Paper 3 of the Interior Observer Cosmological Framework. Beginning from the two founding premises, that the observable universe exists inside a Schwarzschild black hole and that the physics inside the horizon is the same as outside, this paper establishes the framework's input structure. Within the fixed upstream stack (the two premises, the measured mass and radius, the external Barbero-Immirzi parameter, the FIRAS-fixed observer-side temperature normalization, and the active projection branch), Paper 3 introduces no additional continuously fitted cosmological parameter; its only remaining per-epoch input is the observer coordinate, the moment at which the solution is read. Three structural results carry the paper. The Cosmological Invariant Theorem establishes that TIO (R) x R = hbar c sqrt (rₛ/lP) / (4 pi kB) = 1. 1718 x 10²7 K m is an exact algebraic identity at every readable radius, specializing to TIO (RU) = 2. 663 K at the current epoch. The Continuity Theorem proves the transfer function from the observer coordinate to the local transfer set (interior temperature, expansion rate, and characteristic acceleration a0 = 1. 3447 x 10^-10 m/s²) is analytic on the open expansion branch, via a structural lower bound on the Friedmann radicand, and the Monotonicity Corollary shows that set is strictly ordered, so distinct moments give distinct, ordered readouts and the framework is not fine-tuned. The early interior is the continuous mixed-fluid solution of Paper 5, with the clean Israel-junction acceleration ratio of 2 retained as its sharp-transition limit. Reproducibility bundle paper3-v2. 0 (SHA256 a6a8703bac7233359efceab0eb40b4ec63bf5d3dff9c4545893a08de758ca9e6, validator 24/24 PASS) is published at https: //github. com/dfife/io-framework-public/releases/tag/paper3-v2. 0. The empirical confrontation, including the Hubble rate, the matter-radiation equality redshift, and the light-element abundances, is derived on the active branch in Papers 24, 29, 30, and 34. https: //dfife. github. io/index. html v2. 0 (May 2026): Storytelling rebuild onto the active projection branch (Paper 10 / Paper 29: H₀ = 67. 58 km/s/Mpc, Ωₘ = 0. 349, Ωₖ = −0. 046, Ω_Λ = 0. 697, zero-fit). The v1. 7 fitted-H₀ DESI exercise and the Ωᵇ/Ωₘ baryon scan are dropped; empirical confrontation now lives in Papers 24, 29, 30, and 34. The discrete Vaidya-to-OS junction is replaced by Paper 5's continuous mixed-fluid interior, with the clean junction (acceleration ratio 2) retained as its sharp-transition limit (DERIVED/CONDITIONALVERIFIED). The transfer-function table is recomputed on the active branch; the current spatial epoch is the η = 1. 893 row (R = RU), and the Oppenheimer-Snyder (OS) dust-cycloid proper time is kept distinct from the radiation-inclusive master clock tbare (0) = 19. 181 Gyr (Paper 30). The cosmological invariant is corrected to TIO (R) × R = ℏc√ (rₛ/lP) / (4πkB) = 1. 1718 × 10²⁷ K·m, where the factor is the amplification scale, not γBI. zₑq is rehomed to Paper 30 (active value 3810). The Continuity Theorem is promoted to DERIVED/THEOREM (active branch) via a structural lower bound proving Friedmann-radicand positivity over the full open expanding domain η ∈ (0, π), and a Monotonicity Corollary is added (transfer set strictly ordered). The structural anti-fit basis, the Finite Horizon-Readable Quotient (Paper 1 v4. 1 Section 2. 11), is cited in Section 5. An Active Theorem and Definition Ledger (Appendix A) is added. Premises-led abstract restored; hyphens only. Reproducibility bundle: paper3-v2. 0 published (release tag paper3-v2. 0, SHA256 a6a8703bac7233359efceab0eb40b4ec63bf5d3dff9c4545893a08de758ca9e6, commit ac2ab94, validator 24/24 PASS) at https: //github. com/dfife/io-framework-public/releases/tag/paper3-v2. 0. See https: //github. com/dfife/io-framework-public/tree/main for claim-naming convention. v1. 7 (March 2026): Cycloid parameterization correction. The OS cycloid has been corrected from a (η) = (rₛ/2) (1+cos η) (contracting phase) to a (η) = (rₛ/2) (1−cos η) (expanding phase). η=0 is the Big Bang (a=0) ; η=π is maximum expansion (a=rₛ). The Israel junction conditions move from η=0 to η=π; the mathematics is identical, only the epoch label changes. §§2. 4–2. 5 removed: the Vaidya phase duration (τVaidya = 70. 66 Gyr) and full interior lifecycle (τₜotal = 181. 66 Gyr, 148. 3 Gyr elapsed) were computed under the contracting convention and are incompatible with the expanding-phase chronology where the current epoch is at 13. 8 Gyr (12. 4% of the dust half-cycle). Table 1 (transfer function spot-checks) fully recomputed under expanding convention by Codex; T×R invariant verified to machine precision at every row. Current spatial epoch at cycloid τ = 33. 4 Gyr maps to cosmological tₒbs = 13. 8 Gyr via the space-time decoupling. Paper 5 previously superseded the discrete Vaidya junction with a continuous mixed-fluid FRW interior. The Continuity Theorem survives with formula replacement. Δcycloid is downgraded; the purely spatial Δgeometric = x⁴ (1+γ²) = 5. 624 remains the framework definition. Title page reformatted to series standard. See Paper 21 v1. 1 for the full audit. 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: fb = 2γ/x = 0. 313, superseding the BAO-optimized value fb = 0. 25 used in this paper. The BAO scan results reported in §7. 4 remain mathematically valid at fb = 0. 25; annotations throughout mark where the framework's theoretical baryon fraction has changed. Paper 3's BAO-optimized fb = 0. 254 corresponds to the curvature exponent α = 3/2 in the family fb = 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, zₑq = 1758) are unaffected. 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).
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