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. v1.3 - Added missing images. Converted to PDF.