02. Triple Decomposition of Parameter Vector: Structure, Dynamics, and Initial State

Introduction: Three Layers of Building Blueprints

In previous article, we established finite information universe axiom: Universe can be encoded as finite bit string $\Theta \in \{0,1{\}}^{\le I_{\max }}$.

But this raises new question: How should this parameter vector $\Theta$ be organized?

Popular Analogy: Building House Requires Three Types of Information

Imagine building a house requires following three independent types of information:

Type 1: Architectural Blueprint (Structural Information)

• How many floors? How large each floor?
• How are rooms laid out?
• Positions of walls, columns
• Building material types (wood, brick, reinforced concrete)
• This determines “skeleton of house”

Type 2: Construction Rules (Dynamical Information)

• How to pour concrete?
• How to lay bricks? (Laying method, mortar ratio)
• How to wire circuits?
• How to connect water pipes?
• This determines “how house is built”

Type 3: Foundation State (Initial Conditions)

• Is foundation level?
• What is soil bearing capacity?
• How high is water table?
• How to handle existing old buildings?
• This determines “starting state”

Key Insight: These three types of information are logically independent!

• Can build houses on different foundations with same blueprint
• Can realize same blueprint with different construction rules
• But three types of information all necessary

Universe’s situation completely analogous:

$$\boxed{\Theta =({\Theta }_{\text{str}},{\Theta }_{\text{dyn}},{\Theta }_{\text{ini}})}$$

This article will explain in detail:

1. Why need triple decomposition? (Logical independence)
2. What information does each parameter type encode? (Mathematical definition)
3. How to define total information $I_{\text{param}}(\Theta )$? (Bit counting)
4. Encoding redundancy and essential degrees of freedom (Uniqueness question)

Part I: Why Need Triple Decomposition?

Three Levels of Physical Problems

Consider simple system in classical mechanics: Harmonic oscillator.

To completely determine system evolution, need three types of information:

1. System Composition (Structure):

   • Single oscillator? Or coupled oscillator chain?
   • What is mass $m$? What is spring constant $k$?
   • → Corresponds to ${\Theta }_{\text{str}}$ (System’s “hardware specifications”)

2. Equations of Motion (Dynamics):

   • Newton’s second law: $m\ddot{x}=-kx$
   • Or Hamiltonian equations: $\dot{p}=-\partial H/\partial x$
   • → Corresponds to ${\Theta }_{\text{dyn}}$ (System’s “operating rules”)

3. Initial Conditions (Initial State):

   • At $t=0$: Position $x_0$, velocity $v_0$
   • → Corresponds to ${\Theta }_{\text{ini}}$ (System’s “starting point”)

Key Observation:

• Changing initial conditions $(x_0,v_0)$ doesn’t change equations of motion
• Changing mass $m$ doesn’t change initial position $x_0$
• But three together determine trajectory $x(t)$

This separation also exists in quantum field theory:

Mathematical Independence Theorem

Theorem 2.1 (Direct Product Decomposition of Parameter Space):

In QCA framework, universe parameter space can be decomposed as direct product of three subspaces:

$${\mathcal{M}}_{\Theta }={\mathcal{M}}_{\text{str}}\times {\mathcal{M}}_{\text{dyn}}\times {\mathcal{M}}_{\text{ini}}$$

such that:

1. Structural parameter ${\Theta }_{\text{str}}\in {\mathcal{M}}_{\text{str}}$ uniquely determines:

   • Lattice set $\Lambda$
   • Cell Hilbert space ${\mathcal{H}}_{\text{cell}}$
   • Quasi-local algebra $\mathcal{A}(\Lambda ,{\mathcal{H}}_{\text{cell}})$

2. Dynamical parameter ${\Theta }_{\text{dyn}}\in {\mathcal{M}}_{\text{dyn}}$ uniquely determines:

   • QCA automorphism $\alpha :\mathcal{A}\to \mathcal{A}$
   • (After $\mathcal{A}$ is given)

3. Initial state parameter ${\Theta }_{\text{ini}}\in {\mathcal{M}}_{\text{ini}}$ uniquely determines:

   • Initial state ${\omega }_0:\mathcal{A}\to \mathbb{C}$
   • (After $\mathcal{A}$ is given)

4. Three subspaces logically independent: $$\text{Choice of }{\Theta }_{\text{dyn}}\text{ does not depend on }{\Theta }_{\text{ini}}$$ $$\text{Choice of }{\Theta }_{\text{ini}}\text{ does not depend on }{\Theta }_{\text{dyn}}$$ (But both depend on algebra $\mathcal{A}$ determined by ${\Theta }_{\text{str}}$)

Proof Outline:

• ${\Theta }_{\text{str}}$ determines “stage” (algebra $\mathcal{A}$)
• ${\Theta }_{\text{dyn}}$ selects an automorphism from automorphism group $\text{Aut}(\mathcal{A})$
• ${\Theta }_{\text{ini}}$ selects a state from state space $\mathcal{S}(\mathcal{A})$
• Automorphism group and state space are independent structures on $\mathcal{A}$

Why Cannot Merge?

Failed Attempt 1: Merge ${\Theta }_{\text{dyn}}$ into ${\Theta }_{\text{str}}$?

• Counterexample: Same lattice structure $\Lambda$ can define infinitely many different QCA evolution rules
• Example: One-dimensional ${\mathbb{Z}}_L$ lattice can be Dirac-QCA, Ising-QCA, Toffoli-QCA…
• Conclusion: Dynamics not uniquely determined by structure

Failed Attempt 2: Merge ${\Theta }_{\text{ini}}$ into ${\Theta }_{\text{dyn}}$?

• Counterexample: Same Hamiltonian $H$ can have different initial states
• Example: Harmonic oscillator $H=p^2/2m+k{x}^2/2$ can start in ground state $|0⟩$ or coherent state $|\alpha ⟩$
• Conclusion: Initial state not uniquely determined by dynamics

Failed Attempt 3: Merge ${\Theta }_{\text{str}}$ into ${\Theta }_{\text{dyn}}$?

• Counterexample: QCA with different number of lattice points are topologically inequivalent
• Example: ${\mathbb{Z}}_{100}$ and ${\mathbb{Z}}_{101}$ QCA, regardless of evolution rules, are not isomorphic
• Conclusion: Structure is prerequisite for dynamics

Therefore: Triple decomposition is minimal and necessary.

Part II: Definition of Structural Parameter ${\Theta }_{\text{str}}$

Encoded Content: “Blueprint” of Spacetime

Structural parameter ${\Theta }_{\text{str}}$ needs to specify following information:

(1) Spatial Dimension $d$

$$d\in \{1,2,3,4,\dots \}$$

Encoding Method:

• Encode dimension with $\lceil {\log }_2{d}_{\max }\rceil$ bits
• If restrict $d\le 10$ (reasonable physical assumption), need $\lceil {\log }_{2}10\rceil =4$ bits

Physical Meaning:

• $d=1$: Toy model (Dirac chain)
• $d=3$: Our space
• $d=4$: Spacetime unified (Minkowski lattice)
• $d>4$: Extra dimensions (string theory)

(2) Lattice Lengths in Each Direction $L_1,\dots ,L_d$

$$\Lambda =\prod _{i=1}^d\{0,1,\dots ,L_i-1\}$$

Encoding Method:

• If each $L_i\le 2^{64}$ (sufficient for cosmological scales), each direction needs 64 bits
• Total: $64d$ bits

Example:

• $d=3$, $L_1=L_2=L_3=10^{30}$ (observable universe in Planck length units)
• Total number of cells: $N_{\text{cell}}=(10^{30}{)}^3=10^{90}$

(3) Boundary Conditions and Topology

Options:

• Open boundary
• Periodic boundary (topology ${\mathbb{T}}^d$)
• Twisted boundary
• Other topologies ($S^d$, manifold gluing)

Encoding Method:

• Simple case: Use $d\times 2$ bits (each direction two boundaries, 1 bit each)
• Complex topology: Need additional encoding of gluing rules (topological invariants)

(4) Cell Internal Hilbert Space Dimension $d_{\text{cell}}$

$${\mathcal{H}}_x\cong {\mathbb{C}}^{d_{\text{cell}}}$$

Physical Structure (usually decomposed into multiple subsystems):

$${\mathcal{H}}_{\text{cell}}={\mathcal{H}}_{\text{fermion}}\otimes {\mathcal{H}}_{\text{gauge}}\otimes {\mathcal{H}}_{\text{aux}}$$

• ${\mathcal{H}}_{\text{fermion}}$: Fermion degrees of freedom (e.g., ${\mathbb{C}}^2$ spin in Dirac-QCA)
• ${\mathcal{H}}_{\text{gauge}}$: Gauge field registers (e.g., ${\mathbb{C}}^{|G|}$, $G$ is gauge group)
• ${\mathcal{H}}_{\text{aux}}$: Auxiliary qubits (for maintaining unitarity)

Encoding Method:

• Specify dimension of each subsystem (e.g., $\dim {\mathcal{H}}_{\text{fermion}}=4$, $\dim {\mathcal{H}}_{\text{gauge}}=8$)
• Total dimension: $d_{\text{cell}}={\prod }_i\dim {\mathcal{H}}_i$
• Need $\sim {\log }_2{d}_{\text{cell}}$ bits to encode

Example (Standard Model QCA):

• Fermions: 3 generations × 2 spins × 3 colors × 2 (particle/antiparticle) = 36
• Gauge fields: SU(3) × SU(2) × U(1) → ~16 generators
• Auxiliary: Several qubits ensuring reversibility
• Total dimension: $d_{\text{cell}}\sim 2^{10}=1024$

(5) Symmetries and Conservation Laws

Encoded Content:

• Global symmetry group $G_{\text{global}}$ (e.g., $U(1)$, $SU(2)$, Lorentz group)
• Local symmetry group $G_{\text{local}}$ (gauge symmetry)
• Conserved quantity labels (e.g., charge sector, spin sector)

Encoding Method:

• Specify group type (e.g., “SU(3)”) → Encode with finite character set
• Choice of representation (e.g., “fundamental representation”, “adjoint representation”)
• How symmetry acts on ${\mathcal{H}}_{\text{cell}}$

(6) Defects and Non-Trivial Configurations

Optional:

• Topological defects (e.g., cosmic strings, magnetic monopoles)
• Domain walls
• Non-uniform lattice lengths (refinement)

Encoding Method:

• Defect positions: Coordinate list
• Defect types: Finite classification encoding
• (Usually optional in ${\Theta }_{\text{str}}$, not all universes have)

Bit Count of Structural Parameters

Combining above, information content of ${\Theta }_{\text{str}}$:

$$\boxed{|{\Theta }_{\text{str}}|=\underset{\text{dimension}}{\underbrace{4}}+\underset{\text{lattice lengths}}{\underbrace{64d}}+\underset{\text{boundaries}}{\underbrace{2d}}+\underset{\text{cell dimension}}{\underbrace{{\log }_2{d}_{\text{cell}}}}+\underset{\text{symmetries}}{\underbrace{I_{\text{symm}}}}+\underset{\text{defects}}{\underbrace{I_{\text{defect}}}}}$$

Typical Numerical Value ($d=3$, Standard Model):

• Dimension: 4 bits
• Lattice lengths: $3\times 64=192$ bits
• Boundaries: $3\times 2=6$ bits
• Cell dimension: ${\log }_2(1024)=10$ bits
• Symmetries: $\sim 50$ bits (encoding “SU(3)×SU(2)×U(1)” and representations)
• Defects: 0 bits (assuming uniform universe)

Total: $$|{\Theta }_{\text{str}}|\sim 262\text{ bits}$$

(Relative to $I_{\max }\sim 10^{123}$ bits, this is negligible!)

Part III: Definition of Dynamical Parameter ${\Theta }_{\text{dyn}}$

Encoded Content: “Source Code” of Physical Laws

Dynamical parameter ${\Theta }_{\text{dyn}}$ specifies QCA time evolution rules, i.e., automorphism $\alpha :\mathcal{A}\to \mathcal{A}$.

(1) Finite Gate Set $\mathcal{G}=\{G_1,\dots ,G_K\}$

Physical Assumption: Exists a fixed finite gate set, all local unitary evolutions composed from these gates.

Analogy:

• Classical computation: NAND gate is universal (any Boolean function can be composed from NAND)
• Quantum computation: $\{H,T,\text{CNOT}\}$ is universal (can approximate any unitary gate)

QCA Gate Set Requirements:

• Each gate acts on radius $r_0$ neighborhood (e.g., nearest neighbor, next-nearest neighbor)
• Matrix elements are algebraic numbers or finite precision angle parameters
• Preserves locality and reversibility

Encoding Method:

• Gate set $\mathcal{G}$ can be pre-agreed (like choosing programming language)
• Or encode gate set itself in ${\Theta }_{\text{str}}$ (more general)

Example (Dirac-QCA gate set):

• Coin gate: $C(\theta )=\exp (-i\theta {\sigma }_y)$
• Shift gate: $S={\sum }_{x,s}|x+s,s⟩⟨x,s|$
• Parameter $\theta$ is discrete angle (see below)

(2) Circuit Depth $D$

$$U_{{\Theta }_{\text{dyn}}}=U_D\cdots U_2{U}_1$$

Physical Meaning:

• $D$: How many layers of gates needed for one time step
• Analogy: “Loop depth” of program

Encoding Method:

• Use $\lceil {\log }_2{D}_{\max }\rceil$ bits
• If $D\le 10^3$ (sufficiently complex), need $\lceil {\log }_{2}10^3\rceil =10$ bits

(3) Gate Type and Action Region for Each Layer

For layer $\ell$ ($\ell =1,\dots ,D$), need to specify:

Gate Type Index $k_{\ell }\in \{1,\dots ,K\}$:

• Which gate from gate set
• Encoding: ${\log }_{2}K$ bits

Action Region $R_{\ell }\subset \Lambda$:

• Which cells gate acts on
• Encoding: Coordinate list + translational symmetry compression

Example (Translation-invariant QCA):

• Each layer: Apply gate $G_{k_{\ell }}$ to all odd lattice points
• Encoding: Only need to specify $k_{\ell }$ and “odd/even” (1 bit)
• Using symmetry, greatly compressed

(4) Discretization of Continuous Angle Parameters

Many gates contain continuous parameters (e.g., rotation angle $\theta$). Finite information requires these parameters to be discretized.

Discretization Scheme:

$${\theta }_{\ell ,j}=\frac{2\pi n_{\ell ,j}}{2^{m_{\ell ,j}}}$$

where:

• $n_{\ell ,j}\in \{0,1,\dots ,2^{m_{\ell ,j}}-1\}$: Discrete label
• $m_{\ell ,j}$: Precision bit number (e.g., $m=10$ corresponds to $2^{10}=1024$ angles)

Encoding Method:

• For each gate needing angle parameter, encode $n_{\ell ,j}$
• Need $m_{\ell ,j}$ bits
• If uniform precision $m$, each angle needs $m$ bits

Physical Consequences:

• Angle precision $\Delta \theta \sim 2\pi /2^m$
• Propagates to physical constants: $\Delta m_e/m_e\sim \Delta \theta$
• $m=10$: Precision $\sim 0.1\%$ (rough)
• $m=50$: Precision $\sim 10^{-15}$ (close to experimental precision)

Popular Analogy: Imagine adjusting pitch in digital music software:

• Analog knob: Continuous adjustment (infinite precision) → Requires infinite information
• Digital slider: Discrete steps (e.g., 1024 steps) → Only needs 10 bits
• Human ear cannot distinguish adjacent steps → Finite precision sufficient!

Physical measurements similar: Physically distinguishable ≠ Mathematically distinguishable.

(5) Derivation of Effective Coupling Constants

From discrete angle parameters, can analytically derive effective field theory coupling constants:

$$m_e({\Theta }_{\text{dyn}})=f_m({\theta }_1,{\theta }_2,\dots )$$ $$\alpha ({\Theta }_{\text{dyn}})=g^2({\theta }_1^{\prime },{\theta }_2^{\prime },\dots )/4\pi$$

Key Theorem (Article 07 will detail): In continuous limit $a,\Delta t\to 0$, relation between Dirac mass and coin angle:

$$m_{\text{eff}}{c}^2\approx \frac{\theta }{\Delta t}$$

Therefore: $$\Delta m_e/m_e\sim \Delta \theta /\theta \sim 2\pi /2^m\theta$$

If $\theta \sim \pi /4$, $m=50$: $$\Delta m_e/m_e\sim 2\pi /(2^{50}\times \pi /4)=8/2^{50}\sim 10^{-14}$$

(Close to current electron mass measurement precision $\sim 10^{-9}$!)

Bit Count of Dynamical Parameters

$$\boxed{|{\Theta }_{\text{dyn}}|=\underset{\text{depth }D}{\underbrace{10}}+D\times \left(\underset{\text{gate type}}{\underbrace{{\log }_{2}K}}+\underset{\text{action region}}{\underbrace{I_{\text{region}}}}+\underset{\text{angle parameters}}{\underbrace{m\times n_{\text{angles}}}}\right)}$$

Typical Numerical Value (Translation-invariant Dirac-QCA):

• Depth: $D\sim 10$ → 10 bits
• Gate type: $K=5$ → ${\log }_{2}5\approx 3$ bits/layer
• Action region: Translation symmetry → 1 bit/layer
• Angle parameters: 2 angles per layer, precision $m=50$ → $2\times 50=100$ bits/layer

Per Layer: $3+1+100=104$ bits Total: $$|{\Theta }_{\text{dyn}}|\sim 10+10\times 104=1050\text{ bits}$$

(Still much smaller than $I_{\max }$!)

Part IV: Definition of Initial State Parameter ${\Theta }_{\text{ini}}$

Encoded Content: Universe’s “Factory Settings”

Initial state parameter ${\Theta }_{\text{ini}}$ specifies quantum state ${\omega }_0$ at big bang moment ($t=0$).

(1) Physical Choice of Initial State

Classical Cosmology:

• Initial conditions: Matter density ${\rho }_0$, Hubble constant $H_0$, curvature $k$…
• Requires infinite precision real numbers (dilemma of hot big bang theory)

Quantum Cosmology:

• Hartle-Hawking no-boundary proposal: $|{\Psi }_0⟩$ automatically selected by path integral
• Concept of “universe wave function”

QCA Framework:

• Initial state must be some state in ${\mathcal{H}}_{\Lambda }$
• But $\dim {\mathcal{H}}_{\Lambda }=d_{\text{cell}}^{N_{\text{cell}}}\sim e^{S_{\max }}$ (astronomical number)
• Cannot enumerate all states! Need generation algorithm

(2) Finite Depth State Preparation Circuit

Core Idea: Use finite depth quantum circuit to generate initial state from simple reference state.

Reference Product State (trivial product state):

$$|0_{\Lambda }⟩=\underset{x\in \Lambda }{⨂}|0_{\text{cell}}{⟩}_x$$

(Each cell in some fixed “vacuum state” $|0_{\text{cell}}⟩$)

State Preparation Circuit:

$$|{\Psi }_0^{\Theta }⟩=V_{{\Theta }_{\text{ini}}}|0_{\Lambda }⟩$$

where $V_{{\Theta }_{\text{ini}}}$ is finite depth unitary operator constructed from gate set $\mathcal{G}$.

Encoded Content (similar to ${\Theta }_{\text{dyn}}$):

• Circuit depth $D_{\text{ini}}$
• Gate type, action region, angle parameters for each layer

(3) Initial Entanglement Structure

Finite depth circuit can only produce short-range entangled states (Lieb-Robinson bound limitation).

Theorem 2.2 (Lieb-Robinson Bound Constraint on Entanglement):

If state preparation circuit depth is $D_{\text{ini}}$, Lieb-Robinson velocity is $v_{\text{LR}}$, then mutual information of two regions $A,B$ at distance $r>v_{\text{LR}}{D}_{\text{ini}}$ satisfies:

$$I(A:B)\lesssim \exp (-c(r-v_{\text{LR}}{D}_{\text{ini}}))$$

(exponential decay)

Physical Meaning:

• Finite depth circuit → Long-range entanglement limited
• To produce universe-scale entanglement → Need depth $D\sim N_{\text{cell}}$ → Parameter explosion
• Trade-off: Initial state is “locally prepared, globally weakly entangled”

Cosmological Application:

• Cosmic microwave background correlation length $\sim 10^5$ light years
• Observable universe scale $\sim 10^{10}$ light years
• Ratio $\sim 10^{-5}$ → Need circuit depth $D_{\text{ini}}\sim \log N_{\text{cell}}$

(4) Symmetry of Initial State

Using symmetry to compress encoding:

Example 1 (Translation-invariant initial state): $$|{\Psi }_0⟩={\left(V_{\text{local}}\right)}^{\otimes N_{\text{cell}}}|0_{\Lambda }⟩$$

• Apply same local unitary $V_{\text{local}}$ to each cell
• Encoding: Only need parameters of $V_{\text{local}}$ (independent of $N_{\text{cell}}$!)

Example 2 (Ground state or thermal state):

• Ground state: $|{\Psi }_0⟩=|GS(H_{\text{eff}})⟩$ (ground state of some effective Hamiltonian)
• Encoding: Only need to encode $H_{\text{eff}}$ (usually determined by ${\Theta }_{\text{dyn}}$)
• Thermal state: ${\rho }_0=\exp (-\beta H_{\text{eff}})/Z$
• Encoding: Only need temperature $\beta$

Popular Analogy: Imagine factory producing 1000 identical parts:

• Stupid method: Draw separate blueprint for each part → 1000 blueprints
• Smart method: Draw one standard blueprint + “copy 1000 times” instruction → 1 blueprint
• Symmetry is “copy instruction”, greatly compresses information!

Bit Count of Initial State Parameters

$$\boxed{|{\Theta }_{\text{ini}}|\sim 10+D_{\text{ini}}\times (3+I_{\text{region}}+m\times n_{\text{angles}})}$$

Typical Numerical Value (Translation-invariant + short-range entanglement):

• Depth: $D_{\text{ini}}\sim 5$ (short-range entanglement sufficient)
• Per layer: $\sim 104$ bits (same as ${\Theta }_{\text{dyn}}$)

Total: $$|{\Theta }_{\text{ini}}|\sim 10+5\times 104=530\text{ bits}$$

(Still much smaller than $I_{\max }$!)

Part V: Total Information and Finite Information Inequality

Definition of Parameter Information

Definition 2.3 (Parameter Information):

$$\boxed{I_{\text{param}}(\Theta ):=|{\Theta }_{\text{str}}|+|{\Theta }_{\text{dyn}}|+|{\Theta }_{\text{ini}}|}$$

(in bits)

Numerical Estimate (combining previous three parts):

Key Observation: $$I_{\text{param}}\sim 10^3\text{ bits}\ll I_{\max }\sim 10^{123}\text{ bits}$$

Parameter information negligible!

Maximum Entropy of State Space

Definition 2.4 (Maximum Entropy of State Space):

$$\boxed{S_{\max }(\Theta ):={\log }_2\dim {\mathcal{H}}_{\Lambda }=N_{\text{cell}}{\log }_2{d}_{\text{cell}}}$$

Numerical Estimate:

• $N_{\text{cell}}\sim 10^{90}$ (Observable universe in Planck length units)
• $d_{\text{cell}}\sim 10^3$ (Standard Model degrees of freedom)
• $S_{\max }\sim 10^{90}\times 10=10^{91}$ bits

(This is the main part!)

Restatement of Finite Information Inequality

Theorem 2.5 (Finite Information Inequality):

$$\boxed{I_{\text{param}}(\Theta )+S_{\max }(\Theta )\le I_{\max }}$$

Corollary 2.6 (Upper Bound on Number of Cells):

Since $I_{\text{param}}\ll I_{\max }$ (negligible), have:

$$N_{\text{cell}}{\log }_2{d}_{\text{cell}}\lesssim I_{\max }$$

Therefore:

$$\boxed{N_{\text{cell}}\lesssim \frac{I_{\max }}{{\log }_2{d}_{\text{cell}}}}$$

Numerical Value:

• If $d_{\text{cell}}=10^3$, ${\log }_2{d}_{\text{cell}}=10$
• Then $N_{\text{cell}}\lesssim 10^{122}$ cells

Physical Interpretation: Trade-off between spatial resolution (number of lattice points) and internal complexity (cell dimension):

• Want more lattice points → Must reduce cell dimension
• Want more complex cells → Must reduce number of lattice points
• Product (logarithm) constrained by $I_{\max }$

Diagram:

graph TD
    A["Total Information Budget<br/>I_max ~ 10^123"] --> B["Parameter Encoding<br/>I_param ~ 10^3"]
    A --> C["State Space<br/>S_max ~ N × log d"]

    B -.-> D["Almost Negligible"]
    C --> E["Main Constraint"]

    E --> F["Number of Cells N"]
    E --> G["Cell Dimension d"]

    F -.-> H["N ↑ → High Spatial Resolution"]
    G -.-> I["d ↑ → Many Internal DOF"]

    H --> J["Trade-off Relation<br/>N × log d ≤ I_max"]
    I --> J

    style A fill:#ffe6e6
    style C fill:#e6f3ff
    style J fill:#e6ffe6

Part VI: Encoding Redundancy and Uniqueness

Sources of Encoding Non-Uniqueness

Question: Given a universe QCA $\mathfrak{U}$, is parameter $\Theta$ unique?

Answer: Not unique! Multiple ways to encode same universe.

Source 1: Gauge Equivalence

Example (Lattice relabeling):

• Relabel lattice points $\Lambda =\{0,1,\dots ,L-1\}$
• Physically identical, but coordinate representation different
• Encoding of ${\Theta }_{\text{str}}$ may change (if coordinates encoded)

Treatment: Consider as same parameter in equivalence class sense

Source 2: Circuit Equivalence

Example (Quantum circuit optimization): Two circuits $U_1,U_2$ may realize same automorphism: $$U_1^{†}A{U}_1=U_2^{†}A{U}_2,\forall A\in \mathcal{A}$$

But $U_1\ne U_2$ (e.g., differ by global phase)

Treatment: Two circuits encode as same ${\Theta }_{\text{dyn}}$

Source 3: Precision Redundancy

Example (Angle parameter rounding): $\theta =0.5000$ and $\theta =0.5001$ may be physically indistinguishable (finite measurement precision)

Treatment: Define equivalence relation ${\theta }_1\sim {\theta }_2$ when $|{\theta }_1-{\theta }_2|<{ϵ}_{\text{meas}}$

Encoding Uniqueness Theorem

Definition 2.7 (Parameter Equivalence):

Two parameter vectors $\Theta ,{\Theta }^{\prime }$ called equivalent, denoted $\Theta \sim {\Theta }^{\prime }$, if and only if exists quasi-local $C^{\ast }$ algebra isomorphism $\Phi :\mathcal{A}(\Theta )\to \mathcal{A}({\Theta }^{\prime })$ and time bijection $f:\mathbb{Z}\to \mathbb{Z}$, such that:

$$\Phi \circ {\alpha }_{\Theta }^n={\alpha }_{{\Theta }^{\prime }}^{f(n)}\circ \Phi ,\forall n\in \mathbb{Z}$$

and initial state satisfies: $${\omega }_0^{{\Theta }^{\prime }}={\omega }_0^{\Theta }\circ {\Phi }^{-1}$$

Theorem 2.8 (Essential Uniqueness of Parameter Encoding):

Under fixed gate set $\mathcal{G}$ and encoding convention, for each physically distinguishable universe QCA class, exists unique equivalence class representative $\Theta \in {\mathcal{M}}_{\Theta }/\sim$.

Proof Outline:

1. Gauge fixing: Choose standard lattice labeling, standard circuit simplification rules
2. Precision truncation: Round to measurement precision ${ϵ}_{\text{meas}}$
3. Quotient equivalence relation: Select one representative from each equivalence class in ${\mathcal{M}}_{\Theta }/\sim$

Physical Meaning: Although encoding has redundancy, “essential degrees of freedom” are finite and unique.

Effective Parameter Dimension

Definition 2.9 (Effective Parameter Dimension):

$$d_{\text{eff}}:={\log }_2|{\mathcal{M}}_{\Theta }/\sim |$$

Estimate: $$d_{\text{eff}}\approx I_{\text{param}}-I_{\text{redundancy}}$$

where $I_{\text{redundancy}}$ is encoding redundancy (dimension of gauge symmetries)

Numerical Value:

• $I_{\text{param}}\sim 10^3$ bits
• $I_{\text{redundancy}}\sim 10^2$ bits (gauge degrees of freedom, coordinate choices, etc.)
• $d_{\text{eff}}\sim 10^3-10^2\sim 900$ bits

Conclusion: Universe’s “essential free parameters” only about 900 bits!

(Equivalent to a 112 byte file!)

Summary of Core Points of This Article

Logic of Triple Decomposition

$$\boxed{\Theta =({\Theta }_{\text{str}},{\Theta }_{\text{dyn}},{\Theta }_{\text{ini}})}$$

Definition of Parameter Information

$$\boxed{I_{\text{param}}(\Theta )=|{\Theta }_{\text{str}}|+|{\Theta }_{\text{dyn}}|+|{\Theta }_{\text{ini}}|}$$

Finite Information Inequality

$$\boxed{I_{\text{param}}(\Theta )+S_{\max }(\Theta )\le I_{\max }}$$

where:

• $S_{\max }(\Theta )=N_{\text{cell}}{\log }_2{d}_{\text{cell}}$ (Maximum entropy of state space)
• $I_{\max }\sim 10^{123}$ bits (Universe information capacity)

Trade-off Relation

$$\boxed{N_{\text{cell}}\times {\log }_2{d}_{\text{cell}}\lesssim I_{\max }}$$

Physical Picture:

• Trade-off between spatial resolution $N_{\text{cell}}$ and internal complexity $d_{\text{cell}}$
• Parameter information $I_{\text{param}}$ relatively negligible ($\sim 10^3$ vs $10^{123}$)
• Main constraint from state space size

Key Insights

1. Three parameter types logically independent: Structure→Dynamics→Initial state, determined sequentially, but logically separated
2. Parameter information extremely small: $I_{\text{param}}\sim 10^3\ll I_{\max }\sim 10^{123}$
3. Symmetry compression: Using translation/gauge symmetry, greatly reduces encoding cost
4. Discretization necessary: Finite information → Continuous parameters must be discretized (finite precision angles)
5. Essential degrees of freedom finite: After removing redundancy, ~900 bits essential parameters

Next Article Preview: 03. Detailed Explanation of Structural Parameters: Discrete Blueprint of Spacetime

• Construction method of lattice set $\Lambda$
• Tensor product decomposition of cell Hilbert space
• Topology types and boundary conditions
• Representation theory of symmetry groups
• Lattice spacing $a$ and preparation for continuous limit