Relative-Entropy Variational Principle for Semiclassical Gravity with Finite-Resolution Boundaries

We propose a causal-diamond formulation of semiclassical gravity in which a finite-resolution boundary regulator supplies the edge structure for a local Wheeler–DeWitt description. Because the diffeomorphism-invariant Hilbert space does not factorize, each diamond is equipped with a boundary-completed algebra $\mathcal{A}_O$, ensuring that the operational state $\rho_O$ and the semiclassical reference family $\sigma_O[\Lambda]$ are defined on identical operator content. Dynamics are posed as local statistical inference: the relative-entropy functional $S_{\rm rel}(\rho_O\|\sigma_O[\Lambda])$ quantifies the mismatch between data and reference on the fixed algebra. Discretizing gravitational stiffness as a finite response capacity of the boundary completion renders the variational problem well-defined at fixed resolution.The framework synthesizes three previously disconnected foundations: causal diamonds for covariant locality, finite-resolution boundaries for capacity regulation, and relative entropy as the governing variational principle. This synthesis determines the minimal operational axioms required to define subsystems, intrinsic clocks and regulated observables in a background-independent setting.The axioms force the relative-entropy Hessian (the Kubo–Mori metric) to block-diagonalize into orthogonal tensor, vector and scalar sectors in the modular/KMS regime. A quasi-local heat-kernel expansion maps the near-equilibrium response to a matching-scale effective field theory, recovering Einstein stiffness, Yang–Mills susceptibilities, and mass gaps. Topological edge-mode counting fixes an effective internal degeneracy $N \approx 1.23 \times 10^{11}$; combined with Newton’s constant $G$, this calibrates the fundamental matching scale $M_s \approx 3.02 \times 10^{13}$ GeV. The tensor sector, scaling extensively with $N$, yields a plateau universality class with tensor-to-scalar ratio $r \sim 10^{-3}$, while scalar and vector responses remain strictly intensive.Translations between nested diamonds act as completely positive trace-preserving updates that drive non-abelian symmetry data toward commuting Cartan phases. This induces the canonical Lie-theoretical filtration $G \to \mathfrak{g} \to T$. Treating couplings and mass ratios as discrete topological invariants on the fixed history manifold $S^3 \times S^1$, the architecture yields correlated, falsifiable relations for cosmological parameters, electroweak scales, gauge couplings and the lepton mass spectrum, all governed by the single fundamental scale $M_s$.