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

This work formulates semiclassical gravity within a causal-diamond framework where a finite-resolution boundary provides 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 AO, ensuring the operational state ρO and the semiclassical reference family σO[Λ] share identical operator content. Dynamics are posed as local statistical inference: the relative-entropy functional Srel(ρO∥σO[Λ]) quantifies the mismatch between data and reference. This yields the minimal operational axioms defining subsystems, intrinsic clocks, and regulated observables in a finite-resolution, background-independent setting. The topology-locked boundary capacity budget fixes an effective channel multiplicity N≈1.23×1011. Calibrating its coherent fraction to Newton’s constant determines a matching scale Ms≈3.02×1013GeV. In the modular/KMS regime, the relative-entropy Hessian (Kubo–Mori metric) block-diagonalizes into orthogonal tensor, vector, and scalar response sectors. A heat-kernel expansion on the fixed S3×S1 history manifold maps this near-equilibrium response to a matching-scale effective field theory, yielding the Einstein–Hilbert tensor structure, Yang–Mills susceptibilities, and leading mass deformations. Vector and scalar responses remain intensive, while the tensor response scales extensively with coherent channel multiplicity. The fixed modular protocol and quantized boundary currents imply α−1(Ms)=4πk at integer levels k, while the reduced R2 plateau sector yields linked cosmological targets: ns≃0.965, r≃0.0038, and As≃2.1×10−9. Translations between causal diamonds act as completely positive trace-preserving (CPTP) updates. The resulting open-modular Walsh filtration selects the three-dimensional degree-one sector as the algebraic basis for family structure. Treating continuum fields as the structured response of a finite boundary, the framework yields correlated, falsifiable relations for gravitational stiffness, gauge response, plateau cosmology, and threefold matter-sector organization from one minimal operational architecture.