Time Equivalence Class and Generalized Entropy Optimization:\\ Unified Rates, Rigorous Axioms, and Observation-Oriented Closed Framework

We propose and rigorize a unified framework centered on the time equivalence class [T], placing the ``arrow of time,'' ``black hole information,'' ``cosmological redshift/constant,'' and measurable ``delay--phase--spectral shift'' on a common computational/observational platform. First, we provide a precise definition of [T], prove reflexivity, symmetry, and transitivity of the equivalence relation, and clarify covariance criteria with respect to cut families and preservation under state/region changes. Second, we establish the unified rate identity $ \ \rho_{\mathrm{rel}(\omega)=1{2\pi}\,\partial_\omega \phi(\omega)=1{2\pi}TrQ(\omega)\ }, where Q(\omega)=-i\,S^\dagger(\omega)\partial_\omega S(\omega) is the Wigner--Smith time delay operator, \phi(\omega)=\arg\det S(\omega) is the total scattering phase, \rho_{rel}(\omega) is the spectral density defined by spectral shift/relative entropy; this identity arises from Birman--Kreĭn relations and observable phase--delay measurements, with consistent dimensions and invertibility. Third, in the semiclassical regime with Hadamard states and weak curvature, via the chain ``relative entropy monotonicity \Rightarrow QNEC \Rightarrow local GSL/QFC,'' we prove: along a null geodesic family with affine parameter \lambda, S_{gen} is monotonic; reparametrization by [T] gives S_{gen} monotonicity with respect to representative time t, thus the arrow of time emerges as an output property. Fourth, using algebraic embedding/entanglement wedge language, we show ``fixed-projection non-decodability \equiv time-map singularity at event horizon,'' and realize analytic continuation through extremal switching of the island formula, thereby recovering the Page curve. Fifth, we view \Lambda as a global calibration integration constant of [T] (compatible with four-form mechanisms), emphasizing its distinction from the measurable effect of local ``vacuum energy density.'' Finally, we provide three operational verification pathways: dispersion--geometry coupled time delay of order \omega^{-2} in curved spacetime plasma geometrical optics (with null-test protocol), hierarchical Bayesian test of ``redshift--decoherence slope'' for FRBs, and time-delay Bell witness and chiral splitting in cryogenic multi-mode cavities. Appendices include: complete proofs of three equivalence properties; operator--spectral shift derivation of the unified rate identity; proof of main theorems driven by QNEC/GSL; curved spacetime--plasma eikonal expansion (showing conditions for absence of \omega^{-1}$ principal term); gauging, stability, and low-energy constraints of time-field theory.