A registered result in its source context

CELEST.WP5E.WM.01

Original ledger classification: Formal/Identity [E] (exact Q(i)/Q(sqrt2)/cyclotomic mode ledgers, Abel values, tensor-trace splitting, reflection-formula zeta determinants, Quillen arithmetic, delta-1f block constant terms, the full v516 chain, all controls -- sympy/Fraction exact, Wolfram-mirrored; numeric documented: Hurwitz/Lerch and stripped block limits at 40 digits) + [C] (the typed premises TP-REG/TP-Q/TP-NUM/TP-CH) + [O] (the global BCOV integral beyond the fibre zero-mode factor)

Original claim text

THE CONSTRUCTIVE w_m DERIVATION EXECUTED (2026-07-23; the residual [O] of v516/v520, one module, 26 checks, ~1 s; verdict ERFOLG per the frozen preregistration): is the completion normalisation 1/det_j the standard Atiyah-Bott/holomorphic-Lefschetz fixed-point weight AND the Quillen-factorised zeta determinant of the twisted fibre Laplacian -- i.e. COMPUTED rather than declared? YES, from three independent sources. (1) ATIYAH-BOTT (route i): the equivariant mode ledger of the fibre C^2 -- c_n = sum_{a+b=n} i^{j(a-b)} -- equals the closed form 1/((1-q i^j)(1-q i^{-j})) exactly in Q(i) for ALL n <= 120; the closed form is REGULAR at q = 1 (the twisted sectors carry no volume pole) with Abel value (1/2, 1/4, 1/2) = 1/det_j, det_j = det(1-g^j) = (2, 4, 2); the (C,2) Cesaro means converge in exact arithmetic; the fixed-point factor SPLITS OFF EXACTLY at every truncation level d in {7, 12, 25}, for the phased (skeleton) AND the v520-forced unphased (completion) trace alike -- the Abel-limit contact_j = (Q^(0) - Q^(j))/det_j is the v516 contact vector COMPUTED, not declared. (2) ZETA/QUILLEN (routes ii/iii): det_zeta of the g^j-twisted circle Laplacian (eigenvalues (n + j/4)^2) = 4 sin^2(pi j/4) = det_j EXACTLY (Lerch + reflection formula symbolically; Hurwitz certificates ~1e-41); the Quillen split gives det Delta = det_j^2 with |det dbar| = det_j and the REAL POSITIVE holomorphic section unique via the SU(2) conjugation pairing; the delta-1f f1f3 modular block at the (0, b) nodes carries the exact constant term 1/det_b (deviations <= 5.2e-28) -- the v520 transport machinery carries the AB zero-mode weight verbatim. (3) CONSISTENCY CHAIN: the derived weight reproduces the v516 chain NUMBER BY NUMBER -- w_m = sum_j (1-i^{jm})/det_j = (0, 3/2, 2, 3/2) = 4h_m = |mu4| h_m = -4 ch2(T_m); T5 = 0 at every scale; h_2 : h_1 = 4 : 3; the T3 budget 32-8c forces c = 4 = |mu4| uniquely (leftovers 24/16/8/0/-8); per-sector Okubo squares (18, 9, 18)<x,x>^2; total 45<x,x>^2 = (5/4) x 36<x,x>^2; Phi_T3: 32 -> 0, Phi_P: 72 -> 0; psi(A_fix) = +64, psi(contact) = -64. (4) NEGATIVE CONTROLS: 1/det^2 gives w' = (0, 5/8, 1, 5/8) with ratio 8:5, leftovers (T5, T3) = (2, 12) and a Quillen contradiction (det^4 vs measured det^2); 1/|1-i^j| gives w''_1 = 1+sqrt2 notin Q with T5/T3 leftovers exact in Q(sqrt2); the Z2/Eguchi-Hanson anchor PASSES with the same derivation (Abel = zeta = det(1-g) = 4, w = 1/2 = |Z2| h^{A1}, 9<x,x>^2 = (1/4) x 36, both scale teeth clear only at c = 2 = |Z2|); the SO(16)/D8 KILL fires (T5 = 20, T3 = -40); diag(i,i) breaks the zeta = AB identity ITSELF (real (2,4,2) vs complex (2i,4,-2i), no real Quillen section, w_1 != w_3); k = 3, 5 orbifolds: w_m = m(k-m)/2 = k h_m exactly in Q[x]/Phi_k, Dedekind (k^2-1)/12, product k^2 -- the weights wander correctly with the centre count, only k = 4 = |mu4| carries the E8-glue chain. [C] TYPED PREMISES (the mandatory fence): TP-REG (Abel/Cesaro/zeta regularisation IS the definition of the equivariant character), TP-Q (Quillen structure of BCOV F1 = v520 TP-4), TP-NUM (unphased numerator = the v520 ERFOLG-A decision layer under its TP-1..TP-4), TP-CH (the insertion channel (0, j) computes the box-channel weight). [O] the GLOBAL BCOV integral on PT/Z4 beyond the fibre zero-mode factor (the tau-integrated dressing lives in the v518/v520 scaffold); the resolved-ALE route (v514/v517) not built; SEAM.EQUIV.TWISTOR.01 untouched. Discovery provenance: celestial_seam_wp5e_wm_bcov_derivation_probe.py (26/26) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves anywhere.

Original classification detail

ERFOLG (the w_m normalisation DERIVED at probe level): 1/det_j is the Atiyah-Bott/zeta-determinant fixed-point factor, computed from three independent sources (the equivariant mode ledger with Abel value (1/2, 1/4, 1/2); the zeta/reflection determinant 4 sin^2(pi j/4) with the unique real Quillen section; the delta-1f block constant term 1/det_b) and the v516 chain closes number by number (w = 4h, c = 4 = |mu4|, squares (18, 9, 18), total 45<x,x>^2, psi +-64); all negative controls behave (Z2/EH anchor passes at c = 2 = |Z2|, so16 kill fires, diag(i,i) breaks the zeta = AB identity, k = 3/5 weights wander correctly); typed premises TP-REG/TP-Q/TP-NUM/TP-CH; the residual [O] narrows to the global BCOV integral beyond the fibre zero-mode factor; no marker moves

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5E.M2.01; CELEST.WP5E.MEASURE.01; CELEST.WP5E.DELTA1.01; SEAM.EQUIV.TWISTOR.01; v505; v514; v516; v517; v518; v520