2026-07-22 (VI) ( v516 — WP5e back-reaction milestone M2 ``the twisted Kodaira–Spencer measure'' executed (SUCCESS on the preregistered v514 S8.2 criterion, ON THE DECLARED COMPLETION MEASURE — verdict B), promoted as the ledger row CELEST.WP5E.M2.01: the one-loop box coefficient per sector on [formula in source] under the completion contact term [formula in source] — the twisted-sector LOOP contributes the UNPHASED sector trace with the [formula in source] zero-mode normalisation [formula in source] , of which the Atiyah–Bott skeleton kept only the phase-weighted insertion part — (1) the COMPLETION-WEIGHT IDENTITY (the centrepiece): per glue class [formula in source] EXACTLY ( [formula in source] the v502 sector Casimir energies, [formula in source] the McKay/Kronheimer charges, v505 S4.3 index bridge): THE THREE SPHERE AXIONS PAIR THROUGH THEIR OWN ch [formula in source] CHARGES — an exact identity, NO free scale, no fit; (2) PARAMETER-FREE LOCKS: [formula in source] for ANY scale [formula in source] (the ch [formula in source] pattern alone kills [formula in source] ); ratio [formula in source] the [formula in source] -forced leading ratio (REPRODUCED, not fitted); the [formula in source] budget fixes [formula in source] uniquely (leading system det [formula in source] , unique weights [formula in source] ); (3) the SUCCESS TABLE: [formula in source] — a PERFECT Okubo square ( [formula in source] , [formula in source] in EVERY twisted channel: [formula in source] ); total [formula in source] Dedekind [formula in source] Okubo [formula in source] the UNIQUE quartic-free weighting of v505 S3.8 (the rigidity theorem is exactly what the measure lands on); both v508 certificates KILLED ( [formula in source] , [formula in source] ); the v511 slice [formula in source] SUPPLIED EXACTLY ( [formula in source] ) — NO cubic [formula in source] -channel needed ( [formula in source] free [formula in source] ); remainder inside the exchange span ( [formula in source] , [formula in source] ): the BULK axion cancels every channel with the SAME [formula in source] , the twisted propagator carrying the AB zero-mode factor [formula in source] ; the v514 measure chain stays [formula in source] -blind; (4) NEGATIVE CONTROLS: wrong scale [formula in source] leaves [formula in source] — only [formula in source] clears the budget; the [formula in source] shuffle breaks [formula in source] AND [formula in source] ; [formula in source] glue keeps [formula in source] / [formula in source] — the KILL branch FIRES there (Okubo failure inherited: [formula in source] doubly special); [formula in source] degenerates to the [formula in source] target and breaks the conjugation pairing [formula in source] the index bridge; the [formula in source] /Eguchi–Hanson anchor PASSES with the same mechanism ( [formula in source] , coupling scale [formula in source] centre count); (5) the MULTIPLIER NOTE (cross-reference to the parallel [formula in source] strand): the contact weights [formula in source] are REAL RATIONALS with no multiplier system of their own; the [formula in source] residual phases are the SQUARES of the [formula in source] discriminant [formula in source] -phases [formula in source] — orders [formula in source] squaring to [formula in source] , a finite [formula in source] system; Gauss sums [formula in source] , [formula in source] , full Weil sum [formula in source] ; \ the MANDATORY FENCE: the completion reading (the loop of twisted states carries the UNPHASED sector trace with the same zero-mode normalisation) is DECLARED, supported by the [formula in source] modular-completion finding, NOT derived from the BCOV integral; \ the BCOV-integral derivation ( [formula in source] territory); Wolfram extension [formula in source] (verified engine run); NO marker moves — the KILL (leftover independent quartic in ANY sector) does NOT fire, the completion reading stays -declared, SEAM.EQUIV.TWISTOR.01 stays Open, SEAM.EQUIV.01 untouched)
Research changelog · TFPT — Introduction and reading guide
v516_celestial_wp5e_m2_twisted_ks_measure.py ( CELEST.WP5E.M2.01, 23 checks, [formula in source] 1 s; exact sympy/Fraction, no floats; deterministic). S0 replication (roots, class quartics, [formula in source] two routes, bulk Okubo); S1 the index bridge [formula in source] the completion weights (no free scale, parameter-free locks); S2 the contact term (two forms, genuine-contact certificate); S3 the success table
Opening source excerpt. Formulas and the full argument remain in the original document.
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