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

We propose a causal-diamond formulation of semiclassical gravity where 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 ?_?, ensuring the reduced state ?_? and reference family ?_? [Λ] are defined on identical degrees of freedom. Dynamics are defined by an informational principle: the relative-entropy functional quantifies the mismatch ?rel (?_? ∥?_? [Λ]) between data and reference. By discretizing gravitational stiffness (implemented as a finite response capacity of the boundary completion at the diamond waist) rather than the metric degrees of freedom, the variational principle is rendered well-defined. In the modular/KMS regime, the vacuum is at entanglement equilibrium, and the leading dynamics reduce to linear response governed by the Hessian of relative entropy (Kubo–Mori metric). This Hessian decouples tensor, vector and scalar deformations, recovering Einstein stiffness, Yang–Mills susceptibilities and mass gaps. Finite-resolution open updating induces a canonical reduction of non-abelian symmetry data to commuting Cartan phases, yielding a natural three-stage generation hierarchy. A quasi-local heat-kernel expansion maps the near-equilibrium response to a matching-scale EFT, identifying the leading ?² saturation operator and accommodating a spinorial transport structure. Topological edge-mode counting fixes the effective internal degeneracy ?; combined with Newton’s constant ?, this calibrates the matching scale ?_? ∼ 3 × 10¹³ GeV. At ?_?, the framework yields analytic relations for couplings and mass ratios as functions of discrete group-theoretic data of the boundary completion. Identifying ?_? with stiffness saturation places the high-curvature regime in a plateau universality class with ? at the 10⁻³ level. The architecture yields correlated, falsifiable targets governed by a single scale.