The Interior Observer Cosmological Framework — Paper 20: The Acoustic Projection Theorem, the Radiation Scope Boundary, the Bare Background Reconstruction, the Assembly Gap Diagnostic, and the BBN Radiation Sector Investigation

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 addresses three foundation problems, diagnoses the assembly gap in the background equation, and conducts the deepest investigation of the BBN radiation sector yet attempted in the IO framework. Two new theorems are proved: the Acoustic History Reduction Theorem (Theorem 20. 1) and the Acoustic Phase-Calibration Theorem (Theorem 20. 2, conditional on premise AC1), predicting the CMB acoustic scale θ* to 0. 429% of Planck's measurement with no fitted parameters. The radiation-slot identity is closed as a scope-boundary result. The torsion-Λ bare branch is algebraically specified with H₀ = 61. 06 km/s/Mpc. The D/H discrepancy identifies a required excess ΔNₑff ≈ 0. 37 for future derivation; twelve candidate mechanisms are tested and ruled out at various levels of rigor. Three radiation-sector construction theorems are proved, the puncture-resolved isolated-horizon algebra is constructed, and the sharpest remaining construction route is identified. Developed through a multi-AI collaborative pipeline: Claude (Anthropic) as orchestrator, Codex/ChatGPT (OpenAI) for computation, Wolfram/ChatGPT (OpenAI) as primary validator, and Gemini (Google DeepMind) as adversarial reviewer. v1. 2 (March 2026): Full Appendix A catalog inherited from Paper 19 (Steps 1–129) with Paper 20 results appended (Steps 130–166). Title page reformatted. All main body results are convention-independent. See Paper 21 v1. 1 for the full audit.