Original ledger status
QGEO.EMERGE.LIGHT.01
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].
Recorded 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]
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 567Original ledger status
CELEST.WP5DB.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
Original CSV, line 568Original ledger status
CELEST.WP5E.BETA.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts
Original CSV, line 569Original ledger status
SEAM.CLOCK.RIGIDITY.01
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.
Recorded 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
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 570Original ledger status
SEAM.BIT.ORIGIN.01
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.
Recorded 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
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 571Original ledger status
SEAM.BIT.FREEDOM.01
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.
Recorded 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
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 572Original ledger status
CELEST.WP5E.GAMMA.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 573Original ledger status
CELEST.WP5E.EPS2.01
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)
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 574Original ledger status
CELEST.WP5E.DELTA2.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 575Original ledger status
SEAM.TAU.FLAG.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: introduction;tfpt_1;tfpt_3;tfpt_research_contracts
Original CSV, line 576Original ledger status
CELEST.DTERM.NONDERIV.01
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].
Recorded 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
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 577Original ledger status
CELEST.WP5E.EPS1.01
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)
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 578Original ledger status
CELEST.WP5E.M1.01
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)
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 579Original ledger status
CELEST.WP5E.M2.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 580Original ledger status
CELEST.WP5E.M3.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 581Original ledger status
CELEST.WP5E.DELTA1.01
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.
Recorded 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)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts;tfpt_3
Original CSV, line 582Original ledger status
SEAM.EQUIV.MMST.01
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.
Recorded status: Conditional [C] (closed modulo cited theorems: the MMST/OS continuum existence, v336; every computable level pinned [E]; Lean FORM.SEAM.MMST.01)
Dependencies, status detail and source
Canonical status: [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
Location: tfpt_research_contracts;introduction;tfpt_1;tfpt_3;tfpt_4
Original CSV, line 583Original ledger status
SEAM.EQUIV.TWISTOR.01
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].
Recorded status: Open (research route [O]: the Costello-Li construction; preparation [E]/[C] via the CELEST.SEAM.01 work packages v492-v518)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts;tfpt_3
Original CSV, line 584Original ledger status
WOIT.OS.TWISTOR.01
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.
Recorded status: Open (research contract)
Dependencies, status detail and source
Canonical status: 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
Location: tfpt_research_contracts;tfpt_3
Original CSV, line 585Original ledger status
WOIT.THETA.FREE.01
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.
Recorded status: Identity/Lattice (exact sympy + exact Cl(16); 40-digit gap-certified RP inertia certificates)
Dependencies, status detail and source
Canonical status: [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
Location: tfpt_research_contracts;tfpt_3
Original CSV, line 586