Results, with their exact scope.

Original ledger status

AX.P2.01

AX.P2.01

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).

Recorded status: Axiom (declared input; over-determined bootstrap fixed point, Lean-forced)

Original ledger status

E8.GLU.01

E8.GLU.01

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

Recorded status: Lattice

Original ledger status

E8.NUM.01

E8.NUM.01

240 = 16*5*3 and 248 = 240+8 as carrier traces

Recorded status: Lattice

Original ledger status

BOOT.CLASS.01

BOOT.CLASS.01

D5(+)A3 is the UNIQUE familyful cyclic-glue decomposition of E8 (16-dim half-spinor + 3-family A3); glue Z4=|mu4|

Recorded status: Lattice

Original ledger status

ANCHOR.GEN.01

ANCHOR.GEN.01

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}

Recorded status: Identity

Original ledger status

BOOT.01

BOOT.01

bootstrap mu^2-5mu+4=0; g_car=5 three ways; 8=rankE8=h(D5)=phi(30)

Recorded status: Lattice

Original ledger status

GATE.WOLFRAM.01

GATE.WOLFRAM.01

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.

Recorded status: Numerical (second path)

Original ledger status

SEL.THM.01

SEL.THM.01

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

Recorded status: Lattice [L] + selection [P]

Original ledger status

SEAM.EQUIV.CONTINUUM.02

SEAM.EQUIV.CONTINUUM.02

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.

Recorded status: Sharpening [E]/[O] (the order-4 clock + Coxeter-match pin E8 over SO(16); the residual is purely the chiral-edge existence)

Original ledger status

SEAM.EQUIV.CONTINUUM.03

SEAM.EQUIV.CONTINUUM.03

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).

Recorded 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)

Original ledger status

SEAM.MMST.INCLASS.01

SEAM.MMST.INCLASS.01

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.

Recorded status: SEAM.EQUIV.01 reduced to the single S3 lattice-realisation input; MMST in-class verified for the collar [E]/[C]/[O]

Original ledger status

SEAM.S3.LATTICE.01

SEAM.S3.LATTICE.01

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.

Recorded status: S3 realised as an explicit gapped chiral-Majorana lattice model (numerical Chern |C|=1, c_-=8, det K 4->1) [E]/[C]/[O]

Original ledger status

SEAM.S3.CENTRALCHARGE.01

SEAM.S3.CENTRALCHARGE.01

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).

Recorded status: central charge c=8 from finite-size entanglement scaling of the lattice (chiral, c_-=8) [E]

Original ledger status

SEAM.S3.E8CHARACTER.01

SEAM.S3.E8CHARACTER.01

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).

Recorded status: the chiral c=8 content is (E8)_1 (248 currents, 1 primary) via the exact character + mu4/GSO promotion [E]

Original ledger status

SEAM.S3.MODULAR.01

SEAM.S3.MODULAR.01

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).

Recorded status: torus GSD=1 / modular uniqueness -- the genus-1 holomorphy discriminator closes the v344 hard part [E]

Original ledger status

SEAM.S3.RP.01

SEAM.S3.RP.01

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.

Recorded 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]

Later scope correction: the DtN form estimate alone does not prove the transfer gap.

Original ledger status

FORM.SEAM.MMST.01

FORM.SEAM.MMST.01

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.

Recorded status: Formal [F] (Lean kernel; closed modulo the cited MMST/Adamo theorems)

Original ledger status

QGAMB.REDUNDANCY.01

QGAMB.REDUNDANCY.01

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.

Recorded 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)

Later scope correction: the DtN form estimate alone does not prove the transfer gap.

Original ledger status

E8.UNMAPPED.BANDWIDTH.01

E8.UNMAPPED.BANDWIDTH.01

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.

Recorded 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)

Original ledger status

E8.REVERSE.AUDIT.01

E8.REVERSE.AUDIT.01

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.

Recorded 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)