Time Equivalence Class, Observer Projection, and 4D Topological Analogy:\\ From Boundary Time Scale Invariance to Phase Transitions, Fractals, and Exotic Structures

Time Equivalence Class, Observer Projection, and 4D Topological Analogy: From Boundary Time Scale Invariance to Phase Transitions, Fractals, and Exotic Structures Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract Within the unied framework of boundary scatteringtime geometry, this paper systematically characterizes the relationship between time equivalence class and the world picture seen by observers, addressing a natural question: within the same time equivalence class, why do dierent observers provide signicantly dierent descriptions of time and geometric structure? We start from a set of fundamental invariantstime scale mother ruler κ(ω) , relative topological class [K]∈H2(Y, ∂Y ;Z2) , K1 class [u]∈K1(X◦) of scattering family, and generalized entropy variation data Sgen, δ2Srel to dene a unied equivalence relation of timegeometrytopology on the total space Y=M×X◦ . We then introduce the observer prole category Obs , whose elements consist of resolution, coupling structure, and coarse-graining rules, and construct a projection functor FO from the invariant layer to observable time geometry. We prove: FO must factorize through the time equivalence class, meaning all dierences between dierent observers can only arise from multi-scale structures, phase structures, and layers resembling smooth structures, but cannot change the underlying causal order and topological ledger. On this basis, we distinguish and geometrize three types of seeing dierently: (1) Multi-scale self-similarity and fractal-like behavior: dene the action of scale transformation semigroup Rs on time equivalence classes, propose a rigorous denition of multi-scale self-similar time geometry, and provide a solvable one-dimensional scattering model; (2) Phase transitions and phase structure of time geometry: introduce order parameters and critical manifolds for time geometry in parameter space, distinguishing dierent thermodynamic phases on the same equivalence class from topological phase transitions (jumps in [K] or [u] ); (3) 4D topological analogy and exotic time structures: using Freedman's proof of the four-dimensional topological generalized Poincaré conjecture, Donaldson's constraints on smooth fourdimensional manifolds, and the existence of exotic R4 as reference, we propose a picture of topological typesmooth type separation of time geometry and dene a working concept of exotic time structure. Through this we obtain an analogy: time equivalence class corresponds to the topological type of time geometry, while time manifolds seen by dierent observers correspond to dierent smooth/phase structures on the same topological type. 1 Finally, we provide a ve-layer topological relation diagram represented in mermaid, organizing the invariant layer, carrier layer, structure layer, phase/phenomenon layer, and observation/engineering layer into a rigorous conceptual geometric picture. Appendices provide detailed categoried denitions and proofs of time equivalence class and observer projection, analytical derivation of fractals and phase transitions in one-dimensional scattering toy models, and mathematical background synopsis of several theorems and propositions involved in the 4D topological analogy. Keywords: Time Equivalence Class; Modular Time; Observer Projection; Multi-scale Self-similarity; Phase Transition; 4D Topology; Exotic Smooth Structure; Generalized Entropy; K1 Class; Z2 Holonomy  1 Introduction The idea of time equivalence class can be briey described as: given causal structure and boundary scattering background, there exists a mother time scale [τ] such that all physically acceptable time parametrizations are anely equivalent to it, and its calibration is uniformly determined by scattering phase gradient, relative state density, and trace of WignerSmith group delay. The concern of this paper is not the construction of this framework itself, but further philosophical and technical details: even within the same time equivalence class, dierent observers' world picturessubjective experience of time, division of geometry and material phases, distinction between macroscopic and microscopicstill exhibit signicant dierences. What is the root of this dierence? In existing work, time equivalence class has been mainly used to unify: (i) time scale at scatteringspectral shiftgroup delay end; (ii) thermal time and arrow of time at modular ow and generalized entropy end; (iii) Killing time, ADM time, null geodesic ane parameter, and cosmological conformal time at geometric end. This paper extends the view to the projection mechanism of observers, attempting to answer the following questions within a rigorous mathematical framework: 1. Within the same time equivalence class, what structures determine that dierent observers see dierently; 2. Whether this dierence can be understood as fractals, multi-scale self-similarity, phase transitions, or phenomena similar to topological typesmooth type separation in four-dimensional topology; 3. How to construct a unied geometrictopologicalinformation framework incorporating these three types of explanation into the same conceptual picture. In four-dimensional topology, Freedman proved the topological four-dimensional generalized Poincaré conjecture, that any topological four-dimensional homotopy sphere is homeomorphic to S4 , establishing a milestone in 4-dimensional topological manifold classication; while Donaldson's gauge invariants and constraints on intersection forms show dramatic structural dierences between smooth four-dimensional manifolds and topological four-dimensional manifolds, directly leading to the existence of exotic R4 : there exist innitely many smooth manifolds mutually non-dieomorphic but homeomorphic to R4 . [ ? ] In contrast, for dimensions n= 4 , Rn admits no exotic smooth structures. [ ? ] This 2 phenomenon indicates that topologically identical and smooth structure identical are no longer equivalent in 4 dimensions. This paper borrows this picture to propose an analogy: time equivalence class acts as the topological type of time geometry, while various time structures seen by different observers within the same equivalence classincluding fractal-like multi-scale behavior, time experience in dierent thermodynamic phases, and even possible exotic time structurescorrespond to dierent smooth/phase structures on the same time topological type. The main contributions of this paper can be summarized as:  Introduce a set of timegeometrytopology invariants I= (κ(ω),[K],[u], Sgen, δ2Srel) , and dene time geometry equivalence class based on this;  Dene observer prole category Obs and projection functor FO , prove FO must factorize through time equivalence class;  Construct scale transformation semigroup and phase structure on time equivalence class, distinguishing fractal-like behavior, non-topological phase transitions, and topological phase transitions;  Introduce working denition of exotic time structure and make analogy with 4D exotic smooth structures;  Provide a ve-layer topological relation diagram represented in mermaid, organizing the above construction into a unied conceptual framework;  Provide rigorous proofs of several key propositions and detailed derivations of onedimensional solvable models in appendices. Article structure: Section 2 reviews denitions of time scale invariants and time equivalence class; Section 3 formalizes observer prole and projection functor; Section 4 discusses multi-scale structure and fractal-like behavior; Section 5 constructs phase structure and phase transitions of time geometry; Section 6 provides 4D topological analogy and concept of exotic time structure; Section 7 discusses and prospects; Appendices include detailed proofs and model calculations.  2 Time Scale Invariants and Time Equivalence Class This section provides the foundation for this paper's work: time scale mother ruler κ(ω) , topological class [K] , K1 class [u] , generalized entropy variation data, and time equivalence class dened based on these invariants. 2.1 Time Scale Mother Ruler Let M be a Lorentzian manifold with boundary, X◦ the parameter space with singularities removed, Y:= M×X◦ . For each x∈X◦ , given a pair of self-adjoint operators (Hx, H0,x) , dene scattering matrix Sx(ω) on energy window I⊂R . 3 Denition 2.1 (Time Scale Mother Ruler (Denition 2.1)) . On energy window I where BirmanKrein and WignerSmith conditions hold, dene Qx(ω) := −i Sx(ω)†∂ωSx(ω), Φx(ω) := arg det Sx(ω), φx(ω) := 1 2Φx(ω), and let relative state density ρrel,x(ω) be the derivative of relative spectral shift function, then κx(ω) := φ′ x(ω) π=ρrel,x(ω) = 1 2πtr Qx(ω). Call κx(ω) the time scale mother ruler. κ(ω) is a function dened on I×X◦ , invariant under appropriate equivalence transformations of scattering families, thus is a spectralscattering invariant. 2.2 Topological Class [K] , K1 Class [u] , and Z2 Holonomy Let Y:= M×X◦ , ∂Y := ∂M ×X◦∪M×∂X◦ . Denition 2.2 (Unied Relative Topological Class (Denition 2.2)) . In relative cohomology group H2(Y, ∂Y ;Z2) , select a class [K]∈H2(Y, ∂Y ;Z2), whose Künneth decomposition can be written as [K] = π∗ Mw2(TM) + X j π∗ Mµj⌣ π∗ Xwj+π∗ Xρc1(LS), where w2(TM) is the second StiefelWhitney class, µj,wj are one-dimensional Z2 classes, LS is the scattering line bundle, ρ is mod-2 reduction. Denition 2.3 ( K1 Class of Scattering Family (Denition 2.3)) . For each x∈X◦ , dene relative Cayley transform ux:= (Hx−i)(Hx+i)−1(H0,x +i)(H0,x −i)−1, under appropriate restricted conditions ux∈Ures , thus x7−→ ux, X◦→Ures gives a class [u]∈K1(X◦) . Additionally, introduce scattering square-root principal bundle P√s→X◦ , whose holonomy gives Z2 invariant ν√S:π1(X◦)→ {±1}, as projection of [K] onto H1(X◦;Z2) component. 4 2.3 Generalized Entropy Variation Data Choose a point p in M and a family of small causal diamonds Dp,r ⊂M , whose boundary cross-section area A(Σp,r) and volume Vp,r are determined by metric g . Denition 2.4 (Generalized Entropy (Denition 2.4)) . Dene Sgen(p, r) = A(Σp,r) 4Gℏ+Sout(p, r)−Λ 8πG Vp,r Tp,r , where Sout is entropy of external quantum state, Tp,r is appropriately dened eective temperature scale. Hypothesis 2.5 (Generalized Entropy Variation Condition (Postulate 2.5)) . 1. Under xed volume or xed generalized energy constraints, rst-order variation satises δSgen(p, r) = 0; 2. Second-order relative entropy satises δ2Srel(p, r)≥0. In existing work, using weighted light-ray transformations, the above conditions can be proven equivalent to local Einstein equations and HollandsWald gauge energy nonnegativity conditions. This paper treats this as part of timegeometry invariants. 2.4 Time Geometry Equivalence Class We take time parametrization and time geometry as objects and introduce equivalence relation. Denition 2.6 (Ane Equivalence of Time Parameters (Denition 2.6)) . If two time parameters t1, t2 have constants a > 0, b ∈R such that t2=at1+b, then t1, t2 are called anely equivalent, written t1∼aff t2 . Denition 2.7 (Time Geometry Equivalence Class (Denition 2.7)) . Given (M, g) and a set of invariants I:= (κ, [K],[u], Sgen, δ2Srel) . If two sets of time geometry data (g1, t1) , (g2, t2) satisfy: 1. Have the same causal order structure; 2. Corresponding time scale mother rulers κ1, κ2 satisfy κ2(ω) = c κ1(ω) ( c > 0 constant); 3. Topological invariants satisfy [K]1= [K]2 , [u]1= [u]2 , ν√S,1=ν√S,2 ; 4. Generalized entropy variation data are the same or dier only by constant rescaling; then they are said to belong to the same time geometry equivalence class, written [(g1, t1)]time = [(g2, t2)]time. All equivalence classes form the set TimeEq , called the time equivalence class space. This equivalence relation compresses all pure rescaling and topological isomorphism degrees of freedom, but preserves underlying causal order and topological ledger, which is the precise denition of same time equivalence class discussed in this paper.  5 3 Observer Prole and Projection Functor This section formalizes the concept of observer, modeling it as a triple containing resolution, coupling, and coarse-graining, and constructing a projection functor from invariant layer to observable time geometry. 3.1 Observer Prole Denition 3.1 (Observer Prole (Denition 3.1)) . An observer O 's prole is a triple O:= (ΛO, CO,RO), where: 1. ΛO is resolution parameter, describing minimum scales it can resolve in frequency and time domains; 2. CO is coupling structure, describing which degrees of freedom it interacts with (e.g., couples to which boundary regions, which elds, which family of worldlines, etc.); 3. RO is coarse-graining rule, describing partial trace and coarse-grain method for degrees of freedom. Denote the class of all observer proles as Obs . 3.2 Observer's Measurement Window Function For given O and time scale mother ruler κ(ω) , the time quantity actually measurable by observer is typically a convolution: TO:= ZWO(ω; ΛO, CO,RO)κ(ω) dω, where WO is window function determined by prole, encoding frequency band limitation (resolution), coupling weights (which frequencies couple more strongly), and eective weight attenuation caused by coarse-graining. Denition 3.2 (Observer Projection (Denition 3.2)) . Let I= (κ, [K],[u], Sgen, δ2Srel) . For each O∈Obs , dene projection FO:I 7−→ ObsTimeO, where ObsTimeO is structure containing the following data: 1. Observable time scale TO and its local perturbations; 2. Topological information accessible by CO (e.g., whether can measure ν√S(γ) , certain projections of [K] ); 3. Corresponding subjective time indicator (e.g., tsubj based on local Fisher information FQ ); 4. Eective arrow of time and thermodynamic/information-theoretic irreversibility under given coarse-grain. ObsTimeO can be viewed as time geometry seen by that observer. 6 3.3 Categorical Structure and Functor Factorization Denote Inv as category with invariants I as objects and isomorphisms preserving time geometry equivalence class as morphisms, i.e., Obj(Inv) = {I},Mor(Inv) = {ϕ:I → I′|[I]time = [I′]time}. Denote TimeEq as aforementioned time equivalence class space, naturally having discrete category structure: objects are equivalence classes, morphisms are identities. Proposition 3.3 (Projection Factorization (Proposition 3.3)) . For any observer O∈ Obs , there exist unique maps π:Inv →TimeEq, GO:TimeEq →ObsTimeO, such that FO=GO◦π. Proof idea. By denition, if two invariants I,I′ belong to same time geometry equivalence class, there exist ane rescaling and topological isomorphism corresponding their time geometry and invariants. In denition of FO , window function WO and coarsegraining rule RO depend only on O not specic representative, thus FO(I) and FO(I′) dier only by reparametrization absorbable by internal coordinate transformation of O . This means FO is constant on equivalence classes, thus factorizes through quotient map π . Uniqueness comes from universal property of quotient map. Formal proof in Appendix A. □ Physical meaning of Proposition 3.3: all dierences between observers can only come from GO this structure from equivalence class to observable time geometry, but cannot change underlying equivalence class itself. This provides foundation for subsequently attributing dierences to multi-scale structure, phase structure, and exotic structure.  4 Multi-Scale Structure and Fractal-Like Behavior This section introduces action of scale transformation operation Rs on time equivalence class, and denes multi-scale self-similar time geometry to characterize what we intuitively call fractal time. 4.1 Scale Transformation Semigroup Let s > 0 be dimensionless scale parameter, dene scale transformation in frequency domain (Rsκ)(ω) := α(s)κ(β(s)ω), where α(s), β(s) are positive functions satisfying semigroup property Rs◦ Rs′=Rss′. At time geometry level, Rs can correspond to coarse-grain or RG ow, describing eective time scale from high resolution to low resolution. 7 Denition 4.1 (Scale Orbit and Multi-Scale Self-Similarity (Denition 4.1)) . 1. Scale orbit of time equivalence class [τ] is dened as O([τ]) := {[RsI]time :s > 0}, where I is chosen arbitrarily as representative of [τ] . 2. If there exists s= 1 such that [RsI]time = [I]time, then [τ] is called a multi-scale self-similar time equivalence class. In critical systems, xed points of Rs correspond to fractal-like geometry: at each scale, statistical structure of time scale is invariant. 4.2 Observer Scale and Fractal Perception For observer O 's resolution ΛO , can dene operation matching scale transformation sO:= f(ΛO), such that TO∼ZWO(ω)κ(ω) dω=Z˜ WO(ω) (RsOκ)(ω) dω, where ˜ WO is rescaled window function. If [τ] is multi-scale self-similar equivalence class, under appropriate normalization, statistical distribution of TO can remain invariant or exhibit power-law transformation when ΛO changes, corresponding to intuitively fractal time: at coarse and ne levels, time noise structure is similar. Proposition 4.2 (Intra-Equivalence-Class Property of Fractal-Like Behavior (Proposition 4.2)) . If [τ] is multi-scale self-similar time equivalence class, for any two observers O1, O2 , there exists normalization constant c12 >0 such that their observable time scales satisfy TO2≈c12TO1 having same scale exponent in statistical sense. In other words, fractal-like behavior of time is property within equivalence class, not topological property distinguishing equivalence classes. Proof omitted, relies on linear response of Rs xed point and stability of window function family.  5 Phase Structure: Phase Transitions, Topological Phase Transitions, and Time Experience This section introduces phase structure of time geometry, distinguishing parts caused by phase transitions from parts caused by topological jumps in seeing dierently within same equivalence class. 8 5.1 Parameter Space and Phases Let P be physical parameter space (e.g., temperature, coupling strength, density, driving frequency, etc.), each point p∈ P corresponds to a set of invariants I(p) , thus corresponding to time equivalence class [τ(p)] . Denition 5.1 (Phases of Fixed Equivalence Class (Denition 5.1)) . Fix time equivalence class [τ0] , consider P[τ0]:= {p∈ P : [τ(p)] = [τ0]}. On P[τ0] , introduce following equivalence relation: if there exists continuous path γ: [0,1] → P[τ0] connecting p1, p2 , and along path local observable time geometry and thermodynamic functions are all analytic, then p1, p2 are said to belong to same phase. Set of all phases is denoted Π([τ0]) . Thus, dierent thermodynamic phases within same time equivalence class are different elements in Π([τ0]) . 5.2 Non-Topological Phase Transitions and Topological Phase Transitions Denition 5.2 (Non-Topological Phase Transition (Denition 5.2)) . If along some path, thermodynamic or correlation functions exhibit non-analytic behavior, but topological invariants [K],[u], ν√S remain unchanged, it is called a non-topological phase transition. Denition 5.3 (Topological Phase Transition (Denition 5.3)) . If along parameter path γ at some point p∗ , there exists [K](p∗ −)= [K](p∗ +) or [u](p∗ −)= [u](p∗ +) (subscripts indicate two sides of critical point), then a topological phase transition is said to occur at p∗ . Obviously, topological phase transition necessarily leads to time equivalence class change, while non-topological phase transition occurs within same equivalence class. Proposition 5.4 (Phase Transitions and Observer Experience (Proposition 5.3)) . 1. For non-topological phase transitions, observer O 's seen time geometry ObsTimeO can be connected by continuous deformation on two sides of phase, but certain second-order or higher-order responses exhibit non-analyticity; 2. For topological phase transitions, there exists at least one class of topological observables (e.g., ν√S(γ) or vertex moduli) taking dierent values on two sides of phase, in which case time equivalence class changes. This proposition explains: dramatic changes in time experience can have two essentially dierent sources: one is phase transition within equivalence class (e.g., vitrication and aging phenomena), another is topological jump between equivalence classes.  9