One Sphere, Two Physics: Information Causality, the Celestial Sphere, and a Flat-Spacetime Entanglement-Geometry Dictionary V2

Information Causality (IC), the principle that classical channel capacity bounds information gain even with shared entanglement, is shown to enforce the full nonlinear vacuum Einstein equation G_μν = 0 as the unique consistent gravitational dynamics. The derivation proceeds in three stages: (i) IC plus Schoenberg positivity fix the celestial-sphere metric to be the round CP¹ and pin the graviton vertex uniquely to P₂ (cos d) ; (ii) the Dyson resummation with this vertex recovers the Einstein equation order by order; (iii) a non-perturbative completion via the Bondi-Sachs equivalence theorem shows that IC enforces the three boundary conditions (conformal flatness, spin-2 purity, BMS flux balance) that are equivalent to the full nonlinear field equation. The Bondi shear CAB remains as the physical gravitational-wave degree of freedom. No equivalence principle, no AdS/CFT, and no additional axioms beyond IC and the unitarity/positivity conditions it operates with are assumed. Version 2 adds: a non-perturbative Bondi–Sachs proof (Section 25), a representation-theoretic consistency check of the vertex via the Wigner–Eckart theorem and 3j-symbol (Section 20), and explicit statements of anomaly-freedom and helicity identification as prerequisites.