Paper I — Emergent Geometry, Dynamics, and Quantum Structure from Correlation The Unified Information Substrate — UN Theory Formal Derivation of Theorem 1...

This paper presents the formal mathematical derivation of Theorem 1 within the Unified Information Substrate (UN Theory) framework. Beginning from a bivariate correlation kernel I(x,x') defined on an admissible Banach correlation manifold B_adm, it proves that space-time geometry and gravitational dynamics are forced emergent properties of substrate correlations. The emergent Lorentzian metric g_mu_nu[I] is constructed via coincidence-limit mixed partial derivatives of the convex functional F(I) = -ln I. Using a functional chain-rule decomposition for composite action functionals S[I, g[I]], the paper derives the scalar Euler-Lagrange operator E[I] and the Hilbert effective stress-energy tensor T_mu_nu^eff. By applying Noether's Second Theorem to the bivariate Lie derivative of the kernel, the off-shell conservation identity ∇^μ T_mu_nu^eff = -E[I] ∂_ν I is established, which reduces to exact covariant conservation ∇^μ T_mu_nu^eff = 0 on-shell. A Schwinger-DeWitt heat-kernel derivative expansion of the substrate Lagrangian produces the Einstein-Hilbert term with a strictly positive Newton constant a_1(I_0) = 1/(16 π G) > 0, forcing the emergence of the Einstein field equations G_mu_nu(g[I]) = 8 π G T_mu_nu^eff.