Geometry from Information: Master Paper — The Universal GFI Principle for Petrov Type D Black Holes

We present the master framework of the Geometry from Information (GFI) programme. The central result is that the relative entropy Hessian δ²S_rel/δg² on any stationary black hole background of Petrov type D takes the universal form: K_abcd = (1/16πG_N) G_abcd + β W_abcd + O(ℓ_P²/r²) where β = |Ψ₂|² · r²_eff / 60π², and r²_eff = √(-g)/sinθ. The proof rests on three pillars: (I) Bisognano–Wichmann theorem connecting modular flow to Killing flow; (II) Osborn–Shore Ward identity on curved backgrounds; (III) Christensen–Fulling trace anomaly. Four explicit realisations are established: Schwarzschild (DOI: 10.5281/zenodo.19029740), Kerr (DOI: 10.5281/zenodo.19030319), Reissner–Nordström (DOI: 10.5281/zenodo.19032333), and Kerr–Newman (DOI: 10.5281/zenodo.19033062). Key results: (1) Universal GFI Principle: β = |Ψ₂|² r²_eff / 60π² holds for all Petrov type D spacetimes, with unified Weyl scalar Ψ₂ = -(MR-Q²)/R⁴, R = r-ia·cosθ. (2) β_H/T²_H = 1/15 universally for spherical horizons (a=0). (3) This universality breaks for rotating horizons (a≠0): β_H/T²_H becomes θ-dependent. Two open problems remain: OP-1 (non-local remainder) and OP-2 (derivation of G_N).