, the Z2 simple-current crossed product is a LOCAL index-2 extension (LR RMP 7 (1995); Boeckenhauer RMP 8 (1996); BE CMP 197 (1998)); KLM mu = 4/2^2 = 1 => holomorphic => (E8)_1 (v463 pin, E4/eta^8 q^1 = 248). The AGT/AMT 2024/2025-preprint leg (v459) DEMOTED to independent second witness -- the extension now rests on 1995-2001 peer-reviewed journals. [E] index-4 parallel: mu4 glue h(J^k) = {1,1,1} all integer (= v125 isotropy at net level), mu = 16/16 = 1. [E] R1' invariants computed: FHS Chern 1/0, nu = 16, c_- = 8; nu mod 16 = 0 = the Kitaev-E8 class (16-fold way). [E] chain T1->T4 composes (Kitaev/Kapustin-Spodyneiko -> MMST+Boeckenhauer -> LR/BE+KLM -> Dong-Mason+v463). [C] R1 -> R1': collar_realizes replaceable by collar_invariants (quasi-free [C] + gap [E] + class D + c_-=8 [E from P1]); quasi-freeness stays the ONE [C]. [O] SEAM.EQUIV.01 stays [O]. Mixed exact (Wolfram-mirrored) + numerical (FHS Chern)."
    },
    {
      "id": "v470_alpha_inflow_level",
      "cluster": "frontier",
      "descriptionEN": "ALPHA.QUILLEN.INFLOW.01: the alpha^3 level = the COMPUTED bulk Chern invariant + the seam F-normalisation = the affine embedding index k_Y = 5/3. [E] target re-verified: alpha^-1 = 137.0359992168 unique root; Quillen split < 1e-35 (v341/v382). [E] the invariant computed: FHS Chern C(M=1)=1, C(M=3)=0 (the v367/v461 integers, read as the U(1) response). [C] k0 = |C| = 1 under the EM1 reading: the a^3 coefficient EQUALS the computed bulk invariant -- 'why exactly 1' answered by computation + cited inflow (TKNN 1982/ASS 1983 quantisation; Callan-Harvey 1985 inflow; APS/AGDPM/Witten RMP 88 eta=CS part of delta log det; Quillen/Bismut-Freed integrality) -- replacing v435's minimality assumption. [E] k_Y = tr(Y^2)/tr(T3^2) = (5/6)/(1/2) = 5/3 (Ginsparg 1987); (3/5)(41/6) = 41/10 = b1; 41/6 = 20/3 + 1/6. [C] F-normalisation retyped: <JJ> fixed by k_Y once the seam net is (E8)_1 level 1 -- no dial, ZERO independent content (a face of SEAM.EQUIV.01). [E] UNIFICATION: one invertible phase, two responses -- c_- = 8 (feeds SEAM.EQUIV.01/S3) and C = 1 (feeds EM1); closing R1' (v469) feeds both. [O] the bridge lemma stays cited; ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 stays [E]. Mixed exact (Wolfram-mirrored) + numerical (root/Chern)."
    },
    {
      "id": "v471_seam_horizon_replica",
      "cluster": "horizon",
      "descriptionEN": "SEAM.EHMODEL.04: the Seam-Horizon replica chain exercised numerically on a discretized seam collar -- the kernel-identification premise of SEAM.THEOREM.01 (residual of v150/v151/v152) discharged at the finite/model level for the seam's own kernel, with real replica sheets. Replica => EH form on a real operator: dG/d ln m = 2C(gamma) on deficit cones (0.01-0.05%) and replica sheets n=2 (0.8%), n=3 (1.5%); EH coefficient cutoff-independent (0.5% drift) while the zeta-scale mu_lat drifts = the v152 anchor exhibited (no canonical ln(m/mu)=3/4 emerges); BFK/Schur split exact (~1e-16) with the Calderon factor conically clean (pure cut-edge law, tip term 0, power check) and the halves carrying 2C_cone (Kac doubling exact); the v302 transfer masses 6ln(3/2), 6ln3 sit on the calibrated EH line (<=0.05%) and the Perron/attractor mode (m=0) is IR-divergent -- the recovery gap makes the induced-gravity coefficient finite. Honest scope: finite level; SEAM.THEOREM.01 stays open, residual = continuum leg (MMST class, = the SEAM.EQUIV.01 residual) + the anchor. Numerical, Python-only by nature."
    },
    {
      "id": "v472_quillen_detline_moduli",
      "cluster": "frontier",
      "descriptionEN": "ALPHA.QUILLEN.DETLINE.01: the determinant line over the U(1)-twist moduli of the collar model carries curvature = the inflow level -- the finite Quillen/Dai-Freed shadow of the v470 bridge lemma, computed. [E] Bloch level re-verified: BZ Chern C(M=1)=1, C(M=3)=0, C(M=-1)=-1 (the v470/v367 integers). [E] the Quillen object computed: the same collar h(k) in real space with twisted boundary conditions (the twist torus = moduli of flat U(1) connections); the FHS curvature of the DETERMINANT LINE of the occupied frame over that torus = 1 exactly (1e-9) at M = 1, for L = 4 AND L = 6 (size-independent). [E] controls: trivial M=3 -> 0, orientation flip M=-1 -> -1. [E] two tori, one level: twist-moduli integer == Bloch-BZ integer for all three M (Niu-Thouless-Wu PRB 31 (1985), exhibited on the S3 collar model). [E] section globally defined: Fermi gap 2.0 open over the whole twist torus (the Dai-Freed section has no zero). [C] the reading: det-line holonomy over the U(1) moduli = inflow level k0 = |C| = 1 -- the v470 bridge lemma 'delta log det_zeta(seam) = the inflow response' holds verbatim at the finite/model level. [O] not closed: the abstract-seam zeta-determinant identification (continuum leg, = the SEAM.EQUIV.01 face); ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 stays [E]. Numerical, Python-only."
    },
    {
      "id": "v473_entropic_action_bridge",
      "cluster": "frontier",
      "descriptionEN": "GRAV.ENTROPIC.ACTION.01: the entropic-action bridge -- Bianconi's 'Gravity from entropy' (Phys. Rev. D 111, 066001 (2025); arXiv:2408.14391) quantified as an EXTERNAL CANDIDATE for the local volume action behind TFPT's parameter-free Einstein equation (v358/v359, typed 'equation of state, not a from-action quantisation' [O]). [E] carrier Hodge count: her Omega^{0,1,2} fiber on the five-slot carrier C^5 (not 4d spacetime, fiber 1+4+6=11) counts 1+5+10 = 16 = 2^(g_car-1) = dim S+_{D5}, Hodge-folded onto L^even C^5 (vector-space identity only). [E] coefficient match (the one quantitative pin): her weak-coupling L = 3 beta R - alpha L_m matched to 1/(16 pi G) = c3/(2G) fixes beta'_B = c3/6 = 1/(48 pi) exactly; the 6 = 3 blocks x 2 = p2 = |R+(A3)| (echo, not proof). [E] entropy potential: Lambda_G = (1/2 beta)Tr(G - I - ln G) >= 0, quadratic at the fixed point => eps_* = e^{-1/alpha} Q reproduces the v60 branch (3/4 pi^2) e^{-2/alpha} with exact target Tr Q^2 = 32 c3^4 = 1/(128 pi^4) = (2/3) delta_top ([C] until Q constructed). [E] Frullani: her log action = all-scale heat kernel, right genre vs S_rel,chi. [E] R^2 kill test: raw (c3/6)^2 vs Starobinsky 1/(12 c3^7) -- exact gap 3 (8 pi)^9 ~ 1.2e13 (13 orders); direct identification FALSE without a mechanism -- pre-registered. [C] compression conjecture P (G~ g~^-1) P = Delta^1/2 + operator Hodge items open; Lorentzian positivity not automatic (OS route sturdier). [O] nothing closes, zero knobs added. Mixed exact (Wolfram-mirrored) + one quadrature."
    },
    {
      "id": "v474_entropic_hodge_carrier",
      "cluster": "frontier",
      "descriptionEN": "GRAV.ENTROPIC.HODGE.01: the OPERATOR level of the entropic-action bridge (work packages 1 + 4-algebra of v473). [E] CAR => Clifford: ten Gamma's on the carrier Fock space Lambda^* C^5 (dim 2^g_car = 32) satisfy {Gamma_a,Gamma_b} = 2 delta_ab exactly -- the so(10) = D5 Clifford algebra acts on the carrier exterior algebra. [E] S+ = Lambda^even as D5-module: the 45 bilinears commute with (-1)^N, span exactly 45 = dim so(10), zero leak from the 16-dim even subspace -- the half-spinor IS the even exterior algebra WITH its spinor structure (operator level of v2/v44/v197). [E] Hodge-Dirac symbol: c(xi)^2 = -|xi|^2 Id for symbolic xi -- Bianconi's D = d + delta is Clifford multiplication at symbol level. [E] the fold is 5 -> 5bar: traceless u(5) weights of Lambda^4 = negatives of Lambda^1, so the Hodge fold conjugates the vector block -- her fiber 1+5+10 becomes the GUT half-spinor 16 = 1 + 5bar + 10 (the SM carrier decomposition). [E] honest obstruction: Omega^{<=2} is NOT Clifford-invariant (Lambda^2 -> Lambda^3 leak); only Lambda^even is -- the bridge must go through the fold, never degree truncation. [E] Q-target enumeration (the v473 conjecture DECIDED): d k^2 = 32 with integer k has exactly the supports (2,4), (8,2), (32,1) = {|Z2|, rank E8, 2^g_car}; minimal uniform q = c3^2 = 1/(64 pi^2) per Fock mode; the two-form block 10 and the fiber 16 need irrational multipliers => the naive pair-block reading is KILLED; bonus Tr Q^2/delta_top = 32/48 = 2/3. [C] physical Q choice; [O] AP2 compression + R^2 mechanism open, no marker moves. Mixed exact (numpy integer Fock + sympy; Wolfram-mirrored where cheap)."
    },
    {
      "id": "v475_entropic_scalaron_gate",
      "cluster": "frontier",
      "descriptionEN": "GRAV.ENTROPIC.SCALARON.01: the R^2 kill test of the entropic-action bridge EXECUTED (work package 5 of v473) + the explicit Lorentzian-positivity witness. [E] max-symmetric eigenvalues (her Appendix-B flattening verified): R~ g~^-1 has eigenvalues exactly {R; R/4 x4; R/6 x6} on a general diagonal 4d metric, incl. the 6x6 two-form flattening. [E] exact vacuum f(R): L_B(R) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] = 3 beta R + (17/24) beta^2 R^2 + O(R^3) -- the 3 re-derives her Eq. (45) (the v473 pin), 17/24 the exact tensorial R^2 factor. [E] raw scalaron TRANS-PLANCKIAN: with beta' = c3/6 both routes give m_raw^2 = 4608 pi^2/17 Mbar^2, m_raw ~ 51.7 Mbar ~ 1.3e20 GeV -- heavy, not light (TFPT: c3^{7/2} Mbar = 3.06e13 GeV); required enhancement EXACTLY (72/17)(8 pi)^9 ~ 1.69e13 in mass^2 -- v473's interpretation 2 (light G-trace mode) KILLED in naive form. [E] domain: first log pole at R = 384 pi^2 Mbar^2 (trans-Planckian) -- the obstruction is the mass, not the domain. [E] Lorentzian witness: timelike gradient gives eigenvalue 1 - alpha v^2 <= 0 for v^2 >= 1/alpha (ln undefined); spacelike/Euclidean positive -- the v473 caveat exhibited, the OS route sturdier. [C] verdict: interpretation 1 survives (Einstein + Lambda level), 3 (KMS-spectral renormalisation) is the only surviving R^2 route. [O] unchanged: AP2 compression + the R^2 gate; no marker moves. Exact (sympy, Wolfram-mirrored)."
    },
    {
      "id": "v476_entropic_compression_toy",
      "cluster": "frontier",
      "descriptionEN": "GRAV.ENTROPIC.COMPRESS.01: the AP2 compression conjecture of the entropic-action bridge (v473) made WELL-POSED on an exactly solvable model. [N] pure gapped bulk: staggered-mass chain, ground-state covariance a projection (pure bulk, thermal boundary -- the seam situation). [N] compression is thermal (v155 in the bridge context): spec(PCP) strictly inside (0,1), Peschel K finite, S > 0. [N] ORDER OBSTRUCTION (the reading decided): on the pure bulk f(x) = x/(1-x) is singular on spec(C) = {0,1} -- the literal operator-side reading P f(C) P is ILL-POSED while f(PCP) is finite: the identity can only be read STATE-SIDE (build Delta_Sigma from the compressed relative metric), matching Bianconi's own local construction -- AP2 retyped, not weakened. [E] block algebra (symbolic): [C^n]_AA - (C_AA)^n carries the cross block twice -- first order vanishes identically, the mismatch is exactly quadratic in cross-cut correlations (= the entanglement content). [N] gap control at the bounded level: bulk identity f(C_T) = e^{-h/T} exact; mismatch obeys d <= x^2 and falls with the gap; decoupled cut exact -- the two readings converge in the gap-dominated regime, where TFPT operates. [C] retyped conjecture; [O] AP2 open (continuum seam statement), no marker moves. Numerical, Python-only by nature."
    },
    {
      "id": "v477_entropic_scaleflow",
      "cluster": "frontier",
      "descriptionEN": "GRAV.ENTROPIC.SCALEFLOW.01: the scale-flow representation of the entropic action -- v475's surviving interpretation 3 typed precisely: the (72/17)(8 pi)^9 gap collapses to ONE moment condition on the scale weight, satisfied by the SAME KMS moment TFPT already fixes (v36 f0 <-> M^2 = c3^7 Mbar^2); zero new dials, nothing closes. [E] scale-flow identity: -ln A = flat scale-integral of relative heat-kernel actions (Frullani symbolic + 1e-20 quadrature of the chi-form) -- TFPT's S_rel,chi evaluates ONE KMS scale with a weight: the bridge is a choice of scale measure. [E] moment dictionary: weight w(t) dt/t gives 3 beta R mu1 + (17/24) beta^2 R^2 mu2; flat weight = unit moments = the v475 raw coefficients. [E] the one moment condition: demanding m^2 = c3^7 Mbar^2 forces exactly mu2/mu1^2 = (72/17)(8 pi)^9 -- the kill-test number IS a moment ratio, and the closure (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 holds identically: the 13 orders are a scale-measure datum, not a bridge mismatch. [E] v36 consistency: same mass from f0 = 6(4 pi)^2/c3^7 -- one KMS moment, two parametrisations. [C] consistency not derivation (the weight is TFPT input); negative side-record: the exact lattice-BW commutator attempt for AP2 did not hold in simple forms (logged, kept out). [O] unchanged."
    },
    {
      "id": "v478_entropic_continuum_legs",
      "cluster": "frontier",
      "descriptionEN": "GRAV.ENTROPIC.LEGS.01: first computable steps on the two remaining legs of the entropic-action bridge. LEG A (AP2 continuum): [N] the compressed critical interval has S(L) = (c/3) ln L + k with c_est = 1.00089/1.00038/1.00021 (monotone -> Dirac c = 1; gapped control ~6e-8, area law); the Peschel modular Hamiltonian's NN couplings organise on the CHM weight beta(x) = x(L-x)/L (corr 0.87 -> 0.99, plateau drift shrinking); even-range bands vanish identically and the odd tail is 1.8% with parabolic envelope -- the state-side Delta_Sigma (v476's forced reading) flows to the CHM/BW geometric form that TFPT's Einstein derivation consumes (v323/v358): the bridge's modular side and TFPT's gravity side meet in the continuum limit. LEG B (measure): [E] single-scale corollary -- mu2/mu1^2 = e^{t0} exactly, so the v477 condition forces t0 = ln(72/17) + 9 ln(8 pi) = 30.461 (exact closed form); the near-miss to h(E8) = 30 is explicitly DECLINED (v354/v355 discipline); deriving the measure stays open. [C] meeting point; [O] both legs, no marker moves. High-precision numerical (mpmath 50 digits), Python-only by nature."
    },
    {
      "id": "v479_kronheimer_quiver_bridge",
      "cluster": "core",
      "descriptionEN": "SEAM.KRONHEIMER.01: the R3 'combinatorial-to-geometric bridge' of the det K=1 keystone (SEAM.DETK.01/v344) is Kronheimer's theorem (JDG 29 (1989) 665+685) with all finite input data already compiler outputs. [E] Kac marks = null vector of the affine-E8 Cartan AND Perron vector (A delta = 2 delta) = the v312 rewrite-attractor vector; [E] hyper-Kaehler quotient dimension count dim_R X = 4 (the C^2/Gamma geometry), with sum_edges d_i d_j = sum d_i^2 = 120 forced by the null identity; [E] gauge datum = |2I| = 120 = |mu4| h(E8), h(E8) = 30 = 2*3*5; [E] resolution = 8 spheres with unimodular -E8 form (det 1 = Poincare link = the det K=1 face) and 24 = 3*rank deformation parameters. [C] Kronheimer cited, not re-proved (import class of NPW26/MMST/AGT). [O] the residual premise transforms to 'the raw seam supplies the ALE/orbifold datum' -- the same single arrow as SEAM.EQUIV.01; nothing closes."
    },
    {
      "id": "v480_multilocal_four_interval",
      "cluster": "frontier",
      "descriptionEN": "QGEO.MULTILOCAL.01: the four mu4 marks make the seam circle a FOUR-INTERVAL free-fermion geometry -- the class whose multi-interval modular theory is exactly known (Casini-Huerta mixing term; Longo-Xu; Rehren-Tedesco multilocal fermions; KLM: holomorphic <=> trivial multi-interval inclusion, mu=1 -- the same bit as det K=1). On the exactly solvable antiperiodic ring: [E-num] binary clock rho^4 = -1 exactly (the fermionic mu4 clock is the order-8 double-cover lift, the same Z2-lift pattern as 2I over I); [E-num] the primitive-8th-root character projectors decouple the 4-interval covariance at 1e-15 (the multilocal diagonal-cover isomorphism); [E-num] each sector = single-interval quarter-ring kernel with twist lambda_k, spectra match at 1e-14; [E-num] rho C rho^-1 = C at 1e-14 => omega o rho = omega MANIFEST at the state level; [E-num] negative control: one displaced interval re-couples sectors at 1e-2; [N] cross-interval K peaks on the conjugate diagonal (Casini-Huerta mixing shadow). [C] continuum theory cited; [O] the raw-seam premise stays open -- mechanism exhibited, not closed."
    },
    {
      "id": "v481_seesaw_carrier_ladder",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.NUSCALE.02: the absolute neutrino-mass scale under the CARRIER Yukawa normalisation y_nu = y_t -- FLAV.NUSCALE.01's two-parameter (y_nu, M_R) trade-off collapses to ONE parameter, and the required M_R lands INSIDE the compiler's own PS window; the nearby integer c3-rung is explicitly DECLINED at 1-loop (anti-numerology). [P] carrier normalisation (named premise): one SO(10) 16 per family + minimal Yukawa sector (10_H / PS (1,2,2)) => y_nu = y_t at the matching scale. [N] inversion: m_3 = 0.0503 eV demands M_R = 9.3e13 GeV, inside [4.2e13, 2.4e15] (v249), at log_c3(Mbar/M_R) = 3.15 (y_nu = 1 gives the structureless 2.58). [X] integer-rung honesty gate: M_R = c3^3 Mbar predicts m_3 = 0.030 eV (40% low at 1-loop; ADKLR run-down 0.794) -- NEAR, not ON: pin DECLINED; decision computation named (2-loop + PS thresholds). [N] falsifiable band [0.002, 0.115] eV brackets the observation; DESI cuts the window from below; NO floor Sigma = 0.059 eV. [C]/[O] candidate class (genre v467/v468); frozen record untouched; nothing closes."
    },
    {
      "id": "v482_seesaw_rung_decision",
      "cluster": "neutrinos",
      "descriptionEN": "FLAV.NUSCALE.03: the decision computation named by v481, EXECUTED in bracketed form -- the integer rung M_R = c3^3 Mbar is EXCLUDED as an unstructured exact pin (conditional on the carrier premise y_nu = y_t). [E-num] gap: m_3(rung) = 0.0301 eV vs 0.0503 observed, rescue factor x1.670; [E-num] bracketed exclusion: m_t +-3 GeV (>3 sigma), alpha_s >3 sigma, kappa run-down +-10%, PS-leg beta envelope x[0.5,1.5] -- max single shift x1.10, all-favourable combined x1.165, far below x1.670: the 1-loop miss is NOT an RG artifact. [C] named escape recorded and DECLINED: only a third-generation Majorana structure factor r ~ 1.67 revives the rung; post-hoc r = 1.6696 sits 0.18% from g_car/N_fam = 5/3 -- no forcing mechanism, declined per the anti-numerology rule (the t0 ~ 30 pattern). [O] the v481 one-parameter window candidate stands unchanged; frozen record untouched; nothing closes."
    },
    {
      "id": "v483_pillowcase_exact_traces",
      "cluster": "core",
      "descriptionEN": "HYP.PHI0.GEOM.01: the GEOMETRIC side of the phi0 puncture target (HYP.PHI0.PUNCTURE.01/v408) is EXACT -- every twisted/orbifold heat trace of the flat tau = i pillowcase is a t-INDEPENDENT RATIONAL, so the pi^-4 in dtop = 48 c3^4 cannot come from flat orbifold geometry at any order. [E-num] Tr(gamma e^{t Delta}) = 1 exactly for gamma = sigma, rho, rho^3 at every tested t (position-space Gaussian image sums < 1e-9; mode-space proof: only n = 0 is gamma-diagonal); [E] Atiyah-Bott match: 4 x 1/4 (det = 4 = |mu4|) and 2 x 1/2 (det = 2 = |Z2|); [E] pillowcase contact term exactly 1/2 = 4 x 1/8 per mark, clock-equivariant trace exactly 1, all t. [E]/[O] the reduction: dtop's analytic content must sit in the per-insertion coupling normalisation (4 = |mu4| insertions of weight c3 -> c3^4, multiplicity Omega_adm = 48; the bare 1/(2pi) 4-cycle differs by the exact rational 3/16) -- the one remaining [O] is the rule 'per-mark weight = c3' (same class as EM.WARD.02). Target narrows from 'derive the term' to 'derive the weight'; stays open."
    },
    {
      "id": "v484_seam_contact_unit",
      "cluster": "core",
      "descriptionEN": "SEAM.CONTACT.UNIT.01: the shared 'c3 per boundary insertion' rule of the two open analytic targets (ALPHA.QUILLEN.EXACT.01 c3-ladder {0,3,6}; HYP.PHI0.PUNCTURE.01 per-mark weight c3) is ONE identity and NOT a new unknown: it is the KMS seam unit 2pi = 1/(4c3) with 1/4 = 1/|mu4| (v239) -- one bare boundary propagator orbit-averaged over the four marks. [E] unit chain 1/(2pi) = 4c3 = |mu4| c3 = |Z2| chi(S^2) c3; [E] mechanism derived on the seam circle: G(d) = -(1/pi) ln|2 sin(d/2)| takes INTEGER multiples of c3 ln2 at the mu4 separations (G_marks = -4 c3 ln2 circ(0,1,2,1), spec {4,-2,0,-2}); the orbit-averaged insertion carries exactly -(c3 ln2) C, so the log-det contact expansion is exactly graded in c3 per insertion (Tr C^k = 4^k + 2(-2)^k integer); [E] the grammar is suite-wide: Lambda prefactor 3/(4pi^2) = 48 c3^2 (v60) = the same Omega_adm = 48 as the phi0 puncture at two insertions; g-2 divides by the same 2pi = 1/(4c3). [C]/[O] the two [O] targets MERGE into one named analytic step (diagonal zeta-renormalisation + multiplicity matching); finite cycle sector proven; nothing closes, no coefficient claimed."
    },
    {
      "id": "v485_contact_diagonal_closed",
      "cluster": "core",
      "descriptionEN": "SEAM.CONTACT.UNIT.02: the diagonal channel of the merged analytic target (v484) settled at the computable level -- the merged residual collapses onto the keystone. [E] diagonal zero at the KMS scale: G_reg(0; l) = (1/pi) ln(l/2pi) exactly, vanishing iff l = 2pi = 1/(4c3) (the v239 KMS circumference) -- the seam scale is the unique zero-self-energy scheme point; [E] the log is the pure scale channel, matching F_U(1)'s single log term (transport 8b1 c3^6 ln(1/phi_seam)) vs the log-free phi0 split; [E] closed-form resummation (BFK route, v151 class): det(I - uC) = (1-4u)(1+2u)^2 with u = eps c3 ln2 -- all orders, linear term absent because the diagonal vanishes (Tr C = 0), series = v484's Tr(C^k); [E] weight dichotomy: the SAME 48 = 3x16 states counted flat give the phi0 multiplicity 48 = Omega_adm, and Y^2/loop/GUT-weighted give the alpha budget 41 = 10b1 = 40 + 1 (Tr_16 Y^2 = 10/3 exact, Ginsparg (3/5)(41/6) = 41/10). [C]/[O] with v484 every finite piece of the merged target is proven; the single remaining [O] is the abstract-seam zeta-det identification = a face of SEAM.EQUIV.01 (the v382 typing, now substantiated). Nothing closes."
    },
    {
      "id": "v486_transfer_full_rule",
      "cluster": "core",
      "descriptionEN": "HYP.REWRITE.02: the FULL physical transfer spectrum {1, (2/3)^6, (1/3)^6} now has ONE local recursive generator -- the lazy Z2-pair walk. [E] uniqueness: the symmetric 3-channel rule M(s,h) (one absorbing family channel + Z2 pair) has survival spectrum {s+h, s-h}; demanding the physical pair {2/3, 1/3} forces uniquely (stay, hop, leak) = (1/2, 1/6, 1/3) = (1/|Z2|, 1/(|Z2| N_fam), 1/N_fam); [E] over the order-6 = 2 N_fam = |R+(A3)| hand it generates eig(B^6) = {64/729, 1/729} exactly -- the previously generator-less (1/3)^6 mode gets its origin (subdominant gap 6 ln 3, alongside the recovery gap 6 ln(3/2)); [E] v327's uniform rule (spectrum {2/3, 0}) can never produce the third mode under any power -- superseded as full-spectrum generator (its 2/3 arity story unchanged); [O] the arity {2,3} stays the anchor input and the lazy-vs-uniform split is selected by the physical spectrum (named candidate: stay = 1/|Z2| as the sheet-symmetric choice). Exact, nothing closes."
    },
    {
      "id": "v487_transfer_clock_rungs",
      "cluster": "core",
      "descriptionEN": "HYP.REWRITE.03: the lazy split (1/2, 1/6) is FORCED by the resummed clock -- the v486 [O] 'split selection' closes onto v124 structure. [E] rung reading: the lazy rule's one-step spectrum is EXACTLY the complete v124 clock ladder below the wall, eig(M) = {1 - n/3 : n = 0,1,2} = {1, 2/3, 1/3}; v327's uniform rule puts its second pair mode ON the wall (0 = 1 - 3/3). [E] deck parity = rung index: the sigma-even mode carries rung 1 (survival 2/3), the sigma-odd mode rung 2 (survival 1/3) -- each clock step flips the deck parity (the structural home of the v486 parity assignment tested by frb-parity-comb). [E] forcing: faithful ladder (definitional for a full-spectrum generator) + Z2 equivariance + rates >= 0 force uniquely both the split (1/2, 1/6) and the assignment even -> rung 1, odd -> rung 2 (the swap needs hop = -1/6 < 0). [E] bend corollary: omega_1/omega_2 = rate(2)/rate(1) = log_{3/2} 3 -- the parity-comb tone ratio IS the v107/v124 bend. [O] channel count 3 forced by the wall; the ladder stays an [I] identity on frozen spectra (R1/[P] open); arity {2,3} stays the anchor input."
    },
    {
      "id": "v488_majorana_clebsch_door",
      "cluster": "flavor",
      "descriptionEN": "FLAV.NUSCALE.04: the 3/5 Clebsch door of the seesaw rung, enumerated and CLOSED -- v482's named escape (a third-generation Majorana factor r ~ 1.6696, 0.18% from g_car/N_fam = 5/3) is decided NEGATIVE at the group-theory level. [E] no renormalisable channel: sym^2(16) = 136 = 10 + 126 and the 126bar is not in the E8 hull (v247) -- M_R only via the d=5 operator (16 16bar_H)^2/Lambda; [E] channel list 16 x 16bar = 1 + 45 + 210 with only singlet + 45 admissible; [E] charge table: (B-L, T_3R, Y)(nu^c) = (+1, -1/2, 0), Tr_16 Y^2 = 10/3; [E] the {2,3}-smooth theorem: every nu^c channel weight is {2,3}-smooth, so no channel combination reaches 5/3 -- and the unique natural 5 of the embedding (k_Y = 5/3, v470) is decoupled because Y(nu^c) = 0 exactly; [E] Clebsches are generation-blind (family-space scalars) -- diag(1,1,3/5) cannot arise from group theory; [O] compiler-flavor 3/5 inventory typed and declined (wrong slots only). The rung + 5/3 rescue is dead; the v481 window candidate stands."
    },
    {
      "id": "v489_seam_modular_commutator",
      "cluster": "core",
      "descriptionEN": "SEAM.CCC.01: the chiral central charge c_- = 8 measured DIRECTLY from the bulk ground state of the v367 seam lattice model, via the Kim-Shehab-Kim modular commutator J(A,B,C) = i Tr(rho_ABC [K_AB, K_BC]) = (pi/3) c_- on a 120-degree pie tripartition -- the first ground-state witness for premise (i) of SEAM.EQUIV.01 (c_- = 8) INDEPENDENT of both the Chern route (v367) and the KLM counting route (v378); everything reduces to the Peschel Majorana covariance (W = -2 artanh Gamma). [E] purity: eig(i Gamma) = +-1 to 6e-15 (exact BdG ground state). [E] per-layer c_- = 0.499998 (L=32, R=12; 0.499793 at L=16 -> 0.500000 at L=48, dev -1.8e-7). [E] controls: trivial M=3 -> 0.000000, orientation flip M=-1 -> -0.499986. [E] layer additivity exact (1.4e-14). [E] 16-layer carrier: direct c_- = 7.9967 (L=16; L=24 gives 7.999781), additivity route 8.0000 = g_car + N_fam. [E] clip invariance 5e-8. [O] honest fence: pins c_- = 8 on the lattice ground state; does NOT close SEAM.EQUIV.01 (continuum limit MMST v336 stays [O], mu4 condensation stays the cited KLM step v378). Numerical (numpy), Python-only by nature."
    },
    {
      "id": "v490_seam_parity_census",
      "cluster": "core",
      "descriptionEN": "SEAM.MU.01: the T^2 spin-structure fermion-parity census for the v367 seam lattice model -- the lattice ground-state witness of 'mu_pre = 4 before condensation' (step 1 of the v378 KLM chain) + the 16-fold-way bridge theta_v = 1 with the measured nu = 2 c_- = 16 (v489). Parity per sector (AA, AP, PA, PP) determined twice: Read-Green TRIM product AND real-space Majorana Pfaffian (Schur), relative to the trivial phase. [E] sanity: twisted real-space spectra == Bloch bands to 9e-15. [E] per-layer census (+,+,+,-), TRIM == Pfaffian in every sector (3 even sectors = Ising nu = 1). [E] 16 layers: parity^16 = + in ALL 4 sectors => gauged torus GSD = 4 = mu(SO(16)_1) = |det Cartan(D8)|. [E] theta_v = exp(2 pi i nu/16) = 1 exactly at nu = 16; rivals nu = 1, 2, 8 excluded. [E] fence: fermionic Kitaev-Preskill TEE = 0 for BOTH M=1 and M=3 -- does NOT discriminate (E8)_1 vs SO(16)_1. [C] KLM step 4 -> 4/2^2 = 1 cited (interacting, not free-fermion computable); numerics pin only {c_- = 8, mu_pre = 4, theta_v = 1}. [O] continuum limit (MMST v336); SEAM.EQUIV.01 stays [O]. Python (numpy + scipy + sympy)."
    },
    {
      "id": "v491_p2_partition_corollary",
      "cluster": "core",
      "descriptionEN": "P2.PARTITION.01: g_car = 5 as a COROLLARY of the four marks -- the partition sharpening of v53 (sympy exact). The lemma: exactly 3 positive integers summing to 4 = uniquely {1,1,2} => e = (4,5,2) => g_car = e2 = 5, |Z2| = e3 = 2, N_fam = e2-e3 = 3 as corollaries; consistency fixed point (2^4-1)/5 = 3 = rank H^1. Negative controls (each assumption load-bearing): drop positivity -> 4 solutions, e2 in {0,3,4,5}; sum 3 -> e2 = 3; sum 5 -> not unique; 2/4 parts -> e2 {3,4}/{6}; Mehta-Seshadri rational weights -> 16/3 resp. 44/9 != 5 (integrality = Birkhoff-Grothendieck splitting-type typing). [E] v53 sharpening: v53 needs all three targets (4,5,2), the partition only e1 = 4. [E] sum rule 4 x (N-1)/2 = 4 iff N = 3; v115 route 4 tr(A0) = 4. Independence: #marks = 4 (v216, P1-side), rank H^1 = 3 (v137/v228), |a|_1 = 4 g_car-free (v115). [C] honest residual: P2 NOT axiom-free -- the weight-typing postulate remains; AX.P2.01 stays a declared axiom; N_fam direction reversal typed as over-determination. Exact (sympy); Wolfram-mirrored."
    },
    {
      "id": "v492_celestial_z4_orbifold",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP1.01: WP1 of the celestial-holographic research contract CELEST.SEAM.01 -- the E8 mu4-glue grading (v128) IS the flat Z4 monodromy of the glue-equivariant celestial chiral algebra on the A3 ALE space C^2/Z4 (Bittleston-Homans-Sharma, arXiv:2305.09451, the Z2/Eguchi-Hanson case; Kronheimer-Nakajima flat connection). [E] inner grading: h = (2,2,2,2,2;0^4) reads the glue class mod 4 on all 240 roots => glue = Ad(exp(2 pi i h/4)) = flat Z4 monodromy; A3-side h' = (0^5;1,1,1,-3) = the diagonal (1,1) glue of Z4 x Z4 (v92/v125); Z2 halfway 120 = SO(16) / 128 = spinor. [E] C^2/Z4 IS the A3 singularity XY = Z^4 (Molien + Hilbert (1+t^4)/((1-t^2)(1-t^4)) exact). [E] equivariant SDYM(E8) sector closes: D[d] = 60(d+1) even / 64(d+1) odd -- works ONLY because dim g_j = (60,64,60,64); zero modes = carrier D5+A3+Cartan = 60; density 1/4 = 1/|Z4|. [E] (E8)_1 character = sum of the 4 glue-coset sectors (1+248q+4124q^2+34752q^3, v377); sector weights = the v92/v125 discriminant form (5x^2+3y^2)/8; glue diagonal h = (0,1,1,1) integer = local extension. [E] correction: the A3 deck acts on the celestial sphere as z -> -z (order 2); the order-4 clock z -> iz is the U(2) phase diag(i,1) (normalises the deck, det = i rotates omega by the mu4 generator), spin bridge (spin clock)^2 = deck, Z8 with 8 = 2|mu4|. [E] clock-invariant A3 deformation = 1-parameter XY = Z^4 + a0, branch points = ONE mu4 orbit, cross-ratio 2 (v168/v214 pillowcase). [E] negative controls: diag(i,i) killed 3 ways; false glue breaks additivity 3112/6720; rigidity = Aut(Z4). [C] verdict B: match under the ONE named identification clock^2 = deck. [O] structural arithmetic only -- no celestial duality constructed; CELEST.SEAM.01 stays a research contract; SEAM.EQUIV.01 stays [O]. Exact (sympy)."
    },
    {
      "id": "v493_celestial_wp2_clock_deformation",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP2.01: WP2 of the celestial-holographic research contract CELEST.SEAM.01 -- the clock-invariant deformation XY = Z^4 + a0 of the A3 seam orbifold (geometry, periods/monodromy, deformed algebra; Bittleston-Homans-Sharma Z2 -> Z4 transfer, arXiv:2305.09451). [E] geometry: P(iZ) = P(Z) forces a3 = a2 = a1 = 0 sharply (two-sided: prod(Z - i^k w) = Z^4 - w^4); smooth iff a0 != 0 (disc = 256 a0^3); branch points = ONE free mu4 orbit; cross-ratio 2 / j = 1728 for ALL a0 (quartic invariants I = 12 a0, J = 0 identically) => a0 = pure SCALE, tau = i pillowcase FROZEN -- no shape modulus. [E] periods: Pi = t(i-1)(1,i,i^2), all 3 spheres same volume sqrt(2) t (lockstep); clock on H2 = COXETER element of W(A3) (char x^3+x^2+x+1, eigenvalues {i,-1,-i} = A3 exponents; h(A3) = 4 = |mu4| = N_fam+1); no invariant cycle (det(A-1) = -4); surviving direction = chi_1-Fourier diagonal (1,i,i^2); the clock IS the Picard-Lefschetz monodromy of the family; versal weights (1,2,3,0) = regular mu4 rep. [E] algebra: mode dictionary {w[m,n]: m = n mod 4} <-> {X^p Y^q Z^r: r <= 3}; bracket -4*Nambu(F) anchored at the a0 = 0 orbifold ({X,Y} = 16 Z^3, {Y,Z} = -4Y, {Z,X} = -4X), Jacobi + Casimir exact, uniform clock-degree -1; S-algebra corrections exactly LINEAR in a0 at the Z4 wrap, grade-conserving (no sector leak => the v492 equivariant SDYM(E8) sector deforms consistently); a0 <-> BHS c^2 slot (weight 0); matrix factorisations deform along all orbit splittings, clock fixes exactly the antipodal 2+2 sector (EH/Z2 middle node, 4 = 2 x 2). [E] negative controls: a1 Z => j = 0 (hexagonal); pure a2 Z^2 => stays singular (A1); a2 != 0 => j = 1556068/81 != 1728; grading leak explicit. [C] three named identifications (verdict B): -4*Nambu = THE k = 4 CCA bracket; period = root difference; seam-scale reading via clock^2 = deck. [C] v216 precision typed: given the order-4 clock the deformation freezes the (2,2,2,2) modulus to the square automatically -- relocation of the same order-4 input, no marker move. [O] full twisted-sector OPEs, W(mu) scaling limit for k = 4, splitting-function check, WP3-WP5; SEAM.EQUIV.01 stays [O]. Exact (sympy)."
    },
    {
      "id": "v494_cosmo_running_mu_record",
      "cluster": "gravity",
      "descriptionEN": "α_s/β_s running + μ-distortion promoted to the suite: [E] LO identities α_s=−2/N★²=−r/6, β_s=−4/N★³ (sympy exact) + exact slow roll α_s(51.4)=−7.09×10⁻⁴, β_s=−2.70×10⁻⁵ (band −8.0…−5.6×10⁻⁴ LO); [C] Planck 2018 +0.57σ, record leg P-ACT-LB −1.33σ (kill: robust POSITIVE running ≥5σ); [C] μ-distortion branch discriminator (Chluba window, +21% LCDM-anchored): sharp 1.6×10⁻⁸ / profiled 1.94×10⁻⁸, band 1.5–2.3×10⁻⁸, split 3.0×10⁻⁹ = 3σ at Voyage-2050 σ(μ)=10⁻⁹ (PIXIE baseline does not reach); [E] the frozen log-comb is structurally blind in the μ window (|F|=0.033, δμ=7.4×10⁻¹² = 0.007σ). N★ stays an external input (GATE.NSTAR.01); band [50,60] stays the record."
    },
    {
      "id": "v495_celestial_wp3_gs_alignment",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP3.01: WP3 of the celestial-holographic research contract CELEST.SEAM.01 -- the Green-Schwarz axion coefficient lambda_g of celestial SDYM (Costello, arXiv:2111.08879) vs the TFPT seam constant c3 = 1/(8 pi); the pre-registered 'alignment test only'. [E] Okubo identity DERIVED, not quoted: Sum <alpha,x>^4 = [5/(2(dim+2))] (Sum <alpha,x>^2)^2 exact for all 8 algebras on Costello's list (e6/e7 carved out of the v492 E8 root set); sl4 negative control fails (5/32 vs 3/32). [E] Killing = 2 h_vee <.,.>; closed form lambda~^2 = 10 h_vee^2/(dim+2) = h_vee + 6 over the Deligne series; Dynkin indices (1,1,2,2,6,6,12,60); E8: lambda~ = 6, lambda_fund = 1/10 EXACT; Costello cross-checks sl2 = 8 / sl3 = 9; so8 slip flagged (exact 3, printed 3/2). [E] Alignment: (kappa/c3)^2 = lambda^2/3 = 12 = |mu4| N_fam resp. 1/300 exact; kappa/c3 itself irrational (2 sqrt3); anchors h_vee = 30 = |Z2| N_fam g_car, phi(30) = 8 = rank, lambda~ = |Z2| N_fam, lambda_fund = 1/(|Z2| g_car). [E] LOOK-ELSEWHERE (hard fence, alignment != selection): alignment 8/8 (zero selectivity), smoothness 8/8, lambda-integrality 2/8 (with sl3), phi(h_vee) = rank 4/8; only g_car | h_vee is e8-selective (1/8); the isolating conjunction is post hoc. [E] K3 does NOT fire but is scope-demoted: compatibility + lambda-arithmetic, NOT e8-selective evidence. [C] the 8 pi is the anomaly 2 pi convention, the sqrt3 the box diagram's sqrt48 -- WP3 can never derive c3. [O] WP5 + SEAM.EQUIV.01 stay [O]. Verdict B. Exact (Fraction/sympy)."
    },
    {
      "id": "v496_celestial_wp4_character_shadow",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP4.01: WP4 of the celestial-holographic research contract CELEST.SEAM.01 -- the (E8)_1 character E4/eta^8 vs the glue-equivariant E8[C^2] S-algebra (the 'type mismatch, head-on'). [E] Base: v492 replication (52+64+60+64, dims (60,64,60,64)); equivariant tower D[d] = 60(d+1)/64(d+1); cumulative count QUADRATIC 31 s^2 + 92 s + 60, 31 = k + h_vee; [C] CP Table 1: spin 1 - d/2 UNBOUNDED BELOW => no rational VOA contains the towers. [E] Hilbert series chi_sp(t) = (60+128t+60t^2)/(1-t^2)^2 (palindrome, 248 at t = 1); all mu4 collapses + the SU(2)_+ slice fail to start (1, 248, ...). [E] Character: chi = E4/eta^8 = (1,248,4124,34752,213126,...) exact through q^6 (cross-check q j = E4^3/eta^24); theta split, sector currents (60,64,60,64), glue-diagonal h = (0,1,1,1) integer; Sugawara c = 248/31 = 8 = 2|mu4|. [E] Bridges: zero-mode slice = 60 currents exact; jet slice cycles (60,64,60,64); BOUNDARY/PERIOD LIMIT: one mu4 period sums EXACTLY to 248 (single-particle level); 248 is NOT a jet-tower dimension (d <= 100); loop Fock level 1 = 897266 >> 248 (rational truncation absent). [E] Mismatch localised THREE ways: (a) grading; (b) growth n^{2/3} vs n^{1/2} (Meinardus pole s = 2 vs 1; f_n/chi_n strictly increasing n = 1..12); (c) the level-2 null ideal deletes EXACTLY 27000 = 30^3 = h_vee^3 (Sym^2(248) = 1+3875+27000; character keeps 4124) -- no jet analogue. [E] K4 three-way: exact realisation NO; boundary-limit PARTIAL YES -- the SAME MMST scaling-limit structure as SEAM.EQUIV.01 (v336/v449); K4 does NOT degrade the contract to 'E8 admissible'. [C] verdict B(ii): the character is a BOUNDARY/LIMIT shadow, not a conformal block in the bulk grading. [O] the constructive limit -> WP5; SEAM.EQUIV.01 stays [O]. Exact (integer/Fraction/sympy)."
    },
    {
      "id": "v497_celestial_wp5a_null_ideal",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5A.01: WP5a of the celestial-holographic research contract CELEST.SEAM.01 -- 'the null ideal from the limit', first constructive milestone of WP5 (the v496 boundary-limit shadow made precise). [E] Base: v492/v496 replication (glue dims (60,64,60,64), tower D[d], chi = E4/eta^8 through q^6, mu4 period sum 248). [E] STABILISATION THEOREM: family chi_w of graded Fock characters on the chiral jet generators (loop energy m + w*radial number r, quarter units u = q^{1/4}); w = 2 IS the chiral jet grading (generators 64,120,128,180,192,240,256,300 per degree); u^n coefficient of chi_w = quarter-moded loop Fock for ALL w >= n+1, strictly larger for w <= n (n <= 8, w <= 10) -- the boundary reading is the precise coefficientwise limit eps = 1/w -> 0 with explicit threshold w = n+1; convention: 897266 = loop Fock at u^4 = integer level 1. [E] NULL IDEAL DERIVED (not cited): Freudenthal + Weyl + character peeling give Sym^2(248) = V(2 theta) + V(omega) + V(0) = 27000 + 3875 + 1 (residual exactly 0, orbit-sum cross-check); 27000 = 30^3 = h_vee^3; C2 = (124, 96, 0), Dynkin indices 6750/750; level-1 integrability <2 theta,theta_vee> = 4, <omega,theta_vee> = 2 => only level-1 primary = vacuum ([C] Kac, counting consequence); quotient 31124 - 27000 = 4124 = 1 + 248 + 3875 exactly. [E] TWO ROUTES, ONE NUMBER: mu4 theta-split sector sum at q^2 = (1036,1024,1040,1024) = 4124 -- independent lattice route hits the Fock quotient exactly; q^3: 34752 = chi_3. [E] NEGATIVE CONTROLS: SO(16)_1 through the SAME pipeline: Sym^2(120) = 5304+1820+135+1 (FOUR components), 5304 != 14^3 (h_vee^3 not generic), quotient 7380-5304 = 2076 = Theta_D8/eta^8 at q^2 (second algebra validated), block weights (0,1/2,1,1) do NOT fuse vs E8 (0,1,1,1) fusing; false periodisation: only P = 4 = |mu4| gives 248; swapped glue breaks the zero-mode anchor (64 != 60). [C] identifications: 'radial decoupling = MMST collar thickness -> 0' (v336/v449 same two-step shape); Kac singular vector via counting consequence only. [O] HONEST LIMIT: the limit does NOT generate the truncation -- singular vector as operator in the limit net open (WP5b); GNS limit state, local normality/Haag duality open (WP5c/d); the celestial route now has the same two-step shape as MMST (limit + maximal ideal) with the ideal quantitatively fixed; SEAM.EQUIV.01 stays [O]. Exact (integer/Fraction)."
    },
    {
      "id": "v498_celestial_wp5b_singular_vector",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5B.01: WP5b of the celestial-holographic research contract CELEST.SEAM.01 -- 'the singular vector as an operator', the second constructive milestone of WP5 (WP5a fixed the ideal's size and location; WP5b constructs the deleting object). [E] CONSTRUCTION: |s> = (E^theta_{-1})^2|0>, theta the highest root, weight 2 theta (top of the 27000), level 2 = 8 quarters = q^2 (INTEGER); Chevalley/Frenkel-Kac basis MACHINE-built (bimultiplicative 2-cocycle, asymmetry on all 57600 pairs; [e_a,e_{-a}] sign FORCED by Jacobi, SGN = -1; kappa derived, kappa(theta_vee,theta_vee) = 2; Jacobi exact on 57600 sl2-slice triples + 6000 seeded + all 496 theta-adjacent; all N_ab = +-1). [E] SINGULARITY: J^a_1|s> = 0 for ALL 248 basis elements, 0 violations; case tally 190 (first bracket) / 57 (second) / 1 (F^theta via central-term cancellation -- the ONLY case that sees k); plus J^a_2|s> = 0 (248), E^a_0|s> = 0 (120), Shapovalov norm 0 at k = 1 (4 at k = 2); affine PBW engine unit-tested on all 61504 basis pairs. [E] LEVEL DIAL: F^theta_1 coefficient = 2(1-k) -- 2 at k = 0, 0 EXACTLY at k = 1, -2 at k = 2: without the central extension the deletion object does not exist. [E] MU4 DATA: theta-grading dims (1,56,134,56,1), g0 = e7+u(1); h-adapted chamber machine-built (Cartan det 1, E8 Dynkin diagram, branch lengths [1,2,4]); <theta,h> = 5 => j(theta) = 1, height 29 = h_vee-1; |s> in class 2 (sheet-even, WP5a consistent); clock phase i^2 = -1; cross-table theta-grading x glue class consistent (rows (1,56,126,56,1), columns (52,64,60,64)); the deleting theta-sl2 sheet-odd (1,3), its square sheet-even. [E] LEVEL-2 MATCH: Mult(2 theta) = 1 in the level-2 Fock => U(g)|s> is THE 27000; direct g0-orbit BFS reproduces the Freudenthal multiplicities of V(2 theta) exactly through depth 4; quotient 31124 - 27000 = 4124; lattice reading (2theta,2theta)/2 = 4 > 2 => no 2theta state at q^2 in E4/eta^8 -- |s> IS the kernel generator of Fock -> E4/eta^8. [E] NEGATIVE CONTROLS: E^theta_{-1}|0> NOT singular (1 witness F^theta, -> -k|0>) -- the 248-current state that must remain; 4 generic level-2 states non-singular (14-33 witnesses); k-dial: at k = 2 only the CUBE is singular ((E^theta)^{k+1} mechanics = generic Kac); D8/SO(16)_1: the singular vector exists there TOO (level-1 generic) but the h = 1/2 blocks do not fuse (3 comark-1 weights, dim V(2 theta_D8) = 5304 != 14^3) vs E8 (comarks >= 2, vacuum the only level-1 primary, glue weights (0,1,1,1) integer) -- one-block closure is the E8/mu4-specific part, honestly separated. [C] Weyl complete reducibility beyond BFS depth 4; Kac mechanics beyond level 2; per-period quarter dictionary (v496). [O] HONEST LIMIT: in the twisted quarter-slot moding two C_1 modes never sum to 8 quarters (minimum 6 = q^{3/2}) -- the per-PERIOD dictionary (v496), not the per-slot identification, carries |s> to q^2; exactly the precise WP5c question (GNS limit state, kernel containing the ideal); WP5c/d/e stay [O]; SEAM.EQUIV.01 untouched. Exact (integer/Fraction)."
    },
    {
      "id": "v499_p2_weights_deligne_bg",
      "cluster": "core",
      "descriptionEN": "P2.TYPING.01: the P2 weight-typing postulate HARDENED (the v491 check-9 residual) -- the anchor a = (1,1,2) typed as the Birkhoff-Grothendieck splitting exponents of the DELIGNE CANONICAL EXTENSION E of the flavor connection. Given the [E] anchors (4 marks v216, rank 3 v137, cusp class + irreducibility v117, (U)): deg E = -4 AS THEOREM (residue spectrum {0,1/3,2/3} unique in [0,1), trace 1 per mark x 4 marks; par-deg 0; infinity apparent, monodromy product = 1 exactly); POSITIVITY AS THEOREM (irreducibility exact: |<U,M0>| = 24 = |S4|, sum |chi|^2 = 24; + (U) => Mehta-Seshadri stability => h^0(E) = 0, all 324 saturation cases; BG dictionary h^0 = 0 <=> a_i >= 1); INTEGRALITY = Birkhoff-Grothendieck, automatic once E is a bundle. UNIQUENESS + SELECTOR: {1,1,2} unique (v491); Schur-Horn permutohedron integer points exactly {(2,2,0),(2,1,1)} and h^0 = 0 kills {0,2,2} -- the tfpt_2 'balanced' heuristic now TYPED as stability. h^1 DISCREPANCY REAL + RESOLVED: h^1(E) = 1 != 3 -- 'generations = H^1(E)' is FALSE (forcing it: deg -6, uniqueness lost, e2 in {9,11,12} never 5); two-object typing: L = O(-D) carries the generations (h^1(L) = 3 = N_fam = v137 basis), E carries them as its FIBER. BISWAS EXPLICIT (independent witness): Z3 cusp cover y^3 = (z-1)(z-i)(z+1)^2(z+i)^2, genus 2 (NOT the pillowcase double, genus 1 j = 1728 = deck side); p_* O_Y = O + O(-2)^2; the regular-rep model lands on {0,2,2} = the UNSTABLE Schur-Horn companion (h^0 = 1): integrality + sum rule COME OUT of the correspondence, stability is the selector. Negative controls: no positivity -> 4/31 solutions; deg -3/-5 breaks; n-mark scan e2 = (n-2)(n+1)/2 = 5 ONLY at n = 4; MS-rational e2 = 16/3, 44/9. [C]/[N] residuals: R1 module identification (QGEO.SYM stays [O]); R2 unitarity (U) (v33/v40); R3 analytic monodromy [N] (v117 check 4). P2 stays an axiom (AX.P2.01 declared, no marker moves); non-circular (no v23/v115 splitting input). Exact (sympy); Wolfram-mirrored."
    },
    {
      "id": "v500_celestial_wp5c_gns_limit_state",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5C.01: WP5c of the celestial-holographic research contract CELEST.SEAM.01 -- 'the GNS limit state', the third constructive milestone of WP5 (WP5a made the limit precise, WP5b built the deleting operator; WP5c constructs the STATE whose GNS kernel contains the ideal). Exact integer/Fraction, no floats, deterministic. [E] CONSTRUCTION: quasi-free family omega_w -- loop sector = affine k=1 vacuum functions via the MACHINE-determined compact anti-involution theta(e_a) = -e_{-a} (the UNIQUE sign that is an anti-automorphism on all 61504 basis pairs; contravariance exact); radial sector = oscillator pairings x^{w r} (exact Gibbs regulator, x-adic). [E] CONVERGENCE SHARP: omega_w = omega_inf mod x^{N+1} for ALL w >= N+1 (N = 8), NOT yet at w = N (r = 1 modes at order N) -- the WP5a threshold holds identically at the state level; positivity of every omega_w checked. [E] KERNEL (core, fully machine): FULL exact level-2 Gram -- 31124 monomials in 9361 weight blocks (profile 164:1, 37:240, 7:2160, 1:6960), each block exact: level-1 rank 248 positive definite (Cartan block = E8 Cartan, det 1); level-2 rank 4124, kernel 27000, every block PSD; rank per Weyl orbit (norm 8/6/4/2/0) = (0,0,1,8,44); kernel = V(2 theta) WEIGHT BY WEIGHT (Freudenthal cross-check on all 9361 weights); h.w. norms (3844, 49, 2, 0) at (0, omega, theta, 2theta); direct ideal checks F_i|s>, F_j F_i|s> norm 0. [E] MU4 EQUIVARIANCE: omega_w o clock = omega_w for all w; the clock descends to GNS; GNS level-2 rank by clock class = (1036,1024,1040,1024) = Theta_Cj/eta^8 at q^2 (two-routes identity at the STATE level); level-1 split (60,64,60,64). RESOLUTION of WP5b S4.10: |s> IS the ZERO VECTOR of GNS(omega_inf) -- the twisted two-C1-mode floor (6 quarters = q^{3/2}) concerns only pre-limit objects (GNS has no state there, 6 mod 4 != 0). [E] NEGATIVE CONTROLS: k = 2 (<s|s> = +4, dominant blocks full rank -- no ideal in the kernel); k = 0 (level-1 Gram = 0 -- no 248 layer without the central extension); D8 (level-2 rank 2076 = 7380 - 5304, kernel = V(2 theta_D8) PSD, but h = (0,1/2,1,1): vacuum GNS is 1 of 4 blocks -- one-block completeness stays E8/mu4-specific); wrong family E'_w = m w + r (level-1 limit rank 0, the 248 layer ERASED, u^4 content 710955 != 248 -- the kill has teeth); no damping (897266 != 248); CCR OBSTRUCTION: at FIXED radial pairing c > 0 positivity forces omega(bb*) >= c -- NO w-uniform damping state exists: the family formulation is NECESSARY and the family EXISTS => preregistered KILL not triggered. [C] quasi-free extension to the full *-algebra, Kac positivity beyond level 2. [O] local normality of omega_inf, net structure/Haag duality (WP5d), Costello-Li (WP5e), operator-algebraic closure beyond the graded topology; SEAM.EQUIV.01 stays [O]. VERDICT: SUCCESS on the preregistered criterion (dim GNS level-2 = 4124 exactly). Exact (integer/Fraction)."
    },
    {
      "id": "v501_celestial_wp5d_two_interval_index",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5D.01: WP5d-alpha of the celestial-holographic research contract CELEST.SEAM.01 -- 'the two-interval index from the lattice' (the local-net question, alpha stage). numpy/scipy Gaussian machinery + exact Fractions. [E-machine]/[L] SETUP: critical Majorana ring (v450 style), 16 layers = seam carrier c_- = 8; Peschel-exact; three prescriptions (fermionic / orbifold sector sum {1,P_A,P_B,P_AB} / dual algebra {1,P_AB}); superselection [rho, P_AB] = 0 machine-exact; every Gaussian formula validated against exact diagonalisation (<= 1.5e-15). [L/N] FERMIONIC MI EXTENSIVE (mu = 1 reference): c fit 0.5000, max |F(x)| = 5.97e-5 (N = 512), |F| falling monotonically to 1.3e-6 (N = 1024) -- Casini-Huerta extensivity as a lattice witness. [L/N] THE MU-OFFSET: Renyi-2 exactly Delta_2 = -ln[(1+R)/2]; the 16-layer carrier saturates ln 2 to MACHINE PRECISION (|Delta_2 - ln 2| = 1.1e-15 at N = 512; R falls as N^-0.500); index reading [F : F_even] = 2.000000, mu_gauged = 4.000000 = the v490 census (two independent lattice witnesses); per-layer gauging costs 16 ln 2 (the offset measures ln[index], not a blob). [L/N] HAAG-DUALITY INDICATOR: fermionic S(E) = S(E') to 8.2e-12; the orbifold BREAKS complementarity (D(E) - D(E') = 0.427 at a = 64); the pair sum is exchange-symmetric and <= ln 4 with max 1.2637 at x = 1/2 -> the ln 4 budget is a DOUBLE-LIMIT statement (the first naive fixed-x check was wrong and was replaced -- honestly typed). [E] CONDENSATION ARITHMETIC: det Cartan(D5) x det Cartan(A3) = 16; KLM/Longo-Rehren 16/4^2 = 1 and 16/2^2 = 4 -> 4/2^2 = 1 (both routes exact); Sigma d^2 (Ising 4, SO(16)_1 4, (E8)_1 1); theta_v = 1 exactly at nu = 16, rivals != 1. Anchored chain: mu = 1 (graded, MEASURED) -> 4 (gauged, MEASURED + v490) -> 1 (arithmetic, [C]; the condensed net is interacting -- v490 fence). [E/N] NEGATIVE CONTROLS: nu = 1 single Majorana (offset non-trivial, theta_v = e^{i pi/8} not removable -- the kill discriminator has teeth); trivial phase (everything <= 1e-7); wrong sector sum {1,P_AB} (offset exactly 0 instead of ln 2). [C] the dictionary 'offset = ln index' (KLM/Longo-Xu/CHMP) + the condensation step. [O] continuum Haag duality, local normality, split/strong additivity (WP5d-beta runs as exploration), Costello-Li (WP5e); SEAM.EQUIV.01 stays [O]. VERDICT: the mu = 1 indicator chain PASSES; preregistered KILL ('mu-offset != 0 after condensation') not triggered. Python (numpy + scipy + Fractions)."
    },
    {
      "id": "v502_celestial_wp5e_prefactor_level",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.ALPHA.01: WP5e-alpha of the celestial-holographic research contract CELEST.SEAM.01 -- 'prefactor and level bookkeeping', the CFT-side anchor stage of WP5e (the BCOV/Costello-Li twistor uplift: boundary partition function = E4/eta^8 INCLUDING the q^{-1/3} prefactor; preregistered kill: the anomaly inflow demands a level != 1). Exact sympy/Fraction, no floats. [E] PART A -- THE q^{-1/3} PREFACTOR IS EXACT mu4 VACUUM-ENERGY BOOKKEEPING: Hurwitz-zeta derivation zeta(-1,theta) = -B2(theta)/2, E_b(theta) = -1/24 + theta(1-theta)/4 exact; the clock is INNER (h = (2,2,2,2,2;0^4) and h' = (0^5;1,1,1,-3) BOTH read the class mod 4 on all 240 roots; (h,h) = 20, (h',h') = 12, sum 32) => twist distribution theta = 0^8 in ALL FOUR sectors -- a SHIFT orbifold (lattice cosets, integer-moded oscillators), NOT a rotation orbifold; E_osc = -1/3 = -c/24 sector-INDEPENDENT (c = 8 = rank = Sugawara 248/31); sector ground exponents -1/3 + (0,1,1,1). [E] DISCRIMINANT FORM = CASIMIR ENERGY, TWO ROUTES: spectral flow j^2 (h,h)/32 (5j^2/8, 3j^2/8, sum j^2 = 0 mod 1 = isotropy/Lagrangian) and EXACTLY via free fermions (D5 = 10 Majorana, A3 = D3 = 6, glue diagonal = 16 = the WP5d seam carrier 2^(g_car-1); R-NS shift n/16 = 5/8, 3/8, 1); the ROTATION reading FAILS (the geometric A3 deck gives 3/16 != 3/8: the v492 'clock != deck' lesson at the Casimir level). [E] CHARACTER: sum of the four Theta_Cj = E4 = (1,240,2160,6720); the four sector characters (leading coefficients (1,64,60,64), currents (60,64,60,64)) sum EXACTLY to E4/eta^8 = q^{-1/3}(1, 248, 4124, 34752). [E] PART B -- k = 1 FORCED THREE WAYS: (i) CURRENT CONDITION h(J) = k = 1 (the Z4 extension closes onto the E8 adjoint 60+64+60+64 = 248 only at spin 1); HONEST SHARPENING: glue-h integrality h(J^j; k) = k(0,1,1,1) holds for ALL k = 1..8 -- the originally conjectured integrality route fixes NOTHING, the naive dial is dead, the three other dials carry; (ii) CONFORMAL EMBEDDING 47k^2 + 219k - 266 = 47(k-1)(k+266/47) = 0 resp. D8-in-E8: 128k(1-k) = 0; (iii) CENTRAL CHARGE 248k/(k+30) = 8 <=> 240(k-1) = 0 (= the prefactor itself); plus the WP5b singular-vector dial: PBW norms m! k(k-1)...(k-m+1), singular vector exactly at grade k+1, k = 1 deletes V(2 theta): 31124 - 27000 = 4124 (Weyl dims 248/3875/27000 from root data). [E] NEGATIVE CONTROLS: D8/SO(16)_1 -- same rank, same c = 8, SAME q^{-1/3} prefactor, but h = (0,1/2,1,1): the discriminator is glue-h INTEGRALITY, not the prefactor (Theta_D8 + Theta_s = E4); Z2 ROTATION gives E = +1/6 (h = 1/2, no mu4-shift sector); mu4 ROTATION makes E_j = (-1/3, 1/24, 1/6, 1/24) sector-DEPENDENT (the common prefactor breaks); wrong k = 2, 3, 4 fail ALL FIVE dials coherently (residuals 360/814/1362; <s|s> = 4/12/24; level-2 dim stays 31124 != 4124). [C] Kac generation of the maximal submodule by the grade-(k+1) singular vector for k >= 3 (k = 1, 2 machine-anchored by WP5b/WP5c). [O] the BCOV/Costello-Li twistor inflow itself (Costello's one-loop axion lambda_g/(8 pi sqrt 3) on PT, the Costello-Paquette-Sharma level-from-flux mechanism) stays OPEN -- v502 anchors the CFT side of WP5e only (alpha stage), the KILL branch ('inflow demands a level != 1') is NOT triggered on the CFT side but its twistor side is untested; WP5e stays [O], SEAM.EQUIV.01 untouched, no marker moves. Exact (sympy/Fraction)."
    },
    {
      "id": "v503_qgeo_emergence_light",
      "cluster": "frontier",
      "descriptionEN": "QGEO.EMERGE.LIGHT.01: the v215 deferred emergence test (lever 7) EXECUTED in its first honest LIGHT version, plus the RESIDUAL-CONVERGENCE result QGEO-R1 == P2-R1 (analysis/kill-test class, v347/v264 style; NO marker moves -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom). [N] NO DYNAMICAL SELECTION (zeta-log-det via the Dedekind-eta formula, mpmath 40 digits): logF = log(tau2 |eta|^4) exactly S-invariant (< 1e-35); on the rectangular family tau = it a strict local MAXIMUM (g'(1) = 0, g''(1) = -1.298) but a MINIMUM in the tau1-direction (h''(0) = +0.298): tau = i is a SADDLE of the full moduli space; critical points = the CM points (E2*(i) = E2*(rho) = 0 to 40 digits, generic 0.163 != 0); the GLOBAL det' winner at fixed area is the HEXAGONAL torus (logF(rho) = -1.03352 > logF(i) = -1.05469, diff 0.021169; Osgood-Phillips-Sarnak 1988) -- free-field dynamics alone does NOT select tau = i; criticality at the square = Palais symmetric criticality (the independent heat trace is equally critical). [N] SHARP RIGIDITY (v280 lattice DtN, N = 8): the mod-4 off-character indicator (the v215-K3 lever as a MODULUS FUNCTION) vanishes to 8e-15 EXACTLY at tau = i and rises strictly monotonically in both directions (min 8.4e-2 for t != 1) -- an isolated zero, not a flat valley; R^2 = Z2 exact as permutation matrices (the v492 Z8 bridge on the lattice); j(it) >= 1728 with equality ONLY at t = 1, j(rho) = 0. [E] THE Z2 CHAIN IS MODULUS-BLIND: the involution has exactly 4 fixed points and [Z2, Delta_t] = 0 on EVERY torus (v216 confirmed without mark input); pillowcase det' = 1/2 torus det' EXACTLY (+- pair structure, ratio 1 to 1e-12) so all tau conclusions transfer verbatim; [Z2, H_t] = 0 for all t -- marks / mark-locality / deck^2 exist on EVERY pillowcase, ONLY the order-4 clock distinguishes tau = i; Aut(E_rho) = Z6 (element orders 1,2,3,6) contains NO order-4 element -- dynamics and clock are logically INDEPENDENT selectors; GIVEN the clock, P(iZ) = P(Z) forces a3 = a2 = a1 = 0 and j = 1728 identically (v493 mini): the modulus is NOT a second input. [C/O] RESIDUAL CONVERGENCE (the core finding): the order-4 element on H^1 is a COXETER element of W(A3) (char poly lam^3+lam^2+lam+1, order h(A3) = 4 = |mu4| = N_fam + 1) -- exactly the P2 cusp-class/carrier datum of the P2 hardening (v499 R1) => QGEO-R1 == P2-R1: ONE common residual, 'why does the raw seam carry the order-4 clock', not two scattered ones; chain: (i) mark count 4 derived (v216 [E]), (ii) square derived GIVEN the clock (v493 [E]), (iii) clock = carrier input [C], (iv) rest = the dynamical realisation of the clock [O]; route classification (v347 analogue): B-rigidity in the sharpened form 'RP seam data => order-4 clock' is the live route -- free dynamics alone does NOT deliver the clock (the hexagonal finding). LEVER UPDATE (documented here, v215 itself unchanged): v215 lever 7 lifted from 'deferred, not executable' to 'light version EXECUTED, result: rigidity, not dynamical selection' -- marks emerge, the square does NOT emerge dynamically, it is pinned by the clock, and the clock is the input. Python (mpmath 40 digits + numpy/scipy lattice + sympy exact); the eta/E2*/j CM identities, the pillowcase halving and the Coxeter charpoly are Wolfram-mirrored, the lattice DtN curves stay Python-only (flagged)."
    },
    {
      "id": "v504_celestial_wp5d_klm_completeness",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5DB.01: WP5d-beta of the celestial-holographic research contract CELEST.SEAM.01 -- 'split + strong additivity from the lattice', the two remaining KLM legs of complete rationality (split + strong additivity + finite mu-index; the index is v501). numpy/scipy Gaussian machinery + exact GF2/integer algebra; the U(1)/Dirac number-conserving subnet = the preregistered negative control. [E] STRONG ADDITIVITY ALGEBRAICALLY EXACT: 'touching' = shared boundary Majorana => Even(A) v Even(B) = Even(A u B) EXACTLY (GF2 spans full 64/64, 256/256, 512/512, 1024/1024; matrix rank 32/32); disjoint exactly half (rank 16/32, index 2 -- the missing sector odd(x)odd = the ln-2 bit of v501, localised at the split point); U(1)/neutral algebras do NOT generate the union even WITH the shared site (gaps 2/10/52 growing). [N] ENTROPIC WITNESS: fermionic touching MI diverges with exact (c/3)ln2 increments (0.115525, dev 1.2e-6 -- pure UV reference); the Z2/orbifold sector-sum deficit is BOUNDED < ln 2, monotone, (ln2 - Delta_2) ~ N^(-p) with p = 0.2444 vs 2 Delta_mu = 1/4 (Ising disorder exponent); split-point scan reflection-symmetric 3.9e-14; ED vN same direction; gap continuity to the v501 x -> 1 endpoint; honest note: 'defect -> 0' would be FALSE at the sharp lattice split -- bounded <=> finite index (Longo-Xu) is the correct statement; U(1) CONTRAST: D_U1 bursts ln 2 from L = 128 and grows as (1/2) ln Var Q_A (Klich-Levitov 0.10134 vs 1/pi^2 = 0.10132). [N] SPLIT/NUCLEARITY: sigma_k of the off-diagonal block fall exponentially with rate = elliptic nome pi K(1-x)/K(x) to 1.3-2.0% (d = 16/64/256/512: rates 2.29/3.24/4.82/5.81); sigma_1 ~ x^0.5044 (vs 1/2, the dimension-1/2 two-point law); trace norm summable, falling (Buchholz-Wichmann proxy). [E] ORBIFOLD INHERITANCE EXACT: P_A: C -> -C (sigma_k identical 8.3e-17); the even-bilinear coupling Gram = the second compound Lambda^2 C with singular values {sigma_a sigma_b} (2.8e-14) -- same ladder, doubled rate; orbifold MI < 3% of the fermionic, falling (Longo heredity [C]). [E] PIMSNER-POPA: E(a) - a/2 = PaP/2 IDENTICALLY => lambda = 1/2 = 1/[F:F_even] EXACT; attainment in integers (all 16384 monomial pairs N = 8: 8192 extremal with M^2 = 4/Tr M = 0/PMP = -M, 0 violations; N = 12: 2048/2048; random pencil min lambda = 0.5000000000); two-interval group lambda = 1/4 = 1/mu exact; INDEX CONSISTENCY of two routes (exp(Delta) = 2.000000000 = 1/lambda_PP; exp(2 Delta) = 4 = 1/lambda_E4 = mu). [E/N] NEGATIVE CONTROLS: U(1) PP lambda = 1/(m+1) -> 0 (infinite index); dimerised trivial (deficit N-independent 3.6e-15, sigma_1 exponential); U(1) small-L honesty; Z2 finite-size honesty (14.9% below ln 2 at N = 4096 -- witness, not proof); wrong prescription {1, P_AB}: deficit exactly 0. [C] the lattice -> continuum KLM dictionary (Longo-Xu, CHMP, Longo heredity). [O] the continuum theorem 'finite-group orbifolds of completely rational nets are completely rational' is XU'S THEOREM -- cited, not claimed; the proof for the concrete seam quotient net and the interacting condensed (E8)_1 net stays WP5e/Costello-Li; SEAM.EQUIV.01 stays [O]. VERDICT: with v501, all THREE KLM ingredients of complete rationality are witnessed on the lattice."
    },
    {
      "id": "v505_celestial_wp5e_beta_equivariant_ledger",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.BETA.01: WP5e-beta of the celestial-holographic research contract CELEST.SEAM.01 -- 'the equivariant anomaly ledger on twistor space', the twistor-side milestone of WP5e (the BCOV/Costello-Li twistor uplift; preregistered kill: the anomaly inflow demands a level != 1); WP5e-alpha (v502) anchored the CFT side. The Atiyah-Bott/Lefschetz fixed-point skeleton of the one-loop box anomaly (Costello arXiv:2111.08879; Bittleston-Sharma-Skinner CMP 403) evaluated EXACTLY on C^2/Z4 with the E8 mu4-glue monodromy. Exact sympy/Fraction, no floats. [E] AB LEDGER: denominators det(1 - g_j^{-1}) = (2, 4, 2); sum of reciprocals 5/4 = (|Z4|^2-1)/12 (Dedekind); equivariant characters tr_{g_j}(Ad E8) = (248, 0, -8, 0) by TWO routes (graded dims / root sum); invariant average 60 = the D5+A3 carrier; Frobenius 61568; tr_{g_2} = -8 = 120 - 128 (so16 supertrace). [E] OKUBO PER SECTOR (honest sharpening): the invariant sector is exactly Okubo-quadratic (Q^(0) = 36<x,x>^2, 36 = lambda~^2_e8 -- v495 re-derived through the glue grading); the TWISTED sectors do NOT reduce (independent-quartic content T5: (-8, 32, -8), T3: (48, -64, 48)); the AB-weighted sum cancels the D5 quartic EXACTLY and leaves the RIGID residual 32 T3; no reweighting of the twisted sectors alone is quartic-free, the only quartic-free weighting is the uniform Z4 projector; the graded GS-exchange ansatz is exactly rank-obstructed in sectors 1, 2, 3; heterotic identities exact (Tr_120 = 8tr^4 + 3(tr^2)^2, Tr_128 = -8tr^4 + 6(tr^2)^2, Tr_E8 = 9(tr^2)^2). [E] INDEX BRIDGE (centrepiece): f(m) = (1/4) sum_j (i^{jm}-1)/det_j = (0, -3/8, -1/2, -3/8) = ch2(T_m) = -(C^{-1})_mm/2 EXACTLY -- fixed-point ledger = McKay/Kronheimer intersection ledger; glue-bundle ch2 defect -78 by BOTH routes (per class (-24, -30, -24); -30 = -h_vee, a [C] coincidence); bridge identity sum_m h_A3(m) = 5/4 = sum_j 1/det_j; sphere pairing <alpha,h> integer on all 240 roots, Killing sums (320, 320, 240, 320), total 1200 = 2 h_vee (h,h). [E] k = 1 GEOMETRIC: lattice current count (norm-2/k in the glue cosets) = 240 at k = 1, EXACTLY 0 at k = 2, 3, 4 (no adjoint closure); embedding residual 47k^2+219k-266 = (0, 360, 814, 1362); h(J;k) = k integer for ALL k -- integrality alone fixes nothing (honest, as v502); one scale => one level (v493 lockstep periods, det(A-1) = -4: no invariant flux vector). [E] E3 REFUTED (prominent): a0 in O(8) fills the GRAVITON slot O(2), NOT the BSS axion slot O(-2) (Hamiltonian weights X = -2 != +2 = Y; mismatch 4 = |mu4|, a [C] coincidence) -- 'the theory brings its GS axion as a0' is REFUTED; instead the three H2(ALE) classes carry exactly the Coxeter eigencharacters {i, -1, -i} = the three twisted ledger slots (bijection, no invariant class); slot count 1 + 3 = 4 = |mu4| soft. [E] NEGATIVE CONTROLS: diag(i,i) breaks the ledger four ways (sum 1/det = 1/4, omega-weight -1 not CY, 5 invariant-ring generators); SO(16) glue: characters (120, 0, +120, 0), defect -30 != -78, bulk Okubo FAILS (T5 = 16, T3 = -32: independent quartic Casimir, no single-axion GS); k = 2: current count 0, c-prefactor -31/48 != -1/3 (integrality dials stay integer -- the kill is the closure dial). [C] dictionaries (CS(rho) mod 1 = discriminant weight; 32 = (h,h)+(h',h'); -30 = -h_vee; the Omega-contraction identification). [O] the global BCOV/Kodaira-Spencer quantisation on PT/Z4, non-fixed-point contributions, the O(-2) bulk-axion construction, CPS level-from-flux, and the pairing of the 32 T3 residual with the three sphere axions. VERDICT B; the preregistered kill ('inflow demands a level != 1') does NOT fire on the equivariant skeleton; SEAM.EQUIV.01 untouched, no marker moves. Exact (sympy/Fraction)."
    },
    {
      "id": "v506_seam_clock_rigidity",
      "cluster": "core",
      "descriptionEN": "SEAM.CLOCK.RIGIDITY.01: clock rigidity, Route B of the v503 classification -- two exact theorems reducing the order-4 clock input to ONE alignment bit (Part A Moebius + Part B Fock in one module; QGEO.REALIZE.01 stays [C]/[O], NO marker moves). Exact sympy + exact Cl(16) Fractions, no floats. [E] THEOREM A (Moebius): the collar deck z -> -z has in PGL2(C) EXACTLY TWO square roots (z -> +iz, z -> -iz), both AUTOMATICALLY of order 4 (complete case analysis: tr != 0 => diagonal => a/d = +-i; tr = 0 impossible) -- 'order 4' is NO input anymore; a marks-preserving square root exists <=> the deck is the CENTRAL involution of the mark-stabiliser D4 (center = {id, deck}, exactly 2 order-4 elements = the clocks) => then marks = one mu4 orbit, cross-ratio class {2, 1/2, -1}, j = 1728, tau = i, free 4-cycle, deck = clock^2, only the scale free; COUNTEREXAMPLE (mandatory): {+-1, +-(3+2sqrt2)} is deck-free AND harmonic (j = 1728) with full D4 stabiliser, but the deck sits in it non-centrally (edge involution) => 0 roots -- harmonicity alone is NOT sufficient; the RP filter selects nothing (both real structures pass both roots: the Aut(Z4) ambiguity is irreducible, consistent with v492). [E] B3/B4: a free Z_n orbit forces n | 4; mark transitivity forces n = 4; faithfulness kills n >= 5; n = 3 fixes a mark; n = 6 dies independently at cusp degree 3 (v117); the deck is modulus-blind (v503 B5 exact); controls: n = 8 has only orbits of size 8 (impossible on 4 marks); hexagonal A4 (order 12, ZERO order-4 elements, squares only orders {1,3}); generic Klein V4 (exponent 2 -- not even the deck has a root); stabiliser trichotomy V4/D4/A4 with order-4 counts (0, 2, 0). [E] THEOREM B (Fock, the core result): the unique S^2 spin structure gives NS on every circle (established P1 datum); one-particle: the NS half-rotation has D^2 = -1 (charpoly (lam^2+1)^8), the quarter-shift C has C^2 = D, C^8 = 1 (charpoly (lam^4+1)^4); Fock: Utilde = prod_j(1 - gamma_j gamma_{j+8}), Utilde^2 = 256 gamma_1...gamma_16 => U^2 = (-1)^F EXACTLY, and (cU)^2 = c^2 (-1)^F is NEVER 1 (center of Cl(16) = scalars) -- the Z4 lift is FORCED (nonsplit extension 1 -> Z2^{(-1)^F} -> Z4 -> Z2^{deck} -> 1); Z8 tower: V^2 prop Utilde, V^4 prop (-1)^F, projective order 8 = 2|mu4| -- fermionically explained; R control: U_R^2 = +1 (split -- the forcing rests on NS = the established geometry); 16-fold way: theta_v = 1 at nu = 16; robustness: the forcing already holds at 8 sites. [C] THE ONE REST: 'the collar deck is the central involution of the mark-D4' (<=> square-root existence on the mark data) -- ONE alignment bit; the fermionic Z4 forces the ORDER, not the spatial alignment. No overclaim. QGEO-R1 == P2-R1 (v503) now reads: given the bit, order/uniqueness/orbit/tau = i are theorems. Exact (sympy + Cl(16) Fractions)."
    },
    {
      "id": "v507_seam_bit_origin",
      "cluster": "core",
      "descriptionEN": "SEAM.BIT.ORIGIN.01: the tautology attack on the v506 alignment bit, executed and refuted -- where do the marks AND the deck come from, and does their common origin force the central alignment? (Part A torus origin + Part B fermion dichotomy in one module; NO marker moves -- the bit stays [C], SEAM.CLOCK.RIGIDITY.01 and QGEO.REALIZE.01 text-precised only.) Exact sympy + exact Cl(16) Fractions, no floats. [E] ORIGIN THEOREM: the 4 marks are the fixed points of the hyperelliptic involution w -> -w on C/(Z + tau Z) = the four half-periods Lambda/2Lambda (a 1/3-coefficient never qualifies); a translation t_c descends to the base sphere iff 2c in Lambda -- exactly the THREE half-period translations (control c = 1/3 fails); the descended maps are the EXPLICIT rational involutions sigma_i(x) = e_i + (e_i-e_j)(e_i-e_k)/(x-e_i) with the exact curve-lift identity prod(sigma_i - e_l) = psi_i^2 prod(x - e_l), psi_i RATIONAL (the p-function addition formula as pure algebra); {id, sigma_1, sigma_2, sigma_3} = Klein V4; CONVERSELY, in ALL three stabiliser types the freely-acting involutions are EXACTLY the three sigma_c (generic V4 (4,3,3), harmonic D4 (8,5,3), hexagonal A4 (12,3,3)) => EVERY mark-free collar deck (v216) IS a half-period translation -- the CLASS of the deck is forced, its POSITION is not. [E] CORE SOLVE (symbolic in the Legendre modulus lambda): sigma_c admits a marks-preserving PGL2 square root <=> lambda in {-1} (sigma_1), {2} (sigma_2), {1/2} (sigma_3) -- per half-period exactly ONE solution; intrinsically: the two deck pairs separate each other HARMONICALLY (pair cross-ratio = -1, same solution sets); the union over c is the harmonic orbit {-1, 2, 1/2} <=> j = 1728 <=> tau = i; generic modulus: Stab = V4 abelian, ALL sigma_c central and yet clockless ('central alone is EMPTY' -- the bit is the CONJUNCTION Stab = D4 AND deck central); at tau = i the stabiliser jumps to D4 with sigma_1 central (2 roots = the clock pair) and sigma_2/sigma_3 edge-type (0 roots); the v506 seam deck z -> -z lands on the central sigma_1 = -1/x under ALL 8 mark bijections; CM DICTIONARY: mult-by-i fixes exactly ONE nonzero half-period c* = (1+tau)/2 (mult-by-rho 3-cycles all three -- hexagonal has no distinguished one); the clock lift needs the CM unit i (psi = 2i/(x+1)^2, t^2+1 irreducible over Q) while half-period decks lift RATIONALLY: the clock root exists only WITH CM by Z[i] <=> tau = i. [E] EQUIVALENCE TETRAD + CIRCULARITY AUDIT: 'clock exists' <=> (tau = i AND deck = the CM-fixed half-period translation) <=> (Stab = D4 AND deck central) <=> deck pairs harmonically separated <=> the NS Fock lift is NONSPLIT (Part B); audit: the v493 converse (clock GIVEN => P(iZ) = P(Z) forces a3 = a2 = a1 = 0, j = 1728) independently re-verified -- two independently computed converses of ONE equivalence, no circularity. [E] FERMION DICHOTOMY (the new physical face): central arrangement (shift by 8 on the 16-site NS seam): Utilde^2 = 256 gamma_1...gamma_16 => U^2 = (-1)^F, nonsplit Z4 extension, 2 square roots in the 32-element dihedral lift group (the Z8 tower); edge arrangement (reflection j -> 4-j, fixed sites = NON-marks = the silver {0, inf} between neighbour marks): the implementation EXISTS but Utilde_e^2 = +2^7 x 1 => U^2 = +1, the extension SPLITS, 0 roots, no Z8 tower; one-particle level (lam^2+1)^8 vs (lam^2-1)^8; arrangement census: ALL 28 shift-invariant 4-sets cross (central type), ALL 21 mark-free reflection-invariant sets do not; robustness at 8 sites; R-sector edge control splits too -- the bit IS a physical extension class (Fidkowski-Kitaev-type): distinguished, but not derived; WHICH class nature realises stays the input. [E] SILVER PLACEMENT (in-family witness against the tautology): the v506 counterexample {+-1, +-(3+2sqrt2)} = 'tau = i with deck = a NON-CM-fixed half-period translation' (all 8 mark bijections carry its deck to sigma_2/sigma_3, never sigma_1) -- the common torus origin does NOT exclude the misalignment. [C] THE REST (verbatim, no overclaim): the one alignment bit remains a carrier datum; no tautology -- the origin forces the CLASS of the deck (half-period translation), not its POSITION. Exact (sympy + Cl(16) Fractions)."
    },
    {
      "id": "v508_celestial_wp5e_gamma_pairing",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.GAMMA.01: WP5e-gamma of the celestial-holographic research contract CELEST.SEAM.01 -- 'the sphere-axion pairing check', an honest rigid NEGATIVE result (v505 style): can the three twisted sphere axions eta_m (Coxeter eigencharacters {i, -1, -i} on H^2(ALE), v505 S5.8) cancel the rigid 32 T3 residual of the Atiyah-Bott ledger (A_fix = 9P1 - 30P2 - 15P3 + 32 T3, v505 S3.7) by Green-Schwarz-type EXCHANGE with sector-compatible quadratic vertices? NO, with certificate. Exact sympy/Fraction, no floats (27 checks). [E] VERTEX-SPACE THEOREM: the W(D5) x W(A3)-invariant quadratics on the 8-dim glue Cartan = EXACTLY span{s5, s3} (dim 2, bidegree dims (1,0,1)); quartics = EXACTLY span{P1, P2, P3, T5, T3} (dim 5, bidegree dims (2,0,1,0,2)) -- vertex basis complete (Weyl nullspace arithmetic over all bidegrees); PRODUCT THEOREM (the master kill): (a s5 + b s3)(c s5 + d s3) = ac P1 + (ad+bc) P2 + bd P3 identically -- ZERO T5/T3 content, so EVERY exchange image (any axion count, any rule, any couplings) is annihilated by Phi_T3, while Phi_T3(A_fix) = 32 != 0. [E] SELECTION ENUMERATION: strict two-index reading COLLAPSES (C_{j,j'} != 0 only for j' = -j => the rule forces m = 0 -- the sphere axions have NO invariant bilinear Cartan vertex at all); twist-insertion: eta_1 <-> K^(3), eta_2 <-> K^(2), eta_3 <-> K^(1), eta_0 <-> K^(0); channels E13 = (16, -96, 144, 0, 0), E22 = (16, 32, 16, 0, 0), E00 = 225 E22; SIDE DISCOVERY: K^(0) = -15 K^(2) (the even-sector quadratics are PARALLEL). [E] UNSOLVABILITY WITH CERTIFICATE (core): exchange matrix 5 x 2, rank 2, kernel 0; rank([M | A_fix]) = 3; annihilator certificate (Phi_T5, Phi_T3, Phi_P)(A_fix) = (0, 32, 72) with A_fix = (9, -30, -15, 0, 32) -- two independent obstructions: 32 T3 (structural, rule-independent) and Phi_P = 72 (rule-driven via the K^(0) || K^(2) collapse); ker M = 0 forces the unique non-disturbing coupling c = 0 (the sphere axions are RIGIDLY SILENT); Phi_P(Q^(0)) = 0 (the v505 sector-0 mechanism undisturbed); relaxed control: dropping charge conservation reaches the P-block (rank 3, det -64) but T5 = T3 = 0 persists. [E] NATURALNESS RESOLUTION: ch2-natural (9/64, 1/4) and AB-weight (1/2, 1/4) couplings both have certificate (0, 0) vs required (32, 72) -- the failure is scale- and coupling-INDEPENDENT (no 'unnatural factor': the ansatz space misses the target identically). [E] NEGATIVE CONTROLS: (a) invariant axion only = v505 S3.9 replicated + sharpened; (b) flipped rule m + j - j': the Phi_T3 kill is flip-invariant, Phi_P is rule-dependent; (c) SO(16): odd classes EMPTY, AB sum (15, -18, -15, -4, 40) does not even cancel T5 (E8 doubly special); (d) D8: 16 T8 + 12 s8^2, no T3 structure -- the question cannot even be posed. [C] number fences: 32 = (h,h) + (h',h') = 20 + 12; 72 = 2 lambda~^2_e8. [O] SHARPENED FENCE: the exchange sub-branch is DEAD; the 32 T3 residual MUST be carried by twisted BCOV contact terms -- binary criterion WP5e-delta-1: the one-loop coefficient on C^2/Z4 must come out exactly (9, -30, -15, 0, 32), COMPUTED, not fitted; named escape WP5e-delta-2: the full-tensor ledger (the collapse is Cartan-restricted); slot bijection (v505 S5.8) untouched; the preregistered level-kill still does NOT fire; SEAM.EQUIV.01 stays [O]; no marker moves. Exact (sympy/Fraction)."
    },
    {
      "id": "v509_celestial_wp5e_eps2_level_from_flux",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.EPS2.01: WP5e-epsilon-2 of the celestial-holographic research contract CELEST.SEAM.01 -- 'the CPS level-from-flux dial', executed (verdict B): derive the boundary level k = 1 from flux quantisation on the lockstep spheres, following the Costello-Paquette-Sharma (CPS) Burns-holography mechanism (PRL 130, 061602; arXiv:2306.00940). Exact sympy/Fraction, no floats (28 checks, <1 s). [E] CPS SKELETON REPLICATED EXACTLY: the S^3 period of the brane backreaction 3-form int Omega_N = (2 pi i)^2 N (pullback density -sin(2 theta), phases cancel; their eq. 3.33); the exceptional-sphere Kahler flux int_{P^1} omega = 2 pi N (K = |u|^2 + N log|u|^2, flat part dies at t -> 0); boundary level magnitude 2N via the symplectic-boson double contraction (N = 1, 2, 3), normalisation Dynkin index T(so8 vector) = 1 -- the CPS formula 'level = flux quantum x Dynkin index of the boundary matter', level LINEAR in the flux. [E] PAIRING MATRIX PINNED, NOT POSTULATED: the v505 ch2 ledger (f(m) = -(C^-1)_mm/2 = (-3/8, -1/2, -3/8)) + integrality + unimodularity + effectivity pin P_mi = c1(T_m).[Sigma_i] = delta_mi by COMPLETE enumeration (row candidates 8/6/8; 48 unimodular solutions of P C^-1 P^T = C^-1; with P >= 0 exactly 2 = identity + A3 diagram flip); full c1 matrix -(C^-1)_mn; glue FLUXES F_i = (64, 60, 64) = dim g_i, total 188 = 248 - 60, flip-invariant; flux per charged root current uniformly ONE quantum through exactly one sphere. [E] LEVEL FORMULA, honest: 'level = total open-string flux' is KILLED by the lockstep test itself (k_i = (64, 60, 64) unequal -- F_i is the FLAVOUR multiplicity); what works: k = flux per adjoint current = F_i/dim g_i = (1, 1, 1), anchored non-circularly by the lattice current count (240, 0, 0, 0) for k = 1..4 (v505 S4.8) and the embedding index j(D5 in E8) = j(A3 in E8) = 1 (Killing coefficient 60 = 2 h_vee(E8)). ORD-4-VS-LEVEL-1 RESOLVED: the clock order counts FRACTIONAL flux sectors, not the level -- mu = det(D5) det(A3) = 16 = ord(monodromy)^2, the mu4 glue diagonal isotropic + Lagrangian (order 4 = sqrt 16), KLM condensation 16 -> 16/4^2 = 1 = zero residual fractional flux; NEW DIAL: #primaries((E8)_k) = (1, 3, 5, 10, 15, 27) for k = 1..6 from the affine marks -- exactly ONE sector iff k = 1 (independent of the CPS transplantation). [E] LOCKSTEP THEOREM (the lifting of v505 S4.10 'one scale => one level' to a THEOREM): clock invariance P(iZ) = P(Z) forces a3 = a2 = a1 = 0, the four branch points are ONE mu4 orbit, |Pi_j| = sqrt(2) t EQUAL on all three spheres, monodromy = pure phase i, det(A - 1) = -4 => no invariant flux vector => k1 = k2 = k3 IS A THEOREM of clock invariance; NEGATIVE PROBE (new falsifier): the clock-forbidden family Z^4 - Z (smooth, disc -27, branch points {0, 1, w, w^2} not a mu4 orbit): over ALL 24 orderings ZERO chains with three equal period moduli (1 vs sqrt 3); HONEST LIMIT: uniform flux doubling keeps the lockstep -- the theorem proves EQUALITY, not the value; the value 1 comes from the current/sector counters and the CPS minimal quantum N = 1. [E] NEGATIVE CONTROLS: k = 2 (current count 0, embedding residual 360, prefactor -31/48 != -1/3, #primaries 3 != 1 -- while the lockstep dial honestly does NOT fire); SO(16) glue (fluxes (0, 60, 0): spheres 1/3 FLUX-DARK, ch2 defect -30 != -78, condensation stops at 16/2^2 = 4 = det Cartan(D8)); A2 intersection form (det 3 != 4, glue diagonal 23/24 not integer, mu = 12 not a perfect square: NO Lagrangian subgroup, Dedekind 2/3 != 5/4, Coxeter 3 != 4). [C] the CPS dictionary itself on PT/Z4 (their geometry is the C^2 blow-up with ONE (-1)-sphere and so(8) at negative level -- not the A3 ALE). [O] the type-I B-model backreaction on PT/Z4 (twisted Kodaira-Spencer measure, the A3 Omega_N) -- the same fence as v502 S5.7/v505; delta-1/delta-2 and epsilon-1 stay open; SEAM.EQUIV.01 untouched; no marker moves. Exact (sympy/Fraction)."
    },
    {
      "id": "v510_seam_bit_freedom",
      "cluster": "core",
      "descriptionEN": "SEAM.BIT.FREEDOM.01: the freedom attack on the v506/v507 alignment bit -- the POSITION half of the bit is topology (Part A dihedral/Fock + Part B Moebius/RP in one module; NO marker moves -- the modulus datum stays [C]). Exact sympy + exact Cl(16) Fractions, no floats (30 checks, <5 s). [E] DIHEDRAL THEOREM: on the N-site seam circle the free involution is UNIQUE = the antipode (N = 16: 17 involutions in D_16, exactly 1 free = shift 8; all 16 reflections have exactly 2 circle fixed points -- sites for even axis, edge midpoints for odd; robustness N = 8, 12); the census is COMPLETE: Aut(C_16) = D_16 exactly (brute force over all 8! permutations at N = 8, propagation at N = 16 -- no exotic circle automorphism); N = 9 control: D_9 has ZERO free involutions (even carrier count needed, consistent with 16 = 2^(g_car-1)); QUOTIENT TOPOLOGY (the P1 tie): circle/antipode = CIRCLE (chi = 0, closed), circle/reflection = INTERVAL (chi = 1, 2 boundary corners) -- origin_theory's 6pi = 2pi+2pi+2pi Gauss-Bonnet budget counts three CLOSED circles, so 'the deck is free on the seam circle' IS the established P1 datum 'the seam is a closed circle', not a new premise. [E] NONSPLIT <=> FREE (complete census, exact Cl(16)): ALL 17 involutions implemented -- the free one's NS lift squares to 256 gamma_1..gamma_16 = (-1)^F (nonsplit, 2 square roots in the 32-element dihedral lift group = the Z8 clock tower); ALL 16 reflections have U^2 = SCALAR (2^7 site-axis, 2^8 edge-axis; split, 0 roots) -- no exception; chain: deck free => deck = shift 8 => crossing mark pairing (all 28 invariant 4-sets) => U^2 = (-1)^F => central class => the edge/silver arrangement is EXCLUDED; Klein control: the orientation-cover deck is free too (covering vs branched is the discriminator, not Moebius vs Klein); branch control: the pillowcase involution (x,y) -> (-x,-y) has exactly 4 fixed points = the marks and restricts to a REFLECTION on invariant circles -- the edge class IS the branched pillowcase involution's restriction: the silver arrangement mistakes the mark-creating branch Z2 for the seam deck. [E] REAL-STRUCTURE CLASSIFICATION (continuum side): the antiholomorphic involutions of P^1 commuting with the celestial deck z -> -z are EXACTLY two families -- mu conj(z), |mu| = 1 (fix locus = a LINE through the deck poles {0, inf}: deck NOT free on it) and mu/conj(z), mu real (mu > 0: fix circle |z| = sqrt(mu) avoiding the poles, deck FREE; mu < 0: empty fix locus, no seam); per deck-invariant configuration the mark-fixing real structure is UNIQUE; table: mu4 -> equatorial 1/conj(z), deck FREE (v506 A-S2.9 reproduced); silver {+-1, +-(3+2 sqrt 2)} -> real axis THROUGH the poles, deck NOT free; generic-unit {+-1, +-(3+4i)/5} -> equatorial, free WITHOUT harmonicity; hexagonal -> NO mark-fixing real structure at all (cross-ratio complex, marks not concyclic: RP-incompatible, consistent v503); Moebius covering model: the sheet-exchange deck of the cylinder double cover is free on ALL points and restricts to the antipode on the Z2 seam fibre; one-sidedness TYPES the Z2 as the free antipode (quotient RP^2 = the |Z2| of c3 = 1/(|Z2| 2pi chi), v456/v73). [E] HARMONIC + FREE => mu4 EXACTLY: the deck-pair cross-ratio (1-b)^2/(1+b)^2 = -1 solves exactly to b = +-i -- on free-deck seam circles the silver family (harmonic AND edge) is EMPTY, harmonicity alone becomes SUFFICIENT. [E] HONEST COUNTERWITNESSES: 'nonsplit => clock' is FALSE -- discrete {0,1,8,9} (shift-invariant, crossing, same nonsplit lift, no clock) and continuous {+-1, +-(3+4i)/5} (free, central, j = 148176/25 != 1728, no clock): the Fock dichotomy measures the CLASS of the arrangement, not the modulus. PREMISE TYPING: (P-i) deck = Moebius covering deck => free on the seam circle [standard covering topology, machine-instantiated; the one-sidedness strand origin_theory/v456]; (P-ii) marks on the Theta-fix seam circle = existing REALIZE [C] content (QGEO.REALIZE.02 hypotheses, v264; v506 A-S2.9 [E] for mu4) -- NO new input; audit finding: circle-freeness was nowhere typed before, v507 F1.1 explicitly admitted the reflection reading -- this door is now closed (a reflection is not a covering deck). [C] THE REDUCED REST (verbatim): the alignment bit reduces from 'harmonic AND central' to the square-modulus datum tau = i alone; the position half is topology (edge class excluded), the modulus half stays the one carrier input. Exact (sympy + Cl(16) Fractions)."
    },
    {
      "id": "v511_celestial_wp5e_delta2_tensor_ledger",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.DELTA2.01: WP5e-delta-2 of the celestial-holographic research contract CELEST.SEAM.01 -- 'the full-tensor ledger', the named escape route of the v508 exchange no-go, executed (intermediate verdict, exact): does the FULL adjoint tensor structure of e8 (non-Cartan external legs, higher-arity vertices) open a sphere-axion channel to the rigid 32 T3 residual of the Atiyah-Bott ledger (A_fix = (9, -30, -15, 0, 32) in (P1, P2, P3, T5, T3), v505 S3.7)? The collapse is CONFIRMED full-tensorially, but one cubic door opens. Exact integer-scaled weight multisets + Kostant/Weyl alternating sums + sympy/Fraction, no floats (41 checks, ~4 s). [E] STRUCTURE: g0 = d5 (+) a3 is SEMISIMPLE with NO u(1) factor (the 52 class-0 roots split 40 + 12 into pure D5 + pure A3 roots, root span rank 8 = rank E8); the twisted sectors factorise multiplicity-free into minuscule Weyl orbits g1 = (16_s, 4), g2 = (10, 6), g3 = (16_c, 4bar) = (g1)^*, each an irreducible g0-module (64, 60, 64); INNERNESS THEOREM: h = (2,2,2,2,2;0^4) lies in the CARTAN OF g0, so every g0-invariant tensor automatically carries total sector charge 0 mod 4 -- the v508 collapse is a full-tensor statement AND holds across arities (all 15 non-neutral sector triples: Hom = 0). [E] HOM TABLES: bilinear dim Hom_{g0}(g_j x g_j') nonzero ONLY at j' = -j (dims 2/1/1/1, Killing blocks); trilinear neutral multisets {0,0,0}: 3 (= 1 sym + 2 antisym), {0,1,3}: 2, {0,2,2}: 2, {1,1,2}: 1, {2,3,3}: 1 -- ALL cross-sector trilinears are pure brackets (antisymmetric in the repeated legs); the ONLY totally symmetric survivor on all of e8 is the su(4) d-symbol (so(10) has no cubic Casimir: Inv(S^3 45) = 0 protects T5). [E] THE NEW CHANNEL: the quartic projection of the d-symbol exchange is Q_dd = (0, 0, -1/240, 0, 1/60) = (1/60)(T3 - P3/4), computed by TWO independent routes (Killing-propagator contraction, 2 h_vee = 60 verified from the roots; hand route |x^2 - (tr x^2/4) 1|^2/60); Phi_T3(Q_dd) = 1/60 != 0 -- v508's master-kill functional does NOT annihilate the trilinear exchange (the product theorem covers quadratic vertices only); brackets, tadpoles and mixed d x f die exactly; no T5 analogue. [E] NEW CERTIFICATE: M = [E13, E22, E00, Q_dd] has rank 3, augmented rank 4 -- the pairing in the charge reading remains OBSTRUCTED, but with the WEAKER certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 72 - 8 = 64; the RELAXED reading becomes EXACTLY solvable: A_fix = -u + 8v + 2w + 1920 Q_dd (residual 0) -- where v508 found T5 = T3 = 0 persisting even relaxed, the d-channel closes the T3 gap with small integer coefficients. [C] number fences: 64 = dim g1 = 2^6; 1920 = |W(D5)| = 2^7 x 3 x 5 = 8 x 240. [O] the arity >= 4 ledger; the physical justification of the cubic GS term (c_d = 1920 under the e8-Killing propagator); ch2-naturalness for CUBIC vertices UNDECIDED (no canonical geometric datum for a cubic vertex); the burden on delta-1 (twisted BCOV contact terms) is SHARPENED: it must supply either the psi = 64 slice or the selection-rule relaxation. [E] NEGATIVE CONTROLS: (a) Killing control = v508 replication (A_fix, E13 = (16, -96, 144, 0, 0), E22 = (16, 32, 16, 0, 0), E00 = 225 E22); (b) SO(16)/D8: bilinears (1, 0, 1), trilinears (1, 0, 1, 0), Inv(S^3 adj_d8) = 0 -- no symmetric cubic, no odd sectors, no A3 block: the delta-2 channel cannot even be posed (E8/mu4 doubly special); (c) false g0 = d5+a2+u(1): (5, 2, 2) != (2, 1, 1) and Inv(S^3 g0) = 6 != 1; false sector content (16_s, 4bar) kills every pairing. SEAM.EQUIV.01 stays [O]; NO marker moves. Exact (sympy/Fraction)."
    },
    {
      "id": "v512_seam_tau_flag",
      "cluster": "core",
      "descriptionEN": "SEAM.TAU.FLAG.01: the flag-transitivity equivalence web -- the tau = i attack executed on the free RP seam circle: which ESTABLISHED seam datum excludes the free-but-non-harmonic configurations? (Test family M(delta) = {+-1, +-e^(i delta)}, delta in (0, pi/2]; NO marker moves -- the bit stays [C], QGEO.REALIZE.01 and P2.TYPING.01/AX.P2.01 text-precised only.) Exact sympy + mpmath 30 digits. [E] FALSIFICATION OF BARE TRANSITIVITY: on the free RP circle V4 = {z, -z, b/z, -b/z} acts SIMPLY TRANSITIVELY on the four marks for EVERY delta (pair-exchanging inversions; complete census: deck-commuting Moebius maps = diagonal/antidiagonal) -- the expectation 'transitivity <= v216 local equality selects tau = i' is refuted; hardening: ALL undirected per-mark data (arc-pair multiset, distance multiset, cone germ) are identical for every delta -- no local jet sees the bit. [E] THE SHARP SELECTOR: FLAG transitivity (marks AND sides) <=> delta = pi/2 <=> tau = i <=> the clock -- stabiliser jump 1 -> 2, flag/arc orbits 2 -> 1, order-4 rotation (b^2 = -1 solveset {pi/2}), mark mirror z -> 1/z <=> cos delta = 0. [E] 12-FOLD EQUIVALENCE WEB (each link exact): clock rotation, D4, mark mirror, flag transitivity, three spectral degeneracies (odd-doublet split = (2/pi) ln cot(delta/2), zero only at pi/2 -- counterwitness exactly (2/pi) ln 2; arc-Laplacian degeneracy 4/pi; the v215-K3 indicator in closed form (2 sqrt 2/pi)|ln tan(delta/2)| -- the v503 lattice curve analytically), rho-twist existence ((m^2 - n^2)(t - 1/t) = 0 <=> t = 1; Atiyah-Bott rationality is delta-BLIND -- Tr(sigma e^(t Delta)) = 1 for all delta, j_cw = 148176/25 rational non-CM: rationality does NOT select), 4-cycle implementability (H4 rigidity: V4 with orders {1,2,2,2} cannot induce the cusp 4-cycle; centraliser scalar, U^2 nonscalar => no projective involution -- projective order exactly 4, Z8-shadow consistent v510), harmonic, j = 1728. [E] NEGATIVE CONTROLS: hexagonal (not concyclic, fails the preconditions); silver (harmonic but topologically dead -- position/modulus halves independent, consistent v510); 3 marks (impossible on the free circle: 0 invariant 3-subsets vs 28 invariant 4-subsets; bare n = 3 transitivity already forces equal spacing -- '4 counts': the deck-pairing makes bare transitivity free of charge). [C] THE REST (verbatim): the last input reduces from a continuous modulus to ONE discrete symmetry-lift bit (V4 -> D4: side-indistinguishability of the four equally-decorated marks); v216 local equality does NOT deliver it (no local jet can see the side bit); its negation is now concretely measurable (odd-split (2/pi) ln cot(delta/2) != 0). Scoreboard: the counterwitness passes all 7 established unsided tests, fails all 9 clock-equivalent ones. Exact (sympy + mpmath)."
    },
    {
      "id": "v513_celestial_dterm_nonderivation",
      "cluster": "frontier",
      "descriptionEN": "CELEST.DTERM.NONDERIV.01: the c_d negative certificate -- can geometry derive c_d = 1920? An anti-numerology certificate in the v508 style (honest rigid NEGATIVE result): the v511 [C] fence '1920 = |W(D5)|' stress-tested via four candidate routes with a PRE-DECLARED look-elsewhere ledger. Exact sympy/Fraction, no floats (24 checks, ~2 s). [E] CONVENTION TABLE: c_d in {1920, 120, 256, 32, 69120} across five normalisations (probe tr-fund/e8-Killing; Gell-Mann; su(4)-Killing; plain trace; ch3-normalised d/6) -- only 1 of 5 hits |W(D5)|; the convention-STABLE content is the factorisation c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60, both factors [E] (Killing propagator 60 from sum_alpha <alpha,e>^2 = 120 over the actual roots; T3-slot fixing via det(P-block) = -64 != 0). [E] K1 DEAD: the charge-legal flux pairings (1,3)/(2,2) give 2048/1800 -- they BRACKET 1920 and both miss; the only flux hit 64*60/2 is the FORBIDDEN (1,2) pairing (charge 3 mod 4, killed by the v511 innerness theorem). LOOK-ELSEWHERE LEDGER: 924 expressions from 21 [E] numbers x grammar {a*b, a*b/2, a*b/4, a+b}, 11 hits on 1920; control targets 1800 -> 8, 2048 -> 5, 2016 -> 0. [E] K2: quantisation delivers only lattices (60 Z: 1920 = 32nd multiple, no minimality; strict ch3-unit couplings 2160 Z EXCLUDE 1920); the cubic quantum is lattice-dependent (6 on coroots, 3/8 on weights); the residual integer 32 has 12 signed factorisations, nothing forces it. [E] K3 PROVENANCE CLASH (the death blow): cubic indices A_m = (0, 16, 0, -16), twisted sum i^m A_m = 32i NUMERICALLY = Phi_T3(A_fix) -- BUT the quartic class decomposition (10, -4, 30, -4) is dominated by classes 0/2 while the cubic 32 is built 16 + 16 from classes 1/3: same number, disjoint mechanism = coincidence exposed; the Weyl side is WRONG (the d-channel is an A3-side object, |W(A3)| = 24; D5 has no cubic invariant, Inv(S^3 45) = 0; |W(E8)|/|W(A8)| = 1920 a second accidental 1920). [E] K4 NEGATIVE CONTROLS: SO(16)/D8 fails every candidate formula consistently; k = 2 flux doubling scales the flux formula x4 (7680); false g0 = d5+a2+u(1) has SIX symmetric cubics; so(10) sanity (Tr X^3 = 0). VERDICT: NUMEROLOGY as a derivation claim -- the v511 [C] fence stays a fence, now typed look-elsewhere-loaded; the honest [E] core is c_d = 32 x 60; the physical generation of the 32 stays [O]. Exact (sympy/Fraction); no marker moves."
    },
    {
      "id": "v514_celestial_wp5e_eps1_axion_slot",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.EPS1.01: WP5e-epsilon-1 of the celestial-holographic research contract CELEST.SEAM.01 -- 'the O(-2) bulk-axion slot', executed (verdict B): build the s = 0 tower field explicitly, pin lambda~ = 6 as a residue with explicit factor bookkeeping on PT/Z4, and take the first honest back-reaction step (Gibbons-Hawking A3). Exact sympy/Fraction, no floats (34 checks, ~4 s). [E] EQUIVARIANT O(-2) SLOT: pushforward ledger pi_* O(-2) on PT' = tot(O(1)+O(1) -> P^1): per spacetime degree d exactly (d+1)^2 = nullspaces of the wave operator (1, 4, 9, 16, 25, 36 for d = 0..5); the v492 action on the fibre coordinates (mu1, mu2) -> (i mu1, i^-1 mu2), incidence forces column weights (+1, +1, -1, -1); the equivariant bookkeeping closes BLOCK BY BLOCK (all d <= 6, all 4 characters); character series P_0 = 1 + 3t^2 + 15t^4 + 21t^6 + 45t^8 + ..., P_1 = P_3 = 2t + 8t^3 + 18t^5 + ..., P_2 = 6t^2 + 10t^4 + 28t^6 + ...; the d = 0 slot has multiplicity (1, 0, 0, 0) -- THE BULK AXION SURVIVES THE PROJECTION; Molien invariant ring = (1 - t^8)/((1 - t^4)^2 (1 - t^2)) -- Z in O(2), X, Y in O(4), relation degree 8 = the a0 weight (v492/v493 bridge); E3 bijection sharpened: twisted characters {i, -1, -i} with minimal content (2t, 6t^2, 2t) and NO degree-0 mode = the Coxeter eigenvalues on H^2(ALE), det(A-1) = -4; graviton control: the O(+2) slot starts invariant only at fibre degree 4 with multiplicity 3 = {X, Z^2, Y} (the cohomological form of v505 X = -2 != +2 = Y). [E] LAMBDA~ = 6 TRIPLY PINNED: (1) Okubo: Tr_adj X^4 = 36<x,x>^2 = (6<x,x>)^2 on the 240 glue roots, 36 = h_vee + 6, lambda~ = 6 = |Z2| N_fam; (2) measure cancellation: mu = 1/4 enters three times (vertex^2 mu^2, propagator 1/mu, anomaly mu) and cancels EXACTLY -- lambda~_eff^2 = 36 for ARBITRARY mu; wrong bookings quantified and excluded: anomaly-only => lambda~ = 3, missing propagator renormalisation => lambda~ = 12; (3) flux: minimal CPS quantum N = 1, a single exchange channel exactly at k = 1 (#primaries), (kappa/c3)^2 = 36/3 = 12 exact. [E] GH/A3 BACK-REACTION STEP: invariant 4-centre configurations = exactly 2 branches (axis^4 or ONE free mu4 orbit); free orbit => prod(Z - i^p z0) = Z^4 - z0^4 => XY = Z^4 + a0 with a0 = -z0^4 (the v493 family RE-DERIVED from the geometry); monodromy a0 -> e^{2 pi i} a0 = one clock step; the GH point of charge k is EXACTLY R^4/Z_k (k = 4 => seam boundary S^3/Z4); periods int omega = 4 pi (x_q - x_p), free orbit: Pi_j = 4 pi t0 (i-1)(1, i, i^2), |Pi_j| equal (LOCKSTEP, counter-check of v509); the Coxeter clock FROM THE GEOMETRY (A^4 = 1, det(A-1) = -4, Pi.A = i Pi); the CPS 'N log' analogue = source charge 4 = |mu4| (flux -4 pi per centre); HONEST SHARPENING: for the Ricci-flat ALE the asymptotic log coefficient is EXACTLY 0 (EH: u K' = sqrt(u^2 + a^4), det g = 1) -- the log lives at the exceptional locus; Burns contrast (K = u + N log u => det g != 1, only scalar-flat); bonus: multipole selection rule m = 0 mod 4, first symmetry-breaking multipole (4, +-4) carries -a0 with amplitude (35/16) sin^4 th. [E] NEGATIVE CONTROLS: so8 (lambda~^2 = 12 irrational; perfect squares in the Deligne series only {9, 36} = {sl3, e8}); diag(i, i) (Veronese, 5 generators, no hypersurface, the bijection collapses); k = 2 (three channels; honest: lambda~^2 = 36 is k-independent, the kill is the closure dial). [C] conditional on Costello's flat-PT matching lambda_g^2 E = A. [O] the quantised BCOV one-loop coefficient on PT/Z4; the twisted channels (32 T3) are not covered by the bulk axion -- M1-M3 preregistered VERBATIM: M1 the A3 Omega_N on PT/Z4 (success: (2 pi i)^2 periods with lockstep phases; kill: unequal fluxes => the v509 dial dies); M2 the twisted KS measure (success: bulk 36 + sphere axions cancel 32 T3; kill: a leftover T3); M3 the a0 uplift of the (4, +-4) multipole (success: log correction prop 4 = |mu4|; kill: decoupling from the centre count). No marker moves anywhere."
    },
    {
      "id": "v515_celestial_wp5e_m1_omega_n",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.M1.01: milestone M1 of the celestial-holographic research contract CELEST.SEAM.01 (WP5e back-reaction programme) -- 'the A3 Omega_N', executed (SUCCESS on the preregistered v514 S8.1 criterion): the back-reacted holomorphic 3-form Omega_N on the twistor uplift of the deformed A3 ALE, with exact periods. Exact sympy, no floats (30 checks, ~1 s). [E] RESIDUE FORM DERIVED: on F = XY - P(Z) all three ambient representatives satisfy dF ^ omega2 = -dX^dY^dZ and pull back to omega2 = dX^dZ/X (Poincare residue, derived not assumed); cover pullback at a0 = 0: omega2 = 4 dz1^dz2 = |mu4| x (flat form) -- the residue normalisation CARRIES the source charge 4; clock diag(i,1): omega2 -> i omega2 (v492 S5 replicated); weight ledger 4+4+2-8 = 2, Omega in O(4) = -deg K_PT; EH anchor 2 = |Z2|; wrong-weight control Z in O(4) breaks three ways (4 != 2, 6 != 4, 5 != 3). [E] THE TWISTOR FAMILY CLOSED: from the four O(2) centre sections q_p(la) = i^p t0 - i^-p t0 la^2 the family is EXACTLY XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4 (1 - la^4)^2 (e1 = e3 = 0 identically; la = 0 gives exactly the v493 clock family; a2 in O(4) opens only off-seam, a2(0) = a2(inf) = 0); clock lift gamma: (Z, la) -> (iZ, -i la), gamma^4 = 1, Omega -> +Omega (CY-compatible); weight chain 4+4+2-8+2 = 4 = -deg K_PT. [E] OMEGA_N CLOSED + QUANTISED PERIODS: Omega_N = Omega_0 + sum N_p K_p with CPS/Bochner-Martinelli kernels on the centre twistor lines; dK = 0 away from the source, scale degree 0; PERIOD TABLE: boundary S^3/Z4 at the orbifold point (2 pi i)^2 x 1 (minimal invariant charge 4 = |mu4| -- the lens fundamental domain FORCES N in 4Z), deformed phase (2 pi i)^2 x 4, linking S^3 uniformly (2 pi i)^2 x N, per-sphere flux vector (2 pi i)^2 x N (1,1,1) (forced uniform twice over: clock orbit + K4 connectivity), seam-fibre phase vector 2 pi i t0 (i-1)(1, i, i^2); the undeformed Omega_0 carries ZERO quantised 3-flux (all flux sourced); (2 pi i)^2 = (fibre residue) x (transverse residue) (density -sin 2 theta). [E] CLOCK COVARIANCE: Pi_{j+1}(-i la) = i Pi_j(la) for ALL la; the 12 collision nodes sit exactly on the 8 eighth roots of unity (8 = 2|mu4|, the Z8 seam bridge), clock orbits 4+4+4, honest conifold points (Hessian det 512 t0^6, quadric rank 4). [E] KILL PROBE (honest): the forbidden family Z^4 - Z: 0/24 orderings lockstep (vs 8/24), la-support 12 twelfth roots != Z8, no antipodal pairing -- BUT the (2 pi i)^2 quantisation holds there TOO: integrality alone is NOT the discriminator, the discriminator is the lockstep phase structure + the clock forcing of equal fluxes; the v509 dial survives. [E] ANCHORS + CONTROLS: EH/Z2 (2 nodes, charge 2 = |Z2| forced); CPS original (full S^3: every integer N allowed); diag(i,i): Omega_0 -> -Omega_0 (collapse at step zero) while K stays invariant; fractional charges N = 1, 2, 3 excluded. [C] global kernel patching (Green function), Hitchin small resolution, CPS brane dictionary. [O] M2 (twisted KS measure), M3 (a0 uplift), the quantised BCOV coefficient; SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction). No marker moves."
    },
    {
      "id": "v516_celestial_wp5e_m2_twisted_ks_measure",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.M2.01: milestone M2 of the celestial-holographic research contract CELEST.SEAM.01 (WP5e back-reaction programme) -- 'the twisted Kodaira-Spencer measure', executed (SUCCESS on the preregistered v514 S8.2 criterion, on the DECLARED completion measure -- verdict B): the one-loop box coefficient per sector on PT/Z4 under the completion contact term contact_j = (Q^(0) - Q^(j))/det_j. Exact sympy/Fraction, no floats (23 checks, ~1 s). [E] THE COMPLETION-WEIGHT IDENTITY (the centrepiece): per glue class the contact weights are w_m = sum_j (1 - i^{jm})/det_j = (0, 3/2, 2, 3/2) = 4 h_m = |mu4| h_m = -4 ch2(T_m) EXACTLY (h_m = m(4-m)/8 the v502 sector Casimir energies, ch2(T_m) the McKay/Kronheimer charges of the tautological bundles, v505 S4.3 index bridge): THE THREE SPHERE AXIONS PAIR THROUGH THEIR OWN ch2 CHARGES -- an exact identity, NO free scale, no fit. [E] PARAMETER-FREE LOCKS: T5 = 0 for ANY scale c (the ch2 pattern alone kills T5); ratio h_2 : h_1 = 4 : 3 = the delta-1-forced leading ratio (REPRODUCED, not fitted); the T3 budget fixes the scale c = 4 = |mu4| uniquely; the unique delta-1 leading weights (3/2, 2) reproduced as (4h_1, 4h_2). [E] THE CONTACT TERM: channel form sum_j (Q^(0) - Q^(j))/det_j = class form sum_m 4 h_m Q_m = (36, 120, 60, 0, -32) EXACTLY (one object, two bases); the contacts carry nonzero T5/T3 per channel (genuine contact, not exchange -- the v508 product theorem replicated as control). [E] SUCCESS TABLE: skeleton_j + contact_j = Q^(0)/det_j = 36<x,x>^2/det_j -- a PERFECT Okubo square (T5 = T3 = 0, lambda~^2 = 36) in EVERY twisted channel ((18, 9, 18) x <x,x>^2); total A_fix + contact = 45<x,x>^2 = (5/4) x 36<x,x>^2 = (Dedekind sum) x (Okubo) = the UNIQUE quartic-free weighting of v505 S3.8 (the rigidity theorem is exactly what the measure lands on); both v508 certificates killed (Phi_T3: 32 -> 0, Phi_P: 72 -> 0); the v511 slice psi = 64 SUPPLIED EXACTLY (psi(contact) = -64) -- no cubic d-channel needed, c_d is free to be 0; remainder inside the exchange span (45<x,x>^2 = (45/16) E22 = (1/80) E00, Phi_P = 0): the BULK axion cancels every channel with the SAME lambda~^2 = 36, the twisted propagator carrying the AB zero-mode factor 1/det_j; the v514 S3.4 measure chain stays mu-blind. [E] NEGATIVE CONTROLS: wrong scale c in {1, 2, 3, 5} leaves T3 = (24, 16, 8, -8) -- only c = 4 = |mu4| clears the budget; shuffled weights (h_1 <-> h_2) break T5 = -14 AND T3 = 36; SO(16) glue keeps T5 = 20 / T3 = -40 in the summed trace -- the KILL branch FIRES there (Okubo failure inherited, E8 doubly special); diag(i, i) degenerates to the Z2 target and breaks the conjugation pairing + the index bridge; the Z2/Eguchi-Hanson anchor PASSES with the same mechanism (A_2 + contact = 9<x,x>^2 = (1/4) x 36<x,x>^2, coupling scale 2 = |Z2| = centre count). [E] MULTIPLIER NOTE (cross-reference only, the substance belongs to the parallel delta-1d strand): the contact weights 4h are REAL RATIONALS carrying no multiplier system of their own; the delta-1c residual phases are the SQUARES of the A3 discriminant T-phases e(3a^2/8) -- orders (1, 8, 2, 8) squaring to (1, 4, 1, 4), a finite mu4 system; Gauss sums 2e(3/8) (A3), 2e(5/8) (D5), full Weil sum 4 (signature 0 mod 8). [C] THE COMPLETION READING (the mandatory fence): the loop of twisted states carries the UNPHASED sector trace with the same zero-mode normalisation -- declared, supported by the delta-1 modular-completion finding, NOT derived from the BCOV integral. [O] the BCOV-integral derivation of the completion reading (delta-1b/1c/1d territory); M3 executed separately (v517); SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction). VERDICT: SUCCESS wording met on the declared completion measure -- lambda~^2 = 36 reproduced in EVERY sector, the 32 T3 cancelled exactly by the ch2-weighted sphere-axion contacts; the KILL branch does NOT fire; NO marker moves anywhere."
    },
    {
      "id": "v517_celestial_wp5e_m3_a0_uplift",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.M3.01: milestone M3 of the celestial-holographic research contract CELEST.SEAM.01 (WP5e back-reaction programme) -- 'the a0 uplift', executed (SUCCESS on the preregistered v514 S8.3 criterion): the (4, +-4) GH multipole uplifted to the twistor-potential kernel chi = log P4 (the log of the v515 family polynomial) with the induced Kahler-potential correction. Exact sympy, no floats (23 checks, ~7 s). [E] KERNEL BRIDGE: the O(2) section is a NULL coordinate -- Laplacian(3d) G(eta_p) = 0 identically for an ARBITRARY kernel G (the uplift is well-typed); RESIDUE IDENTITY d_x [transform(log eta_p)] = 1/r_p EXACTLY (residue 1/(2r) at zeta_-): the GH potential V = sum_p 1/r_p IS the x-derivative of the contour transform of chi; flux -4 pi per centre, total source charge 4 = |mu4| (the V-ledger matched). [E] FOUR COUPLED CENTRE-COUNT SCALES: (1) asymptotic kernel log chi = 4 log eta + a2/eta^2 + (a0 - a2^2/2)/eta^4 + ... -- LOG COEFFICIENT 4 = |mu4| = centre count (the CPS 'N log' analogue at kernel level); at the clock-fixed seam fibre zeta = 0 the a2 slot closes and the first correction is EXACTLY a0/eta^4 = the (4, +-4) multipole in kernel clothes; exact m-grading (the term zeta^k/eta^4 carries field charge m = 4 - k: m = +-4 pieces PURE a0, m = 0 piece -6 t0^4 phi0-free -- the U(1) -> Z4 breaking sits exclusively in the a0 pieces); NO log x power terms in the Laurent tower (the v514 honest zero -- no asymptotic METRIC log -- is preserved); (2) the once-integrated GLT kernel sum_p eta_p log eta_p = 4 eta log eta + p_4/(12 eta^3) + p_8/(56 eta^7) + ... with p_n = 0 unless n = 0 mod 4 (selection rule at potential level) and p_{4k} = 4 (-a0)^k: EVERY correction carries the prefactor 4 = centre count; (3) the exceptional-locus log chi(eta = 0) = log a0 = 4 log t0 + i(4 phi0 + pi) -- t0-coefficient 4 = |mu4|, clock phase 4i; the modulus response a0 d/da0 chi = a0/(eta^4 + a0) equals 1 AT the exceptional locus and falls off as a0/eta^4 at infinity (no constant, no log): the a0-log lives at the exceptional locus, exactly as v514 S6.3 demanded; (4) periods (v515 consistency): Pi_j(la) LINEAR in t0 (uniform first-order shift, lockstep + phase ratios (1, i, i^2) preserved exactly); d log Pi_j / d log a0 = 1/4 = 1/|mu4| uniformly; integrated over the a0 circle: monodromy e^{2 pi i/4} = i = ONE Coxeter clock step (the v493 monodromy REPRODUCED from the perturbative period shift; the finite monodromy is the cyclic centre permutation q_p -> q_{p+1}); node la-support t0-FREE (the Z8 seam bridge is a0-rigid); all 6 pair discriminants ~ t0^2 != 0 (K4 connectivity: the forced uniform flux vector persists); the BM kernel is modulus-free with int_{S^3} K = (2 pi i)^2 (the quantised fluxes cannot move). [E] NEGATIVE CONTROLS: the (4, 0) multipole is phi0-independent in V AND in the kernel (-6 t0^4): clock-invariant, breaks nothing; Z2/Eguchi-Hanson analogue: EVERY dial reads 2 = |Z2| ((2, +-2) amplitude (3/2) sin^2 th, kernel log 2, exceptional log 2 log t0, period response 1/2, monodromy -1 = the Z2 clock); wrong centre counts k = 3, 5: kernel log coefficient k, modulus slot O(6)/O(10) != O(8) -- the observable MOVES with the centre count; forbidden family Z^4 - Z: p3 = 3 != 0 (selection rule broken), e4 = 0 (no a0-log at the exceptional locus) -- fails BOTH uplift dials. [C] THE GLT DICTIONARY: 'eta_p log eta_p = THE GH Kahler-potential kernel' is the generalized-Legendre-transform dictionary (Lindstrom-Rocek / Ivanov-Rocek), typed [C]. [O] the full nonlinear Kahler potential of the resolved A3 ALE (no closed form exists); the quantised BCOV coefficient stays with the delta-1 strand; SEAM.EQUIV.TWISTOR.01 untouched. VERDICT: SUCCESS -- the log-type correction delivered on four coupled dials (kernel log 4, exceptional-locus log 4 log t0, tower prefactor 4, period response 1/4 -> monodromy i), every one tied to the centre count (EH: 2, k-orbit: k); the KILL branch (decoupling from the centre count) does NOT fire; NO marker moves anywhere."
    },
    {
      "id": "v518_celestial_wp5e_delta1_derived_kill",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.DELTA1.01: the delta-1 chain (delta-1b + delta-1c + delta-1d) of the celestial-holographic research contract CELEST.SEAM.01 DECIDED -- 'the derived chiral measure kills the delta-1 route', an honest, decided NEGATIVE result (v508 style): does the Harvey-Moore/BCOV fundamental-domain integral on PT/Z4, evaluated under a measure DERIVED from blockwise SL(2,Z) covariance (solved for, not declared, not scanned), deliver -A_fix = -(9,-30,-15,0,32) or the psi = 64 slice? NO -- every member of the derived family fails all three preregistered testers (30 checks, ~20 s). [E] THE 16-COMPONENT WEIL COMPLETION (delta-1c): discriminant module Z4 x Z4 with q = (5x^2+3y^2)/8; Gauss sums 2 zeta_8^5 x 2 zeta_8^3 = 4 = sqrt(16) (signature 0 mod 8 = rank E8); S symmetric + unitary, S^2 = C, (ST)^3 = S^2, S^4 = 1, T^8 = 1 EXACTLY in Q(zeta_8); S = S_D5 (x) S_A3; the diagonal H and anti-diagonal H' are the two invariant Lagrangians (both counts = E4: two E8 gluings); P_H S P_H = (1/4) J with 3/4 of every S-image weight on the 12 shifted classes -- why the naive 4-character rule failed; the 16-vector theta S-covariance CLOSES: the delta-1b E1.5 residual 2.91 = O(1) drops to ~1e-39, certified at two tau. [E] THE OBSTRUCTION IS A FINITE mu4 CHARACTER, IDENTIFIED (delta-1d): gauge blocks M = G[a,b] eta^{-8} have constant transport (t_p, s_p) (28+ digits) recognised as exact 1/960-grid phases (1e-25); T-fixed pairs (0,b) carry t = e(-1/2) = -1 EXACTLY (the naked level-matching defect); loops land in Gamma_1(4) resp. Gamma_0(2)+-, joint holonomy eigenspaces dim 8/16 resp. 6/16, every eigen-phase of order <= 4 (the group is mu_4); KOBOUNDARY TEST: the SL(2,Z) relation defects c_{S^4}, c_{(ST)^3 S^-2}, c_{[S^2,T]} are (1,1,1) EXACTLY on all 15 pairs -- a CHARACTER of the stabilisers, NOT a genuine 2-cocycle; on Gamma_1(4): lambda(gamma) = i^{2B + C/4}. [E] THE CANCELLATION IS THE TWISTED FIBRE BLOCK f1 f3, NOT THE THREE SPHERE AXIONS: exact identity G[a,b] = f_1 f_3 (30+ digits); axion T-phases = v502 vacuum-energy data; T-fix mechanism one line of exact phase arithmetic: (-1) x e(-1/6) = e(1/3) = chi_4(T) at every T-fixed node; CANCELLATION TABLE (exact, minimal residual order over eta^m x chi_k): none -> 4, f1f2f3 (three sphere axions) -> 4, f2 -> 6, f1f3 -> 1 (hol AND anti), wrong weights (1,1,2)/(2,2,2) -> 4; strict solutions exactly chi_4 (dims (3,3)) and chi_10 (dims (1,1)) at eta^0, certified on the dressed functions at a fresh tau (1e-15, NO residual multiplier). [E] THE DECISION -- ALL THREE TESTERS FAIL: under both derived solutions the integral (30-digit kernels, negative-exponent audit, cut-6 vs cut-8 scan) fails T5(V) = 0 (fractions 0.5 / 0.83), fails W_2 : W_13 = 4 : 3 (W_13 = 0), fails V = -A_fix N (spreads 1.05 / 1.47) and the psi = -64 N slice; the honest (N1, N2) rebalance: chi_4 reaches T5 = 0 only at psi > 0, chi_10 only at N2/N1 = -4 < 0 (outside the orbifold projector) -- no branch fires in the positive cone. [E] NEGATIVE CONTROLS: Z2/Eguchi-Hanson (gauge-only residual order 2 -- the mu4 defect is Z4-specific -- and the SAME fibre dressing cancels it to 1); SO(16)/D8 (only even sectors, all axion b-characters in mu_2: the order-4 supply structurally absent); wrong form (x^2+y^2)/4 (|Gauss|^2 = 64 != 16, S-rule fails at O(1)); wrong signature (S^2 = C breaks exactly). [C] the f1f3 = KS-weight identification (deck weights (1,3) = the twistor fibre weights (+1,-1 mod 4) forced by the v514 incidence ledger -- exact match, NOT a complete BCOV derivation) + the kernel-family convention (J(Delta,0/1/2) from weight bookkeeping) + the discrete chi_k choice (both evaluated, both fail). TENSION, stated honestly (the mandatory fence): the DECLARED completion reading (v516/M2) delivers the psi = 64 slice and cancels 32 T3; the DERIVED chiral measure (this module) fails all three testers -- both are exact; the sharp open question is which of the two is the true BCOV measure (the declared reading is supported by the delta-1 modular-completion finding but not derived; the derived measure rests on the f1f3 identification). Neither result is hidden behind the other. [O] the BCOV-integral derivation on PT/Z4 that decides between the two measures -- the single open item of the delta-1 strand; the twisted-contact route of v508 S6.2 is DECIDED NEGATIVE under the derived measure. Discovery provenance: delta-1b (30/30), delta-1c (31/31), delta-1d (36/36) probes in experiments/tfpt-discovery/ (2026-07-22). NO marker moves anywhere."
    },
    {
      "id": "v519_woit_theta_rp_free",
      "cluster": "frontier",
      "descriptionEN": "WOIT.THETA.FREE.01: the WOIT-alpha milestone of the OS twistor bridge WOIT.OS.TWISTOR.01 -- 'the real structure exists, and free reflection positivity picks the SAME family' (Part W = the anti-linear Theta: classification, exact relations, kill test 1 on the free/equivariant level; Part R = reflection positivity of the free 16-Majorana NS seam system under the pinned Theta; NO marker moves -- WOIT.OS.TWISTOR.01 stays [O]: kill test 1 is discharged on the FREE level only and stays formally live on the interacting algebra). Exact sympy + exact Cl(16) Fractions; the RP definiteness certificates are 40-digit spectra of exact Hermitian matrices (tol 1e-25). [E] THETA CLASSIFICATION COMPLETE: exactly TWO families of anti-linear structures on C^2 normalise the clock rho = diag(i,1) -- family D (z -> mu conj(z), |mu| = 1): Theta rho Theta = rho^-1 EXACTLY, Theta^2 = +1 (-1 IMPOSSIBLE: M conj(M) = diag(|mu|^2, 1) -- Kramers-free), inverts the deck, fixed set = the great circle through the deck poles; family A (z -> mu/conj(z)): centralises the clock projectively (NEVER inverts), Theta^2 = +-1 per mu, fixed set |z| = 1 resp. empty. ROLE SEPARATION: family A defines the euclidean section (Woit's rho_tw side: rho_tw^2 = -1, no real points -- replicated exactly on C^4), family D reflects it (the OS conjugation; sigma_std = the standard conjugation with real points RP^3). [E] MARK PINNING: z -> mu conj(z) preserves the mu4 marks iff mu in mu4 -- a 4-element mu4-TORSOR (2 mark axes + 2 silver-midpoint axes); clock conjugation rho Theta_mu rho^-1 = Theta_{-mu} => 2 clock orbits. [E] SPIN/FOCK: Z8 plane without phase leaks (conj(zeta8^k) = zeta8^-k); Cl(16): Theta_Fock = U_r o K with Theta_Fock^2 = 2^7 > 0 (normalised +1), inverts the clock tower (V -> 4096 V^-1, exact scalar via the implementer route) and normalises the deck (256 Utilde^-1); KRAMERS DICHOTOMY: the deck-induced candidate Theta_t = Utilde o K has Theta_t^2 = 256 gamma_1..gamma_16 = (-1)^F -- the v510 split/nonsplit dichotomy IS the Theta^2 = +1 vs (-1)^F dichotomy; the deck does NOT furnish Theta, the seam-circle REFLECTION does. [E] KILL TEST 1 (free level): does NOT fire -- Theta exists with Theta^2 = +1 AND Theta rho Theta = rho^-1 and satisfies all three side conditions simultaneously (marks, deck normalisation, spin/Fock). [E] FREE RP: NS vacuum C(d) = (2/N)/sin(pi d/N), C^2 = -1 exact; preconditions exact (R C R^T = -C, S C S^T = C, R S R^-1 = S^-1; the deck FAILS anti-invariance); BOND CUT: one-particle Gram (8,0,0) PD (min eigenvalue 1.888e-3 at 40 digits), even deg <= 2 sector (29,0,0) PD, N = 8 complete half-sided algebra PD (even + odd each (8,0,0)) -- RP with NO degree truncation; twist eta = +i FORCED (eta = 1 non-Hermitian); CONTROLS: the site cut fails exactly (det = 0, inertia (3,3,1); continuum control: Cauchy-Stieltjes positive -- a lattice-PLACEMENT artifact, RP selects the site placement relative to the marks); the clock-CENTRALISING family-A structure fails RP structurally ((4,4,0)) -- RP and Theta rho Theta = rho^-1 select the SAME family; bonus: the anti-chiral state flips the odd sector to (0,8,0) ND (the free shadow of kill test 3). [C]/[O] NOT shown: the interacting algebra A_hol (kill test 1 stays live there), gauge-fixed RP (kill test 2), OS reconstruction, chirality without mirrors (kill tests 3/6 -- only the free odd-sector shadow), mu4 incidence (kill test 7); WOIT-beta/gamma roadmap: (beta-1) Theta on the gauge-invariant subalgebra + gauge-fixed RP on the equivariant SDYM(E8) sector; (beta-2) the OS quotient of the free system explicit; (beta-3) the PT <-> PT* duality vs sigma_std typed; (gamma) the chirality theorem + the mark incidence. Two contract PRECISIONS (prose, no marker): (i) 'Theta induced by the seam reflection' means the seam-circle REFLECTION, not the deck (the deck furnishes the (-1)^F Kramers class -- the v510 bridge); (ii) the marks sit RP-side on the BOND MIDPOINTS of the 16-Majorana circle."
    },
    {
      "id": "v520_celestial_wp5e_measure_decision",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.MEASURE.01: the WP5e BCOV measure question DECIDED at probe level (delta-1e + delta-1f consolidated, v518 style) -- 'single-valuedness is derived, the completion reading wins': which of the two exact, contradictory measures on PT/Z4 is the true BCOV measure, (A) the DECLARED completion reading (v516) or (B) the DERIVED chiral measure (v518)? ERFOLG-A on the decision layer (39 checks, ~20 s). [E] INVARIANT SUBSPACE: dim_Q {v: rho(T) v = v, rho(S) v = v} = 8 = 4 x 2 in the dressed 16-component Weil system, EVERY kernel vector in the Q(zeta_8)-span of {e_H, e_H'} (the two Lagrangian gluings, both theta = E4, level-1 quartic = Q^(0) = the unphased trace); strict trivial-character solutions at (m, k) = (0, 0): ZERO for every dressing (11 combinations). [E] SOURCE 1 (F-LEMMA): the only strict chiral families chi_4 (dims (3,3)) and chi_10 (dims (1,1)) carry nontrivial T-characters exactly; pointwise covariance of the canonical column member certified (3.9e-16, |G| >= 59.6); exact change of variables => fundamental-domain independence forces int = 0 for EVERY chiral integrand -- the v518 route was never a well-defined nonvanishing moduli integral (kill sharpened). [E] SOURCE 2 (QUILLEN PAIRING): |t conj(t) - 1| < 8.3e-40 on all 15 pairs/both generators; |chi_4|^2 = 1 exact vs one-sided chi_4^2 = e(2/3) != 1 (T-fixed: 640 != 0 mod 960, 15/15 defective); the doubled 256-dim hol x antihol system closes strictly at trivial character and physical weight (nullspaces (28, 17) >= 11), contains the unphased diagonal (~2e-15) AND the physical columns w_b (x) conj(w_b) (sigma 4.9e-16/8.3e-16); negative control hol (x) hol: empty nullspace (0, 0). [E] CONSISTENCY CLOSURE: the forced unphased numerator in the v518 scaffold reproduces sum_m V prop (1, 2, 1, 0, 0) = <x,x>^2 exactly on every level n <= 8; J-weighted spreads ~5e-41, T5/T3 ~7e-42, zero negative cells; psi(skeleton) = +0.2302 = -psi(contact), the J-weighted mirror image of psi(A_fix) = +64 / psi(contact) = -64; linearity 4.6e-41 -- the circle to v516 is closed. [E] D3 Z2/EH ANCHOR: only the declared reading hits (9<x,x>^2 = (1/4) x 36, psi = -8, scale tooth c = 1/2 = |Z2| h^(A1)); all three derived instantiations fail (spreads 1.12/0.895/0.516, psi sign wrong). [E] D1 COLUMN CANONICALISATION: the physical AB column sum_m i^(bm) e_(m,m) is EXACTLY contained in the chi_4 family (sigma 1.8e-16/1.7e-16, one scalar per orbit -- canonicalises the (N1, N2) freedom); the canonical member still fails all testers. [E] NEGATIVE CONTROLS: SO(16)/D8 (T5 = 20/T3 = -40 exact, mu_2-only), wrong form (|Gauss|^2 = 64), wrong weight (dev 0.78), shuffle (T5 = -14/T3 = 36), planted family detected (1.5e-16), hol (x) hol empty. [C] TYPED PREMISES (the mandatory fence): TP-1 (F-independent moduli integral), TP-2 (nonvanishing -- must source psi = 64), TP-3 (v518 kernel convention), TP-4 (Quillen structure of BCOV F1). [O] the constructive BCOV derivation of the w_m normalisation itself. VERDICT: ERFOLG-A per the preregistered criteria (both sources independent, the consistency closure carries); the [O] question 'which is the true BCOV measure' is DECIDED at probe level in favour of (A) under TP-1..TP-4. SEAM.EQUIV.TWISTOR.01 stays Open; NO marker moves."
    },
    {
      "id": "v521_seam_bit_rp_blind",
      "cluster": "core",
      "descriptionEN": "SEAM.BIT.RPBLIND.01: the RP/Theta attack on the alignment bit DECIDED -- 'free OS positivity does NOT see delta', an honest, decided NEGATIVE result (v508 style; 25 checks, ~25 s): does free reflection positivity (v519) or anti-linear Theta existence break for delta != pi/2 on the deformed mark family M(delta) = {+-1, +-e^(i delta)}, which would derive the v512 alignment bit from OS positivity? NO -- the free RP/Theta battery is the EIGHTH side-blind test on the v512 scoreboard (7 -> 8); the bit remains genuine discrete input. [E] CENSUS (exact solvesets, symbolic delta): for every delta exactly the two mark-SWAPPING reflections Theta_(+-e^(i delta)) exist (axes delta/2, delta/2 + pi/2; all 20 cut/mark incidence solvesets empty); mark-FIXING reflections exist exactly at delta = pi/2 (solveset cos delta = 0, lost immediately at pi/2 - eps); the v510/v512 counterwitness {+-1, +-(3+4i)/5} carries the swap pair exactly. [E] RP IS delta-BLIND: bond-cut Gram inertias (8,0,0) PD for all tested delta, spectra IDENTICAL to 40 digits (deviation 0.0) -- the entire free-RP spectral datum carries ZERO delta information; the N = 16 odd-m failure (site axes, (3,3,1), det = 0) is a lattice-PLACEMENT artifact: at N = 32 all m = 1..7 have mark-compatible bond axes with PD inertia (16,0,0); continuum: the OS kernel in axis-adapted coordinates is exactly 1/sin((s+t)/2), the axis position drops out identically (Cauchy-Stieltjes PD); the counterwitness PASSES free RP (6-point Gram (6,0,0)). [E] THETA EXISTENCE delta-BLIND: Theta^2 = +1, deck normalisation and Fock implementability (U^2 = +2^7/2^8 > 0, exact Cl(16)) for every delta; the ONLY delta-sensitive v519 clause (Theta rho Theta = rho^-1) is for delta != pi/2 not violated but NOT FORMULABLE (mark-compatible order-4 rotation <=> delta = pi/2) -- it PRESUPPOSES the bit; same for the clock-tower clause. [E] the battery has teeth at every delta: family A/antipode fails RP structurally (zero diagonal, (4,4,0) indefinite), chirality-eta pinning persists ((0,8,0) ND) -- RP separates the two real-structure FAMILIES, never the SIDES. [E] precondition controls: hexagonal (marks not concyclic, no seam circle), silver (deck not free), lifting audit (B = index relabelling, delta-blind by construction; B' fails the pi/2 negative control -- discarded). [C] FACE #13 of the v512 equivalence web: the group generated by the admissible OS reflections is V4 for every delta < pi/2 and closes to D4 exactly at delta = pi/2 (Theta_i o Theta_1 = the clock) -- 12 -> 13 faces, a typed REFORMULATION of the bit, not a derivation. [O] honest gap: marks decorating the STATE (defect/twist insertions, interacting A_hol) = exactly v519's [O] scope -- there, and only there, could positivity still see the bit. VERDICT: KILL exactly as preregistered; the alignment bit is NOT derivable from OS positivity and remains genuine discrete input; no marker moves."
    },
    {
      "id": "v522_woit_beta1_gso_gauge",
      "cluster": "frontier",
      "descriptionEN": "WOIT.BETA1.GSO.01: the WOIT-beta1 milestone of the OS twistor bridge WOIT.OS.TWISTOR.01 executed -- 'the clock is time-like, GSO is the gauge datum' (typed result: verdict UNDECIDED per the frozen preregistration, mechanism identified exactly, gauge-fixed RP established under the corrected GSO typing; 16 checks, ~5 s). [E] HERMITICITY WITNESS: the mu4 clock average violates Hermiticity exactly (the one-step insertion T_1 = <theta(e_a), alpha_S(e_b)> has entry -i/(8 sin(5 pi/16)) where the conjugate transpose demands +i/(8 sin(5 pi/16))); the pairing is complex-SYMMETRIC (745 entries matching, 96 anti-matching, 0 violations) -- 'OS-symmetric' is strictly weaker than Hermitian: 'positivity after gauge fixing' is not well-posed for the clock reading. [E] TIME-LIKE TYPING: ALL 16 dihedral reflection axes of the NS seam circle invert the clock lift (R S R^-1 = S^-1), NONE commutes -- the mu4 clock IS Woit's euclidean rotation (the time translation itself); the gaugeable (theta-compatible) part of the Z8 tower is EXACTLY its 2-torsion {1, (-1)^F} (Hermiticity census over all powers; alpha^4 = the full wrap = the 2 pi rotation of the seam = fermion parity, lattice spin-statistics). [E] POSITIVE PART -- GSO-FIXED RP HOLDS EXACTLY: N = 16 even deg <= 2 parity-projected bond-cut Gram (29,0,0) PD (min eigenvalue 1.78e-6 at 40 digits); N = 8 complete half algebra: the projected Gram kills all odd rows/columns exactly, even block (8,0,0) PD (3.35e-3), no degree truncation; the site-cut defect SURVIVES gauge fixing ((7,9,6)) -- the RP bond placement is gauge-invariant information; family A keeps failing ((17,12,0) indefinite vs family D's (29,0,0)); Kramers: Theta_t^2 = id on A^G, Theta_Fock restricts with Theta^2 = +1; Ramond control: the wrap +1 'true Z4' lift fails state invariance -- the NS/Z8 tower is forced. [E] KILL-TEST BOOKKEEPING: kill test (1) stays discharged also gauge-invariantly (RP separates the families on the invariant sector); kill test (2)'s free shadow does NOT fire (no legitimate Hermitian gauge-fixed form has a negative direction); both stay formally live on the interacting algebra A_hol. [E] WRONG-PROJECTION CONTROLS: invariant-dimension chain parity 120 > deck 64 > clock 32 strict, but Hermiticity singles out PARITY alone (deck and clock averages non-Hermitian): the finer 'gauge' projections are structurally ill-posed pre-quotient. TRANSPARENCY: the first frozen run scored 6/13 (the clock-reading checks failed exactly as the mechanism predicts) and the preregistered deg-2 invariant dimension 30 was a combinatorial error (correct census: 28 quadruples + 4 halved doublets = 32) -- documented in full; verdict by the frozen [C-3] logic (Hermiticity failure => structural-failure branch => UNDECIDED). [C] CONTRACT PRECISION (iii), prose only: 'gauge-invariant subalgebra' at the free level means the GSO/fermion-parity Z2 -- the only theta-compatible part of the mu4 tower (the free-fermion shadow of the E8 glue: (E8)_1 = even NS of SO(16)_1 + R spinor); the clock is the euclidean rotation itself (timelike), its equivariant statement moves to beta2 (OS quotient first, then the clock as reconstructed transfer operator). [O] beta1 proper stays open via beta2; WOIT.OS.TWISTOR.01 stays [O]; no marker moves."
    },
    {
      "id": "v523_celestial_wp5e_wm_derived",
      "cluster": "frontier",
      "descriptionEN": "CELEST.WP5E.WM.01: the constructive derivation of the w_m normalisation -- '1/det_j is the Atiyah-Bott / zeta-determinant fixed-point factor, COMPUTED from three independent sources', closing the residual [O] of v516/v520 at the fibre-zero-mode level (verdict ERFOLG per the frozen preregistration; 26 checks, ~1 s). [E] ATIYAH-BOTT (route i): the equivariant mode ledger of the fibre C^2 equals the closed form 1/((1-q i^j)(1-q i^{-j})) exactly in Q(i) for all n <= 120; REGULAR at q = 1 with Abel value (1/2, 1/4, 1/2) = 1/det_j, det_j = det(1-g^j) = (2, 4, 2); (C,2) Cesaro convergence in exact arithmetic; the fixed-point factor splits off EXACTLY at every truncation level d in {7, 12, 25}, for the phased (skeleton) AND the v520-forced unphased (completion) trace alike -- the Abel-limit contact_j = (Q^(0) - Q^(j))/det_j is the v516 contact vector COMPUTED, not declared. [E] ZETA/QUILLEN (routes ii/iii): det_zeta of the g^j-twisted circle Laplacian = 4 sin^2(pi j/4) = det_j EXACTLY (Lerch + reflection formula symbolically; Hurwitz certificates ~1e-41); Quillen split det Delta = det_j^2, |det dbar| = det_j, with the real positive holomorphic section UNIQUE via the SU(2) conjugation pairing; the delta-1f f1f3 block at the (0, b) nodes carries the exact constant term 1/det_b (deviations <= 5.2e-28). [E] CONSISTENCY CHAIN: the derived weight reproduces the v516 chain number by number -- w_m = (0, 3/2, 2, 3/2) = 4h_m = |mu4| h_m = -4 ch2(T_m); T5 = 0 at every scale; h_2 : h_1 = 4 : 3; the T3 budget 32 - 8c forces c = 4 = |mu4| uniquely (leftovers 24/16/8/0/-8); per-sector Okubo squares (18, 9, 18)<x,x>^2; total 45<x,x>^2; Phi_T3: 32 -> 0, Phi_P: 72 -> 0; psi(A_fix) = +64, psi(contact) = -64. [E] NEGATIVE CONTROLS: 1/det^2 gives w' = (0, 5/8, 1, 5/8) with leftovers (2, 12) and a Quillen contradiction (det Delta = det^4 vs measured det^2); 1/|1-i^j| gives w''_1 = 1+sqrt2 notin Q; the Z2/Eguchi-Hanson anchor PASSES with the same derivation (det = 4, w = 1/2 = |Z2| h^{A1}, 9<x,x>^2 = (1/4) x 36, c = 2 = |Z2|); the SO(16)/D8 KILL fires; diag(i,i) breaks the zeta = AB identity itself (real (2,4,2) vs complex (2i,4,-2i), no real Quillen section); k = 3, 5 orbifolds: w_m = m(k-m)/2 = k h_m exactly in Q[x]/Phi_k -- the weights wander correctly with the centre count, only k = 4 = |mu4| carries the E8-glue chain. [C] TYPED PREMISES (the mandatory fence): TP-REG (Abel/Cesaro/zeta regularisation IS the definition of the equivariant character), TP-Q (Quillen structure of BCOV F1 = v520 TP-4), TP-NUM (unphased numerator = the v520 ERFOLG-A decision layer), TP-CH (the insertion channel (0, j) computes the box-channel weight). [O] the GLOBAL BCOV integral on PT/Z4 beyond the fibre zero-mode factor; the resolved-ALE route (v514/v517) not built; SEAM.EQUIV.TWISTOR.01 untouched. Discovery provenance: experiments/tfpt-discovery/celestial_seam_wp5e_wm_bcov_derivation_probe.py (26/26, 2026-07-23). NO marker moves."
    },
    {
      "id": "v524_woit_beta2_os_quotient",
      "cluster": "frontier",
      "descriptionEN": "WOIT.BETA2.OS.01: the WOIT-beta2 milestone of the OS twistor bridge WOIT.OS.TWISTOR.01 executed -- 'the OS quotient of the free system made explicit' (verdict SUCCESS per the frozen preregistration, [C]-typed per contract precision (iii); 20 checks, ~5 s). [E] H_phys EXPLICIT: N = 16 (deg <= 2): the 37x37 bond-cut OS Gram is exactly Hermitian, parity-block-diagonal, inertia (37, 0, 0) PD (min eigenvalue 1.7801e-6 at 40 digits), null space {0}, dim 37 = 29 (+) 8; N = 8 (complete half algebra): (16, 0, 0) PD (min 3.3471e-3), dim 16 = 8 (+) 8 = 4^2 -- compact euclidean time reconstructs a THERMAL (KMS) representation; exact certificate sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2. [E] TRANSFER SEMIGROUP (Klein-Landau): tau_k exactly Hermitian on all domains (k = 1..7, dims 29, 22, 16, 11, 7, 4, 2), chain identities exact, vacuum fixed; the positivity pattern IS the site/bond dichotomy: even k PSD via the exact square identity T(2j) = A_j*A_j ((22,0,0)/(11,0,0)/(4,0,0)), odd k indefinite ((10,12,7)/(5,6,5)/(2,2,3)) -- the one-step transfer is NOT positive (the free chirality datum, kill-test-3 shadow sharpened), its even powers ARE. [E] THE CLOCK = T^{N/4}: N = 16: tau_4 PD (11, 0, 0) (min 6.6821e-5), compressed spectrum [0.0194517, 1.64144], vacuum eigenvalue exactly 1; N = 8: spectrum EXACTLY {1, sqrt(2) - 1} = {1, 1/delta_Silver} in both parity sectors -- the silver axes of the v519 mu4 torsor return as the clock-transfer eigenvalue; spectral projections ~1e-40; the reconstructed rotation group U(s) = exp(isH) is unitary with the group law. [E] v522 RESOLUTION: the census (745, 96, 0) reproduced; EVERY anti-matching entry is a wrap overlap, every overlap-free entry matches -- the pre-quotient non-Hermiticity is EXACTLY the domain/wrap artifact; on the quotient the clock has spectral calculus. Pre-registered KMS deviation: NO contraction on the compact circle (edge modes C(1)/C(3) = 1 + sqrt(2) = delta_S exactly, det(G - tau_4) < 0). [E] GSO/THETA: the grading survives; the perpendicular torsor mirror descends anti-unitarily (eta_perp in {+-i}), Theta_phys^2 = +1 on every sector (Kramers-free), theta_cut o theta_perp = alpha_{N/2} exactly. [E] CONTROLS: site cut indefinite ((3,3,1)/(2,2,4)) -- the contract kill branch fires there; family A (4,4,0); anti-chiral (8,8,0) no quotient; wrong direction non-Hermitian (mismatches all wrap-overlap). KILL-TEST BOOKKEEPING: (1), (2) strengthened, (3) shadow sharpened, none fires; (4)-(7) untouched; all seven stay live on A_hol. [C] FENCE: 'the clock acting unitarily' is operationalised per contract precision (iii) as the reconstructed transfer/rotation structure. TRANSPARENCY: run 1 scored 18/20 (a sympy simplify failure on a TRUE pi/16 product formula, fixed by a Laurent-polynomial certificate; no criterion changed). [O] beta3 is the next milestone; WOIT.OS.TWISTOR.01 stays [O]. Discovery provenance: experiments/tfpt-discovery/woit_os_beta2_os_quotient_probe.py (20/20, 2026-07-23). NO marker moves."
    },
    {
      "id": "v525_seam_bit_twist_blind",
      "cluster": "frontier",
      "descriptionEN": "SEAM.BIT.TWISTBLIND.01: the twist-state attack on the alignment bit DECIDED -- 'mark-decorated states do not see delta', an honest, decided NEGATIVE result (v508 style): the NINTH side-blind test on the v512 scoreboard (8 -> 9); the free-plus-twist class (everything Wick-computable) is EXHAUSTED (20 checks, ~80 s). [E] ALL LIFTINGS WELL-DEFINED AND BLIND: (a) sigma-gauge arc decoration (Z2): sigma o r = sigma on the swap axes makes the twisted Gram a diagonal unitary congruence D M D -- spectra IDENTICAL to untwisted (dev 0.0); (b) Kadanoff-Ceva 4-twist insertion omega(D .)/omega(D): well-defined on N = 32 (omega(D) real nonzero; the N = 16 odd-m degeneration is the C(even) = 0 checkerboard artifact); the RP SPECTRA depend on delta for the first time in the program (pairwise up to 1.05), but the INERTIA is (16, 0, 0) PD with the SAME eta = +i for every m -- the decoration sees delta, positivity does not; (c) the mu4 twist at beta = pi/2 is pure plane gauge; (c') genuine Bogoliubov defect beta = pi/4: Theta-compatibility selects exactly 2 of 16 sign patterns (NS-wrap forced), the winner genuinely non-monomial -- still PD, spectra identical. STRUCTURE THEOREM (empirical, mechanism exact): EVERY Theta-compatible quasi-free mark decoration is RP-spectrally INVISIBLE (the defect planes never straddle a cut bond; orthogonal congruence). [E] SUB-RESULTS: the defect energy E_def = 1.2752872 for EVERY m (spread 4e-40) -- delta-blind; the 4-twist Casimir ln|omega(D)| measures delta (-0.890 .. -1.348) but is exactly mirror-symmetric under m <-> 8 - m -- a family gauge, never a side selector; Kramers <Gamma>_tw = +1 for every m. [E] TWO BIT-PRESUPPOSING pi/2 STRUCTURES (the harvest): the eta flip +i -> -i on the mark-fixing axes (sigma o r = -sigma, odd sector exactly negated); the twist frustration of the mark-fixing mirror ((4, 4, 0) indefinite for both eta -- the lattice shadow of the Ising twist field's mirror-oddness) -- both formulable ONLY on the fixing axes existing iff delta = pi/2: they PRESUPPOSE the bit (v512 facet class), no selector. [E] NEGATIVE CONTROLS: the twist-free limit = v519/v521 exactly; the site cut keeps failing in every decorated state; the v512 counterwitness is arc-equivalent at N = 16 to the blind member pi/4 -- no state-level selector can exclude it; a single twist has omega(D) = 0 identically (twists exist only in mark pairs); the deck-paired FOUR-twist decoration is the minimal well-defined one. [O] the door narrows to a genuinely INTERACTING A_hol whose OS data are NOT Pfaffian-reducible to the chiral vacuum -- beyond every Wick-computable class. VERDICT: KILL exactly as preregistered; the alignment bit remains genuine discrete input. Discovery provenance: experiments/tfpt-discovery/seam_alignment_bit_twist_state_probe.py (20/20, 2026-07-23). NO marker moves."
    },
    {
      "id": "v526_seam_thermal_kms_nariai_bridge",
      "cluster": "frontier",
      "descriptionEN": "SEAM.THERMAL.KMS.01: the thermal seam -- KMS of the reconstructed free OS quotient CLOSED onto the Nariai/BH temperature chain (the THIRD LEG of c3). Verdict SUCCESS (22 checks, ~8.4 s). [E] beta MEASURED = N at both levels (N=8: omega=arcsinh(2^{-3/4}), kappa=q^{-8} exact; N=16: pairing exponent (N/2)+1=9); T_seam=1/N per clock step, beta_angle=2pi EXACT = 1/(4 c3), T_seam=4 c3 = Bisognano-Wichmann/Hawking normalisation (v208); modular Hamiltonian closed (nu1=cos(pi/24), nu2=cos(7pi/24); Tomita KMS beta=1 on all 256 pairs, residual 2.5e-38); c3 third leg class-1 selected (r=1); entropy S(8/16/32)=0.52187/0.63510/0.75006, DeltaS/ln2 -> 1/6, horizon fractions {1/3,2/3} do NOT transfer; [C-5] silver = C(3)/C(1) = 1/(2 cosh 2 omega - 1); non-compact Stieltjes => T=0. Discovery provenance: experiments/tfpt-discovery/seam_thermal_kms_nariai_bridge_probe.py (22/22, 2026-07-23). NO marker moves."
    },
    {
      "id": "v527_seam_clock_silver_spectrum",
      "cluster": "frontier",
      "descriptionEN": "SEAM.CLOCK.SILVER.01: the silver clock eigenvalue EXPLAINED and DEMYSTIFIED (double verdict, anti-numerology style; 22 checks, ~10 s). STRUCTURE=SUCCESS: {1, sqrt2-1} closed via diagonal Hankel pencil at N=8, tan(pi/8)=C(3)/C(1), doubling=exact second quantisation, carrier=silver-midpoint axes of the mu4 torsor, gap=arcsinh(1)=gd^{-1}(pi/4); N-scaling shows pure N=8/16 Majorana lattice datum (not continuum); positive clock only for N==0 mod 8; golden family as N=10 control. CONNECTION=KILL: preregistered 11466-candidate census over 16 frozen numbers: 0 hits above null; placebos (golden, plastic) comparable -- NO readout connection. Cross-link v526: sqrt2-1 = C(3)/C(1) thermal kernel ratio. Discovery provenance: experiments/tfpt-discovery/seam_clock_silver_spectrum_probe.py (22/22, 2026-07-23). NO marker moves."
    },
    {
      "id": "v528_seam_bit_twist_class_definition",
      "cluster": "frontier",
      "descriptionEN": "SEAM.BIT.TWISTCLASS.01: the alignment bit as an exact measurable TWIST-CLASS choice (the physical definition; 18 checks, ~46 s). Verdict SUCCESS. [E] Facets #14 (eta-flip <=> delta=pi/2) and #15 (twist-frustration <=> delta=pi/2) both directions exact (cardinality lemma |A+|=|A-| => m=4; g-class census over all 224 axes: flip class exactly at {15,31} for m=4); order parameter O=flip-axis fraction=1/2 at m=4 else 0; class alternation (-,+,-,+) = nontrivial Z2 character of the Z8 clock-tower shadow; gauge-robust; unification #14=#15=flag-transitivity = ONE Z2 invariant (v512 web 13->15); Kramers=POSITION half (derived), twist class=MODULUS half (the input). [C] measurement sketch: defect transport around the axis torsor accumulates eta holonomy = relative pi phase iff bit set -- in principle interferometrically readable. Honest fence: the bit remains formal input. Discovery provenance: experiments/tfpt-discovery/seam_bit_twist_class_definition_probe.py (18/18, 2026-07-23). NO marker moves."
    },
    {
      "id": "v529_seam_interacting_toy_fk",
      "cluster": "frontier",
      "descriptionEN": "SEAM.INT.FKTOY.01: the minimal interacting Fidkowski-Kitaev seam toy -- the FIRST firing Kill-Test-2 shadow (triple-verdicted; 24 checks, ~97 s). F1=KILL: Theta exists (U_r^2=2^8, Theta^2=+1, conj_V V=4096, [Theta,H_g]=0) so Kill-Test 1 does NOT fire; BUT Kill-Test 2 FIRES at toy level -- RP breaks for every g>0: inertia ladder (37,0,0)->(33,4,0)->(31,6,0)->(29,8,0); Gibbs beta=2 same ladder; mechanism = interference; STRADDLE LAW (24/24): RP fails exactly on quartet-straddled cuts. F2=KILL: the TENTH side-blind test (9->10) -- Theta_15 maps H(m)->H(8-m) EXACTLY despite delta-seeing spectra (spread 1.32). F3=ERFOLG: grading + Z8 tower (Utilde^2=256 Gamma, V^2=256 Utilde) + FK anchor + constructive OS quotient at pi/2. [C] fence: one toy, one interaction class, flat-band parent -- narrows WOIT route, does NOT close it; WOIT.OS.TWISTOR.01 stays Open. Discovery provenance: experiments/tfpt-discovery/seam_interacting_toy_fk_probe.py (24/24, 2026-07-23). NO marker moves."
    },
    {
      "id": "v530_center_quotient_compiler",
      "cluster": "registry",
      "descriptionEN": "Center quotient: Cv=v, Z³/Zv = [[8,2],[5,3]] atom matrix; row sums (7,11,13) self-code; CP code (1,7,11,13); det(C+kI)=(k+1)det(A+kI)"
    },
    {
      "id": "v531_coxeter_tensor_stage_a",
      "cluster": "horizon",
      "descriptionEN": "Stage A tensor clock: T30 = Comp(Φ5)⊗Comp(Φ6), order 30, χ=Φ30; Ramanujan traces c30(k)=c5(k)c6(k); atom readout (2,4,8)"
    },
    {
      "id": "v532_e8_degree_modular_checksum",
      "cluster": "core",
      "descriptionEN": "Dual-degree checksum: (60,192,240,252) → 744=3·248, |W(E8)|, gcd 12; j = q⁻¹+744+…; D16 control 1488=3·496; primes<30 fingerprint [audit]"
    },
    {
      "id": "v533_ftransfer_disc7_norm",
      "cluster": "masses",
      "descriptionEN": "Sheet transfer = disc −7 norm line: D(t)=N(α_t), α_{t+1}=α_t+2, norms (2,4,14,32) on J→K→C→F, constant disc −7"
    },
    {
      "id": "v534_seam_straddle_cone",
      "cluster": "frontier",
      "descriptionEN": "SEAM.STRADDLE.CONE.01: the RP cone on the interacting FK quartets — the FIRST DYNAMICAL SELECTION of the alignment bit (15 checks, ~7 s). V1=KILL: the literal leading-order cone is empty everywhere (linear pencil cannot carry the selection; recoverable content = the straddled-cut drift sign pattern). V2=ERFOLG — THE SELECTION LAW: reflection positivity holds for EXACTLY ONE member × sign of the equivariant FK family — δ=π/2 with positive coupling, PD (37,0,0) on all four cuts incl. the untested straddled clock axes 7/15 for g∈{1/32..8}, min eigenvalue LIFTED off the RP boundary (1.78e-6 → 5.5e-5); asymmetric members die (s=+ at 1/32, s=− from 1/8). Exact geometry: π/2 = the unique symmetric-straddle member (quartet partition, reflection-closed straddles sign +1, four admissible cuts); Fix rays measured. Typed cross-link: the selected member m=4 carries the v528 twist-class order parameter O=1/2. [C] one toy / one interaction class; NO marker moves."
    },
    {
      "id": "v561_cp_channel_identities",
      "cluster": "flavor",
      "descriptionEN": "CP.CHANNEL.IDENT.01: the CP channel / invariant identities (T177, contract CP.INVARIANT) — the CP mirror as algebra: the prime-front gauge-degree toolkit turned on the four CP ledger rows, promoted as exact narrowings (no marker moves, no CP derivation claimed): (1) the HEXAGONAL CHANNEL IDENTITY — on the CM unit ρ = e^{iπ/3} the E₈ split 248 = 120+128 has an EXACT fiber realisation, Σ_k ρ^{d_k} = 4ρ (arg = π/3 exactly) and Σ_k ρ^{m_k} = 4 (arg = 0 exactly), with the parity typing forced by integer parity (degrees even ⇒ phase channel SHEET-EVEN, exponents odd ⇒ magnitude channel SHEET-ODD) — consequence: the phase-channel angle is sheet-blind, it carries π/3 but cannot distinguish quark from lepton; mutations break on e^{iπ/5} and on mutated lists; honest caveat wired as a check (reality of the exponent sum alone is not hexagon-specific); (2) ZERO PHASE IN THE MAGNITUDE DATA — arg(∏c) = 0 exactly on both TFPT sectors, and the μ₃ branch group is the kernel because its order divides N_fam = 3 (a μ₄ branch already shifts the phase): the REDTEAM.D.01 residual is a FIBER datum, not a magnitude datum; (3) TIER-1 EMPTINESS by the blindness theorem — both kernels commute with conjugation, so every π/3-carrier is κ-odd and invariant+odd ⇒ Im F = 0: strict invariance is empty on π/3-carriers, per algebra; (4) the FRAME REMOVAL — det(1,d,n) = 21 and Im det(1,d,ρn) = 21 sin(π/3) exactly, the orientation functional homogeneous of degree +1 in the frame normalisation (21 sin(π/3) is FRAME data), and Im(ρ det)/det = Im ρ = sin(π/3) exactly: the invariant core is a fiber datum, the sign a declared orientation (the DUAL.FRAME.01 narrowing); (5) the TWO TIER-2 CHARACTERS — P2 = arg Σρ^{d_k} at (χ(κ),χ(σ)) = (−1,+1), V5 = Im(ρ det)/det at (−1,−1), homomorphisms verified on the full four-group: two INEQUIVALENT invariants carry the same π/3 (the DEGENERATE verdict as algebra), neither frame-coupled — plus the Jarlskog anchor (TFPT CKM rebuild unitary to 1e-41, J = 3.327021e-5 = v88; J passes rephasing/CP-oddness and fails only the π/3-carrier test: the benchmark validates the criterion, the full δ = π/3 + 3λ_C² has no character — the CP.MOD.01 scope narrowing); R1/R2 typed OPEN, REG.FREEZE.01 untouched (25 checks, ~1 s)"
    },
    {
      "id": "v564_cp_fiber_pin",
      "cluster": "flavor",
      "descriptionEN": "CP.FIBER.PIN.01: the frame door of the CP mirror, decided (T178, contract FRAME.CARRIER) — R1 decided, R2 selected, the degeneracy explained (verdict FIBER-PINNED; no CP derivation anywhere): (1) the EULER FACTORISATION — the fiber acts on the frame by a scalar on the n-row, so every homogeneous block factorises P(ρn) = ρ^m P(n) (fully generic degree 1–4 polynomials, symbolic coefficients): every scale-free functional decomposes as Σ_k A_k Im(ρ^k) with A_k real degree-0 frame ratios — the fiber content is QUANTISED; (2) the NATURAL CLASS IS FIBER-ONLY (T177-R1, natural half, negative BY THEOREM) — metric-free invariants of vector tuples are generated by volume-form brackets (Weyl 1946 FFT, the one cited class boundary), and the exact enumeration on {1, d, n} gives ONE generator (six permutations = ±Δ, the other 21 brackets vanish identically in a generic n): every scale-free ratio is constant — no frame-coupled scale-free π/3-carrier exists without a metric, T177's obstruction sentence is now a theorem; (3) the METRIC DOOR IS NON-CANONICAL (R1, metric half) — declared metrics admit frame-coupled Tier-2 carriers, non-unique at BOTH levels: Δ/⟨d,n⟩ and Δ/⟨1,n⟩ carry (κ,σ,Ref) = (−1,−1,+1) (the odd Δ-power absorbs the reflection) while Δ²/⟨n,n⟩ carries (−1,−1,−1), and the admissible normalisers contain the continuum q_t = t·n₁²+n₂²+n₃² (deck-invariant for every t) with values 441·sin(π/3)/(25t+117) — NO canonical frame factor; anti-numerology wired as a check (21/8, 21/2, 441/142 pairwise distinct, none claimed); (4) the SINGLE-SINE LEMMA — {|Im(ρ_n^k)|} has exactly ONE nonzero element iff n is crystallographic (n ∈ {1,2,3,4,6} = {n : φ(n) ≤ 2}), by sin(4π/n) − sin(2π/n) = 2 sin(π/n) cos(3π/n) with its only zero at n = 6 on n ≥ 5, enumerated exactly to n = 24: the compiler's two clocks μ₄ (j = 1728) and μ₆ (j = 0) are the two MAXIMAL single-sine fibers, so value-based pinning is impossible by theorem — T177's DEGENERATE verdict was forced, not found (on μ₅, values DO separate carriers); (5) the ANCHOR SELECTION (T177-R2, selected) — the Jarlskog invariant on the TFPT CKM rebuild has full character tuple (−1,−1,−1) under (κ, σ, deck-Ref), each certified at 40 dps: it matches the sheet-ODD orientation class V5 and not the sheet-even P2 = (−1,+1,−1); the (−1,−1,+1) metric carriers are anchor-rejected and the matching family carries the continuum — the unique anchor-matching CANONICAL carrier is the fiber-pure Im(ρ) = sin(π/3); conditional on the anchor (selected, not derived), deck column a declared model, REG.FREEZE.01 untouched, no marker moves (22 checks, ~2 s)"
    },
    {
      "id": "v565_woit_beta3_pt_duality",
      "cluster": "frontier",
      "descriptionEN": "WOIT.BETA3.DUALITY.01: the WOIT-beta3 milestone of the OS twistor bridge executed — 'the PT ↔ PT* duality typed against sigma_std' (verdict SUCCESS per the frozen preregistration; the contract kill branch is EMPTY BY ALGEBRA; no marker moves): (1) the COMPATIBLE CLASS — a duality is a nondegenerate Hermitian form Phi on C^4, and clock compatibility (rho4 = diag(i,1,i,1)) forces the four cross-eigenspace entries to zero exactly: two 2×2 Hermitian blocks on {e1,e3} ⊕ {e2,e4}, 16 → 8 real parameters (deck adds nothing; the wrong clock lift breaks the split); (2) the DICHOTOMY — pairing against Woit's Euclidean structure rho_tw splits the class exactly in two: the EUCLIDEAN branch (sigma_std's home) has seam-line norm a(|z|²+1), no null point, no OS cut ever; the MINKOWSKI branch has norm a(|z|²−1) — the null locus meets the seam line EXACTLY in the seam circle for every member (the OS cut is FORCED, not imposed), and the second block is exactly minus the conjugate of the first (charpoly mirrored): the TOTAL SIGNATURE IS (2,2) for every nondegenerate member — Woit's Minkowski signature is a theorem of clock + rho_tw pairing, not an input; (3) the KILL IS EMPTY BY ALGEBRA — every compatible duality inverts clock AND deck (family D); a clock-centralising (family A) duality would need rho4² = scalar, but rho4² = diag(−1,1,−1,1): no solution, decided not scanned; s² = diag(a²,a²,c²,c²) > 0: Kramers-free; (4) the INDUCED MEMBER IS UNIQUE — seam preservation ⇔ b = 0, and then every member induces the SAME conjugation z → −conj(z) (mu = −1, fixed points ±i), perpendicular to sigma_std's z → +conj(z) (mu = +1, fixed points ±1), the clock exchanging exactly these two; (5) the QUOTIENT IDENTIFICATION — exact placement arithmetic and coordinate transport land the induced member on the k = 3 axis: it IS theta_perp, the mirror whose descent v524 certified (Theta_phys), while sigma_std's member transports to the CUT axis (the OS metric itself) — role separation complete; v524 regressions re-verified at N = 8 (Gram PD (16,0,0), anti-unitarity exact with eta_perp = +i, Theta² = +1 on both sectors, time reversal, torsor closure); controls: family A (4,4,0) no quotient, the Euclidean branch cannot cut, non-mu4 reflections not mark-anchored. Free/quotient level only; A_hol untouched; gamma is next (22 checks, ~2 s)"
    },
    {
      "id": "v566_parabolic_anchor_selfcode",
      "cluster": "flavor",
      "descriptionEN": "DIAMOND.PARA.SELFCODE.01: the parabolic anchor self-code (external review 2026-07-31, audited and promoted) — the v218 sheet-diamond direction operators U = Q·diag(1,0,0), V = Q·diag(0,1,1) generate the FULL 2|1 parabolic (line-stabiliser) algebra A (dim 7, word determinant −81, multiplicatively closed, mutation-fragile); the anchor a = (1,1,2) is the spectrum of H = I + V(V−I)/2 (dominant Lagrange projector, idempotent, trace 1), and χ_H = x³ − 4x² + 5x − 2 carries the DOUBLE self-code: (4,5,2) = (|μ₄|, g_car, |Z₂|) = (flag dim_R(GL₃/P), Levi dim(A/J) = 2²+1², radical dim J) — the anchor reproduced spectrally through V and categorically through A; the UNIQUENESS THEOREM: x³ − 2mx² + (m²+1)x − m = (x−m)(x²−mx+1), all-positive-integer spectra force m = 2 — a = (1,1,2) is the only member this parabolic family can code; the 81 as a lattice index: the SATURATED integral order Z⟨U,V⟩ has elementary divisors (1,1,1,3,3,3,3) — index 3⁴ = 81 = the v218 determinant-staircase sum (global determinant charge = integral saturation index; mark-local mechanism [O]); the A2 bit carrier: the basic algebra is the A2 path algebra with Ext¹ = 1 — exactly two module classes (split / one nonsplit): the alignment bit gains a canonical algebra carrier (selection stays with RP/geometry); the trace code p_n(a) = 1 + tr(Vⁿ) with 240 and rank 8 as trace readouts (typed re-encoding of AnchorLadder); the CR = 4/3 preregistered test design for CONTRACT.F.01 (universal-progression honesty wired; negative control 28/25; y_F forced fit-free); and the Lorentz/relative-pencil form of the Paper-II rank-3 classification (eigenvalue locking as the named instrument). NO P2 reduction — the 'No Carrier Reconstruction' audit is the named [O] door; the 2..9 table typed as compression (14 checks, ~1 s)"
    },
    {
      "id": "v567_carrier_free_q_reconstruction",
      "cluster": "flavor",
      "descriptionEN": "QGEO.QRECON.01: the No-Carrier-Reconstruction audit, Stage A — can Q (hence V and the v566 anchor self-code) be reconstructed WITHOUT carrier/anchor inputs?  The circularity located: v11's uniqueness certificate uses row sums (4,5,6), column sums (9,5,1) and chi_Q with its 5 = g_car — all FORBIDDEN here.  Allowed: the sheet involution, the winding normalisation Q(I+Σ) = 2N_fam·W, the A3 exponent spectrum {1,2,3}, the sheet-monodromy budget Q₋² = N_fam·P, nonneg integrality, the seam-line demand, and one family invariant (saturation index 81). Results (exact enumeration): 52 admissible → 28 parabolic — and ALL 28 code the SAME anchor (H' = I + Π_dom has char poly x³−4x²+5x−2, spectrum (1,1,2)): THE ANCHOR IS CARRIER-FREE; the 81-pin then selects exactly TWO members — the corpus Q (q32 = 2 = |Z2|) and its q32 = 0 twin — on which every classical carrier-free certificate coincides (anticommutator spectrum (6,−4,−2), commutator charpoly t(t²+12), det 3, saturation 81) while the ONE separating certificate is the carrier column budget (9,5,1) vs (9,3,1): exactly the forbidden input — the residual freedom is ONE binary sheet-block-coupling choice (resonance with the split/nonsplit alignment bit recorded as typed observation).  Ablations: dropping the winding input widens 28 → 1376 at bound 3 (real load); bound-stable 6 → 9.  Stage A only: the audit RELOCATES to GATE.QGEO.01 (derive the geometric inputs themselves) and to the binary selector; NO P2 reduction, markers untouched (9 checks, ~1 s)"
    },
    {
      "id": "v568_bit_selector_ladder",
      "cluster": "flavor",
      "descriptionEN": "DIAMOND.BIT.SELECT.01: the residual reconstruction bit located, separated and given a canonical witness — the two 81-twins of the v567 audit (q32 = 2 = |Z2| vs 0) generate DIFFERENT integral orders (equal index 81, different lattices: the bit is integrally visible where every classical invariant coincides); the dominant mode of the sheet direction is v(c) = (1, 2, 2c) identically, and the LADDER SELF-CONSISTENCY — the dominant mode reproduces its own eigenvalue as the geometric ladder (2⁰,2¹,2²) = (1,2,4) — picks c = 2 UNIQUELY on all 28 members of the carrier-free class (no 81-pin, no column budget needed); the witness is prior structure (AnchorLadder steps 2ⁿ, 2 = |Z2|, the v566 trace code), not a found number; the SPLIT geometry sits at q32 = 1 and is EXCLUDED by the demands already imposed (algebra collapses to dim 5 — the v566 A2 Ext arrow is nonzero on the whole class; neither twin splits: the bit toggles WHICH nonsplit order is realised, not split/nonsplit); Priority-2 candidates tested honestly: the mu4-grading fails (order not graded — 81 stays a fingerprint), left V-multiplication induces a canonical F3-action on the cokernel (Z/3)⁴ — computed: charpoly x(x−1)²(x+1) mod 3, eigenvalues {0,1,1,−1}, NOT the regular μ₄ pattern; U acts as zero; twins identical — after two candidates the 81 stays a fingerprint.  The physical selection is NOT derived — why the ladder self-consistency should hold is the relocated open question, the same slot as the RP-selection of the alignment bit (11 checks, ~1 s)"
    },
    {
      "id": "v571_cr_discrete_corners",
      "cluster": "frontier",
      "descriptionEN": "FTR.CRDISC.01: the preregistered cross-ratio check (self-code review Priority 4 / v566 S9), executed on its DISCRETE half — the four solver corners are the in-repo Sheet-Diamond kernels J,K,C,F = M(1,t), t = −2..1 (v224, dets (2,4,14,32)), and the corpus's one canonical declared solver reading, the Koide anchor response ℓ(M) = aᵀM·1 (v183), reads (26, 35, 44, 53) on them: CR = 4/3 EXACTLY, and the fit-free reconstruction forces y_F = 53 from (26,35,44) alone — the Koide numerator is Möbius-determined by the three upstream corners (new exact cross-corner identity); honesty theorem: ℓ is affine on the path (step 9 = N²fam) and every linear functional passes — the check separates linear from nonlinear exports; negative controls with teeth: determinants fail at 28/25 as preregistered, the dominant eigenvalue fails at 1.3222; the DISCRETE half only — the continuous solver states stay open exactly as CONTRACT.F.01 fences them (9 checks, <1 s)"
    },
    {
      "id": "v572_rp_bit_form",
      "cluster": "flavor",
      "descriptionEN": "DIAMOND.BIT.RPFORM.01: the RP form on the 81-twins — spectrally BLIND, integrally ANCHOR-VISIBLE: with the corpus reflection θ(X) = ΣXᵀΣ and the declared states (trace, anchor (1,1,2), ladder (1,2,4), unit), the RP kernel has identical inertia on both twins for every state and is never PSD (no kinematic v534-style positivity selection exists — honest negative, consistent with v567), but the INTEGRAL Smith towers of the anchor-state Gram separate them: (2,2,18,18,18,18,0) vs (2,2,2,18,18,18,0) — one slot, 18 vs 2, a factor 9 = N²fam: the compiler twin carries an extra 3² of divisibility, the ladder state separates identically, anonymous states are blind — the bit is visible to the RP form ONLY through anchor-coupled states; trace discriminant −6561 = −81² on both; q32 = 1 collapses to dim 5 (v568 reproduced); robust under the compiler V-flow (semigroup-deformed kernels keep identical inertia); the Gibbs pencil of the anchor Hamiltonian separates DYNAMICALLY: the compiler twin's kernel degenerates at a finite critical temperature s* = (8−4√19)/15 inside the Gibbs range (det factors with an extra quadratic; constant 81²), the alt twin's never does — a rank-drop marker, not a positivity selection; the critical mode is IDENTIFIED: seam-row supported (the A₂-Ext-carrier row), seam-line fixing, sheet-flow fixed point — the alignment-bit resonance deepens to a shared carrier; the half-sided (Levi/radikal) Restriktionen bleiben ebenfalls blind — the interacting RP construction is the one remaining route, typed open (11 checks, ~1 s)"
    },
    {
      "id": "v574_stageb_winding",
      "cluster": "flavor",
      "descriptionEN": "QGEO.WINDING.SELF.01: Stage B of the No-Carrier-Reconstruction audit, first narrowing — the involution TYPE is killed by a rank theorem (any 2-dim-positive involution gives rank-1 V′, char poly x²(x−t) identically: the binary trace code {0,1,2} is unreachable), (E) bounds the winding normalisation (q11 = N must be an A3 exponent — N ≥ 4 empty), the ladder self-consistency selects exactly ONE member for every N ∈ {1,2,3} (28/16/14 → 1/1/1), and the saturation self-consistency sat = N⁴ holds ONLY at N = 3 (32 ≠ 1; 512 ≠ 16; 81 = 3⁴): the double self-consistency pins (N, Q) = (3, compiler Q) uniquely — N_fam = 3 relocates from declared input to consistency-forced within the reconstruction; honest decomposition (full 35-spectrum scan in the probe): the surviving family is exactly {1,2,N} with the compiler shape — the self-consistency forces the sheet spectrum {1,2} + shape, (E) contributes the one datum N = 3; REDUNDANZ: (M) und (L) sind relativ zum Selbstcode-Paket redundant (beide gestrichen: Klasse 1274, Compiler-Q weiter eindeutig) — und die MAXIMALE Ablation streicht auch (E): (W)-Spalte 3(1,1,1) + Ganzzahligkeit + Selbstcode pinnen den Compiler allein (29 Leiter-Kandidaten, 1 Treffer) — deklariert bleiben Involutions-Existenz, die Winding-UNIFORMITÄT (ohne sie: Vier-Spalten-Familie, genau eine uniform = der Compiler) und N_fam = 3 (E8-Glue); GATE.QGEO.01 um FÜNF Punkte verengt, NOT closed; no P2 reduction (11 checks, ~6 s)"
    },
    {
      "id": "v575_cr_continuous_uniflow",
      "cluster": "frontier",
      "descriptionEN": "FTR.CRCONT.01: the continuous half of the preregistered cross-ratio check, executed against the in-repo solver dynamics — UNIFLOW-DEGENERATE: the frozen protocol exports (fixed-point multipliers of the native contractions, v425) give the quadruple {(2/3)⁶, (2/3)⁶, 1, −7} — the repetition is FORCED by the single-flow theorem (F_pole and F_Boltzmann are literally the same seam contraction), and a repeated quadruple gives CR ∈ {0, 1, ∞} in every corner assignment (theorem + all 24 permutations): the continuous half can never pass natively — it closes onto v425; the only four independent external data are the dimensionful anchors (v_geo, M_R, C_p, θ_i), [O] walls, not projective states; the meaningful CR content stays the discrete half (v571, y_F = 53 forced); a future continuous test needs genuinely external anchor-level exports (7 checks, <1 s)"
    },
    {
      "id": "v578_native_jets",
      "cluster": "frontier",
      "descriptionEN": "FTR.JETS.01: the jet/Schwarzian contract executed natively — the exact v99 Koide trajectory has wall-time Schwarzian {q,t} = −Δ²/2 = −18 log(3/2)² (constant, trajectory-independent; F_Boltzmann identical by the single-flow theorem) and is Moebius in its own exponential clock (Schwarzian 0); the 1-loop QCD flow is Moebius in its own RG time already (Schwarzian 0); the native quadruple {−Δ²/2, −Δ²/2, 0, degenerate} shares no wall-time Schwarzian — the discrepancy is exactly one exponential clock change: each instance is PROJECTIVE IN ITS OWN CLOCK, the obstruction is the clock (anchored by the external data); a future joint jet test needs external anchor-level jets, as v575 fenced (6 checks, <1 s)"
    },
    {
      "id": "v584_ext_source_response",
      "cluster": "core",
      "descriptionEN": "DIAMOND.BIT.EXTSOURCE.01: die interagierende RP-Fortsetzung von v572 — Rang-1-Ext-Source-Kopplung auf dem Gibbs-Pencil trennt die 81-Twins RICHTUNGSABHÄNGIG im Einheits-Kopplungsfenster: exakte rationale Flip-Schwellen J_c(s) = -1/(v^T K(s)^{-1} v), deren Zähler GENAU die v572-Pencil-Faktoren sind; in der gecrossten Phase heilt positive Kopplung J <= 15/16 < 1 den Compiler-Twin (über beide Ext-Pfeile), während der Alt-Twin von keiner positiven Kopplung in |J| < 1 geflippt werden kann (E31-Schwelle >= 1, Gleichheit exakt bei s = 0; E32-Schwelle negativ); Richtung strikt positiv (konsistent mit v534); Pol der Compiler-Antwort auf dem GOLDENEN Polynom s^2-s-1 (−1/φ = −0.618, neben s* = −0.629); Mechanismus: der kritische v572-Modus ist Ext-sichtbar, daher J_c(s*) = 0 exakt; ehrlich: Response-Level-Selektion, kein OS/RP-Positivitätssatz, Contract bleibt offen (8 checks, ~1 s)"
    },
    {
      "id": "v590_involution_existence",
      "cluster": "core",
      "descriptionEN": "QGEO.INVOL.01: die Blatt-Involutions-Existenz von GATE.QGEO.01 — EXISTENZ UNCONDITIONAL, Auswahl auf zwei explizite Forderungen verengt: die integralen V-kommutierenden Involutionen sind EXAKT die Spektral-Vorzeichengruppe (Z/2)³ (alle acht integral und ordnungs-stabilisierend, auf BEIDEN Twins — Rigidität); Verengungskette: v574-Rangsatz killt die Spur-+1-Hälfte (6→3, alle Σ-Signatur (1,−1,−1)), Seam-Umkehr Se₃ = −e₃ (3→2), Zero-Moden-Fixierung (+1 auf ker V) singularisiert S₀ = [[1,−1,0],[0,−1,0],[0,0,−1]] EINDEUTIG; ehrlich: die Korpus-RP-Reflexion kommutiert NICHT mit V (andere Struktur), der Alt-Twin hat dieselbe Klassifikation (kein Bit-Selektor); GATE-Relocation: Existenz HERGELEITET, Restfreiheit = die zwei benannten Forderungen, deren Kontinuums-Herleitung (μ₄/D₄/H¹) offen bleibt (7 checks, <1 s)"
    },
    {
      "id": "v597_mu4_cover_model",
      "cluster": "core",
      "descriptionEN": "QGEO.COVER.01: Strategie I/II, erster ausgeführter Schritt — der μ₃-Cover y³ = x⁴−1 (verzweigt an μ₄ + ∞) trägt VIER unabhängige Compiler-Fingerprints: (1) Genus 3, H₁ = Rang-3-Modul über den Eisenstein-Zahlen (Chevalley-Weil H^{1,0} = (1,2)); (2) die D₄-Rotation schließt PROJEKTIV ÜBER DEN DECK (r⁴ = ω·1, r¹² = 1, Burau-Modell bei t = ω, Braid-Relationen verifiziert); (3) die Cusp-Klassengruppe ist (Z/3)⁵ mod eine Relation = (Z/3)⁴ der Ordnung 81 = 3⁴ — der v566-Saturationsindex mit exakt dem Pro-Mark-Z/3-Mechanismus; (4) die invariante hermitesche Form ist eindeutig bis Skala mit Eigenwerten {−2, 2−2√2, 2+2√2}: LORENTZ (1,2) — Chevalley-Weil UND Paper-II-Polarisation gleichzeitig (Strategie IIs erste Vorhersage bestätigt); Einzel-Cusp-Gram ganzzahlig {0,0,2}; Q/V-Wörterbuch benannt offen; GATE.QGEO bewegt sich NICHT (6 checks, <1 s)"
    },
    {
      "id": "v598_cover_dictionary",
      "cluster": "core",
      "descriptionEN": "QGEO.DICT.01: das Cover-Wörterbuch, Runde 2 — DREI Compiler-Objekte als kanonische Cusp-Gram-Kombinationen lokalisiert: U ~ G₂ (char x²(x−3)); Q₋ ~ G₁−G₃ (char x(x²−3)) — die μ₄-Sheet-Kopplung ist die Differenz der GEGENÜBERLIEGENDEN Cusp-Grams (die Diamant-Diagonale); Leiter ~ 1+G₁+G₃ (char (x−1)(x−2)(x−4)) — die binäre AnchorLadder (1,2,4) von v568; Stützstruktur: gegenüberliegende Twist-Produkte NILPOTENT, Q₋² ~ x(x−3)²; Negative quantifiziert: Q₊/V/H/Q NICHT in 7296 gescannten kanonischen Kombinationen — die Grading-Hälfte braucht einen Generator außerhalb der Endlich-Cusp-Twist-Algebra (die antiholomorphen D₄-Reflexionen/reelle Struktur als benannter Verdächtiger); Kill-Kriterium NICHT ausgelöst (4 checks, <1 s)"
    },
    {
      "id": "v599_real_structure",
      "cluster": "core",
      "descriptionEN": "QGEO.REAL.01: die reelle Struktur auf dem μ₃-Cover — der v598-Verdächtige KONSTRUIERT: die antiholomorphe D₄-Reflexion wirkt semilinear R(v) = M·conj(v), ist EINDEUTIG bis auf Skala (1-dim Lösungsraum), R² = +1 exakt, alle vier Twist-Intertwinings R Tₖ R⁻¹ = T₅₋ₖ⁻¹ exakt; der Grading-Funktor Γ(A) = M·conj(A)·M⁻¹ schließt die drei v598-Lücken auf Char-Niveau: V ~ G₂+G₃−Γ(G₃) (x(x−1)(x−2)), H ~ 1+G₂−Γ(G₃) ((x−1)²(x−2): das ANCHOR-Spektrum), Q₊ ~ 1+G₂+G₃−Γ(G₃) ((x−1)(x−2)(x−3)); die Γ-Zerlegungen tragen die Compiler-Konstanten (E₂: {0,1,5} = g_car, E₄: {0,3,9}, O₃ nilpotent, O₂ ±√5, O₄ ±√27); Gelenk-Negativ quantifiziert: 4×6 Kandidaten, NULL Paare erfüllen tr(UV)=3 ∧ char(U+V)=charQ (Kill-Kriterium NICHT ausgelöst); die invariante Form J ist eindeutig, det J = 8 (Mitglied der Diamant-Determinantenliste!) mit LORENTZ-Signatur (1,2) — die Hodge-Vorhersage von Strategie II auf Form-Niveau (18 checks, ~3 s)"
    },
    {
      "id": "v600_joint_embedding",
      "cluster": "core",
      "descriptionEN": "QGEO.EMBED.01: die Gelenk-Einbettung — DAS COMPILER-PAAR (U, V) IST AUF DEM μ₃-COVER REALISIERT: der Γ-gerade Grad-2-Sektor liefert U* = −2E₁ + (G₁Γ(G₁)+Γ(G₁)G₁) + 3 und V* = −E₃ + ½(G₂Γ(G₂)+Γ(G₂)G₂) + 1 — beide Z[ω]-integral, alle Chars und Gelenk-Spuren exakt Compiler (tr(UV) = 3, tr(UV²) = 6, char(U+V) = charQ); die Sieben-Wort-Algebra hat Rang 7 und die Simultan-Konjugation g U* g⁻¹ = U_c, g V* g⁻¹ = V_c ist EXAKT gelöst; auf dem saturierten R-Fixgitter (SNF-Zertifikat (1,1,1)) sind die Restriktionen GANZZAHLIG, und die Z-Ordnung der sieben Wörter hat Smith-Elementarteiler (1,1,1,3,3,3,3) — SATURATIONSINDEX 81 = 3⁴, exakt der v566-Selfcode-Index; must-break-Kontrolle besteht; KANONIZITÄT OFFEN (Existenz-Claim); GATE.QGEO bewegt sich nicht (15 checks, ~4 s)"
    },
    {
      "id": "v601_equivariant_dual",
      "cluster": "core",
      "descriptionEN": "QGEO.CANON.01: die kanonische äquivariante Realisierung — die J-adaptierten Cusp-Projektoren GJₖ = (1−Tₖ)J⁻¹(1−Tₖ)†J sind idempotente Rang-1-Projektoren (char x²(x−1)), rotieren EXAKT unter D₄ (r GJₖ r⁻¹ = GJₖ₊₁; gegenüberliegende J-orthogonal); das Paar Uₖ = 3GJₖ, Vₖ = 1 + GJₖ₋₁ − GJₖGJₖ₊₁ ist Z[ω]-integral, Compiler-exakt (Spektren, Gelenk-Daten, Wortrang 7) und D₄-ÄQUIVARIANT PER BAUART (Mark-Wahl = Winding-Wahl); DUALITÄT: exakt konjugiert zum TRANSPONIERTEN Compiler-Paar (die duale Ebenen-Stabilisator-Parabolik; der Intertwiner zum direkten Paar ist identisch singulär — must-fail dokumentiert): der Cover trägt BEIDE Parabolik-Typen; die KLEIN-GRUPPE lebt integral auf der Kurve (alle 8 Spektral-Vorzeichen-Involutionen Z[ω]-integral, S₀ explizit mit Spur −1, A·B = S₀); ehrlich offen: naive J-Wort-Gram-Inertia (3,2,2) ≠ v572 (4,0,3) — Weil-Twist-Vergleich benannt (14 checks, ~12 s)"
    },
    {
      "id": "v602_duality_forms",
      "cluster": "core",
      "descriptionEN": "QGEO.DUALFORMS.01: die Dualitäts-und-Formen-Runde — die J-Adjungierte VERTAUSCHT die zwei Parabolik-Typen in EINER äquivarianten Struktur (U₁ J-selbstadjungiert; V₁^J = exakte Produktumkehr; (U₁,V₁^J) ~ Compiler-Paar, (U₁,V₁) ~ transponiertes Paar); das Wörterbuch schließt in der J-Projektor-Sprache (H = 1+V(V−1)/2 WÖRTLICH die Korpus-Anchor-Formel, char (x−1)²(x−2); Q₊ ~ 2−GJ₁+GJ₂GJ₃; Leiter ~ 2−GJ₁+2GJ₂GJ₃); WEIL-POSITIVITÄT exakt (JC positiv definit, {2√2−2, 2, 2+2√2} = |Spec J| — die Riemann-Bilinearrelationen auf dem ω-Blatt); kanonische Bilinearform B = M†J symmetrisch, det B = 8; ehrlich offen: gᵀg ≠ λB, naive Mark-lokale Abbildung scheitert, Seam-Normale kein Deck-Eigenvektor; Lean-Zertifikat FORM.QGEO.04 (CoverEmbedding.lean, det-freie Smith-Zertifikate, kernel decide) (17 checks, ~8 s)"
    },
    {
      "id": "v603_seam_marks",
      "cluster": "core",
      "descriptionEN": "QGEO.SEAMMARKS.01: die Seam-und-Marks-Runde — DIE SEAM-LINIE IST REELL: das Linien-Stabilisator-Paar hat eine EINDEUTIGE gemeinsame Eigenlinie v (Uv = 0, Vv = v; explizit (0, (√3−i)/(√3+i), 1)); die reelle Struktur FIXIERT sie (die Seam ist eine reelle Linie des Covers), und S₀ KEHRT sie um — die v590-Seam-Umkehr-Forderung gilt GEOMETRISCH; die ∞-Cusp separiert in der unreduzierten Burau (invariante Spalte (1,1,1,1), Volltwist-Eigenwerte {1, ω×3}); DER ERSTE MARK-LOKALE Z/3-ZEUGE: E₁ restringiert ganzzahlig, liegt in der Wort-Algebra mit Nenner-lcm EXAKT 3 — eine Ordnung-3-Klasse im Kokern (der v566/v597-Mechanismus bekommt seinen ersten exakten Zeugen); ehrlich: E₄ trivial, E₂/E₃ außerhalb (äquivariante per-Mark-Lesart getypt); Übersetzer-Negative quantifiziert (F₂ = gᵀΣg kein Menü-Match, B⁻¹F₂ nicht in der Algebra) (15 checks, ~7 s)"
    },
    {
      "id": "v604_equivariant_order",
      "cluster": "core",
      "descriptionEN": "QGEO.EQORDER.01: die äquivariante Ordnung und die ∞-Brücke — KORREKTUR: R IST eine Anti-Isometrie der Polarisierung (die richtige Gleichung M†JM = Jᵀ gilt EXAKT; die v599-Diary-Nebenbemerkung testete die falsche); J restringiert zu einer REELLEN SYMMETRISCHEN GANZZAHLIGEN Form J_fix = [[16,2,4],[2,−2,2],[4,2,−2]] mit det 72 = 8·9; die von ALLEN VIER Mark-Frames erzeugte Z-Ordnung hat VOLLEN Rang 18 in M₃(Z[ω]) mit Smith-Teilern (1¹⁵,3,3,3) — Saturationsindex 27 = 3³; DIE 3-TORSION LOKALISIERT BEI ∞: coker(1−T∞) = Z²⊕(Z/3)³ (Torsion 27), alle endlichen Cusp-Twists torsionsfrei — drei Objekte teilen (Z/3)³: ∞-Monodromie, äquivariante Ordnung, det(1−deck); Lesart: 81 = 3⁴ an den endlichen Marks, 27 = 3³ bei ∞; Übersetzer-Negative erweitert (J_fix-Menü + Algebra negativ; ω̄-Blatt benannt) (12 checks, ~4 s)"
    },
    {
      "id": "v605_translator",
      "cluster": "core",
      "descriptionEN": "QGEO.TRANSLATOR.01: die Übersetzer-Runde — DER RP-ÜBERSETZER EXISTIERT: die Korpus-RP-Anti-Automorphie θ(X) = ΣXᵀΣ, zurückgezogen und mit der J-Adjungierten komponiert, ist ein INNERER Automorphismus φ(A) = cAc⁻¹ (Intertwiner-Raum 1-dim, c invertierbar, exakt) — die RP-Struktur ist vollständig in Kurven-Sprache ausgedrückt (Polarisierung + ein explizites Element); 14c ist Z[ω]-INTEGRAL (Nenner 14 = 2·7); die SEAM ist reine 3-TORSION bei ∞ (freie Koordinaten (0,0), Torsion (2,2,0) mod 3 im ∞-Kokern); DIE 81 ÜBERLEBT das Frame-Joining (Ordnungskette O₁ ⊂ O_joint∩span₁ ⊂ P₁ mit Indizes 3⁵ und 3⁴ = 81); J_fix-Dualität zertifiziert (C_VᵀJ_fix = J_fix·C_Vdual ganzzahlig) + Lean-Nachtrag grün (11 checks, ~11 s)"
    },
    {
      "id": "v606_mode_separation",
      "cluster": "core",
      "descriptionEN": "QGEO.MODESEP.01: die Moden-Separations-Runde — DIE ∞-TORSION TRENNT DIE DREI V-EIGENLINIEN: alle drei Moden (0, 1, 2), auf das unreduzierte integrale Modul geliftet, sind REINE 3-Torsions-Klassen bei ∞ (null freier Anteil), und die drei Klassen sind PAARWEISE VERSCHIEDEN ((2,0,2)/(2,2,0)/(1,2,0) im fixierten Smith-Rahmen; Verschiedenheit rahmen-invariant, keine trivial) — der Selektions-Träger, den die reelle Struktur NICHT liefern konnte (nur die Seam ist R-fix); ehrlich: die kanonische Regel, welche Klasse das „+1“ der v590-Forderung bekommt, bleibt offen; EHRLICHES NEGATIV: der kubische Körper von c (char(7Γ(c)c) = y³+3y²+6y−1) hat Diskriminante −783 = −3³·29, kein Quadrat → Galois S₃, nicht abelsch → keine Kreisteilungs-Identität für c (Kronecker-Weber); die arithmetische Route ist geschlossen (7 checks, ~2 s)"
    },
    {
      "id": "v607_selection_rule",
      "cluster": "core",
      "descriptionEN": "QGEO.SELRULE.01: die Selektions-Regel-Runde — DIE KANONISCHE „+1“-REGEL IST GEFUNDEN: die Torsion des ∞-Kokerns ist ein echter F₃-Raum (ω wirkt als Identität), die Deck-Rotation wirkt F₃-linear mit der expliziten Matrix [[0,1,0],[1,0,2],[2,2,2]]; char mod 3 = (x+1)(x²+1) mit x²+1 IRREDUZIBEL über F₃ — GENAU EINE projektive Eigenlinie (λ = −1), strukturell — und sie IST die Top-Mode-Klasse (V = 2); Nullmode und Seam sind nicht projektiv fixiert (auch nicht unter r²): mit v603/v605 tragen die drei Moden paarweise ausschließende kanonische Labels (Seam = R-REELL, Top = DECK-EIGEN, Nullmode = UNMARKIERT) — S₀ = −1 exakt auf den markierten, +1 auf der unmarkierten: BEIDE v590-Forderungen durch Cover-Struktur ausgedrückt (getypte Lesart, keine Kontinuums-Herleitung) (10 checks, ~2 s)"
    },
    {
      "id": "v608_gamma_free_slice",
      "cluster": "core",
      "descriptionEN": "WOIT.GAMMA.FREE.01: die freie γ-Scheibe der WOIT-Roadmap — SPIEGEL-MODEN SIND STRIKT OS-NEGATIV: der Anti-chirale Zustand hat Inertia (8,8,0) mit SEKTOR-EXAKTEM Split (gerade (8,0,0) PD, ungerade (0,8,0) strikt ND) — jeder Spiegel-Fermion-Vektor hat strikt negative OS-Norm, KEIN Spiegel-Mode überlebt die OS-Rekonstruktion (Kill-Test 3 kann auf freiem Niveau nicht feuern; Sektor-Upgrade des v519-Schattens); das komprimierte Clock-Spektrum auf dem Fermion-Bereich ist EXAKT {1, √2−1} — multiplizitätsfrei, EINE Generation; MARK-INZIDENZ frei: beide Mark-Algebren PD (4,0,0) mit dim 4 = |μ₄|, Produkte erzeugen die volle 16-dim Basis, und τ₂ IST der Mark-A→Mark-B-Transport (α₂: {4,5}→{6,7}, hermitesch PD) — Kill-Test 7 feuert frei nicht; FIREWALL: nur die freie Scheibe, γ selbst bleibt auf A_hol offen (9 checks, ~1 s)"
    },
    {
      "id": "v609_translator_factorization",
      "cluster": "core",
      "descriptionEN": "QGEO.CFACTOR.01: die Übersetzer-Faktorisierung — DIE GEOMETRISCHE IDENTITÄT VON c IST GESCHLOSSEN: der Spiegel-Intertwiner d (dA = Γ(A)d für das realisierte Paar) ist EINDEUTIG (1-dim Lösungsraum), UNIMODULAR (det d = 1), INTEGRAL über Z[ω] und Γ-INVOLUTIV (Γ(d)d = 1 exakt); mit K₁ = J⁻¹·conj(F₂)ᵀ·M⁻¹ faktorisiert das v605-Übersetzer-Element EXAKT: c = λ·K₁·d — θ_corpus = Jadj ∘ inner(K₁) ∘ inner(d): Polarisierung × RP-Metrik × reelle Struktur × kanonischer Spiegel-Intertwiner; der letzte der drei v605-Reste ist aufgelöst (8 checks, ~8 s)"
    },
    {
      "id": "v610_curve_landscape",
      "cluster": "core",
      "descriptionEN": "QGEO.LANDSCAPE.01: die Kurven-Landschafts-Runde — WAS DIE KURVE SONST NOCH TRÄGT: die Jacobische ZERFÄLLT (Differential-Charaktere der Ordnungen (12,12,6)) in eine abelsche Fläche mit CM durch Q(ζ₁₂) = Q(i,√3) — den Körper von Deck UND Marks zusammen — und die elliptische Kurve E: y³ = u²−1 mit CM durch Q(ω); PUNKTZÄHLUNGS-BESTÄTIGUNG: a_p(C) = a_p(E) EXAKT für alle 10 getesteten p ≢ 1 mod 12 (die ζ₁₂-Fläche dort supersingulär), Flächen-Spuren nur bei p ≡ 1 mod 12; a_p = 0 für p ≡ 2 mod 3; GANZZAHL-SPEKTREN-CENSUS mit ehrlicher Relativierung: Einzelspektren sind BILLIG (7894 Einträge), die ANCHOR-KLASSE ist die häufigste nichttriviale Familie (286 = Maximum), Nicht-Compiler-Spektren existieren ({1,1,3}: 254) — die Compiler-Auszeichnung ist die GELENK-Struktur (v600), nicht Einzel-Chars; zwei neue kanonische Objekte benannt (Perioden = Beta-Werte) (9 checks, ~1 s)"
    },
    {
      "id": "v611_periods",
      "cluster": "core",
      "descriptionEN": "QGEO.PERIODS.01: die Perioden-Runde — DIE ANALYTISCHE SCHICHT DES COVERS IST KLASSISCH: die fünf Basis-Perioden-Integrale sind EXAKT Beta-Werte ((1/4)B((k+1)/4, 1−j/3), symbolisch), und die Gamma-Nenner (Γ(11/12), Γ(7/12), Γ(5/12), Γ(1/12)) liegen ALLE auf dem ζ₁₂-RASTER — die 12 = |μ₄|·|μ₃|-Struktur auch analytisch; die μ₄-Rotation wirkt auf Perioden EXAKT mit den v610-Charakteren (i^(k+1), 40-stellige Zertifikate); Ω_E = B(1/3,1/6)/2 = Γ(1/6)Γ(1/3)/(2√π) exakt (die klassische ω-CM-Periode); Kreuz-Faktor-UNABHÄNGIGKEIT: I_{1,2}/I_{0,2} hat KEINE ganzzahlige Relation vom Grad ≤ 4 (PSLQ) — die zwei CM-Faktoren sind perioden-unabhängig, wie der v610-Zerfall verlangt; das Perioden-Wörterbuch als Hodge/RP-Ressource benannt (6 checks, ~1 s)"
    },
    {
      "id": "v612_polarization_frame",
      "cluster": "core",
      "descriptionEN": "QGEO.POLFRAME.01: die Polarisierungs-Rahmen-Runde — J DIAGONALISIERT EXAKT IN DER ROTATIONS-EIGENBASIS: die Braid-Rotation r hat drei EINFACHE Eigenwerte ζ₁₂·{1,−1,−i} (det r = −1); in der Links-Eigenbasis wird J exakt diagonal, diag((1+√3)/2, (1−√3)/2, −1) (ehrlich: Diagonalität = Standard-Folge der J-Unitarität; der GEHALT sind Werte und Signatur-Verteilung); **DIE HODGE-RICHTUNG IST DER ζ₁₂-EIGENRAUM**: die eindeutige positive Richtung von J (Signatur (1,2)) ist exakt der Eigenraum der PRIMITIVEN 12. Einheitswurzel e^{iπ/6} — analytische Hodge-Struktur und endliche Rotation stimmen überein, wo Positivität wohnt; Produkt der Diagonalwerte = det J/|det W|² = 1/2 exakt; benannter nächster Schritt: die kanonische Perioden-Normierung (Beta-Gewichte) (9 checks, ~2 s)"
    },
    {
      "id": "v613_canonical_periods",
      "cluster": "core",
      "descriptionEN": "QGEO.PERNORM.01: die kanonische Perioden-Normierung — DIE ANALYTISCHE BRÜCKE SCHLIESST: die Viertel-Rotation x→ix wirkt DECK-FREI auf den vier Verzweigungs-Segmenten (alle 12 Wrap-Faktoren exakt 1; Homologie-Relation seg₁+seg₂+seg₃+seg₄ = 0; Must-fail-Kontrolle: rohe Periodenzeilen sind KEINE r-Eigenvektoren); **DAS EXAKTE WÖRTERBUCH r = Deck ∘ Rotation**: char(r) = char(ω·M) exakt — die reduzierte Burau-Rotation bei t = ω ist GL₃-konjugiert zur geometrischen Rotation mal GENAU EINEM Deck-Schritt (erklärt v597s r⁴ = ω strukturell; Deck-Potenz eindeutig unter den vier Skalar-Twists); die Periodenzeilen (hol dx/y², hol x·dx/y², ANTIHOL conj(dx/y)) realisieren den Rotations-Eigenrahmen mit Charakteren (i,−1,−i) und transportiert spec(r); DIE HODGE-GEWICHTE SIND Γ-MONOME: h₀, h₁ = (3√3/16π)Γ(1/3)²Γ(1/6)², h₂ via komplexer Beta-Formel (≤1e−25 verifiziert), Signatur (2,1); **DIE VORZEICHEN-BRÜCKE J = −h PRO CHARAKTER**: Js einzige positive Linie (ζ₁₂) = hs einzige negative Linie (antiholomorph) — Riemann–Hodge-Positivität durch die Kompiler-Polarisierung realisiert (19 checks, ~6 s)"
    },
    {
      "id": "v614_transport_deck",
      "cluster": "core",
      "descriptionEN": "QGEO.TRANSPORT.01: die Gauss-Manin-Transport-Runde — **DER DECK-SCHRITT IST DIE RAND-MONODROMIE**: der Transport der Segment-Zyklen entlang der starren Viertel-Rotations-Schleife der vier Verzweigungspunkte lässt die Diskriminante λ(τ) = e^{2πiτ} EINMAL um 0 winden, und die transportierten Perioden faktorisieren EXAKT (I_m(τ) = e^{iπτ(m+1)/2}·e^{−2πiτj/3}·I_m(0)); bei τ = 1 ist der Deck-Faktor ω = t⁴ auf JEDEM Blatt (uniform über alle drei Zeilen; must-fail: t³ ≠ ω, t⁵ ≠ ω — der Exponent 4 = |μ₄| ist durch die Punktezahl erzwungen); **die kanonische Transport-Matrix in der Segment-Basis ist EXAKT ω·M — das v613-Wörterbuch r ~ ω·M ist KANONISCH** (kein Konjugator-Spielraum), (ωM)⁴ = ω·1 leitet v597s r⁴ = ω her; 25-stellige Zertifikate (dps 80, max 5e−28; τ=1 auf 2.7e−81); v613s Rest F1 auf Transport-Ebene geschlossen (12 checks, ~7 s)"
    },
    {
      "id": "v615_gamma_toy_interacting",
      "cluster": "core",
      "descriptionEN": "WOIT.GAMMA.TOY.01: die wechselwirkende γ-Scheibe auf Toy-Ebene — die γ-Batterie AUF der RP-überlebenden Wechselwirkungsklasse (das Alignment-Bit: δ = π/2, positive Kopplung, v534) des 16-Majorana-FK-Seam-Toys: **DER WECHSELWIRKENDE SPIEGEL-FLIP** — beim erzwungenen Twist η = +i sind die chiralen Grams PD (37,0,0) auf allen vier Cuts (g ∈ {1/32..8}, v534-Regression), und der SPIEGEL-Zustand spaltet sektor-exakt auf JEDEM Cut und JEDEM g: gerade (29,0,0) PD, ungerade (0,8,0) STRIKT NEGATIV DEFINIT (max. Eigenwert ≤ −1.5e−3, vom Nullpunkt weg beschränkt) — **Kill-Test (3) kann auf der überlebenden Klasse nicht feuern** (v608-Upgrade von g = 0 aufs volle Gitter); der Mark-Transport existiert wechselwirkend (α₄: Mark-A → Mark-B exakt 16/16; Gram und Transport-Form hermitesch PD bei jedem g — Kill-Test (7) feuert nicht); das ungerade Transfer-Spektrum ist bei jedem g SIMPEL (8 positive Eigenwerte, min. Abstand 2.3e−5; ehrlich: Beinahe-Verdopplung bei g = 8 ohne Schließen); Kontroll-Mitglied m = 2 indefinit — Bit-Selektion und Chiralität KO-LOKALISIEREN; Kill-Tests (3)/(6)/(7) bleiben formal auf A_hol (11 checks, ~2 s)"
    },
    {
      "id": "v616_su2_internal_kinematic",
      "cluster": "core",
      "descriptionEN": "WOIT.SU2.KIN.01: das interne SU(2), kinematische Separation auf Twistor-Ebene — die erste ausgeführte Scheibe des letzten unberührten γ-Kill-Tests (6): auf ℂ⁴ = ℍ² ist **Woits euklidische Struktur wörtlich die interne j-Richtung** (ρ_tw = right-j, v519/v565-Konventionen); das interne su(2) (right-Sp(1)) erhält JEDE Twistor-Faser (trivial auf der Raumzeit S⁴), kommutiert EXAKT mit der gesamten Raumzeit-Quaternionen-Wirkung, Schnitt = {0}: **SEPARATION — der kinematische Zweig von Kill-Test (6) feuert nicht**; **DIE CLOCK FAKTORISIERT: RHO4 = left-u ∘ right-u (u = e^{iπ/4}) EXAKT** — Raumzeit-Halbviertel mal interner Halbviertel-Phase; der ALE-Deck ist rein raumzeit-seitig (DECK4 = left-i), RHO4² = DECK4 ∘ right-i; die komplexe Wahl BRICHT das interne SU(2) auf U(1), und der v519-Mark-Torsor IST die μ₄-Untergruppe des internen U(1) (∘ σ_std); σ_std fixiert die euklidische Struktur und invertiert die interne Ladung; dynamische Hälfte bleibt offen auf A_hol [C] (17 checks, <1 s)"
    },
    {
      "id": "v617_seam_cover_forcing",
      "cluster": "core",
      "descriptionEN": "QGEO.SEAMFORCE.01: die Seam-Forcing-Runde — **DIE KONFORMEN SEAM-AXIOME ERZWINGEN DEN μ₃-COVER** (konstruktive Scheibe der Bedrock-Prämisse QGEO.SYM.01): ℤ₄-MÖBIUS-RIGIDITÄT — jede zyklisch ℤ₄-symmetrische 4-Punkt-Konfiguration auf ℙ¹ ist Möbius-äquivalent zu μ₄ (Multiplier ±i erzwungen; Cross-Ratio-Zensus = harmonischer Orbit {2, ½, −1}; must-fail: 4/3 nicht im Orbit); GEWICHTS-ZENSUS — unter allen 81 μ₃-Monodromie-Vektoren sind genau 4 clock-äquivariant, die Mark-Äquivalenz killt das alternierende Paar: **Gewichte UNIFORM erzwungen, j ∈ {1,2} = das konjugierte Blatt-Paar (v613)**; j = 1 gibt EXAKT y³ = x⁴−1, volle ∞-Verzweigung (v603), Riemann–Hurwitz Genus 3 exakt; die Seam-Geometrie passt: Bond-Reflexion z ↦ −iz̄ = das (k,5−k)-Muster von v599s R, Schnittpunkte = die zwei v519-Cut-Punkte; Bedrock-Rest verengt auf (i) die Zyklisch-3-Wahl und (ii) die physisch↔konform-Identifikation [C] (14 checks, <1 s)"
    },
    {
      "id": "v620_cyclic_n_census",
      "cluster": "core",
      "descriptionEN": "QGEO.NCENSUS.01: der zyklisch-N-Zensus — **DER SEAM LÄSST N ∈ {3,5} ALS PRIMITIVE LORENTZ-ORDNUNGEN ZU, UND DIE COMPILER-KONSTANTEN PINNEN N = 3** (der erste v617-Bedrock-Rest, konditional gemacht): volle ∞-Verzweigung ⟺ gcd(N,4) = 1 ⟺ N ∈ {3,5}; Genus-Tabelle 1,3,3,6,7 (N = 2..6, Riemann–Hurwitz exakt); primitive Blatt-Dimension 3 genau für N ∈ {3,5,6}; **Chevalley–Weil: N = 3 hat BEIDE primitiven Blätter Lorentz ((1,2)/(2,1) — die v599/v613-Signatur), N = 5 ein Lorentz-Paar, N = 6 KEIN primitives Lorentz-Blatt (sein Lorentz-Gehalt = die Ordnung-3-Charaktere: N = 6 REDUZIERT auf N = 3)**; das Pinning: det_ℤ(1−deck) = Φ_N(1)³ = N³ — 27 für N = 3 (EXAKT die maschinell geprüfte v597-Konstante) vs. 125 für N = 5; Saturation 81 = 3⁴ vs. 625 = 5⁴: **N = 3 eindeutig**; der Rest wandert eine Ebene tiefer: warum der Compiler 3-adisch ist = N_fam = 3 (E8-Glue-verankert) (9 checks, <1 s)"
    },
    {
      "id": "v621_interacting_semigroup",
      "cluster": "core",
      "descriptionEN": "WOIT.GAMMA.SEMI.01: die wechselwirkende Klein–Landau-Halbgruppe — **DIE CLOCK ÜBERLEBT DIE WECHSELWIRKUNG ALS POSITIVER TRANSFER, GENAU AUF DEN μ₄-KOMMENSURABLEN SCHRITTEN**: bei g = 0 reproduziert sich das v524-Muster bei N = 16 exakt (7 Schritte hermitesch, Dimensionen 29,22,16,11,7,4,2; gerade Schritte PD; ungerade k = 1,3,5 indefinit mit EXAKT NULL Einteilchen-Diagonale; k = 7 paritäts-trivial); **bei g > 0 überlebt exakte Hermitizität GENAU auf {4, 6, 7}** und geht auf {1,2,3,5} verloren (dev > 0.1) — die Halbgruppe KONTRAHIERT aufs μ₄-Fenster; **die Clock (k = 4) bleibt hermitesch PD (11,0,0) über das ganze Gitter g ∈ {1/32..8}**; Mechanismus exakt: die Survivor-Wechselwirkung ist α₄-invariant EXAKT und nicht α₁/α₂-invariant (Quartett-Stabilisator {0,4,8,12}); k = 5 zeigt: Quartett-Einbettung allein schützt nicht (gerade Parität UND Alignment nötig); Lesart: die rekonstruierte Rotationsgruppe des wechselwirkenden Toys ist der CLOCK-TURM, nicht der volle Kreis (6 checks, ~2 s)"
    },
    {
      "id": "v622_seam_identification",
      "cluster": "core",
      "descriptionEN": "QGEO.SEAMID.01: die Seam-Identifikations-Runde — **DER PHYSISCHE SEAM IST DER KONFORME SEAM** (der zweite v617-Bedrock-Rest, auf Kernel+Wörterbuch-Ebene geschlossen): **DIE KERNEL-IDENTITÄT — der v519-Seam-Kernel c(d) = (2/N)/sin(πd/N) (ungerade d, Null für gerade) ist EXAKT die antiperiodische (NS) chirale Moden-Summe auf dem 16-Punkte-Kreis, für ALLE d = 1..15 inklusive der geraden Nullen: das Flat-Band IST das NS-Spektrum** (geschlossene Form Σ sin((2j+1)x) = sin²(8x)/sin(x) exakt); must-fail: die Ramond-Summe reproduziert den Kernel NICHT (NS erzwungen); die Site-Karte θ_a = (2a+1)π/16 legt die vier Mark-Bond-Mittelpunkte EXAKT auf μ₄ (kein Site trifft eine Mark — v519s Präzision (ii) wörtlich); Gruppen-Wörterbuch exakt (α₁ = Rotation, Clock = z→iz, refl_map(k) = Spiegelung an Achse (k+1)π/16, Cuts ↦ {π/4, π/2, 3π/4, π}); **DAS STRADDLE-GESETZ IST GEOMETRIE: durch-Mark-Achsen = exakt die v534-straddled Cuts {7,15}**; v617s Bond-Reflexion = Cut k = 11 mit dem v599-(k,5−k)-Muster; Kontinuums-Rest benannt: die Covering-Ebene (10 checks, ~1 s)"
    },
    {
      "id": "v623_covered_seam",
      "cluster": "core",
      "descriptionEN": "QGEO.COVERSEAM.01: die Covered-Seam-Runde — **DAS GITTER-SKELETT DER COVERING-EBENE** (der benannte v622-Rest, erste ausgeführte Scheibe): der auf den μ₃-Cover gehobene Seam ist **EIN 48-Site-NS-Kreis** für die uniformen Gewichte (j = 1, 2) und ZERFÄLLT in 3 Kreise für triviale UND alternierende Gewichte — **ein neuer, unabhängiger Kill des alternierenden Paars (Seam-Konnektivität)**; der explizite Walk schließt nach exakt 48 Schritten (3 Basis-Umläufe, 12 Mark-Kreuzungen); **DIE GEHOBENE CLOCK: L⁴ = Deck und L¹² = id auf ALLEN 48 Punkten mit echter Ordnung 12 — v597s r⁴ = ω und v614s Rand-Monodromie sind GITTER-KOMBINATORIK**; die v622-Kernel-Identität generalisiert wörtlich auf N = 48; **DAS 12-RASTER: die NS-Moden spalten unter dem Deck in drei 16er-Klassen mit Offsets {1/6, 1/2, 5/6}, die 1/2-Klasse = die BASIS-NS-Frequenzen EXAKT** (der ungetwistete Sektor IST der Basis-Seam); der fermionische Deck-Lift erfüllt Deck³ = −1 (Spin-Doppel, (−1)^F-Resonanz), Bilinear-Eigenwert = ω exakt (dort leben die t = ω/ω̄-Blätter); die 12 Mark-Kreuzungen bilden einen einzigen 12-Zyklus unter der Clock; Rest benannt: die wechselwirkende Orbifold-Aussage (9 checks, ~4 s)"
    },
    {
      "id": "v624_external_lattice_audit",
      "cluster": "core",
      "descriptionEN": "EXTREV.LATTICE.01: das externe-Review-Gitter-Audit — drei behauptete Ableitungen maschinell geprüft, ein Folgetest ehrlich umgetypt: **(A) DER VIERMARKEN-ABSCHLUSS [E, konditional]: unter der D_{d+1}+A_{d−1}-Glue-Architektur erzwingen DREI unabhängige Selektoren d = 4** (Disc-Gruppen-Zensus mit Negativ-Kontrollen d = 3,5,8; die Norm-Gleichung (d+1)/4+(d−1)/d = 2 mit eindeutiger Lösung; der Pythagoras-Selektor (d−1)²+d² = (d+1)² — das (3,4,5)-Tripel aus Zyklen, Marken, Träger); mit (1,1,2) als einziger 3-Teil-Partition von 4 und c₃ = 1/(8π): **die Zwei-Eingaben-Darstellung reduziert auf EINEN Grenzgrad d = 4 plus π** (ehrlich: konditional auf die Architektur); (B) DER ANCHOR-BYTECODE [E]: χ = (t−1)²(t−2), Rekursion p_{n+2} = 3p_{n+1}−2p_n, Newton, 240+8 = 248; **(C) DIE LORENTZ-KONGRUENZ [E]: PᵀJ_det P = J_fix EXAKT (det P = −6)** — Primzahl-Determinantenform und Cover-Polarisierungsgitter sind DIESELBE rationale quadratische Form (Index 6; 36 = 6²; 6 = p₂(a)); der Review-Folgetest zu B = M†J ist SCHLECHT GESTELLT (B komplex-symmetrisch, det 8 — must-fail ehrlich); Suzuki arXiv 2606.09096 + 2607.24830 web-verifiziert für den PRIME.WEIL.OPERATOR-Vertrag (14 checks, <1 s)"
    },
    {
      "id": "v626_e8_code",
      "cluster": "core",
      "descriptionEN": "E8.CODE.01: die E8-Code-Runde — **E8 IST EIN FEHLERKORRIGIERENDER CODE** (Theorem, nicht Metapher). [E] Der erweiterte Hamming-Code [8,4,4]: 16 Codewörter, Gewichte {0:1, 4:14, 8:1}, selbst-dual; Construction A liefert ein gerades unimodulares Rang-8-Gitter = E8 (Eindeutigkeit); Schalen 240/2160 = v625-Theta; Einzelbit-Fehler exhaustiv korrigierbar (16×8 Fälle, eindeutige nächste Codewörter). [C] Compiler-Bezüge getypt: 8 = Rang, 4 = d = |μ₄|, 16 = Halbspinor, 14 = 2×7 (Parabolik-Dimension) (7 checks, ~2 s)"
    },
    {
      "id": "v628_orbifold_casimir",
      "cluster": "core",
      "descriptionEN": "QGEO.ORBCAS.01: die Orbifold-Casimir-Runde — **DIE DREI DECK-MODENKLASSEN DER GEDECKTEN NAHT TRAGEN EXAKT DIE ℤ₃-TWIST-CASIMIR-DATEN** (erste quantitative Scheibe des v623-Orbifold-Rests). [E, exakt] Geschlossene Form E₀(ν,N) = −cos(π(2ν−1)/2N)/(2 sin(π/2N)); 1/N-Koeffizient exakt π(6ν²−6ν+1)/12; Sektor-Tabelle (1/72, −1/24, 1/72) für ν = (1/6, 1/2, 5/6): Twisted-Paar entartet, Verhältnis −1/3, Twist-Gap 1/18 exakt. [C] Der verbleibende Rest ist die WECHSELWIRKENDE Aussage (Twist-Feld-OPE, Crossing, RP) (5 checks, ~1 s)"
    },
    {
      "id": "v629_root_incidence",
      "cluster": "core",
      "descriptionEN": "E8.INCIDENCE.01: der Wurzel-Inzidenz-Zensus — **DIE NAIVE 48×5-INZIDENZ AUF DIE 240 E8-WURZELN IST GEKILLT, DER POSITIV-REST IST SCHARF** (der Kill-Test des Reviews, erste Hürde ausgeführt). [E, KILL] Das compiler-kanonische Ordnung-12-Element (μ₄-Clock × Familien-3-Zyklus) wirkt NICHT frei: Orbits 19×12 + 3×4 statt 20 freier ζ₁₂-Orbits. [E, positiv] Die μ₄-Clock allein wirkt FREI mit EXAKT 60 Orbits — 60 = D_start als freier Orbit-Zählwert auf den Wurzeln; der 3-Zyklus fixiert exakt 12 Wurzeln (= gehobene Marks, v623) (6 checks, ~1 s)"
    },
    {
      "id": "v632_ftransfer_pgl2",
      "cluster": "core",
      "descriptionEN": "FTR.PGL2.01: die F_transfer-PGL₂-Runde — **DIE VIER EXTERNEN TRANSFER-SOLVER TEILEN EINE GEMEINSAME MÖBIUS-AKTION AUF DER DISKRIMINANTE-−7-NORMLINIE, WO SIE ÜBERHAUPT WIRKEN — DAS KILL-KRITERIUM ‚VIER INKOMPATIBLE ARITHMETISCHE AKTIONEN‘ IST NICHT ERFÜLLT** (die v533-Präregistrierung in wörtlicher Gruppenform ausgeführt). [E] Diskrete Hälfte: die v183-Koide-Lesart induziert die Schritt-Aktion +9 = N_fam² (parabolisch, exakt mit der Basis-Translation T verflochten, Φ∘T = g∘Φ); CR = 4/3 exakt; Möbius-Transport erzwingt die F-Ecke 53 fit-frei. [E] Kontinuierliche Hälfte: g_pole hyperbolisch (2/3)⁶, g_Boltzmann = g_pole identisch (v425-Single-Flow), g_QCD parabolisch und EXAKT zu T konjugiert (Translation +7/(2π) in 1/α), g_relic = Identität; Uniflow-Degeneration (v575) und Own-Clock-Äquivalenz (v578) reproduziert; 5 Must-fail-Kontrollen (25 checks, ~1 s)"
    },
    {
      "id": "v633_orbit60_quotient",
      "cluster": "core",
      "descriptionEN": "E8.ZIQUOTIENT.01: die μ₄-Quotienten-Runde — **DER QUOTIENT DER 240 E8-WURZELN NACH DER FREIEN μ₄-CLOCK IST DAS HERMITESCH-UNIMODULARE ℤ[i]-E8 MIT DEM 60-HYPEREBENEN-SYSTEM VON ST31; DIE 60→6-KASKADEN-ROUTE AUF DEM QUOTIENTEN IST GEKILLT**. [E] H(v,v) = 2 auf allen 60 Repräsentanten; ℤ[i]-Spann = volles E8 (Covolumen 256), hermitesch-unimodulare Basis unter den Reps (det Gram = 1); σ auf den 60 Linien: 19×3 + 3 fix (= die J-Orbits der 12 σ-fixen Wurzeln); alle 60 Ordnung-2-Reflexionen erhalten die Wurzeln, Linien-Gruppe der Ordnung 11520 = 46080/4 (ST31-Fingerprint; ST32 durch 40 Hyperebenen gekillt). [E, KILL] Kaskaden-Harttest: Gram-Rang 4, beide Quotienten-Graphen regulär, keine kanonische Paar-Regel — getyptes Negativ (17 checks, ~3 s)"
    },
    {
      "id": "v634_st31_structure",
      "cluster": "core",
      "descriptionEN": "E8.ST31.01: die ST31-Struktur-Runde — **G31 IST DER VOLLE UNITÄRE STABILISATOR (EXAKT GEZÄHLT: 46080), DIE GRADE (8,12,20,24) SIND MOLIEN-EINDEUTIG, σ = c⁴ UND J = c⁹ EXAKT — UND DIE NUMEROLOGIE |G31| = |W(D5)|·|W(A3)| TRÄGT KEINE STRUKTUR: DREIFACH GEKILLT** (keine Isomorphie: Zentren 4 vs 1; kein (semi)direktes Produkt: die Vereinigung aller kleinen Normalabschlüsse hat Ordnung 64; W(D5) bettet in KEINE Rang-4-Gruppe ein — minimale treue Dimension 5 > 4; W(A3) = S4 bettet ein). Echte Struktur BERECHNET: G31 = (Z4∘2^{1+4})·Sp₄(2); Wurzel-Stabilisator = G(4,2,3) (Steinberg), Linien-Stabilisator = Z4 × G(4,2,3); EHRLICH: die v629-Uhr c ist NICHT das Springer-reguläre 12-Element (Zensus 19×12+3×4, |C(c)| = 72 ≠ 288) — der freie 20er-Kamm gehört der regulären 12-Uhr, einer anderen Konjugationsklasse; Lean-Anker: die Ordnungs-Identitäten in G31Orders.lean maschinell bewiesen (56 checks, ~10 s)"
    },
    {
      "id": "v638_code_semantics",
      "cluster": "core",
      "descriptionEN": "E8.CODESEM.01: die Code-Semantik-Runden (Kill + Stufe 2) — **DIE EINDEUTIGE ÄQUIVARIANTE HAMMING-PLATZIERUNG C* IST REED–MULLER RM(1,3) AUF AG(3,2) UND TRÄGT EIN EXAKTES WÖRTERBUCH**: Informationsbits = 1 pro μ₄-Paar (3 Familien + 1 Anker; das eindeutige minimale Z6-Orbit der Informationsmengen), Syndrom = In-Paar-Bit + Paar-Zeiger mit σ-3-Zyklus (Anker = Summe der Familien), Decode = Projektion wörtlich auf der ganzen Einzel-Fehler-Klasse (3840/3840, exakte Fraction-Arithmetik) — **und ALLES stirbt für jede nicht-äquivariante Platzierung** (nur 2/30 tragen die Struktur; naive v626-Platzierung tot für πσ; Kill-Kontrolle feuert) (27 checks, ~3 s)"
    },
    {
      "id": "v639_twist_ope",
      "cluster": "core",
      "descriptionEN": "QGEO.TWISTOPE.01: die Twist-Feld/OPE-Runde — **DIE WECHSELWIRKENDE SCHEIBE DER ℤ₃-ORBIFOLD-AUSSAGE STEHT AUF DEM ABELSCHEN VERTEX-NIVEAU**: h_σ = 1/36 aus DREI exakten Routen (Konventionen versöhnt, η(1−η)/2-Falle exakt aufgelöst), Gitter-Zweipunkt 2Δ = 2/9 auf 0.05–0.11%, Crossing EXAKT symmetrie-geschützt (1.4e−14), geschlossene Vierpunkt-Form [w(1−w)]^{−2Δ} < 0.1% nach N^{−2/3}-Extrapolation, OPE c₁ = 2/9 modellgebunden 0.19% (modellfrei ±15%-Band offen, getypt); **RP differenziert: ℤ₂-Klasse exakt PSD (Maschinerie validiert), naive ℤ₃-Paarung N-stabil verletzt (−4.5e−3) → der parafermionische Klein-Twist ist das benannte Offen-Item**; GATE.QGEO bewegt sich nicht (25 checks, ~1 s)"
    },
    {
      "id": "v645_klein_rp",
      "cluster": "core",
      "descriptionEN": "QGEO.KLEINRP.01: der parafermionische Klein-Twist — **DER ℤ₃-TWIST-SEKTOR IST REFLEXIONSPOSITIV auf dem Naht-Doppel-Gitter unter der Klein-Paarung (Spiegelung ∘ LADUNGSKONJUGATION, ω^{k·q}-Nullmoden-Phasen, η = (1,1) EINDEUTIG aus 36)** — das benannte Offen-Item von QGEO.TWISTOPE.01 ist auf Gitter-OS-Ebene GELÖST: alle geladenen Klein-Blöcke exakt reell-symmetrisch UND PSD (worst 2.9e−12), N-stabil 48…384, beide Achsen, **volle gemischte OS-Gram {1, σ₁, σ₂, Dipole} ohne Sektor-Projektion PSD**; die naive Paarung reproduziert ihre N-stabile Verletzung (−4.5e−3, must-fail); Kontinuums-OS/Modularität getypt offen; GATE.QGEO bewegt sich nicht (17 checks, ~30 s)"
    },
    {
      "id": "v646_rm13_reverse",
      "cluster": "core",
      "descriptionEN": "E8.RM13REV.01: die RM(1,3)-Rückwärts-Lesung — sind Compiler-Zahlen CODE-Operationen? **EHRLICH ZWEIGETEILT: die σ-fixierten Syndrome tragen EXAKT die Anker-Zerlegung (Leader-Gewichte (0,1,1,2), Norm 6, Anker = Familiensumme P3 = P0+P1+P2 — die AUSWAHL ist äquivarianz-spezifisch, die Gewichte selbst generisch für jede [8,4,4]-Platzierung), aber die Bytecode-Folge p_n = 2+2ⁿ hat KEINE Code-Lesart (0/11 Familien; 18/27/81 ohne Treffer — das Bingo ist beerdigt)**; Kill-Kontrolle feuert SELEKTIV (naive Platzierung verliert q-Universalität + σ-Aktion, behält generische Gewichte und sogar den Zähler 12 — ehrlich umgetypt) (20 checks, ~1 s)"
    },
    {
      "id": "v647_st31_degree24",
      "cluster": "core",
      "descriptionEN": "E8.ST31DEG.01: die ST31-Grad-24/-20-Runde — **DAS μ₄-ZENTRUM IST EINE POTENZ BEIDER REGULÄRER UHREN (w⁶ = ±J mit χ_w = x⁴+ix²−1 exakt, primitive ζ₂₄; u⁵ = ±J zentral), UND DIE UHR-FRAGE c = w² IST PER PARITÄTSSATZ GETÖTET** — die 12-Zyklen eines Quadrats entstehen PAARWEISE aus 24-Zyklen, census(c) = 19×12 + 3×4 hat UNGERADE Zahlen: unmöglich selbst in S₂₄₀ (beweisfest); u⁴ wirkt frei mit Zensus {5:48}, 48 = 240/5 exakt, Linien {5:12}; die Gleichheit 48 = Naht-Sites bleibt getypte BEOBACHTUNG (25 checks, ~4 s)"
    },
    {
      "id": "v649_discipline_audit",
      "cluster": "frontier",
      "descriptionEN": "META.DISCIPLINE.01: das Disziplin-Audit — die Suite parst SICH SELBST und zertifiziert ihre Prozess-Disziplin als eingefrorene Mindestwerte: **214 Ledger-Zeilen dokumentieren Kills/ehrliche Negative/Retypes, 1469 Must-fail/Kill/Scramble-Vorkommen in 233 von 643 Modulen, 47 Contract-Rows (29 mit benannten Kill-Kriterien; ausgeführt: FTR.PGL2 v533→v632, W1-Kette →v643), deterministisches Replay 12/12 (deklarierte Stichprobe v55–v648, zwei frische Läufe), 5 Klassik-Anker gegen die Literatur nachgerechnet (Jacobi r₄, Construction A → 240 Minimalvektoren, ST31-Grade, Suzuki-Konstanten 25+ Stellen, ξ-Funktionalgleichung), 54 committete Lean-Beweismodule**; ehrliche Grenze [C]: Prozess-Disziplin, nicht Wahrheit der Physik — die offenen Gates bleiben offen (278 Klasse-O-Rows) (17 checks, ~200 s)"
    },
    {
      "id": "v650_orbifold_modular",
      "cluster": "core",
      "descriptionEN": "QGEO.ORBMOD.01: die Modular-Daten des ℤ₃-Orbifolds der Naht auf dem Gitter — **ALLE NEUN Z[g,h] TREFFEN |θ[g/3,h/3]/η|² AUF < 1 %, DIE MODULARE S-KOVARIANZ IST GEMESSEN MIT SAUBERER N⁻⁴-RATE (das N⁻²-Gitterartefakt ist selbst S-kovariant und KÜRZT SICH), T IST DER EXAKTE ℤ₆-DEHN-TWIST (g,h) → (g, g+h) × (−1)^F auf Maschinenpräzision** — die getwisteten Charaktere tragen h_σ = 1/36 (rel 5.6e−4) mit dem parafermionischen 1/6-Gap, die v628-Casimir-Tafel reproduziert pro Deck-Klasse, und **deck³ = (−1)^F schließt EXAKT als ℤ₆-Spur-Arithmetik** (inkl. Cover-Faktorisierung; positive Buchführung = ℤ₆-Ladungsprojektion, Grundklasse c = 0 in jedem Sektor); chirale Phasen/Kontinuums-T/OS getypt offen, GATE.QGEO bewegt sich nicht (17 checks, ~25 s)"
    },
    {
      "id": "v651_orbifold_assembly",
      "cluster": "core",
      "descriptionEN": "QGEO.ORBASM.01: der volle ℤ₃-Orbifold-Zusammenbau der Naht — **H²(ℤ₃, U(1)) = 0 PER MASCHINEN-ENUMERATION (27 Kozykel, alle symmetrisch: KEINE diskrete Torsion), UND VON SECHS KANDIDATEN ÜBERLEBEN {ganzzahliges Spektrum, eindeutiges Vakuum, S gemessen, T exakt} GENAU {B, C3}** — der naive ℤ₃ (A) stirbt an EXAKTER T-Verletzung 3.5e−3 (= sein Ladungs-3-Gehalt), C1 an S-Stau, C2 hat kein Vakuum, A' zusätzlich am falschen Twist-Exponenten (5/18 statt 1/9); **die Naht-Daten diskriminieren B von C3: B hält jeden gemessenen c = 0-Sektor-Grund und die Klein-Phase η = (1,1) — DER NAHT-ZUSAMMENBAU IST B** (ℤ₆-Deck-Quotient mit uniformer Ladungs-0-Projektion; datengestützt [C], v652 härtet strukturell); die chirale Phase e^{−2πi·ã·b} GEMESSEN (17 checks, ~35 s)"
    },
    {
      "id": "v652_orbifold_arf",
      "cluster": "core",
      "descriptionEN": "QGEO.ORBARF.01: die Arf/GSO-Härtung des B-vs-C3-Diskriminators — **DIE 6⁶-KLASSIFIKATION ALLER SEKTOR-CHARAKTER-VERDRAHTUNGEN LÄSST GENAU {B, C3} ÜBRIG, DER EIN-FOCKRAUM-PIN ERZWINGT B, UND DIE μ₆-DEFEKT-LEITER MISST DIE VON C3 EXZIDIERTEN ZUSTÄNDE DIREKT** — Leiter-Exponenten 2Δ = (1/18, 2/9, 1/2) auf < 8e−4, exakte ladungsaufgelöste Minima (0,1,4,9,4,1)/36; **C3s Odd-Sektor-Vorhersagen 85/18 und 81/18 verfehlen die Messung um Faktor 85.0 und 9.0** (der Defekt-String ist exakt ladungsneutral: C3 exzidiert Zustände, die auf der Gitter-Naht mit positiver Klein-Norm existieren); der v639-2/9-Read allein ist B-vs-C3-BLIND (beide stimmen auf geraden Sektoren exakt überein) — das neue τ-Datum: OPE-Exponent 1/18 (rel 2.9e−3); Bond-Defekt-Prämisse → v653 (15 checks, ~30 s)"
    },
    {
      "id": "v653_bond_defect",
      "cluster": "core",
      "descriptionEN": "QGEO.BONDDEF.01: die Bond-Defekt-Prämisse wird ein GITTER-SATZ — **TWIST-SEKTOR = DECK-RANDBEDINGUNG = BOND-DEFEKT, KANONISCH UND EXAKT (unitäre Äquivalenz A_r = √3·R·P_r auf allen Moden, in ℚ(ζ₆) exakt, bei 16/48 UND 48/144), MIT D_can = L⁴ UND D_can³ = −1 ALS OPERATOR-IDENTITÄTEN** (die Viertel-Uhr FIXIERT den kanonischen Sektions-Lift); die Defekt-Position ist REINE EICHUNG (symbolischer Teleskop-Satz in allen 16 freien Bond-Phasen) mit geometrisch gepinnter Eichklasse (Blatt-Reduktion = eine uniforme Twist-Einheit pro Mark; Kozykel-Träger = Mark-Menge); **Alternativen-Zensus vollständig: die Basiskette hat KEIN anderes ℤ₃, Traces/Holonomie/v628-Casimir töten alles außerhalb — inkl. der EHRLICHEN isospektralen NN-Kette (existiert auf 7.8e−16! rein spektral nicht ausschließbar, aber keine ℤ₃-Twist-Realisierung, und die gemessenen String-Daten diskriminieren sie)**; Kontinuums-CFT + Naht-Identifikation bleiben [C] (18 checks, ~40 s)"
    },
    {
      "id": "v654_st31_degree8",
      "cluster": "core",
      "descriptionEN": "E8.ST31D8.01: die ST31-Grad-8-Runde + der vereinheitlichte d/4-SATZ — **JEDE Springer-reguläre d-Uhr von G31 erfüllt x^{d/4} = ±i·Id, einen ERZEUGER von μ₄ (nie ±1; eine Formel aus der Exponenten-Systematik: alle Exponenten ≡ 3 mod 4); DIE UMKEHRUNG IST EXAKT BEI d ∈ {8, 20, 24}, UND DIE EINZIGE AUSNAHME BEI d = 12 IST DIE COMPILER-UHR-KLASSE** (1280 Extra-Elemente = Klasse von c und Galois-Zwilling; χ_c = (x−i)²(x²+ix−1) = die HALBE reguläre 12er-Springer-Box, alle Eigenwerte λ³ = −i — exakt darum ist c³ = J³ zentral: die Compiler-Uhr-Klassen sind die EINZIGEN nicht-regulären Uhren in ganz G31, deren d/4-Potenz das Zentrum erzeugt); **Ordnung-8-Landschaft komplett (10560 Elemente, 480 regulär = 2 Galois-Klassen, χ = (x²+i)² exakt) und der ehrliche KILL: „frei ⇒ regulär“ ist FALSCH** (4320 freie 8er nichtzentral; schon bei Ordnung 4: 510, keins regulär); Grade = freie Ordnungen mit 4 | d, d > 4 (30 checks, ~10 s)"
    },
    {
      "id": "v664_look_elsewhere",
      "cluster": "frontier",
      "descriptionEN": "META.LOOKELSEWHERE.01: das Look-Elsewhere-Audit — das Bingo-Budget der getypten [C]-Koinzidenzen, quantifiziert — **N = 42 STRUKTUR-SLOTS, K = 19 TREFFER, 6 KILLS ALS 0-TREFFER-TRIALS IM BUDGET; GLOBAL p = 5.08e−3 IM KONSERVATIVSTEN VON DREI NULL-MODELLEN (das Ensemble überlebt die ehrliche Korrektur), PSEUDO-THEORIE-KONTROLLE: nur 0.2 %/0.3 % von 2000 größengematchten Listen erreichen Trefferzahl/p — ABER KEINE EINZELBEOBACHTUNG IST FÜR SICH SIGNIFIKANT**: suggestiv (0.01…0.1) nur 60 = D_start, 12 = Marks, G31-Grade, 48 = Ω_adm; unauffällig (p > 0.1): 6 = p₂(a) (p = 0.169) und die CODESEM-Zahlen-Batterie (p = 0.593 — der v646-Befund quantifiziert); 14/16 FORCED (ihr Wert ist das Theorem, nicht die Zahl); **Re-Typisierungen als datierte Ledger-Notes AUSGEFÜHRT (EXTREV.LATTICE.01, E8.CODESEM.01, E8.CODE.01)**; Seed 20260802, 100k MC, exakte Poisson-Binomial-Tails (14 checks, ~2 s)"
    },
    {
      "id": "v679_orbifold_continuum_os",
      "cluster": "core",
      "descriptionEN": "QGEO.ORBOS.01: die erste Kontinuums-OS-Scheibe des Naht-Orbifolds B — **KORRELATOR-KONVERGENZ MIT VERSTANDENEN RATEN: alle sechs σ-Kanal-Observablen teilen die uniforme Roh-Rate 0.659 = die vorhergesagte Fisher-Hartwig-2/3-Branche, der τ-Kanal-Kontrast misst 1.298 vs 4/3 (Kanal-Gesetz ρ = 2(1−β)² − 2β²: die Raten sind VERSTANDEN, nicht gefittet); nach EINEM FH-Term liegen die Limites < 0.05% an den CFT-Werten; RP ÜBERLEBT DEN LIMES: λ_min-Extrapolationen weg von Null (σ 1.36e−3, Marge 7×; τ 2.80e−4, Marge 358×; gemischter OS-Gram 1.34e−3, Marge 31×; must-fail η = −1 kippt)**; Cluster in Raum UND euklidischer Zeit exakt (2Δ = 0.222220 vs 2/9 auf 8e−6, Zylinder-/cosh-Formen punktgenau, Zeit-Gap = ε-Level 0.99933 vs 1); Charaktere 1/36, 1/9, 5/18, 4/9 auf 1e−12 mit gemessener N⁻²-Rate 2.000; **getypt: E0/E2/E3/E4 auf Konvergenz-Ebene GEMESSEN, E1 nur diskret exakt; der formale Limes-Rest ist benannt (uniforme Schranken, GNS-Limes-Hilbertraum, Operator-Konvergenz, R-Sektoren, B-Assembly) — GATE.QGEO bewegt sich NICHT** (19 checks, ~3 s)"
    },
    {
      "id": "v689_gaussian_code_bridge",
      "cluster": "core",
      "descriptionEN": "E8.GAUSSCODE.01: die Gauß-Codebrücke (Review-Vertrag E8.GAUSSIAN.CODE.01) — **E8(Z[i])/(1+i) ≅ F₂⁴ EXAKT (Smith-Normalform; Standardmodell: |det| = 256 = 2⁸, SNF [1⁴,2⁴] — präsentationsunabhängig): der 240 = 15×16-Census (Nullklasse BEWEISBAR leer, jede Klasse = 4 der 60 G31-Linien), σ = c⁴ wirkt auf dem Quotienten als RM(1,3)-Familienpermutation (3-Zyklus F1→F2→F3 + Fix-Anker, exakt die v638-Info-Bit-Aktion), Rest-Eichung = Zentralisator in GL(4,F₂) der Ordnung 18, Koordinatenblock = F1+F2+F3-Label**; alle must-fail-Kontrollen feuern: naive Platzierung behält den Census, verliert aber die σ-Semantik (Zeuge off-lattice); nicht-πJ-Platzierung: kein Z[i]-Modul; Z[i]⁴: trivialisiert (16 Vektoren, 11 Klassen leer); exakte Ganzzahl/Fraction-Arithmetik; kein Marker-Move (26 Checks, ~3 s)"
    },
    {
      "id": "v690_quartic_half",
      "cluster": "core",
      "descriptionEN": "E8.QUARTICHALF.01: die quartische Invarianten-Restriktion (Review-Vertrag E8.QUARTIC.HALF.01) — ist G31 der holomorphe μ4-Schatten des E8-Compilers? AM LEBEN, mit dem ehrlichen Split: **die Verschwinde-Hälfte {2,14,18,30} ist WAHR aber TRIVIAL (μ4-Orbitfaktor, symbolisch bewiesen: W(Jα) = −i·W(α) ⇒ jeder J-Orbit trägt t^d(1+(−i)^d+(−1)^d+i^d) = 0 für d ≢ 0 mod 4 — gilt auch für 6,10,22,26); DIE SUBSTANZ: F8|, F12|, F20|, F24| sind G31-invariant (60/60 Spiegel exakt), ALGEBRAISCH UNABHÄNGIG (exakte Jacobi-Determinante ≠ 0), Grade Molien-eindeutig mit Produkt 46080 = |G31| ⇒ per Chevalley [zitiert] BASISINVARIANTEN: die holomorphe Restriktion des W(E8)-Rings erzeugt den VOLLEN G31-Ring**; die Jacobi-Determinante trägt die 60 Spiegellinien (Grad 60 = Σ(d_i−1), punktverifiziert 60/60; volle Faktorisierung = zitierter Satz, benanntes Residual); Kontrollen: J′ = RJRᵀ bricht die informative Hälfte, das wurzelerhaltende J″ erhält alles (der Satz gilt der W(E8)-Konjugationsklasse); keine Floats; kein Marker-Move (22 Checks, ~1 s)"
    },
    {
      "id": "v700_orbit60_census",
      "cluster": "core",
      "descriptionEN": "E8.ORBIT60.01: der ORBIT60-Census — die 60 μ4-Linien von E8 als kombinatorisches Objekt: Quotientengraph, σ/Uhr/Coxeter-Dynamik auf dem ORBIT-RAUM und ein STRIKTER Park-Test für die Kaskade 60 → 6 — **alle fünf Routen adjudiziert: (i) Grad-Regeln TOT (alle Quotientengraphen regulär); (ii) σ-Struktur TOT (Orbitgrößen {3,1}, keine 2-Blöcke); (iii) INVOLUTIONS-ZEUGE EXISTIERT NICHT: der volle Involutions-Census der Liniengruppe LG (Ordnung 11520; 375 Involutionen, 21 Klassen) zeigt KEIN Element vom Zykeltyp 2²⁷1⁶ (Fixlinien-Census {0:120, 4:180, 12:15, 16:60}); (iv) W(E8)-Coxeter steigt nicht ab; (v) MUST-FAIL-Numerologie-Kontrolle feuert: regelfreies lexikographisches Paar-Entfernen reproduziert 60−2n in exakt 27 Schritten, > 10⁷⁰ gleich-erfolgreiche Sequenzen — die Zählung trägt keine Struktur**; A5-Einzeiler: Ordnung-5-Elemente existieren (2304), aber keine sichtbare natürliche A5-Aktion (⟨σ, Ordnung-5⟩ hat Ordnung 960); GEPARKT per Review-Protokoll; deterministisch (18 Checks, ~1 s)"
    },
    {
      "id": "v705_chain_deck_sector",
      "cluster": "core",
      "descriptionEN": "Original v705 D1 physical NS circle: 48-site antiperiodic shift, order-12 clock, projective deck cube -1 and three 16-mode deck sectors."
    },
    {
      "id": "v715_qgeo_car_continuum",
      "cluster": "core",
      "descriptionEN": "QGEO.CARLIMIT.01: QGEO-CAR — die quasifreie CAR-Route zum formalen Orbifold-Kontinuumslimes (GATE.QGEO), erste Scheibe — **QGEO-CAR-RATES-SUMMABLE: alle 6+1 Deck/Twist-Kanäle sind Schatten-Norm-Cauchy auf ε-separierten Konfigurationen (N bis 3072/6144) mit geometrisch summierbaren Raten; die FH-Renormierung ist FIT-FREI GITTER-HERGELEITET (Barnes-G, 8.8e-7); der RP-Kegel degeneriert nicht entlang der Leiter; alle vier Kill-Kriterien frei**; der benannte Rest: das Majoranten-Lemma (L1)–(L3) mit Konstanten + klassischer Beweisroute — benannt, nicht behauptet: GATE.QGEO bewegt sich NICHT (22 Checks, ~10 s)"
    },
    {
      "id": "v722_phys_ramified_ns_r",
      "cluster": "core",
      "descriptionEN": "GNET.RAMIFIED.01: PHYS-Moonshot-Scheibe (T2/G_net) — ist die Ramond-Projektion (1+i)-adisch? — **RAMIFIED-SEES-NSR: (1+i)L liegt im geraden/NS-Untergitter (Z-Basis-Beweis), die 240 Wurzeln fallen in 15 × 16 Klassen mit REINER Parität (7 NS + 8 R), Parität ist ein Homomorphismus F₂⁴ → F₂ — die NS/R-Graduierung IST der Paritäts-Charakter von E8(Z[i])/(1+i) = F₂⁴**; erster exakter Zeuge für die Gruppoid-Lesart von G_net (die EINE verzweigte Kante des Turms ist die Gauß-Primzahl (1+i)); Kontrollen feuern: Z[i]⁴-Census ≠ 15×16, gescrambelte Labels brechen die Reinheit (9 Checks, ~1 s)"
    },
    {
      "id": "v723_phys_modular_clock",
      "cluster": "core",
      "descriptionEN": "FTRANSFER.STT.01: PHYS-Moonshot-Scheibe (T3/F_transfer) — kann Thermal-Time den externen Uhren-Vertrag ersetzen? Der Konjugationsklassen-KILL-Test — **STT-KILLED + INTERNAL-PAIR-CONFIRMED: die F_QCD-1-Loop-Uhr ist PARABOLISCH und nicht diagonalisierbar (einzelner Jordan-Block), und exp eines reell-diagonalisierbaren Generators ist diagonalisierbar (symbolisches Lemma) — kein konstant-reskalierter Modularfluss erzeugt die QCD-Uhr; der Pol-Multiplikator ist EXAKT (2/3)⁶ = 64/729 (hyperbolisch, dimensionslos ⇒ intern)**; getypte Konsequenz: der externe Uhren-Vertrag BESTÄTIGT und geschärft zum Anker-Census {eine Einheit (v_geo-Klasse), ein Winkel θᵢ, ein Gitter-O(1)} — T3s Uhren-Wand == T1s Einheiten-Wand, einmal gezählt (11 Checks, ~1 s)"
    },
    {
      "id": "v724_phys_t3b_modular_flows",
      "cluster": "core",
      "descriptionEN": "FTRANSFER.COCYCLE.01: T3-B — konditionierte Modularflüsse aus EINEM internen Mutterzustand, der zweite Thermal-Time-KILL — **T3B-DEAD: mit target-freien Graph-Projektoren, EINEM globalen KMS-Maßstab Δ = 6·log(3/2) und exakter Connes-Kozykel-Komposition (1.7e-15 entlang J–K–C–F) kommen alle vier konditionierten Flüsse LOXODROMISCH/KOMPLEX heraus — die eingefrorenen v578/v632-Solver-Klassen (hyperbolisch/parabolisch/degeneriert) entstehen NICHT aus konditionierten Modularflüssen eines internen Mutterzustands**; die Sonde verweigert ehrlich den NEEDS-LINDBLAD-Ausweg (v238-GKSL ebenfalls all-loxodromisch); mit v723 sind BEIDE Thermal-Time-Routen geschlossen: der externe Uhren-Vertrag steht (12 mechanische Checks, ~1 s)"
    },
    {
      "id": "v725_phys_vgeo_torsor_audit",
      "cluster": "core",
      "descriptionEN": "VGEO.TORSOR.01: T1-SCHLUSS — der Kalibrations-/Skalen-Torsor-Audit der v_geo-Schnittstelle — **TORSOR-SCHLIESSUNG: die v_geo-Schnittstelle ist als R₊-Skalen-Torsor STRUKTURELL GESCHLOSSEN — (1) vollständige Export-Tabelle O_i = r_i·v_geo^{d_i}, (2) Dimensionsmatrix Rang 1 (konditional [A] auf dem Flavor-Block, benannt), (3) alle Konsistenz-Bedingungen λ-homogen (Maschinenform des No-Unit-Satzes v153), (4) Zwei-Anker leave-one-out kompatibel (G vs ρ_Λ: 0.11%), (5) Kalibrations-Satz in Torsor-Sprache formuliert**; FENCE: Struktur-Schließung, KEINE Skalen-Ableitung (per No-Unit unmöglich); Moonshot BENANNT, nicht ausgeführt: H_EW = ln(M̄/v_EW) = 37.1776 (21 Checks, ~1 s)"
    },
    {
      "id": "v726_phys_car_pp_index",
      "cluster": "core",
      "descriptionEN": "GNET.PPINDEX.01: PHYS-T2-Hauptscheibe — Pimsner-Popa/Watatani-Index 4 auf der CAR-Leiter, mit HERGELEITETER μ₄-Wirkung und Ramond-Nullmoden-Test — **T2-SLICE-GO: die optimale PP-Konstante ist EXAKT ¼ (expliziter Rang-1-Minimierer + Pinching-Satz) und der Watatani-Index EXAKT 4 via expliziter Quasi-Basis (Σvv* == 4·1, Gewichte ERZWUNGEN — uniforme Gewichte brechen die Skalarität, Z₂-Mittel gibt Index 2); μ₄ NICHT eingesetzt, sondern aus der Uhr L = S^{N/12} hergeleitet (Ordnung 4 von der NS-Spinstruktur erzwungen); der Ramond-Sektor degeneriert undressed (Index 2) und das v679-Klein-Dressing stellt Index 4 ZUSTANDSERHALTEND wieder her ([ρ,U′] = 0)**; das (1+i)-Wörterbuch zu v722 schließt; GNET.RAMIFIED.01-Vertragskandidat gedruckt (57 Checks, ~7 s)"
    },
    {
      "id": "v736_orbit_packet",
      "cluster": "core",
      "descriptionEN": "E8.ORBITPACKET.01: das Gaußsche ORBIT-PACKET-Lemma — **ORBIT-PACKET-EXACT: Burnside 7 = (15+3+3)/3; die kanonische 5×48-Partition (Fixblock + 4 bewegte Blöcke à 12 Gaussian Lines, label-frei kanonisch — Packete = 1+4 = 5 = g_car, 240 = g_car×48); Anchor (1,1,2) aus den Fixsyndrom-Leader-Gewichten (0,1,1,2), Leader-Paar e₆/e₇ = v638-Anchor-Paar; q-Normalform 240 = 16·15, 248 = 240+8**; Zirkularitäts-Fence: P1/P2 kommen als EXACT-INTERNAL-RECONSTRUCTION zurück (Rekonstruktion, KEINE Ableitung); σ-Klassen-Spezifität: die 160 freien Ordnung-3-Elemente von G31 geben Burnside 5, nicht 7; Lean-Begleitmodul OrbitPacket.lean (34 Theoreme, kernel-checked) (26 Checks, ~1 s)"
    },
    {
      "id": "v737_seam48_intertwiner",
      "cluster": "core",
      "descriptionEN": "E8.SEAM48.01: der Seam-48-Intertwiner-Test — der schärfste Struktur-Kill des Review-Morgens — **SEAM48-DEAD: die 48×5-Inzidenzmatrix ist NICHT identisch 1 (Eintrags-Verteilung 0:96 / 1:64 / 2:64 / 3:16), repräsentanten-unabhängig (0 von 2304 Ordnung-5-Elementen von G31) und die Must-Fail-Kontrolle feuert (0 von 25 Zufalls-Operatoren)**; wertvolle Residuen: die 4 bewegten Blöcke tragen die v623-Charaktertafel EXAKT, der Fixblock weicht genau um die 12 Lifted Marks ab; 5-Perioden-Lemma u⁵ = ±J exakt; 240 = 48·5 bleibt Faktorisierung OHNE Koordinatisierung (24 Maschinerie-Checks, ~4 s)"
    },
    {
      "id": "v745_qgeo_car_l1_sector_lemma",
      "cluster": "core",
      "descriptionEN": "QGEO.CARL1.01: das (L1)-Sektor-Lemma der CAR-Kontinuums-Majorante — **L1-SECTOR-PROVEN-MODULO-ELEMENTARY: ‖K_2N − K_N‖_{S1} ≤ B_q(ε)/N für alle 6 undressed Sektoren auf dem festen Gitter beweisnah — die Besetzungs-Bogen-Modensummen sind EXAKTE geometrische Summen (exakte Summation statt Euler–Maclaurin), die gesamte 1/N-Struktur ist der Subgitter-Einbettungs-Offset, führender Term = exakte Matrix (iu/2MN)·f_q′(s) (Ratio → 1 auf 6e-4), γ = 2 HERGELEITET, 39744 + 108 Ungleichungen mit 0 Verletzungen, r₁ = 1 scharf**; Status getypt: PROVEN-MODULO-ELEMENTARY mit expliziten Konstanten; benannte Residuen (a) Gitter-Sup [elementar], (b) L2 uniform FH/RH [hart-technisch], (c) L3-Uniformität [mittel]; GATE.QGEO bewegt sich nicht (11 Checks, ~1 s)"
    },
    {
      "id": "v746_phys_gnet_local_functor",
      "cluster": "core",
      "descriptionEN": "GNET.LOCALNET.01: der G_net-Lokal-Netz-Funktor — **GNET-LOCAL-FUNCTOR-ALIVE: die Netz-Axiome sind EXAKT (Isotonie, graduierte Lokalität, Uhr-Kovarianz auf dem Quotienten-Kreis); der Watatani-Index der lokalen Fixpunkt-Inklusion ist LOKAL EXAKT 4 für jedes Intervall ℓ ≥ 2 — die [D,A]-Index-Aussage ist intervall-lokal, nicht nur global; Ramond = Halblinien-/Bond-Defekt (solitonisch; lokaler Footprint, Leckage √(2/N))**; 4 Must-Fail-Kontrollen feuern; fehlende Kontinuums-Sätze BENANNT, nicht behauptet (GNS-Limes-Netz, Index-Kontinuität Pimsner–Popa/Longo, Longo-Q-System-Identifikation zu GATE.METRIC.11, getwistete DHR/Longo–Roberts-Sektoren, Haag-Dualität/Split) (16 Checks, ~2 s)"
    },
    {
      "id": "v752_projective_hamming_incidence",
      "cluster": "core",
      "descriptionEN": "E8.PROJHAMMING.01: das projektive Hamming-Inzidenzlemma auf V = L/(1+i)L — der symplektische F2^4 des unimodularen hermiteschen E8-Gitters, und die v737-Auflösung — **PROJ-HAMMING-EXACT: h = H/4 ist Z[i]-wertig mit hermitescher Gram-Determinante 1 (Z[i]-UNIMODULAR); h̄ alternierend (doubly-even), nicht-entartet ⇒ (V, h̄) symplektisch mit expliziter Darboux-Basis; |H_y| = 7, Paarschnitte 3; B Bᵀ = 4I + 3J EXAKT, char. Polynom (x−7)(x−2)⁹(x+2)⁵ — die Multiplizitäten (1, 9, 5) sind die SYMPLEKTIK-Signatur; K² = (4/49) I + (45/49) Π₀ exakt (K selbst ehrlich nicht psd); NS/R-χ = h̄(·, y0) für den EINDEUTIGEN σ-fixen Punkt y0 = FSUM (das Anker-Bit), 112 + 128 = 240, +8 Cartan: 248 = 120 + 128; DIE v737-AUFLÖSUNG: Verteilung 0:96/1:64/2:64/3:16 exakt abgeleitet (16 × 3×5-Label-Tafel) UND die 5×48-Koordinatisierung existiert via genau EINEN ū-invarianten Spread (Kill erklärt, nicht revidiert)**; zwei ehrliche Korrekturen an der Review-Herleitung dokumentiert; Lean-Begleitmodul ProjectiveHamming.lean (32 Theoreme) (36 Checks, ~5 s)"
    },
    {
      "id": "v757_fano_disc7_bridge",
      "cluster": "core",
      "descriptionEN": "QGEO.FANODISC7.01: die preregistrierte Fano→Diskriminante(−7)-Brücke — die Dreizyklus-Hälfte real, die Paritäts-Hälfte stirbt arithmetisch — **FANO-DISC7-DECORATIVE (Bruch exakt bei Bedingung 2): das NS/R-Paritäts-Bit hat KEINEN arithmetischen Träger in O_{−7} — 2 SPALTET in Q(√−7) (−7 ≡ 1 mod 8, O ⊗ Z₂ = Z₂ × Z₂), jeder 2-adische Idealquotient hat 2-Rang ≤ 2 < 3 (vollständiger Zensus aller Norm-8/16-Ideale): die Code-F₂-Strukturen sind arithmetisch UNREALISIERBAR; DIE REALE HÄLFTE (COND1, exakt): äquivariante Lifts existieren für beide kanonischen Differenzmengen-Fano-Modelle, die kanonische Ordnung-3-Wirkung ist x → 2x (der Differenzmengen-Multiplikator, 2 = der Transfer-Schritt), jeder Lift schickt den eindeutigen σ̄-Fixpunkt auf 0 = red(α₀) = die C-Ecke (Norm 14 = dim G2)**; die Sieben-Korrelation dauerhaft als DEKORATIV getypt; Kontrollen feuern; der eine [FAIL] = der preregistrierte COND2-Bruch (26 Checks, ~1 s)"
    },
    {
      "id": "v774_arf_spinor_compiler",
      "cluster": "core",
      "descriptionEN": "ARF.SPINORCOMPILER.01: der Arf/Spinor-Compiler auf V = L/(1+i)L = F2^4 (Priorität 1 des Arf-Programms) — **ARF-SPINOR-EXACT (46/46, ~2 s, null Kills, beide Must-fail-Kontrollen feuern): der Paritäts-Lift iota: V ≅ C_even(5) trägt beta = hbar EXAKT in allen 256 Zellen (Crosslink 1); Brute force über alle 2^16 Funktionen: GENAU 16 quadratische Verfeinerungen; Arf-Zensus 6 + 10 mit |Sp(4,2)| = 720 (voller 65536-Matrix-Zensus) und treuer+surjektiver S6-Wirkung; der eingefrorene Selektor (sigma-Invarianz 4 -> q(A) = 1: 2 -> q(F_Sigma) = 0: 1) wählt ein EINDEUTIGES q*, und die Gewichtsform q_wt IST q* (Offset 0; alle 16 Verfeinerungen = q* + hbar(., c) bijektiv); Stab(q*) ≅ S5 liefert 16 = 1 + 5bar + 10 auf Verfeinerungen UND Worten; das K6-Duaden-Modell ist eine Sp/sigma-äquivariante Bijektion auf E(K6) (Split 5 + 10); B = I + A_KG(6,2) in allen 225 Zellen, Charpoly (x-7)(x-2)^9(x+2)^5, Ovoid-Eigenvektor EXAKT (die 6 Indikatoren spannen den (-2)-Raum); alle 6 voll-isotropen Spreads tragen q*-Signatur (1, 2); die Hyperladungs-Tafel X = (-2,-2,-2,3,3) reproduziert das v310/v14-Wörterbuch (Werte UND Multiplizitäten); Code-Polynom-Momente (16, 0, 120, 0) mit der Kette 40/41/240/248 (v2-Master-Moment 120 = 5!); Pati–Salam 8 + 8 mit B-L = 1 - 2 n_c/3 exakt; chi_NSR = das Anker-Bit a.  SEMANTIK-TRENNUNG: A3^Fam (Familienzyklus) != A3^PS (3+2-Slot-Split) — dasselbe sigma, zwei Register.  ZAUN: die Code->Materie-Lesart ist NICHT beansprucht — v775 tötet sie auf Wurzel-Niveau.  Lean-Begleitmodul ArfSpinorCompiler.lean (23 Kernel-Theoreme, lake build grün)"
    },
    {
      "id": "v775_gaussian_class_d5_purity",
      "cluster": "core",
      "descriptionEN": "ARF.ROOTCLASS.01: der GEFEUERTE physikalische Kill des Arf-Compiler-Programms — **ROOTCLASS-MIXED (18/18, alle 3 Kontrollen feuern; FROZEN_SPEC SHA-256 58d9be9b… gehasht VOR allen Wurzeldaten): KEINE D5⊕A3/SU(5)-Konvention macht auch nur EINEN vollen Satz von 15 Gauß-Klassen rein — die einbettungs-UNABHÄNGIGE Zählschranke (jede Einbettung hat genau 128 Spinor-Wurzeln, 240 > 128) erzwingt >= 7 Adjoint-seitige Klassen für JEDE D5⊕A3-Einbettung; der erschöpfende Konventions-Scan (56 Splits x 240 Zuordnungen x 2 Chiralitäten) findet NULL reine Konventionen; alle 8 Spinor-Klassen mischen je 8 verschiedene 16-Gewichts-Paare (bestes Match des eingefrorenen linearen Wörterbuchs 8/128; X-Purity sekundär 0/15).  KONSEQUENZ (Zaun, bindend): die Code->Materie-Lesart darf in KEINEN Hauptclaim — einschließlich der 'leerer-Briefkasten = rechtshändiges Neutrino'-Lesart; die Arf-Mathematik von v774 bleibt unberührt.  Kontrollen: falsche Ladungsvektoren, LCG-Scramble, falsche symplektische Form (alle feuern)"
    },
    {
      "id": "v776_boundary_hamming_uniqueness",
      "cluster": "core",
      "descriptionEN": "ARF.BOUNDARY.CODE.01: Rand-Code-Eindeutigkeit — **BOUNDARY-CODE-UNIQUE (31/31): die nicht-zirkuläre Kette Rand -> Code -> E8 -> F2^4 -> 5 Slots SCHLIESST mit g_car = 5 als OUTPUT — Zensus 135 -> 30 -> 14 -> 2 -> 1 ohne Hamming/E8-Input; das Flag-Datum ist EIN deck-invariantes Z2 (Produkt der vier Seiten-Orientierungen; alle drei vordeklarierten Formalisierungen wählen denselben Code, alle 16 Flag-Zuordnungen entscheiden nur über sum(s) mod 2); die Rand-Eichgruppe (Zentralisator, Ordnung 12) macht das flaglose Zweifach pure Eichung; Construction A -> E8 EXAKT (240/2160-Theta, Cartan det 1, Arme 1-2-4; Post-Selektions-Check: Gewicht-4-Träger == die deployte v638-C*-Liste); q* eindeutig mit der rand-kanonischen Anker-Benennung; Pakete 1 + 4 -> g_car = 5.  EHRLICHE RESIDUEN: R1 die Universums-Prämisse (doppelt-gerade Selbstdualität — die Zirkularität ist VERLAGERT und GESCHRUMPFT, nicht entfernt), R2 Deck-Platzierungs-Blindheit (SEMANTISCH, getypt — die Wrong-Deck-Kontrolle feuert nicht), R3 Anker-Benennung (bbar(F_Sigma, A) = 1).  KEIN P2-Entfernungs-Claim; die neue Vertragsform (Evenness/Selbstdualität aus den Naht-Axiomen ableiten) als ARF.BOUNDARY.CODE.01 [O] registriert"
    },
    {
      "id": "v777_ftransfer_clock_jets",
      "cluster": "core",
      "descriptionEN": "FTRANSFER.CLOCKS.01 AUSGEFÜHRT + die gefaserte Verfeinerung von CONTRACT.F.01, EIN Modul aus zwei Probes — **NO-COMMON-CONNECTION + FTRANSFER-FIBERED-CARRIES (24 + 14 Checks, 0 Fails): der präregistrierte K1-Kill EXAKT wie der eingefrorene Prior erwartet (Prereg-YAML 880224f7… byte-verifiziert; Datentabellen VOR dem ersten Lauf kuratiert + eingefroren: Saikawa–Shirai g_*s 8ae2a4fe…, PDG-2024 alpha_s 2ca6332d…); alle vier Wörterbücher konstruktions-valide (Koide-Schema-Check 571x, Washout-Drift 1.15e-4, QCD-Overlay 0.83%, Relikt-Steigung 0.0602); der Kill sympy-exakt {-Δ²/2, -Δ²/2, 0} — ein Pass bräuchte GENAU den verbotenen nicht-affinen Möbius-Clock (K1-Duty exhibiert).  Die diskrete Schicht koinzidiert EXAKT (CR = 4/3 symbolisch, Deck-Einheit |Δσ| = 1, unit-parabolische Klasse — v632 gitter-explizit); die Obstruktion ist der KONSTANTE Naht-Kozykel mit Werten exakt {0, ±Δ²/2} (Antisymmetrie + Kozykel-Identität exakt, d/dσ ≡ 0 auf 50 Digits); Klassifikation = der vordeklarierte Thermal-vs-RG-Marker; Relikt degeneriert unklassifiziert; alle Kontrollen inkl. Möbius-Transplantat feuern; das float64-Artefakt des ersten Groupoid-Laufs offengelegt, Schranke unberührt.  Die Jet-Route von CONTRACT.F.01 schließt auf Physik-Clock-Ebene; nächster benannter proof-grade Schritt: die Coset-Zuordnung aus KMS/modularer Clock-Typisierung ableiten"
    },
    {
      "id": "v778_qgeo_gridsup_logsum",
      "cluster": "core",
      "descriptionEN": "QGEO.GRIDSUP.01: Rest (a) des (L1)-Majoranten-Lemmas VOLL geschlossen in der korrigierten (1 + ln M)-Form, EIN Modul aus zwei Probes — **GRIDSUP-SPLIT-CLOSED-LOG-LAW + FOURIER-LOGSUM-CLOSED (19/19 + 19/19): der v745-Rest (a) spaltet vierfach — (a1) das Kontinuums-Supremum GESCHLOSSEN (Taylor 5/8 symbolisch + Hüllkurven symbolisch; ~1.26e6 Zensus-Ungleichungen, 0 Verletzungen), (a2) Konfigurations-S1 GESCHLOSSEN mit explizitem B_conf(ε) (Packung + Nachbar-Leiter + ζ(2)/ζ(3)), (a3) der Verfeinerungs-Rest GESCHLOSSEN mit RHO(ε) via exakten csc³-Integralen (uniform über ALLE M <= N/2), (a4) der führende Teil ist ehrlich LOG-divergent (sprung-getrieben, Taper-kontrolliert) — die wörtliche v715-Klausel 'ein B(ε) für ALLE Gitter-Verfeinerungen' ist bei r1 = 1 FALSCH und wird zu (1 + ln M) repariert; die überlebende Ungleichung auf THEOREM-Niveau zertifiziert: exakte ν-getwistete Zirkulanten-Diagonalisierung + zwei exakte zyklische Summationen-nach-Teilen + harmonischer Vergleich: ||F_q^(M)(ε)||₁ <= c0(q,ε) + (J(ε)/4) ln(M+3), a fortiori (J/π) ln M, J = 2|f_q'(ε)|, alle Konstanten exakte csc-Integrale/Hüllkurven; 162-Fälle-Zensus bis M = 6144 mit 0 Verletzungen in allen sieben Klauseln; die reparierte (L1)-Klausel ||D_N^(M)||₁ <= [B0 + (J/2) ln(M+3)]/N auf allen 252 Transfer-Fällen verifiziert; Araki/Powers-Størmer-Summierbarkeit in expliziter geschlossener Form.  EHRLICH: die gemessene Steigung ist J/π² (Spike-Interferenz) — die Schranke ist valide, NICHT scharf.  Residuen (b) L2 / (c) L3 unberührt und offen; GATE.QGEO bewegt sich NICHT"
    },
    {
      "id": "v779_gnet_gns_arf",
      "cluster": "core",
      "descriptionEN": "GNET.GNSLIMIT.01: der G_net-Kontinuums-Strang, EIN Modul aus zwei Probes — **GNS-PRECURSORS-COHERENT + GNET-ARF-NO-CORRESPONDENCE (27/27 + 27/27): die Haar-Skalierungs-Isometrie intertwinet EXAKT (<= 2.2e-16), die kommutierenden Index-4-Erwartungs-Quadrate sind exakt (<= 1.5e-14), der kanonische Zustand ist Cauchy geometrisch ~1/4 pro Verdopplung, der Index-Zensus ist maschinen-exakt auf JEDER Stufe ohne Degradation, der getwistete Haag-Dualitäts-Defekt = 0 mit dim B'/dim A' = 4 = Watatani-Index (l = 1: 3); 10-Abschnitts-Kontinuums-Paper-Skelett.  Das EXAKTE Dimensions-Gesetz d_c(l) = 4^(l-1) + 2^(l-1) cos(πc/2) macht die C[Z4]-Quasi-Basis-Obstruktion exakt 2^-l (asymptotisch restauriert — der endliche Vorläufer der GATE.METRIC.08/11-Identifikation) und ERKLÄRT die l = 1-Anomalie (d_2(1) = 0); explizite 185-Element-Quasi-Basis bei l = 2 (sum v v* = 4·1 exakt); sektor-aufgelöste uniforme Split-Schranken entlang des Turms.  DAS EHRLICHE NEGATIV, erschöpfend: das 16 = 1 + 5 + 10-Zähl-Echo ist REAL und exakt, aber NICHT geometrisch (F2-Rang-Zertifikat 3 vs 4, GL(4,2)-Zensus 0/20160, kein nichttrivialer interner Ordnung-3-Träger); K_matter regiert den Turm-Gewichts-Transport NICHT (x5 = (0.414, 0.586) vs (1/3, 2/3)); die zwei Vieren (Index 4 = |mu4| vs Kanal 4/49) bleiben in ihren Registern.  Alle fünf Kontrollen feuern"
    },
    {
      "id": "v781_one_object_clock_census",
      "cluster": "core",
      "descriptionEN": "E8.ONEOBJECT.01: der d = 4-Forcing-Zensus + der 60-Linien-Uhren-Kill, EIN Modul aus zwei Probes — **ONE-OBJECT-PARTIAL + CLOCK-FIBER-DEAD (21/21 + 23/23): das TFPT-Glue-Muster (zyklische isotrope Verklebung, Geradheit q = 0 mod 2Z, Unimodularität |H|² = 4d) lässt eine gerade unimodulare DIAGONALE Verklebung von A_{d−1} ⊕ D_{d+1} GENAU für d = 4 zu (Zensus d = 2..12, exakte Brüche; bei d = 4 exakt die zwei v92-Lagrangians k -> k und k -> 3k; explizites E8-Zertifikat mit 240 Norm-2-Vektoren; d = 9 stirbt an der Isotropie; beide Relaxations-Kontrollen feuern) — Korollare am erzwungenen d: g_car = d+1 = 5, N_fam = d−1 = 3, Rang 2d = 8, Index 4 = d = |mu4|; EHRLICHKEITS-GATE: c3 = 1/(2πd)|_{d=4} = 1/(8π) ist eine korpus-legitime UMSCHREIBUNG (v23-Anker), KEINE neue P1-Ableitung.  DER UHREN-KILL: die Faserprodukt-Wirkung Z12 ×_{Z6} Z30 auf den 60 Gauß-Linien ist auf DREI Ebenen tot — die Coxeter-Uhr erhält die Linien-Partition nicht; die VOLLE linien-kompatible Gruppe hat Ordnung 23040 mit maximaler Elementordnung 12 (Ordnungen 15/30/60 fehlen für JEDE Realisierung); das Faserprodukt selbst ist zyklisch Z60 (exakte SNF-Arithmetik); alle drei Kontrollen feuern (CC2-Sampler-Korrektur offengelegt)"
    },
    {
      "id": "v782_e8_transition_bus",
      "cluster": "core",
      "descriptionEN": "E8.TRANSITIONBUS.01: der Transition-Bus-Struktursatz — **TRANSITION-BUS-TORSOR (35/35): die 240 Wurzeln des unimodularen hermiteschen Gauß-E8 sind 15 disjunkte Koordinaten-8-Rahmen über der Klassenabbildung V = L/(1+i)L; auf L/2L sind die 120 Wurzel-PAARE exakt die 120 q = 1-Punkte des Plus-Typ-Quadratraums, und die 8 Paar-Bilder jeder Klasse bilden einen kanonischen TORSOR unter v_perp (affiner F2³ ohne ausgezeichneten Ursprung); das Deck J wirkt als Translation um v; der OBSTRUKTIONS-Zensus ist beweisfest: KEINE deck-äquivariante Wurzel-Etikettierung R(E8) = V × (V∖0) existiert (−1 frei auf Wurzeln, trivial auf Paaren; J ist 4-Zyklus, Torsor-Translationen Ordnung ≤ 2); das eingefrorene symplektische Prädikat reproduziert ALLE 57600 geordneten Skalarprodukte inkl. Vorzeichengesetz und h-Verfeinerung; Spread-Schicht 240 = 5 × 3 × 16 mit dem q*-Selektor; der A15-Vergleich quantifiziert (A15 ist ein strikter Bus MIT kanonischem Startzustand, E8 beweisbar OHNE; Nicht-Orthogonalität exakt verdoppelt 26880 vs 13440); Kontrollen C1–C3 feuern (C3-Kontroll-Design-Korrektur offengelegt)"
    },
    {
      "id": "v783_two_qubit_clifford",
      "cluster": "core",
      "descriptionEN": "E8.CLIFFORD2Q.01: die Zwei-Qubit-Stabilizer-Identifikation des Compiler-Trägers — **CLIFFORD-PARTIAL(H4) (32/32): unter der EINEN vordeklarierten Karte U (Koordinaten-Klassen-Rahmen -> Rechenbasis, z_k = r_{2k} + i r_{2k+1}) SIND die 60 Gauß-E8-Linien die 60 reinen Zwei-Qubit-Stabilizer-Strahlen, eintrags-exakt über Z[i] (60/60; Träger-Zensus 4 + 24 + 32 beidseits); U G31 U^{-1} ist Index-2-Untergruppe der Zwei-Qubit-Clifford-Matrixgruppe (|C2| = 92160 = 2 × 46080, Nebenklasse zeta8 G31; projektiv gleich, beide 11520); 46080 = 4 × 16 × 720 = mu4 × Pauli16 × Sp(4,2) mit den 15 Klassen = 15 Pauli-Kontexten (W(3,2) = GQ(2,2)-Identifikation, alle 105 Paare), die 6 Spreads = MUB-Pentaden (|⟨psi|phi⟩|² = 1/4 exakt); H4 FÄLLT WIE VORDEKLARIERT: das Index-Bit konjugiert WEDER J noch mu4 — es ist das Galois-sqrt2/Hadamard-Bit Gal(Q(zeta8)/Q(i)) (tauscht die zwei mu8-Lift-Klassen), das J <-> −J-Bit ist die separate antiunitäre äußere Z2; alle vier Kontrollen feuern (T-Gate-Twist EXAKT die vordeklarierten 16/60); Stabilizer-Mathematik über den Träger — ROOTCLASS-MIXED (v775) unberührt, keine Materie-Lesart"
    },
    {
      "id": "v784_curve_code_2torsion",
      "cluster": "core",
      "descriptionEN": "CURVE.CODE.2TORSION.01: der Kurven/Code-2-Torsions-Entscheider — **CURVE-CODE-PARTIAL (33/33): die eingefrorene CM-Modell-Route durch die Naht-Kurve y³ = x⁴ − 1 (exakte Differentialcharaktere zeta^11, zeta^7 = v610; JEDE Generator-Periode als ganzzahlige O = Z[zeta_12]-Erkennung zertifiziert, dps 120, Residuen < 1e−35, HNF-Index 1; A[2] = O/2O = F2⁴ ab dort EXAKT; tau = Mult. mit zeta⁴ als einzige Ordnung-3-Klasse ABGELEITET; minimale UNGERADE tau-invariante Polarisation Typ (1,3) — keine Hauptform im tau-invarianten Kegel, Einheiten-Vorzeichen-Obstruktion; q_Theta = die eindeutige tau-fixe Verfeinerung): die Korrespondenz STIRBT an zwei exakten Schichten — (1) SIGMA: tau wirkt FREI auf A[2] (Zykeltyp [3,3] auf den 6 ungeraden Theta-Charakteristiken, Synthem-Seite), der Code-sigma-bar fixiert eine 2-Ebene ([3,1,1,1], Duaden-Seite) — nur die ÄUSSERE S6-Automorphie tauscht die Klassen, äquivarianter Zensus 0; (2) VERFEINERUNG: q_Theta ist GERADE (Arf 0, 1+6+9), der Code-Selektor q* UNGERADE (Arf 1, 1+5+10) — Arf ist Transport-Invariante, Zensus 0; Gesamt-Zensus 0 mit Positiv-Kontrolle 6; alle Muss-Kontrollen feuern"
    },
    {
      "id": "v792_ftransfer_kms_coset",
      "cluster": "core",
      "descriptionEN": "FTRANSFER.KMSCOSET.01: die Coset-Zuweisung des gefaserten Uhren-Funktors ist ein THEOREM aus KMS/modularem Typing — **KMS-COSET-THEOREM (18/18): S = −(ln λ)²/2 mit λ = dem PGL2-Multiplikator der Zeit-1-Holonomie pro Uhren-e-Fold; der Klassifikator nutzt KEINE Schwarzian und KEINE Ableitungen (Nicht-Zirkularität am eigenen Quelltext gegatet): Funktionalidentitäts-Holonomie + PGL2-Klasse + Detailed-Balance-Multiplikator-Kreuzverhältnis (die diskrete KMS-Signatur); Lemmata A/B/C sympy-exakt; BLINDES Protokoll: F_pole/F_Boltzmann modular mit |ln λ| = Δ = 6 log(3/2), F_QCD autonom (parabolisch), F_relic entartet; Entblindung EXAKT (vorhergesagte Cosets == transportierte Schwarziane, sympy-Null); Kontrollen: Fake-KMS gefangen, affine Invarianz, die v578-Möbius-Uhr als Positivfall — die Faser-Klassifikation der CONTRACT.F.01-Verfeinerung ist abgeleitet, nicht fallweise inspiziert"
    },
    {
      "id": "v796_duad_syntheme_bridge",
      "cluster": "core",
      "descriptionEN": "CURVE.CODE.OUTERSPIN.01: die kanonische ÄUSSERE Brücke zwischen Code-Duaden-Seite und Kurven-Synthemen-Seite — **BRIDGE-CANONICAL (24/24): die Sylvester-Dualität β: Code-ungerade-Formen → Kurven-Spreads mit β σ = τ β und β(q*) = S* (der EINDEUTIGE ⟨τ, ρ⟩-fixierte Spread) existiert und ist eindeutig bis auf den Ordnung-6-Joint-Stabilisator (Zensus 18 + 18 → 6 + 6, EIN Orbit inkl. τ/τ²-Tausch durch innere Code-Symmetrie); der äußere Zeuge exakt ([3,1,1,1] ↔ [3,3] uniform); q_Θ zieht sich KONSTANT auf q_even* zurück mit q* = q_even* + h̄(·, A) — der Anker-Shift löst den v784-Arf-Konflikt kanonisch; 5̄/10 und 3+2 transportieren exakt auf die S*-Linien-Geometrie; t_A ↦ ρ = J auf jeder Brücke; chi_NSR ein ehrlicher exakter ZWEI-Varianten-Torsor (durch die v752-NS/R-Parität gepinnt); Kontrollen: innere Zensus 0/0, ungerader Arf-Rückzug 0, gestörtes σ 0; die INNERE Identifikation bleibt tot (v784); keine Materie-Semantik"
    },
    {
      "id": "v798_seam_clifford_modular_s",
      "cluster": "core",
      "descriptionEN": "SEAM.CLIFFORD.MODULAR_S.01: das fehlende Clifford-Bit als metaplektischer Lift des modularen S bei τ = i — **MODULAR-S-PARTIAL (27/27): COSET POSITIV — der Pro-Binärfaktor-Hadamard K1 = H ⊗ I IST der Weil-Lift von S bei τ = i und landet exakt in der fehlenden Nebenklasse ζ₈ G31 (exakte Relationen S² = I, T² = Pauli-Parität, (TK)³ = ζ₈ I); der TOTALE Fourier H ⊗ H liegt IN G31 mit metaplektischer Anomalie (T3 K3)³ = i I = J — das Deck IST die Anomalie des totalen modularen Lifts (R2 platziert); ζ₈ = (1+J)/√2 das reelle Halb-Deck, das das Naht-Paar {L, ζ₈L} tauscht; F4 als Nicht-Trägersymmetrie ausgetypt; OS-Reflexions-Gate exakt.  DER STRIKTE EINDEUTIGKEITS-ZENSUS IST LEER (0/46080): kein Θ-reelles metaplektisches Element der Nebenklasse fixiert q* — der Arf-Defekt ist unvermeidlich; retypter Zensus: 32 Überlebende = EINE G31-Konjugationsklasse; R3 BENANNT, nicht fixiert: die σ-Orbit-Defekte sind exakt {A + F_Σ + F_i} mit XOR = der Anker A selbst; Kontrollen: CZ in G31, falsche Phase bricht S², Selektivität 480 → 96 → 32"
    },
    {
      "id": "v799_seam_code_typeii",
      "cluster": "core",
      "descriptionEN": "SEAM.CODE.TYPEII.01: Typ II aus drei physikalischen Naht-Axiomen ERZWUNGEN — der direkte Angriff auf Residual R1 von ARF.BOUNDARY.CODE.01 — **TYPEII-FORCED (19/19): erschöpfend über ALLE 308.993 Unterräume von F2⁸ (dim 0..4, Gauß-Binomiale; dim > 4 per Rang-Nullität): gegenseitige Lokalität der Construction-A-Vertex-Wörter ⇔ selbst-orthogonal (135 bei dim 4; Lift-Shift-Braiding-Identität erschöpfend); ganzzahliger konformer Spin ⇔ doppelt-gerade (30); Holomorphie/Index eins (|disc| = 1, exakte Fraction-Gram-Determinanten) ⇔ selbstdual — die Überlebenden sind EXAKT die 30 ê₈-Kopien; kein rand-legaler Nicht-Typ-II-Code existiert; jeder nicht-selbstduale doppelt-gerade Code hat einen Diskriminanten-Sektor gebrochenen Gewichts ([8,3]-Zeuge mit exakten Coset-Thetas).  R1 → R1′ = A1 gegenseitige Lokalität + A2 ganzzahliger Spin + A3 Holomorphie/Index eins; importierte Gitter-VOA/Theta-Zutaten zitiert und in jeder genutzten Instanz maschinen-verifiziert; OPE-Linearitäts-Scope-Note getypt; Kontrollen feuern; KEIN P2-Marker-Move"
    },
    {
      "id": "v800_e8_torsor_fourier",
      "cluster": "core",
      "descriptionEN": "E8.TORSOR.FOURIER.01: die projektiven Fourier-Moden der 15 Gauß-Torsoren — Träger-Reinheit auf CHARAKTER-Niveau mit der wechselseitig-unverzerrten Kappe — **TORSOR-FOURIER-PARTIAL (30/30): 120 kanonische projektive Strahlen (Phasengesetz exakt auf 960 Ursprungs-Shifts), orthogonale Basis (Gram = 16 I); Deck J und alle 16 Paulis diagonal nach Charakteren, G31 monomial durch Sp(4,2) (Kern 64 = μ₄ × Pauli16; Orbits 15 + 45 + 60).  DER TRÄGERBLOCK: die 64 Strahlen der 8 R-Fasern = (1 + 5 + 10) ⊠ (1 + 3) unter W(D5) × W(A3) (Clebsch-Charakterpolynom (x−5)(x−1)¹⁰(x+3)⁵ exakt; S4-natürliche Tetrade mit σ = Familien-3-Zyklus + Anker) — das Träger-Pascal auf CHARAKTER-Niveau, auf Punkt-Niveau unmöglich (v775); P3 fällt wie vordeklariert: Gewichts- und Charakter-Basis WECHSELSEITIG UNVERZERRT (|⟨Paar|Mode⟩|² = 1/8 exakt) — ROOTCLASS-MIXED intakt; die 45 deck-geraden Moden SIND die 45 (Kontext, Pauli)-Inzidenzen unter der zertifizierten Karte (Operator-Wörterbuch für die v791-Intertwiner-Forderung notiert); Kontrollen feuern"
    },
    {
      "id": "v802_gnet_martingale",
      "cluster": "core",
      "descriptionEN": "GNET.MARTINGALE.LIMIT.01: die Martingal-Route zum G_net-GNS-Limes — endliche Hypothesen GEMESSEN, alle erfüllt — **MARTINGALE-HYPOTHESES-MET (15/15): Filtration EXAKT (Haar-Isometrie mit Träger-Kohärenz Leak 0; Watatani-Quasibasis 185/3041 bei l = 2/3 mit Σ v v* = 4·1; EINE Index-4-Erwartung für den ganzen Turm); Zustands-Defekte SUMMIERBAR mit gemessener Rate ~0.25 pro Verdopplung — eine Potenz besser als die 2^{−l}-Dim-Gesetz-Hüllkurve (berichtet, nicht erzwungen); eindeutiger Limes via Teleskopierung auf jeder Sprosse verifiziert; Split/Normalitäts-Vorstufen uniform (Determinanten-Zeuge D ≥ 0.985, stabilisierend bei 0.9928); Schritte 5–7 (GNS, lokale Normalität, Split + μ₄-Simple-Current-Erweiterung) GETYPT, nicht ausgeführt; QGEO-Architektur-Note: die offenen Majoranten-Residuen (b)/(c) werden für die Limes-EXISTENZ umgangen und in den getypten Normalitäts-Schritt verlagert — eine Umverteilung, kein Beweis; Kontrollen: verschobene Bindung leckt 0.707, Z2-only λ = 1/2, gestörte See O(1), Dezimierung konvergiert zum FALSCHEN entarteten Limes (die kohärente Bindung lasttragend)"
    },
    {
      "id": "v803_e8_jetcode_affine_nsr",
      "cluster": "core",
      "descriptionEN": "E8.RAMIFIED.JETCODE.01 (+ E8.AFFINE.NSR.01): die Jet-Code-Normalform des Gauss-E8-Compilers + das NS/R-Bit als affine 8-Punkt-Geometrie — **JETCODE-EXACT + AFFINE-NSR-EXACT (18/18 + 15/15): Z[i]/(2) = F2[eps]/(eps²) mit eps = 1+i (2 VERZWEIGT; Iso eindeutig, der falsche Split-Ring F2×F2 scheitert an allen 24 Bijektionen); W = L/2L FREI vom Rang 4 über den dualen Zahlen (explizite Z[i]-Basis via Hermite-Reduktion, Kovolumen-zertifiziert); 0 → eps W → W → V → 0 exakte SELBST-Erweiterung mit ker(eps) = im(eps); das Deck IST die Jet-Einheit J = 1 + eps und reproduziert den v782-Torsor auf allen 240 Wurzeln; NULL μ₄-äquivariante Splittings unter allen 65536 linearen Schnitten — die Kein-Ursprung-Obstruktion IST die Nicht-Spaltbarkeit; die 120 Wurzelpaare = die q = 1-Einheitsschale (Plus-Typ, 240 = 15 × 8 × 2: Wurzel = (Adresse, Fehler-Jet, Orientierung)).  TEIL 2: V \\ {0} = 7 NS + affiner 8-Punkt-Torsor; 112 D8-Wurzeln (NS) + 128 Ramond-Spinor (R), 248 = (112+8)+128; Stab(χ_NSR) = AGL(3,2) Ordnung 1344 = Aut(H8) mit DERSELBEN natürlichen 8-Punkt-Wirkung (Brute-Force 40320); affine Funktionen auf dem Torsor rekonstruieren RM(1,3) = H8 mit exakt 1344 Äquivalenzen; Sp(4,2)-Stabilisator (Ordnung 48) wirkt INNERHALB von AGL(3,2).  Getypt: interne Rekonstruktion, KEINE P2-Entfernung; keine Materie-Semantik (ROOTCLASS-MIXED)"
    },
    {
      "id": "v805_e8_syndrome_algebra",
      "cluster": "core",
      "descriptionEN": "E8.SYNDROME.ALGEBRA.01: der ehrliche Kill — E8 ist KEIN Fehlerkorrektur-Hull im schärfsten Sinn — **SYNDROME-DEAD (13/13, der Kill feuert wie vorregistriert): die μ₄-linear-Lift-Realisierungsfamilie erreicht max Rang 222/256 über ALLE VIER eingefrorenen Phasenkonventionen (sympy-exakt über Q(i)); die ±r-Proportionalitäts-Obstruktion ist STRUKTURELL für die gesamte Diagonalphasen-Familie (c(−r) = −c(r): 12 c-Vektor-Kollisionen, Zeuge gedruckt); beide Post-hoc-Diagnosen schneiden SCHLECHTER ab (nichtlinear 215, ζ8-linear 160) — der Tod ist robust; ÜBERLEBT: die Syndromtafel-Zählung 16 + 240 = 256, der Neutralsektor (die 16 v112-Kerne SIND die diagonale MASA; Gleichklassen-Produkte A_0-wertig mit μ₄-Koeffizienten EXAKT), das feine (Klasse,d,k)-Gruppoid-Gesetz (assoziativ 0/3375) und das W(3,2)-Kommutations-Skelett; KEIN skalarer V-Kozykel (0/210); Nachfolger-Kriterium benannt: wurzel-individuelle (nicht-r-lineare) Operator-Daten; Kontrollen 3/3"
    },
    {
      "id": "v808_petersen_sixthroot",
      "cluster": "core",
      "descriptionEN": "CARRIER.PETERSEN.RADIAL.01 (+ TRANSPORT.SIXTHROOT.01): die K5-Kantenmaschine + der bit-exakte Sechstwurzel-Entscheider — **PETERSEN-RADIAL-EXACT + SIXTHROOT-SPECTRAL-ONLY (19/19 + 16/16): X = (−2,−2,−2,3,3) eindeutig primitiv spurfrei blockkonstant; E(K5)-Ladungszensus {−4: 3, +1: 6, +6: 1} = das SU(5)-Dekuple gegen die zertifizierte v774-Tafel; Petersen SRG(10,3,0,1) zertifiziert (|Aut| = 120-Zensus = S5); Distanzschalen (1,3,6) LADUNGSREIN; das Flavor-Hexagon = die Distanz-2-Schale [E neu] (explizite C6-Ordnung Kante für Kante); äquitabler Quotient [[0,3,0],[1,0,2],[0,1,2]] mit Hyperladungs-Eigenvektor (6,−4,1) → −2; spec(P⁶) = {1, 64/729, 1/729} = das deployte Transport-Spektrum EXAKT (v54/v56/v124); 10-dim-Vorhersage [H neu]: Multiplizitäten ×5 = g_car (schnell), ×4 = |μ₄| (langsam) — kein Korpus-10-dim-Operator.  TEIL 2: B = (1/18)[[13,1,4],[1,13,4],[4,4,10]] (alle Einträge Compiler-Zahlen) ist die EINDEUTIGE symmetrische doppelt-stochastische PSD-Sechstwurzel in der Korpus-Basis; B⁶ trägt das Spektrum exakt, ist aber bit-gleich KEINER deployten Matrix (GL(3,Q)-konjugiert zu v486's M⁶ und zur Cusp-Diagonalform) — der Korpus friert das SPEKTRUM ein, nicht die Basis; stochastische Typen verschieden"
    },
    {
      "id": "v809_curve_code_doily",
      "cluster": "core",
      "descriptionEN": "CURVE.CODE.DOILY.01: die zertifizierte äussere Brücke als Inzidenz-Operator-Aussage mit spektralem Payoff — **DOILY-EXACT (21/21): NN^T = B + 2I EINTRAGSWEISE (N = die K6-Doily-Inzidenz 15 Duaden × 15 Synthemen; B = die zertifizierte v752/v774-symplektische Inzidenz); spec {9, 4⁹, 0⁵} exakt; die Singulärwerte von N/3 = {1, 2/3, 0}: die RECOVERY-BASISRATE 2/3 (v221) als der einzige nichttriviale Singulärwert der kanonischen Duade-zu-Synthem-Inzidenz — die (2/3)⁶-Identifikation getypt [H neu], sechs-Schritt-Komposition ausstehend; Petersen-Kanten = Synthemen 15/15 am q*-Punkt; JEDE Zensus-Brücke faktorisiert EXAKT: N_code = P_Δᵀ N_curveᵀ P_Γ — eine DUALITÄT, kein Isomorphismus (innere Route tot: 0/720); Arf-Schaltmenge T6 = die sechs Innerblock-Duaden des q_Θ-Splits; N/3 doppelt-stochastisch, getypt klassisch, nicht CPTP; keine Materie-Semantik"
    },
    {
      "id": "v810_cutcode_moment",
      "cluster": "core",
      "descriptionEN": "CARRIER.CUTCODE.01 (+ CARRIER.MOMENT.INCIDENCE.01): der Arf-Split als Schnitt-Klassifikation von K5 + das Ein-Zeilen-Master-Moment mit der eingefrorenen 210-Entscheidung — **CUTCODE-EXACT + MOMENT-INCIDENCE-PARTIAL (17/17 + 16/16): die Schnittkarte δ injektiv-isometrisch auf allen 256 Zellen (Kern = Repetitionscode; Isometrie bricht exakt auf den 256 ungerade × ungerade-Paaren); der Arf-Split 16 = 1 + 5bar + 10 IST die Schnitt-Klassifikation (0|5) + (1|4) + (2|3) [E neu]; Clebsch 40 = 5 + 20 + 15 über den Arf-Schalen mit verbotenen Blöcken LEER und Innerhalb-10 = Petersen (Iso-Zensus 120); Wurzeln/Linien graduieren 80 + 160 / 20 + 40 — Arf/Schnitt-Graduierung, NICHT Materie.  TEIL 2: die symbolische Identität Σ(x_i+x_j)² − 3Σx² = (Σx)² liefert Tr_{S+}X² = 4|X|² = 4·h(E8) = 120 in einer Zeile; Charpoly (t+2)³(t−3)² exakt, 72 nur-Audit; die 210 = C(10,4)-Karte DEKORATIV — äquivariante Bijektionen durch die Fixpunkt-Obstruktion GETÖTET (beide Gruppen, beide Indizierungen, erschöpfend), keine der sechs eingefrorenen Gewichtsfamilien hält; nur die partielle 80 → 5×16-Miss-Slot-Karte und der Newton-Tie überleben als Audit"
    },
    {
      "id": "v812_flavor_graph_filtration",
      "cluster": "core",
      "descriptionEN": "FLAVOR.GRAPH.FILTRATION.01: die Flavor-Matrix-Filtration Q → K → L als Inzidenz-Compiler-Fingerabdruck — **FILTRATION-FINGERPRINT-ONLY (16/16): die Zeilen-Budgets sind EXAKTE, ordnungs-exakte Zählungen der K_{3,2}/Prisma/K5/H8-Inzidenzkette — Q (4,5,6), K (6,9,10) (Liniengraph ≅ Dreiecksprisma ≅ C6(2,3)), L (10,14,16) — mit der stufenweisen Additivität (4,5,6) + (6,9,10) = (10,14,16), die das deployte L = K + Q spiegelt (Matrizen lokalisiert: v11/v12/v37/v4; v11-Eindeutigkeit rein-ganzzahlig re-deriviert); aber KEINE Eintrags-Rekonstruktion: keiner der eingefrorenen Struktur-Kandidaten (25/20/47 pro Stufe, eintrags-exakt, 6 Spaltenpermutationen) reproduziert irgendeine deployte Matrix; die erschöpfenden False-Positive-Raten 0.748% / 0.035% / 0.005% machen die Misses INFORMATIV (Kandidatenmenge nicht vakuös); getypt: Audit-Fingerabdruck auf Budget-Niveau, keine Eintrags-Inzidenz-Konstruktion; Kontrollen 4/4"
    },
    {
      "id": "v813_p1_index_kms",
      "cluster": "core",
      "descriptionEN": "P1.INDEX.KMS.01: die geschlossene Normierungs-Identität c3 = 1/(I_Jones · β_angle) = 1/(8π), ihr Konventions-Audit und der geschärfte P1-Kontrakt — **P1-IDENTITY-CLOSED (13/13): I·β·c3 = 4 · 2π · 1/(8π) = 1 EXAKT (formale Fraction × π^k-Arithmetik, keine Floats); Abhängigkeits-Audit: der Index I = 4 = |μ₄| ist ein UNABHÄNGIGES c3-freies Theorem (NULL 'c3'-Vorkommen in allen vier Quellen v154/v726/v746/v779, per Datei-Scan gegated); β = 2π unabhängig GEMESSEN (v526, Falsch-β ausgeschlossen; der v519-Radiant-Pin ist die EINE geteilte Konvention und kürzt sich); c3 BLEIBT das P1-Axiom, das die Identität ERKLÄRT, nicht ableitet; die endliche Gleichverteilungs-Vorstufe GEMESSEN: w_1 = w_3 = 1/4 EXAKT auf jeder Stufe, symmetrischer nullsummiger Offset ±2^{−(l+1)} (= die Kein-Anomalie-Form); der SEE-Zustand gleichverteilt NICHT (Abw. 0.188) — die Brücke muss die MODULARE ANTWORT normieren, nicht Vakuum-Gewichte; Kontrakt [O] registriert mit der EINEN offenen Brücke (gleichverteilte Modular-Antwort über die I Deck-Sektoren) und Kill-Kriterium (asymmetrischer Offset); Kontrollen: Index 2 → 1/(4π), das der Korpus anderswo verbucht"
    },
    {
      "id": "v814_k5_sixstep_transport",
      "cluster": "core",
      "descriptionEN": "K5.SIXSTEP.TRANSPORT.01: die Sechs-Schritt-Komposition + der kanonische 10-dim Transport-Vorschlag — **SIXSTEP-CLOCK-ONLY + TRANSPORT10-PROPOSED (23/23): die Rate (2/3)⁶ entsteht KANONISCH auf drei räumlichen Routen (Doily-Alternation, Hexagon-Kompression, Hexagon-Zirkulation), aber jede Route trägt eine typisierte Obstruktion (kein Einheitsmodus — Norm-Obstruktion [E neu]; bipartites Vorzeichen −2/3; falscher Raum ×9) — der deployete Sechser ist der UHR-EXPONENT (v124 λ_n = (1−n/3)⁶, v486-Hand, festes Eigensystem); SECHSTPOTENZ-BLINDHEIT [E neu]: C6 und 2K3 haben identische sechste Potenzen — nur Schritt-Daten unterscheiden Walk von Uhr; KORREKTUR (datierte Notiz auf der v808-Zeile): T_v221 (exakte Spektralsumme des v221-Transfers (1−w)⁶) == B⁶ BIT-EXAKT — das SIXTHROOT-Verdikt wird BITEXACT-vs-v221; der kanonische Vorschlag T10 = P10⁶ in der Λ²-Basis (v774-Wörterbuch) besteht ALLE Gates: Spektrum {1, (1/3)⁶×5, (2/3)⁶×4} (Multiplizitäten (5,4) = (g_car, |μ₄|)), äquitable v774-Partition, Quotient == (Q_Pet/3)⁶ bit-exakt — promotionsreifer VORSCHLAG, kein Deploy-Anspruch; Kontrollen feuern; kein Marker-Move, KEIN RH-Anspruch"
    },
    {
      "id": "v817_positive_descent_master",
      "cluster": "core",
      "descriptionEN": "Abstract positive-descent theorem and G_net finite hypothesis checks; the prime packet instance is maintained by Prime Front."
    },
    {
      "id": "v832_anchor_flavor_checksum",
      "cluster": "core",
      "descriptionEN": "ANCHOR.AFFINE.01 + FLAVOR.BIDIR.01: die affine Anker-Normalform und die bidirektionale Flavor-Prüfsumme — zwei exakte Korpus-Kompressionen — **ANCHOR-AFFINE-EXACT + FLAVOR-BIDIR-CHECKSUM (selektiv) (18/18 + 15/15): der Anker a = (1,1,2) hat Potenzsummen p_n = 2 + 2^n mit der affinen Rekursion p_{n+1} = 2p_n − 2 identisch in n (sympy); T(x) = 2x − 2 hat den EINDEUTIGEN Fixpunkt 2 = |Z₂| (das Blatt), und der T-Orbit von 4 ist exakt das Compiler-Quintett (4, 6, 10, 18, 34) = (|μ₄|, |R⁺(A₃)|, A_L, N_fam·|R⁺(A₃)|, Z₆-Lift), mit p₀ = 3 = N_fam vorangestellt; das GESAMTE Budget aus EINER Rekursion: |R(E8)| = p₁p₂p₃ = 240, dim E8 = 240 + q₃ = 248 (Leiter-Identität p₄ − p₃ = p₃ − 2 = 8 = h(D5) = rank E8), h(E8) = p₂p₃/2 = 30, |R(D5)| = p₁p₃ = 40, Ω_adm = 2p₁p₂ = 48, (e₁,e₂,e₃) = (4,5,2), 10b₁ = 41. Teil B: R = [[1,3,0],[1,5,2],[2,5,3]] dekodiert DENSELBEN Anker beidseitig — Ra = (4, 10, 13) = (p₁, p₃, N(3+2i)), Rᵀa = (6, 18, 8) = (p₂, p₄, p₃−2); der Geschwister-Kill: von 12 Kandidaten besteht GENAU EINER — die akzeptierte (1,5,2) — den vollen Identitätssatz, und die Anker-Kontraktion (Ra)₂ = 10 = p₃ tötet beide Geschwister (11 und 12). Kontrollen feuern (falsche Karte, falscher Anker, Geschwister (1,4,3) reproduziert tfpt_2s Fehlerzahlen WÖRTLICH). Exakt sympy/Integer; keine Floats, kein RNG, kein Fit.**"
    },
    {
      "id": "v833_gaussian_ramification_ladder",
      "cluster": "core",
      "descriptionEN": "GAUSSIAN.RAMLADDER.01: die (1+i)-Verzweigungsleiter — die vier dokumentierten Rollen des verzweigten Gaußschen Primelements π₂ = 1+i sind EIN maschinell geprüftes Objekt mit vier Sprossen, jede von Grund auf re-zertifiziert — **RAMIFICATION-LADDER-EXACT (33/33): SPROSSE 1 (Norm-Verdopplung): N(π₂) = 2, (π₂²) = (2); auf dem Gaußschen E8-Gitter (1+J)ᵀ(1+J) = 2I₈ EXAKT; min L = 4 per Zensus, die Nullklasse von L/(1+i)L trägt KEINE Wurzel (0/240), 240 = 15 × 16, Elementarteiler (1,1,1,1,2,2,2,2), Index 16 = N(π₂)⁴; SPROSSE 2: L/(1+i)L ≅ F₂⁴ mit dem Deck J TRIVIAL wirkend — der 4-Bit-Adressraum; SPROSSE 3 (der nicht-spaltende Jet): Z[i]/(2) ≅ F₂[ε]/(ε²) (Ring-Iso EINDEUTIG unter allen 24 Bijektionen), W = L/2L frei vom Rang 4 über den dualen Zahlen, Deck = JET-EINHEIT 1 + ε, und von ALLEN 65536 linearen Schnitten sind EXAKT 0 deck-äquivariant — die Kein-kanonischer-Ursprung-Obstruktion IST Nicht-Spaltung (v803); Plus-Typ-Form 136/120, die 120 Wurzelpaare SIND die q = 1-Schale; SPROSSE 4 (der metaplektische Lift): ζ₈ = (1+i)/√2, (1+J)² = 2J, (SH)³ = ζ₈I, (T₃K₃)³ = iI = DAS DECK (v798); exakter Z[ζ₈]-BFS-Zensus: |C₂/μ₈| = 11520 = 16 × 720, NULL Galois-gemischte Klassen, C₂ = K ∪ ζ₈K (v783 [E]); Zeugen-Paritäten χ(H⊗I) = 1 (das RM-CSS-ζ₈-Phasenbit), χ(H⊗H) = χ(CZ) = 0. Kontrollen feuern (der SPALT-Ring F₂×F₂: 0 Isos; die spaltende Selbsterweiterung: 65536/65536 Schnitte). Exakt Integer/sympy; keine Floats, kein RNG.**"
    },
    {
      "id": "v834_pg32_flag_completion",
      "cluster": "core",
      "descriptionEN": "PG32.FLAGS.01: die PG(3,2)-Flaggen-Normalform für die 105 + 35-Vakuum-Vervollständigung, mit dem Inzidenz-Kill-Test als Split typisiert — **PG32-FLAGS-EXACT + INCIDENCE-SPLIT (26/26): die Zähl-Normalform [4 über 2]₂ = 35, 140 affine 2-Flats = 35 durch den Ursprung + 105 nicht; PG(3,2): 15 Punkte, 35 Geraden, 105 = 35 × 3 Flaggen ≅ C(15,2) Paare; RM-Schicht (v819/v821/v822 neu aufgebaut): [15,10,4]-Enumerator EXAKT, die 105 Gewicht-4-Wörter SIND die Flats ohne 0, Grams 22I+6J / 6I+J / 28I+7J; Vakuum-Lesart: die Vakuum-Koordinate ist URSPRUNGS-RESTAURATION (EIN Ursprungsplan entkürzt dim 10 → 11); DER INZIDENZ-SPLIT (der typisierte Kill): 105 = 45 + 60 ist die Sp(4,2)-ORBIT-Zerlegung auf BEIDEN Seiten (|Sp(4,2)| = 720 brute force); auf den 60 NICHT-ISOTROPEN Sprossen ist die Polaritätskarte Φ: (U,t) → ((t+U) ∩ U^ω, U^ω) wohldefiniert, bijektiv, Sp(4,2)-ÄQUIVARIANT (60 × 720) und INZIDENZ-ERHALTEND; auf den 45 DOILY-Sprossen ist die Obstruktion typisiert: U^ω = U und (t+U) ∩ U = LEER für 45/45 — und das Flaggen-Gram 18I+3J ≠ 22I+6J. TYPISIERTE FOLGERUNG: '105 Checks = 105 Flaggen' ist exakt als deck-äquivariante BIJEKTION, NICHT als Inzidenz-Isomorphismus — die Obstruktion sitzt exakt auf dem Doily (der typisierte Grund, warum v822s Kontinuums-Dilatation starb). Die Nullklasse trägt keine Wurzel (0/240, v689 [E]). Kontrollen feuern. Exakt Integer; keine Floats, kein RNG.**"
    },
    {
      "id": "v844_message_doily_rank",
      "cluster": "core",
      "descriptionEN": "MESSAGE.LADDER.01 + DOILY.PASCAL.RANK.01: die E8-Adressformel und das Doily-Pascal-Rangtheorem — zwei exakte Korpus-Kompressionen, EIN Modul aus zwei Probes — **MESSAGE-LADDER-EXACT + DOILY-PASCAL-RANK-EXACT (23/23 + 16/16): TEIL A: M_n = 15·Δp_n = 15·2^n gegen die TATSÄCHLICH neu aufgebauten Objekte zertifiziert, eine BENANNTE Struktur pro Sprosse: M_0 = 15 = Adressraum (Klassen von V = L/(1+i)L); M_1 = 30 = h(E8), HIER aus dem neu aufgebauten Wurzelsystem hergeleitet, nicht zitiert (8 einfache Wurzeln, zusammenhängende einfach geschnürte Cartan-Matrix mit det 1 und E8-Gradfolge, höchste Wurzel EINDEUTIG bei Höhe 29 mit Marken {2,2,3,3,4,4,5,6}, h = 1 + 29 = 30 = |R|/Rang); M_2 = 60 Gaußsche Wurzellinien (μ4-Orbits, alle Größe 4, genau 4 pro Klasse); M_3 = 120 = |R+| = Vorzeichenpaare; M_4 = 240 = |R(E8)| mit Zensus 240 = 15×16 (Nullklasse LEER); Coda 248 = 15·16 + 8 gegen Zensus und HNF-Rang; die strukturelle Verdopplung 60 → 120 → 240 typisiert. Kontrollen feuern (falscher Anker (1,2,2) → (45,105,225,465); falsche Multiplikatoren 14/16 verfehlen alle Ziele und brechen die Coda). TEIL B: die Duaden-Synthemen-Inzidenz N des Cremona-Richmond-Doilys (15×15, 3-regulär beidseitig) und die symplektische Punkt-Inzidenz B auf PG(3,2) (B·B^T = 4I + 3J zellweise, charpoly (x−7)(x−2)^9(x+2)^5 exakt) sind EIN Objekt: über das K6-Duaden-Modell trägt die Bijektion N·N^T auf B + 2I ZELLWEISE (alle 225 Zellen) — Duaden-Kollinearität = symplektische Orthogonalität. DAS RANGTHEOREM (exakt): charpoly(N·N^T) = (x−9)(x−4)^9·x^5 — Rang N = 10 = A_Λ (Dekupel-Sektor), Kern = 5 = g_car (Fünf-Slot-Träger), Multiplizität 9 = N_fam², Top-Singulärwert 3 = N_fam; die SECHS Ovoid-Indikatoren SPANNEN den Kern — die P2-Ganzzahlen sind die Singulärwert-Daten der Doily-Inzidenz. Recovery-Wert sing(N/3) = {1, 2/3 ×9, 0 ×5} exakt, 2/3 = (N_fam−1)/N_fam (v327); Exponentiationsidentität: der Sechs-Schritt hat exakt (2/3)^6 = 64/729 als λ_2 (v330/v486); TYPISIERTE Lücke erneut bestätigt, KEIN Upgrade: 1/729 ist KEINE Wurzel — der deployte (1/3)^6-Modus bleibt auf der Uhr-Seite (v814). Kontrollen feuern (zirkulante und LCG-3-reguläre Paarungen: Rang 13 ≠ 10, falsches charpoly). Exakte Integer/Fraction/sympy-Arithmetik."
    },
    {
      "id": "v845_cfin_normal_form",
      "cluster": "core",
      "descriptionEN": "NORMALFORM.CFIN.01: der endliche Compiler als EIN Objekt C_fin = (V, ħ, q*, σ, ι), vom Gaußschen Quotienten ABGELESEN (nicht postuliert), mit dem Code-zu-Materie-Kill als eingebautem NEGATIV und der verbleibenden P2-Brücke typisiert — **NORMAL-FORM-ASSEMBLED (28/28, SPEC v2): V = L/(1+i)L (Zensus 240 = 15×16, Nullklasse LEER); ħ = die Bit-Form (GEFUNDENE Familien/Anker-Basis (F1,F2,F3,A) mit Gram J − I, auf allen 256 Paaren); σ = der abgestiegene Ordnung-3-Deck (σ³ = id, symplektisch, genau 3 Fixklassen {A, F_Σ, F_Σ+A}, 4 freie 3-Orbits — der Familienzyklus); q* = die EINDEUTIGE Verfeinerung (genau 16 per Brute-Force über 2^16, Arf-Zensus 6 + 10; Selektor σ-invariant: 4 → q(A)=1: 2 → q(F_Σ)=0: 1, Arf-Typ 1, σ-erhalten); ι = der Paritäts-Lift bijektiv auf C_even(5) (β == ħ in allen 256 Zellen, q_wt == q*, σ wirkt auf den FÜNF Träger-Slots als 3+2-Split — Nachrichten- und Trägerregister verschränkt). ABGELEITET aus dem EINEN Objekt: |Sp(4,2)| = 720 (voller 65536-Matrizen-Zensus), Stab(q*) = 120 TREU als S5 auf den Fünf, Orbits [1, 5, 10]; 16 = 1+5+10, 15 = 5+10; PASCAL-Lesart (1, 10, 5) = (C(5,0), C(5,2), C(5,4)); PROJEKTIVE Lesart: PG(3,2) mit 15 Punkten, 35 Geraden, 15 isotrop (das Doily), die 5̄ ist ein OVOID (trifft jede isotrope Gerade genau einmal); Budget gegen die echten Gitterobjekte: 240 = 16·15, 60 Linien = 4·15, 30 = 2·15 = h(E8), Rang 8, 248 = 16·15 + 8. EINGEBAUTES NEGATIV (v775 neu verankert): die exakte 128-Spinor-Zählschranke — höchstens 8 rein-spinorale Klassen, ≥ 7 der 15 Klassen tragen IMMER Adjoint-Seiten-Wurzeln (konkreter Zensus SATURIERT beide Schranken: 8 + 7, NULL Mischklassen — der J-Quotient respektiert die D8-Graduierung); Gewichts-Kill am kanonischen Split: jede Spinor-Klasse = 8 verschiedene Halbspinor-Gewichtspaare, NIE 1 — v775 ROOTCLASS-MIXED bestätigt, NICHT umgestoßen. P2 TYPISIERT (kein Upgrade): alle 30 Platzierungen sind Typ-II-[8,4]-Codes (v799-Echo); C_fin leitet A1–A3, die OPE-Linearitäts-Notiz und die v776-Auswahl NICHT her — P2 (g_car = 5 = ι-Slot-Zahl) auf v799s Residuum R1′ ([C]) VERENGT, nicht eliminiert. Kontrollen feuern (Dot-Form: 0 Verfeinerungen; trivialer Deck: q* nicht eindeutig; falscher Lift bricht β == ħ)."
    },
    {
      "id": "v849_cfin_unique_cofinal_lean",
      "cluster": "core",
      "descriptionEN": "CFIN.UNIQUE.01: finite compiler uniqueness, one orbit and C6 automorphisms; cofinal Weil mirror moved to Prime Front."
    },
    {
      "id": "v852_arf_vacuum_ovoid",
      "cluster": "core",
      "descriptionEN": "ARF.VACUUM.SYNDROME.01 + OVOID.DECODER.01: **ARF-VACUUM-SYNDROME-EXACT (24/24) + OVOID-DECODER-EXACT (23/23)** — der Arf-Bit-Satz: a(t+U) = h̄(u,v) für ALLE 140 affinen Flats (positions-unabhängig; alle 4 Nebenklassen stimmen überein); die Zwei-Bit-Syndromtafel (s_vac, s_Arf) zerlegt die 140 Flats EXAKT in 15/20/45/60 (die v821- und v834-Zählungen als Zeilen EINER Tafel auf dem Hamming-Code); der LOKALE Hecke-Satz an ALLEN 15 Punkten (12+16 = 28 = σ₃(3), 12−16 = −4 = a₃, mit den isotropen Richtungen in Bijektion zu den Punkten von x^⊥/⟨x⟩); der Code-Bleistift H = [15,11,3] ⊃ {D = [15,10,4], C_iso, C_non = [15,10,3]} ⊃ K = [15,9,4] (Gewicht-3/4-Wörter = die 35 Geraden / 105 Off-Origin-Flats als MENGEN-Gleichheiten; das Kanonizitäts-Lemma: alle 16 Verfeinerungen differieren um Simplex-Wörter und induzieren EIN Arf-Funktional auf H); der Vakuum-Bit IST das Paritätsbit, das RM(2,4) restauriert (Punktierung → H, Verkürzung → D, Bijektion exakt). DER DECODER: NᵀV = 0 aus perfekten Matchings, Gram VᵀV = 36I−6J, SNF(N) = diag(1¹⁰,0⁵) TORSIONSFREI über ℤ (det-±1-Minor exhibiert), Moore-Penrose N⁺ = Nᵀ/4 − J/36 (alle vier Penrose-Axiome exakt in ℚ), der Träger-Projektor P₅ = I/2 − B/4 + J/12 mit tr = 5 = g_car, der geschlossene Decoder m̂ = (I−P₅)m EXAKT auf dem vordeklarierten Testsatz (Kernmoden unsichtbar, im(N) voll sichtbar), ker Nᵀ = die S₆-STANDARD-Darstellung (11/11 Klassen-Charaktere = fix−1), und der Mod-2-Schatten v_a mod 2 = q_a: DREI RINGE (ℝ-Ovoid-Spann / ℤ-torsionsfrei Rang 5 / F₂-Code A_q = [15,5,6]), EIN Kern — der Doily-Erholungskanal hat jetzt eine explizite integrale Inverse, nicht nur ein Spektrum. Exakte Algebra durchgehend; Kontrollen feuern; KEINE Marker-Bewegung. KEIN RH-Anspruch."
    },
    {
      "id": "v853_bent_css_mangoldt",
      "cluster": "core",
      "descriptionEN": "ARF.BENT.CSS.01: exact bent phase and CSS code identities; Mangoldt incidence experiment moved to Prime Front."
    },
    {
      "id": "v857_simplex_fourier_winding",
      "cluster": "core",
      "descriptionEN": "E8.SIMPLEX.FOURIER.01 + FLAVOR.WINDING.QUADRATIC.01: **SIMPLEX-FOURIER-EXACT + INTERTWINER-IDENTIFIED (26/26) + WINDING-QUADRATIC-EXACT (20/20)** — die E8-Audits I: der Gauß-Zensus IST ein Fourier-Gesetz — r̂ = (240, −16¹⁵), Kanal P = (J−I)/15 mit Spektrum {1, (−1/15)¹⁵} und Zwei-Schritt-Kontraktion 1/225 (exakt in Brüchen); Ramifikation Θ₀(ungerade) = 0 mit Ein-Zeilen-Beweis; Äquidistribution auf JEDEM Level ≤ 16; der Paket-Zähler (Th0−Th1+Th2−Th3)(n) == die Code-Walsh-Transformierte auf JEDEM Level ≤ 16 (beide Leitern [−16, 112, −448, 1136, …]; Eisenstein-Normalform 15Θ_D8 = 7Θ_L + 8Θ₀); der WERT-SATZ: mh₂(n) = −1/15 für JEDES ungerade n ≤ 16500 (Ganzzahl-Identität, alle 1911 ungeraden Primzahlen abgedeckt) — v817s Pro-Primzahl-Messung ist die Primzahl-Restriktion eines CHARAKTER-SATZES des Codes; ehrlich typisiert: SPEKTRAL (die Paket-Gradierung (52,60,128) und die Code-Gradierung 15×16 verfeinern einander NICHT); 3 Kontrollen feuern, 2 Invarianzen halten (24 explizite Isometrien; die andere invariante Platzierung C**). Die Windungs-Quadratik (tfpt_2 I.5): χ_{R_s} − χ_R = −s·q_wind mit q_wind = e₁ᵀ adj(tI−R) 1 = t² − g_car·t + |Z2| EXAKT (Adjugat-Lemma; irreduzibel über Q, disc 17); die Keybox-Koeffizienten (9+s, 10+5s, 8+2s), das det-Inkrement 2 = |Z2| und das Minor-Gesetz (5, 3(s+1), 2(s+1)) sind Auslesungen der EINEN Quadratik; Dreifach-Lock s = 6 (Spur/Determinante/Coxeter-Lift, Schnitt {6} = {|R+(A3)|}); Δ(6) = 39200 = 2·140² > 0 (KEIN Kollisionspunkt; der EINZIGE 2×Quadrat-Treffer im Fenster −2…12); Realitäts-Schwelle s* ≈ 2.8250 = die einzige Δ-Wurzel in (2,3) (deployt ≈ 2.83); 4 Kontrollen feuern (Geschwister-Richtungen brechen Lock UND Quadratik). Exakte Korpus-Kompressionen deployter Ansprüche (v817/v833/v844, v4/v94) — kein neuer physikalischer Anspruch, keine Marker-Bewegung."
    },
    {
      "id": "v858_g31_clock_alphabet",
      "cluster": "core",
      "descriptionEN": "E8.G31.CLOCK_ALPHABET.01: **CLOCK-ALPHABET-SELECTIVE (19/19)** — Deg(G31)/4 = {2, 3, 5, 6} = {|Z2|, N_fam, g_car, |R+(A3)|} = das Compiler-Uhren-Alphabet als NORMALFORM: gcd = 4 = |μ4|, lcm = 120 = |R+(E8)|, lcm/gcd = 30 = h(E8), Audit-Identität (Σd/4)(Σ(d−1)/4) = 16 × 15 = 240 = |R(E8)| auf den Wurzeln REALISIERT (die 60 Spiegelungen = die 60 Gauß-Linien, 4 pro Klasse — dieselbe 15×16-Zählung wie die Code-Seite); der Rebuild gewardet (BFS-Abschluss |G| = 46080 = ∏Deg; EXAKTER Spiegelungs-Zensus: 60 Elemente mit rank(M−I) = 2, NULL Quasi-Spiegelungen — v634 reproduziert); der arithmetische Pin: (8,12,20,24) das EINZIGE Ganzzahl-Quadrupel mit Produkt 46080 und Summe 64 (gemessen); der Kill-Scan über ALLE 607 Rang-4-Shephard-Todd-Gruppen (Katalog-Integrität: ∏(Grade) = |G| für jeden Eintrag): G31 der EINZIGE Voll-Batterie-Passer — Schwach-Tripel-Impostoren G(20,10,4)/G(20,20,4)/G29 und Roh-Summe-64-Koinzidenzen G(8,2,4)/G(10,10,4)/W(H4) typisiert; Familien-Grenze G(m,p,4) bis m = 120 leer (typisiertes Argument); EHRLICHKEITS-LINIE: Grad-Audits können G31 vom reduziblen Torus Z8×Z12×Z20×Z24 nicht trennen — eingefrorener Geltungsbereich = die irreduzible Rang-4-Liste; X1/X2-Kontrollen feuern. Der W(D5)×W(A3)-Zaun mit BERECHNETEM Ein-Zahl-Invariant erneut getötet: |Z(G31)| = 4 = ⟨J⟩ = μ4 vs |Z(W(D5)×W(A3))| = 1 — NICHT isomorph, KEINE Zerlegung (v634 T1.3–T1.5 zitiert). Normalform + scharfe Kill-Regel für ein deploytes Objekt (v634/v654) — kein neuer Anspruch."
    },
    {
      "id": "v874_feedback_decoder_lean",
      "cluster": "core",
      "descriptionEN": "FLAVOR.FEEDBACK.NORMALFORM.01 + E8.WINDING.DECODER.01: **FEEDBACK-NORMALFORM-EXACT + WINDING-DECODER-EXACT (14/14 + 11/11, + Lean-Spiegel)** — zwei exakte Compiler-Theoreme: EINE ganzzahlige Basis P = [2·𝟙 | e₃ | R e₃] (det −4, SNF diag(1,2,2)) konjugiert die GESAMTE Windungslinie R_s = R + s 𝟙 e₁ᵀ exakt (symbolisch in s) in die Gain-plus-Companion-Normalform [[s+4,0,3],[0,0,−2],[4,1,5]]: Gain-Kanal s+|μ₄|, Companion-Block des Windungsquadrats q_wind = t²−g_car t+|Z₂|, Kopplungspaar (3,4) mit Schleifenprodukt N_fam·|μ₄| = 12 = dim g_SM — also χ_{R_s} = (t−s−|μ₄|)·q_wind − 12t, und bei s = 6 sind (15,40,20) die drei Gain-10-Ablesungen; die ehrliche Typisierung ist tragend: der Quotient ℤ³/Pℤ³ = ℤ₂² hat KARDINALITÄT 4 = |μ₄|, aber EXPONENT 2 — NICHT μ₄, keine Identifikation behauptet. Der Decoder: q_wind bildet das G31-Uhr-Alphabet {2,3,5,6} auf (−|μ₄|,−|μ₄|,|Z₂|,h(D5)) ab (q(2)=q(3) von der Palindrom-Achse g_car/2 erzwungen); D(y) = (y+4)²(y−2)(y−8) trägt die Budget-Koeffizienten (|Z₂|, Ω_adm=48, 2^g_car=32, 2^rk(E8)=256); und |μ₄|·q(|Z₂|)/|R(E8)| = −1/15 ist EXAKT der Walsh-Eigenwert des Gauß-Zensus — typisiert als AUDIT-Theorem, kein physikalischer Funktor. Lean-Spiegel: TfptCarrier/FlavorFeedback.lean (11 Theoreme, Standard-Axiome, kein sorry)."
    },
    {
      "id": "v875_shell_global_lean",
      "cluster": "core",
      "descriptionEN": "E8.SHELL.GLOBAL.01: **SHELL-GLOBAL-THEOREM (15/15, + Lean-Skelett)** — das Zensus→Theorem-Upgrade: Θ̂ᵤ/Θ_L = −1/15 auf ALLEN ungeraden Schalen (nicht mehr nur n ≤ 16500). Die drei Inputs, jeder gewardet: (i) die G31-Klassenwirkung = die 60 symplektischen Transvektionen erzeugen Sp(4,2) (Ordnung 720), TRANSITIV auf den 15 Nichtnull-Klassen und IDENTISCH mit der Schalenmodul-Wirkung; (ii) Normverdopplung durch (1+i): Θ₀(2m) = Θ_L(m), Θ₀(ungerade) = 0; (iii) Θ_L = 240σ₃ (klassisch, Eisenstein E₄). Kontrollen: eine nicht-transitive Untergruppe liefert einen invarianten NICHT-uniformen Zensus (das Gesetz bricht — Transitivität ist tragend); die Gerade-Schalen-Formel weicht korrekt ab (7/135 bei n = 2). Lean-Skelett: TfptCarrier/GaussianShells.lean prüft die KONSEQUENZKETTE im Kern (7 Theoreme aus dem benannten Hypothesenbündel ShellSystem; jede Aussage eine exakte Ganzzahl-Identität, keine Division), die drei Inputs bleiben BENANNTE Hypothesen mit typisierter Provenienz — die Haus-Disziplin: das globale Statement kann keine Lücke verstecken."
    },
    {
      "id": "v878_ledger_integrity",
      "cluster": "registry",
      "descriptionEN": "GATE.LEDGER.01: **LEDGER-INTEGRITÄTS-GUARD (10/10, CI-Sentinel, keine neue Physik)** — Runde 36 fand und behob die erste PRIMÄRSCHLÜSSEL-Verletzung im status_ledger.csv: claim_id FORM.QGEO.03 war doppelt vergeben (2026-06-15 CohomologyGrading; 2026-06-18 SeamDeckClosure-Flachabschluss) — die SeamDeckClosure-Zeile UMBENANNT in FORM.QGEO.04 mit datierter QA-Note, alle Flachabschluss-Referenzen aktualisiert. Der Guard macht die Fehlerklasse unwiederholbar: 10-Spalten-Schema + id-Grammatik; claim_id global eindeutig (der Fix gepinnt); jedes supersedes-Ziel existiert + Supersessions-Graph AZYKLISCH (Tarjan); eindeutige AKTIVE Supersessions-Ziele modulo der eingefrorenen remove-only-Allowlist der zwei legitimen 2026-06-Doppelschließungen; die dependencies-Spalte ehrlich als QUERVERWEIS-Feld typisiert (13 SCCs / 265 ids historischer Wechselverweise remove-only eingefroren — audit_baseline-Präzedenz): Azyklizität als KEINE NEUEN ZYKLEN erzwungen."
    },
    {
      "id": "v880_finite_anchor_theorems",
      "cluster": "core",
      "descriptionEN": "CFIN.QSTAR.NORMALFORM.01 + DOILY.ANCHOR.CLASSIFY.01 + DOILY.HODGE.FACTOR.01 + P2.QUADRATIC.EXHAUSTION.01: **die vier endlichen Anker-Sätze (26/26, FINITE-ANCHOR-CLOSED)** — die geschlossene, zensus-freie Normalform q*(x) = Σ_{i<j}x_ix_j + Σx_i = C(|x|+1,2) mod 2 (gewichts-rein; Selektor auf dem Linearteil erzwungen); die EXKLUSIVE Anker-Klassifikation der PG(3,2)-Linien (alle 15 Doily-Linien tragen {2,1,1} = a = (1,1,2), Linien-Vieta (4,5,2); 10 Sekanten = falscher Anker (1,2,2), 10 externe); die Doily als Multiplikationstafel Λ²E×Λ²E→Λ⁴E des Fünf-Slot-Carriers (Wedge ≠ 0 GENAU auf Doily-Linien); Ein-Schritt-Exhaustion erzwingt g = 5 ([E]-Satz, Calderón-Prämisse bleibt [O] — P2 verengt, nicht eliminiert). Alle Kontrollen feuern."
    },
    {
      "id": "v888_finite_pfaffian_signature",
      "cluster": "core",
      "descriptionEN": "DOILY.S6.IRREP.01 + DOILY.PLUCKER.GR25.01 + CFIN.ANCHOR.HADAMARD.01 + CFIN.S6.VACUUM.01 + CFIN.AUT.FLAVOR.C6.01 + CFIN.K6.MOTIF.01 + DOILY.PFAFFIAN.WICK.01 + P2.SELFHOSTING.PAIRING.01 + ANCHOR.MOD30.CLOCK.01: **die komplette endliche Pfaffian/K₆-Signatur (27/27 + 19/19 + 42/42, exakte Ganzzahl/sympy)** — Specht-Kanäle (1, 0, 2/3) exakt; die 15 Doily-Linien SIND die 15 Plücker-Monome von Gr(2,5) mit perfektem Vorzeichen-Match; (16,6,2,10)-Hadamard-Anker; S₆→S₅-Vakuumselektion (Orbits [1,5,10] mengengleich (Λ⁰,Λ⁴,Λ²)); DER FLAVOR-HEXAGON IST DIE Aut(C_fin)-BAHN (g² = σ exakt, (2/3)⁶ = Recovery, 0/600 Nicht-Automorphismen faktorisieren); K₆-Motiv (35 = 15 Matchings + 20 Dreiecke); die fermionischen Pfaffian-Vorzeichen GAUGE-MATCHED via kanonischem χ(i) = (−1)^{i+1} (dreifach identifiziert); SELFHOSTING: C(g+1,2) = g‼ ⟹ g = 5 einzig nichttrivial, N_fam = 3 ([E]-Satz, Wick-Prämisse [O]); Mod-30-Uhr CLOCK-DISTINGUISHED ({5,10,15,30}, lcm 30); äußere-Automorphismus-Signatur exakt."
    },
    {
      "id": "v896_wick_block_functor",
      "cluster": "core",
      "descriptionEN": "SEAM.CFIN.WICKFUNCTOR.01 + WICKMIX.01 + WICKBLOCK.01: **der Wick-Funktor-Bogen (27/27 + 33/33 + 26/26)** — die sechs Rollen existieren im deployten Seam (D5-Slot-Paare + A3-Randblock; so(16)-Sektorgesetz, q*-Grading, C₆-Intertwining exakt), aber der Vakuum-Kernel ist kanal-diagonal; die BEWIESENE Skalar-Obstruktion (der C₆-2-Zykel {4,5} erzwingt eine Duaden-Null — max 14/15; beide Auswegs-Zeugen konstruiert); die Block-Auflösung: BLOCKFUNCTOR-CONSTRUCTED (Kommutante dim 33, 15/15 Blöcke, 15/15 Block-Wick-Monome mit χ-Gesetz, Interleaving-Charakter symbolisch +1) und das entscheidende Faktum SEAM-DIAGONAL (der deployte Seam hat NULL Kreuzblöcke — die Wert-Ebene wartet auf einen physikalischen Misch-Mechanismus; [O]-Prämisse unbewegt)."
    },
    {
      "id": "v898_kms_schur_mixing",
      "cluster": "core",
      "descriptionEN": "SEAM.CFIN.KMSMIX.01 + SCHURMIX.01: **der kanal-mischende Zustand, auf den der Block-Wick-Funktor wartet (30/30)** — der C₆-kovariante mischende KMS-Zustand EXISTIERT (u = 1, t = 1/8, β = 1: alle 15 Kreuzblöcke am Floor, Grading 5/5 + 10/10, alle 15 Wick-Monome mit kanonischen Pf4-Vorzeichen; β = 2 robust); EXAKT (sympy): die Schur-Elimination des Randes ERZEUGT alle 10 Träger-Duaden aus dem bloßen Diagonal-Zustand (A_eff = κ·A_CC + (t²m/(1−m²))·V J₃ Vᵀ, uniformer 3J-Block) — auf beiden Fronten (RH-Port, Wick-Funktor) lebt die echte Korrelation im ELIMINIERTEN Rand; die 15-Signatur-Tabelle ist die registrierte Vorhersage; RP-THETA-OPEN benannt; die [O]-Prämisse bleibt unbewegt."
    },
    {
      "id": "v903_seam_rp_exclusion",
      "cluster": "core",
      "descriptionEN": "SEAM.STATE.DERIVATION.01 + CFIN.MIXING.NORMALIZATION.01 + CFIN.MINIMAL.MEDIATOR.01 + CFIN.TWISTED.RP.01 + CFIN.RP.DILATION.01 + CFIN.GAP.PENCIL.01: **die Seam-RP/Modular-Exklusion + zwei tote Lesarten (25/25 + 23/23 + 20/20 + 20/20 + 23/23 + 18/18)** — strikte RP erzwingt t = 0 unter BEIDEN räumlichen Reflexionen (odd-Gram-Eigenwerte EXAKT ±|a_J| der {4,5}-Duade auf 2.2e-16: OS-Positivität ⟺ a_J = 0 ⟺ der v898-Mischungs-Floor FÄLLT — RP und Mischung schließen sich auf dieser Familie gegenseitig aus); u ≥ t erzwungen (u = 0 RP-ausgeschlossen, Runde-57-Lesart korrigiert); 2π-KMS-Locus = der Punkt (t=0, β=2π); der komplette getwistete OS-Zensus EXKLUSIONÄR (48 Kandidaten, 16 involutiv, 6 admissibel, 0/6 mit strikter RP bei t > 0); die Dilations-Route GESPALTEN (der v898-M2-Parent ist θ_abT-marginaler Zeuge AUF dem Kegelrand, die Schur-Kompression trägt die VOLLE Pfaffian-Mischung mit t²·3m/(1−m²) = 1/200 EXAKT; die strikte Kollar-Route familienweit exakt obstruiert, 30 Defekteinträge ALLE mit Betrag 2t); t_gap = 0.230949 THEOREM-FÖRMIG (Pf(A_dep + t·A_int) = −(t−1)(3t²−1)·q(t) EXAKT, q = 9t³+21t²−t−1 irreduzibel, Sturm-Eindeutigkeit auf (0, 1/4], Kern-Dimension exakt 2, drei kritische Flächen = Projektionen EINER Varietät, KEINE Wurzel ist 1/8); DIE ZWEI TOTEN LESARTEN als erstklassige Negative: 't = 1/8 ist ein Compiler-Wert' ist eine DEKORATION (55/55 t bestehen alle eingefrorenen v898-Gates, erlaubte Menge (0, ∞), PIN-ABSENT), 'N_fam = 3 = minimaler Mediations-Rang' ist WIDERLEGT (Rang 1 genügt per Vorzeichengesetz + Ganzzahl-Zeuge, deployter S-Rang exakt 2; die 3 pint Rand-Reinheit = Dimensions-Buchhaltung); die [O]-Prämisse von v898 bleibt UNBEWEGT."
    },
    {
      "id": "v908_seam_equilibrium_wiring",
      "cluster": "core",
      "descriptionEN": "SEAM.CFIN.MARGINAL.STABILITY.01 + CFIN.NESS.PARENT.01: **die Seam-Gleichgewichts-Verdrahtung (22/22 + 25/25) — der Physik-Abschluss der Runde-60-Fragen** — strikte 2-Zykel-RP ist auf der GANZEN C6-kovarianten Klasse UNMÖGLICH (das Sitz-Gesetz reduziert kovariant EXAKT auf M1 = diag(a_J, −a_J), det = −a_J²: Inneres LEER — der v903-Marginal-Zeuge ist weder Limes kovarianter strikter Gleichgewichte (es gibt keine) noch isoliert: das ganze s = 0-Stratum ist marginal, sup(RP-Marge × Mischung) = 0 EXAKT); der einzige strict-enabling Ausweg ist die kovarianz-BRECHENDE X-Koordinate (exakter linearer Preis 2ε, Margen-Exponent 2) mit dem FLOOR-EXCHANGE als schärfstem exakten Objekt: Pf4(ε) = (ε − 1/200)(ε + 1/200) — das kanonische Vorzeichen-Gate IST die negative effektive RP-Marge, Crossover exakt am 1/200-Floor; die Strict-Collar-Obstruktion ist ein ZWEI-SITZ-LINEARGESETZ der Kopplung (exakter Rang 12, Kern 12 = die {J, Z}-Koordinaten des 24-dim kovarianten Kopplungsraums), die deployte Seam-Verdrahtung ist PURE-I = eine MAXIMAL obstruierte kovariante Richtung (erklärt das v903-Gesetz ‚30 Einträge, Magnitude 2t' exakt), und der Kern enthält GLEICHGEWICHTS-Zeugen: V_J/V_Z bestehen strikte θ_S-RP in exakter Arithmetik und tragen die VOLLE kanonische Pfaffian-Mischung mit J-Koordinate EXAKT t²·3m/(1−m²) = 1/200 — KEIN NESS NÖTIG: bei endlicher Größe gilt stationär ⟹ σ ≡ 0 exakt (‚NESS mit positiver Entropieproduktion' ist kategorien-unanwendbar) und Drive kauft KEINE RP — die minimale Entropieproduktion, bei der die Mischungs-Gates öffnen, ist EXAKT NULL; SEAM.CFIN.NESS.PARENT.01 schließt als NOT-NEEDED; der registrierte nächste Kontrakt SEAM.STATE.WIRING.SELECTOR.01 (ist PURE-I compiler-erzwungen?) ist in v911 GESCHLOSSEN: Deployment-Wahl, kein Compiler-Theorem; die [O]-Prämisse von v898/v903 bleibt unbewegt."
    },
    {
      "id": "v911_wiring_freedom",
      "cluster": "core",
      "descriptionEN": "SEAM.STATE.WIRING.SELECTOR.01 (+ THETA.SELECTOR.01 + WIRING.GAUGE.RP.01): **das Verdrahtungs-Freiheits-Theorem (30/30 + 26/26 + 21/21) — die CXXXIX-Frage geschlossen, der v908-registrierte Kontrakt beantwortet: DEPLOYMENT-WAHL, kein Compiler-Theorem** — der exakte Gröbner-Zensus bestimmt GENAU EINE admissible 3-dim Verdrahtungs-Komponente mod Regel-Eichung (C_rot); Z/X sind eichinvariant durch Orientierungspropagation ausgeschlossen; pure-I ist INNERER Punkt, durch die ganzzahlige Eichung (⊕₅J2)⊕I6 mit pure-J verbunden — PURE-I IST DEPLOYMENT-REPRÄSENTANT; θ-CONVENTIONAL: keine kompilerseitige Demand zeichnet das θ_S-Frame aus (fünf Demand-Klassen silent-on-angle), der deployte Parent BESTEHT strikte Kragen-RP exakt im ganzzahlig konjugierten Frame — die Runde-58/59-Obstruktion ist FRAME-RELATIV; RP-FRAME-COVARIANT: die strict-collar RP-Grams transformieren exakt kovariant, der pure-I-Ausschluss ist eine Frame-Aussage (Option 2), KEIN Verdrahtungs-No-Go; kanonischer Satz eingefroren (sha 79950314), VERBOTEN: „pure-I ist compiler-erzwungen“; die [O]-Prämisse von v898/v903 bleibt [O]; NULL RH-Gehalt."
    },
    {
      "id": "v957_ccc_crossover_kinematics",
      "cluster": "horizon",
      "descriptionEN": "CCC.SEAM.KINEMATICS.01 (Runden DL–DLV, 2026-08-24): **das CCC-Crossover-Kinematik-Wörterbuch (28/28 + 29/29 + 18/18 + 21/21 + 14/14 + 7/7, sechs Probes byte-exakt eingebettet)** — (1) Penroses reziproke Eichung Ω·τ(Ω) = −1 ist das Element τ = ρ²σ des bewiesenen μ4-Stabilisators D4 (τ kehrt die Clock um = Z2-Sheet-Parität; der Crossover-Kreis trägt alle vier Marken); (2) die Clock allein lässt auf der 2D-Seam genau EINEN konformen Modul, das reziproke Z2 killt ihn (Liouville-exakt; so(3,1)-Kommutanten 2/0); (3) Stinespring-Dilatation des v221-Transports exakt (V†V = 1, ‖Tⁿ−P⋆‖ = (2/3)^{6n}, globaler Overlap invariant — Entropie-Reset = reduzierter Kanal, keine Informationsvernichtung); (4) der kinematische 2D→3D→4D-Uplift: Seam-Sphäre = Hopf-Basis (stereographisch = z1/z2 exakt), Crossover X = S³/Z4 = Linsenrand des korpuseigenen A3-ALE (Deck frei; Marken = 4 verschlungene Faserkreise in 2 Sheet-Orbits, Gauss-Linking 1; so(4,1)-Deck-Kommutant = 4 Killing/0 Boosts — Yamabe-Starrheit ohne Zusatzbedingung, global via Obata-Familie), 4D-Brücke = Kegel mit Inversion als Äon-Tausch, Lorentz-Schatten Einstein R_ab = 3g_ab mit raumartigem S³-Crossover und Ω̂·Ω̌ = −1; (5) KMS-Schließung: Detailed Balance exakt, spec(−ln T) = {0, 6ln(3/2), 6ln3}, T = e^{−H_mod}, β_angle = 2π = 1/(4c3) = Gibbons–Hawking auf der Faser — diskreter Takt = modulare Zeit (freies Level); (6) Holonomie: Pol-Strings tragen die volle μ4 (i), jeder Seam-String exakt die 2-Torsion −1 = v522s GSO/Fermionparitäts-Wrap ([C]-Korrespondenz); (7) Selektor-Notwendigkeit: zulässige Spiegelachsen von M(δ) = V4 generisch, D4 GENAU bei δ = π/2; Majorana-Operator-Ebene mit Fermion-Vorzeichen: Clock-Achsen Θ-real NUR bei k = 4 — die Uniqueness-Hälfte des v534-Selektors HERGELEITET (Survival + Vorzeichen zitiert); (8) der eingefrorene Gate-6-Kernel: Top-Hat-Scheibe θ_max ≈ 1.16°, Raten ln(729/64)/ln729 (Verhältnis 2.709511), κ aus v526 hergeleitet (Band deklariert, Kontrast ≤ 3.3e−4) — speist die preregistrierte Suche ccc-crossover-disc (keine Daten berührt). Zyklische Lesart bleibt [C], WOIT.OS.TWISTOR.01 bleibt [O]."
    },
    {
      "id": "v971_markov_embedding_generator",
      "cluster": "frontier",
      "descriptionEN": "DYN.MARKOV.EMBED.01 (finite Kernel-Hälfte, Review-Runde 2026-08-27 „drei Bedeutungen von Dynamik“): **Q = log T des v221-Seam-Transferkernels ist ein EXAKTER klassischer Markov-Generator** — sympy-exakter Spektral-Log mit exp(Q) = T, Zeilensummen symbolisch exakt 0, Off-Diagonalen exakt {ln(9/8), 2 ln 3, 2 ln 3} = {0.117783, 2.197225, 2.197225} > 0, Diagonalen −ln(81/8)/−ln(81/8)/−4 ln 3; Uniformisierungs-Zertifikat Q + (4 ln 3)·I ≥ 0 elementweise ⇒ e^{tQ} positivitätserhaltend doppelt-stochastisch für ALLE t ≥ 0 (T einbettbar, jede Wurzel T^{1/n} stochastisch); spec(Q) = −spec(H_mod) = {0, −6 ln(3/2), −6 ln 3} (Gap = Δ), Detailed Balance zur Gleichverteilung. NEGATIV-Kontrolle: zweiter Eigenwert −1/10 ⇒ det = −1/200 < 0 — beweisbar NICHT einbettbar (Einbettbarkeit ist nicht generisch; die RP/OS-Spektralpositivität ist der ermöglichende Fakt). FIREWALL: schließt Relaxation→Halbgruppe (a)→(b) NUR auf finitem Kernel-Niveau; die lokale/größenkonsistente unitäre Dilatation (b)→(c) ist der separate offene Kontrakt DYN.UNITARY.DILATION.01; die geteilte F_transfer-Relaxationsform ist eine universelle KONTRAKTIONSKLASSE, keine gemeinsame physikalische Uhr (v723/v724/v777)."
    },
    {
      "id": "v972_seam_interaction_front",
      "cluster": "core",
      "descriptionEN": "SEAM.INT.ODDSPLIT.01 [E] + SEAM.INT.OSAVOID.01 [E] + SEAM.STATE.RPMIXING.01 [E] (+ Updates an SEAM.INT.FKTOY.01 / WOIT.OS.TWISTOR.01 / SEAM.STATE.DERIVATION.01; Konsolidierung dreier Explorationsrunden vom 2026-08-27, alle drei Sonden byte-exakt eingebettet mit gepinnten Quell-SHA-256 und wörtlich ausgeführt, 20/20 + 18/18 + 25/25, Verdikte CLASS_EMPTY_ON_BASIS + OS_VIABLE_ON_AVOIDING + RP_ADMITS_MIXING): **DIE SEAM-INTERAKTIONSFRONT** — (1) Die hinreichende Jaffe–Pedrocchi-Bedingung für Majorana-Reflexionspositivität (arXiv:1406.1384; Wei–Li–Xiang arXiv:1601.01994) wird auf dem v529-FK-Toy ENTSCHEIDBAR gemacht: die Kreuzungs-Koeffizientenmatrix B ist EINDEUTIG (Links-Monom-Paare bijektiv auf Kreuzungsmonome), also ist „B hermitesch UND −B PSD\" ein endlicher Check; der freie FK-Hamiltonian wird auf allen 8 Bond-Cuts zertifiziert (Twist η = −i erzwungen); die Konfusionsmatrix gegen gemessene RP über 35 Zellen ist PERFEKT (23 zert.&PD / 12 unzert.&fail / 0 FP / 0 FN); der μ4-kovariante Quartik-Raum (dim 464) ist auf der Basis LEER (und auf 64 geseedeten Kombinationen; der Träger-Ausschluss pinnt den vollen 1820-dim Raum) — DIE ENTDECKUNG: das v529-Straddle-Gesetz VERFEINERT sich zum ODD-SPLIT-GESETZ: JP sagte RP-Überleben auf ZENTRIERT gestraddelten Cuts voraus (m = 4-Modell, Achsen 7/15, 2+2-Spiegelsplit), die Direktmessung bestätigt ((37,0,0) bei g = 0.5/1/2); die v529-Ankerleiter exakt reproduziert. (2) Die quartett-MEIDENDE Cut-Familie des uhr-invarianten m = 4-Mitglieds erzeugt die VOLLE gerade Seam-Algebra (Rang 15, dim 32768 = 2¹⁵; Einzel-Cut-Mitglieder m = 2/6 mit typisiertem Defizit EXAKT 1), und OS/GNS auf dem meidenden Cut k = 3 bei g = 1 ist POSITIV (min +4.43e−5, GNS 37/37, Distanzen 1.044/1.386/1.650 monoton), während die gestraddelte Kontrolle ehrlich scheitert (−6.94e−2); die μ4-Uhr bildet das meidende Paar {3, 11} exakt auf sich ab — das Kill-Test-2-Gegengewicht VERENGT, hebt nicht auf. (3) Die v903-Exklusivität ist STRAHL- und SPIEGELUNGS-relativ, nicht slice-weit: auf dem vollen 33-dim C6-kovarianten Mixing-Slice sieht θ_abT genau EINE Koordinate (a_J, Kernwert −1/(1+c²) = −0.8240271368 bei c = tanh(1/2)), 32/33 unsichtbar ⇒ 32-dim MARGINALE RP-Hyperebene {a_J = 0}; die θ_S-Defektabbildung hat Rang 16 ⇒ 17-dim erstordnungs-neutraler Unterraum mit STRIKTEN finite-s-Zeugen (W_S bei s = 1/8, λ_min 0.1558 — genau die Richtung, die θ_abT tötet); der v903-Anker reproduziert (a_J = +0.04068013). Modul-eigene Exaktsektion S0 (sympy/GF(2), keine Sonden-Importe): JP-Entscheidbarkeits-Lemma am 4-Majorana-Toy (9 paarweise verschiedene Kreuzungsmasken ⇒ eindeutiges B; Falschvorzeichen-Toy SCHEITERT), Odd-Split-Arithmetik (zentriert ⇔ 2+2 exakt auf Z₁₆; der ungerade 1+3-Block [[0,b],[b*,0]] erzwingt b = 0 — die Obstruktion; Vorzeichen-Flip-Mutant CAUGHT), GF(2)-Erzeugungs-Buchhaltung (Paarrang 15, 2¹⁵ = 32768; Innerhalb-Hälften-Rang 14 — Defizit exakt 1), Fenster-Arithmetik (33 = 24 + 9, 32 + 1, 33 − 16 = 17; Kernwert symbolisch == −cosh(1/2)²/cosh(1)) plus 4 Verdikt-Bars mit Kipp-Mutanten. EHRLICH: KEINE Marker-Bewegung; SEAM.EQUIV.01 bleibt [O]; der v529-Pflichtzaun (EIN Toy, EINE Wechselwirkungsklasse, [C] Flachband-Parent) wird von allen drei Resultaten GEERBT; JP ist nur hinreichend; der S1-Zensus ist sektor-trunkiert. KEIN RH-Claim."
    },
    {
      "id": "v973_seam_route_narrowing",
      "cluster": "core",
      "descriptionEN": "SEAM.EQUIV.MMST.01 [O-Update] + SEAM.EQUIV.TWISTOR.01 [O-Update] + SEAM.EQUIV.SKELETON.01 [E, NEU] + SEAM.EQUIV.01 [O-Update] (Seam-Adjudikationswelle, 2026-08-27): **die Seam-Routen-Verengung — drei Probes byte-exakt eingebettet (19/19 + 20/20 + 19/19; SPEC e6f392cbcd0d3ae1 / 0875f39fe1d0fa0f / 451bd470b98a9d6d), plus die modul-eigene Exaktsektion S0** — (1) MMST-ABDECKUNG ZERLEGT: der zitierte Osborne–Stottmeister-Satz (CMP 398 (2023) 219, arXiv:2107.13834) DECKT die bilineare Hälfte des Seam-Ziels — so(16)₁-Fermion-Bilinear-WZW-Ströme konvergieren STARK auf dem endlichen Teilchenkern (Thm 5.1 + Cauchy–Schwarz Gl. 229 inkl. Majorana-Algebren), Virasoro/Koo–Saleur (Thms A/4.11/4.16), Korrelatoren (Thms B/C), polynomiale Fehler Error² ≤ C·2^(−2δN), δ ∈ [0,2] — und deckt vier BENANNTE Punkte NICHT: N1 die μ₄/GSO-Erweiterung SO(16)₁ → (E8)₁ als NETZ-Aussage (die 128 Spinor-Ströme sind keine Bilineare), N2 der Twist-/Spinfeld-Sektor (vom Paper explizit verschoben), N3 das einseitige chirale Randnetz eines 2+1D-Bulks, N4 der Holomorphie-Selektor det K = 1 vs 4; NEUE MESSUNG: die Rand-Korrelator-Konvergenzrate auf dem v367-Collar p = 3.48 (Richardson 3.1–4.2, präregistriertes Band [0.5, 4.0]), Fehler 6.3e-2 → 2.7e-6 über Nx = 16..256; die M = 3-Kontrolle konvergiert NICHT zum chiralen Ziel (min rel Abw 0.836); die Abdeckungs-Gates exakt-ganzzahlig (248 = 120 + 128, 240 Wurzeln = 112 + 128, det Cartan 4 → 1, Index-Arithmetik 2·2 = 4 = |μ₄|).  (2) TWISTOR-MASS-RIGIDITÄT: auf dem berechenbaren v508/v515-Sektor hat die 18-Parameter-Deformation der deklarierten Maßdaten (Kontakt-Normierung, 4 Sektorgewichte, Residuen-Normierung, 4 Linking-Ladungen, 4 Zentrumspositionen) die exekutierten Constraint-Ränge C1 Uhr 9 / C2 Lockstep+Ganzzahligkeit 8 / C3 Ladung 2 / C4 Null-Ideal 2, gemeinsamer Rang 13, 4 Eichrichtungen ⇒ der Moduli-Kern ist EXAKT 1 = der uniforme Sektorgewichts-Shift dv = (1,1,1,1), abgebildet auf das ungetwistete Okubo-Quadrat Q⁰ = 36(1,2,1,0,0), von jedem exekutierten Zertifikat-Funktional PER KONSTRUKTION annihiliert (exakt flach zu allen Ordnungen); Anker re-deriviert (A_fix = (9,−30,−15,0,32), Zertifikat (0,32,72), ψ(A_fix) = 64); Ablations-Leiter (voll,−C1,−C2,−C3,−C4,−C1&C2) = (1,3,1,2,3,10); ehrlich: der W₂:W₁₃ = 4:3-Tester ist NOT-TRANSCRIBABLE.  (3) OS-SKELETT GETEILT: ALLE fünf WOIT-Killtest-Hauptinvarianten (α: erzwungenes η = +i, Θ² = +1, Bond-RP; β1: GSO-2-Torsion; β2: OS-Quotient existiert + KMS λ_max > 1; β3: Perp-Descent; γ: Spiegel-Flip, multiplizitätsfrei, Mark-Inzidenz) gelten WÖRTLICH auf der p+ip-Collar-Kante; ALLE Divergenzen kontroll-typisiert und auf EINEN Mechanismus zurückgeführt (das exakte chirale Schachbrett C(even) = 0 des Twistor-Gitters vs K(even) bis 0.1736 des Collars — bestätigt v519-R4 unabhängig von der zweiten Route); BONUS: die obere Kante realisiert den anti-chiralen Zustand EXAKT (max|K_top + K_bot| = 7.5e-16); der triviale M = 3-Collar verliert das Skelett ZUERST bei GAMMA.MARK.INCIDENCE — das Skelett folgt der topologischen Phase.  Modul-eigene Exaktsektion S0: Abdeckungs-Arithmetik exakt (Wurzel-Zensus, Cartan-Determinanten, Index); die Eindeutigkeits-Lineare-Algebra als exakte Rationale auf einem unimodular gemischten 18-Raum (Ränge (9,8,2,2) → 13 → Kern 1; Ablations-Leiter exakt; der C4-Drop-Mutant öffnet den Kern auf 3 — CAUGHT); die Skelett-Mechanismen exakt (bipartit S H S = −H ⇒ ungerade Funktionen verschwinden auf geraden Distanzen; P H P = −H auf dem Streifen ⇒ K_top = −K_bot exakt); Verdikt-Bars mit Kipp-Mutanten (ein falsches COVERAGE_FULL wird vom nichtleeren N1–N4-Gate VERWEIGERT; Hash-Mutation CAUGHT).  EHRLICH: die Abdeckungs-Gates sind ausführbare TRANSKRIPTIONEN einer menschlichen Literatur-Adjudikation; die Rate SUPERKONVERGIERT (p > δ_max = 2) gegen die zitierte OBERE Schranke — Band kalibrierungs-informiert; die Sektor-Eindeutigkeit ist KEINE BCOV-Herleitung; Skelett-Treue pro Test typisiert.  KEINE Marker-Bewegung: MMST und TWISTOR bleiben [O], der Parent bleibt [O]; die einzige neue Zeile ist SEAM.EQUIV.SKELETON.01 [E]."
    },
    {
      "id": "v974_alpha_faces_computed",
      "cluster": "core",
      "descriptionEN": "ALPHA.QUILLEN.EXACT.01 [O-Update] + SEAM.CONTACT.UNIT.01/.02 + EM.WARD.01 / ALPHA.QUILLEN.INFLOW.01 + ALPHA.QUILLEN.FACES.01 [E, NEU] (Seam/Alpha-Promotionswelle 2026-08-27): **beide benannten Faces des Alpha-Targets tragen jetzt GERECHNETE Zeugen** — drei Probes byte-exakt eingebettet, sha256-gepinnt und wörtlich ausgeführt (21/21 MATCH_MODULO_LOCAL_FACTOR + 20/20 PUMP_QUANTIZED_MATCHES_CHERN + 18/18 CHAIN_SHADOW_OK) plus die modul-eigene Exaktsektion S0 (sympy/Fraction/mpmath, keine Probe-Importe). (1) Das BFK-Face AUSGEFÜHRT: die Burghelea–Friedlander–Kappeler-Klebekonstante ist EXAKT 2^−N (symbolisch hergeleitet N = 1..4, allgemeine Intervalllängen, Laurent-Polynom-Identität); masselose Reassemblierung det′_ζ(Δ_S¹) = l² = 2^−N·∏(2a_j)·(l/N)·det′(R₀) mit det′(R₀) = N·l/∏a_j; an der KMS-Skala l = 2π = 1/(4c₃): R₀ = 16c₃·circ(2,−1,0,−1) (16c₃ = |μ₄|·4c₃ pro Intervall), det′R₀ = 2¹⁶c₃³, Produkt (2π)²; Sprungmatrix R(m) = (G_m|_marks)⁻¹ exakt; gegen die v484-Markenmatrix G_marks = −4c₃ln2·circ(0,1,2,1) KEINE globale Proportionalität (Kerne in VERTAUSCHTEN μ₄-Kanälen: Eich-Nullmode vs. Vorzeichendarstellung), aber kanalweise EXAKT (gleiche μ₄-Fourier-Eigenbasis, |spec|-Multimengen {0,2,2,4}, Dublett-Verhältnis 4/ln2 mit KÜRZENDEM c₃, Tr B^k = 4^k + 2·2^k spiegelt Tr C^k = 4^k + 2(−2)^k — der Vorzeichen-Flip IST der Kern-Tausch); der v485-Skalenkanal EXAKT reproduziert über die KMS-Einheit (log det′-Differenz = (1/(4c₃))·G_reg(0;l)). (2) EM1 GERECHNET: gepumpte Ladung == Rand-Spektralfluss == Streda == Chern == 1 pro Flussquant auf dem v367-Kragenmodell (Wannier-Winding-Residuum 0.0; Streda ΔN = C·n_φ exakte Ganzzahlen; der Konjugations-Mutant kippt ALLE VIER Lesarten simultan, der Falschband-Projektor bricht die Quantisierung gitterinstabil); Buchhaltung exakt: 16 Majoranas = 8 komplexe Kopien, c₋ = 8 = g_car + N_fam, k_Y = 5/3, (1/k_Y)·41/6 = 41/10 = b₁. (3) Die endliche Quillen-Identität ZWEISEITIG: Berry-Integral == Poincaré–Lelong-Divisor-Integral == +1 über dem Twist-Torus (beide Referenzen, beide Größen), Differenz eine EXAKTE Form (3.3e−16), Übergangs-Winding == Chern (Dai–Freed); typisiert konditional auf die Quasi-frei-Rigiditäts-Prämisse (gemessen, nicht bewiesen — SEAM.STATE.RPMIXING.01). S0 leitet alles unabhängig her: die BFK-Geschlossenformen aus den ζ_R-Werten, den Alpha-Anker bei 50 Stellen (8b₁ = 164/5 exakt, EINDEUTIGE Wurzel im Scanfenster, α⁻¹ = 137.03599921684071, Split-Residuum 6.4e−58), die Inflow-Buchhaltung als exakte Brüche und ein probe-freies FHS-Chern-Zertifikat. Mutanten: Hash-Mutation vom sha256-Pin gefangen; die falsche KMS-Einheit (2π → 4π) bricht die Skalenidentität um EXAKT −2ln2; der b₁-Mutant gefangen. KEINE Marker-Bewegung: das Target bleibt [O] — Restinhalt typisiert als {Rigiditäts-Prämisse} + {Kanal-Tausch-Frage (das geschärfte Diagonal-Zeta-Face)} + {Kontinuums-Bismut–Freed-Identifikation}; α⁻¹ bleibt [E] unabhängig davon."
    },
    {
      "id": "v975_dimension_selector_4d",
      "cluster": "core",
      "descriptionEN": "DIMENSION.SELECTOR.4D.01 [C, NEU] (Master-Routen-Welle 2026-08-27): **der konditionale 4D-Dimensionsselektor** — 16/16 sympy-exakt. Unter dem benannten TFPT-kompatiblen Axiomensatz A1 (d > 2: propagierende lokale Eichfelder), A2 (dimensionslose Yang-Mills-Kopplung), A3 (reelle selbstduale/anti-selbstduale 2-Form-Sektoren: der euklidische Hodge-Stern als involutiver Endomorphismus auf 2-Formen), A4 (chirale Weyl-Darstellungen: gerades d) ist d = 4 auf dem exakten Scanfenster d = 2..12 EINDEUTIG und MINIMAL — und der Selektor ist ÜBERDETERMINIERT: A2 allein = {4} ([g_YM] = (4−d)/2 symbolisch erzwungen, einzige Nullstelle 4) und A3 allein = {4} (expliziter 6×6-Stern auf Λ²(R⁴), *² = I, Eigenwerte ±1 mit Multiplizitäten 3/3), während A1 & A4 allein {4, 6, 8, 10, 12} geben — Chiralität selektiert 4 NICHT allein (ehrliche Ablations-Leiter). Weyl-Chiralität zertifiziert durch exakte Clifford-Algebren d = 2..7 (gerade d: γ* antikommutiert, γ*² = I, spurfrei; ungerade d: das Produkt ist ZENTRAL und skalar). Signatur ehrlich typisiert: lorentzisch ist *² = −1 auf 2-Formen — der reelle Split ist eine EUKLIDISCHE/OS-seitige Aussage. Mutanten gefangen ([g] = (6−d)/2 selektiert {6}; der 'A4 allein'-Overclaim verweigert). FIREWALL: μ₄ und g_car gehen NIRGENDS ein — 'μ₄ ⇒ 4D' bleibt verboten (DIMENSION.UPLIFT.FIREWALL.01); die Axiom-Provenienz aus {c₃, g_car} ist die OFFENE Hälfte (Kill-Kriterium der Ledger-Zeile); kein 4D-Konstruktionskontrakt wird bewegt."
    },
    {
      "id": "v976_seam_lift_birkhoff",
      "cluster": "frontier",
      "descriptionEN": "TRANSFER.HIDDEN.CIRCULATION.01 [E, ausgefuehrte Haelfte] + EB-No-Go fuer DYN.UNITARY.DILATION.01 (External-Review-Runde 2026-08-27): **die Birkhoff-Faser des Seam-Einzelschritts B ist exakt eine Gerade, ihr verborgener Parameter ist der Signum-Charakter, und die deployete Kraus-Dilatation ist entanglement breaking** — B = (1/2)I + (1/18)P12 + (2/9)P13 + (2/9)P23 bit-exakt (B^6 = v221-Kernel), Kern der Zerlegungsabbildung 1-dim (I + C123 + C132 = P12 + P13 + P23 = J), vollstaendige Familie w_t mit t in [0, 1/18]; w_t-hat(sgn) = 6t, Populationsdynamik exakt t-frei, kein Faser-Mitglied chiral (w(C123) = w(C132) erzwungen). DISKRIMINATOR: Ein-Schritt-Kohaerenzdynamik = links-regulaere S3-Darstellung, Spektrum {1, 6t, 2/3 x2, 1/3 x2} exakt; der Dreiecks-Ringstrom ist exakter Eigenobservable Psi_t(A) = 6t*A (populationsblind: EIN Schritt + EIN Strom-Readout misst t, ohne Ancilla; Tomographie-Rang 6). KONTRAST-QUBIT: spec(Choi^T2) = {-3t/2, 1/6+3t/2, 1/3+3t/2, 1/2-3t/2} symbolisch; KRITIKALITAET lambda_min = (1 - lambda2 - lambda3 - 6t)/4 mit 2/3 + 1/3 = 1 EXAKT (t = 0 genau auf der EB-Grenze; Mutations-Spektren verfehlen um -+1/12 — nicht generisch); quadrierter Kanal PPT auf dem ganzen Range (Endpunkt-Minima 1/9, 1/12) => Qubit-EB-Index 1 (t = 0) / 2 (t > 0) [PPT<=>EB in 2x2 + Lami-Giovannetti, Literatur-typisiert]. FULL-LEVEL: lambda_min(Choi(Psi_t^n)^T2) = -(1/3)(2/3)^n t-UNABHAENGIG (exakt fuer n in {1,2,6}, Float n = 1..8) — auf 3 Leveln nie EB. DAS NO-GO: die deployete Kraus-Dilatation K_ij = sqrt(T_ij)|i><j| ist measure-and-prepare => entanglement breaking [HSR] => Quantenkapazitaet 0; C_0(T) = 0 exakt; sie liegt NICHT in der Birkhoff-Faser. NACHRICHT BINAER: p* = (1/2, 1/2, 0) KKT-exakt optimal fuer T UND B (C(T) = 0.0083580191 bit, C(B) = 0.4890415237 bit, Ratio 1.71 %). [C]-Beobachtung: Gewichte (9,1,4,4)/18 = Quadrate/18 = die v652-Defektleiter-Werte g^2/18 (Multiplizitaet nur bei g = 2 deckend; der Funktor ist der OFFENE Kontrakt TRANSFER.DEFECT.BIRKHOFF.01). FIREWALL: endliche Matrixalgebra + Literatur-typisierte Kanaltheorie; die Faser-Punkt-Auswahl bleibt [O]."
    },
    {
      "id": "v977_transfer_unistochastic_wilson",
      "cluster": "frontier",
      "descriptionEN": "TRANSFER.COHERENT.WILSON.01 [E, ausgefuehrte Haelfte] (External-Review-Runde 2026-08-27, Begleiter zu v976): **B ist unistochastisch — exakter SU(3)-Lift mit dem Wilson-Code 5-12-13, der Center-Bruecke und einem klassisch unsichtbaren Orientierungsbit** — Winkel von B erzwungen (sin^2 th13 = 2/9, sin^2 th12 = 1/14, sin^2 th23 = 2/7), cos delta = -4/sqrt(65) eindeutig, e^{i delta} = (-4 +- 7i)/sqrt(65), Liftmenge = {U, U*} bis auf Rephasierung; U^dagger U = I, det U = 1, |U_ij|^2 = B_ij entrywise exakt [Coherification iff unistochastic, Literatur]. J = Im(Plaquette) = 1/27 EXAKT; Re jeder Plaquette durch B fixiert (Unitaritaet) => das gesamte rephasierungsinvariante Wilson-Datum ist klassisch bis auf das Z2-ORIENTIERUNGSBIT sgn J (U* liftet dasselbe B mit J = -1/27; Observable = das Plaquette-Interferenz-Readout). DER CODE: Q_{12;12} = (-5 + 12i)/324, normiert (-5 + 12i)/13 mit 5^2 + 12^2 = 13^2 (Atome 5 = g_car, 12 = dim g_SM, 13 = Delta_Q = N(3+2i)); alle neun Plaquettes = vier Gauss-Phasentypen, Normen exklusiv {10, 13, 130, 169} = {2*5, 13, 2*5*13, 13^2}; jede Im = +-1/27; vier Fundamental-Plaquettes erzeugen alle neun exakt. CENTER-BRUECKE: disc chi_A = 65 = det(A-I)^2 + (det A/2)^2 = 4^2 + 7^2 fuer die v530-Atommatrix A = [[8,2],[5,3]]; e^{i delta} = (-det(A-I) +- i det A/2)/sqrt(disc) EXAKT = die Transportphase; -4 + 7i = (1+2i)(2+3i) (v738-Turmprimzahlen, Normen 5 und 13); die v530-Geschwister A_K/A_Q VERFEHLEN die Quadratsummen-Identitaet (die Bruecke selektiert A). POTENZGESETZ: J_n^2 = (1-b)^2(1+2b-3a^2)/108, jedes B^n unistochastisch (U_6 gebaut, |U_6|^2 = T exakt); |J_6| = 0.0951088904 = 0.9884006 J_max; der uniforme Limes liftet zur Fourier-Matrix F3 mit J_max exakt; J_n RATIONAL NUR bei n = 1 (3(1+2b-3a^2) = 1 exakt => J = 1/27) — der gaussisch-pythagoreische Code selektiert den Einzelschritt. KEIN FESTER MIKROSCHRITT: B^2 != |U^2|^2 exakt; Dirichlet-Rekurrenz-Zeuge n* = 967909 (min diag |U^n*|^2 = 0.999999 waehrend B^n uniform bis 5.6e-17) — U_6 != U_1^6 (Clock-vs-Walk kohaerent geschaerft). ZENSUS (harter Kill bestanden): 238 gueltige Familienpunkte, 4 mit rationalem J, GENAU EINER traegt den vollen Code = der deployete Punkt (2/3, 1/3) (0.42 % << 10 %). PMNS-NEAR-MISS bleibt NEGATIV (Zweigphase 240.2551 Grad vs deployed 240 Grad; -1/27 != -0.02965; alle Winkel differieren). FIREWALL: sgn-J-Auswahl, CP-/Zeit-Lesarten und der A->Winkel-Intertwiner bleiben [O]."
    },
    {
      "id": "v983_simple_current_generator",
      "cluster": "frontier",
      "descriptionEN": "SEAM.SIMPLECURRENT.GENERATOR.01 [O-Update: der Generator IDENTIFIZIERT, sein Gitter-Arithmetik-Schatten vollstaendig] (Externe Master-Objekt-Review Welle 2, 2026-08-28): **der eine Glue-Vektor lambda = (omega_s, omega_f) erzeugt den gesamten 128er-Spinorsektor — exakt auf Gitterebene** — |omega_s|^2 = 5/4, |omega_f|^2 = 3/4, |lambda|^2 = 2, konformes Gewicht h_lambda = 1 (ganzzahlig => triviale Selbstmonodromie); [lambda] hat ORDNUNG VIER in der Diskriminantengruppe Z4 x Z4 von D5 (+) A3 (4 lambda in L0, lambda und 2 lambda nicht); lambda . L0 in Z => isotrope Verklebung, geklebtes Gitter GERADE; det = 16/4^2 = 1 => UNIMODULAR Rang 8 => E8 (Klassifikation [C]; explizites Zertifikat v1). COSET-WURZEL-ZENSUS [52, 64, 60, 64] (Summe 240; +8 Cartan = 248 = (45+15) + (16,4) + (10,6) + (16bar,4bar)); die ungeraden Fusionspotenzen tragen 64 + 64 = 128 = der VOLLE fehlende Spinorsektor — EIN Feld erzeugt alle vier Cosets. KILL-TESTS FEUERN: Vektorklassen-Glue (Norm 7/4) und Fehlklassen-Glue (Norm 9/4) brechen beide die Geradheit. HOLOMORPHIE-SCHATTEN: det 1 => ein Sektor vs det(D8) = 4 => SO(16)_1 hat vier (nur der Gitter-Schatten von N4). FIREWALL: die analytische Haelfte (Implementierer-Konvergenz S1/S2, Crossed-Product/Skalierungs-Vertauschung S3, operatoralgebraische Holomorphie S4) bleibt [O]; 1+1D rein und raus; keine Marker-Bewegung."
    },
    {
      "id": "v984_markov_qca_dilation",
      "cluster": "frontier",
      "descriptionEN": "DYN.UNITARY.DILATION.01 [O-Update: das DISKRETE Bein konstruktiv realisiert] (Externe Master-Objekt-Review Welle 2, 2026-08-28): **das exakte Kollisionsmodell — B^n als reduzierte Dynamik eines exakt unitaeren, strikt lokalen QCA-Bandes** — der Kollisionsschritt V = C (U_B (x) I) mit dem mod-3-Rekord C|j,k> = |j, j+k> ist unitaer auf System + Ancilla; die STINESPRING-IDENTITAET Phi(rho) = Tr_A[V (rho x |0><0|) V^dag] = sum_j P_j U_B rho U_B^dag P_j gilt als symbolische Operatoridentitaet (Kraus-Vollstaendigkeit sum K_j^dag K_j = I); diag Phi(diag p) = B p exakt mit verschwindenden Nebendiagonalen; SECHS frische Ancillas geben diag Phi^6 = B^6 p = T p exakt — das v977-Makro-Gate als reduzierte Dynamik eines Radius-1-Kollisionsbandes. MUST-FAIL FEUERT: der geschlossene Unitaer hat |U_B^2|^2 != B^2 (31/54 vs 0.2450) — die Dilatation ist notwendig. TYPISIERTE BEMERKUNG (Probenbefund GEGEN den Review-Text): eine WIEDERVERWENDETE mod-3-Addierer-Ancilla gibt diag = B^2 p weiterhin exakt (der Rekord j1+i bleibt unterscheidbar) — frische Ancillas braucht man fuer die Kohaerenzen, nicht fuer die Populationen. Der konjugierte Zweig V* dilatiert dasselbe B (sgn J = +-1/27 bleibt klassisch unsichtbar; Auswahl bleibt [O]). FIREWALL: nur das diskrete endliche Bein; quasi-lokale Hamilton-Dilatation, OS-Fortsetzung zu e^{-itH} und jede 3+1D-Lesart bleiben [O]; keine Marker-Bewegung."
    },
    {
      "id": "v985_quillen_channel_swap",
      "cluster": "frontier",
      "descriptionEN": "ALPHA.QUILLEN.EXACT.01 [O-Update: die CHANNEL-SWAP-Facette auf endlicher Ebene durch mu4-Charakter-Graduierung aufgeloest] (Externe Master-Objekt-Review Welle 2, 2026-08-28; v974-Anker read-only): **die Deck-Graduierung loest den v974-Kernel-Swap exakt** — das Jump-Muster circ(2,-1,0,-1) hat mu4-Eigenwerte [0, 2, 4, 2] (Kernel k = 0), das Mark-Muster circ(0,1,2,1) hat [4, -2, 0, -2] (Kernel k = 2) — der v974-Befund symbolisch; der DECK-SWAP r -> r+2 (P0 <-> P2, P1 <-> P3) bildet das Jump-|Spektrum| EXAKT auf das Mark-Muster ab — nach dem Swap sitzen beide Kernel im selben Kanal, der graduierte Vergleich ist wohldefiniert; der UNGERADE Swap r -> r+1 passt NICHT (Must-Fail feuert); die graduierte det'-Differenz ist c3-FREI: gr(jump) - gr(mark) = log(ln2/4) exakt; das kanalweise Einheitsverhaeltnis ist UNIFORM = 4/ln2 in jedem gematchten Nicht-Null-Kanal (das v974-Doublet-Verhaeltnis zur vollen Uniformitaet erweitert). FIREWALL: nur endliche 4x4-mu4-Zirkulanten; Kontinuums-Bismut-Freed, KMS-Rigiditaet, die 2^-4/Quillen-Metrik-Kompensation und die Exaktheit von F_U(1) bleiben [O]; alpha^-1 unberuehrt; keine Marker-Bewegung."
    },
    {
      "id": "v986_nu_scalaron_rung",
      "cluster": "frontier",
      "descriptionEN": "FLAV.NUSCALE.05 [C]/[N] Kandidat (Externe Master-Objekt-Review Welle 2, 2026-08-28): **die Skalaron-Spur-Sprosse M_R = N_fam M_scal = 3 c3^{7/2} Mbar — ein strukturierter Punkt IM v481-Fenster, 1.8 % von der 1-Loop-Anforderung** — 3 M_scal = 9.180e13 GeV vs das v481-geforderte 9.346e13 GeV (dieselbe eingefrorene 1-Loop-Inversion, y_nu = y_t [P]) — Abweichung -1.8 % (der Review sagte 'etwa 1 %'; der ehrliche Wert ist 1.8), m_3 am Kandidaten = 0.0512 eV (+1.9 %, innerhalb der >50 %-v482-RG-Enveloppe); EXAKTE Leiterposition 3 c3^{7/2} = (3 sqrt c3) c3^3 mit 3 sqrt(c3) = 3/sqrt(8 pi) = 0.598413, 0.26 % von der 3/5 der v482/v488-Geschichte — ADJAZENZ EHRLICH TYPISIERT: v488 toetete 3/5 als SO(10)-CLEBSCH; die Skalaron-Spur-Lesart ist eine ANDERE Mechanismusklasse (HONEST RETYPE Welle 3: Tr I = 3 ist locker, Spec(I)={1,1,1}; Operator-Lesart ist Q_+ Spec{1,2,3}; Textur DATA_CONSTRAINS_TEXTURE) und bleibt selbst [O]; ~20x schaerfer als die abgelehnte Ganzzahl-Sprosse c3^3 Mbar (0.019 vs 0.401 in m_3); dasselbe M_scal traegt bereits f_a = M_scal/128 (v185) — kein neues Skalenatom. KILL-TEST: Sigma m_nu (NO) = 0.059 eV; eine robuste Schranke darunter oder ein >~5 %-Shift der geforderten M_R toetet den Kandidaten. FIREWALL: y_nu = y_t bleibt [P]; KEINE Seesaw-Schliessung; der v481-Fensterkandidat und das v488-Negativ unberuehrt; kein PMNS-Claim."
    },
    {
      "id": "v987_os_dilation_package",
      "cluster": "frontier",
      "descriptionEN": "DYN.UNITARY.DILATION.01 [O-Update: OS/Dilatation-Leiter ausgefuehrt] (Promote-Runde 2026-08-28): **Kernel-OS-Fortsetzung exakt + groessen-uniformes freies Band + Kongruenz-Familie + Quanten-SWAP** -- B = P1+(2/3)P2+(1/3)P3 > 0, H = -log B PSD (Luecke log(3/2)), T(t) Semigruppe im stochastischen Kegel, U(t) = e^{-itH} mit Wick-Anker U(-in) = B^n (n=1..6); Must-Fail: v971-Negativkern hat keinen reellen Halbschritt.  Freies Band: Gap log(3/2) UNIFORM in L; D2: Per-Zelle-Kanal = B bei L=1,2; LR-Radius 1.  Interagierend: T_eps = K^{1/2} B^(x L) K^{1/2} SPD fuer jede getestete Diagonalkopplung; einseitiger Mutant nicht symmetrisch; Gap-Plateau ~0.18.  SWAP: d(theta) ~ theta^2, CPTP, LR-Radius 1.  FIREWALL: kontinuierliche Feld-OS + Zweigwahl bleiben [O].  Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v988_psi_lambda_reduction",
      "cluster": "frontier",
      "descriptionEN": "SEAM.SIMPLECURRENT.GENERATOR.01 [O-Update: S3 exakt + S1/S2-Skelett gemessen + Lemma REDUZIERT, nicht geschlossen] (Promote-Runde 2026-08-28): **S3 gekreuztes Produkt / Turm-Austausch EXAKT** (Natuerlichkeit; nicht-aequivarianter Mutant bricht); S1 Fenster-Konvergenz + Sektor-Kollaps gemessen (Raten ~N^{-1.85}); S2 quantitativ: gefenstertes HS beschraenkt (Shale ~0.376) + globales HS^2 = 3.605+0.203 log N (topologisch) vs Saettigung (trivial); Cauchy auf dem phasenfixierten Band; Energiehuelle f_n < C^{n+1}(n!)^2 (Sonde C=2.86).  VERDIKT: restliches Lemma REDUZIERT auf zitierte quasi-freie Saetze (Shale-Stinespring/Araki/Ruijsenaars) + MMST-Identifikation -- REDUKTION, NICHT ABSCHLUSS.  Offener Rest = MMST-Identifikation.  Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v989_lattice_gate_battery",
      "cluster": "frontier",
      "descriptionEN": "TFPT4D.LATTICE.ACTION.01 + CHIRAL4D.NOMIRROR.01 [O-Update: T1-T7-Geschirr ausgefuehrt + euklidischer T2-Kill + Hamilton-Route raeumt T2; Kontrakte bleiben offen] (Promote-Runde 2026-08-28): **T4 exakte SM-Anomalie-Buchhaltung** (Drop-L-Mutant feuert); T5/T6 2D-Overlap Index=q, GW-Kreis; T7 Z2-Wuerfel kappa_4 != 0.  EUKLIDISCHER T2-KILL (Z2 N_t-exakt): Overlap-|det D_ov|^2 fehlt bei N_t=2, erscheint bei N_t=4 (det-only -0.249); Wilson-Kontrolle ueberall PSD -- auf Overlap-Zeitnichtlokalitaet gepinnt.  HAMILTON-ROUTE raeumt T2 konstruktionsbedingt (Gauss-Hilbert, H=H^dag).  Chirales Gauss-Zensus: Ein-Exponent genau dann q_L=q_R; Zwei-Exponent immer (Gitter-Dirac-Verdopplung).  Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v990_wj_transduction_mechanism",
      "cluster": "frontier",
      "descriptionEN": "FTRANSFER.GENERATING.01 + OBS.TRANSDUCTION.01 [O-Update: Inhalts-Schatten + endliches W[J] + Transduktions-Schatten; Kontrakte bleiben offen] (Promote-Runde 2026-08-28): **Z6-Zentrum 6Y-Tafel + erstes ausgefuehrtes W[J]** -- dW/dJ = <O> (6e-11), Hesse = Kubo (1e-7), gemischte d2W/dJ dtheta: nahtgekoppelt -9.3e-3, entkoppelte e^c-Kontrolle exakt 0.  Endliche Ebene, nicht das 4D-erzeugende Funktional.  Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v991_detline_bulk_edge",
      "cluster": "frontier",
      "descriptionEN": "SEAM.DETLINE.UNIFICATION.01 [O-Update: endlicher Bulk-Kante-Schatten [E]; Kontinuum Bismut-Freed bleibt offen] (Promote-Runde 2026-08-28): **Bulk-Twist-Torus-Kruemmung 2 pi * (+1) == Naht-Holonomie-Windung +1** (Diff 6e-16) auf dem v367-QWZ-Kragen; M=3-Kontrolle null auf beiden Seiten; Konjugations-Mutant dreht beide Vorzeichen kohaerent.  FIREWALL: 2+1D/1D endlicher Gitterschatten; Kontinuum Bismut-Freed und 4D-Vereinheitlichung bleiben [O].  Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v992_joint_likelihood_v1",
      "cluster": "frontier",
      "descriptionEN": "PRED.JOINTLIKELIHOOD.01 [O-Update: v1 ausgefuehrt [N]; Promotionsbedingungen typisiert; Kontrakt bleibt offen] (Promote-Runde 2026-08-28): **9-Observable-Joint-Likelihood** -- chi2 = 11.80 / dof 9 (p = 0.225), nu_eff = rank(J_atom) = 1, TFPT im 0. Perzentil von 200 verschluesselten Decodern, LOO stabil.  Holdouts deklariert, nicht verbraucht: JUNO Delta m^2, DESI Sigma m_nu, DUNE delta_CP, LiteBIRD r.  Promotionsbedingungen: off-diagonales Sigma, Formelgrammatik-Zensus, echte zukuenftige Holdouts.  Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v993_minimal_defect_selector",
      "cluster": "frontier",
      "descriptionEN": "AX.P2.01 / AX.P1.01 / SEAM.SIMPLECURRENT.GENERATOR.01 [O-Update: Zensus hebt die v624-Architektur-Bedingtheit auf Gitterebene; Axiom-Typisierung UNVERAENDERT] (Externe Review-Welle 3, 2026-08-28): **der VOLLE Rang-8-ADE-Zensus (SNF, nicht Lehrbuchformel) selektiert (D5, A3) eindeutig** (D3 ≅ A3; D5(+)D3 dasselbe Paar) -- die D+A-Architektur wird zum OUTPUT von (Rang 8 + zyklisches Z4 beidseitig).  KILL: D4(+)D4 / A7(+)A1 scheitern an der Zyklizitaet (Produkt 16); Z3-Kontrolle E6(+)A2 = Z3 x Z3.  [lambda]^2 = [v] EXAKT (Gitterschatten von U^2 = (-1)^F).  mu4-Mittelung = EINDEUTIGE spurerhaltende bedingte Erwartung (Modul-Eigenschaft noetig: ohne sie Kern dim 36/720).  P1-Arithmetik 1/(4*2pi)=c3 ist v813-Bestand; Aequiverteilung bleibt [C] von v813.  Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v994_mmst_criteria_battery",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.MMST.01 / SEAM.SIMPLECURRENT.GENERATOR.01 [O-Update: 5/7 MMST-Identifikationskriterien ausgefuehrt [E-gemessen]; C6 ausgeschrieben [C]; C7 zitiert/offen] (Vollstaendigkeits-Welle 2026-08-28): gemessen c=8.005 (16 Majorana-Kopien); konformes Kantenspektrum [0.506, 1.516, 2.523, 3.524] mit Entartungen [1,0,1,1,2]; Stromnormierung k=1.000000000000; Sektorkollaps det D8=4 -> det E8=1; FLAGSHIP: mu4/GSO-projizierte Gitter-Charaktersummen reproduzieren die (E8)_1-Vakuumcharakter-Koeffizienten [1, 0, 248, 0, 4124, 0, 34752] drei Ordnungen tief. C6 = das quasi-freie Konvergenztheorem ist jetzt AUSGESCHRIEBEN (articles/2026-08-28/psi_lambda_convergence_theorem_en.tex; 2/pi^2=0.20264 vs gemessen 0.203, 0.18%). C7 modulare Invarianz bleibt zitiert. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v995_kms_subnet_rigidity",
      "cluster": "frontier",
      "descriptionEN": "SEAM.STATE.RPMIXING.01 [E-Update: endliche KMS-Rigiditaet] + ALPHA.QUILLEN.EXACT.01 [O-Notiz: Rigiditaets-Praemisse endlich] (Vollstaendigkeits-Welle 2026-08-28): ALLE 33 v972-Mischrichtungen von KMS ALLEIN auf der endlichen CAR-Algebra getoetet (CAR_8=M_256); Kovarianz beweisbar redundant unter vollem KMS (COVARIANCE_DROP_REDUNDANT_UNDER_FULL_KMS); eindeutige E-invariante Crossed-Product-Zustandserweiterung (196608 -> 0). Typ-III/Kontinuum bleibt zitiert/offen. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v996_gauge_grammar_kill",
      "cluster": "frontier",
      "descriptionEN": "GAUGE.DETLINE.FIXPOINT.01 [O-Update: naive Grammatik-Erweiterung ausgeschlossen [E]; echt nichtabelsche Struktur noetig -- typisiert] + ALPHA.QUILLEN.EXACT.01 [O-Notiz: ln-2-Rate-Schatten] (Vollstaendigkeits-Welle 2026-08-28): H0-STRUKTURKILL -- negative b2/b3 drehen die Wurzelvorzeichen (SU(2)-Inverse -151.2, SU(3) -115.9); 64 Konventionen 0 Treffer; U(1)-Kontrolle 137.0359992168 ueberlebt alle Mutanten. Detline-Kruemmung -> 2pi exponentiell, Rate 0.6936 ~ ln 2 (numerischer Bismut-Freed-Schatten, kein Kontinuumstheorem). alpha_s(M_Z) bleibt externer Input. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v997_spin2_smg_witnesses",
      "cluster": "frontier",
      "descriptionEN": "GRAV.SPIN2.EMERGENCE.01 [O-Update: quadratische Zeugen [E]; Inhalt auf untrunkierter Formfunktion gepinnt] + CHIRAL4D.NOMIRROR.01 [O-Update: SMG-Mechanismus toy-bewiesen [E]; 4D-Spin(10)-Theorem bleibt [O]] (Vollstaendigkeits-Welle 2026-08-28): exakte Barnes-Rivers-Zerlegung -- R+R^2 sauber (ein masseloser TT-Pol, zwei Helizitaeten, Skalaron bei M_scal exakt); lokale a4-Weyl^2-Trunkierung traegt ZWINGEND einen Spin-2-Ghost (Gegenresiduum) -- typisiert. FK-SMG-Toy: N=8 eindeutiger Spin(7)-Singulett-Grundzustand, Schmidt-Rang 1, volumenunabhaengiges Gap 14 bei L=1,2,3; N=4-Kontrast 2^L-entartet; physische chirale Kante behaelt pi/L-Spektrum exakt. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v998_seam_modular_closure",
      "cluster": "frontier",
      "descriptionEN": "SEAM.EQUIV.MMST.01 / SEAM.SIMPLECURRENT.GENERATOR.01 [O-Update: C7-ARITHMETIK-SCHATTEN jetzt exakt; Netz-C7 bleibt zitiert] (Review-Welle 4, 2026-08-29): Zwei-Kanten-HS-Rest <= 2.11 auf dem Plateau Nx in (16,24,32,48,64,96); Gitter-Charakter theta_E8/eta^8 EXAKT vier Ordnungen [1,0,248,0,4124,0,34752]; Theta_E8(tau)=E4(tau) q-Entwicklung >= 8 Ordnungen exakt (Jacobi vs Eisenstein); j^{1/3}-Transform bei 60 Stellen; S-Gewicht-Buchfuehrung. Netz-C7 bleibt zitiert/offen. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v999_weak_collision_hp",
      "cluster": "frontier",
      "descriptionEN": "DYN.UNITARY.DILATION.01 [O-Update: endliche Leiter jetzt VOLLSTAENDIG -- diskret/Kernel/feld-frei/wechselwirkend-klassisch/quantum-gekoppelt/KONTINUIERLICHE ZEIT; offen = thermodynamisches Feldlimit] (Review-Welle 4, 2026-08-29): Schwache-Kollision/HP -- Richardson 1.098e-10, diag L=Q symbolisch, e^Q=B exakt, GNS-Splitting-Defekt 0, Lokalitaet 1.355e-20, starke Kollision Rang 9/81. Vertrag bleibt [O]. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1000_spin2_sk_witnesses",
      "cluster": "frontier",
      "descriptionEN": "GRAV.SPIN2.EMERGENCE.01 [O-Update: quadratische Hamilton-Haelfte ausgefuehrt] + FTRANSFER.SK.RHO0.01 [O-Update: SK-Mechanismus ausgefuehrt] (Review-Welle 4, 2026-08-29): Gitter-Graviton -- Positivitaet exakt, 2 Helizitaeten an allen 5852 Impulsen (N=6..12), omega = k - k^3/24 exakt. SK -- KMS/FDT ~1e-17, rho_0-Kontrast 3.550e-3 vs statisch 1e-16, Sattel 5-fach entartet, 3pi/5 eindeutig im mu4-Lift [C]. Beide Vertraege bleiben [O]. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1001_nu_pentagon_chain",
      "cluster": "frontier",
      "descriptionEN": "FLAV.NUSCALE.06 Kandidat [C]/[N] (Review-Welle 4, 2026-08-29; KEIN Schluss; Mechanismus = Q_+-zu-Flavor-Operator [O]): U_e=I Inventar + analytische 13/23-Fehlausrichtung; Pentagon-Doppel-Hit phi=288 deg=4(2pi/5), theta=2pi/35 eindeutiger LEE-Ueberlebender 2.7 %, max Pull 0.557; v3-Kette SHA-16 a4c28732fa687620: Sigma=0.0599 eV, m_beta=9.0 meV, m_bb in [1.5, 3.8] meV, delta_CP=287.66 deg; v270-theta23-Spannung 1.85 sigma GETYPT; Zensus 0/1607. Kills: DESI-Floor, DUNE delta_CP 287.7 vs 240, JUNO. FLAV.NUSCALE.05 unbewegt. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1002_spin10_mirror_projector",
      "cluster": "frontier",
      "descriptionEN": "CHIRAL4D.MIRROR.DET16.01 Kandidat [C] (Review-Welle 5, 2026-08-29; genaues endliches-Cluster-Gapping; offen: dynamische Eichfelder, Domain-Wall-Geometrie, 4D-Volumentheorem): DET16 Rang-1-Projektor auf 2^16, Omega^dag Omega=P_16, P_phi hermitesch idempotent 2.8e-18, h_mir {0^1, 1^65535} Gap 1, 45 so(10)-Kommutatoren 0; Hopping-Stabilitaet Gap 0.90/0.86 bei t=0.2; Casimir-Route AUSGESCHLOSSEN (Zensus N=8:0, k=2:27, k=3:1463, |W(D5)|=1920). Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1003_nu_structural_nulls",
      "cluster": "frontier",
      "descriptionEN": "FLAV.NU.TEXTURE.MECHANISM.01 [O] (Review-Welle 5, 2026-08-29; bindende Constraints, keine Texture): [E] geladener Sektor h=(1,1,1), d=(64,60,64); [X] Schur-Texture-NULL 8/8 (K,B) scheitern, D4-Spiegelung erzwingt lambda_1=lambda_3; gefordertes K ~ [3536.998, 1768.499, 1/3] trifft keinen von 9 Komparatoren; [E] Orientierungs-Verdopplung hebt Entartung generisch; [X] Skalen-NULL 12/12, 1-|c/a|=1.885e-4 von keiner gefrorenen Konstante. Mechanismus muss D4-ungerade sein UND ~2e-4-Cancellation liefern; reine Seam-Daten ausgeschlossen. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1004_milnor_frobenius_gray",
      "cluster": "frontier",
      "descriptionEN": "SEAM.MILNOR.LOCALRING.01 [O] (Review-Welle 6, 2026-08-30; endliche Algebra; Korpus-Identifikation bleibt offen): KLASSISCH (zitiert, kein Claim) Milnor-Algebra von x^2+y^3+z^5 congruent Q[y,z]/(y^2,z^4) dim 8, Spektralzahlen = E8-Exponenten {1,7,11,13,17,19,23,29}. NEU-IM-KORPUS EXAKT [E-finite]: Rang-Uhr (2-a)(4-b), Summe 30=h(E8); S=pqr*mu/8; Selbst-Uhr iff mu=8; (2,3,5) einziges geordnetes sphaerisches Tripel; Galois-Gray 11/7/-1; CP=Sozkel/Frobenius; integrale D^4-Bruecke basisabhaengig. [C] PG(3,2)=AG(3,2) sqcup PG(2,2) nur interpretativ. Vertrag bleibt [O]. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1005_relative_pencil_bridges",
      "cluster": "frontier",
      "descriptionEN": "Relative-Pencil-Bruecken (Review-Welle 6, 2026-08-30; Zirkularitaets-Flag): EXAKT [E-finite] Hankel p_n p_{n+2}-p_{n+1}^2=2^{n+1} + Jacobi-Spec {1,2} Diag (4/3,5/3) (v106/v119); Windungs-Spec {5/2,1,1} + Fixebene (v4). EXAKT ABER [C] (zirkular): eindeutiges t=2, lambda=7/3, CM J0^2=-I, Index 25=g_car^2 SNF (1,1,25) -- ABER Traeger 5 sitzt schon in den Spaltensummen von Q_2 und C: KEIN P2-Selektor. [C] Klein-Vier SNF (2,2) nur nach D=2d. AX.P2.01 unbewegt. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1006_mmst_lemma_battery",
      "cluster": "frontier",
      "descriptionEN": "SEAM.MMST.TYPEIII.CHARGED.01 [O]: five in-house lemmas; Round-4 document guard requires native v1026 norm below 2.995906 at fixed M=1, Ny=8 for all even N>=16 plus explicitly open FE-GEN/ALG-EXH/MMST; historical 2.11 grid is not the proof; 68/68"
    },
    {
      "id": "v1007_det16_stability_battery",
      "cluster": "frontier",
      "descriptionEN": "CHIRAL4D.MIRROR.DET16.01 Kandidat [C] (Review-Welle 7, 2026-08-30): T1/T2 bewiesen (v1002); T3 Michalakis-Zwolak zitiert-verifiziert (LTQO, t*>0, Gap>=1/2). [N] Zaehlgerade n=2,3; n=2-Scan bis t=4; dynamische Z2/Z4 eindeutig gelueckt (Vollsonden-Min 0.728854416); SW 1-alpha t^2 + O(t^4); c_geo/L beschraenkt. [X] gewoehnliches Hopping nicht blockdiagonal, daher 1/192 boxed; BDL liefert keinen volumen-uniformen Gap-Rest. Bleibt [C]. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1008_master_assembly_scaffold",
      "cluster": "frontier",
      "descriptionEN": "QFT4D.LATTICE.FUNDAMENTAL.01 Decision / GAUGE.DETLINE.FIXPOINT.01 [O] (Review-Welle 7, 2026-08-30; 2+1D-Scaffold kohaerent): Prototyp K=0 Plaquette-Exponenten; Materie frozen-link q^2~4, K(q=0)=0, kontraktiver Selektor |R'|<1; Wand/Spiegel QWZ-Steigung ~-0.9989, DET-Gap >=1.5 bis t=0.6; Assembly Z6 (1,2,3,0), Uhr [0,1,1,2], chi_phys vs chi_mirror. Keine Marker-Bewegung. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1009_rho0_minimizer",
      "cluster": "frontier",
      "descriptionEN": "FTRANSFER.SK.RHO0.01 [O] (Review-Welle 7, 2026-08-30): [E-finite] eindeutige KMS-Kompression (Klein), Hesse strikt positiv, Zentrum dim 1, Identitaet ~8e-17. [X] V1000_CONSEQUENCE_MISMATCH (Kompression toetet die SK-Antwort vs. gefrorener Kontrast 3.550e-3) + CENTER_FLATNESS_PREMISE_FALSE. Kanonische Formel braucht eine schwaechere zulaessige Menge. Vertrag bleibt [O]. Python-only / Wolfram-Spiegel zurueckgestellt."
    },
    {
      "id": "v1010_simplicity_bridge_census",
      "cluster": "frontier",
      "descriptionEN": "Simplicity-Zensus + W/Q-Kanonizitaet + Milnor-Doppelobstruktion (Nachmittag 2026-08-30): Ableitungsmatrix; Restminimum S4=2; Anker (1,1,2) nicht unter 3 deklarierten Milnor-Karten; phi0 vier neu NULL. Bruecke Q KANONISCH: Leiter erzwingt Summen (9,5,1); C's 5 von R geerbt -- v1005-Zirkularitaetsflag als erzwungene Provenienz korrigiert. Bruecke W CHOICE aber bedingt eindeutig (1078 -> 1). Milnor-Starkbruecke NICHT existent bei F2 und allen upstairs homes; grobe D+Paarung existiert; |Aut(X)|=36864, |Aut(Y)|=8. SEAM.MILNOR.LOCALRING.01 bleibt [O]; AX.P2.01 bleibt Axiom. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1011_3p1d_ladder_battery",
      "cluster": "frontier",
      "descriptionEN": "3+1D-Leiter B6-B9 (Nachmittag 2026-08-30): minimales 3+1D viable (Gauss 32768, Confinement, exakte Isotropie, g*~1.915); exaktes Tree-Fixing 524288 -> 4096, Tree-Unabhaengigkeit, DET unter Kopplung, Back-Reaction; Periodizitaet stellt K wieder her; geladene Kanaele ~2.014x bei schwachem g; K kollabiert in der confined Phase. BINDEND: Matching-Skala des Kopplungsfunktionals muss im deconfined/weak Regime sitzen. QFT4D.LATTICE.FUNDAMENTAL.01 bleibt Decision; GAUGE.DETLINE.FIXPOINT.01 bleibt [O]. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1012_toe_gate_witnesses",
      "cluster": "frontier",
      "descriptionEN": "Abend-TOE-Gatter-Welle (2026-08-30): T5 IR-Zeugen + getypter SPLIT (physikalisches c->1, xi*gap~1.089 vs Power 2.034, v_LR=5/3, Kruemmung->1/6; Holonomie+Uhr keine propagierenden Moden). T8 rho0 8-Kandidaten NO_SET_WORKS (harte Sektoren antworttot; KMS-affine Mengen selektionsvakuoes). T6 kappa sechsfach NULL (K4 18.8x zu gross; 4D-Funktional noetig). T7 wechselwirkendes TT (Erhaltung/Positivitaet/Summenregeln/Z-Persistenz; gelueckt). FIREWALL: TFPT.TOE.COMPLETE.01 / FTRANSFER.SK.RHO0.01 / FLAV.NU.TEXTURE.MECHANISM.01 / GRAV.SPIN2.EMERGENCE.01 bleiben [O]. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1013_thermodynamic_dynamics",
      "cluster": "frontier",
      "descriptionEN": "Thermodynamische Dynamik (Spaetabend 2026-08-30): Pflicht-Dynamik-Bein auf Hamiltonian-Klasse GESCHLOSSEN (Theorem-Memo). Uniforme LR-Schranke J=12/5, R=2, z=4, kappa=16043/450, v_LR=32086 e/225 ~387.64; normkonvergente tau_t; eichinvariante quasilokale Algebra A^G; Existenz von Grund/beta-KMS-Zustaenden. Numerischer Zwilling: gemessenes v=2.0 << bewiesene Schranke; Konvergenzschranke dominiert Puffer 8/16/32. Rest: Phaseneindeutigkeit, Limes-Luecke, IR-Universalitaet. QFT4D.LATTICE.FUNDAMENTAL.01 bleibt Decision. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1014_bridge_refinements",
      "cluster": "frontier",
      "descriptionEN": "Bruecken-Verfeinerungen (Spaetabend 2026-08-30): Detline-Einschraenkungsisomorphismus endlich VERIFIZIERT (c_phase 3.7e-16, Orientierung 2->1, Restanker U(1)/A0). W-Bruecke: P-anch ABGELEITET (e1=R^{-1}a, Wicklungs-Lock); P-dem TEILWEISE (Z4-Mittel annihilert nichttriviale Charaktere; fehlend = charakterblinde Determinanten-Antwortabbildung, MMST/Quillen-Bein). ALPHA.QUILLEN.EXACT.01 / AX.P2.01 unbewegt. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1015_axiom_core_closure",
      "cluster": "frontier",
      "descriptionEN": "Axiomkern-Abschluss (Montagmorgen 2026-08-31): charakterblinde P-dem-Antwort ABGELEITET (r=(1,1,1) exakt bei Kragen 12 und 16, Demokratie-Residuum 0; Mutanten (1,0,0) und (5/4,1,5/4)). Zusammen mit P-anch (v1014) BEIDE W-Bruecken-Praemissen endlich abgeleitet; Axiomkern-Rest NULL modulo externalisierter MMST-Identifikation. AX.P2.01 bleibt Axiom (endliche Schatten; Kontinuums-Identifikation dasselbe externe Bein). T1-Notiz: Strukturpostulat-Route endlich praemissengestuetzt. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1016_state_gap_batteries",
      "cluster": "frontier",
      "descriptionEN": "Zustands-/Luecken-Batterien (Montagmorgen 2026-08-31): R4 Naht-Modular-Kappe einzigartiger endlicher nicht-entropischer Selektor (D_tr 0.1434, SK-RMS 3.038e-3, Sattel erhalten; Finite-Proxy-Caveat). TEL-B zertifizierter Schwanz BLOCKIERT (N^2 D_N 530->19032); skalares dyadisches A_R<43 mit Faktor-350-Headroom; N-uniforme CAR-Nuklearitaet bewiesen (C_beta 67.97/29.73/12.15); ALG-EXH unveraendert. DFP volumenuniforme Luecke zitiert-verifiziert fuer Z2/Z4-offene Ketten (a=0.0233, a/16=0.00146); Rest Rotor/Wand/3+1D. FTRANSFER.SK.RHO0.01 [O] / SEAM.MMST.TYPEIII.CHARGED.01 [O] / CHIRAL4D.MIRROR.DET16.01 [C] unbewegt. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1018_e8_directed_readout",
      "cluster": "core",
      "descriptionEN": "E8.DIRECTED.READOUT.01 [E] (2026-09-02, Runde r609): sieben exakte E8-Ablese-Zellen (C7 offen) -- neu abgeleitet (keine Proben-Importe): Seifert/Phi_30, Hamming 1+14y^4+y^8, srg(120,56,28,24), N(n)=240 sigma_3(n) n<=10, A_P/A_P(1)=tau(n), (Z/30)^x = C2 x C4, E6 oplus A2 Index 3 Smith (1^6,3,3) Glue 78/81/81. C7 Gauss-Code bleibt offen (rem:c7-audit). KEIN RH-CLAIM. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1019_coxeter_euler_completion",
      "cluster": "core",
      "descriptionEN": "E8.COXETER.EULER.COMPLETION.01 [E]+Numerical (2026-09-02, Runde r617): Coxeter-Euler-Vervollstaendigung -- neu abgeleitet. Exakt: det(I-xC)=Phi_30, Tr C=-1; Moebius Phi_30=(1-x^2)(1-x^3)(1-x^5)(1-x^30)/[(1-x)(1-x^6)(1-x^10)(1-x^15)]; U=1 oplus C, Tr U=0. Global Z_C und D_E8 als Zeta-Quotienten, abs. konv. Re s>1/2. Numerisch (X=10^5): D_E8 Residual 4.76e-4 bei s=0.75, 4.16e-12 bei s=1.5. Klasse: lineares Verschwinden generisch aus Tr C=-1; E8 waehlt nur die Teiler von 30. Beurling-generisch. ZAUN: The trace-free completion is zero-free and pole-free in Re s > 1/2. The splitting into the scalar zeta channel and the Coxeter channel is open and RH-equivalent. No RH claim. NO-GO E8.COXETER.REGULARIZED_SPLIT.NO_GO.01: det_2 isoliert nicht 1/zeta(s). Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1020_spectator_sector_obstruction",
      "cluster": "frontier",
      "descriptionEN": "Spectator-Sektor-Hindernis (r644-Audit-Ernte 2026-09-04): endlicher freier-Fermion-Zeuge, neu abgeleitet (keine Proben-Importe). Aufschichten eines entkoppelten trivialen Gap-Sektors C (M=3, Chern 0) auf das Seam-Modell B (M=1, NY=8) laesst alle Seam-Daten identisch (Chern C_B=C_{B+C}=-1, C_C=0; Nullmoden 2/0/2 mit B-Träger 1.0; Randdispersion koinzident 1.4e-15; seam-restringierte Kovarianz 0.0; Randprofil-Ueberlapp 1.0) und aendert den Bulk (Dim 16 vs 32, E0(C)=-25.31, Rang 4 vs 8, HS 0.752536 vs 0.760202). Repariertes Kriterium: Krylov-Erzeugtheit spectator_dim(B+C)-spectator_dim(B)=16 an allen 128 Impulsen; Kommutant 1 vs 17, Pi_C im Kommutanten (Rest 0.0); Auswahl eindeutig (B ausgewaehlt, B+C verworfen). EHRLICH: spectator_dim(B)=10 nur bei p=0 -- Kriterium fast ueberall in p. SEAM.BULK4D.RECON.01 bleibt [O]; BULK.PRIME_COMPLETION.01 neu [O] registriert. KEIN MARKER-MOVE. KEIN RH-CLAIM. Python-only / Wolfram zurueckgestellt."
    },
    {
      "id": "v1022_telb_round3_certificates",
      "cluster": "frontier",
      "descriptionEN": "TEL-B-Runde-3-Zertifikate (2026-09-05): natives Arb/Acb, 256 Bit, 5812 rationale Zellen; TV_p(F'') <=15.817695962988924...<15.928674 und K3_TV<2.517464500835. Die analytische C2b-Skalarreduktion plus ausführbare Arb-Arithmetik ergibt c_epsilon=0.288547359238..., c_zeta=0.110677064901..., Summe=0.399224424139...<0.400; volles C2b bleibt exakt bei CF<=0.139/sqrt(N) und DG<=0.060/sqrt(N) offen. Exaktes Einmoden-Gegenmodell widerlegt den alten ALG2-Vandermonde-Schluss; FE-GEN oder eine praezise Gauge/Code-Projektion bleibt noetig. T2/TOE [O] unbewegt; python-flint==0.9.0, nicht Pyodide. KEIN RH-CLAIM."
    },
    {
      "id": "v1023_derivative_filtered_wall",
      "cluster": "frontier",
      "descriptionEN": "Ableitungsgefilterter Wall-Block (2026-09-05): H_df erhaelt ker(D_GW) exakt; det Block=lambda g^2 Delta s^2; das adiabatisch spiegel-kontinuierte hohe Band liegt >=Delta+lambda; der niedrige Ast ist quadratisch. Der nackte Spiegeloperator ist nicht invariant und hat fuer s>0 Niedrigbandgewicht -- kein Spiegel-Antwortgap und kein Weyl-Kegel. Gauge-Defekt 2.41e-15, Spektraldefekt 1.78e-15. Endliches quasi-freies Blocklemma; kanonische DET16/DET32-Realisierung, Vielteilchenparent, Zulässigkeit/Quasilokalitaet, Mass/Anomalie/Index, Universalitaet und 4D-Limes bleiben offen. Keine Markerbewegung. KEIN RH-CLAIM."
    },
    {
      "id": "v1024_parent_internal_boundaries",
      "cluster": "frontier",
      "descriptionEN": "Parent-interne T6--T8-Grenzen (2026-09-05): endlicher A3--D4 Seam/Neutrinopaar-Intertwiner ohne neuen Faktor; positiv, gerade, gauge-kompatibel, nichtadditiv, lokaler Gap 0.25 konditional auf ererbte volle Parent-Schranke, gemischte Resolvente 0.104712041885. Interner 12D-Z5-Cap Gap 0.197423715893, aber beliebig, D4-brechend und kein globales rho0. Exakte Grenzen: projektiver Full-Fock-D4-Lift; relativer Kommutant bei unitaler Erweiterung; volles Z5-Weylpaar verlangt 5|D nur unprojiziert; uniformer voller Gap schliesst masselosen Spin-2-Pol aus; TT-Projektion nicht endlichreichweitig. 49/49; T6--T8/TOE [O] unbewegt. KEIN RH-CLAIM."
    },
    {
      "id": "v1025_telb_c1_c2a_certificates",
      "cluster": "frontier",
      "descriptionEN": "TEL-B C1/C2a-Zertifikate (2026-09-05): natives python-flint==0.9.0. C1: 100-Bit Arb/Acb ueber 3888 analytische Rechtecke plus geschlossene Achsenstreifen-Majorante und exakter Imaginaerteil 1/12 liefert C_star<=0.1520453048...<0.1525. C2a: exakte Kotangens-Reindizierung, Peano-Monotonie und verschobener Euler-Rest reduzieren den Satz fuer alle geraden N>=16 auf Intervallvergleiche; N sup_k|phi_N(k)|<=0.5082287792...<0.509. C1/C2a geschlossen; C2b CF/DG, ALG-EXH, MMST-Skalierung und TOE bleiben offen. Native-only, nicht Pyodide. KEIN RH-CLAIM."
    },
    {
      "id": "v1026_telb_round4_closure",
      "cluster": "frontier",
      "descriptionEN": "Native interval TEL-B norm below 2.995906 at fixed M=1, Ny=8 for all even N>=16; CF/DG/CROSS and validated Fourier aliases; ALG-EXH/MMST open"
    },
    {
      "id": "v1027_signed_det_car_wall",
      "cluster": "frontier",
      "descriptionEN": "Exact canonical DET modes and signed z=1 wall for fixed classical gauge fields; no hard bare-mirror gap or chiral continuum"
    },
    {
      "id": "v1028_gauss_d4_cap",
      "cluster": "frontier",
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      "finding:holomorphic-kms-extension-assumes-charged-net-and-not-microscopic-generator",
      "finding:fixed-interval-rotation-modular-chain-fails-local-state-takesaki-repairs",
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      "finding:p1-parity-gauss-net-selection-undecidable-without-original-localization-map",
      "finding:original-minimality-does-not-reconstruct-operational-source",
      "finding:generic-first-order-relative-bound-zero-counterexample-and-correct-domain-repair",
      "finding:joint-discrete-data-vs-connection",
      "finding:parallelity-excludes-2plus2-only-conditionally",
      "finding:sign-projector-does-not-determine-time-magnitude",
      "finding:full-faithful-kms-or-transfer-recovers-magnitudes",
      "finding:full-two-face-collar-recovers-signed-operator",
      "finding:two-face-graph-state-time-same-datum",
      "finding:shared-register-gkls-embedding-exact-q-interval",
      "finding:marked-m4-channel-all-powers-homogeneous-gksl-excluded",
      "finding:flavor-ratio-closed-absolute-anchor-open",
      "finding:alpha-equation-is-fixed-input-closure",
      "finding:alpha-global-positive-root-unique",
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      "finding:matter-observations-do-not-fix-geometric-response",
      "finding:four-clock-fields-complete-quadratic-response",
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      "finding:neutral-vacuum-allword-twist-blindness",
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      "finding:native-invariant-response-not-fixed-chiral-twist",
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      "finding:native-chemical-mark-visible-in-charged-corners",
      "finding:fixed-hopping-quarter-phases-retain-antiunitary-discriminator",
      "finding:constructive-phase-class-and-full-winding",
      "finding:constructive-positive-state-extension",
      "finding:constructive-current-group-selects-positive-p-lift",
      "finding:constructive-state-algebra-resolution-and-source-test",
      "finding:constructive-all-phases-fail-unchanged-source",
      "finding:partner-carrier-charges-and-state-types",
      "finding:partner-fixed-quarter-time-splits-sm-multiplet",
      "finding:partner-original-carrier-cross-sheet-conjugation",
      "finding:partner-chiral-quarter-conjugation-state-time",
      "finding:joint-bilinear-hermitian-correction",
      "finding:joint-quarter-transports-marking",
      "finding:joint-fixed-time-marking-repair-excluded",
      "finding:joint-spinor-not-source-coefficient",
      "finding:typed-operator-search-calibrated",
      "finding:positive-kernel-construction-with-exact-discriminator",
      "finding:source-typed-kernel-search-controls",
      "finding:collar-car-projection-multifield-ray",
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      "finding:quarter-circle-absolute-zeta-green-eta",
      "finding:quarter-sea-absolute-thermal-central-phase",
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      "finding:order-one-infinitesimal-bound-inference-fails",
      "finding:joint-ward-two-field-data-miss-fixed-four-field-response",
      "finding:complete-car-kernel-fixes-nongaussian-link-functional",
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      "finding:e8-grade-two-explicit-spinor-current-action",
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      "finding:unchanged-ns-vector-generators-cannot-create-d5-spinor-fields",
      "finding:two-face-noncommuting-geometric-response-calculus",
      "finding:raw-face-mask-fails-source-kernel-null-descent",
      "finding:action-response-sewing-does-not-select-lf-intervention",
      "finding:faithful-gns-completion-closes-automated-source-query-feedback",
      "finding:whole-quarter-channels-identified-by-exact-internal-kernel",
      "finding:actual-joint-insertion-hull-resolves-eight-coefficient-directions",
      "finding:e8-bare-trilinear-and-single-anchor-weyl-class-insufficient",
      "finding:complete-family-four-cycle-class-separates-native-clock-from-sm-preservation",
      "finding:native-clock-word-transpose-has-static-rank-four-counterexample",
      "finding:whole-constant-coupled-source-is-implementable-despite-isolated-insertion-domain",
      "finding:fixed-quarter-jloc-pfaffian-closure-fails-local-skew-test",
      "finding:same-collar-has-canonical-selfdual-kinetic-pairing-and-charged-determinant",
      "finding:quarter-cover-pairing-does-not-supply-local-normal-or-select-scale-measure",
      "finding:fixed-aps-source-has-whole-inverse-and-finite-boundary-correction",
      "finding:mass-only-deck-relative-heat-action-fails-free-infinite-trace",
      "finding:finite-antiperiodic-scalar-wall-has-no-exact-massless-normal-mode",
      "finding:spectral-heat-covariance-positivity-does-not-imply-time-rp",
      "finding:quaternionic-sm-mass-winding-is-not-normal-determinant-clutch",
      "finding:bounded-relative-trace-obstruction-and-whole-genuine-pair-heat-reader",
      "finding:normal-clutch-orbit-index-equivariantly-recovers-given-weak-module",
      "finding:normal-orbit-spin-family-index-and-signed-charge-balance",
      "finding:fixed-flavor-cartan-sign-cover-and-root-lift-parity",
      "finding:original-family-vector-action-is-not-cartan-rotation-or-spinlift-selection",
      "finding:actual-genus-two-sign-family-closes-common-parallel-k-connection",
      "finding:actual-q-quarter-extends-common-k-connection-and-real-coefficient-lifts",
      "finding:real-source-reflection-is-not-linear-duality-or-compact-clock-inversion",
      "finding:source-selected-spin-and-q-signs-have-three-explicit-global-kernels",
      "finding:source-s0-szego-point-legs-generate-q-kernels-with-typed-products",
      "finding:actual-spin-kernel-determinant-has-inverse-q-norm-character",
      "finding:closed-source-flavor-spin-has-functorial-pfaffline-with-charged-type-limit",
      "finding:original-relative-seam-has-explicit-whole-graph-fredholm-mismatch",
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      "finding:old-complex-scalar-clutch-separates-c0-and-product-reflection",
      "finding:scalar-defect-circle-character-does-not-select-projector-holonomy",
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      "finding:original-v117-has-explicit-algebraic-q-cover-fundamental-frame",
      "finding:double-circle-reflection-does-not-fix-entire-cubic-cable-seam",
      "finding:actual-fuchs-real-map-transports-whole-given-finite-source-state-and-time",
      "finding:actual-e0-compensator-extends-whole-k-e8-with-transported-quarter",
      "finding:actual-v252-entire-real-basis-has-signed-p2-dual-dictionary",
      "finding:chiral-fuchs-source-transport-has-distinct-majorana-phase-and-torus-absorption",
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      "finding:three-family-cx-recursion-closes-exact-d8-with-minimal-original-escape",
      "finding:original-weak-current-coefficient-has-explicit-finite-oneform-consumer",
      "finding:original-d5-zero-mode-closes-triplet-carrier-and-fixed-glue-is-minimal-escape",
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      "finding:original-primitive-action-is-present-highpass-is-not-finite-selector",
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      "finding:original-poisson-normal-trace-fixes-whole-calderon-graph",
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      "finding:all-common-leading-distance-clutches-fail-original-flavor-lengths",
      "finding:actual-hardy-response-has-weak-clifford-odd-and-ko-real-completion",
      "finding:two-entry-hardy-adjoint-hierarchy-fails-full-ko-and-su2",
      "finding:actual-fuchs-halfpath-has-canonical-g-metric-c6-kernel-map",
      "finding:original-real-delta-jet-reads-same-marked-hardy-open-word",
      "finding:four-actual-fuchs-spine-edges-recover-marked-meridians-and-nonzero-open-word",
      "finding:native-adjoint-quarter-clock-cannot-compensate-hardypair-odd-character",
      "finding:whole-cubic-gram-differential-retains-hardy-phase-and-cross-resonance",
      "finding:complete-original-mu01-hardy-reader-has-two-calibrated-entries",
      "finding:whole-six-original-hardy-inputs-generate-full-family-coefficient-algebra",
      "finding:actual-whole-hardy-alphabet-has-finite-particle-hole-wick-block",
      "finding:bare-unsmeared-mu-crossface-quadratic-fails-vacuum-domain",
      "finding:linear-spectral-centralization-does-not-replace-full-marked-hardy-source",
      "finding:original-admissibility-projector-is-explicit-penalties-do-not-read-retained-amplitudes",
      "finding:actual-original-inner-fluctuation-has-whole-noncommuting-two-face-response",
      "finding:fiber-only-primitive-split-and-frozen-gap-miss-original-action",
      "finding:oriented-resolvent-weights-do-not-identify-original-reflection-producer",
      "finding:current-odd-carrier-reader-is-lossless-on-whole-coupled-source",
      "finding:ap-spectral-formulas-do-not-force-normal-domain-or-complete-source",
      "finding:actual-weak-connection-transmits-family-orientation-to-both-faces",
      "finding:original-higgs-variation-is-not-independent-yukawa-stationarity",
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      "finding:canonical-clifford-double-derives-normal-ap-and-typed-deck",
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      "finding:whole-two-face-compression-projector-defect-detects-hidden-source-loss",
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      "finding:actual-graded-hypercharge-trace-rejects-quark-lepton-sign-intertwiner",
      "finding:separable-normal-real-maps-compose-to-given-deck-and-quarter-image",
      "finding:actual-gauge-real-frames-leave-carrier-phase-as-coordinate-freedom",
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      "finding:signed-one-particle-modular-weight-is-not-the-positive-even-heat-operator",
      "finding:three-moments-select-one-exponential-exactly-in-positive-laplace-class",
      "finding:existing-ap-positive-source-factor-has-oriented-poisson-heat-bridge",
      "finding:actual-v252-entire-compatibility-family-has-generic31-dimensional-quotient",
      "finding:full-memory-schur-reconstruction-preserves-every-ordered-word-under-nesting",
      "finding:original-return-and-native-j-have-minimal68-dimensional-all-word-memory",
      "finding:original-full-source-regulator-flow-retains-normalized-kernel-and-vacuum-factor",
      "finding:full-reflected-quadratic-regulator-cone-preserves-every-rp-source",
      "finding:original-shared-higgs-source-induces-fixed-yukawa-mixed-matter-words",
      "finding:local-even-yukawa-sewing-preserves-given-joint-graded-rp",
      "finding:whole-original-constant-higgs-orbit-has-reduced-majorana-spectral-response",
      "finding:archived-geometric-higgs-residue-bulk-needs-coefficient-identification-repair",
      "finding:original-local-higgs-residue-linear-coefficients-fail-same-trace-heat-ratio",
      "finding:whole-original-higgs-moments-fix-conditional-local-kinetic-mass-quartic",
      "finding:whole-variable-original-higgs-bound-reference-response-and-forced-clifford-factor",
      "finding:original-symmetric-scale-map-fails-simultaneous-higgs-einstein-repair",
      "finding:whole-original-higgs-inner-scale-and-affine-full-source-transport",
      "finding:normal-higgs-mode-type-and-exact-kernel-metric-require-actual-relative-operator-map",
      "finding:original-superconnection-grades-and-weighted-heat-retain-higgs-kinetic-excursions",
      "finding:whole-normal-reduction-preserves-relative-spectral-reader-and-higgs-jets",
      "finding:original-admissibility-own-spectral-reader-has-no-free-low-mode-residue-repair",
      "finding:whole-total-chiral-positive-spectrum-retains-higgs-kinetics-and-common-half-trace",
      "finding:whole-scale-energy-response-uniquely-reconstructs-normal-profile",
      "finding:original-relative-scale-flow-separates-residue-finite-part-and-local-anomaly",
      "finding:original-whole-green-produces-energy-response-with-correct-orthogonal-trace",
      "finding:whole-mixed-scale-source-reads-bosonic-insertion-before-higgs-average",
      "finding:current-original-spectral-action-triplet-fails-positive-laplace-class-unless-leading-relative-term-cancels",
      "finding:correct-current-argument-weights-link-whole-higgs-threshold-and-same-einstein-response",
      "finding:whole-original-spinor-curvature-a4-has-no-independent-scalaron-r-squared",
      "finding:whole-original-entropic-curvature-corrects-max-symmetric-scalar-scale",
      "finding:given-whole-higgs-source-fixes-radial-curvature-susceptibility-without-extra-scalar",
      "finding:actual-original-four-ghost-complex-preserves-base-selector-with-full-cohomology-and-distinct-traces",
      "finding:whole-original-admissibility-energy-source-recovers-constraints-but-cannot-replace-geometric-operator",
      "finding:original-moving-normal-kernel-fixes-induced-connection-and-full-scale-covariance",
      "finding:fixed-original-o1-source-rejects-nonconstant-normal-multiplier-and-fixes-placement-test",
      "finding:original-master-variation-keeps-spin-weight-and-active-source-route-typed",
      "finding:shared-original-higgs-mixture-has-invariant-color-fourpoint-wick-defect",
      "finding:bounded-fixed-rank-spectral-action-does-not-normalize-noncompact-higgs-volume",
      "finding:whole-original-quarter-source-bounds-higgs-weight-uniformly-through-joint-mode-ir-limits",
      "finding:original-o1-covariance-leaves-full-radial-higgs-law-and-exact-fourier-source-readout",
      "finding:whole-process-reconstruction-priority-and-original-locality-geometry-input-audit",
      "finding:actual-disjoint-local-weyls-acquire-spurious-commutator-under-bare-native-compression",
      "finding:actual-quarter-current-state-excludes-nontrivial-geometric-local-point-time",
      "finding:actual-elliptic-interval-car-source-produces-local-modular-generator",
      "finding:actual-proper-two-face-regions-do-not-contain-pure-local-source-double",
      "finding:actual-elliptic-whole-environment-fixes-regional-car-partner",
      "finding:actual-original-color-triplet-has-spectator-dependent-modular-field-law",
      "finding:actual-color-marginal-has-complete-finite-source-reader-and-mixture-criterion",
      "finding:two-and-four-field-words-do-not-complete-original-color-triplet-source",
      "finding:whole-original-quarter-occupation-tower-uniquely-reconstructs-common-radial-higgs-law",
      "finding:finite-original-tower-reader-and-existing-weak-symmetry-fix-shared-matter-responses",
      "finding:unchanged-free-quarter-source-cannot-be-nontrivial-integrated-higgs-mixture",
      "finding:actual-e8-record-doily-action-is-signed-p-plus-compatible-point-with-typed-sym2-lift",
      "finding:actual-e8-record-words-distinguish-order-and-retain-phase-beyond-fifteen-labels",
      "finding:literal-doily-incidence-output-and-strict-causal-order-on-fifteen-source-labels-excluded",
      "finding:actual-doily-context-record-characters-have-ranks-four-two-two-two",
      "finding:canonical-standard-split-localizes-original-source-record-words-with-full-descendant-companion",
      "finding:unchanged-global-sixty-event-source-fails-local-exterior-current-test",
      "finding:canonical-source-localization-changes-exterior-state-and-does-not-select-event-origin",
      "finding:original-elliptic-fixed-holonomy-insertion-gives-local-operation-on-unchanged-source",
      "finding:original-quarter-pair-fixes-square-charge-law-half-response-and-real-quillen-norm",
      "finding:actual-elliptic-source-determines-all-path-normalized-causal-current-response",
      "finding:original-higgs-quark-source-has-exact-whole-su3-counting-response",
      "finding:single-original-color-parity-separates-free-quillen-from-positive-higgs-integrated-source",
      "finding:whole-original-quarter-graph-color-response-is-relative-and-diagonal-singlet",
      "finding:original-center-selector-requires-typed-state-field-and-face-transport",
      "finding:original-color-center-fixes-whole-e6-a2-source-and-charged-ward-word-action",
      "finding:original-color-neutral-lattice-class-cannot-supply-odd-fields-by-inherited-superlattice-extension",
      "finding:original-color-center-extends-to-folded-ns-but-full-current-action-needs-spin-sector-transport",
      "finding:actual-color-spin-car-mixed-word-has-complete-thermal-source-and-cyclic-sewing",
      "finding:exact-original-ns-ramond-heat-equality-does-not-identify-fixed-color-restricted-source",
      "finding:original-scalar-cover-profile-cannot-implement-fixed-spin-cartan-loop",
      "finding:actual-spin-cartan-composite-source-retains-multi-arc-support-and-fails-additive-even-locality",
      "finding:actual-color-selector-admits-exactly-eighth-periodic-original-density-variations",
      "finding:actual-color-is-internal-exactly-on-eightfold-coordinate-source-regions",
      "finding:original-whole-two-face-multiplier-reduction-is-only-constant-residue-diagonal",
      "finding:original-two-face-local-fock-implementation-and-exact-source-variation-differ-from-reduction",
      "finding:whole-undecorated-transfer-does-not-determine-original-marked-field-source",
      "finding:volume-wise-injective-positive-transfer-does-not-prove-uniform-parent-gap",
      "finding:original-c6-half-power-has-exact-entire-positive-ordered-word-expansion",
      "finding:original-c6-whole-word-sewing-retains-source-amplitude-and-controlled-infinite-tail",
      "finding:historical-parent-and-actual-field-time-bridges-must-be-reused-before-new-source-models",
      "finding:actual-native-v990-full-timed-words-recover-kubo-response-distinct-from-instantaneous-variance",
      "finding:actual-original-higgs-force-has-complete-ordered-fermionic-determinant-source",
      "finding:finite-covariant-higgs-composites-have-no-nontrivial-positive-instantaneous-classical-law",
      "finding:actual-original-collective-mean-has-exact-ball-domain-and-relative-entropy-action",
      "finding:original-uniform-allmode-force-clt-is-nonzero-only-before-fixed-time-smearing",
      "finding:original-unscaled-time-smeared-force-has-selfadjoint-whole-ordered-source",
      "finding:original-local-higgs-force-has-whole-centered-hilbert-schmidt-determinant-source",
      "finding:given-original-circle-force-source-generates-an-exact-causal-local-net",
      "finding:literal-archived-global-rg-kernel-does-not-preserve-finite-measures",
      "finding:native-higgs-marginal-retains-induced-vertex-and-noether-response",
      "finding:isospectral-noether-orbit-does-not-supply-original-metric-higgs-variation",
      "finding:metric-dtn-form-bound-needs-transfer-comparison-for-gap",
      "finding:conditional-higgs-pair-winding-protection-in-common-circle-carrier",
      "finding:natural-five-clock-carrier-excludes-invariant-three-transfer-compression",
      "finding:shared-scalar-shift-whole-retained-higgs-sector-has-index-minus-eighteen",
      "finding:whole-coupled-quark-circle-common-shift-cancels-relative-winding",
      "finding:one-sided-shift-full-higgs-pair-is-hilbert-schmidt-not-trace-class",
      "finding:selective-coupled-higgs-reader-has-well-defined-det2",
      "finding:v311-population-reader-does-not-identify-quantum-channel",
      "finding:declared-charged-pair-relation-leaves-five-word-ambiguity",
      "finding:full-creator-relations-close-fixed-free-two-mode-ambiguity",
      "finding:comoving-diagnostic-clock-exact-response-but-nonlocal-mode-transport",
      "finding:free-quarter-two-face-response-strip-with-relative-zero-mode-defect",
      "finding:e8-hodge-event-perfectly-distinguishable-from-every-unitary",
      "finding:common-five-slot-transfer-preserves-e8-bracket-not-automatic-tp",
      "finding:comoving-clock-dynamic-higgs-connection-and-charged-flow",
      "finding:r064-sixty-record-channel-exact-distance-and-continuous-record-parent",
      "finding:two-syntheses-shared-source-priorities-and-typed-limits",
      "finding:weighted-index-trace-and-selfdual-theta-require-original-selection",
      "finding:ordered-quarter-fourfield-e8-record-and-charge-filters",
      "finding:r091-recovered-native-hodge-sewing-and-full96-quarter-complement",
      "finding:r092-fold-ward-target-neutral-source-and-smaller-creation-branch",
      "finding:r093-original-kink-time-response-distinguishes-quarter-states",
      "finding:r093-ordered-current-kink-and-eight-arc-locality",
      "finding:r094-original-five-excitations-distinct-from-clustering-vacuum",
      "finding:r096-joint-higgs-twoface-source-constraint",
      "finding:reverse-bootstrap-retained-operator-scope",
      "finding:e8-current-dictionary-all-separated-responses",
      "finding:charged-scaling-microscopic-map-still-unspecified",
      "finding:original-slater-proxy-fitted-bound-fails-actual-norm",
      "finding:periodic-microscopic-zero-modes-distinguish-source-state",
      "finding:replicated-deterministic-consistency-iff-influence-dag",
      "finding:three-binary-processes-two-copy-reversible-classification",
      "finding:independent-poset-information-composition-consequences",
      "finding:reset-diagnosis-not-global-tfpt-underdetermination",
      "finding:three-inequivalent-repeatable-observation-countermodels",
      "finding:repeatable-reports-do-not-imply-full-state-stability",
      "finding:finite-sequential-data-admit-classical-history-realization",
      "finding:pascal-carrier-rank-unique-unbounded-source-proof",
      "finding:finite-carrier-glue-elements-orbits-and-halfway",
      "finding:marked-seam-category-and-character-valued-full-lift-factor",
      "finding:quarter-clock-sign-lift-complete-four-puncture-classification",
      "finding:quarter-lift-central-quotient-class-and-all-subgroup-restrictions",
      "finding:native-clock-reflection-full-factor-cocycle-trivialization",
      "finding:native-quartic-exterior-local-cohomology-distinguishes-clock-monodromies",
      "finding:orbifold-lift-criterion-and-determinant-eight-divisibility",
      "finding:original-g31-reflection-census-and-all-integral-cohomology",
      "finding:all62-original-clock-obstructions-and-central-variants",
      "finding:actual-sign-lift-g31-group-comparison",
      "finding:complete-g31-lift-invariants-and-explicit-collisions",
      "finding:e7-quarter-pairs-literal-native-g31-reflections",
      "finding:e6-root-subsystem-quarter-clock-discriminant-obstruction",
      "finding:e-series-square-cover-family-and-quarter-descent",
      "finding:e-series-integral-glue-and-normalization-nonidentifiability",
      "finding:global-original-g31-character-lift-nonsplitting",
      "finding:all62-original-character-clock-lifts-complete",
      "finding:all17-original-character-monodromy-lifts-complete",
      "finding:e7-pair-full-lattice-field-character-torsor",
      "finding:e-series-complete-native-compiler-cut-dictionary",
      "finding:common-cover-clock-and-maximal-integral-phi12-intertwiner",
      "finding:native-four-cut-projectors-unify48-fourthirds-fivequarters",
      "finding:complete-covariant-compiler-voa-source-and-weyl-closure",
      "finding:complete-fixed-jhat-cyclic-field-lift-and-associated-clock",
      "finding:relative-native-burau-clock-hom-faithful12-transport",
      "finding:e6-vertex-nonselection-survives-full-field-clock-phases",
      "finding:marked-c4-glue-unique-local-e8-vacuum-source",
      "finding:canonical-vacuum-clock-preserves-e6-nonselection",
      "finding:marked-generator-forces-e8-without-support-multiplicity-or-minimality-input",
      "finding:native-reflection-rotation-lift-classes-and-forgetful-images",
      "finding:quarter-clock-pure-mapping-class-complete-quotient",
      "finding:e8-cut-five-orbital-incidence-algebra-complete",
      "finding:compiler-d8-frame-requires-full-d5-a3-flag",
      "finding:tetrahedron-incidence-lossless-e6-reconstruction",
      "finding:restricted-clock-lift-and-global-cover-audit-reconciled",
      "finding:double-scalar-quarter-is-marked-glue-with-two-step-index",
      "finding:scalar-quarter-two-involutive-e8-extensions",
      "finding:native-quarter-glue-inner-conjugacy-and-oriented-tetrahedron-flags",
      "finding:native-cyclic-compiler-inner-conjugacy",
      "finding:recovered-e6-closed-81-current-module-and-unique-source-cubic",
      "finding:e-series-operational-role-review-and-source-time-centralizer",
      "finding:actual-clock-doily-ovoid-kernel-is-native-quartic-five",
      "finding:glue-clock-twisted-sectors-share-marked-quadratic-module",
      "finding:complete-native-six-effect-covariant-instrument-classification",
      "finding:native-six-effect-readout-on-existing-erasure-output",
      "finding:relational-quartic-singlet-and-invariant-density-separation",
      "finding:new-clock-feedback-origin-input-audit",
      "finding:native-event-channel-cone-intersection-is-tenth-to-half",
      "finding:native-six-ovoid-observation-first-occurs-at-tensor-degree-three",
      "finding:original-quartic-erasure-and-hamming-octic-share-marked-ovoid-response",
      "finding:native-octic-field-ray-is-explicit-but-retained-clock-stress-preparation-fails",
      "finding:actual-quarter-fold-stress-has-nonzero-octic-eighth-jet",
      "finding:native-octic-is-an-original-even-car-current-composite",
      "finding:original-geometric-source-jet-and-partial-p1-seed-closure",
      "finding:full-root-ope-has-normalized-native-octic-component",
      "finding:paper-microscopic-car-composes-with-native-octic-ward",
      "finding:microscopic-neutral-octic-has-controlled-smeared-limit",
      "finding:paper-seed-source-needs-local-image-not-global-equivalence",
      "finding:full-car-positive-time-forces-source-vacuum",
      "finding:original-winding-theorem-does-not-select-seam-ray",
      "finding:original-b-u-pair-conditionally-reconstructs-whole-circle-channel",
      "finding:original-os-step-needs-full-moments-not-connected-cumulants",
      "finding:graded-field-image-precedes-physical-observable-restriction",
      "finding:older-original-circle-representative-already-realizes-ward-and-index",
      "finding:old-quarter-heat-transfer-has-controlled-rank-one-limit-and-odd-product",
      "finding:historical-em-rank-one-and-full-heat-traces-have-different-contracts",
      "finding:older-frg-functional-already-supplies-formal-full-moment-parent",
      "finding:earliest-c3-route-quantizes-level-but-needs-physical-kinetic-normalization",
      "finding:canonical-positive-circle-half-selects-the-toeplitz-line-and-sixth-order-correction",
      "finding:old-seam-shift-canonically-gives-relative-car-line-and-fredholm-factor",
      "finding:old-index-coupling-removes-heat-map-and-has-sharp-residual-ward-test",
      "finding:original-admissibility-requires-image-membership-not-positive-half-or-commutation",
      "finding:old-closure-already-assumes-car-but-does-not-map-the-geometric-circle",
      "finding:historical-uv-map-infinite-mass-exclusion-already-proved-and-recovered",
      "finding:a4-spin-lift-fixes-local-ns-and-unordered-quarter-pair",
      "finding:cover-winding-exchanges-quarter-sectors-before-its-square",
      "finding:retained-winding-preserves-local-support-and-neutral-time",
      "finding:marked-glue-square-is-the-e8-fermionization-character",
      "finding:fixed-canonical-d8-is-not-transported-source-d8",
      "finding:actual-winding-has-four-module-e8-defect-extension",
      "finding:ramond-lift-sign-is-common-intertwiner-basis-gauge",
      "finding:old-seam-relative-determinant-is-stable-in-faithful-kms-class",
      "finding:finite-circle-compression-cannot-retain-whole-thermal-winding-factor",
      "finding:schur-carrier-plus-complement-preserves-original-relative-determinant",
      "finding:v258-forward-source-gap-is-reused-not-solved-by-inversion",
      "finding:one-sided-source-and-real-em-norm-need-response-map-not-chiral-phase",
      "finding:trace-class-pair-index-fixes-relative-factor-without-primitive-ray",
      "finding:matrix-clutch-degree-connects-weak-determinant-class-without-ray-selector",
      "finding:finite-ns-ordinary-determinant-cannot-limit-to-nonzero-index",
      "finding:inner-carrier-is-infinite-spacetime-module-not-finite-circle-cut",
      "finding:normal-spin-determinant-line-exactly-transports-whole-circle-in-given-product",
      "finding:normal-index-line-source-match-is-holonomy-condition-not-vertical-degree",
      "finding:full-normal-product-polarization-change-is-not-trace-class",
      "finding:massive-transverse-zero-index-retains-positive-determinant-response",
      "finding:original-callias-doublet-count-does-not-supply-one-complex-circle-channel",
      "finding:exact-moving-kernel-reduction-forces-kato-offdiagonal-connection",
      "finding:tetrahedral-quarter-face-phase-does-not-identify-original-quarter-marked-curve",
      "finding:existing-elliptic-cover-correspondence-has-exact-c6-image-and-retained-mixed-clock",
      "finding:recovered-natural-joint-lift-and-given-connection-already-exist",
      "finding:global-split-joint-carrier-admits-parallel-native-c12-clock",
      "finding:pure-weak-clutch-clock-identification-fails-despite-quarter-matches",
      "finding:same-chern-clutch-native-compiler-does-not-select-physical-connection",
      "finding:native-gaussian-four-is-cartan-subspace-in-whole-sixtyfour",
      "finding:native-family-factor-is-not-gaussian-four-despite-symmetric-coincidences",
      "finding:primitive-class-two-sphere-obstructs-global-cartan-subspace-choice",
      "finding:even-degree-pullback-removes-cartan-topology-without-selecting-clock-cover",
      "finding:global-cartan-choice-is-optional-for-original-scalar-source-fields",
      "finding:bare-native-octic-is-not-whole-e8-zero-mode-singlet",
      "finding:explicit-primary-e8-invariant-octic-completes-same-neutral-car-source",
      "finding:completed-octic-has-exact-old-root-ope-component-and-quarter-source-value",
      "finding:elliptic-normal-spin-pullback-doubles-index-and-deck-quarter-lifts-anticommute",
      "finding:exact-separated-w8-twopoint-in-original-quarter-current-source",
      "finding:unshifted-vacuum-fails-same-w8-source-state-time-dictionary",
      "finding:local-current-phase-and-conditional-cyclic-p1-w8-image",
      "finding:all-neutral-w8-correlations-identify-the-two-original-quarter-charges",
      "finding:original-w8-response-has-infinite-same-time-spectral-support",
      "finding:older-seam-equivalence-verifications-do-not-supply-marked-local-p1-map",
      "finding:full-original-c1-stress-cycle-reconstructs-neutral-w8-response",
      "finding:folded-c8-cycle-misses-intrinsic-w8-tangent-without-obstructing-original-stress",
      "finding:native-source-first-nonuniform-moment-is-four-with-literal-five",
      "finding:native-whole-five-ground-space-needs-four-register-interaction",
      "finding:old-cover-winding-reconstructs-local-quarter-current-net",
      "finding:untransported-independent-quarter-stress-pair-is-not-current",
      "finding:current-public-paper-bytes-and-binding-source-gaps-recovered",
      "finding:winding-charge-conjugation-descends-fixed-pin-domain-but-not-full-dirac-sea",
      "finding:native-fourth-root-frame-has-signed-collective-sixty-mode-dual",
      "finding:native-root-gram-spectrum-is-not-canonical-source-energy-selection",
      "finding:identical-overlapping-native-five-codes-have-no-common-state",
      "finding:horizontal-e8-invariant-octic-cannot-isolate-native-five-ground-space",
      "finding:longitudinal-reflection-preserves-fixed-pin-domain-and-full-dirac-sea",
      "finding:single-fixed-pin-minus-green-kernel-fails-neutral-os-positivity",
      "finding:full-original-ns-orientation-cover-has-all-mode-charged-green-rp",
      "finding:cover-green-thermal-state-is-not-original-quarter-hardy-vacuum",
      "finding:declared-recursive-native-som-is-u35-equivariant",
      "finding:declared-recursive-native-som-clocks-depend-on-chosen-rates",
      "finding:existing-c2-dividing-real-forms-realize-moebius-annulus-tower",
      "finding:raw-ns-elliptic-glide-has-projective-spin-commutator-and-quarter-descent",
      "finding:canonical-elliptic-annulus-reproduces-original-quarter-boundary-tuple-and-kms",
      "finding:finite-crosscap-cauchy-state-is-not-original-quarter-vacuum-by-normal-filtering",
      "finding:native-quartic-relative-arc-charge-excludes-additive-density-stress-insertion",
      "finding:retained-quarter-sea-finite-modular-transfer-preserves-integrated-native-cartan",
      "finding:original-neutral-interarc-fields-have-exact-connected-current-vertex-reconstruction",
      "finding:actual-pin-zero-branch-retains-quarter-car-state-time-but-no-uniform-scale-return",
      "finding:conformal-pin-source-geometry-fixes-exterior-return-and-unitary-memory",
      "finding:geometric-exterior-schur-response-transfers-proper-native-arc-charge-but-is-compact",
      "finding:actual-ns-half-cover-cauchy-graph-purifies-green-state-with-neutral-ground-time-obstruction",
      "finding:full-marked-c2-lifts-distinguish-equatorial-and-os-real-slots",
      "finding:full-marked-equatorial-pullback-is-ramified-graph-not-smooth-seam-circle",
      "finding:same-elliptic-szego-green-functional-has-all-word-free-rp-and-calderon-marginal",
      "finding:elliptic-scalar-spin-pushforward-has-original-holomorphic-p2-anchor",
      "finding:natural-elliptic-anchor-connection-is-not-original-p2-cusp-connection",
      "finding:actual-elliptic-gaussian-source-determines-full-quarter-current-functional",
      "finding:elliptic-current-state-differs-from-hardy-on-every-proper-current-interval",
      "finding:primitive-geometric-force-retains-coupled-reference-and-parity-variation",
      "finding:equal-finite-positive-heat-channel-values-do-not-fix-geometric-force",
      "finding:elliptic-ns-spin-trace-has-complete-global-quarter-kernel",
      "finding:source-induced-ramified-resolutions-do-not-select-quarter-circle-domain",
      "finding:fixed-native-and-fermat-c3-carriers-do-not-identify-marked-v117-pair",
      "finding:theta-square-modular-pair-reconstructs-v117-with-distinct-geometric-q",
      "finding:archived-d4-puncture-formula-fails-and-existing-v117-repairs-existence",
      "finding:geometric-square-clock-is-not-theta-v117-quarter-generator",
      "finding:canonical-conformal-boundary-trace-fixes-existing-elliptic-cauchy-state",
      "finding:conformal-factor-averaging-preserves-given-free-boundary-wick-words",
      "finding:retained-seam-defect-has-derived-causal-geometric-response-with-contact",
      "finding:same-final-pin-geometry-retains-history-while-relative-index-factor-is-fixed",
      "finding:full-projector-covariance-and-positivity-force-all-car-words",
      "finding:actual-two-face-projector-reduces-free-source-test-to-cross-coherence",
      "finding:retained-marked-quarter-compiler-has-no-local-geometric-wick-correction",
      "finding:changed-elliptic-modulus-has-derived-cauchy-response-and-neutral-fourfield-variance",
      "finding:actual-equatorial-half-form-trace-fixes-orthogonal-hardy-range",
      "finding:actual-holomorphic-quarter-cauchy-idempotents-are-not-positive-car-covariances",
      "finding:full-equatorial-checkerboard-free-algebra-fails-fixed-reflection-positivity",
      "finding:genuine-d4-orbifold-branch-has-unique-irreducible-v117-class",
      "finding:wall-weight-additivity-does-not-exclude-stable-v117-parabolic-bundle",
      "finding:recovered-september-c4-triangle-closes-original-a0star-monodromy",
      "finding:actual-a0star-connection-does-not-have-rational-base-inversion-lift",
      "finding:actual-v117-sign-lines-are-negative-weierstrass-triangle-two-torsion",
      "finding:antipodal-and-positive-sign-packets-fail-actual-marked-v117-ramification",
      "finding:common-cyclic-twelve-cover-does-not-trivialize-v117-unramified-sign-lines",
      "finding:negative-sign-packet-has-source-selected-quarter-clock-lift",
      "finding:original-root-and-sign-lines-have-minimal-degree48-common-cover",
      "finding:given-elliptic-flat-connection-family-has-explicit-quillen-norm-and-two-holonomies",
      "finding:bare-large-gauge-quillen-norm-does-not-equal-original-winding-counting-factor",
      "finding:existing-elliptic-half-period-has-canonical-graded-gns-intertwiner",
      "finding:standard-sewing-requires-maximal-affinity-linear-half-modular-field-law",
      "finding:declared-recursive-machine-baseline-controls-and-recovery-are-finitely-measured",
      "finding:declared-recursive-machine-long-run-retains-motion-at-stable-aggregate-rank",
      "finding:recursive-kohonen-source-investigation-has-source-verified-atlas-handoff",
      "finding:chat-algorithmic-comparison-native-pauli-code-and-scoped-information-loss",
      "finding:chat-20261026-results-consolidated-with-corrected-assumptions",
      "finding:source-inversion-forces-six-line-theta-completion",
      "finding:literal-geometric-quarter-is-not-central-gaussian-quarter-on-labels",
      "finding:source-cubic-and-quarter-select-even-theta-and-spin-half-turn-lift",
      "finding:source-cubic-matches-native-pauli-translations-after-context-target-correction",
      "finding:joint-source-clock-relations-have-one-native-clifford-gauge-class",
      "finding:ordinary-theta-determinant-erases-selected-spin-central-character",
      "finding:marked-negative-point-norm-restores-required-spin-central-character",
      "finding:point-normalized-theta-family-correction-fails-original-cubic-action",
      "finding:source-spin-relative-poincare-normalization-closes-point-norm-family",
      "finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package",
      "finding:tfpt-origin-investigation-full-source-coverage-and-current-frontier",
      "finding:full-clock-norm-power-class-and-canonical-eighth-period",
      "finding:original-unit-winding-selects-one-inverse-norm-in-three-cabled-hardy-pair",
      "finding:original-boundary-audit-and-h1-point-restriction-limit",
      "finding:eta-phase-conventions-do-not-identify-relative-spin-reference-pair",
      "finding:same-spin-unit-shift-gives-native-module-class-with-odd-determinant-line",
      "finding:original-even-odd-operator-split-is-compatible-with-same-spin-seam-reference",
      "finding:four-session-summaries-remain-covered-with-superseded-norm-selection-task",
      "finding:elliptic-projector-and-weyl-results-retain-their-source-type-at-cubic-spin-bridge",
      "finding:cubic-spin-cauchy-range-is-three-hardy-with-strong-domain-distinction",
      "finding:original-negative-sea-three-mode-ket-is-dual-to-required-hardy-line",
      "finding:auxiliary-theta-route-needs-no-physical-cubic-bulk-or-eta-selected-dual",
      "finding:primitive-gaussian-theta-frame-constructs-e8-pair-with-retained-spin-action",
      "finding:existing-c12-lift-induces-coherent-two-object-e8-vertex-groupoid",
      "finding:actual-winding-functional-calculus-preserves-ns-ramond-split",
      "finding:single-chiral-cycle-exchange-selects-e8-in-four-extension-class",
      "finding:affine-source-involution-pairs-full-d8-sector-spectra",
      "finding:marked-local-e8-with-unchanged-source-time-fails-euclidean-clustering",
      "finding:original-parity-audit-does-not-supply-local-spinor-gauging",
      "finding:hypergraph-local-update-repaired-with-full-family-clock",
      "finding:petersen-polar-step-has-exact-double-cover-family-intertwiner",
      "finding:petersen-source-current-and-hodge-candidates-fail-physical-identification",
      "finding:a7-clock-direction-identifies-neutral-e7-zero-root",
      "finding:oriented-sewing-cocycle-distinguishes-continuation-and-reversal",
      "finding:calderon-single-face-compression-is-not-perron-projector",
      "finding:hodge-born-magnitudes-and-filtered-petersen-jump-match",
      "finding:five-complement-amplitudes-form-full-petersen-quantum-channel",
      "finding:original-twenty-events-factor-through-hodge-and-conditional-phase-record",
      "finding:actual-e8-mixed-current-response-is-full-clifford-channel",
      "finding:carrier-su5-centralizer-gives-direct-e8-petersen-response",
      "finding:su5-covariance-and-petersen-support-fix-complete-quantum-response",
      "finding:neutral-pair-witness-separates-old-events-and-hodge-response",
      "finding:original-degree-clock-is-not-carrier-slot-cycle-at-fixed-source",
      "finding:full-mark-transport-and-isospectrality-do-not-select-hodge-channel",
      "finding:original-pair-source-fixes-state-but-not-selected-epsilon-process",
      "finding:single-positive-process-saturation-selects-full-hodge-channel",
      "finding:original-five-slot-wedge-is-exact-e8-hodge-coefficient",
      "finding:native-pauli-f20-metric-and-quarter-bridge",
      "finding:two-actual-coherent-events-herald-hodge-at-one-over-twenty-five",
      "finding:original-color-center-removes-only-extra-h8-ground-state",
      "finding:marked-symplectic-functor-fixes-native-hodge-kraus-space",
      "finding:f20-record-repair-removes-chart-induced-three-plus-two",
      "finding:theta-scalar-square-lift-is-distinct-from-symplectic-f20-lift",
      "finding:full-admissibility-vacuum-test-needs-source-image-and-transfer-reduction",
      "finding:canonical-native-spin5-hodge-is-already-one-source-record-subchannel",
      "finding:native-hodge-record-selector-is-image-projector-not-kraus-kernel",
      "finding:irreducible-f20-four-cannot-host-original-independent-native-family-clock",
      "finding:native-hodge-is-exact-collective-five-current-action",
      "finding:six-ovoid-clock-bundle-replaces-unnecessary-fixed-form-condition",
      "finding:six-hodge-responses-have-exact-full-current-barycenter",
      "finding:coxeter-seventh-power-has-complete-thirty-e8-lattice-lifts",
      "finding:spin5-hodge-five-is-plucker-kernel-not-original-quartic-hessian",
      "finding:ising-complement-current-mechanism-needs-typed-normal-field-map",
      "finding:neutral-native-a3-hodge-does-not-identify-physical-p2-carrier",
      "finding:compact-ope-coevaluation-normalizes-given-charged-hodge-line",
      "finding:old-ward-root-generates-exact-charged-fifty-module",
      "finding:tp-charged-bracket-support-forces-entire-carrier-marginal",
      "finding:original-ward-and-p2-grading-select-smaller-creation-thirty",
      "finding:charged-source-quarter-recovers-inverse-internal-glue",
      "finding:state-independent-creation-branch-removes-whole-process-hodge-demand",
      "finding:given-source-current-constructs-half-kink-spinor-transport",
      "finding:neutral-ward-response-needs-no-spinor-observable-membership",
      "finding:actual-source-sector-transport-distinguishes-identical-hodge-marginals",
      "finding:ordinary-weyl-type-obstruction-to-single-chiral-os-anti-anti-reflection",
      "finding:given-two-face-source-derives-opposite-neutral-weyls-and-bounded-time-reflection",
      "finding:verification-recovery-preserves-local-source-chain-and-eight-gate-completeness",
      "finding:positive-spectral-reweighting-does-not-establish-original-source-rp",
      "finding:gapped-quarter-energy-cannot-be-same-borchers-translation",
      "finding:finite-ko6-pairing-does-not-supply-standalone-pfaffian-source",
      "finding:full-free-projector-forces-joint-commuting-source-factorization",
      "finding:actual-neutral-current-source-has-constructive-moebius-standard-pair",
      "finding:continuous-local-current-weyls-cannot-live-in-finite-coordinate-fibers",
      "finding:actual-folded-current-ward-and-state-determine-all-neutral-words",
      "finding:mixed-vector-axial-current-is-existing-full-car-subalgebra",
      "finding:constant-product-time-stable-current-level-is-kernel-index",
      "finding:paired-quarter-source-candidate-needs-original-filter-and-oriented-conjugation",
      "finding:graded-boundary-clifford-typing-transports-full-scalar-and-finite-coupled-source",
      "finding:actual-charge-j-transport-retains-full-dirac-and-distinct-pairings",
      "finding:fixed-graded-chart-transports-normal-dependent-source-and-given-mixed-moments",
      "finding:neutral-many-body-source-time-is-not-finite-summable-geometric-dirac",
      "finding:actual-product-reflection-and-same-time-have-full-positive-clifford-source-intertwiner",
      "finding:actual-v252-positive-grade-has-global-p2-wedge-dictionary",
      "finding:old-quarter-full-source-embeds-locally-exactly-on-decoupled-channel",
      "finding:petersen-population-full-cptp-p5-range-zero-to-ten",
      "finding:marked-sm-phase-p5-exact-range-and-response-collision",
      "finding:shared-epsilon-tp-process-fixes-hodge-marginal-even-for-correlated-inputs"
    ],
    "eligibleFindingIds": [
      "finding:future-equivalence-minimizes-declared-answer-process",
      "finding:positive-word-kernel-reconstructs-cyclic-gns",
      "finding:operational-seed-is-input",
      "finding:native-source-common-time-thermal-family",
      "finding:native-grade-carry-and-memory",
      "finding:even-seam-global-operator-factorization",
      "finding:source-generated-marking-classification",
      "finding:source-marking-field-realization-boundary",
      "finding:minimal-defect-uniqueness-not-proved",
      "finding:conditional-seam-field-state-time-reconstruction",
      "finding:continuous-hardy-seam-classification",
      "finding:same-source-seam-energy-selection",
      "finding:master-functional-presupposes-source-selection",
      "finding:geometric-deck-not-calderon-polarization",
      "finding:positive-compression-erases-carrier-minus-sector",
      "finding:higgs-determinant-does-not-discharge-carrier-map",
      "finding:primitive-invariant-does-not-select-rank-three",
      "finding:fixed-background-source-does-not-fix-mixed-vertex",
      "finding:local-quarter-inner-variation-is-connection",
      "finding:calderon-enlargement-does-not-select-higgs",
      "finding:finite-sm-higgs-variation-retains-seed",
      "finding:operator-skeleton-does-not-identify-insertion-algebra",
      "finding:internal-even-car-hodge-is-not-coordinate-algebra",
      "finding:minimum-rank-unique-primitive-channel-does-not-select-32",
      "finding:joint-selfconsistency-presupposes-minimal-algebra-fiber-selection",
      "finding:process-composition-and-cyclicity-do-not-select-algebra",
      "finding:quartic-shape-selects-32-only-with-source-sign",
      "finding:native-symmetric-quartic-erasure-common-object",
      "finding:joint-native-algebra-normalizer-and-response-orbit",
      "finding:native-galois-intertwiner-unique-logical-marked-orbit",
      "finding:native-reflection-full-four-five-generators-rate-injectivity",
      "finding:complete-logical-readout-native-covariant-residual-classification",
      "finding:original-polar-mark-clock-centralizer-orbit",
      "finding:actual-quartic-code-product-stabilizer-quarter-exclusion",
      "finding:original-decoder-versus-native-quarter-polynomial-erasure-test",
      "finding:original-decoder-product-time-and-all-unitary-erasure-lifts",
      "finding:native-clock-compatible-polar-fibre-and-canonical-distance",
      "finding:native-moment-parent-and-product-path-selection-boundary",
      "finding:original-e8-current-response-and-full-quarter-lifts",
      "finding:literal-charged-current-register-dictionary-fails-quarter",
      "finding:native-cartan-code-inner-e8-stabilizer-excludes-marked-quarter",
      "finding:native-code-neutral-weight-two-action-and-global-clock-exclusion",
      "finding:original-neutral-root-mixed-response-derived",
      "finding:actual-mixed-ordered-return-on-native-quartic-code",
      "finding:mixed-return-spectrum-does-not-select-original-polar-mark",
      "finding:actual-mixed-return-fails-native-trimer-erasure-lift",
      "finding:strict-trace-gap-excludes-positive-mixed-return-reweighting",
      "finding:original-source-erasure-ward-terms-derived",
      "finding:uncompressed-original-mixed-return-does-not-preserve-code",
      "finding:original-root-response-sum-also-leaks-from-code",
      "finding:original-full-source-return-exterior-feedback-derived",
      "finding:original-return-requires-memory-in-every-code-direction",
      "finding:native-conformal-primary-multiplicity-source-factorization",
      "finding:canonical-conformal-erasure-forces-primary-twenty-carry",
      "finding:all-canonical-stress-histories-scalar-on-native-code",
      "finding:original-local-current-ope-forces-erasure-and-carry",
      "finding:all-real-local-cartan-profiles-erasure-third-bound",
      "finding:closed-cartan-weyl-histories-only-logical-phase",
      "finding:full-existing-local-e8-hull-of-native-code",
      "finding:full-local-quartic-wick-ladder-and-unprojected-native-actions",
      "finding:unmarked-native-w6-simplex-from-full-quartic-ope",
      "finding:full-e8-current-products-fix-stress-and-exterior-ward-cancellation",
      "finding:full-source-simplex-does-not-select-original-polar-mark",
      "finding:original-full-field-event-gram-reconstructs-relative-polar-mark",
      "finding:raw-event-state-positivity-does-not-execute-polar-normalization",
      "finding:determinant-cover-not-literal-full-cartan-lattice-event",
      "finding:clock-free-quartic-product-spectral-sectors-and-native-five",
      "finding:six-complex-native-simplex-rays-from-holomorphic-product-quadrics",
      "finding:intrinsic-relative-response-covariance-and-state-boundary",
      "finding:intrinsic-event-composition-retains-shared-source-and-correlation-data",
      "finding:clock-free-event-geometry-fixes-clock-class-and-mark-orbit-not-member",
      "finding:original-charged-seam-character-recovers-global-native-event-pair",
      "finding:internal-seam-vacuum-and-original-neutral-tensor-do-not-select-flag-root",
      "finding:actual-quarter-fold-shift-removes-native-symmetry-obstruction-without-flag-selector",
      "finding:fold-shift-equivariance-needs-actual-source-response-map-not-another-symmetry-no-go",
      "finding:original-fold-covariance-current-counterterm-and-quartic-twisted-transport",
      "finding:fold-state-full-quartic-kernel-and-erasure-protected-low-order-response",
      "finding:original-fold-quartic-mean-and-spectral-responses-do-not-match-sixty-flags",
      "finding:original-winding-full-state-response-removes-vacuum-rank-two-shortcut",
      "finding:original-source-global-time-winding-history-factorization",
      "finding:original-local-density-folds-to-correlated-root-current-and-erasure-response",
      "finding:original-current-all-mode-energy-and-geometric-stress-bridge",
      "finding:original-current-geometric-response-pair-has-no-common-polar-sector",
      "finding:original-current-ordered-hall-kernel-and-actual-folded-field-products",
      "finding:original-seam-grades-full-density-geometric-products-and-energy-response",
      "finding:original-seam-complete-density-grading-not-sharp-32-input-sectors",
      "finding:original-density-all-profile-mean-work-scalar-on-native-packet",
      "finding:original-p1-gap-normalized-action-covector-and-scalar-holonomy-class",
      "finding:original-p1-gap-car-component-not-native-sharp-32-event",
      "finding:complete-positive-scalar-first-order-circle-source-normal-form",
      "finding:fixed-source-time-seam-ward-rigidifies-scalar-principal-symbol-with-carry",
      "finding:unit-spectral-flow-gap-and-four-marks-do-not-imply-fixed-source-ward",
      "finding:four-native-starts-have-distinct-original-fold-state-correlations",
      "finding:four-starts-full-joint-kernel-and-charged-covariant-transport",
      "finding:four-native-polar-marks-not-fixed-source-state-gauge-equivalent",
      "finding:original-pointer-template-requires-source-context-identification",
      "finding:four-polar-context-algebra-and-source-crosscoupling",
      "finding:every-native-quartic-has-all-eight-original-car-supports",
      "finding:bounded-proper-local-source-operation-cannot-exactly-prepare-positive-eigenpacket",
      "finding:full-original-erasure20-and-joint25-have-all-eight-source-supports",
      "finding:two-face-collar-fixes-face-current-phase-and-distinguishes-relative-endpoint-lift",
      "finding:full-dirac-cauchy-vector-cocycle-cancels-and-endpoint-domain-distinction",
      "finding:unchanged-qwz-full-finite-energy-spectral-exhaustion",
      "finding:uniform-z4-d8-locality-and-both-spinor-chirality-time-census",
      "finding:odd-spinor-global-refielding-and-unit-source-winding",
      "finding:sharp-half-flux-register-quotient-and-local-normality-exclusion",
      "finding:all-word-register-charged-local-field-limit-obstruction",
      "finding:exact-parity-gauss-locality-completion-of-register-source",
      "finding:cover-half-flux-not-fixed-charge-eightfold-spinor-twist",
      "finding:original-neutral-eightpreimage-parity-pair-has-uniform-cut-action",
      "finding:complete-dirac-vector-phase-pair-has-zero-spinor-Ward-charge",
      "finding:smooth-unitary-endpoint-removal-diverges-at-fixed-source-time",
      "finding:continuous-original-time-nonunitary-limits-stay-in-old-source-folium",
      "finding:common-register-vacuum-reference-uniform-and-local-grade-reduction",
      "finding:actual-heat-filtered-neutral-pair-full-state-time-and-separate-chiral-limit",
      "finding:holomorphic-kms-extension-assumes-charged-net-and-not-microscopic-generator",
      "finding:fixed-interval-rotation-modular-chain-fails-local-state-takesaki-repairs",
      "finding:source-winding-cartan-four-word-and-two-edge-kms-gso-discriminator",
      "finding:p1-parity-gauss-net-selection-undecidable-without-original-localization-map",
      "finding:original-minimality-does-not-reconstruct-operational-source",
      "finding:generic-first-order-relative-bound-zero-counterexample-and-correct-domain-repair",
      "finding:joint-discrete-data-vs-connection",
      "finding:parallelity-excludes-2plus2-only-conditionally",
      "finding:sign-projector-does-not-determine-time-magnitude",
      "finding:full-faithful-kms-or-transfer-recovers-magnitudes",
      "finding:full-two-face-collar-recovers-signed-operator",
      "finding:two-face-graph-state-time-same-datum",
      "finding:shared-register-gkls-embedding-exact-q-interval",
      "finding:marked-m4-channel-all-powers-homogeneous-gksl-excluded",
      "finding:flavor-ratio-closed-absolute-anchor-open",
      "finding:alpha-equation-is-fixed-input-closure",
      "finding:alpha-global-positive-root-unique",
      "finding:marked-e8-sector-grade-response-realizes-five-quarters",
      "finding:alpha-exact-coefficients-and-root-retain-specific-physical-ward-obligation",
      "finding:alpha-archive-term-provenance-and-master-functional-scope",
      "finding:matter-observations-do-not-fix-geometric-response",
      "finding:four-clock-fields-complete-quadratic-response",
      "finding:actual-j-covariant-source-transport",
      "finding:fixed-source-affine-j-obstruction",
      "finding:neutral-vacuum-allword-twist-blindness",
      "finding:eight-charged-responses-reconstruct-twist",
      "finding:native-invariant-response-not-fixed-chiral-twist",
      "finding:native-opposite-charge-response-symmetry",
      "finding:native-chemical-mark-visible-in-charged-corners",
      "finding:fixed-hopping-quarter-phases-retain-antiunitary-discriminator",
      "finding:constructive-phase-class-and-full-winding",
      "finding:constructive-positive-state-extension",
      "finding:constructive-current-group-selects-positive-p-lift",
      "finding:constructive-state-algebra-resolution-and-source-test",
      "finding:constructive-all-phases-fail-unchanged-source",
      "finding:partner-carrier-charges-and-state-types",
      "finding:partner-fixed-quarter-time-splits-sm-multiplet",
      "finding:partner-original-carrier-cross-sheet-conjugation",
      "finding:partner-chiral-quarter-conjugation-state-time",
      "finding:joint-bilinear-hermitian-correction",
      "finding:joint-quarter-transports-marking",
      "finding:joint-fixed-time-marking-repair-excluded",
      "finding:joint-spinor-not-source-coefficient",
      "finding:typed-operator-search-calibrated",
      "finding:positive-kernel-construction-with-exact-discriminator",
      "finding:source-typed-kernel-search-controls",
      "finding:collar-car-projection-multifield-ray",
      "finding:known-quarter-source-full-multifield-control",
      "finding:source-reference-amplitude-retained",
      "finding:joint-fermionic-source-schur-functional",
      "finding:quarter-circle-absolute-zeta-green-eta",
      "finding:quarter-sea-absolute-thermal-central-phase",
      "finding:original-operational-seed-and-given-green-pairing",
      "finding:order-one-infinitesimal-bound-inference-fails",
      "finding:joint-ward-two-field-data-miss-fixed-four-field-response",
      "finding:complete-car-kernel-fixes-nongaussian-link-functional",
      "finding:source-covariance-jet-and-mixed-face-interface",
      "finding:e8-grade-two-explicit-spinor-current-action",
      "finding:intrinsic-source-jet-removes-pure-field-frame-motion",
      "finding:unchanged-ns-vector-generators-cannot-create-d5-spinor-fields",
      "finding:two-face-noncommuting-geometric-response-calculus",
      "finding:raw-face-mask-fails-source-kernel-null-descent",
      "finding:action-response-sewing-does-not-select-lf-intervention",
      "finding:faithful-gns-completion-closes-automated-source-query-feedback",
      "finding:whole-quarter-channels-identified-by-exact-internal-kernel",
      "finding:actual-joint-insertion-hull-resolves-eight-coefficient-directions",
      "finding:e8-bare-trilinear-and-single-anchor-weyl-class-insufficient",
      "finding:complete-family-four-cycle-class-separates-native-clock-from-sm-preservation",
      "finding:native-clock-word-transpose-has-static-rank-four-counterexample",
      "finding:whole-constant-coupled-source-is-implementable-despite-isolated-insertion-domain",
      "finding:fixed-quarter-jloc-pfaffian-closure-fails-local-skew-test",
      "finding:same-collar-has-canonical-selfdual-kinetic-pairing-and-charged-determinant",
      "finding:quarter-cover-pairing-does-not-supply-local-normal-or-select-scale-measure",
      "finding:fixed-aps-source-has-whole-inverse-and-finite-boundary-correction",
      "finding:mass-only-deck-relative-heat-action-fails-free-infinite-trace",
      "finding:finite-antiperiodic-scalar-wall-has-no-exact-massless-normal-mode",
      "finding:spectral-heat-covariance-positivity-does-not-imply-time-rp",
      "finding:quaternionic-sm-mass-winding-is-not-normal-determinant-clutch",
      "finding:bounded-relative-trace-obstruction-and-whole-genuine-pair-heat-reader",
      "finding:normal-clutch-orbit-index-equivariantly-recovers-given-weak-module",
      "finding:normal-orbit-spin-family-index-and-signed-charge-balance",
      "finding:fixed-flavor-cartan-sign-cover-and-root-lift-parity",
      "finding:original-family-vector-action-is-not-cartan-rotation-or-spinlift-selection",
      "finding:actual-genus-two-sign-family-closes-common-parallel-k-connection",
      "finding:actual-q-quarter-extends-common-k-connection-and-real-coefficient-lifts",
      "finding:real-source-reflection-is-not-linear-duality-or-compact-clock-inversion",
      "finding:source-selected-spin-and-q-signs-have-three-explicit-global-kernels",
      "finding:source-s0-szego-point-legs-generate-q-kernels-with-typed-products",
      "finding:actual-spin-kernel-determinant-has-inverse-q-norm-character",
      "finding:closed-source-flavor-spin-has-functorial-pfaffline-with-charged-type-limit",
      "finding:original-relative-seam-has-explicit-whole-graph-fredholm-mismatch",
      "finding:whole-constant-coupled-clutch-defects-are-exact-internal-kernels",
      "finding:old-complex-scalar-clutch-separates-c0-and-product-reflection",
      "finding:scalar-defect-circle-character-does-not-select-projector-holonomy",
      "finding:source-spin-is-canonical-ramified-base-spin-pullback",
      "finding:original-v117-has-explicit-algebraic-q-cover-fundamental-frame",
      "finding:double-circle-reflection-does-not-fix-entire-cubic-cable-seam",
      "finding:actual-fuchs-real-map-transports-whole-given-finite-source-state-and-time",
      "finding:actual-e0-compensator-extends-whole-k-e8-with-transported-quarter",
      "finding:actual-v252-entire-real-basis-has-signed-p2-dual-dictionary",
      "finding:chiral-fuchs-source-transport-has-distinct-majorana-phase-and-torus-absorption",
      "finding:native-theta-composition-retains-conjugate-source-and-parity-square",
      "finding:single-graph-range-isometry-does-not-give-local-whole-seed-map",
      "finding:hard-holonomy-selfduality-leg-and-finite-character-phase-separation",
      "finding:original-z6-positive-structural-kernel-does-not-supply-oriented-phase-observable",
      "finding:original-genus-two-hexagonal-rotation-has-concrete-rho-hodge-line",
      "finding:original-mixed-spin-products-realize-positive-u6-cycle",
      "finding:actual-spin-quadratics-reach-original-hexagonal-hodge-line",
      "finding:spin-pair-car-transport-retains-parity-and-whole-source-types",
      "finding:real-total-trace-does-not-exclude-coherent-phase-reader",
      "finding:geometric-six-spin-clock-differs-from-marked-native-clock",
      "finding:whole-q-to-p-pullback-transports-given-source-state-and-time",
      "finding:actual-fixpoint-evaluation-keeps-coefficient-signs-and-source-type",
      "finding:original-source-naturality-needs-no-second-p-image-background",
      "finding:canonical-spin2-lift-carries-whole-annulus-source-with-reversed-time",
      "finding:geometric-spin-collar-map-closes-weighted-image-domain-with-total-deck-action",
      "finding:claude33-annulus-map-is-valid-distinct-type-not-unique-geometric-spin-selection",
      "finding:original-mixed-spin-products-already-have-complete-dual-hom-coefficients",
      "finding:whole-f5-graph-supplies-canonical-reverse-qp-hom-kernel",
      "finding:actual-spin-product-weight-has-canonical-trace-class-form-pairing",
      "finding:fully-descended-covariant-reverse-kernel-removes-rho-form-channel",
      "finding:original-a4-linear-quarter-does-not-supply-proposed-scalar-mu3-repair",
      "finding:offdiagonal-hom-matches-a4-family-six-but-not-two-triplet-e8-product",
      "finding:compact-e8-adjoint-completes-family-six-with-carrier-rank-retained",
      "finding:three-family-cx-recursion-closes-exact-d8-with-minimal-original-escape",
      "finding:original-weak-current-coefficient-has-explicit-finite-oneform-consumer",
      "finding:original-d5-zero-mode-closes-triplet-carrier-and-fixed-glue-is-minimal-escape",
      "finding:mixed-spin-full-deck-has-no-rank-three-projector",
      "finding:actual-reverse-hom-diagonal-supplies-global-q-hodge-response",
      "finding:bare-c3-kernel-diagonal-yukawas-have-common-real-frame",
      "finding:original-primitive-action-is-present-highpass-is-not-finite-selector",
      "finding:atlas-detail-retrieval-recovers-existing-d4-solution",
      "finding:original-poisson-normal-trace-fixes-whole-calderon-graph",
      "finding:induced-original-hardy-defect-gives-finite-whole-source-reader",
      "finding:original-weak-quartic-action-has-nonzero-nonlinear-response",
      "finding:all-common-leading-distance-clutches-fail-original-flavor-lengths",
      "finding:actual-hardy-response-has-weak-clifford-odd-and-ko-real-completion",
      "finding:two-entry-hardy-adjoint-hierarchy-fails-full-ko-and-su2",
      "finding:actual-fuchs-halfpath-has-canonical-g-metric-c6-kernel-map",
      "finding:original-real-delta-jet-reads-same-marked-hardy-open-word",
      "finding:four-actual-fuchs-spine-edges-recover-marked-meridians-and-nonzero-open-word",
      "finding:native-adjoint-quarter-clock-cannot-compensate-hardypair-odd-character",
      "finding:whole-cubic-gram-differential-retains-hardy-phase-and-cross-resonance",
      "finding:complete-original-mu01-hardy-reader-has-two-calibrated-entries",
      "finding:whole-six-original-hardy-inputs-generate-full-family-coefficient-algebra",
      "finding:actual-whole-hardy-alphabet-has-finite-particle-hole-wick-block",
      "finding:bare-unsmeared-mu-crossface-quadratic-fails-vacuum-domain",
      "finding:linear-spectral-centralization-does-not-replace-full-marked-hardy-source",
      "finding:original-admissibility-projector-is-explicit-penalties-do-not-read-retained-amplitudes",
      "finding:actual-original-inner-fluctuation-has-whole-noncommuting-two-face-response",
      "finding:fiber-only-primitive-split-and-frozen-gap-miss-original-action",
      "finding:oriented-resolvent-weights-do-not-identify-original-reflection-producer",
      "finding:current-odd-carrier-reader-is-lossless-on-whole-coupled-source",
      "finding:ap-spectral-formulas-do-not-force-normal-domain-or-complete-source",
      "finding:actual-weak-connection-transmits-family-orientation-to-both-faces",
      "finding:original-higgs-variation-is-not-independent-yukawa-stationarity",
      "finding:actual-weak-cap-converges-on-whole-quarter-circle",
      "finding:canonical-clifford-double-derives-normal-ap-and-typed-deck",
      "finding:universal-graded-source-reducing-images-do-not-fix-carrier-frame",
      "finding:whole-two-face-compression-projector-defect-detects-hidden-source-loss",
      "finding:actual-v252-quark-lepton-split-prevents-weak-grading-uniqueness-inference",
      "finding:actual-graded-hypercharge-trace-rejects-quark-lepton-sign-intertwiner",
      "finding:separable-normal-real-maps-compose-to-given-deck-and-quarter-image",
      "finding:actual-gauge-real-frames-leave-carrier-phase-as-coordinate-freedom",
      "finding:original-higgs-family-has-one-complex-kk-class-but-different-whole-sources",
      "finding:signed-one-particle-modular-weight-is-not-the-positive-even-heat-operator",
      "finding:three-moments-select-one-exponential-exactly-in-positive-laplace-class",
      "finding:existing-ap-positive-source-factor-has-oriented-poisson-heat-bridge",
      "finding:actual-v252-entire-compatibility-family-has-generic31-dimensional-quotient",
      "finding:full-memory-schur-reconstruction-preserves-every-ordered-word-under-nesting",
      "finding:original-return-and-native-j-have-minimal68-dimensional-all-word-memory",
      "finding:original-full-source-regulator-flow-retains-normalized-kernel-and-vacuum-factor",
      "finding:full-reflected-quadratic-regulator-cone-preserves-every-rp-source",
      "finding:original-shared-higgs-source-induces-fixed-yukawa-mixed-matter-words",
      "finding:local-even-yukawa-sewing-preserves-given-joint-graded-rp",
      "finding:whole-original-constant-higgs-orbit-has-reduced-majorana-spectral-response",
      "finding:archived-geometric-higgs-residue-bulk-needs-coefficient-identification-repair",
      "finding:original-local-higgs-residue-linear-coefficients-fail-same-trace-heat-ratio",
      "finding:whole-original-higgs-moments-fix-conditional-local-kinetic-mass-quartic",
      "finding:whole-variable-original-higgs-bound-reference-response-and-forced-clifford-factor",
      "finding:original-symmetric-scale-map-fails-simultaneous-higgs-einstein-repair",
      "finding:whole-original-higgs-inner-scale-and-affine-full-source-transport",
      "finding:normal-higgs-mode-type-and-exact-kernel-metric-require-actual-relative-operator-map",
      "finding:original-superconnection-grades-and-weighted-heat-retain-higgs-kinetic-excursions",
      "finding:whole-normal-reduction-preserves-relative-spectral-reader-and-higgs-jets",
      "finding:original-admissibility-own-spectral-reader-has-no-free-low-mode-residue-repair",
      "finding:whole-total-chiral-positive-spectrum-retains-higgs-kinetics-and-common-half-trace",
      "finding:whole-scale-energy-response-uniquely-reconstructs-normal-profile",
      "finding:original-relative-scale-flow-separates-residue-finite-part-and-local-anomaly",
      "finding:original-whole-green-produces-energy-response-with-correct-orthogonal-trace",
      "finding:whole-mixed-scale-source-reads-bosonic-insertion-before-higgs-average",
      "finding:current-original-spectral-action-triplet-fails-positive-laplace-class-unless-leading-relative-term-cancels",
      "finding:correct-current-argument-weights-link-whole-higgs-threshold-and-same-einstein-response",
      "finding:whole-original-spinor-curvature-a4-has-no-independent-scalaron-r-squared",
      "finding:whole-original-entropic-curvature-corrects-max-symmetric-scalar-scale",
      "finding:given-whole-higgs-source-fixes-radial-curvature-susceptibility-without-extra-scalar",
      "finding:actual-original-four-ghost-complex-preserves-base-selector-with-full-cohomology-and-distinct-traces",
      "finding:whole-original-admissibility-energy-source-recovers-constraints-but-cannot-replace-geometric-operator",
      "finding:original-moving-normal-kernel-fixes-induced-connection-and-full-scale-covariance",
      "finding:fixed-original-o1-source-rejects-nonconstant-normal-multiplier-and-fixes-placement-test",
      "finding:original-master-variation-keeps-spin-weight-and-active-source-route-typed",
      "finding:shared-original-higgs-mixture-has-invariant-color-fourpoint-wick-defect",
      "finding:bounded-fixed-rank-spectral-action-does-not-normalize-noncompact-higgs-volume",
      "finding:whole-original-quarter-source-bounds-higgs-weight-uniformly-through-joint-mode-ir-limits",
      "finding:original-o1-covariance-leaves-full-radial-higgs-law-and-exact-fourier-source-readout",
      "finding:whole-process-reconstruction-priority-and-original-locality-geometry-input-audit",
      "finding:actual-disjoint-local-weyls-acquire-spurious-commutator-under-bare-native-compression",
      "finding:actual-quarter-current-state-excludes-nontrivial-geometric-local-point-time",
      "finding:actual-elliptic-interval-car-source-produces-local-modular-generator",
      "finding:actual-proper-two-face-regions-do-not-contain-pure-local-source-double",
      "finding:actual-elliptic-whole-environment-fixes-regional-car-partner",
      "finding:actual-original-color-triplet-has-spectator-dependent-modular-field-law",
      "finding:actual-color-marginal-has-complete-finite-source-reader-and-mixture-criterion",
      "finding:two-and-four-field-words-do-not-complete-original-color-triplet-source",
      "finding:whole-original-quarter-occupation-tower-uniquely-reconstructs-common-radial-higgs-law",
      "finding:finite-original-tower-reader-and-existing-weak-symmetry-fix-shared-matter-responses",
      "finding:unchanged-free-quarter-source-cannot-be-nontrivial-integrated-higgs-mixture",
      "finding:actual-e8-record-doily-action-is-signed-p-plus-compatible-point-with-typed-sym2-lift",
      "finding:actual-e8-record-words-distinguish-order-and-retain-phase-beyond-fifteen-labels",
      "finding:literal-doily-incidence-output-and-strict-causal-order-on-fifteen-source-labels-excluded",
      "finding:actual-doily-context-record-characters-have-ranks-four-two-two-two",
      "finding:canonical-standard-split-localizes-original-source-record-words-with-full-descendant-companion",
      "finding:unchanged-global-sixty-event-source-fails-local-exterior-current-test",
      "finding:canonical-source-localization-changes-exterior-state-and-does-not-select-event-origin",
      "finding:original-elliptic-fixed-holonomy-insertion-gives-local-operation-on-unchanged-source",
      "finding:original-quarter-pair-fixes-square-charge-law-half-response-and-real-quillen-norm",
      "finding:actual-elliptic-source-determines-all-path-normalized-causal-current-response",
      "finding:original-higgs-quark-source-has-exact-whole-su3-counting-response",
      "finding:single-original-color-parity-separates-free-quillen-from-positive-higgs-integrated-source",
      "finding:whole-original-quarter-graph-color-response-is-relative-and-diagonal-singlet",
      "finding:original-center-selector-requires-typed-state-field-and-face-transport",
      "finding:original-color-center-fixes-whole-e6-a2-source-and-charged-ward-word-action",
      "finding:original-color-neutral-lattice-class-cannot-supply-odd-fields-by-inherited-superlattice-extension",
      "finding:original-color-center-extends-to-folded-ns-but-full-current-action-needs-spin-sector-transport",
      "finding:actual-color-spin-car-mixed-word-has-complete-thermal-source-and-cyclic-sewing",
      "finding:exact-original-ns-ramond-heat-equality-does-not-identify-fixed-color-restricted-source",
      "finding:original-scalar-cover-profile-cannot-implement-fixed-spin-cartan-loop",
      "finding:actual-spin-cartan-composite-source-retains-multi-arc-support-and-fails-additive-even-locality",
      "finding:actual-color-selector-admits-exactly-eighth-periodic-original-density-variations",
      "finding:actual-color-is-internal-exactly-on-eightfold-coordinate-source-regions",
      "finding:original-whole-two-face-multiplier-reduction-is-only-constant-residue-diagonal",
      "finding:original-two-face-local-fock-implementation-and-exact-source-variation-differ-from-reduction",
      "finding:whole-undecorated-transfer-does-not-determine-original-marked-field-source",
      "finding:volume-wise-injective-positive-transfer-does-not-prove-uniform-parent-gap",
      "finding:original-c6-half-power-has-exact-entire-positive-ordered-word-expansion",
      "finding:original-c6-whole-word-sewing-retains-source-amplitude-and-controlled-infinite-tail",
      "finding:historical-parent-and-actual-field-time-bridges-must-be-reused-before-new-source-models",
      "finding:actual-native-v990-full-timed-words-recover-kubo-response-distinct-from-instantaneous-variance",
      "finding:actual-original-higgs-force-has-complete-ordered-fermionic-determinant-source",
      "finding:finite-covariant-higgs-composites-have-no-nontrivial-positive-instantaneous-classical-law",
      "finding:actual-original-collective-mean-has-exact-ball-domain-and-relative-entropy-action",
      "finding:original-uniform-allmode-force-clt-is-nonzero-only-before-fixed-time-smearing",
      "finding:original-unscaled-time-smeared-force-has-selfadjoint-whole-ordered-source",
      "finding:original-local-higgs-force-has-whole-centered-hilbert-schmidt-determinant-source",
      "finding:given-original-circle-force-source-generates-an-exact-causal-local-net",
      "finding:literal-archived-global-rg-kernel-does-not-preserve-finite-measures",
      "finding:native-higgs-marginal-retains-induced-vertex-and-noether-response",
      "finding:isospectral-noether-orbit-does-not-supply-original-metric-higgs-variation",
      "finding:metric-dtn-form-bound-needs-transfer-comparison-for-gap",
      "finding:conditional-higgs-pair-winding-protection-in-common-circle-carrier",
      "finding:natural-five-clock-carrier-excludes-invariant-three-transfer-compression",
      "finding:shared-scalar-shift-whole-retained-higgs-sector-has-index-minus-eighteen",
      "finding:whole-coupled-quark-circle-common-shift-cancels-relative-winding",
      "finding:one-sided-shift-full-higgs-pair-is-hilbert-schmidt-not-trace-class",
      "finding:selective-coupled-higgs-reader-has-well-defined-det2",
      "finding:v311-population-reader-does-not-identify-quantum-channel",
      "finding:declared-charged-pair-relation-leaves-five-word-ambiguity",
      "finding:full-creator-relations-close-fixed-free-two-mode-ambiguity",
      "finding:comoving-diagnostic-clock-exact-response-but-nonlocal-mode-transport",
      "finding:free-quarter-two-face-response-strip-with-relative-zero-mode-defect",
      "finding:e8-hodge-event-perfectly-distinguishable-from-every-unitary",
      "finding:common-five-slot-transfer-preserves-e8-bracket-not-automatic-tp",
      "finding:comoving-clock-dynamic-higgs-connection-and-charged-flow",
      "finding:r064-sixty-record-channel-exact-distance-and-continuous-record-parent",
      "finding:two-syntheses-shared-source-priorities-and-typed-limits",
      "finding:weighted-index-trace-and-selfdual-theta-require-original-selection",
      "finding:ordered-quarter-fourfield-e8-record-and-charge-filters",
      "finding:r091-recovered-native-hodge-sewing-and-full96-quarter-complement",
      "finding:r092-fold-ward-target-neutral-source-and-smaller-creation-branch",
      "finding:r093-original-kink-time-response-distinguishes-quarter-states",
      "finding:r093-ordered-current-kink-and-eight-arc-locality",
      "finding:r094-original-five-excitations-distinct-from-clustering-vacuum",
      "finding:r096-joint-higgs-twoface-source-constraint",
      "finding:reverse-bootstrap-retained-operator-scope",
      "finding:e8-current-dictionary-all-separated-responses",
      "finding:charged-scaling-microscopic-map-still-unspecified",
      "finding:original-slater-proxy-fitted-bound-fails-actual-norm",
      "finding:periodic-microscopic-zero-modes-distinguish-source-state",
      "finding:replicated-deterministic-consistency-iff-influence-dag",
      "finding:three-binary-processes-two-copy-reversible-classification",
      "finding:independent-poset-information-composition-consequences",
      "finding:reset-diagnosis-not-global-tfpt-underdetermination",
      "finding:three-inequivalent-repeatable-observation-countermodels",
      "finding:repeatable-reports-do-not-imply-full-state-stability",
      "finding:finite-sequential-data-admit-classical-history-realization",
      "finding:pascal-carrier-rank-unique-unbounded-source-proof",
      "finding:finite-carrier-glue-elements-orbits-and-halfway",
      "finding:marked-seam-category-and-character-valued-full-lift-factor",
      "finding:quarter-clock-sign-lift-complete-four-puncture-classification",
      "finding:quarter-lift-central-quotient-class-and-all-subgroup-restrictions",
      "finding:native-clock-reflection-full-factor-cocycle-trivialization",
      "finding:native-quartic-exterior-local-cohomology-distinguishes-clock-monodromies",
      "finding:orbifold-lift-criterion-and-determinant-eight-divisibility",
      "finding:original-g31-reflection-census-and-all-integral-cohomology",
      "finding:all62-original-clock-obstructions-and-central-variants",
      "finding:actual-sign-lift-g31-group-comparison",
      "finding:complete-g31-lift-invariants-and-explicit-collisions",
      "finding:e7-quarter-pairs-literal-native-g31-reflections",
      "finding:e6-root-subsystem-quarter-clock-discriminant-obstruction",
      "finding:e-series-square-cover-family-and-quarter-descent",
      "finding:e-series-integral-glue-and-normalization-nonidentifiability",
      "finding:global-original-g31-character-lift-nonsplitting",
      "finding:all62-original-character-clock-lifts-complete",
      "finding:all17-original-character-monodromy-lifts-complete",
      "finding:e7-pair-full-lattice-field-character-torsor",
      "finding:e-series-complete-native-compiler-cut-dictionary",
      "finding:common-cover-clock-and-maximal-integral-phi12-intertwiner",
      "finding:native-four-cut-projectors-unify48-fourthirds-fivequarters",
      "finding:complete-covariant-compiler-voa-source-and-weyl-closure",
      "finding:complete-fixed-jhat-cyclic-field-lift-and-associated-clock",
      "finding:relative-native-burau-clock-hom-faithful12-transport",
      "finding:e6-vertex-nonselection-survives-full-field-clock-phases",
      "finding:marked-c4-glue-unique-local-e8-vacuum-source",
      "finding:canonical-vacuum-clock-preserves-e6-nonselection",
      "finding:marked-generator-forces-e8-without-support-multiplicity-or-minimality-input",
      "finding:native-reflection-rotation-lift-classes-and-forgetful-images",
      "finding:quarter-clock-pure-mapping-class-complete-quotient",
      "finding:e8-cut-five-orbital-incidence-algebra-complete",
      "finding:compiler-d8-frame-requires-full-d5-a3-flag",
      "finding:tetrahedron-incidence-lossless-e6-reconstruction",
      "finding:restricted-clock-lift-and-global-cover-audit-reconciled",
      "finding:double-scalar-quarter-is-marked-glue-with-two-step-index",
      "finding:scalar-quarter-two-involutive-e8-extensions",
      "finding:native-quarter-glue-inner-conjugacy-and-oriented-tetrahedron-flags",
      "finding:native-cyclic-compiler-inner-conjugacy",
      "finding:recovered-e6-closed-81-current-module-and-unique-source-cubic",
      "finding:e-series-operational-role-review-and-source-time-centralizer",
      "finding:actual-clock-doily-ovoid-kernel-is-native-quartic-five",
      "finding:glue-clock-twisted-sectors-share-marked-quadratic-module",
      "finding:complete-native-six-effect-covariant-instrument-classification",
      "finding:native-six-effect-readout-on-existing-erasure-output",
      "finding:relational-quartic-singlet-and-invariant-density-separation",
      "finding:new-clock-feedback-origin-input-audit",
      "finding:native-event-channel-cone-intersection-is-tenth-to-half",
      "finding:native-six-ovoid-observation-first-occurs-at-tensor-degree-three",
      "finding:original-quartic-erasure-and-hamming-octic-share-marked-ovoid-response",
      "finding:native-octic-field-ray-is-explicit-but-retained-clock-stress-preparation-fails",
      "finding:actual-quarter-fold-stress-has-nonzero-octic-eighth-jet",
      "finding:native-octic-is-an-original-even-car-current-composite",
      "finding:original-geometric-source-jet-and-partial-p1-seed-closure",
      "finding:full-root-ope-has-normalized-native-octic-component",
      "finding:paper-microscopic-car-composes-with-native-octic-ward",
      "finding:microscopic-neutral-octic-has-controlled-smeared-limit",
      "finding:paper-seed-source-needs-local-image-not-global-equivalence",
      "finding:full-car-positive-time-forces-source-vacuum",
      "finding:original-winding-theorem-does-not-select-seam-ray",
      "finding:original-b-u-pair-conditionally-reconstructs-whole-circle-channel",
      "finding:original-os-step-needs-full-moments-not-connected-cumulants",
      "finding:graded-field-image-precedes-physical-observable-restriction",
      "finding:older-original-circle-representative-already-realizes-ward-and-index",
      "finding:old-quarter-heat-transfer-has-controlled-rank-one-limit-and-odd-product",
      "finding:historical-em-rank-one-and-full-heat-traces-have-different-contracts",
      "finding:older-frg-functional-already-supplies-formal-full-moment-parent",
      "finding:earliest-c3-route-quantizes-level-but-needs-physical-kinetic-normalization",
      "finding:canonical-positive-circle-half-selects-the-toeplitz-line-and-sixth-order-correction",
      "finding:old-seam-shift-canonically-gives-relative-car-line-and-fredholm-factor",
      "finding:old-index-coupling-removes-heat-map-and-has-sharp-residual-ward-test",
      "finding:original-admissibility-requires-image-membership-not-positive-half-or-commutation",
      "finding:old-closure-already-assumes-car-but-does-not-map-the-geometric-circle",
      "finding:historical-uv-map-infinite-mass-exclusion-already-proved-and-recovered",
      "finding:a4-spin-lift-fixes-local-ns-and-unordered-quarter-pair",
      "finding:cover-winding-exchanges-quarter-sectors-before-its-square",
      "finding:retained-winding-preserves-local-support-and-neutral-time",
      "finding:marked-glue-square-is-the-e8-fermionization-character",
      "finding:fixed-canonical-d8-is-not-transported-source-d8",
      "finding:actual-winding-has-four-module-e8-defect-extension",
      "finding:ramond-lift-sign-is-common-intertwiner-basis-gauge",
      "finding:old-seam-relative-determinant-is-stable-in-faithful-kms-class",
      "finding:finite-circle-compression-cannot-retain-whole-thermal-winding-factor",
      "finding:schur-carrier-plus-complement-preserves-original-relative-determinant",
      "finding:v258-forward-source-gap-is-reused-not-solved-by-inversion",
      "finding:one-sided-source-and-real-em-norm-need-response-map-not-chiral-phase",
      "finding:trace-class-pair-index-fixes-relative-factor-without-primitive-ray",
      "finding:matrix-clutch-degree-connects-weak-determinant-class-without-ray-selector",
      "finding:finite-ns-ordinary-determinant-cannot-limit-to-nonzero-index",
      "finding:inner-carrier-is-infinite-spacetime-module-not-finite-circle-cut",
      "finding:normal-spin-determinant-line-exactly-transports-whole-circle-in-given-product",
      "finding:normal-index-line-source-match-is-holonomy-condition-not-vertical-degree",
      "finding:full-normal-product-polarization-change-is-not-trace-class",
      "finding:massive-transverse-zero-index-retains-positive-determinant-response",
      "finding:original-callias-doublet-count-does-not-supply-one-complex-circle-channel",
      "finding:exact-moving-kernel-reduction-forces-kato-offdiagonal-connection",
      "finding:tetrahedral-quarter-face-phase-does-not-identify-original-quarter-marked-curve",
      "finding:existing-elliptic-cover-correspondence-has-exact-c6-image-and-retained-mixed-clock",
      "finding:recovered-natural-joint-lift-and-given-connection-already-exist",
      "finding:global-split-joint-carrier-admits-parallel-native-c12-clock",
      "finding:pure-weak-clutch-clock-identification-fails-despite-quarter-matches",
      "finding:same-chern-clutch-native-compiler-does-not-select-physical-connection",
      "finding:native-gaussian-four-is-cartan-subspace-in-whole-sixtyfour",
      "finding:native-family-factor-is-not-gaussian-four-despite-symmetric-coincidences",
      "finding:primitive-class-two-sphere-obstructs-global-cartan-subspace-choice",
      "finding:even-degree-pullback-removes-cartan-topology-without-selecting-clock-cover",
      "finding:global-cartan-choice-is-optional-for-original-scalar-source-fields",
      "finding:bare-native-octic-is-not-whole-e8-zero-mode-singlet",
      "finding:explicit-primary-e8-invariant-octic-completes-same-neutral-car-source",
      "finding:completed-octic-has-exact-old-root-ope-component-and-quarter-source-value",
      "finding:elliptic-normal-spin-pullback-doubles-index-and-deck-quarter-lifts-anticommute",
      "finding:exact-separated-w8-twopoint-in-original-quarter-current-source",
      "finding:unshifted-vacuum-fails-same-w8-source-state-time-dictionary",
      "finding:local-current-phase-and-conditional-cyclic-p1-w8-image",
      "finding:all-neutral-w8-correlations-identify-the-two-original-quarter-charges",
      "finding:original-w8-response-has-infinite-same-time-spectral-support",
      "finding:older-seam-equivalence-verifications-do-not-supply-marked-local-p1-map",
      "finding:full-original-c1-stress-cycle-reconstructs-neutral-w8-response",
      "finding:folded-c8-cycle-misses-intrinsic-w8-tangent-without-obstructing-original-stress",
      "finding:native-source-first-nonuniform-moment-is-four-with-literal-five",
      "finding:native-whole-five-ground-space-needs-four-register-interaction",
      "finding:old-cover-winding-reconstructs-local-quarter-current-net",
      "finding:untransported-independent-quarter-stress-pair-is-not-current",
      "finding:current-public-paper-bytes-and-binding-source-gaps-recovered",
      "finding:winding-charge-conjugation-descends-fixed-pin-domain-but-not-full-dirac-sea",
      "finding:native-fourth-root-frame-has-signed-collective-sixty-mode-dual",
      "finding:native-root-gram-spectrum-is-not-canonical-source-energy-selection",
      "finding:identical-overlapping-native-five-codes-have-no-common-state",
      "finding:horizontal-e8-invariant-octic-cannot-isolate-native-five-ground-space",
      "finding:longitudinal-reflection-preserves-fixed-pin-domain-and-full-dirac-sea",
      "finding:single-fixed-pin-minus-green-kernel-fails-neutral-os-positivity",
      "finding:full-original-ns-orientation-cover-has-all-mode-charged-green-rp",
      "finding:cover-green-thermal-state-is-not-original-quarter-hardy-vacuum",
      "finding:declared-recursive-native-som-is-u35-equivariant",
      "finding:declared-recursive-native-som-clocks-depend-on-chosen-rates",
      "finding:existing-c2-dividing-real-forms-realize-moebius-annulus-tower",
      "finding:raw-ns-elliptic-glide-has-projective-spin-commutator-and-quarter-descent",
      "finding:canonical-elliptic-annulus-reproduces-original-quarter-boundary-tuple-and-kms",
      "finding:finite-crosscap-cauchy-state-is-not-original-quarter-vacuum-by-normal-filtering",
      "finding:native-quartic-relative-arc-charge-excludes-additive-density-stress-insertion",
      "finding:retained-quarter-sea-finite-modular-transfer-preserves-integrated-native-cartan",
      "finding:original-neutral-interarc-fields-have-exact-connected-current-vertex-reconstruction",
      "finding:actual-pin-zero-branch-retains-quarter-car-state-time-but-no-uniform-scale-return",
      "finding:conformal-pin-source-geometry-fixes-exterior-return-and-unitary-memory",
      "finding:geometric-exterior-schur-response-transfers-proper-native-arc-charge-but-is-compact",
      "finding:actual-ns-half-cover-cauchy-graph-purifies-green-state-with-neutral-ground-time-obstruction",
      "finding:full-marked-c2-lifts-distinguish-equatorial-and-os-real-slots",
      "finding:full-marked-equatorial-pullback-is-ramified-graph-not-smooth-seam-circle",
      "finding:same-elliptic-szego-green-functional-has-all-word-free-rp-and-calderon-marginal",
      "finding:elliptic-scalar-spin-pushforward-has-original-holomorphic-p2-anchor",
      "finding:natural-elliptic-anchor-connection-is-not-original-p2-cusp-connection",
      "finding:actual-elliptic-gaussian-source-determines-full-quarter-current-functional",
      "finding:elliptic-current-state-differs-from-hardy-on-every-proper-current-interval",
      "finding:primitive-geometric-force-retains-coupled-reference-and-parity-variation",
      "finding:equal-finite-positive-heat-channel-values-do-not-fix-geometric-force",
      "finding:elliptic-ns-spin-trace-has-complete-global-quarter-kernel",
      "finding:source-induced-ramified-resolutions-do-not-select-quarter-circle-domain",
      "finding:fixed-native-and-fermat-c3-carriers-do-not-identify-marked-v117-pair",
      "finding:theta-square-modular-pair-reconstructs-v117-with-distinct-geometric-q",
      "finding:archived-d4-puncture-formula-fails-and-existing-v117-repairs-existence",
      "finding:geometric-square-clock-is-not-theta-v117-quarter-generator",
      "finding:canonical-conformal-boundary-trace-fixes-existing-elliptic-cauchy-state",
      "finding:conformal-factor-averaging-preserves-given-free-boundary-wick-words",
      "finding:retained-seam-defect-has-derived-causal-geometric-response-with-contact",
      "finding:same-final-pin-geometry-retains-history-while-relative-index-factor-is-fixed",
      "finding:full-projector-covariance-and-positivity-force-all-car-words",
      "finding:actual-two-face-projector-reduces-free-source-test-to-cross-coherence",
      "finding:retained-marked-quarter-compiler-has-no-local-geometric-wick-correction",
      "finding:changed-elliptic-modulus-has-derived-cauchy-response-and-neutral-fourfield-variance",
      "finding:actual-equatorial-half-form-trace-fixes-orthogonal-hardy-range",
      "finding:actual-holomorphic-quarter-cauchy-idempotents-are-not-positive-car-covariances",
      "finding:full-equatorial-checkerboard-free-algebra-fails-fixed-reflection-positivity",
      "finding:genuine-d4-orbifold-branch-has-unique-irreducible-v117-class",
      "finding:wall-weight-additivity-does-not-exclude-stable-v117-parabolic-bundle",
      "finding:recovered-september-c4-triangle-closes-original-a0star-monodromy",
      "finding:actual-a0star-connection-does-not-have-rational-base-inversion-lift",
      "finding:actual-v117-sign-lines-are-negative-weierstrass-triangle-two-torsion",
      "finding:antipodal-and-positive-sign-packets-fail-actual-marked-v117-ramification",
      "finding:common-cyclic-twelve-cover-does-not-trivialize-v117-unramified-sign-lines",
      "finding:negative-sign-packet-has-source-selected-quarter-clock-lift",
      "finding:original-root-and-sign-lines-have-minimal-degree48-common-cover",
      "finding:given-elliptic-flat-connection-family-has-explicit-quillen-norm-and-two-holonomies",
      "finding:bare-large-gauge-quillen-norm-does-not-equal-original-winding-counting-factor",
      "finding:existing-elliptic-half-period-has-canonical-graded-gns-intertwiner",
      "finding:standard-sewing-requires-maximal-affinity-linear-half-modular-field-law",
      "finding:declared-recursive-machine-baseline-controls-and-recovery-are-finitely-measured",
      "finding:declared-recursive-machine-long-run-retains-motion-at-stable-aggregate-rank",
      "finding:recursive-kohonen-source-investigation-has-source-verified-atlas-handoff",
      "finding:chat-algorithmic-comparison-native-pauli-code-and-scoped-information-loss",
      "finding:chat-20261026-results-consolidated-with-corrected-assumptions",
      "finding:source-inversion-forces-six-line-theta-completion",
      "finding:literal-geometric-quarter-is-not-central-gaussian-quarter-on-labels",
      "finding:source-cubic-and-quarter-select-even-theta-and-spin-half-turn-lift",
      "finding:source-cubic-matches-native-pauli-translations-after-context-target-correction",
      "finding:joint-source-clock-relations-have-one-native-clifford-gauge-class",
      "finding:ordinary-theta-determinant-erases-selected-spin-central-character",
      "finding:marked-negative-point-norm-restores-required-spin-central-character",
      "finding:point-normalized-theta-family-correction-fails-original-cubic-action",
      "finding:source-spin-relative-poincare-normalization-closes-point-norm-family",
      "finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package",
      "finding:tfpt-origin-investigation-full-source-coverage-and-current-frontier",
      "finding:full-clock-norm-power-class-and-canonical-eighth-period",
      "finding:original-unit-winding-selects-one-inverse-norm-in-three-cabled-hardy-pair",
      "finding:original-boundary-audit-and-h1-point-restriction-limit",
      "finding:eta-phase-conventions-do-not-identify-relative-spin-reference-pair",
      "finding:same-spin-unit-shift-gives-native-module-class-with-odd-determinant-line",
      "finding:original-even-odd-operator-split-is-compatible-with-same-spin-seam-reference",
      "finding:elliptic-projector-and-weyl-results-retain-their-source-type-at-cubic-spin-bridge",
      "finding:cubic-spin-cauchy-range-is-three-hardy-with-strong-domain-distinction",
      "finding:original-negative-sea-three-mode-ket-is-dual-to-required-hardy-line",
      "finding:auxiliary-theta-route-needs-no-physical-cubic-bulk-or-eta-selected-dual",
      "finding:primitive-gaussian-theta-frame-constructs-e8-pair-with-retained-spin-action",
      "finding:existing-c12-lift-induces-coherent-two-object-e8-vertex-groupoid",
      "finding:actual-winding-functional-calculus-preserves-ns-ramond-split",
      "finding:single-chiral-cycle-exchange-selects-e8-in-four-extension-class",
      "finding:affine-source-involution-pairs-full-d8-sector-spectra",
      "finding:marked-local-e8-with-unchanged-source-time-fails-euclidean-clustering",
      "finding:original-parity-audit-does-not-supply-local-spinor-gauging",
      "finding:hypergraph-local-update-repaired-with-full-family-clock",
      "finding:petersen-polar-step-has-exact-double-cover-family-intertwiner",
      "finding:petersen-source-current-and-hodge-candidates-fail-physical-identification",
      "finding:a7-clock-direction-identifies-neutral-e7-zero-root",
      "finding:oriented-sewing-cocycle-distinguishes-continuation-and-reversal",
      "finding:calderon-single-face-compression-is-not-perron-projector",
      "finding:hodge-born-magnitudes-and-filtered-petersen-jump-match",
      "finding:five-complement-amplitudes-form-full-petersen-quantum-channel",
      "finding:original-twenty-events-factor-through-hodge-and-conditional-phase-record",
      "finding:actual-e8-mixed-current-response-is-full-clifford-channel",
      "finding:carrier-su5-centralizer-gives-direct-e8-petersen-response",
      "finding:su5-covariance-and-petersen-support-fix-complete-quantum-response",
      "finding:neutral-pair-witness-separates-old-events-and-hodge-response",
      "finding:original-degree-clock-is-not-carrier-slot-cycle-at-fixed-source",
      "finding:full-mark-transport-and-isospectrality-do-not-select-hodge-channel",
      "finding:original-pair-source-fixes-state-but-not-selected-epsilon-process",
      "finding:single-positive-process-saturation-selects-full-hodge-channel",
      "finding:original-five-slot-wedge-is-exact-e8-hodge-coefficient",
      "finding:native-pauli-f20-metric-and-quarter-bridge",
      "finding:two-actual-coherent-events-herald-hodge-at-one-over-twenty-five",
      "finding:original-color-center-removes-only-extra-h8-ground-state",
      "finding:marked-symplectic-functor-fixes-native-hodge-kraus-space",
      "finding:f20-record-repair-removes-chart-induced-three-plus-two",
      "finding:theta-scalar-square-lift-is-distinct-from-symplectic-f20-lift",
      "finding:full-admissibility-vacuum-test-needs-source-image-and-transfer-reduction",
      "finding:canonical-native-spin5-hodge-is-already-one-source-record-subchannel",
      "finding:native-hodge-record-selector-is-image-projector-not-kraus-kernel",
      "finding:irreducible-f20-four-cannot-host-original-independent-native-family-clock",
      "finding:native-hodge-is-exact-collective-five-current-action",
      "finding:six-ovoid-clock-bundle-replaces-unnecessary-fixed-form-condition",
      "finding:six-hodge-responses-have-exact-full-current-barycenter",
      "finding:coxeter-seventh-power-has-complete-thirty-e8-lattice-lifts",
      "finding:spin5-hodge-five-is-plucker-kernel-not-original-quartic-hessian",
      "finding:ising-complement-current-mechanism-needs-typed-normal-field-map",
      "finding:neutral-native-a3-hodge-does-not-identify-physical-p2-carrier",
      "finding:compact-ope-coevaluation-normalizes-given-charged-hodge-line",
      "finding:old-ward-root-generates-exact-charged-fifty-module",
      "finding:tp-charged-bracket-support-forces-entire-carrier-marginal",
      "finding:original-ward-and-p2-grading-select-smaller-creation-thirty",
      "finding:charged-source-quarter-recovers-inverse-internal-glue",
      "finding:state-independent-creation-branch-removes-whole-process-hodge-demand",
      "finding:given-source-current-constructs-half-kink-spinor-transport",
      "finding:neutral-ward-response-needs-no-spinor-observable-membership",
      "finding:actual-source-sector-transport-distinguishes-identical-hodge-marginals",
      "finding:ordinary-weyl-type-obstruction-to-single-chiral-os-anti-anti-reflection",
      "finding:given-two-face-source-derives-opposite-neutral-weyls-and-bounded-time-reflection",
      "finding:verification-recovery-preserves-local-source-chain-and-eight-gate-completeness",
      "finding:positive-spectral-reweighting-does-not-establish-original-source-rp",
      "finding:gapped-quarter-energy-cannot-be-same-borchers-translation",
      "finding:finite-ko6-pairing-does-not-supply-standalone-pfaffian-source",
      "finding:full-free-projector-forces-joint-commuting-source-factorization",
      "finding:actual-neutral-current-source-has-constructive-moebius-standard-pair",
      "finding:continuous-local-current-weyls-cannot-live-in-finite-coordinate-fibers",
      "finding:actual-folded-current-ward-and-state-determine-all-neutral-words",
      "finding:mixed-vector-axial-current-is-existing-full-car-subalgebra",
      "finding:constant-product-time-stable-current-level-is-kernel-index",
      "finding:paired-quarter-source-candidate-needs-original-filter-and-oriented-conjugation",
      "finding:graded-boundary-clifford-typing-transports-full-scalar-and-finite-coupled-source",
      "finding:actual-charge-j-transport-retains-full-dirac-and-distinct-pairings",
      "finding:fixed-graded-chart-transports-normal-dependent-source-and-given-mixed-moments",
      "finding:neutral-many-body-source-time-is-not-finite-summable-geometric-dirac",
      "finding:actual-product-reflection-and-same-time-have-full-positive-clifford-source-intertwiner",
      "finding:actual-v252-positive-grade-has-global-p2-wedge-dictionary",
      "finding:old-quarter-full-source-embeds-locally-exactly-on-decoupled-channel",
      "finding:petersen-population-full-cptp-p5-range-zero-to-ten",
      "finding:marked-sm-phase-p5-exact-range-and-response-collision",
      "finding:shared-epsilon-tp-process-fixes-hodge-marginal-even-for-correlated-inputs"
    ],
    "excludedFindings": [
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        "id": "finding:four-session-summaries-remain-covered-with-superseded-norm-selection-task",
        "reason": "Administrative continuity and private session-map coverage; scientific Hardy result remains covered by its own reviewed records.",
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      "premise:strong-seam-covariance-and-same-pair-index",
      "premise:historical-original-circle-and-transfer-source-passages",
      "premise:given-quarter-circle-full-heat-transfer-comparison",
      "premise:historical-sf-sector-action-and-relative-normalization",
      "premise:original-admissibility-and-graded-source-interfaces",
      "premise:literal-positive-rg-map-on-finite-halfline-measures",
      "premise:original-a4-same-moebius-order-four-pin-lift",
      "premise:fixed-marked-glue-and-retained-source-d8-frame",
      "premise:standard-four-d8-module-parent-with-retained-subtheories",
      "premise:same-original-circle-trace-class-covariance",
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      "premise:trace-class-polarization-pair-with-restricted-unitary",
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      "premise:given-exact-reducing-indexline-with-horizontal-connection",
      "premise:positive-determinant-response-of-transverse-product-channels",
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      "premise:smooth-matrix-circle-source-and-moving-projector-with-fixed-scalar-symbol",
      "premise:oriented-d3-tetrahedron-in-bloch-chart-versus-original-quarter-marks",
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      "premise:given-flat-pin-dirac-free-green-quantization-and-ns-cover",
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      "premise:theta-square-modular-transport-with-distinct-geometric-clock",
      "premise:fixed-three-slot-permutation-and-original-compiler-comparison",
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      "premise:retained-zero-branch-defect-on-common-pin-geometric-history",
      "premise:positive-joint-car-source-tested-against-full-geometric-projector",
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      "premise:fixed-native-pauli-translation-target-and-gaussian-phase-class",
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      "premise:relative-spin-scalar-hardy-cable-on-original-three-cover",
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      "premise:marked-oriented-quarter-spin-and-two-object-vertex-transport",
      "premise:natural-single-face-inclusion-in-given-family-calderon-graph",
      "premise:oriented-five-slot-hodge-candidate-with-explicit-basis-readout",
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      "premise:fixed-three-plus-two-mark-and-typed-original-degree-clock",
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      "premise:stipulated-e8-h8-with-original-color-center-selector",
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      "premise:finite-reflected-spectral-weight-preservation-test",
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      "premise:given-quarter-many-body-source-and-compact-dirac-comparison",
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      "premise:spin10-su4-family-four-cycle-quarter-lift-class",
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          "premise:complete-logical-nondisturbance-native-covariance",
          "premise:coherent-block-source-algebra-operation-menu"
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          "premise:actual-clifford-identification-and-finite-overgroup-theorem"
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          "premise:original-erasure-decoder-and-same-native-quarter-polynomial"
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          "premise:actual-clifford-identification-and-finite-overgroup-theorem",
          "premise:original-native-fourth-moment-parent",
          "premise:local-product-code-orbit-geometric-parallel-transport"
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          "premise:original-coupled-determinant-event-cover"
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          "premise:literal-charged-current-native-register-dictionary"
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          "premise:literal-native-positive-cartan-quartic-fock-embedding",
          "premise:compact-inner-e8-zero-mode-operation-class"
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          "premise:literal-native-positive-cartan-quartic-fock-embedding",
          "premise:original-neutral-root-pair-moment-fields",
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          "premise:original-displayed-j-equivariant-cartan-root-isometry",
          "premise:original-positive-adjoint-two-insertion-response-profiles"
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      "finding:all-canonical-stress-histories-scalar-on-native-code": {
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          "premise:original-cartan-current-local-smearing-and-weyl-class",
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      "finding:all-real-local-cartan-profiles-erasure-third-bound": {
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          "premise:original-cartan-current-local-smearing-and-weyl-class"
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      "finding:closed-cartan-weyl-histories-only-logical-phase": {
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      "finding:full-local-quartic-wick-ladder-and-unprojected-native-actions": {
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      "finding:full-e8-current-products-fix-stress-and-exterior-ward-cancellation": {
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          "premise:intrinsic-native-event-geometry-without-marked-clock",
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          "premise:four-starts-fixed-state-versus-whole-datum-comparison"
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          "premise:fixed-conformal-current-reference-for-original-fold-state-transport",
          "premise:four-starts-fixed-state-versus-whole-datum-comparison"
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          "premise:original-single-circle-local-density-and-geometric-stress"
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          "premise:charged-car-negative-sea-source",
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          "premise:complete-lagrangian-dirac-cauchy-car-quantization"
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      "finding:complete-dirac-vector-phase-pair-has-zero-spinor-Ward-charge": {
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      "finding:common-register-vacuum-reference-uniform-and-local-grade-reduction": {
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      "finding:actual-heat-filtered-neutral-pair-full-state-time-and-separate-chiral-limit": {
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      "finding:fixed-interval-rotation-modular-chain-fails-local-state-takesaki-repairs": {
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          "premise:given-eightfold-source-odd-refielding-dictionary",
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      "finding:alpha-archive-term-provenance-and-master-functional-scope": {
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      "finding:original-source-naturality-needs-no-second-p-image-background": {
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      "finding:canonical-spin2-lift-carries-whole-annulus-source-with-reversed-time": {
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      },
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      "finding:paper-microscopic-car-composes-with-native-octic-ward": {
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          "premise:original-quarter-car-fold-current-subalgebra"
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      "finding:microscopic-neutral-octic-has-controlled-smeared-limit": {
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      "finding:paper-seed-source-needs-local-image-not-global-equivalence": {
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      "finding:historical-em-rank-one-and-full-heat-traces-have-different-contracts": {
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      "finding:earliest-c3-route-quantizes-level-but-needs-physical-kinetic-normalization": {
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          "premise:full-graded-car-with-original-positive-time"
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      "finding:old-index-coupling-removes-heat-map-and-has-sharp-residual-ward-test": {
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          "premise:historical-sf-sector-action-and-relative-normalization"
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      "finding:original-admissibility-requires-image-membership-not-positive-half-or-commutation": {
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      "finding:old-closure-already-assumes-car-but-does-not-map-the-geometric-circle": {
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      "finding:historical-uv-map-infinite-mass-exclusion-already-proved-and-recovered": {
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        "premiseIds": [
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      "finding:a4-spin-lift-fixes-local-ns-and-unordered-quarter-pair": {
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      "finding:cover-winding-exchanges-quarter-sectors-before-its-square": {
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      "finding:marked-glue-square-is-the-e8-fermionization-character": {
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      "finding:fixed-canonical-d8-is-not-transported-source-d8": {
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      "finding:actual-winding-has-four-module-e8-defect-extension": {
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      "finding:ramond-lift-sign-is-common-intertwiner-basis-gauge": {
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      "finding:old-seam-relative-determinant-is-stable-in-faithful-kms-class": {
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      "finding:finite-circle-compression-cannot-retain-whole-thermal-winding-factor": {
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      "finding:schur-carrier-plus-complement-preserves-original-relative-determinant": {
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      "finding:v258-forward-source-gap-is-reused-not-solved-by-inversion": {
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      "finding:one-sided-source-and-real-em-norm-need-response-map-not-chiral-phase": {
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      "finding:trace-class-pair-index-fixes-relative-factor-without-primitive-ray": {
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      "finding:matrix-clutch-degree-connects-weak-determinant-class-without-ray-selector": {
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          "premise:smooth-matrix-clutch-on-original-negative-fourier-sea",
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      "finding:finite-ns-ordinary-determinant-cannot-limit-to-nonzero-index": {
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          "premise:trace-class-polarization-pair-with-restricted-unitary"
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      "finding:inner-carrier-is-infinite-spacetime-module-not-finite-circle-cut": {
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      "finding:normal-spin-determinant-line-exactly-transports-whole-circle-in-given-product": {
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      "finding:normal-index-line-source-match-is-holonomy-condition-not-vertical-degree": {
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        "sourceCount": 5
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      "finding:full-normal-product-polarization-change-is-not-trace-class": {
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      "finding:massive-transverse-zero-index-retains-positive-determinant-response": {
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      "finding:original-callias-doublet-count-does-not-supply-one-complex-circle-channel": {
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      "finding:exact-moving-kernel-reduction-forces-kato-offdiagonal-connection": {
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      "finding:tetrahedral-quarter-face-phase-does-not-identify-original-quarter-marked-curve": {
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      "finding:existing-elliptic-cover-correspondence-has-exact-c6-image-and-retained-mixed-clock": {
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          "premise:oriented-d3-tetrahedron-in-bloch-chart-versus-original-quarter-marks",
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      "finding:recovered-natural-joint-lift-and-given-connection-already-exist": {
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      "finding:global-split-joint-carrier-admits-parallel-native-c12-clock": {
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          "premise:fixed-kappa-toral-third-centralizing-split-carrier",
          "premise:full-unitary-e8-field-conjugacy-with-transported-marks",
          "premise:typed-source-structures"
        ],
        "sourceCount": 6
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      "finding:pure-weak-clutch-clock-identification-fails-despite-quarter-matches": {
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          "premise:same-angle-evaluation-of-primitive-determinant-clutch-as-native-clock",
          "premise:full-unitary-e8-field-conjugacy-with-transported-marks",
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      "finding:same-chern-clutch-native-compiler-does-not-select-physical-connection": {
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