The Interior Observer Cosmological Framework Paper 21 - The AC1 Derivation, the Puncture Construction, and the BBN Branch Assignment

Paper 21 of the Interior Observer (IO) Cosmological Framework. 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 derives AC1 (the acoustic angular scale Jacobian) from the existing Papers 15–20 stack, upgrading Paper 20's Theorem 20. 2 to DERIVED/THEOREM status within the reduced scalar/longitudinal acoustic sector. The zero-parameter prediction θₚred = 0. 599° (Planck: 0. 597°, residual 0. 429%). Part II investigates the structural origin of the D/H radiation deficit, discovering the SU (2) Geometric Weight (Zₚunc = 1. 4221), proving the Optical Filtration Theorem (direct reduced RT/BY optical readout of θ is blind to noncentral SU (2) puncture data), and deriving the Absolute Local Casimir Load Fₐbs = 0. 36879 matching the D/H target to 0. 066% precision (STRUCTURALLY SUPPORTED CONSTRUCTION). Nineteen routes (eighteen Pᵣesp plus freeze-in) are tested and killed. The surviving target is a non-separable, channel-dependent radiation response operator. https: //dfife. github. io/index. html v2. 0 (May 2026): Major scope reduction and retitle. The paper is restructured around the three theorem-grade results that are not superseded by downstream papers: the AC1 Derivation (Part I, closing Paper 20 Theorem 20. 2’s AC1 dependency), the Puncture Construction (Part II, deriving the L₁ and L₂ puncture loads used by Paper 22 v1. 5/v1. 6 and the operator-algebra theorems consumed by Papers 22 and 32), and the BBN Branch Assignment (Part III, Theorem 21. L typing BBN as local bulk thermodynamics that evaluates at TIO). Removed: the Fₐbs / ΔNₑff hunt (former §12A, superseded by Paper 22 v1. 5’s rate-dressing resolution as documented in v1. 5 history) ; the six-eliminated-alternatives investigation (former §8) and the nineteen Pᵣesp routes (former §12B), all tied to a problem the v1. 5 wrapper correction eliminated; No-Go 21. K (former §12 opening) since the radiation gap no longer exists; the BBN scorecard at Fₐbs (former §12A. 4, χ² = 3. 05, superseded by the active χ² = 0. 80) ; Theorem 21. A H₀ Observable-Class Jacobian (bare-branch infrastructure, retired by Paper 29) ; Theorem 21. C Assembly Gap Decoupling (closed by Paper 29) ; Theorem 21. E Baryon Freeze-In (independently dead, ωb, geom already below BBN target) ; Theorem 21. F Tensor Coupling Classification (Fₐbs infrastructure) ; Theorem 21. X λ=4 Conditional Uniqueness (Fₐbs infrastructure) ; Theorem 21. Bb No Dynamic Scattering (step toward ΔNₑff = 0) ; §13 Connection to Open Foundation Problems (retired with the radiation investigation) ; and all “SUPERSEDED: bare branch retired by Paper 29” bracket inserts in active body text. The active L₁ = 0. 22417 and L₂ = 0. 13805 puncture-load values are unchanged; the AC1 closure proof is unchanged; Theorem 20. 2’s upgrade target is DERIVED/CONDITIONALVERIFIED through AH1–AH7 + P1–P6 + RT/BY + B1–B5 (the four upstream premise packages, all theorem-grade in their respective papers) ; the surviving operator-algebra theorems (21. G, 21. I, 21. J, 21. L, 21. O, 21. Y, 21. Ba) are unchanged. Section numbering compressed; abstract and §1. 1 Claims and Scope rewritten to match the new structure. Reproducibility bundle: paper21-v2. 0 (release tarball SHA256 335d72254f3954cdda1a9db8ab47d886b089060b322e7bc26641669770915bdd; validator 16/16 PASS; bundle includes R4/FIRAS dependency audit confirming R4 does not enter the active Paper 21 BBN scorecard). v1. 7 (May 2026): Reproducibility bundle release. The Paper 21 v1. 7 reproducibility bundle is published at https: //github. com/dfife/io-framework-public/releases/tag/paper21-v1. 7 (commit 6f7117bf, tarball SHA256 661c2f406b37c1f009a6e2ad6d324f1d5173ae3c3b556c5e9dcbfb7ad992b019). The bundle ships scripts and frozen JSON outputs for the live Paper 21 numerical claims: the singleton puncture-load values L₁ = 0. 22417 and L₂ = 0. 13805, the reduced-sector AC1 acoustic closure check, the corrected PRyMordial wrapper convention (YPCMB), the corrected BBN scorecard arithmetic, and the live operator-algebra theorem artifacts. Validator script 07ᵥalidateₑxpectedₒutputs. py reports totalchecks=12 / passcount=12 / failcount=0. A kappa-style structural audit was run on Paper 21 v1. 6 in parallel with the bundle build and found no hidden continuous fitted parameter in the active claims; the full audit report is included in the bundle. New §15 Code and Data Availability section added; bibliography entry 31 added for the bundle. The v1. 7 update notes that L₁ and L₂ are derived in this paper, while their downstream use as Paper 22 rate-dressing amplitudes inherits Paper 22 GMP/TBS and Paper 25 WMR conditional bridge premises and is scoped accordingly there. No physics, theorem, or numerical changes from v1. 6. v1. 6 (May 2026): Cross-paper reference update to reflect current Paper 22 published state. All references to "Paper 22 v1. 3" updated to "Paper 22 v1. 5" (nine total locations) since Paper 22 has subsequently been corrected through v1. 4 (PRyMordial wrapper YPBBN -> YPCMB correction, May 2026) and v1. 5 (GMP/TBS premise-status corrections labeling Theorems 22. 23-22. 25 as DERIVED/CONDITIONAL on GMP + TBS, May 2026). The cited Paper 22 chi² (D/H + Yₚ) value is updated from 1. 50 (Paper 22 v1. 3 published, using YPBBN output) to 0. 80 (Paper 22 v1. 5 corrected, DERIVED/CONDITIONAL on GMP + TBS). The Paper 21 v1. 5 corrected scorecard (Yₚ +0. 70 sigma, D/H -0. 55 sigma, chi² = 0. 80) is unchanged and now matches Paper 22 v1. 5 exactly, confirming cross-paper consistency. Additionally, the Part II claims subheader (Section 1. 1, claims 4-7) and the Not-claimed list have been contextualized with explicit v1. 5/v1. 6 supersession framing, so readers encountering claims 4-7 about DeltaNₑff and Pᵣesp understand they are scoped to Part II's superseded investigation, not active framework targets. No Paper 21 numerical values, theorems, or no-go results change in v1. 6; this is a reference-hygiene and supersession-clarity update only. The Part II header callout is updated from 'SUPERSEDED BY PAPER 22 v1. 3' to 'SUPERSEDED BY PAPER 22 v1. 5' (the v1. 6 sweep updated body references but initially missed the bold callout itself, caught in adversarial review). The Master Mathematical Reference table source labels for D/H and Yₚ are updated from plain 'Paper 12' to 'Paper 12 / Paper 21 v1. 6 corrected wrapper' to disambiguate that the displayed values are wrapper-corrected, not historical Paper 12 values. Bibliography entry 30 is updated to cite IO Framework Conventions v2. 0 (unified: observational, labeling, versioning, and cross-paper reference conventions; https: //dfife. github. io/data/conventionsᵥ2. md), the new canonical reference deployed in May 2026 that consolidates the prior Observational Conventions v1. 0 (now Section 1 of v2. 0) with formal labeling, versioning, and cross-paper reference conventions in Sections 2-4. The v1. 0 file remains live at its original URL for backward compatibility with already-published papers including Paper 21 v1. 5; in-text references to "Observational Conventions v1" remain accurate as a historical citation. Paper 21's distinctive results (operator algebra construction, nineteen killed Pᵣesp routes, puncture load derivations L₁ and L₂, AC1 closure work, optical filtration, nontraciality, branch assignment) are unaffected. v1. 5 (May 2026): PRyMordial output index correction (YPBBN -> YPCMB) and amplitude framework alignment to Paper 22 v1. 3 / Paper 24 v2. 2 standard (epsilonw = 0. 012301, epsilonₙ = 0. 023842). Paper 21 v1. 4 BBN scorecard reflected an older amplitude branch and read PRyMordial output index 4 (YPBBN) instead of index 3 (YPCMB). Corrected scorecard: Yₚ = 0. 24782 (+0. 70 sigma), D/H = 2. 510 x 10^-5 (-0. 55 sigma), Li-7/H = 5. 36 x 10^-10 (+12. 20 sigma), chi² (D/H + Yₚ) = 0. 80. Part II's nineteen-route radiation investigation is superseded by Paper 22 v1. 3's rate-dressing paradigm: the v1. 4 D/H tension at -3. 9 sigma (chi² approximately 15. 6) that motivated the Fₐbs = 0. 36879 / DeltaNₑff approximately 0. 37 hunt shrinks to -0. 55 sigma (chi² = 0. 80) under the corrected wrapper plus modern amplitudes, removing the phenomenological need for ΔNₑff approximately 0. 37. Paper 22 v1. 3, Theorem 22. 23, derives the active resolution via the j=1 weak channel and j=2 nuclear channel rate-dressing construction (epsilonw = Kgauge x L₁, epsilonₙ = (langle K rangle / 10) x L₂) using exactly the puncture loads L₁ = 0. 22417 and L₂ = 0. 13805 derived in Paper 21. The nineteen no-go results in Part II remain as documented research provenance constraining the search space; the L₁ and L₂ puncture-load derivations remain load-bearing inputs to Paper 22's rate-dressing construction. Observational denominators per IO Framework Observational Conventions v1 (https: //dfife. github. io/data/observationalconventionsᵥ1. md). Paper 21's distinctive results (operator algebra construction, nineteen killed Pᵣesp routes, puncture load derivations L₁ and L₂) are unaffected by this correction. v1. 4 (April 2026): Appendix replaced with Paper 20 v1. 5 clean + Paper 21 results. AC1 closed. Open/Closed tracking. v1. 3 (April 2026): Full Schur branch appendix correction. All inherited appendix steps (29, 56, 58, 60, 107, 122) and open-problem Item 16 updated to Schur definitive branch (H₀ = 68. 91, Paper 29). Paper 19 intermediate supersession notes updated. Bare branch (H₀ = 61. 06) references in §§7, 13, 14 marked SUPERSEDED (Paper 29). Title page standardized. All body theorems (AC1 derivation, SU (2) geometric weight, sky-slot inheritance, optical filtration, Fₐbs) are branch-independent. Version 1. 2 includes cycloid parameterization correction, clean Appendix A catalog (Steps 1–207) with Papers 18–20 status updates applied, and Wolfram/Gemini hostile review passes.