A registered result in its source context

CELEST.WP5E.GAMMA.01

Original ledger classification: 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)

Original claim text

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.

Original classification detail

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 as recorded

CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; SEAM.EQUIV.01; v492; v493; v495; v502; v505