A registered result in its source context

CELEST.WP5E.BETA.01

Original ledger classification: 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)

Original claim text

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.

Original classification detail

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 as recorded

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