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

This work formulates a finite-resolution, background-independent framework for local semiclassical gravity and examines its phenomenological consequences across cosmology and particle physics. The underlying construction is formulated on a causal diamond equipped with a boundary-completed algebra, reflecting the non-factorization of the diffeomorphism-invariant Hilbert space. This algebra supplies the edge data needed for a local Wheeler–DeWitt description and is shared by the operational state and semiclassical reference family, whose relative entropy defines the local response. The entropic boundary capacity, with its topological term set by the Euler curvature charge, fixes an effective channel multiplicity N ≈ 1.23 × 10¹¹. Calibrating the coherent tensor response to Newton's constant then gives the matching scale M_s ≈ 3.02 × 10¹³ GeV. In the equilibrium KMS regime, the leading relative-entropy Hessian separates into tensor, vector, and scalar response blocks, which a quasi-local heat-kernel expansion connects to the effective theory at the matching scale. The finite–continuum bridge uses the noncommutative-geometric algebra A_F ≃ ℂ ⊕ ℍ ⊕ M₃(ℂ) with a common trace convention to map finite responses to Standard-Model observables. Once the boundary capacity, Newton calibration, and finite internal structure are fixed, the primary projections require no sector-specific continuous retuning. In cosmology, one-resolution saturation of the scalar-curvature response sets the scalaron pole M_R = M_s and fixes its curvature stiffness λ_R² = M_P²/(12M_s²) ≈ 5.42 × 10⁸. The isotropic boundary response gives the coherence duration 𝒩* = 18π, yielding n_s ≈ 0.9646 and r ≈ 0.0038. All continuous dimensional scales cancel from the primordial scalar amplitude, leaving the finite-capacity relation A_s = 81π/N ≈ 2.08 × 10⁻⁹. Extensions of the same response give exploratory late-time targets for vacuum energy, structure growth, and the acceleration scale. In particle physics, single-pixel scalar saturation provides linked electroweak consistency tests, while the vector Hessian fixes correlated gauge responses. At the matching scale the construction gives α_s⁻¹ = 12π, α_em⁻¹ = 36π, and sin² θ_W = 7/18. The same finite geometry carries a Lie-theoretical filtration G → 𝔤 → T, whose successive accessibility levels organize the charged-lepton mass hierarchy, while canonical anticommutation relations provide the fermionic operator structure supporting the quark Yukawa hierarchy. Finally, finite-mesh simulations test the main analytical mechanisms numerically across the boundary, cosmological, gauge, and flavor sectors, including the modular filtration that selects the threefold generation carrier. The resulting cross-sector consistency makes the construction usefully fragile and directly testable across its correlated phenomenological projections.