QCA Axiomatization: Rigorous Foundation of Five-Tuple Definition

In previous section, we saw intuitive picture of QCA universe—discrete spacetime, finite-dimensional states, local evolution. Now we give rigorous mathematical definition.

Why Axiomatization?

From Intuition to Rigor

In history of physics, axiomatization always transforms vague intuition into precise mathematics:

Example 1: Euclidean Geometry

• Intuition: “Obvious” properties of points, lines, surfaces
• Axiomatization: Five axioms → entire geometry

Example 2: Quantum Mechanics

• Intuition: Wave-particle duality, measurement collapse
• Axiomatization: Hilbert space + unitary evolution + Born rule

Example 3: QCA Universe

• Intuition: Discrete lattice points, local jumps, quantum superposition
• Axiomatization: Five-tuple $(\Lambda ,{\mathcal{H}}_{\text{cell}},\mathcal{A},\alpha ,{\omega }_0)$

graph LR
    A["Physical Intuition<br/>Vague Concepts"] --> B["Mathematical Axioms<br/>Precise Definitions"]

    B --> C["Rigorous Derivation<br/>Theorem Proofs"]

    C --> D["Testable Predictions<br/>Experimental Verification"]

    style A fill:#d4a5a5
    style B fill:#ffd93d
    style C fill:#6bcf7f
    style D fill:#6bcf7f

Benefits of Axiomatization:

1. Eliminate Ambiguity: Every concept has precise definition
2. Logical Self-Consistency: Derive from axioms, avoid circular reasoning
3. Testability: Clearly distinguish assumptions from conclusions
4. Universality: Axioms apply to all systems satisfying conditions

Five Components of Five-Tuple

Universe QCA object is defined as: $${\mathfrak{U}}_{\text{QCA}}=(\Lambda ,{\mathcal{H}}_{\text{cell}},\mathcal{A},\alpha ,{\omega }_0)$$

Let’s go deep into each component.

Component 1: Discrete Space $\Lambda$

Graph Theory Foundation

Definition 1.1 (Countable Connected Graph): $\Lambda$ is countable set, carrying undirected connected graph structure:

• Vertex Set: $\Lambda$ (countably infinite or finite)
• Edge Set: $E_{\Lambda }\subset \Lambda \times \Lambda$
• Symmetry: $(x,y)\in E_{\Lambda }\Rightarrow (y,x)\in E_{\Lambda }$
• No Self-Loops: $(x,x)\notin E_{\Lambda }$
• Connectivity: Path exists between any $x,y\in \Lambda$

Graph Distance: $$\text{dist}(x,y)=\min \{n:\text{exists path of length }n\text{ connecting }x,y\}$$

Closed Ball: $$B_R(x):=\{y\in \Lambda :\text{dist}(x,y)\le R\}$$

Local Finiteness Assumption: For all $x\in \Lambda$ and $R\in \mathbb{N}$, $|B_R(x)|<\infty$.

graph TD
    A["Graph Λ"] --> B["Vertices=Lattice Points"]
    A --> C["Edges=Neighbor Relations"]

    B --> D["ℤᵈ Lattice<br/>Standard Example"]
    C --> E["dist(x,y)<br/>Shortest Path Length"]

    E --> F["Closed Ball B_R(x)<br/>All Points Within Radius R"]

    F --> G["Locally Finite<br/>|B_R(x)|<∞"]

    style A fill:#ffd93d
    style D fill:#6bcf7f
    style G fill:#6bcf7f

Standard Lattice ${\mathbb{Z}}^d$

Most Common Choice: $\Lambda ={\mathbb{Z}}^d$ ($d$-dimensional integer lattice)

One-Dimensional ($d=1$): $\Lambda =\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$

• Neighbor relation: $(x,x+1)$
• Distance: $\text{dist}(x,y)=|x-y|$

Two-Dimensional ($d=2$): $\Lambda ={\mathbb{Z}}^2$ (planar lattice points)

• Neighbor relation: $(x,y)$ with $(x\pm 1,y)$, $(x,y\pm 1)$
• Distance (Manhattan): $\text{dist}((x_1,y_1),(x_2,y_2))=|x_1-x_2|+|y_1-y_2|$

Three-Dimensional ($d=3$): $\Lambda ={\mathbb{Z}}^3$ (spatial lattice points)

• Corresponds to discretization of physical space
• Cubic lattice structure

graph TD
    A["Dimension d"] --> B["d=1<br/>One-Dimensional Chain"]
    A --> C["d=2<br/>Two-Dimensional Plane"]
    A --> D["d=3<br/>Three-Dimensional Space"]

    B --> E["...─○─○─○─○─..."]
    C --> F["Square Lattice Network"]
    D --> G["Cubic Lattice"]

    H["Physical Space<br/>ℝ³"] -.->|Discretization| D

    style A fill:#ffd93d
    style D fill:#6bcf7f
    style H fill:#d4a5a5

Translation Symmetry

Definition 1.2 (Translation Action): For $\Lambda ={\mathbb{Z}}^d$, translation ${\tau }_a:\Lambda \to \Lambda$ is defined as: $${\tau }_a(x):=x+a,a\in {\mathbb{Z}}^d$$

Properties:

• ${\tau }_a\circ {\tau }_b={\tau }_{a+b}$ (group property)
• ${\tau }_0=\text{id}$ (identity)
• ${\tau }_a^{-1}={\tau }_{-a}$ (inverse)

Physical Meaning: Space invariant under translations → momentum conservation (Noether theorem)

Why Discrete? Three Reasons

Reason 1: UV Cutoff

Continuous space ${\mathbb{R}}^d$ has UV divergences: $$\int \frac{d^{d}k}{(2\pi {)}^d}\frac{1}{k^2}\sim {\int }_0^{\infty }{k}^{d-3}dk\to \infty (d\ge 3)$$

Discrete space naturally cuts off: $$\sum _{x\in \Lambda }\to \int \frac{d^{d}x}{a^d},a=\text{lattice spacing}$$ Maximum momentum $k_{\max }=\pi /a$ (Brillouin zone boundary).

Reason 2: Information Finiteness

Continuous space infinite-dimensional per point → infinite information. Discrete space $\Lambda$ countable + finite-dimensional ${\mathcal{H}}_{\text{cell}}$ → finite information per finite volume.

Reason 3: Quantum Gravity Hint

Planck scale ${\ell }_P=\sqrt{\hslash G/c^3}\approx 10^{-35}$ m suggests spacetime discreteness.

graph LR
    A["Continuous Space ℝᵈ"] --> B["UV Divergences<br/>Infinite Information<br/>Planck Limit?"]

    C["Discrete Space Λ"] --> D["Natural Cutoff<br/>Finite Information<br/>Quantum Gravity Friendly"]

    B -.->|Paradigm Shift| D

    style A fill:#ff6b6b
    style B fill:#ff6b6b
    style C fill:#ffd93d
    style D fill:#6bcf7f

Component 2: Cellular Hilbert Space ${\mathcal{H}}_{\text{cell}}$

Finite-Dimensional Quantum State Space

Definition 2.1 (Cellular Space): ${\mathcal{H}}_{\text{cell}}$ is finite-dimensional complex Hilbert space: $${\mathcal{H}}_{\text{cell}}\cong {\mathbb{C}}^d,d\in \mathbb{N}$$

Each lattice point $x\in \Lambda$ carries a copy: $${\mathcal{H}}_x\cong {\mathcal{H}}_{\text{cell}}$$

Inner Product: $$⟨\psi |\varphi ⟩=\sum _{i=1}^d\overline{{\psi }_i}{\varphi }_i$$

Orthonormal Basis: $$\{|i⟩:i=1,\dots ,d\},⟨i|j⟩={\delta }_{ij}$$

Physical Interpretation: Local Degrees of Freedom

Value of $d$ determines quantum degree of freedom dimension of each lattice point:

Example 1: Spin

• $d=2$: Spin-1/2 (up $|\uparrow ⟩$, down $|\downarrow ⟩$)
• $d=3$: Spin-1 ($|1⟩,|0⟩,|-1⟩$)

Example 2: Occupation Number

• $d=2$: Fermion (empty $|0⟩$, occupied $|1⟩$)
• $d=N$: Boson (truncated $|0⟩,|1⟩,\dots ,|N-1⟩$)

Example 3: Standard Model

• $d=18$: 3 colors × 2 flavors × 3 generations?
• Gauge degrees of freedom on edges (see Component 3)

graph TD
    A["Cellular Dimension d"] --> B["d=2<br/>Spin-1/2 or Fermion"]
    A --> C["d=3<br/>Spin-1 or Qubit+Auxiliary"]
    A --> D["d=18<br/>Quark Color+Flavor?"]

    B --> E["Simplest QCA"]
    C --> F["Medium Complexity"]
    D --> G["Standard Model Candidate"]

    style A fill:#ffd93d
    style E fill:#6bcf7f
    style G fill:#ff9999

Why Must Be Finite-Dimensional?

Theorem 2.2 (Finite-Dimensional Necessity): If $\dim {\mathcal{H}}_{\text{cell}}=\infty$, then total Hilbert space $\mathcal{H}$ on infinite volume $\Lambda$ is non-separable (uncountable basis), causing:

1. Cannot define tensor product structure of local algebras
2. Quasi-local $C^{\ast }$ algebra ill-defined
3. QCA evolution cannot guarantee continuity

Proof Idea: $$\dim \mathcal{H}=\prod _{x\in \Lambda }\dim {\mathcal{H}}_x={\infty }^{\infty }=2^{{\aleph }_0}$$ Non-separable space, cannot expand with countable basis.

Physical Meaning:

Quantum degrees of freedom of universe at each lattice point must be finite!

This is core constraint of QCA ontology.

Component 3: Quasi-Local $C^{\ast }$ Algebra $\mathcal{A}$

Construction of Infinite Tensor Product

Finite Volume Algebra: For finite set $F⋐\Lambda$ ($⋐$ means finite subset): $${\mathcal{H}}_F:=\underset{x\in F}{⨂}{\mathcal{H}}_x\cong {\mathbb{C}}^{d^{|F|}}$$ Bounded operator algebra: $${\mathcal{A}}_F:=\mathcal{B}({\mathcal{H}}_F)$$ ($\mathcal{B}$ means all bounded operators)

Embedding Map: If $F\subset G⋐\Lambda$: $${\iota }_{F,G}:{\mathcal{A}}_F\hookrightarrow {\mathcal{A}}_G,A\mapsto A\otimes 1_{G\setminus F}$$

Local Algebra: $${\mathcal{A}}_{\text{loc}}:=\bigcup _{F⋐\Lambda }{\mathcal{A}}_F$$ Set of all local operators.

Quasi-Local $C^{\ast }$ Algebra: $$\mathcal{A}:={\overline{{\mathcal{A}}_{\text{loc}}}}^{\Vert \cdot \Vert }$$ Completion with operator norm $\Vert \cdot \Vert$.

graph TD
    A["Single Lattice Point<br/>𝒜_{x}=ℬ(ℂᵈ)"] --> B["Finite Set F<br/>𝒜_F=ℬ(⊗_{x∈F}ℋ_x)"]

    B --> C["All Finite Sets<br/>𝒜_loc=⋃_F 𝒜_F"]

    C --> D["Norm Completion<br/>𝒜=𝒜̄_loc"]

    E["Local Operators"] --> C
    F["Quasi-Local Operators"] --> D

    style A fill:#d4a5a5
    style C fill:#ffd93d
    style D fill:#6bcf7f

Concept of Support

Definition 3.1 (Operator Support): For $A\in {\mathcal{A}}_{\text{loc}}$, there exists smallest finite set $F⋐\Lambda$ such that $A\in {\mathcal{A}}_F$, call $F$ the support of $A$: $$\text{supp}(A):=F$$

Physical Meaning: $\text{supp}(A)$ is lattice point set where operator $A$ “truly acts”.

Examples:

• Single lattice point operator: $\text{supp}({\sigma }_x^z)=\{x\}$
• Nearest neighbor interaction: $\text{supp}({\sigma }_x^x{\sigma }_y^x)=\{x,y\}$

Basic Properties of $C^{\ast }$ Algebra

Definition 3.2 ($C^{\ast }$ Algebra): $\mathcal{A}$ is $C^{\ast }$ algebra, satisfying:

1. Algebra: $A,B\in \mathcal{A}\Rightarrow A+B,AB\in \mathcal{A}$
2. Conjugation: $A\in \mathcal{A}\Rightarrow A^{\ast }\in \mathcal{A}$
3. Norm: $\Vert AB\Vert \le \Vert A\Vert \Vert B\Vert$
4. $C^{\ast }$ Identity: $\Vert A^{\ast }A\Vert =\Vert A{\Vert }^2$

Why $C^{\ast }$ Algebra?

$C^{\ast }$ algebra is correct framework for “non-commutative topological spaces”:

• Commutative $C^{\ast }$ algebra $\leftrightarrow$ Compact Hausdorff space (Gelfand duality)
• Non-commutative $C^{\ast }$ algebra $\leftrightarrow$ “Quantum space”

Quasi-local $C^{\ast }$ algebra $\mathcal{A}$ of QCA is non-commutative geometry of “infinite lattice quantum configuration space”.

graph LR
    A["Classical Configuration Space<br/>Compact Topological Space X"] <==> B["Commutative C* Algebra<br/>C(X)"]

    C["Quantum Configuration Space<br/>(Non-Commutative)"] <==> D["Non-Commutative C* Algebra<br/>𝒜"]

    A -.->|Quantization| C
    B -.->|Non-Commutativization| D

    style A fill:#d4a5a5
    style D fill:#6bcf7f

Component 4: QCA Evolution $\alpha$

$C^{\ast }$ Algebra Automorphism

Definition 4.1 (QCA): Map $\alpha :\mathcal{A}\to \mathcal{A}$ is called quantum cellular automaton of radius at most $R$ if:

Axiom QCA-1 ($\ast$-Automorphism): $$\alpha (AB)=\alpha (A)\alpha (B),\alpha (A^{\ast })=\alpha (A{)}^{\ast },\alpha (1)=1$$ and $\alpha$ is bijective and continuous.

Axiom QCA-2 (Finite Propagation Radius): There exists $R\in \mathbb{N}$ such that for any finite $F⋐\Lambda$ and $A\in {\mathcal{A}}_F$: $$\text{supp}(\alpha (A))\subset B_R(F):=\bigcup _{x\in F}{B}_R(x)$$

Axiom QCA-3 (Translation Covariant): For automorphism ${\theta }_a:\mathcal{A}\to \mathcal{A}$ induced by translation ${\tau }_a$: $$\alpha \circ {\theta }_a={\theta }_a\circ \alpha ,\forall a\in {\mathbb{Z}}^d$$

graph TD
    A["QCA Axioms"] --> B["Axiom 1<br/>*-Automorphism"]
    A --> C["Axiom 2<br/>Finite Propagation R"]
    A --> D["Axiom 3<br/>Translation Covariant"]

    B --> E["Preserve Algebra Structure<br/>Unitarity"]
    C --> F["Causal Light Cone<br/>Information Propagation Speed"]
    D --> G["Spatial Homogeneity<br/>Momentum Conservation"]

    E --> H["Quantum Evolution"]
    F --> H
    G --> H

    H --> I["QCA Universe Evolution α"]

    style A fill:#ffd93d
    style I fill:#6bcf7f

Physical Meaning of Finite Propagation

Intuitive Picture: Operator $A_x$ supported on single point $\{x\}$, after $n$ steps of evolution: $$\text{supp}({\alpha }^n(A_x))\subset B_{nR}(x)$$

Information Propagation Speed: Let time step $\Delta t$, lattice spacing $a$, then maximum information propagation speed: $$v_{\max }=\frac{R\cdot a}{\Delta t}$$

In continuous limit $a,\Delta t\to 0$ with $c:=a/\Delta t$ fixed, if $R=1$: $$v_{\max }=c$$ This is exactly speed of light!

graph TD
    A["n=0<br/>Operator at x"] --> B["n=1<br/>Spreads to B_R(x)"]
    B --> C["n=2<br/>Spreads to B_2R(x)"]
    C --> D["After n Steps<br/>Spreads to B_nR(x)"]

    E["Discrete Light Cone<br/>dist≤nR"] --> D

    F["Continuous Limit<br/>|x-y|≤ct"] -.-> E

    style A fill:#6bcf7f
    style D fill:#ffd93d
    style F fill:#ff9999

Schumacher-Werner Theorem

Theorem 4.2 (Structure Theorem, Schumacher-Werner 2005): Any $\alpha$ satisfying Axioms QCA-1 to QCA-3 can be written as: $$\alpha ={\theta }_s\circ \beta$$ where $\beta$ is block-local QCA, ${\theta }_s$ is some spatial translation.

Block-Local QCA: There exists finite set $F_0⋐{\mathbb{Z}}^d$ and unitary $u\in {\mathcal{A}}_{F_0}$ such that: $$\beta (A)=\prod _{a\in {\mathbb{Z}}^d}{\theta }_a(u)\cdot A\cdot \prod _{a\in {\mathbb{Z}}^d}{\theta }_a(u{)}^{\ast }$$ (product finite in some order)

Physical Interpretation: QCA evolution = periodically apply same local unitary $u$ on each translation block $F_0+a$.

This is similar to Trotter decomposition: $$e^{-it(H_1+H_2)}\approx (e^{-i\Delta t{H}_1}{e}^{-i\Delta t{H}_2}{)}^{t/\Delta t}$$

Iteration and Time Evolution

Definition 4.3 (Iteration): Define integer iterations: $${\alpha }^n:=\underset{n\text{ times}}{\underbrace{\alpha \circ \cdots \circ \alpha }},n\in \mathbb{N}$$ $${\alpha }^0:=\text{id},{\alpha }^{-n}:=({\alpha }^{-1}{)}^n$$

Time Parameterization: Discrete time step $n\in \mathbb{Z}$ corresponds to physical time: $$t=n\Delta t$$

Continuous Limit: When $\Delta t\to 0$ and ${\alpha }^n\to e^{-iHt/\hslash }$ (in appropriate sense), recover continuous time evolution.

Component 5: Initial Universe State ${\omega }_0$

Definition of State

Definition 5.1 (State): State on $C^{\ast }$ algebra $\mathcal{A}$ is linear functional $\omega :\mathcal{A}\to \mathbb{C}$ satisfying:

1. Positivity: $\omega (A^{\ast }A)\ge 0,\forall A\in \mathcal{A}$
2. Normalization: $\omega (1)=1$

Physical Meaning: $\omega (A)$ is expectation value of observable $A$.

Pure States and Mixed States

Pure State: Cannot be decomposed into non-trivial convex combination of other states.

GNS Representation: For pure state $\omega$, there exists Hilbert space ${\mathcal{H}}_{\omega }$, representation ${\pi }_{\omega }:\mathcal{A}\to \mathcal{B}({\mathcal{H}}_{\omega })$ and cyclic vector $|{\Omega }_{\omega }⟩\in {\mathcal{H}}_{\omega }$: $$\omega (A)=⟨{\Omega }_{\omega }|{\pi }_{\omega }(A)|{\Omega }_{\omega }⟩$$

Mixed State: Can be written as convex combination of pure states: $$\omega =\sum _i{p}_i{\omega }_i,p_i\ge 0,\sum _i{p}_i=1$$

graph TD
    A["State ω"] --> B["Pure State<br/>Indecomposable"]
    A --> C["Mixed State<br/>Convex Combination"]

    B --> D["GNS Representation<br/>ℋ_ω, |Ω_ω⟩"]
    C --> E["Density Matrix<br/>ρ=Σ_i p_i|ψ_i⟩⟨ψ_i|"]

    D --> F["Quantum State |ψ⟩"]
    E --> F

    style A fill:#ffd93d
    style F fill:#6bcf7f

Initial Condition ${\omega }_0$

Definition 5.2 (Initial Universe State): ${\omega }_0:\mathcal{A}\to \mathbb{C}$ is quantum state of universe at $n=0$.

Time Evolution: In Heisenberg picture, state evolves with time: $${\omega }_n(A):={\omega }_0({\alpha }^{-n}(A)),n\in \mathbb{Z}$$

Schrödinger Picture: If ${\omega }_0$ has GNS representation $({\mathcal{H}}_0,{\pi }_0,|{\Omega }_0⟩)$ and $\alpha$ can be unitarily realized as $U:{\mathcal{H}}_0\to {\mathcal{H}}_0$: $$|{\Omega }_n⟩=U^n|{\Omega }_0⟩$$

Translation-Invariant States

Definition 5.3 (Translation-Invariant): State $\omega$ is called translation-invariant if: $$\omega ({\theta }_a(A))=\omega (A),\forall a\in {\mathbb{Z}}^d,A\in \mathcal{A}$$

Physical Meaning: Universe homogeneous in space, no special position.

Example 1: Vacuum State Vacuum $|0⟩$ of quantum field theory is translation-invariant.

Example 2: Thermal Equilibrium State Gibbs state at temperature $T$ is invariant under translations.

graph LR
    A["Translation-Invariant State<br/>ω∘θ_a=ω"] --> B["Spatial Homogeneity<br/>No Special Point"]

    B --> C["Physical Symmetry<br/>Momentum Conservation"]

    D["Vacuum State |0⟩"] --> A
    E["Thermal State e^{-βH}"] --> A

    style A fill:#ffd93d
    style C fill:#6bcf7f

Complete Definition of Five-Tuple

Universe QCA Object

Definition 5.4 (Universe QCA Object): Five-tuple $${\mathfrak{U}}_{\text{QCA}}=(\Lambda ,{\mathcal{H}}_{\text{cell}},\mathcal{A},\alpha ,{\omega }_0)$$ is called universe QCA object if satisfies:

1. $\Lambda$ is countably infinite connected graph, locally finite
2. ${\mathcal{H}}_{\text{cell}}\cong {\mathbb{C}}^d$ finite-dimensional
3. $\mathcal{A}$ is quasi-local $C^{\ast }$ algebra $\overline{{\bigcup }_{F⋐\Lambda }{\mathcal{A}}_F}$
4. $\alpha :\mathcal{A}\to \mathcal{A}$ satisfies Axioms QCA-1 to QCA-3
5. ${\omega }_0:\mathcal{A}\to \mathbb{C}$ is normalized state (initial condition)

Hierarchical Structure of Five-Tuple

graph TD
    A["𝔘_QCA"] --> B["Λ<br/>Spatial Structure"]
    A --> C["ℋ_cell<br/>Local Degrees of Freedom"]
    A --> D["𝒜<br/>Observable Algebra"]
    A --> E["α<br/>Evolution Law"]
    A --> F["ω₀<br/>Initial Condition"]

    B --> G["Geometry/Topology"]
    C --> G

    D --> H["Quantum Structure"]
    C --> H

    E --> I["Dynamics"]
    F --> I

    G --> J["Physical Theory"]
    H --> J
    I --> J

    style A fill:#ffd93d
    style J fill:#6bcf7f

Hierarchical Interpretation:

From Axioms to Physics: Three Key Corollaries

Corollary 1: Causal Structure Emergence

Proposition (detailed proof in next section): From QCA axioms, event set $E=\Lambda \times \mathbb{Z}$ naturally induces causal partial order: $$(x,n)⪯(y,m)⟺m\ge n\text{ and }\text{dist}(x,y)\le R(m-n)$$

Physical Meaning: Causal structure is not pre-given, but naturally emerges from QCA locality.

Corollary 2: Unitarity and Information Conservation

Theorem (Unitary Realization): If ${\omega }_0$ is faithful translation-invariant state, then there exists GNS representation $({\mathcal{H}}_0,{\pi }_0,|{\Omega }_0⟩)$ and unitary operator $U:{\mathcal{H}}_0\to {\mathcal{H}}_0$: $${\pi }_0(\alpha (A))=U{\pi }_0(A)U^{†}$$

Physical Meaning: QCA evolution preserves information → quantum information conservation → reversibility.

Corollary 3: Lieb-Robinson Bound

Theorem (Lieb-Robinson Bound): For local operators $A,B$ supported on $X,Y⋐\Lambda$ with $\text{dist}(X,Y)=r$: $$\Vert [{\alpha }^n(A),B]\Vert \le C\Vert A\Vert \Vert B\Vert e^{-\mu (r-vn)}$$ where $v=O(R)$ is effective light speed, $\mu >0$.

Physical Meaning: Even in discrete time, quantum information propagation still has speed upper bound similar to relativity.

graph TD
    A["QCA Axioms<br/>Finite Propagation R"] --> B["Corollary 1<br/>Causal Partial Order (E,⪯)"]
    A --> C["Corollary 2<br/>Unitary Evolution U"]
    A --> D["Corollary 3<br/>Lieb-Robinson Bound"]

    B --> E["Emergent Relativity<br/>Light Cone Structure"]
    C --> F["Information Conservation<br/>Reversibility"]
    D --> G["Effective Light Speed v<br/>Causal Constraint"]

    E --> H["Continuous Limit<br/>Lorentzian Spacetime"]
    F --> H
    G --> H

    style A fill:#ffd93d
    style H fill:#6bcf7f

Popular Analogy: QCA as “Universe Operating System”

Computer Operating System Analogy

Let’s use operating system analogy for five-tuple:

Deep Analogy:

Discrete Space $\Lambda$ = Memory addresses

• $\Lambda ={\mathbb{Z}}^3$ ~ 3D memory layout
• Local finiteness ~ finite neighbors per address

Cellular Space ${\mathcal{H}}_{\text{cell}}$ = Register size

• $d=2$ ~ 1 bit ($|0⟩,|1⟩$)
• $d=256$ ~ 1 byte

Quasi-Local Algebra $\mathcal{A}$ = Instruction set architecture

• Local operators ~ single instructions (involve few registers)
• Quasi-local algebra ~ entire instruction set (can combine infinitely)

QCA Evolution $\alpha$ = Clock cycle

• Single step $\alpha$ ~ one CPU clock
• Finite propagation $R$ ~ instructions can only access adjacent cache

Initial State ${\omega }_0$ = BIOS/boot program

• Determines system startup state

graph LR
    subgraph "QCA Universe"
    A1["Λ<br/>Lattice Space"]
    A2["ℋ_cell<br/>Local States"]
    A3["α<br/>Evolution"]
    end

    subgraph "Computer Operating System"
    B1["Memory Addresses<br/>Discrete Space"]
    B2["Registers<br/>Finite Bits"]
    B3["CPU Clock<br/>Discrete Time"]
    end

    A1 -.->|Analogy| B1
    A2 -.->|Analogy| B2
    A3 -.->|Analogy| B3

    C["QCA=Universe Operating System"] --> A1
    C --> A2
    C --> A3

    style C fill:#ffd93d

Key Insight:

Just as operating system runs on discrete memory addresses, finite registers, discrete clock cycles,

universe also “runs” on discrete lattice points, finite quantum states, discrete time steps!

Chess Game Analogy

Another popular analogy: Chess.

Finite Propagation Radius $R$:

• Pawn: $R=1$ (can only move 1 square)
• Knight: $R=\sqrt{5}$ (L-shaped move)
• Rook/Queen: $R$ can be large (diagonal/straight lines)

Translation Covariance: Chess rules same everywhere on board (except boundary special rules).

Unitarity: Modern chess irreversible, but reversible chess (each move can be undone) similar to unitarity of QCA.

Summary: Unified Picture of Five-Tuple

QCA five-tuple $(\Lambda ,{\mathcal{H}}_{\text{cell}},\mathcal{A},\alpha ,{\omega }_0)$ is most concise mathematical definition of universe:

graph TD
    A["Five-Tuple Axioms"] --> B["Space Λ+Local DOF ℋ_cell"]
    B --> C["Quantum Configuration Space 𝒜"]

    A --> D["Evolution Law α"]
    D --> E["Finite Propagation R"]
    E --> F["Causal Structure Emergence"]

    A --> G["Initial State ω₀"]

    C --> H["Observables"]
    F --> I["Relativistic"]
    G --> J["Universe History"]

    H --> K["Physical Predictions"]
    I --> K
    J --> K

    style A fill:#ffd93d
    style K fill:#6bcf7f

Core Points:

1. $\Lambda$: Determines spatial dimension and topology
2. ${\mathcal{H}}_{\text{cell}}$: Determines local quantum degrees of freedom (must be finite!)
3. $\mathcal{A}$: Contains all observables
4. $\alpha$: Encodes physical laws (finite propagation → causality)
5. ${\omega }_0$: Initial condition of universe

Philosophical Revelation:

These five components completely determine a universe.

No other “hidden variables”, no “background spacetime”.

Universe = QCA, everything emerges from this!

Next Step: Emergence of Causal Structure

Next section will prove: From finite propagation radius $R$ of QCA, how to strictly derive causal partial order $(E,⪯)$ on event set $E=\Lambda \times \mathbb{Z}$.

We will see:

• Geometric relation ${\le }_{\text{geo}}$ from definition of $R$
• Statistical causality ${⪯}_{\text{stat}}$ from definition of correlation functions
• Theorem: Two are equivalent ${\le }_{\text{geo}}={⪯}_{\text{stat}}$

This will reveal: Relativistic causal structure is not assumption, but inevitable result of QCA locality!

This is one of deepest insights of QCA paradigm—causal light cone of continuous spacetime originates from finite propagation of discrete QCA!