One Sphere, Two Physics: Information Causality, the Celestial Sphere, and a Flat-Spacetime Entanglement-Geometry Dictionary V4

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. On this shared sphere the kernel E (d) = −cos d, the antipodal image of the conformal overlap, is the common form from which the boundary identities follow. Built on Schoenberg/Gaussian states, the first-order entanglement variation reproduces the linearized Einstein equation (verified, δS/δ⟨K⟩ →1) and the second order gives the positive canonical energy (Fisher structure, slope→2. 0) ; in AdS3/CFT2 the full nonlinear correspondence holds, while in realistic flat spacetime the entropy alone yields only these two orders. We show the finite curvature coefficient is topological (Gauss-Bonnet, −1/90 derived), isolating it from the entanglement-entropy divergence. 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 (breaking theamp2 barrier), Schoenberg positivity pins the three-graviton vertex uniquely to V = (3E2 −1) /2, and the conformal factor equals the measurement-angle operator by an exact SL (2, C) continuation. In the scalar-zonal limit, the Dyson series collapses by Funk– Hecke to a closed form with a graviton pole; an independent lattice simulation reproduces the predicted non-Gaussian deviation aboveamp2, with the spin-2 pole matched quantita- tively. However, the fullS-matrix requires an extended vertex. We show that IC provides the exact geometric premises (Lorentz invariance viaCP¹, massless spin-2 via the ℓ = 2 sector) for Weinberg’s soft graviton theorem, which fixes the extended vertex up to a single coupling constant c, yielding V = −c∆S². The Dyson resummation of this vertex yields Gℓ = G0, ℓ/ (1 −c), preserving the massless1/ℓ (ℓ+ 1) long-range structure and predicting a simultaneous pole for all allowed spins at the critical couplingc→1. This extended vertex 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 Gµν = 0 in the 4D bulk. The logical chain from Information Causality to the full Einstein equation is mathematically closed. 1 D. Süß One Sphere, Two Physics Changes in Version 4: The zonal obstruction is derived: the pure P₂ vertex cannot reproduce the Weinberg soft-graviton theorem (Funk-Hecke diagonality generates no ℓ ≥ 3 content, and the zonal soft kernel has a₂ < 0). Section 20's uniqueness claim is accordingly scoped to the scalar-zonal class, and the failure is recorded in Section 27. A non-zonal spin-2 extension (Section 25) is constructed: IC provides the geometric premises for Weinberg's theorem, which fixes an extended vertex V = −cΔₒℂ up to a single coupling constant c. Its Dyson resummation yields G_ℓ = G₀, ℓ/ (1−c), preserving the massless long-range structure and satisfying the soft-theorem amplitude ratios exactly. The three structural assumptions of the extension (perturbative factorization A_ℓ = V_ℓ G₀, ℓ S_ℓ; the ℓ (ℓ+1) ↔ p² dictionary; the tensor-harmonic bridge placing non-zonal multipole content in the source data) are stated explicitly in Section 25. The Bondi-Sachs reconstruction (Section 26) is rebuilt on the extended vertex; the Open section adds the interpretation of the critical coupling c → 1 as an explicitly open point. All Version-3 corrections are retained.