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) = H²ₘ (z) + H²_Λ, bare × √Δ + H²ₖ (z) × Δ^ (−1/x²) where Δ = x⁴ (1+γ²) = 5. 624, x = rₛ/RU = 1. 519, and γBI = 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⟩ = Kgeom + Kgauge = 4 ln (x) + ln (1+γ²) = ln (Δ), where Kgauge is derived from the covariant Laplacian on the horizon boundary (an algebraic identity of the SU (2) gauge structure) and Kgeom 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 (MU = 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, Ωₖ = −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 Kgeom = 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.