Theory of Categorical Equivalence\\ Between Computational Universe and Physical Universe:\\ Reversible QCA, Unified Time Scale,\\ and Complexity Geometric Invariants

Theory of Categorical Equivalence Between Computational Universe and Physical Universe: Reversible QCA, Unified Time Scale, and Complexity Geometric Invariants Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In previous works on the “computational universe” series Ucomp = (X, T,C,I), we constructed complexity geometry and information geometry at discrete level, obtaining under unified time scale scattering mother scale control manifold (M, G) and task information manifold (SQ, gQ), such that complexity distance and information distance are characterized respectively by geodesic distances of Gand gQin continuous limit. However, to truly achieve unification of “computational universe = physical universe”, geometric correspondence alone is insufficient: we need categorical equivalence between two “universe categories,” i.e., existence of mutually inverse functors on appropriate subclasses such that objects of physical universe and computational universe can correspond one-to-one, with geometric invariants such as complexity and time scale preserved under this correspondence. This paper introduces a physical universe category PhysUniv, whose objects are physical universe models satisfying unified time scale assumption Uphys = (M, g, F, κ, S), where (M, g) is spacetime manifold with metric (or more general causal structure), Fis matter field content, κ(ω) is unified time scale density, S(ω) is scattering data. Morphisms are geometric mappings preserving causal structure, unified time scale, and scattering structure. We simultaneously review computational universe category CompUniv, whose objects are computational universe objects satisfying finite information density and locality axioms, with morphisms being simulation mappings with complexity upper bounds. On this basis, we select two subcategories: one is PhysUnivQCA consisting of physical universes realizable by reversible quantum cellular automata (QCA), another is CompUnivphys consisting of computational universes physically realizable under unified time scale. Reversible QCA is simultaneously a local discrete dynamical system and a physical system satisfying unified time scale controllable scattering structure, thus becoming bridge connecting two categories. This paper defines two core functors: 1 1. Functor F:PhysUnivQCA →CompUnivphys, mapping physical universe Uphys through QCA discretization to computational universe Ucomp; 2. Functor G:CompUnivphys →PhysUnivQCA, reconstructing from local reversible computational universe its continuous limit spacetime manifold, unified time scale, and scattering data. Under set of explicit technical axioms (QCA universality, local reversibility, unified time scale consistency, and appropriate continuous limit existence), we prove:  Fand Gare quasi-inverse at object level, i.e., for any Uphys ∈PhysUnivQCA, there exists natural isomorphism ηUphys :Uphys ≃ −−→ G(F(Uphys)), for any Ucomp ∈CompUnivphys, there exists natural isomorphism ϵUcomp :F(G(Ucomp)) ≃ −−→ Ucomp;  Fand Gpreserve simulation structure at morphism level: geometric invariants such as complexity geometry (metric Gand geodesic distance), unified time scale density κ(ω), and scattering phase are stable under categorical equivalence. Thus obtaining main theorem: on physically realizable subclass, physical universe category and computational universe category are equivalent in categorical sense, they are merely different presentations of same “unified time scale–complexity geometry–scattering structure” object from continuous and discrete perspectives. Keywords: Computational universe; Physical universe; Category theory; Categorical equivalence; Quantum cellular automata; Unified time scale; Complexity geometry; Scattering theory 1 Introduction Previous works have systematized the “computational universe” idea:  At discrete level, a computational universe is quadruple Ucomp = (X, T,C,I), where Xis configuration set, Tis one-step update relation, Cis single-step cost, Iis information quality function. First two works constructed discrete complexity geometry and discrete information geometry on this basis.  Based on unified time scale scattering mother scale κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), we introduced physical time scale into computational universe, making its singlestep cost a function of group delay matrix, proving that in refinement limit, discrete complexity distance converges to geodesic distance on control manifold (M, G).  At task-aware information geometry level, we constructed information manifold (SQ, gQ), writing on joint manifold EQ=M×SQjoint action AQfor time– information–complexity, geometrizing “optimal algorithms” as “minimal worldlines.” 2 Although these results achieved unification from discrete computation to continuous geometry, “physical universe” still appears in relatively external position: it is used to provide unified time scale and scattering data, but not yet juxtaposed with computational universe at categorical level. To truly give rigorous meaning to “universe is computation,” we need to introduce two “universe categories”:  A category PhysUniv with physical theory objects as objects;  A category CompUniv with computational universe objects as objects. And give equivalence on appropriate subclass PhysUnivQCA ≃CompUnivphys. This paper’s task is to construct these two categories, relevant subcategories, and functors F, G connecting them, proving categorical equivalence under constraints of unified time scale and complexity geometry. Section 2 defines physical universe category and QCA-realizable subcategory. Section 3 reviews computational universe category and defines physically realizable subcategory. Section 4 constructs QCA discretization functor Ffrom physical universe to computational universe, Section 5 constructs continuous limit functor Gfrom computational universe to physical universe. Section 6 states and proves main theorem of categorical equivalence, discussing invariance of complexity geometry and unified time scale under this equivalence. Appendices provide detailed axioms, QCA construction, and categorical proofs. 2 Physical Universe Category and QCA-Realizable Subcategory This section constructs physical universe category PhysUniv, selecting within it subcategory PhysUnivQCA realizable by reversible QCA. 2.1 Physical Universe Objects Our working physical universe object is multiple object with spacetime geometry, matter fields, and unified time scale structure. Definition 2.1 (Physical Universe Object).A physical universe object is quintuple Uphys = (M, g, F, κ, S), where: 1. (M, g) is four-dimensional Lorentzian manifold or more general causal manifold, M is spacetime manifold, gis metric or equivalent causal structure; 2. Fis matter field content defined on (M, g) (such as gauge fields, fermion fields), typically solution space of field equations or operator algebra; 3 3. κ(ω) is unified time scale density: for selected class of scattering processes, its scattering phase derivative, spectral shift function derivative, and group delay trace satisfy κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω); 4. S(ω) is frequency-resolved scattering matrix family of corresponding scattering processes, satisfying standard scattering theory axioms (unitarity, analyticity, etc.). We only consider universe objects satisfying “distinguishable unified time scale,” i.e., there exists at least one family of scattering processes such that above mother formula holds and κ(ω) is non-degenerate. 2.2 Physical Universe Morphisms In category PhysUniv, morphisms should be mappings preserving causal structure, unified time scale, and scattering features. Definition 2.2 (Physical Universe Morphism).Given two physical universe objects Uphys = (M, g, F, κ, S), U′ phys = (M′, g′,F′, κ′,S′), a morphism f:Uphys →U′ phys consists of following data: 1. A smooth mapping fM:M→M′, locally bijective in causal sense (preserving timelike causal order); 2. Pushforward fF:F → F′between field contents, covariant with fM; 3. Preservation of unified time scale and scattering data: there exists frequency transformation fω: Ω →Ω′, such that κ′(ω′) = κ(ω),S′(ω′)≃S(ω) when ω′=fω(ω), where “≃” denotes equivalence under gauge transformation and isomorphism. With physical universe objects as objects and physical universe morphisms as morphisms, we form category PhysUniv. 2.3 QCA-Realizable Subcategory PhysUnivQCA We care about those physical universe objects that can be realized or approximated by reversible quantum cellular automata (QCA). Definition 2.3 (QCA-Realizable Physical Universe).A physical universe object Uphys is called QCA-realizable if there exist: 1. A lattice set Λ ⊂Mand its embedding i: Λ ,→M, forming uniform covering of M at large scales; 4 2. Finite-dimensional local Hilbert space Hxat each lattice point and global Hilbert space H=O x∈Λ Hx; 3. A QCA evolution operator U:H → H satisfying locality and reversibility, such that:  Its Lieb–Robinson light cone approximates causal structure of (M, g) at large scales;  Its scattering matrix family SQCA(ω) approximates S(ω) in appropriate limit, unified time scale density κQCA(ω) consistent with κ(ω). All QCA-realizable physical universe objects and morphisms induced by QCA simulation form subcategory denoted PhysUnivQCA ⊂PhysUniv. 3 Computational Universe Category and Physically Realizable Subcategory This section reviews definition of computational universe category, selecting within it physically realizable subcategory. 3.1 Computational Universe Category CompUniv A computational universe object is Ucomp = (X, T,C,I), satisfying axioms of finite information density, local update, (generalized) reversibility, and cost additivity. Morphisms are simulation mappings: Definition 3.1 (Simulation Mapping Recalled).If there exist f:X→X′and constants α, β > 0, monotone function Φ, such that 1. Step preservation: (x, y)∈T⇒(f(x), f(y)) ∈T′; 2. Cost control: for any path γ:x→y, there exists γ′:f(x)→f(y) such that C′(γ′)≤αC(γ) + β; 3. Information fidelity: I(x)≤Φ(I′(f(x))); then fis a simulation mapping from Ucomp to U′ comp. With computational universe objects as objects and simulation mappings as morphisms, we form category CompUniv. 5 3.2 Physically Realizable Subcategory CompUnivphys We need to select those computational universe objects realizable under unified time scale and QCA framework. Definition 3.2 (Physically Realizable Computational Universe).Computational universe object Ucomp = (X, T,C,I) is called physically realizable if there exist: 1. A QCA system (Λ,Hx, U), and configuration encoding mapping e:X→ H to normalized basis vector subset; 2. A family of control parameters θ∈ M and scattering matrix family S(ω;θ), such that one-step evolution of Ucorresponds to some control step size; 3. Single-step cost C(x, y) can be written as discrete integral of unified time scale density: C(x, y) = ZΩx,y κ(ω;θ) dµx,y(ω); 4. Complexity geometry in refinement limit is approximated by geodesic distance of some control manifold (M, G), as stated in previous theorem on Riemannian limit. All physically realizable computational universe objects and morphisms induced by physically realizable simulation form subcategory CompUnivphys ⊂CompUniv. 4 Functor Ffrom Physical Universe to Computational Universe: QCA Discretization This section constructs functor F:PhysUnivQCA →CompUnivphys, mapping each QCA-realizable physical universe object to computational universe object. 4.1 Object Level: QCA Discretization Construction Given Uphys = (M, g, F, κ, S)∈PhysUnivQCA, by definition there exist lattice Λ ⊂M and QCA system (Λ,Hx, U). 1. Configuration set X: Select set of normalized basis vectors Bxfor each Hx, let X=Y x∈Λ Bx, i.e., set of all basis tensor product labels. Any x∈Xcorresponds to basis vector |x⟩∈H. 6 2. One-step update relation T: Define T={(x, y)∈X×X:⟨y|U|x⟩ = 0}. If Udecomposes into fundamental gate sequence with time step ∆t, we can define Tby this step size as “one physical time step” update relation. 3. Single-step cost C: Using unified time scale density κ(ω) and corresponding scattering matrix S(ω), assign cost to each (x, y)∈T C(x, y) = ZΩx,y κ(ω) dµx,y(ω), where Ωx,y and spectral measure µx,y given by local scattering structure of QCA. For non-adjacent pairs (x, y)/∈Tlet C(x, y) = ∞. 4. Information quality function I: According to task, choose appropriate observation operator family, translating task information on physical field content Fto I:X→Ron configuration space X. For example, for output distribution of given scattering experiment, define I(x) as negative relative entropy or likelihood relative to some target distribution. From locality and reversibility of QCA we can verify: Ucomp = (X, T,C,I) satisfies computational universe axioms and is physically realizable. Definition 4.1 (Object Mapping).Let F(Uphys)=(X, T,C,I). 4.2 Morphism Level: Discrete Simulation of Physical Universe Morphisms Given physical universe morphism f:Uphys →U′ phys, from its realization at QCA level, we get unitary mapping or isometric embedding fH:H → H′between Hilbert spaces, thereby inducing basis vector level mapping fX: X→X′. We require fXto satisfy: 1. Step preservation: if (x, y)∈Tand ⟨y|U|x⟩ = 0, then (fX(x), fX(y)) ∈T′, corresponding to ⟨fX(y)|U′|fX(x)⟩ = 0; 2. Cost control: there exist α, β > 0, such that for any path γits image fX(γ) has cost satisfying C′(fX(γ)) ≤αC(γ) + β; 7 3. Information fidelity: influence of physical morphism on scattering output controlled by fF, thereby inducing monotone function Φ on task information, such that I(x)≤Φ(I′(fX(x))). This is precisely the condition for simulation mapping. Definition 4.2 (Morphism Mapping).Let F(f) = fX:F(Uphys)⇝F(U′ phys). 4.3 Functoriality Proposition 4.3. Above definition gives covariant functor F:PhysUnivQCA →CompUnivphys. Proof in Appendix A.1. Key is verifying: identity morphism maps to identity simulation mapping, morphism composition at discrete level corresponds to composition of simulation mappings, and complexity and information control parameters satisfy simulation conditions after composition. 5 Functor Gfrom Computational Universe to Physical Universe: Continuous Limit Reconstruction This section constructs functor G:CompUnivphys →PhysUnivQCA, starting from physically realizable computational universe, reconstructing continuous physical universe object under constraints of unified time scale and complexity geometry. 5.1 From Discrete Control to Spacetime Manifold Given Ucomp = (X, T,C,I)∈CompUnivphys, by assumption there exist control manifold (M, G) and QCA realization. We first construct spacetime manifold (M, g) from control– complexity geometry. 1. Control manifold Mis parameter space for “space + internal degrees of freedom,” already equipped with Riemannian metric G. 2. Using unified time scale density κ(ω;θ), we can construct effective spacetime metric or causal structure (M, g) on R× M, e.g., in simplified case let M=Rt× M, g =−c2(θ)dt2+Gab(θ)dθadθb, where c(θ) related to κ, ensuring light cone structure consistent with Lieb–Robinson light cone of QCA. More generally, we can jointly define Lorentz-type metric using causal structure (update direction) of complexity graph and unified time scale, such that “reachability relation” and “nonzero propagation velocity” correspond to causal structure of g. 8 5.2 Reconstruction of Scattering and Unified Time Scale By physically realizable assumption, computational universe has QCA realization U, whose scattering matrix family SQCA(ω;θ) realizes frequency domain characteristics of computational update. Using previous unified time scale mother formula, we define κ(ω;θ) = 1 2πtr QQCA(ω;θ), QQCA(ω;θ) = −iS† QCA∂ωSQCA. Taking this as unified time scale density of physical universe, define scattering data S(ω;θ) = SQCA(ω;θ), thereby satisfying scattering and time scale structure axioms of physical universe object. 5.3 Embedding of Field Content and Information Quality Configuration information Xand task information Iof computational universe can be embedded into physical field content Fthrough local operators of QCA, e.g., viewing Xas set of expectation values of certain local operators or boundary condition set. More precise approach is constructing local operator algebra net A(O)⊂ B(H), treating configuration information and task information as functions of these operators under QCA evolution. From categorical structure perspective, we only need to ensure existence of embedding from (X, I) to F, such that comparison relation of information quality is preserved under this embedding (monotone homomorphism). 5.4 Object and Morphism Mappings Definition 5.1 (Object Mapping).Given Ucomp ∈CompUnivphys, let G(Ucomp)=(M, g, F, κ, S), where (M, g) and (κ, S) constructed respectively from control–complexity geometry and QCA scattering structure, Fis field content generated by QCA local operator algebra. Definition 5.2 (Morphism Mapping).Given simulation mapping f:Ucomp ⇝U′ comp, its corresponding QCA realization induces mappings between control manifolds, spacetime manifolds, and scattering data fM:M→M′, fF:F → F′, fω: Ω →Ω′, define G(f) = (fM, fF, fω) : G(Ucomp)→G(U′ comp). Proposition 5.3. Above definition gives covariant functor G:CompUnivphys →PhysUnivQCA. 9