Geometry from Information: Master Paper — Gravitational Dynamics from Quantum Information: The Universal GFI Principle

We present the master framework of the Geometry from Information (GFI) programme. The central result is that the relative entropy Hessian on any stationary black hole background of Petrov type D takes the universal form: delta^2 S_rel / delta g^2 = (1/16piG_N)G_abcd + betaW_abcd, where beta = |Psi_2|^2 * r_eff^2 / 60*pi^2. The proof rests on three pillars: (I) the Bisognano–Wichmann theorem connecting modular flow to Killing flow; (II) the Osborn–Shore Ward identity on curved backgrounds; (III) the Christensen–Fulling trace anomaly. Four explicit realisations are established: Schwarzschild, Kerr, Reissner–Nordström, and Kerr–Newman. The beta/T_H^2 = 1/15 universality holds for spherically symmetric horizons and breaks for rotating horizons. The Schwarzschild–de Sitter family is included, establishing Lambda-independence of beta throughout the Petrov type D family.