Original ledger status
GATE.UWALL.12
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
Recorded status: Identity [I] (index-11 congruence + parity + reductions) + [P]/[A] (establishment of the two line pairings)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
Original CSV, line 227Original ledger status
GATE.METRIC.09
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
Recorded status: Identity/Lattice [I]/[L] (duality, projector, orbits, fusion) + [P] (the one conformal-net step)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
Original CSV, line 228Original ledger status
HOR.CLOCK.12
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
Recorded status: Identity [I] (exact rigidity + closed-form ratio + audit coefficient) + [P] (finite-weight absorption)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_horizon_readouts
Original CSV, line 229Original ledger status
GATE.UWALL.13
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
Recorded status: Identity [I] (atom identities + new closure) + [P] (the one lift map)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
Original CSV, line 230Original ledger status
FLAV.QGEO.07
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
Recorded status: Identity/Lattice [I]/[L] (exact cohomology action + parity match) + [P] (module identification)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_2
Original CSV, line 231Original ledger status
HOR.CLOCK.13
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
Recorded status: Identity [I] (model theorem + det'-clean bend + localisation) + [P] (measure identification)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_horizon_readouts
Original CSV, line 232Original ledger status
GATE.METRIC.10
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.]
Recorded status: Identity [I] (exact census) + [P] (scoping conclusion)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
Original CSV, line 233Original ledger status
GATE.UWALL.14
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
Recorded status: Identity [I] (cusp-frame anchor reading + equivalences) + [P] (the one assignment)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
Original CSV, line 234Original ledger status
SEAM.EHMODEL.01
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
Recorded status: Identity [I] (model mechanism exact) + [A] (the theorem proper stays open)
Dependencies, status detail and source
Canonical status: 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
Location: origin_theory
Original CSV, line 235Original ledger status
SEAM.EHMODEL.02
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
Recorded status: Identity [I] (BFK/Kac algebra exact) + [A] (the theorem proper stays open)
Dependencies, status detail and source
Canonical status: 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
Location: origin_theory
Original CSV, line 236Original ledger status
SEAM.EHMODEL.03
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
Recorded status: Identity [I] (dimensional bookkeeping exact) + [A] (the anchor stays declared)
Dependencies, status detail and source
Canonical status: 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
Location: origin_theory;tfpt_horizon_readouts
Original CSV, line 237Original ledger status
ANCHOR.VGEO.02
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.]
Recorded status: Identity [I] (scaling bookkeeping) + [A]/[O] (closed metrology: R_+ unit torsor; one calibration)
Dependencies, status detail and source
Canonical status: 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
Location: origin_theory;tfpt_research_contracts
Original CSV, line 238Original ledger status
GATE.METRIC.11
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.]
Recorded status: Identity/Lattice [I]/[L] (the extension theorem exact) + [P] (the seam-side identification)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_5_redteam;tfpt_research_contracts;origin_theory
Original CSV, line 239Original ledger status
GATE.METRIC.12
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
Recorded status: Identity [I] (compression contraction + Wick) + [P] (the one shared free-field premise)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts;origin_theory
Original CSV, line 240Original ledger status
GATE.METRIC.13
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
Recorded status: Identity/Lattice [I]/[L] (DtN + character identity exact) + [P]/[A] (free-bulk premise + net existence)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_5_redteam;tfpt_research_contracts;origin_theory
Original CSV, line 241Original ledger status
GATE.METRIC.14
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
Recorded status: Identity/Lattice [I]/[L] (symbol homogeneity + rigidity count + |Z2| arithmetic) + [P] (the gapped flow reaching the fixed point)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_5_redteam;tfpt_research_contracts;origin_theory
Original CSV, line 242Original ledger status
GATE.METRIC.15
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)
Recorded status: Identity [I] (operator-dimension count) + [P] (basin/convergence of the gapped flow)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts;origin_theory
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GATE.METRIC.16
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
Recorded status: Identity [I] (signed cumulant inversion + Wick functoriality + PF dichotomy) + [P] (the scope premise: v56 acts on the full cone)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
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GATE.METRIC.17
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'
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
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GATE.METRIC.18
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)
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
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