Emergent Geometry, Dynamics, and Quantum Structure from Correlation

We present the first complete derivation of the Einstein field equations and covariant stress‑energy conservation from a single bivariate correlation kernel I(x,x') defined on a smooth four‑dimensional manifold. The emergent metric gμν[I] is obtained from the coincidence‑limit mixed partial derivatives of the convex informational potential F(I)=−ln I, and its variation is governed by a distributional response operator Kμν(x; y, y') acting on admissible kernel fluctuations in the Banach manifold B_adm. Using the functional chain rule for composite maps S[I, g[I]], we derive the scalar Euler‑Lagrange operator E[I] and the Hilbert effective stress‑energy tensor T_eff. Applying Noether’s Second Theorem to the bivariate Lie derivative of I(x,x') yields the off‑shell identity ∇μT_effμν = −E[I]∂νI, which reduces to exact covariant conservation on‑shell. A Schwinger–DeWitt derivative expansion of the non‑local substrate Lagrangian produces the Einstein–Hilbert term with strictly positive coefficient a₁(I₀)=1/(16πG), guaranteeing attractive gravity. Variation of the effective action yields Gμν = 8πG T_eff, completing Theorem 1: the Hessian information geometry induced by I(x,x') necessarily satisfies the Einstein field equations with a conserved effective stress tensor.