The Interior Observer Cosmological Framework: Paper 3 — The Two-Phase Interior, the Cosmological Invariant, and the Observer Coordinate

Paper 3 of the Interior Observer Cosmological Framework. Derives the two-phase Vaidya-to-Oppenheimer-Snyder interior structure from Israel junction conditions with zero free parameters: metric continuity, extrinsic curvature continuity, and the equation-of-state acceleration ratio ä_rad/ä_dust = 2. The junction geometry motivates τ_Vaidya = r_s/c and the geometric identity τ_rad/τ_max = 2/π. The total interior lifecycle is 181.66 Gyr; the observable universe has completed 81.6%.
The Continuity Theorem proves the IO transfer function is an analytic law valid at every interior epoch. The cosmological invariant T_CMB × R_U = ħcγ/(4πk_B) is an algebraic identity. The observer coordinate τ_obs is the irreducible minimum input all physical theories require — the framework is not fine-tuned.
Two errors in Paper 1 are identified and corrected: (1) Ω_k normalization used a = r_s instead of a = R_U, corrected from −0.057 to −0.130; (2) DESI observable identification (D_V/r_d compared against D_M/r_d data at z ≥ 0.510). Both independently verified by Gemini. Core predictions (T_CMB = 2.663 K, ρ_Λ, a₀ = 1.345 × 10⁻¹⁰ m/s², γ derivation) are completely unaffected.

With H₀ as a single fitted parameter, the IO geometric expansion history achieves χ² = 7.57 (AIC = 9.57) versus ΛCDM χ² = 8.83 (AIC = 14.83) — statistically preferred with five fewer parameters. The IO matter-radiation equality z_eq = 1758 is a new zero-parameter prediction testable against Matter Power Spectrum data from Euclid and DESI.
v1.4 correction: The Ω_b sensitivity analysis in §7.4 used Ω_r (total radiation including neutrinos) instead of Ω_γ (photons only) in the baryon loading formula R_b, making R_b 41% too small and shifting the χ² minimum from f_b ≈ 0.25 to f_b ≈ 0.41. Corrected results: zero-parameter minimum is χ² = 14.05 at Ω_b/Ω_m = 0.25, with Ω_b = 0.049 — consistent with ΛCDM's fractional baryon density. The 1-parameter fit (H₀ = 62.3, χ² = 7.57) is unaffected. See Paper 4 Appendix A for the full diagnosis.
v1.5 correction:

The two-phase Vaidya-to-Oppenheimer-Snyder model used the Vaidya null dust metric for the radiation-dominated phase. Paper 5 demonstrates through independent symbolic tensor analysis (Wolfram/ChatGPT 5.3) that Vaidya null dust is anisotropic radial streaming, fundamentally incompatible with the isotropic thermal bath required for CMB acoustic oscillations and Big Bang nucleosynthesis. The radiation phase is correctly modeled as a continuous mixed-fluid closed FRW interior containing both radiation (p = ρ/3) and dust (p = 0), which naturally transitions through matter-radiation equality without a sharp junction. The Israel junction conditions analysis, the Continuity Theorem, the cosmological invariant, the dust-phase Friedmann equation, and the DESI BAO fit (χ² = 7.57) are all unaffected by this correction — they use only the dust-phase geometry. See Paper 5 for the full analysis.

Companion to Paper 1 (DOI: 10.5281/zenodo.18854813), Paper 2 (DOI: 10.5281/zenodo.18868612), Paper 4 (DOI: 10.5281/zenodo.18883069), and Paper 5 (DOI: 10.5281/zenodo.18889865).

v1.6 correction: Baryon sector annotations per Paper 12 (Baryon Dictionary Principle, DOI: 10.5281/zenodo.18936508). Paper 12 derived the baryon fraction from the framework's geometry: f_b = 2γ/x = 0.313, superseding the BAO-optimized value f_b = 0.25 used in this paper. The BAO scan results reported in §7.4 remain mathematically valid at f_b = 0.25; annotations throughout mark where the framework's theoretical baryon fraction has changed. Paper 3's BAO-optimized f_b = 0.254 corresponds to the curvature exponent α = 3/2 in the family f_b = 2γ/x^α; Paper 12's Baryon Dictionary Principle selects α = 1. The main results of this paper (two-phase interior, Continuity Theorem, cosmological invariant, observer coordinate, DESI BAO methodology, z_eq = 1758) are unaffected.