Universe as Quantum Discrete Cellular Automaton: Axiomatic Characterization

Universe as Quantum Discrete Cellular Automaton: Axiomatic Characterization Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 19, 2025 Abstract Within the frameworks of Quantum Cellular Automata (QCA), quasi-local C∗ - algebras, and causal sets, we construct an axiomatic system for "Universe = Single Quantum Discrete Cellular Automaton". Specically, we use a countable connected graph Λ as the discrete space; a nite-dimensional local Hilbert space Hcell and the quasi-local algebra A on its innite tensor product to describe local quantum degrees of freedom; a ∗ -automorphism α:A→A with nite propagation radius and its unitary implementation U to describe discrete time evolution; and an initial cosmic state ω0 to describe the universe at time n= 0 . We prove that under these axioms, the naturally induced relation on the event set E= Λ ×Z constitutes a locally nite partial order, thereby yielding a discrete causal set whose local niteness strictly corresponds to the nite propagation radius condition of the QCA, aligning with the "locally nite partial order" structure in causal set theory. Furthermore, we dene the "Universe QCA Object" UQCA = (Λ,Hcell,A, α, ω0) , provide an equivalence theorem of "QCA Locality ⇐⇒ Local Finiteness of Causal Partial Order", and construct a 1D Dirac-type QCA in the single-particle limit, demonstrating how to recover the continuous Dirac equation in appropriate scaling limits. Finally, we discuss the expression of observation, entropy, and the arrow of time within this framework, as well as relationships with causal set quantum gravity and discretization schemes for quantum eld theory. Keywords: Quantum Cellular Automata; Quasi-local C∗ -Algebra; Discrete Universe Model; Causal Set; Discrete Time Dynamics; Lattice Implementation of Dirac Equation 1 Introduction & Historical Context Quantum Cellular Automata (QCA) can be viewed as "discrete-time, local unitary dynamics on innite quantum lattices". Their representative axiomatization appeared in the work of SchumacherWerner on reversible QCA: dening QCA as a translation-invariant ∗ -automorphism on a quasi-local algebra with a strictly nite propagation radius, ensuring that each evolution step propagates support only within a nite neighborhood. Subsequently, Arrighi, Farrelly, and others systematically reviewed structural theorems, computability, quantum simulation, and continuum limits of QCA, demonstrating the 1 broad applicability of QCA as a tool for discretized quantum eld theory and topological phase simulation. ([arXiv][1]) On the other hand, the causal set program represented by Sorkin and Surya proposes that spacetime ontology can be replaced by a locally nite partial order set, where the partial order encodes causal structure and local niteness encodes discreteness, satisfying the "order + number ∼ geometry" program. In this scheme, causal partial order and local niteness jointly characterize a discrete "proto-spacetime", with continuous Lorentzian manifolds being merely limit interpolations. ([Living Reviews][5]) Existing research on QCA mainly treats it as a "simulation tool" or "discretization scheme": to approximate a given continuous quantum eld or condensed matter system, and converge back to continuous theory in appropriate limits. This paper chooses the reverse perspective: not treating QCA as a numerical approximation of a continuous universe, but dening the "Universe Ontology" itself as a QCA object satisfying certain axioms; causal structure and "spacetime" are derived from the locality and time iteration of this object, rather than being pre-given. More specically, the core questions of this paper are: 1. In the context of QCA, how to dene "The Universe" as a global object that simultaneously contains space, local degrees of freedom, dynamics, and initial state? 2. Can we derive a locally nite partial order on the event set starting from QCA locality and graph structure, thereby aligning with the fundamental structure of causal set theory? 3. In this axiomatic system, how to abstract observation, entropy, and the arrow of time, and recover continuous relativistic eld equations, such as the Dirac equation, in appropriate limits? Focusing on these questions, this paper constructs the Universe QCA Object UQCA , proves that its induced event set (E, ⪯) is a locally nite partial order, and provides the continuous limit construction of Dirac-type QCA, demonstrating how to obtain the lattice version of the standard Dirac equation and its limit in the single-particle sector. 2 Model & Assumptions This section provides the basic mathematical structures and axiomatic assumptions used in this paper, including discrete space, local Hilbert space, quasi-local C∗ -algebra, and the denition of QCA, and on this basis denes the "Universe QCA Object". 2.1 Discrete Space and Graph Structure Let Λ be a countable set, serving as the "lattice set" or "discrete space". Assume Λ carries an undirected connected graph structure with edge set EΛ⊂Λ×Λ , satisfying: 1. (x, y)∈EΛ⇒(y, x)∈EΛ ; 2. No self-loops (x, x)∈EΛ . Based on this, dene the graph distance dist : Λ ×Λ→N∪ {0} as the number of edges in the shortest path connecting x and y (or +∞ if no path exists). The connectivity assumption ensures nite distance between any x, y ∈Λ . For any R∈N and x∈Λ , dene the closed ball BR(x) := {y∈Λ : dist(x, y)≤R}. 2 Assume each BR(x) is a nite set. This assumption holds automatically on standard lattices Zd . When Λ = Zd , dene the translation action τa: Λ →Λ as τa(x) := x+a , where a∈Zd . This structure will be used later when discussing spatial homogeneity. 2.2 Local Hilbert Space and Quasi-Local C∗ -Algebra For each lattice site x∈Λ , associate a nite-dimensional Hilbert space Hx , and assume there exists a xed nite-dimensional Hilbert space Hcell such that Hx≃ Hcell ≃Cd holds for all x∈Λ , where d∈N is the dimension of local degrees of freedom of the cell. For any nite subset F⋐Λ , dene the nite volume Hilbert space HF:= O x∈F Hx, and the corresponding bounded operator algebra is AF:= B(HF). If F⊂G⋐Λ , there is a natural embedding ιF,G :AF,→ AG, A 7→ A⊗1G\F, where 1G\F is the identity operator on Nx∈G\FHx . Let Aloc := [ F⋐Λ AF, dene the quasi-local C∗ -algebra as A:= Aloc |·|, where |·| is the operator norm. The support supp(A)⊂Λ of an element A∈ Aloc is dened as the minimal nite set F such that A∈ AF . For general A∈ A , support can be dened as the closure of supports of local operators approximating A . This construction is consistent with the standard quasi-local algebra formalism for quantum spin systems and quantum lattice systems. ([SpringerLink][7]) 2.3 States and GNS Representation On the C∗ -algebra A , a state is a positive, normalized linear functional ω:A → C, ω(A∗A)≥0, ω(1) = 1. Physically, ω(A) gives the expectation value of observable A . By the GNS construction, for any state ω , there exists a triple (πω,Hω,Ωω) , where πω:A→B(Hω) is a ∗ -representation, Ωω∈ Hω is a cyclic vector, such that ω(A) = ⟨Ωω, πω(A)Ωω⟩, A ∈ A, and the set {πω(A)Ωω:A∈ A} is dense in Hω . 3 2.4 Heisenberg Picture Denition of QCA Adopting the algebraic denition from SchumacherWerner and subsequent works, QCA is viewed as a ∗ -automorphism on a quasi-local algebra with nite propagation radius and commuting with translations. ([arXiv][1]) Denition 2.1 (Quantum Cellular Automaton) Let R∈N . A map α:A → A is called a Quantum Cellular Automaton with radius at most R , if it satises: 1. α is a ∗ -automorphism of the C∗ -algebra, i.e., for any A, B ∈ A and λ∈C , α(AB) = α(A)α(B), α(A∗) = α(A)∗, α(1) = 1, and α is bijective and continuous. 2. Locality: For any nite F⋐Λ and any AF∈ AF , there exists a nite set G⊂Λ such that α(AF)∈ AG , and satisfying G⊂BR(F) := [ x∈F BR(x). 3. If Λ carries translation action τa , there exists a corresponding translation automorphism θa:A→A , such that for all a and A∈ A , α◦θa=θa◦α. If there exists some nite R such that the above conditions hold, α is called a local QCA. From α , integer iterations can be dened αn:= α◦ · · · ◦ α | {z } n times , n ∈Z, where α0= id , α−n:= (α−1)n . 2.5 Schrödinger Picture and Unitary Implementation In the Schrödinger picture, states evolve with time. Given QCA α , for any state ω , dene the state after step n as ωn:= ω◦α−n, n ∈Z. In an appropriate GNS representation, QCA can be implemented by a unitary operator. Let ω be a faithful and α -invariant state, i.e., ω◦α=ω , and ω(A∗A) = 0 ⇒A= 0 . Then in the GNS representation (πω,Hω,Ωω) , there exists a unique unitary operator U:Hω→ Hω such that πω(α(A)) = Uπω(A)U†, A ∈ A, and UΩω= Ωω . This result is a standard conclusion of GNS formalism application; proof is given in Appendix B. In specic models, a unitary operator U is often directly specied on a given Hilbert space H , setting α(A) := U†AU , and then verifying it satises locality and translation symmetry conditions. 4 2.6 Universe QCA Object Based on the above structures, dene the "Universe QCA Object", summarizing space, local degrees of freedom, dynamics, and initial state into a quintuple. Denition 2.2 (Universe QCA Object) A set of data UQCA = (Λ,Hcell,A, α, ω0) is called a Universe QCA Object if it satises: 1. Λ is the vertex set of a countable innite connected graph with graph distance dist , and for any x∈Λ, R ∈N , the ball BR(x) is nite. 2. Hx≃ Hcell is a nite-dimensional Hilbert space, and A is its quasi-local C∗ -algebra. 3. α:A→A is a QCA with some nite radius R . 4. If Λ carries translation action, α commutes with the corresponding translation automorphism. 5. ω0:A → C is a normalized state, called the initial cosmic state; for n∈Z , dene ωn:= ω0◦α−n. In this denition, Λ and Hcell x the "type of discrete space and local degrees of freedom of the universe", α xes the "dynamical laws", and ω0 xes the "initial conditions". 3 Main Results (Theorems and alignments) This section presents the main results: the causal structure derived from the Universe QCA Object, the equivalent characterization of its local niteness and QCA locality, and the continuous limit of Dirac-type QCA. 3.1 Event Set and Causal Reachability Relation In the Universe QCA Object, the natural event set is E:= Λ ×Z, where (x, n) denotes the event at lattice site x at time step n . For e= (x, n)∈E , denote spatial coordinate as sp(e) = x , and time as tm(e) = n . The nite propagation radius of QCA determines the "discrete light cone". Let the propagation radius of α be R . For any n < m and local operator By supported on a single point {y} , by locality we have suppαm−n(By)⊂BR(m−n)(y). Dene the geometric relation (x, n)≤geo (y, m)⇐⇒ m≥n, dist(x, y)≤R(m−n). To obtain causal relations, we need to link statistical correlation with geometric light cones. Dene: Denition 3.1 (Causal Reachability Relation) For (x, n),(y, m)∈E , we say (x, n)⪯(y, m) 5 if: 1. m≥n ; 2. There exists a local operator Ax∈ A{x} supported on {x} , and By∈ A{y} , and some state ω , such that ωαn(Ax)αm(By)=ωαn(Ax)ωαm(By). That is, a local perturbation at x at time n can produce a statistical inuence at y at time m . 3.2 Theorem 1: QCA Causal Structure is a Locally Finite Partial Order Theorem 3.2 (Partial Order and Local Finiteness) In any Universe QCA Object UQCA , the relation ⪯ induced by Denition 3.1 is equivalent to the geometric relation ≤geo , i.e., (x, n)⪯(y, m)⇐⇒ (x, n)≤geo (y, m). Thus: 1. (E, ⪯) is a partial order set (reexive, antisymmetric, transitive); 2. (E, ⪯) is locally nite, i.e., for any e1, e2∈E , the causal interval I(e1, e2) := {e∈E:e1⪯e⪯e2} is a nite set. Therefore, (E, ⪯) shares the same structural type as "locally nite partial order" in causal set theory. The proof idea relies on two points: First, if (x, n) is within the geometric light cone of (y, m) , there exist local operators and states creating non-trivial statistical correlations; Second, if the distance condition is not met, operator supports can be decomposed into disjoint regions, leading to factorization of expectation values and no causal inuence. Local niteness is given by the niteness of each ball BR(x) in the graph structure and the nite dierence in time steps. Detailed proof in Appendix A. 3.3 Theorem 2: Equivalent Characterization of QCA Locality and Locally Finite Partial Order Theorem 3.2 gives the construction from QCA to causal sets. The reverse statement is: Given a discrete time dynamics and a locally nite partial order on the event set, the propagation radius of QCA can be reconstructed from the "light cone interface" of this partial order. Theorem 3.3 (Equivalent Characterization) Let α:A→A be a C∗ -algebra automorphism, Λ be a connected graph, E= Λ ×Z . The following two are equivalent: 1. There exists R < ∞ , such that for any nite F⋐Λ and AF∈ AF , supp(α(AF)) ⊂ BR(F) ; 2. There exists a relation ⪯ such that (E, ⪯) is a locally nite partial order, and satises: if (x, n)⪯(y, m) and m=n+ 1 , then dist(x, y)≤R , and for all (x, n) , its one-step future reachable point set is contained in BR(x)× {n+ 1} . In other words, the nite propagation radius condition of QCA is equivalent to the locally nite partial order structure on the event set where "causal links per time step 6 only connect nite neighborhoods". This equivalent characterization restates the QCA denition of SchumacherWerner and subsequent works in the language of causal sets as "discrete time + locally nite partial order + automorphism on quasi-local algebra". 3.4 Theorem 3: Continuous Limit of Dirac-Type QCA QCA is widely used as a discretization framework for continuous quantum eld theory. Below we present the continuous limit result for 1D Dirac-type QCA, demonstrating how the standard Dirac equation is obtained within the Universe QCA Object. Consider Λ = Z , each site carrying Hx≃C2 , basis vectors denoted |x, ↑⟩,|x, ↓⟩ . Dene the time step evolution operator U:= S◦R, where 1. Local spin rotation Rx= e−iθσy= cos θ1−i sin θ σy acts independently on each site; 2. Conditional translation S|x, ↑⟩ =|x+ 1,↑⟩, S|x, ↓⟩ =|x−1,↓⟩. This U is a unitary operator with propagation radius R= 1 , corresponding to 1D Dirac-type QCA. Denote the single-particle state at time step n as |ψn⟩=X x∈Zψ↑ n(x)|x, ↑⟩ +ψ↓ n(x)|x, ↓⟩, evolution equation is |ψn+1⟩=U|ψn⟩. Let lattice spacing ε > 0 , dene continuous coordinates X=εx , T=εn . Take rotation angle scaling as θ=εm , where m > 0 is constant. If assuming the wavefunction is smooth in the limit ε→0 , satisfying ψ↑ n(x)≈ψ↑(X, T), ψ↓ n(x)≈ψ↓(X, T), then retaining up to rst order terms in Taylor expansion, we obtain the following continuous equations: ∂Tψ↑=−∂Xψ↑−m ψ↓, ∂Tψ↓=∂Xψ↓+m ψ↑. Denoting Ψ=(ψ↑, ψ↓)T as a two-component spinor, the above can be written as i∂TΨ = −iσz∂X+mσyΨ, which is a standard form of the 1D Dirac equation. This result shows that in appropriate scaling limits, the Dirac-type QCA in the Universe QCA Object produces dynamics consistent with the continuous Dirac equation in the single-particle sector. Detailed derivation in Appendix C. 7 4 Proofs This section provides the proof framework for the main theorems; full details are in appendices. 4.1 Overview of Proof for Theorem 3.2 We need to prove two points: 1. Causal relation ⪯ is equivalent to geometric relation ≤geo ; 2. On this basis, (E, ⪯) is a locally nite partial order. Causal Reachability within Geometric Light Cone If (x, n)≤geo (y, m) , then m≥n and dist(x, y)≤R(m−n) . By locality, for any By supported on {y} , suppαm−n(By)⊂BR(m−n)(y), so this support contains degrees of freedom near x . Choose Ax and By such that αm−n(By) does not commute with Ax near x , and pick a state ω capable of resolving this noncommutativity, then ωαn(Ax)αm(By)−ωαn(Ax)ωαm(By) can be constructed to be non-zero, thus (x, n)⪯(y, m) . This construction utilizes the nite dimensionality of local Hilbert spaces and freedom to choose Pauli-type operators. Non-Causality Outside Geometric Light Cone If m≥n and dist(x, y)> R(m− n) , then the support of αm−n(By) is completely contained in some nite set G⊂Λ , and x /∈G . Select a nite set F⋐Λ containing x and disjoint from G , then Ax∈ AF, αm−n(By)∈ AG, F ∩G=∅. Under tensor decomposition AF∪G∼ =AF⊗ AG , they act on dierent factors. For any product state ωF⊗ωG , ωαn(Ax)αm(By)=ωF(. . . )ωG(. . . ), general states can be constructed by limits of product states, yielding factorization. Thus (x, n)⪯ (y, m) . Combining the above two points gives equivalence of ⪯ and ≤geo . Partial order properties follow directly from reexivity, antisymmetry, and transitivity of ≤geo . Local niteness: For e1= (x, n)⪯e2= (y, m) , if m < n the interval is empty. If m≥n , any e= (z, k)∈I(e1, e2) satises n≤k≤m, dist(x, z)≤R(k−n),dist(z, y)≤R(m−k), thus dist(x, z)≤R(m−n),dist(y, z)≤R(m−n), i.e., z∈BR(m−n)(x)∩BR(m−n)(y) , which is nite; and k can only take nitely many integer values, so I(e1, e2) is a nite set. Full technical details in Appendix A. 8 4.2 Overview of Proof for Theorem 3.3 The construction from nite propagation condition of QCA to locally nite partial order is done in Theorem 3.2. Reverse direction: Assume (E, ⪯) is a locally nite partial order, and there exists integer R such that for any (x, n) , its one-step future {(y, n + 1) : (x, n)⪯(y, n + 1)} satises dist(x, y)≤R . Using this condition, one can dene "neighborhood propagation bound" for each time step, characterizing the propagation radius of automorphism α as no more than R . Specically, examine the set of all lattice points that may have correlations after one step of evolution on any nite region F , and use local niteness to ensure this set remains nite. Further prove that for any AF∈ AF , support of α(AF) is contained in BR(F) , obtaining the nite propagation radius condition. Complete proof requires formalizing the relationship between "statistical correlation" and "support containment", see Appendix A. 4.3 Overview of Proof for Theorem 3.4 Proof of continuous limit for Dirac-type QCA is based on standard scaling limit analysis. Key steps are: 1. Scale discrete space and time coordinates as X=εx, T =εn , and let rotation angle θ=εm scale with ε ; 2. Perform rst-order Taylor expansion for cos θ, sin θ and wavefunction ψ↑,↓ n(x±1) ; 3. Substitute expansions into discrete evolution equations ψ↑ n+1(x) = cos θ ψ↑ n(x−1) −sin θ ψ↓ n(x−1), ψ↓ n+1(x) = sin θ ψ↑ n(x+ 1) + cos θ ψ↓ n(x+ 1) replacing terms with continuous functions and derivatives, ignoring ε2 and higher terms; 4. Obtain system of rst-order partial dierential equations, rearrange into spinor form to get 1D Dirac equation. This construction aligns with typical methods in quantum walks and QCA simulation of Dirac equations. Full expansion calculation in Appendix C. 5 Model Apply This section demonstrates the application of Universe QCA Object in specic models, focusing on the Dirac-type QCA example and its embedding in the Universe QCA framework. 5.1 1D Dirac-Type Universe Section Let Λ = Z , Hcell ≃C2 , A be the corresponding quasi-local algebra. Choose Dirac-type QCA evolution α(A) = U†AU , where U=S◦R as described before. Take a translationinvariant ground state ω0 as initial state, e.g., spin-up lled state, KMS state, or Gaussian state. Then the quintuple UDirac QCA = (Z,Hcell,A, α, ω0) 9 B Appendix B: Proof of QCA Unitary Implementation Theorem This appendix proves that QCA can be implemented by a unique unitary operator in the GNS representation of an α -invariant faithful state. Let α:A→A be a QCA, ω be a faithful and α -invariant state, i.e., ω◦α=ω , and ω(A∗A) = 0 ⇒A= 0 . Let (πω,Hω,Ωω) be its GNS representation. B.1 B.1 Construction and Isometry of Operator U Dene linear operator on dense subspace D0:= {πω(A)Ωω:A∈ A} U0:D0→ Hω, U0πω(A)Ωω:= πω(α(A))Ωω. For any A, B ∈ A , we have ⟨U0πω(A)Ωω, U0πω(B)Ωω⟩=⟨πω(α(A))Ωω, πω(α(B))Ωω⟩=ωα(A)∗α(B). Using α is a ∗ -automorphism and ω◦α=ω , ωα(A)∗α(B)=ωA∗B=⟨πω(A)Ωω, πω(B)Ωω⟩. Thus U0 preserves inner product on D0 , is an isometric linear operator. Since D0 is dense in Hω , U0 uniquely extends to bounded operator U:Hω→ Hω , and U is isometric, i.e., ⟨Uϕ, Uψ⟩=⟨ϕ, ψ⟩, ϕ, ψ ∈ Hω. B.2 B.2 Surjectivity and Unitarity Need to prove U is surjective. For any B∈ A , since α is bijective, there exists A∈ A such that B=α(A) . Thus πω(B)Ωω=πω(α(A))Ωω=U0πω(A)Ωω∈Ran(U0). Therefore Ran(U0) contains {πω(B)Ωω:B∈ A} , which is dense in Hω . Since U is continuous extension of U0 , Ran(U) is closure of Ran(U0) , both closed and dense, thus Ran(U) = Hω , U is surjective isometry, i.e., unitary operator. B.3 B.3 Conjugate Action Implements α For any A, B ∈ A , Uπω(A)U†πω(B)Ωω=Uπω(A)U†πω(B)Ωω. Note that U†πω(B)Ωω 16 is some vector, and by denition can be written as linear combination of GNS vectors. More convenient is to calculate directly on D0 : Uπω(A)U†πω(B)Ωω=Uπω(A)U†πω(B)Ωω =Uπω(A)πω(α−1(B))Ωω =U0πω(Aα−1(B))Ωω =πω(α(Aα−1(B)))Ωω =πω(α(A)B)Ωω =πω(α(A))πω(B)Ωω Since {πω(B)Ωω:B∈ A} is dense in Hω , above shows Uπω(A)U†=πω(α(A)) holds on entire Hilbert space. Finally, UΩω=Uπω(1)Ωω=πω(α(1))Ωω=πω(1)Ωω= Ωω, showing GNS vector is U -invariant. This completes proof of unitary implementation theorem. C Appendix C: 1D Dirac-Type QCA and Continuous Limit of Dirac Equation This appendix gives specic construction and continuous limit derivation of 1D Dirac-type QCA. C.1 C.1 Model Denition and Discrete Evolution Equation Take Λ = Z , each site carries Hx≃C2 , basis vectors |x, ↑⟩,|x, ↓⟩ . Overall Hilbert space formally H=O x∈Z Hx. Dene local spin rotation operator Rx= e−iθσy= cos θ1−i sin θ σy, global rotation R:= O x∈Z Rx. Dene conditional translation operator S action on single particle basis as S|x, ↑⟩ =|x+ 1,↑⟩, S|x, ↓⟩ =|x−1,↓⟩. Time step evolution operator is U:= S◦R. U is unitary with propagation radius R= 1 . 17 In single particle sector, write |ψn⟩=X x∈Zψ↑ n(x)|x, ↑⟩ +ψ↓ n(x)|x, ↓⟩, discrete evolution is |ψn+1⟩=U|ψn⟩. Expanding gives recurrence relation: ψ↑ n+1(x) = cos θ ψ↑ n(x−1) −sin θ ψ↓ n(x−1), ψ↓ n+1(x) = sin θ ψ↑ n(x+ 1) + cos θ ψ↓ n(x+ 1). C.2 C.2 Continuous Limit and Taylor Expansion Introduce lattice spacing ε > 0 , dene continuous variables X=εx, T =εn. Assume smooth functions ψ↑,↓(X, T) exist such that ψ↑ n(x)≈ψ↑(X, T), ψ↓ n(x)≈ψ↓(X, T) at X=εx, T =εn . Let rotation angle scale with ε as θ=εm, where m > 0 is constant. First order expansion for cos θ and sin θ : cos θ= cos(εm)≈1−1 2ε2m2, sin θ= sin(εm)≈εm. First order expansion for spatial translation and time step: ψ↑ n(x±1) ≈ψ↑(X±ε, T)≈ψ↑(X, T)±ε∂Xψ↑(X, T), ψ↓ n(x±1) ≈ψ↓(X±ε, T)≈ψ↓(X, T)±ε∂Xψ↓(X, T), ψ↑ n+1(x)≈ψ↑(X, T +ε)≈ψ↑(X, T) + ε∂Tψ↑(X, T), ψ↓ n+1(x)≈ψ↓(X, T +ε)≈ψ↓(X, T) + ε∂Tψ↓(X, T). Substitute into discrete evolution equation, keep rst order terms in ε . For spin-up component: ψ↑(X, T) + ε∂Tψ↑(X, T)≈1−1 2ε2m2ψ↑(X−ε, T)−εm ψ↓(X−ε, T). RHS expands to 1−1 2ε2m2ψ↑(X, T)−ε∂Xψ↑(X, T)−εmψ↓(X, T)−ε∂Xψ↓(X, T). 18 Ignoring ε2 terms, get ψ↑(X, T) + ε∂Tψ↑(X, T)≈ψ↑(X, T)−ε∂Xψ↑(X, T)−εm ψ↓(X, T). Subtract ψ↑(X, T) from both sides, rearrange to ∂Tψ↑(X, T) = −∂Xψ↑(X, T)−m ψ↓(X, T). For spin-down component: ψ↓(X, T) + ε∂Tψ↓(X, T)≈εm ψ↑(X+ε, T) + 1−1 2ε2m2ψ↓(X+ε, T). RHS expands to εmψ↑(X, T) + ε∂Xψ↑(X, T)+ψ↓(X, T) + ε∂Xψ↓(X, T), Ignoring ε2 terms, get ψ↓(X, T) + ε∂Tψ↓(X, T)≈ψ↓(X, T) + ε∂Xψ↓(X, T) + εm ψ↑(X, T). Subtract ψ↓(X, T) from both sides, rearrange to ∂Tψ↓(X, T) = ∂Xψ↓(X, T) + m ψ↑(X, T). Write in spinor form. Dene Ψ(X, T) := ψ↑(X, T) ψ↓(X, T), then ∂TΨ(X, T) = −σz∂XΨ(X, T)−m σxΨ(X, T). Multiply by i and rearrange i∂TΨ(X, T) = −iσz∂X+mσyΨ(X, T), where −mσx and mσy can be interchanged by basis redenition. This is a standard form of 1D Dirac equation. C.3 C.3 Relation to Universe QCA Object In Universe QCA framework, 1D Dirac-type model corresponds to UDirac QCA = (Z,Hcell,A, α, ω0), where α(A) = U†AU . This object induces event set E=Z×Z and causal partial order ⪯ , QCA propagation radius R= 1 corresponds to "maximum propagation speed", and Dirac equation in continuous limit is the eective description of this universe at low energy and long wavelength scales. This example shows: under the premise that the universe is dened as a QCA object, through appropriate scaling and coarse-graining, continuous relativistic eld theory can be reproduced within the internal construction, providing concrete mathematical support for rationality of "Universe as Quantum Discrete Cellular Automaton". 19