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

Time Equivalence Class and Generalized Entropy Optimization: Unied Rates, Rigorous Axioms, and Observation-Oriented Closed Framework Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract We propose and rigorize a unied 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 denition of [T] , prove reexivity, 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 unied rate identity ρrel(ω) = 1 2π∂ωϕ(ω) = 1 2πTr Q(ω), where Q(ω) = −i S†(ω)∂ωS(ω) is the WignerSmith time delay operator, ϕ(ω) = arg det S(ω) is the total scattering phase, ρrel(ω) is the spectral density dened by spectral shift/relative entropy; this identity arises from BirmanKren 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 ⇒ QNEC ⇒ local GSL/QFC, we prove: along a null geodesic family with ane parameter λ , Sgen is monotonic; reparametrization by [T] gives Sgen 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 xed-projection non-decodability ≡ 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 Λ as a global calibration integration constant of [T] (compatible with fourform mechanisms), emphasizing its distinction from the measurable eect of local vacuum energy density. Finally, we provide three operational verication pathways: dispersiongeometry coupled time delay of order ω−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 unied rate identity; 1 proof of main theorems driven by QNEC/GSL; curved spacetimeplasma eikonal expansion (showing conditions for absence of ω−1 principal term); gauging, stability, and low-energy constraints of time-eld theory. Keywords: Time Equivalence Class; Modular Time; Generalized Entropy; QNEC; QFC; QES/Island Formula; WignerSmith Time Delay; BirmanKren Spectral Shift; Plasma Geometrical Optics; Hierarchical Bayesian  1 Introduction and Historical Context The generalized entropy Sgen =A/4Gℏ+Sout in semiclassical quantum gravity connects geometry and information. Non-perturbative reconstruction of the Page curve relies on quantum extremal surfaces and the island formula; QNEC and (local) QFC relate nulldirectional entropy deformation to stress-energy tensors and quantum focusing, and can be derived from relative entropy monotonicity. On the other hand, TomitaTakesaki modular theory views the modular ow σω s of a statealgebra pair (A, ω) as intrinsic time, providing mathematical support for the perspective that time emerges from informationcausal structure. On the scattering side, the WignerSmith delay operator and total scattering phase ϕ are directly measurable, and the Kren spectral shift function ξ(ω) couples with det S(ω) via the BirmanKren formula. This paper closes these supporting points into a unied whole: governing time through the equivalence class structure of [T] , and bringing phasedelayspectral shift into a common experimentally accessible gauge through the unied rate .  2 Model and Assumptions 2.1 Time Equivalence Class and Covariance Objects : Globally hyperbolic (M, g) ; region family R ; statealgebra pairs {(A(R), ωR)}R∈R ; cut cluster of null geodesic families C . Denition 2.1 (Equivalence (Denition 2.1)) . For xed (A(R), ωR,C) , we say T1∼T2 if there exist an outer automorphism Φ∈Out(A(R)) and a strictly monotonic f:R→R such that Φ◦σωR s◦Φ−1=σωR f(s) and E(C;T1) = E(C;T2), where E represents the generalized entropy monotonicity/extremal structure (quantum expansion signature and null set) along null generators of C . Proposition 2.2 (Three Properties (Proposition 2.2)) . ∼ is an equivalence relation; for completely positive trace-preserving maps induced by state/region homotopy deformations, if the outer conjugacy class of relative modular ow is invariant, then [T] is preserved. Under cut family renement, if Θq≥0 , then E is preserved, and [T] is said to be covariant under that null geodesic family. Proof in Appendix A.1A.2. 2 2.2 Unied Rate and Dimensional Consistency WignerSmith and Phase : Q(ω) = −i S†∂ωS , Tr Q=∂ωarg det S=∂ωϕ . Spectral Shift and Phase : BirmanKren: det S(ω) = e−2πi ξ(ω)⇒∂ωϕ=−2π ξ′(ω) . Unied Rate : ρrel(ω) = −ξ′(ω) = 1 2π∂ωϕ(ω) = 1 2πTr Q(ω). The denition of ρrel adopts local window variation, so that S(ρ|σ) = Rρrel(ω) dω ; dimensions are consistent with ∂ωϕ , Tr Q . Numerical inversion is regularized using KramersKronig and phase unwrapping. Derivation and inversion details in Appendix A.3. 2.3 Domain of Assumptions and Failure Conditions Hadamard states, weak curvature, local Rindler approximation and controlled deformations; geometric congurations restricted to quantum light sheets/event horizons and other GSL-applicable situations. Failure domains: strong curvature neighborhoods, nonHadamard states, UV-dominated deviations, etc.  3 Main Results (Theorems and Alignments) 3.1 Theorem 3.1 (Arrow of Time = Output of Generalized Entropy Monotonicity) Within the domain of Section 2.3, along a specied quantum light sheet with ane parameter λ : dSgen/dλ≥0 . For any t∈[T] , if t=f(λ) with f′>0 , then dSgen/dt≥0 . Proof chain and failure conditions detailed in Appendix B. 3.2 Theorem 3.2 (Black Hole Information: Fixed-Projection NonDecodability and Island Formula Analytic Continuation) Using algebraic embedding A(I+)⊂ A(D) to express external observations; xed projection corresponds to irreversible CPTP dimensionality reduction maps, information loss is merely non-decodability under that projection. Island formula through saddle-point switching (QES) is equivalent to analytic continuation within [T] , expanding the reconstructable domain ⇒ Page curve recovery. Appendix C provides explicit isomorphism in JT scenarios. 3.3 Proposition 3.3 (Redshift = Time Unit Rescaling) (1 + z)=(k·u)e/(k·u)o is a covariant expression of rhythm ratio, equivalent to FRW a0/ae . The increment is manifested in the unied rate directly relating this ratio to observational inversion of ϕ′(ω) and ρrel . 3 3.4 Theorem 3.4 (Time Holography and Time Quantum Error Correction) JLMS and entanglement wedge nesting imply: [T(p)] = Π(s, γp) ; the projection family of [T] satises KnillLaamme conditions on code subspace C , thus time-selection errors can be corrected by equivalence class redundancy. Appendix D provides modular Berry curvature and measurable phases of path dependence. 3.5 Theorem 3.5 ( Λ as GaugeIntegration Constant and FourForm Discretization) Trace-free/Unimodular and four-form mechanisms make Λ an integration constant; in the thermal time gauge its semantics is a global calibration of [T] , compatible with neardiscrete spectrum induced by axionfour-form quantization. Appendix E provides action and variation.  4 Proofs 4.1 Arrow of Time (Theorem 3.1) Relative entropy monotonicity ⇒ ANEC/QNEC; combined with quantum Raychaudhuri gives quantum expansion non-increase in null direction, Sgen monotonicity. Strict monotonicity of t=f(λ) gives monotonicity for any representative time. Failure domains include strong curvature and non-Hadamard states. Details in Appendix B. 4.2 Coordinate-Independent Formulation of Black Hole Information (Theorem 3.2) Characterize non-decodability using relative entropy and Petz recovery; extremal switching of island formula is equivalent to changing representative of [T] and expanding entanglement wedge. In JT model, demonstrate this equivalence using replica geometry. 4.3 Unied Rate (Equation 2.2) BirmanKren: det S=e−2πiξ ⇒∂ωϕ=−2πξ′ ; multi-channel Tr Q=∂ωϕ . Dene ρrel =−ξ′ to obtain the stated identity. Numerical inversion controls noise amplication using phase unwrapping and KramersKronig regularization.  5 Model Application: Two Minimal Models 5.1 1+1 Dimensional CFTRindler Single-channel scattering S=eiϕ ⇒Q=∂ωϕ ; decompose relative entropy into spectral window integrals to verify ρrel = (2π)−1∂ωϕ . KMS scale consistent with modular ow rescaling. 4 5.2 JT Gravity + Free Field Page transition corresponds to QES saddle-point switching; analytic continuation in [T] converts external projection non-decodability to expanded reconstruction domain decodability. Appendix C provides equation family and schematic curves.  6 Engineering Proposals (Magnitudes, Systematics, and Null-Tests) 6.1 Deep Space Multi-Frequency Links: Time Delay in Curved SpacetimePlasma Eikonal Geometrical Optics : Static weak eld Φ/c2≪1 , isotropic plasma: n(ω, x) = q1−ω2 p(x)/ω2,dt dℓ≃1 c1 + 2Φ c21 + ω2 p 2ω2. Path variation gives ∆t(ω)=∆tShapiro +Zω2 p 2ω2 dℓ c+ZΦ c2 ω2 p ω2K(x)dℓ c+O(ω−4). Conclusion : Dominant coupling term is ω−2 rather than ω−1 ; ω−1 only possible in anisotropic media or strong non-adiabatic uids. Magnitudes and Systematics : Ka/X (832 GHz) at impact angles 5◦−15◦ can achieve picosecond-level dierences; main systematic errors are ionospheric/heliospheric modeling and hardware nonlinearity. Null-Tests : Geometric commutation (impact angle ip) should preserve ω−2 scaling while changing geometric weight sign; day-night dierence for same geometry cancels ionospheric principal term. Facilities : DSN X/Ka and DSAC stability links. 6.2 FRB RedshiftDecoherence Slope Hierarchical Bayesian Model : W∼W0(1 + z)α(ν/ν0)−β , host/IGM/instrument components in hierarchical priors; selection function based on channelization threshold and DM z inversion uncertainty. Power : Catalog-1 and subsequent localized samples at N∼102−103 can distinguish the null hypothesis α= 0 at 5σ ; null-test with randomized z should wash out slope. 6.3 Cryogenic Multi-Mode Cavity: Time Delay Bell Witness and Chiral Splitting BellTime Delay Witness : W=|TrQA⊗TrQB−TrQAB| , classical ≤0 , quantum coupling W>0 . Chiral Splitting : Controlled micro-distortion and gravityelectromagnetic weak coupling cause ∆tchiral ∝ω linear splitting. 5 Noise Budget : Contributions from phase white noise, TLS 1/f , thermally induced frequency shifts, mechanical microphonics, counting dead time, etc., and target subpicosecond sensitivity are provided in Appendix G with formulas/tables.  7 Time-Field Theory: Gauging, Stability, and LowEnergy Constraints Introduce clock 1-form uµ=∂µT √−∂αT ∂αT, construct minimal action LT=M2 T 2hc1∇µuν∇µuν+c2(∇µuµ)2+c3aµaµi+V(T) + Lmatter(ψ;uµ), using Stückelberg to handle reparametrization redundancy T7→ f(T) . Ghost-free and causally stable domain, PPN and gravitational Cherenkov constraints give feasible subspace for (ci) ; key distinction from æther/khronon is: the preferred direction here is merely a gauge representative of equivalence class, observability concentrated on rate invariants ∂ωϕ , Tr Q rather than anisotropy.  8 Discussion (Consistency, Boundaries, and Connections)  Consistency : Main theorems are strictly restricted to Hadamard/weak curvature and quantum light sheet/event horizon geometry; outside the domain, only conjecture strength is maintained.  Black Hole Information : Algebraicisland formula isomorphism avoids coordinate dependence; JT scenarios provide checkable instances.  Λ : As a gauge integration constant, not equal to vacuum energy density; compatible with four-form discretization/axion mechanisms.  Veriability : Deep space links and FRB pipelines provide reproducible implementation elements and null-tests; cryogenic cavity experiments provide indoor repeatable verication platforms.  9 Conclusion We establish the rigorous equivalence structure of [T] and the unied rate, provide common semantics for arrow of time, black hole information, redshift, and Λ , and ground the unication of timecausalityinformation at the data level through implementable 6 observational/experimental pathways. This framework closes under the triple criteria of provablequantiableveriable.  Acknowledgements, Code Availability Thanks to public literature and materials on QNEC/QFC, QES/island formula, modular theory, spectral shiftscattering theory, and curved spacetime plasma geometrical optics. Appendices provide inversion from phase data to ρrel and minimalist implementation protocol for FRB hierarchical Bayesian.  References [1] Bousso, Fisher, Leichenauer, Wall. Quantum focussing and inequalities, Phys. Rev. D 93 (2016). [2] Faulkner et al. Modular Hamiltonians for deformed half-spaces and the ANEC, JHEP (2016). [3] Engelhardt & Wall. Quantum extremal surfaces, JHEP 01 (2015) 073. [4] Almheiri et al. Replica wormholes and the entropy of Hawking radiation, JHEP 05 (2020) 013. [5] Connes & Rovelli. Von Neumann algebra automorphisms and the thermal time hypothesis, (1994). [6] Smith. Lifetime matrix in collision theory, Phys. Rev. 118 (1960). (Wigner Smith) [7] Birman & Kren. On wave and scattering operators, Sov. Math. Dokl. (1962). [8] Perlick. Ray optics in a plasma on a GR spacetime, and related works. [9] Bisnovatyi-Kogan & Tsupko. Gravitational lensing in plasma, reviews (2015 2022). [10] Rogers. Frequency-dependent lensing in plasma, MNRAS 451 (2015). [11] JLMS and successors: boundarybulk relative entropy equivalence. [12] Standard cosmology texts for redshiftprojection identity. [13] DSN/DSAC capability briefs. [14] CHIME/FRB Collaboration. The rst catalog, ApJS 257 (2021).  7 A Proofs of Equivalence Relation and Unied Rate A.1 Three Properties and Outer Conjugacy (A.1) Group action of G= Out(A)⋊Diff+(R) and order preservation give reexivitysymmetry transitivity; preservation under state/region homotopy classes guaranteed by outer conjugacy invariance. A.2 Cut Family Covariance (A.2) Within QNEC sucient-necessary domain, cut renement corresponds to subalgebra restriction, Sgen order property preserved. A.3 BirmanKren ⇒ Unied Rate (A.3) det S=e−2πiξ ⇒∂ωϕ=−2πξ′ ; multi-channel Tr Q=∂ωϕ , yielding ρrel =−(ξ′) = (2π)−1Tr Q . Phase unwrapping and KramersKronig provide robust inversion. B Relative Entropy ⇒ QNEC ⇒ Local GSL Form B.1 Data processing inequality and subalgebra restriction; B.2 Second-order formula for modular Hamiltonian under half-space deformation derives ANEC/QNEC; B.3 Quantum Raychaudhuri combined gives Θq≤0 , hence dSgen/dλ≥0 . C Algebraic Proposition of Black Hole Information and JT Example C.1 Fixed projection corresponds to irreversible CPTP, Petz recovery quanties nondecodability; C.2 Island formula saddle-point switching equivalent to changing representative in [T] and expanding entanglement wedge; C.3 JT scenario: replica geometry saddle-point switching example diagram for spectral window projection of ϕ′(ω) . D Modular Berry and Time Quantum Error Correction D.1 [T(p)] = Π(s, γp) = P expRγpAmod ; D.2 KnillLaamme condition satisfaction and threshold under time-selection error noise model. E Λ as GaugeIntegration Constant and Four-Form E.1 Action and variation; 8 E.2 Near-discrete spectrum under quantization/axion coupling; E.3 Semantic correspondence with equivalence class calibration zero point. F FRB Hierarchical Bayesian Pipeline (Reproducible Implementation Skeleton) Minimalist ow for likelihood, priors, selection function, posterior-predictive, and nulltests. G Cryogenic Cavity Experiment Noise Budget Contribution formulas, representative parameters, and conguration table for achieving sub-picosecond for phase white noise, TLS 1/f , thermally induced frequency shifts, mechanical microphonics, readout dead time.  Symbol Table [T] ; σω s ; ϕ(ω) ; Q(ω) ; ρrel ; Θq ; λ ; C . 9