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. 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.