One Sphere, Two Physics: Information Causality, the Celestial Sphere, and a Flat-Spacetime Entanglement-Geometry Dictionary V3

We identify the qubit Bloch sphere, the relativistic celestial sphere, and the projective line CP¹ as one and the same manifold, proven by the rank-one factorization of a null vector, and show that the antipodal map on this sphere is simultaneously the EPR singlet condition and the BMS boundary-matching condition. We establish that the positivity Information Causality (IC) imposes on the correlation kernel on S² and the positivity Lorentz unitarity imposes on celestial correlators are two physical sources of one mathematical condition, Schoenberg positivity of zonal functions, bridged by an antipodal translation whose origin is the singlet structure. Built on Schoenberg/Gaussian states, the first-order entanglement variation reproduces the linearized Einstein equation and the second order gives the positive canonical energy; in AdS₃/CFT₂ the full nonlinear correspondence holds, while in realistic flat spacetime the entropy alone yields only these two orders. We then assemble a flat-spacetime entanglement-geometry dictionary whose nonlinear term is a Dyson-resummed propagator, and show that three constraints make it concrete: IC forces non-Gaussian states, Schoenberg positivity pins the three-graviton vertex uniquely to V = P₂ (cos d), and the conformal factor equals the measurement-angle operator by an exact SL (2, ℂ) continuation. From the closed resummation the nonlinear Einstein equation is recovered order by order. We then elevate this perturbative recovery to a non-perturbative theorem via the Bondi-Sachs formalism: IC enforces the three structural conditions (conformal flatness, spin-2 purity, BMS flux-balance) that the Bondi-Sachs theorem requires to reconstruct the full, nonlinear, non-perturbative vacuum Einstein equation G_μν = 0 in the 4D bulk. What is proven and what remains open are stated explicitly throughout; the principal remaining step is independent verification by the community. Changes in Version 3: •The non-Gaussian bend (Section 22) is now computed via exact diagonalization of an anharmonic φ⁴ chain, replacing an earlier mean-field surrogate. Slope corrected from 2. 19 to ~2. 02; non-Gaussianity certified by c₄ < 0. •The Ward identity test (Section 23) has been upgraded from a trivial tracelessness check to a full trilinear tensor contraction, proving that gauge legs decouple exactly on physical states. •The spin-2 pole calibration (Section 24) now uses an exact Sherman-Morrison closed form (zero free parameters, machine-precision agreement), replacing an earlier two-parameter fit that carried a spurious factor of 2. •The celestial three-point coupling closed form (Section 20, Eq. 17) is corrected to −√ (2/35).