One Sphere, Two Physics: Information Causality, the Celestial Sphere, and a Flat-Spacetime Entanglement–Geometry Dictionary

This research identifies the qubit Bloch sphere, the relativistic celestial sphere, and the projective line $CP^1$ as a single manifold, proven via rank-one factorization of null vectors. It establishes that Information Causality (IC) on $S^2$ and Lorentz unitarity on celestial correlators are distinct physical manifestations of Schoenberg positivity, bridged by a canonical antipodal translation derived from the EPR singlet structure.The work provides a rigorous analytical derivation of the Einstein Field Equations (EFE) as a "State Equation of Information Causality." Key results include:1. First-order variation: Reproduction of the linearized Einstein equation ($\delta S/\delta \langle K \rangle \to 1$).2. Second-order variation: Recovery of positive canonical energy (Fisher structure).3. Nonlinear Resummation: Assembly of a flat-spacetime entanglement-geometry dictionary using a Dyson-resummed propagator, where the three-graviton vertex is uniquely pinned to $V = (3E^2 - 1)/2$ by Schoenberg positivity.4. Topological Invariance: Derivation of the curvature coefficient ($-1/90$) as a Gauss-Bonnet invariant.The theory is validated against lattice simulations, quantitatively matching the predicted non-Gaussian deviations and the spin-2 graviton pole.