The Interior Observer Cosmological Framework: Paper 10 - The Complete Holographic Friedmann Equation — Zero Free Parameters, One Physical Principle

We derive the complete Friedmann equation for an observer inside a Schwarzschild black hole — the Interior Observer (IO) Friedmann equation: H²(z) = [H²_m(z) + H²_Λ,bare] × √Δ + H²_k(z) × Δ^(−1/x²) where Δ = x⁴(1+γ²) = 5.624, x = r_s/R_U = 1.519, and γ_BI = 0.2375. The energy projection √Δ was derived in Paper 9 from the spectral norm of the Ashtekar-Barbero connection. The curvature projection Δ^(−1/x²) is derived here via a modular Hamiltonian theorem in the thermodynamic limit of N ~ 10¹²² horizon cells. The modular energy decomposes as ⟨K⟩ = K_geom + K_gauge = 4 ln(x) + ln(1+γ²) = ln(Δ), where K_gauge is derived from the covariant Laplacian on the horizon boundary (an algebraic identity of the SU(2) gauge structure) and K_geom follows from the Conformal Modular Principle — the framework's one irreducible physical principle, uniquely isolated by a no-go theorem excluding all local QFT alternatives.With one input (M_U = 4.50 × 10⁵³ kg) and zero fitted parameters, this equation produces: H₀ = 67.58 km/s/Mpc, θ* = 0.5964° (exact), ℓ₁ = 220 (exact), Age = 13.54 Gyr, Ω_k = −0.046 (matching the Planck CMB-only curvature anomaly), and Pantheon+ Δχ² = −1.8 (better than ΛCDM). Ten derivation attempts for the curvature projection were subjected to adversarial multi-AI review and killed before the modular Hamiltonian route survived. Nine independent QFT approaches were proven to give zero contribution (the No-Go Theorem), establishing that K_geom = 4 ln(x) is a gravitational measure effect — the unique survivor selected by diffeomorphism invariance.This is Paper 10 of the Interior Observer series. https://dfife.github.io/index.html v1.7 (April 2026): Paper 29 branch consistency audit. The i.i.d. curvature projection Δ^(−1/x²) in §5 and the projection map in §9 are superseded by the Schur complement P_k = exp(−Δ/2) (Papers 12–13). The observational confrontation in §8 and conclusion in §12 carry the retired mixed branch (H₀ = 67.58); the Schur definitive branch (H₀ = 68.91, Ω_m = 0.336, Ω_k = −0.006) is the active prediction (Paper 29). All component theorems (§4, §6, §10) are unaffected. v1.6 (March 2026): Inline revision markers removed. Content is now clean and currently accurate; version history documents all changes. No content changes. v1.5 (March 2026): Cycloid parameterization correction. The OS cycloid in §2 corrected from a(η) = (r_s/2)(1+cos η) to a(η) = (r_s/2)(1−cos η) (expanding phase); η updated from 0.196 to η_s = 1.893. K_geom reference epoch rebased: ln[a(0)⁴/a(η_s)⁴] → ln[a(π)⁴/a(η_s)⁴], since a(π) = r_s in the expanding convention. The numerical value K_geom = 4 ln x = 1.672 is unchanged. "Temporal decoupling factor" replaced with "spatial decoupling factor" throughout. The modular Hamiltonian derivation, curvature suppression Δ^(−1/x²), and all K_Λ results are convention-independent. Title page reformatted to series standard. See Paper 21 v1.1 for the full audit. v1.4 - Paper 19 correction annotations. The per-source universal projection architecture (§9) is superseded by Paper 19's global readout architecture: H²_obs = √Δ × [(8πG/3)(ρ_m + ρ_r + ρ_Λ) − k/a²], where √Δ acts as a single boundary-Hamiltonian readout factor on the entire bare constraint, not as separate projectors on individual source terms. The curvature projector Δ^(−1/x²) is superseded by the Schur complement P_k = exp(−N/2) with N still open. Core results (CMP, K_geom, K_gauge, modular energy decomposition) are unaffected. v1.3 - Added missing images. Converted to PDF.