The Interior Observer Cosmological Framework — Paper 25: The Weak-Sector Identity Pin — Quadratic Thermal Covariance, the Channel-Budget Equation, and the V-vs-V' Class-Membership Theorem

Beginning from two premises — (1) the observable universe exists inside a Schwarzschild black hole, and (2) the physics inside the horizon is the same as outside — this paper resolves the V-vs-V' class-membership problem: whether the weak-sector BBN dressing coefficient reads V (α) = Kgauge = ln (1+γ²) or V' (α) = 2γ. Conditional on three physical identification premises (H1–H3), the paper proves V = Kgauge exactly, resolving the WMR premise from Paper 22. The central result is the Quadratic Thermal Covariance Theorem: the physical weak freeze-out rate, represented as a centered two-time KMS correlator on the bridge CCR algebra, is bilinear in the bridge Riesz vector F₀ for all time separations. This structurally excludes V' = 2γ (no linear bridge channel exists in the centered quasi-free KMS state) and excludes V'' = 2 (1+γ²) by direct computation on the constructed extension. The paper corrects Paper 22's weak amplitude from εw = Kgauge × √L₁ to εw = Kgauge × L₁. PRyMordial-verified results: D/H (−0. 61σ), Yₚ (+0. 68σ), Li-7/H (+0. 55σ), χ² = 1. 13, with all three primordial abundances within 1σ of observation and zero fitted parameters. The observable-class taxonomy is clarified: BDP reads V' because it evaluates a Wilson line (one-point amplitude) ; the weak rate reads V because it evaluates |M|² (two-point function in a thermal state). Twenty-nine killed derivation routes are documented. Contact: david@fife. cc https: //dfife. github. io/index. html v1. 3 (May 2026): Reproducibility bundle release. The Paper 25 v1. 3 reproducibility bundle is published at https: //github. com/dfife/io-framework-public/releases/tag/paper25-v1. 3 (commit 4a68f85c, tarball SHA256 1e3ee6cbacadb6c0820b8ff60bcc5a6022b9094103f714a943d6a64723003119). The bundle ships scripts and frozen JSON outputs reproducing the active weak-sector identity pin scorecards: the corrected YPCMB BBN scorecard arithmetic (D/H sigma=-0. 57, Yₚ sigma=+0. 68, Li-7/H sigma=+0. 55, chi² (D/H + Yₚ + Li-7) = 1. 089), the V/V’/V’’ branch comparator arithmetic, and channel-budget equation checks. Validator script reports totalchecks=22 / passcount=22 / failcount=0. A kappa-style structural audit was run on Paper 25 v1. 2 in parallel with the bundle build and found no unlabelled continuous fitted parameter in the active claim stack; the H1-H3 premise package remains visible exactly where v1. 2 marks it as conditional. The active weak amplitude epsilonw = Kgauge * L₁ is correctly cited as DERIVED/CONDITIONAL on H1-H3 plus upstream L₁. The active BBN scorecard uses the Paper 24 final-push YPCMB path; private kinetic-runner Li-7 rows are not the active lithium scorecard. The V’ exclusion is structural via the centered two-time correlator theorem (Paper 25 Theorem 25. 12), with the chi² = 401. 74 catastrophe serving as a numerical backstop, not the proof. New §11 Code and Data Availability section added; bibliography entry 10 added for the bundle. No physics, theorem, or numerical changes from v1. 2. v1. 2 (May 2026): PRyMordial output index correction (YPBBN -> YPCMB), observational denominator alignment to IO Framework Observational Conventions v1, and amplitude alignment to the Paper 22 v1. 4 standard (epsilonw = ln (1+gamma²) x L₁ = 0. 012300778733811872, epsilonₙ = (1. 72704/10) x L₂ = 0. 02384221534546833). Paper 25 v1. 1 support rows inherited Paper 22 Round10's YPBBN wrapper convention; the wrapper now reports YPCMB / PRyMresults () 3 and retains YPBBN only as an audit field. Corrected aligned scorecard: D/H = 2. 510 x 10^-5 (-0. 57 sigma), Yₚ = 0. 24772 (+0. 68 sigma), Li-7/H = 1. 751 x 10^-10 (+0. 55 sigma), chi² (D/H + Yₚ + Li-7) = 1. 089 (was 1. 13 in v1. 1). Linear-branch comparison: chi² (3-obs) = 1. 99 (was 2. 11 in v1. 1) ; quadratic still beats linear by margin 0. 90. V’ branch chi² (3-obs) = 401. 74 (was 412. 85), remaining catastrophically excluded. The central theorem of this paper - the Quadratic Thermal Covariance Theorem - structurally proves that the physical weak rate is bilinear in the bridge field (a two-point function, not a one-point amplitude), which is the rate-vs-amplitude distinction underlying why the aligned quadratic branch is the correct physical branch and underlying why YPCMB (the helium fraction relevant to observational compilations) is the correct observational comparison. Observational denominators per IO Framework Observational Conventions v1 (https: //dfife. github. io/data/observationalconventionsᵥ1. md). Paper 25's distinctive results (V-vs-V’ Class-Membership Theorem, Quadratic Thermal Covariance, WMR closure, seven new HIO theorems, twenty-nine killed routes, channel-budget equation) are unaffected by this correction. v1. 1 (April 2026): Full appendix from Paper 24 v2. 1 clean + Paper 25 enriched results (Steps 352–381). Open/Closed tracking added (WMR closed). Author block updated. Rosetta terminology retired. v1. 1 (April 2026): Full appendix from Paper 24 v2. 1 clean + Paper 25 enriched results (Steps 352–381). Open/Closed tracking added (WMR closed). Author block updated. Rosetta terminology retired.