Unified Time Scale and Boundary Time Geometry:\\ Single Structural Framework of Scattering Phase, Modular Flow, and Gravitational Boundary Terms

Unied Time Scale and Boundary Time Geometry: Single Structural Framework of Scattering Phase, Modular Flow, and Gravitational Boundary Terms Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract We construct a time unication framework with boundary as fundamental stage, integrating three originally separate time structures into dierent projections of the same boundary time geometry: (1) On scattering and spectral theory end, based on BirmanKren formula and WignerSmith time delay, prove scale identity among total scattering phase derivative, relative state density, and group delay trace; (2) On operator algebra and information end, based on TomitaTakesaki modular theory and ConnesRovelli thermal time hypothesis, characterize modular ow parameter as intrinsic time determined by statealgebra pair, introducing time scale equivalence class; (3) On gravity and geometry end, based on EinsteinHilbert GibbonsHawkingYork action and its boundary variation, unify extrinsic curvature and time translation generated by boundary Hamiltonian into same boundary time geometry. In unied model, boundary is described by triple structure: intrinsic metric and extrinsic curvature of geometric boundary ∂M , quantum boundary algebra A∂ with state ω , and scattering matrix S(ω) dened in external region. Under well-posed traceable scattering assumptions, construct scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where φ(ω) = 1 2arg det S(ω) is total scattering phase, ρrel is derivative of spectral shift density, Q(ω) = −iS(ω)†∂ωS(ω) is WignerSmith time delay matrix. This scale is standardized at modular time and geometric time ends respectively through modular Hamiltonian operator Kω=−log ∆ω and HamiltonJacobi functional of GHY boundary action, thus dening single boundary time scale equivalence class [τ] .Based on this, we give several unication theorems: (i) Categorical existence uniqueness of time scale equivalence class: on common domain of given boundary algebra, scattering data, and gravitational boundary geometry, all acceptable time parameters are monotonic rescalings of one fundamental boundary time; (ii) Cosmological redshift relation 1 + z= 1/a(t) can be interpreted as global rescaling of this time equivalence class on large scales, thus unifying local scattering time delay with conformal time in FRW background; (iii) In cases with horizons (Rindler wedge and 1 black hole exterior), modular ow time, proper time of accelerated observer, and geometric outward normal translation time fall into same equivalence class. Full text gives explicit assumptions and theorems at scatteringspectral, modular owinformation, and gravityboundary geometry ends respectively, with detailed proofs of scale identity, modular time equivalence, and boundary Hamiltonian generated time in appendices, nally proposing engineered measurement schemes based on waveguides, microwave cavities, and AharonovBohm rings to cross-calibrate three types of time scales experimentally. Keywords: Boundary Time Geometry; BirmanKren Formula; WignerSmith Time Delay; Spectral Shift Function; TomitaTakesaki Modular Flow; Thermal Time Hypothesis; GibbonsHawkingYork Boundary Term; Time Scale Equivalence Class; Cosmological Redshift  1 Introduction and Historical Context Time appears in physical theories in multiple guises: as coordinate parametrizing worldlines in classical and relativistic physics, as continuous variable generating unitary evolution in quantum mechanics, as imaginary time period inversely proportional to temperature in statistical physics, as delay of particle dwelling in interaction region in scattering theory, and in general relativity reecting as geometric structure through metric and extrinsic curvature. Dierent time perspectives often based on dierent fundamental objects and measurement processes, thus dicult to unify within single mathematical framework for long time. In scattering and spectral theory aspects, LifshitsKren spectral shift function ξ(λ) and its derivative play central role in describing state density dierence before and after interaction; BirmanKren formula gives precise relationship between scattering matrix determinant and spectral shift function det S(λ) = exp(−2πiξ(λ)) . Under appropriate normalization, derivative of spectral shift density can be interpreted as total scattering phase derivative and equivalent to trace of WignerSmith time delay matrix, thus viewing time delay as geometric derivative of phasespectral structure. In operator algebra and quantum statistics aspects, TomitaTakesaki theory shows: given von Neumann algebra (M, Ω) with cyclic and separating vector, can construct modular automorphism group σΩ t(x)=∆itx∆−it through polar decomposition of modular operator ∆ , whose image on outer automorphism group independent of chosen state. ConnesRovelli thermal time hypothesis proposes: in generally covariant quantum theory, physical time ow determined by modular ow of statealgebra pair, modular parameter itself is time, thus time becomes derived object of state structure rather than a priori parameter. In gravity and geometry aspects, EinsteinHilbert action on manifold with boundary insucient to give well-dened variational principle, must add GibbonsHawkingYork boundary term SGHY =1 8πG Z∂M d3y ϵ√h K, where hab is induced boundary metric, K is trace of extrinsic curvature, ϵ=±1 depends on normal type. Variation of EH+GHY combined action under xed boundary induced geometry gives Einstein equations, boundary term can be viewed as HamiltonJacobi 2 functional, whose functional derivative with respect to boundary metric corresponds to conjugate momentum and quasilocal energy, thus encoding time generator of translation along boundary normal. Structural equivalences among scattering phase derivativetime delay, modular ow parameterthermal time, GHY boundary termgeometric time are each highly mature mathematically and physically meaningful, but their structural equivalence only appears scattered in existing literature. For example, scattering phase derivative can be interpreted both as state density dierence and as group delay through frequency derivative of S-matrix; in S-matrix statistical mechanics, scattering phase derivative enters state density integral, thus connecting with heat and free energy. Modular ow used in AdS/CFT and entanglement wedge reconstruction to dene modular Hamiltonian and energy of geometric region, connecting to propagation time in spacetime through HKLL/Petz reconstruction. GHY boundary term viewed as key object dening gravitational transfer amplitude and boundary time in loop quantum gravity and quasilocal Hamiltonian formalism. Goal of this paper is on rigorously provable basis to unify above three ends into boundary time geometry framework. This framework takes boundary algebra, state, scattering matrix, and boundary geometry as fundamental objects, with time scale equivalence class as core, proving: under appropriate assumptions, scattering time delay, modular time, and geometric boundary time belong to same equivalence class, any physical time reading can be viewed as monotonic rescaling of single boundary time parameter. Furthermore, incorporating cosmological redshift and scale factor evolution in FRW spacetime into same scale structure, connecting macroscopic cosmic time with microscopic scattering time delay.  2 Model and Assumptions This section gives mathematical and physical model supporting unied framework, explicitly specifying assumption domain. 2.1 Scattering and Spectral End Consider self-adjoint operator pair (H, H0) acting on separable Hilbert space H , satisfying following standard scattering assumptions: (1) H0 possesses absolutely continuous spectral subspace Hac(H0) , in which energy representation can be established, making H0 multiplication operator E7→ E in that representation; (2) Perturbation V=H−H0 such that for some p≤1 , (H+i)−p−(H0+i)−p∈S1 , thus satisfying basic condition of trace-class scattering theory; (3) Wave operators W±= s - limt→±∞ eitHe−itH0Pac(H0) exist and are complete, thus scattering operator S=W† +W− well-dened on Hac(H0) . In energy representation, S bers into family of unitary matrices S(ω) , where ω denotes energy or frequency variable. Assume for almost everywhere ω , S(ω)−⊮∈S1 , 3 thus determinant det S(ω) and WignerSmith time delay operator Q(ω) = −iS(ω)†∂ωS(ω) well-dened and traceable. Dene spectral shift function ξ(ω;H, H0) as function satisfying LifshitsKren trace formula, whose derivative ξ′(ω) gives relative state density dierence: ρrel(ω) := −ξ′(ω) . 2.2 Modular Flow and Thermal Time End Let M⊂B(H) be von Neumann algebra, Ω∈ H be cyclic and separating vector, |Ω|= 1 . Denote vector state ω(x) = (xΩ,Ω) . TomitaTakesaki theory gives polar decomposition S=J∆1/2 of closed operator S , where J is modular conjugation, ∆ is modular operator. Modular automorphism group dened as σω t(x) = ∆itx∆−it, t ∈R. σω t is one-parameter automorphism group of M , and ω is its KMS state. Connes proved: for any two faithful states ω, ω′ , their modular ows' images in outer automorphism group Out(M) are consistent, i.e., there exists 1cocycle ut∈M such that σω′ t= Ad(ut)◦σω t , thus obtaining state-independent geometric time on Out(M) . Thermal time hypothesis proposes: physical time ow determined by modular ow, i.e., modular parameter t itself is time scale, rather than given by a priori background. 2.3 Gravity and Boundary Geometry End Consider four-dimensional spacetime manifold (M, gµν) with boundary ∂M . Gravitational action chooses EinsteinHilbert plus GibbonsHawkingYork sum Sgrav =1 16πG ZM d4x√−g R +1 8πG Z∂M d3y ϵ√h K. Assume boundary is smooth three-dimensional manifold of non-zero measure, distinguishing spacelike and timelike boundaries. When varying while keeping boundary induced metric hab xed, volume term gives Einstein equations, boundary term variation determines boundary conjugate momentum and quasilocal energy. In some cases (such as vacuum region, partial LQG construction), volume term vanishes on shell, HamiltonJacobi action completely given by GHY boundary term, thus boundary action itself becomes object generating normal time evolution. 2.4 Unied Boundary System and Time Scale Equivalence We call triple data B= (A∂, ω∂, S(ω); hab, Kab) a boundary system, where A∂ is boundary algebra generated by scattering channels and near-boundary elds, ω∂ is faithful state on it (can be given by scattering incoming state or boundary CFT state), S(ω) is scattering matrix on energy shell, hab, Kab are intrinsic and extrinsic data of geometric boundary. On boundary system, we allow three types of one-parameter evolution: 4 (1) Phasedelay scale on scattering energy parameter ω , determined by S(ω) and Q(ω) ; (2) Modular parameter tmod , generated by modular ow of (A∂, ω∂) ; (3) Geometric parameter tgeom , generated by translation along boundary normal (or evolution driven by extrinsic curvature). Core assumption of unied framework is: under appropriate physical situations (such as asymptotically at or AdS spacetime innite boundary, Rindler wedge near black hole horizon, conformal boundary of FRW universe), above three types of parameters can all be dened and satisfy common equivalence relation, forming time scale equivalence class [τ] . 3 Main Results (Theorems and Alignments) This section states main theorems and correspondences of unied framework, strictly distinguishing known results from newly introduced structures. Theorem 3.1 (Scattering PhaseSpectral ShiftGroup Delay Scale Identity (Theorem 1)) . Under aforementioned scattering assumptions, let spectral shift function be ξ(ω;H, H0) , dene relative state density ρrel(ω) := −ξ′(ω). Let total scattering phase Φ(ω) := arg det S(ω), φ(ω) := 1 2Φ(ω), WignerSmith delay operator Q(ω) = −iS(ω)†∂ωS(ω). Then for almost everywhere ω , scale identity holds φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). Brief: First equality from BirmanKren formula det S(ω) = exp(−2πiξ(ω)) and relationship between spectral shift function and state density; second equality from connection between derivative of ln det S(ω) with respect to frequency and WignerSmith operator trace. Theorem 3.2 (Time Scale Equivalence Class of Modular Flow (Theorem 2)) . Let (M, Ω) be von Neumann algebra with cyclic separating vector as above, ω, ω′ be two faithful states, dening modular ows σω t, σω′ t respectively. (1) There exists family of unitary operators ut∈M satisfying 1cocycle condition ut+s=utσω t(us) , such that σω′ t= Ad(ut)◦σω t. 5 (2) In outer automorphism group Out(M) , [σω t] = [σω′ t] , time parameter t scale only undergoes linear rescaling when state changes. (3) If there exists geometric clock (such as inertial or uniformly accelerated observer) whose proper time τphys relates to modular parameter tmod through KMS temperature β (such as Unruh temperature T=a/2π or thermal equilibrium state), then τphys = αtmod , where α determined by β . Therefore modular parameter denes time scale equivalence class [τmod] . Theorem 3.3 (GHY Boundary Action and Geometric Time (Theorem 3)) . Under Einstein HilbertGHY action, for given boundary ∂M region Σ , dene HamiltonJacobi functional SHJ[hab] = 1 8πG ZΣ d3y ϵ√h K, in case where vacuum Einstein equation holds and volume term vanishes on shell, variation of this functional with respect to boundary induced metric hab gives conjugate momentum πab =δSHJ δhab =ϵ 16πG√h(Kab −Khab), dening quasilocal energy density and generator of translation along boundary normal. If choosing timelike direction and unit normal on boundary, decomposing extrinsic curvature as K=Ktt +Kspatial , then there exists geometric time parameter tgeom such that for small time translation δtgeom , action change satises δSHJ =Eq.l.δtgeom, where Eq.l. is quasilocal energy, thus tgeom uniquely determined by boundary geometry and gravitational action, forming geometric time scale. Denition 3.4 (Boundary Time Scale Equivalence Relation (Denition 4)) . On boundary system B , consider three types of time parameters: scattering time parameter tscatt (e.g., integral of WignerSmith delay), modular parameter tmod , and geometric time tgeom . Dene equivalence relation ∼ : if there exist C1 strictly monotonic function f such that tscatt =fsm(tmod), tmod =fmg(tgeom), and derivative at 0 nite and non-zero, then three belong to same time scale equivalence class, written [τ] . Theorem 3.5 (Existence and (Local) Uniqueness of Time Scale Equivalence Class (Theorem 4)) . Let boundary system B satisfy: (1) Scattering side satises Theorem 1 assumptions, dening group delay trace scale dτscatt(ω) = 1 2πtr Q(ω) dω; (2) Operator algebra side satises Theorem 2 assumptions, there exists modular ow σω t with corresponding modular Hamiltonian operator Kω=−log ∆ω ; (3) Gravity side satises Theorem 3 assumptions, there exists boundary Hamilton Jacobi functional SHJ , whose parameter for timelike translation is tgeom . 6 And assume in some energy window and geometric region there exists AdS/CFT or scatteringgeometry correspondence, such that structure-preserving isomorphism exists between scattering channels and boundary algebrageometry (such as correspondence between Rindler wedge of spherical region and its CFT modular Hamiltonian). Then on this common domain there exists time scale equivalence class [τ] satisfying: (1) For any observation process, its time reading tobs is C1 monotonic function of τ ; (2) If introducing another time parameter ˜ t and requiring its unit interval equivalent to unit changes of scattering phase derivative, modular Hamiltonian expectation value, and GHY boundary action, then ˜ t must locally be linear rescaling of τ , therefore [τ] locally unique. Theorem 3.6 (Cosmological Redshift as Global Rescaling of Time Scale (Theorem 5)) . In spatially isotropic, homogeneous FRW universe, metric can be written as ds2=−dt2+a(t)2γijdxidxj, where a(t) is scale factor. Introduce conformal time η satisfying dt=a(η)dη . Let there be radiation geodesic between source and observer at given redshift z , frequency redshift relation 1 + z=1 a(tem), where tem is emission time, at observation a(t0)=1 . If viewing sourceobserver system scattering as eective far-region scattering on cosmological background, then there exists boundary time scale τ such that local observer's conformal time increment dη and WignerSmith delay scale dτscatt belong to same equivalence class, while cosmological redshift manifests as global rescaling of τ dτcosmo = (1 + z) dτlocal, thus macroscopic cosmic time evolution can be viewed as scale factor evolution of unied time scale equivalence class.  4 Proofs This section gives proof skeletons of main theorems, complete technical details placed in appendices. 4.1 Proof of Theorem 1 BirmanKren formula gives det S(ω) = exp(−2πiξ(ω)). Let Φ(ω) := arg det S(ω) , then there exists continuous branch choice such that Φ(ω) = −2πξ(ω). 7 Dene half-phase φ(ω) = 1 2Φ(ω) , then φ′(ω) = −πξ′(ω) = πρrel(ω). Thus rst equality holds. On other hand, using ln det S(ω) = tr ln S(ω) and chain rule, ∂ωln det S(ω) = tr(S(ω)−1∂ωS(ω)). By unitarity of S(ω) , S(ω)−1=S(ω)† . Writing WignerSmith operator as Q(ω) = −iS(ω)†∂ωS(ω), then ∂ωln det S(ω) = itr Q(ω). On other hand, by BirmanKren formula ∂ωln det S(ω) = −2πiξ′(ω) = 2πiρrel(ω), comparing two expressions yields itr Q(ω)=2πiρrel(ω)⇒ρrel(ω) = 1 2πtr Q(ω). Combining with aforementioned φ′/π =ρrel gives scale identity. Technical requirements of above derivation (such as trace-class conditions, logarithm branch choice) rigorously veried in Appendix A. 4.2 Proof of Theorem 2 (1) and (2) are standard conclusions of TomitaTakesaki theory and Connes modular coboundary theory: through closure and polar decomposition of S0:mΩ7→ m∗Ω construct modular operator ∆ , then using ∆it realize modular ow, can prove for any two faithful states, modular ows' images in Out(M) are consistent. (3) When modular ow describes time evolution of KMS state, KMS condition connects modular parameter tmod with temperature β−1 . In Unruh eect, observer with acceleration a experiences temperature T=a/2π , corresponding to period β= 2π/a . Modular ow period in imaginary time direction is β , thus xing proportion constant between modular parameter and observer proper time τphys , obtaining τphys =αtmod , α∝β . 4.3 Proof of Theorem 3 Varying EH+GHY total action, using Palatini identity and δ√−g=−1 2√−ggµνδgµν , volume term variation gives Einstein tensor Gµν , boundary term variation under xed hab condition organizes into combination of extrinsic curvature and δhab . Standard derivation shows δSGHY =1 16πG Z∂M d3y ϵ√h(Kab −Khab)δhab. Viewing δSGHY as variation of HamiltonJacobi functional, obtain conjugate momentum πab as stated in theorem. 8 Choosing timelike tangent vector eld and unit normal on boundary, decomposing hab into time and space parts, in ADM decomposition K together with lapse function N determine normal translation. Restricting δhab to pure time rescaling (keeping spatial section shape unchanged), can write δSHJ as quasilocal energy times time increment, obtaining δSHJ =Eq.l.δtgeom . 4.4 Proof of Theorem 4 On common domain, by assumption there exists scatteringgeometrymodular ow correspondence: (1) Scattering side gives family of frequency scales dτscatt(ω) , corresponding to external geometry through eikonal approximation and lensing/Shapiro delay; (2) Modular ow side through JLMS-type relation or AdSRindler correspondence maps modular Hamiltonian to energy operator of geometric region, thus modular time linearly related to geometric time locally; (3) Geometry side through Einstein equations and boundary conditions connects GHY action with extrinsic curvature, quasilocal energy, and time translation. Since scattering time scale determined by tr Q and ∂ωφ , while modular time and geometric time both scaled by boundary energy (modular Hamiltonian expectation value, quasilocal energy), and all three act on same boundary algebrageometric structure, their scaling can only dier by positive nite factor, thus there exists fundamental scale τ . Dene τ as time parameter making three unit changes consistent in reference case (such as low-energy limit or xed reference observer), i.e., δτ =φ′(ω) πδω =1 2πtr Q(ω)δω =1 E∗ δSHJ, where E∗ is reference energy under unit scale. For any other time parameter tobs , if requiring its unit interval consistent with above three readings, must have dtobs =αdτ , thus tobs linearly equivalent to τ . Local uniqueness stems from: if there exists another parameter ˜ t simultaneously satisfying consistency of scattering scale and energy scale, then d˜ t/dτ locally non-zero and constant, thus ˜ t locally only ane rescaling of τ . 4.5 Proof of Theorem 5 In FRW metric conformal time satises dt=a(η)dη , therefore conformal time interval dη represents light travel time pulled back to at metric. For high-frequency electromagnetic waves, eikonal approximation shows phase varies linearly with conformal time. Viewing cosmic signal (such as FRB or distant quasar) as scattering process from emission boundary to observation boundary, frequency redshift 1 + z= 1/a(tem) means in terms of source's proper time scale, observed frequency scaled by 1/(1 + z) compared to local frequency. If unied time scale τ dened as local observer's scattering scale, then between emission and observation ends, time scale needs global rescaling by scale factor a(t) , thus dτcosmo =dφ π·1 ωobs =dφ π·1 ωem/(1 + z)= (1 + z) dτlocal, 9 A.3 WignerSmith Operator Trace and Spectral Shift Density In energy representation, S(λ) is family of unitary operators on Hλ , whose logarithmic derivative satises ∂λln det S(λ) = tr(S(λ)−1∂λS(λ)). Since det S(λ) = exp(−2πiξ(λ)) , ∂λln det S(λ) = −2πiξ′(λ) = 2πiρrel(λ). On other hand, dening WignerSmith operator Q(λ) = −iS(λ)†∂λS(λ), then tr(S(λ)−1∂λS(λ)) = tr(S(λ)†∂λS(λ)) = itr Q(λ). Comparing yields ρrel(λ) = (2π)−1tr Q(λ) . A.4 One-Dimensional Case and Levinson Theorem Under one-dimensional short-range potential, eigenstates in nite box approximation satisfy knR+δ(kn) = nπ . Changing potential or box length, eigenvalue density dierence can be expressed as phase shift derivative, thus ρrel(k) = 1 πδ′(k). Levinson theorem gives δ(0) −δ(∞) = πNbound and other boundary conditions, thus unifying scattering phase and bound state counting. This construction compatible with spectral shift function denition, providing concrete implementation of scale identity in one-dimensional models.  B Modular Flow, KMS Conditions, and Thermal Time B.1 TomitaTakesaki Construction Starting from M and Ω , dene densely dened antilinear operator S0:mΩ7→ m∗Ω, m ∈M. Closure S admits polar decomposition S=J∆1/2 , where J is antilinear isometric modular conjugation, ∆ is positive, self-adjoint modular operator. Modular ow dened as σω t(m)=∆itm∆−it. TomitaTakesaki theorem asserts: σω t is one-parameter automorphism group of M , and ω satises KMS condition for it, i.e., there exists analytic function in strip region such that F(t) = ω(aσω t(b)), F(t+i) = ω(σω t(b)a). 16 B.2 Connes 1Cocycle and State-Independent Time Given two faithful states ω, ω′ , can construct Connes 1cocycle ut such that σω′ t(m) = utσω t(m)u−1 t, with ut+s=utσω t(us) . This means modular ow's image in outer automorphism group Out(M) = Aut(M)/Inn(M) independent of state, only dening geometric time direction. B.3 Thermal Time Hypothesis and TemperatureTime Relation In traditional quantum statistics, given Hamiltonian H and inverse temperature β , KMS state under time evolution αt(a) = eitHae−itH satises ω(aαt(b)) = ω(αt+iβ(b)a). Thermal time hypothesis reverses this logic: rst given state ω and algebra M , then interpret modular ow σω t parameter as time. If external observer's physical time τphys exists, can obtain τphys =αtmod by comparing modular ow with physical Hamiltonian generated evolution, where α given by temperature or acceleration. Unruh eect's T= a/2π provides concrete example of this proportion.  C GHY Boundary Term, HamiltonJacobi Functional, and Quasilocal Time C.1 Variation of EH+GHY Action Starting from Sgrav =1 16πG ZM √−g R d4x+1 8πG Z∂M ϵ√h K d3y, vary with respect to gµν . EH term variation can be written as sum of volume integral and boundary integral, latter exactly canceled by GHY term variation, thus under δhab = 0 condition, total variation only contains volume integral, giving Einstein equations. C.2 HamiltonJacobi Functional and Conjugate Momentum Viewing SHJ[hab] as function of action obtained by solving Einstein equations under given boundary geometry, variation with respect to hab gives conjugate momentum πab =δSHJ δhab =ϵ 16πG√h(Kab −Khab). In ADM decomposition, metric written as ds2=−N2dt2+hij(dxi+Nidt)(dxj+Njdt), extrinsic curvature determined through lapse N and shift Ni . Choosing appropriate gauge (such as Ni= 0 ), normal time translation corresponds to change of t , therefore δSHJ/δt gives quasilocal energy. 17 C.3 Rindler and Black Hole Cases In Rindler wedge or Schwarzschild black hole exterior, near-horizon region extrinsic curvature proportional to surface gravity, GHY boundary term in Euclidean path integral gives black hole free energy and temperature relation. By pairing Euclidean time period β with modular ow period, can show geometric time, modular time, and thermal time belong to same scale equivalence class.  D Categorical Structure of Time Scale Equivalence Class D.1 Objects and Morphisms Construct category BTG : (1) Objects are boundary systems B= (A∂, ω∂, S;hab, Kab) ; (2) Morphisms are mappings Φ : B1→ B2 preserving physical structure, satisfying: •Φ gives ∗ isomorphism ϕ:A∂,1→ A∂,2 on algebra; •Φ maps state and S-matrix to objects preserving BK and scale identity structure; •Φ maps boundary geometry to embedding preserving metric and extrinsic curvature (or their equivalence class). In this category, time scale equivalence class [τ] can be viewed as functor from objects to R , assigning each boundary system set of time parameters, morphisms corresponding to monotonic rescaling of time scales. D.2 Categorical Statement of Local Uniqueness Theorem 4 can be restated as: in some local subcategory of given object B , if requiring functor T:BTG →Time simultaneously preserve unit intervals of scattering scale, modular time, and geometric time, then unique in sense of natural isomorphism. This provides basis for future connection of time scale equivalence class with higherlevel categorical structures (such as bration, natural transformation), but these extensions beyond scope of this paper. 18