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 baryon fraction Ω_b/Ω_m = 0.41, consistent with the IO geometric dark matter framework (a₀ = c²/r_s replacing particulate dark matter), brings zero-parameter χ² to 14.08, AIC-equivalent to ΛCDM. The IO matter-radiation equality z_eq = 1758 is a new zero-parameter prediction testable against Matter Power Spectrum data from Euclid and DESI.

Paper 4 objective precisely defined: derive Ω_b/Ω_m from the geometric dark matter framework to achieve zero fitted parameters against BAO data.

Companion to Paper 1: https://doi.org/10.5281/zenodo.18854813 and Paper 2: https://doi.org/10.5281/zenodo.18868612