EBOC: Eternal-Block Observer-Computing Unified Theory

Author: Auric Date: 2025-10-18

Abstract

Objective. We propose EBOC (Eternal-Block Observer-Computing): a geometry-information unified framework without explicit global time, merging the timeless causal encoding of Eternal-Graph Cellular Automata (EG-CA) with the program semantics and observation-decoding of Static-Block Universe Cellular Automata (SB-CA) into a single formal system, with verifiable information laws and constructive algorithms. This paper treats the static block $X_f$ and the eternal graph edge shift $(Y_G,\sigma )$ as equivalent dual representations; the exposition below primarily uses $X_f$, but every conclusion can be equivalently restated on $(Y_G,\sigma )$ via finite-type presentation/encoding equivalence.

Three Pillars.

1. Geometric Encoding (Graph/SFT): The universe as a static block $X_f\subset {\Sigma }^{{\mathbb{Z}}^{d+1}}$ satisfying local rule $f$; its causality/consistency characterized in parallel by an eternal graph $G=(V,E)$ and a subshift (SFT).
2. Semantic Emergence (Observation = Decoding): Observation = factor decoding. The decoder $\pi :{\Sigma }^B\to \Gamma$ reads the static block leaf-by-leaf according to an acceptable foliation (cross-leaf reading advancing from layer $c$ to $c+b$ along $\tau$), outputting a visible language; “semantic collapse” is the information factorization from base configuration to visible record.
3. Information Constraint (Non-increase Law): Observation does not create information: $$K(\pi (x{|}_W))\le K(x{|}_W)+K(\pi )+O(1),$$ and under causal thick boundary conditions, provides conditional complexity upper bounds consistent with Brudno entropy limits.

Unified Metaphor (RPG Game). The universe as an RPG with infinite storyline: the game data and evolution rules are already written ($(X_f,f)$), “choices” (apparent free will) must be consistent with storylines (determinism). Leaf-by-leaf reading unlocks chapters according to a preset rhythm $b$; “choice” is to select a representative among compatible branches and exclude incompatible ones, without adding or removing information from the ontological “ROM”.

Core Object. $$\mathcal{U}=(X_f,G,\rho ,\Sigma ,f,\pi ,\mu ,\nu ),$$ where $X_f$ is the space-time SFT, $G$ is the eternal graph, $\rho$ provides the acceptable leaf family (level sets of the primitive integral covector ${\tau }^{\star }$, $⟨{\tau }^{\star },\tau ⟩=b\ge 1$), $\pi$ is the decoder, $\mu$ is a shift-invariant ergodic measure, and $\nu$ is a universal semimeasure (used only for typicality weighting). Equivalently, all results can be expressed using the edge shift $Y_G$ of $G$ and its path shift $(Y_G,\sigma )$; observation/information laws hold equally for both representations.

1. Introduction and Motivation

Traditional CA present “evolution” via global time iteration; the block/eternal-graph perspective gives the entire spacetime at once, with “evolution” being merely the narrative path obtained by leaf-by-leaf reading. The dynamical view depends on a time background and is difficult to make background-independent; the static view lacks observational semantics. EBOC unifies the two via “geometric encoding × semantic decoding × information laws”: SFT/graph structure ensures consistency and constructiveness; factor maps provide visible language; complexity/entropy characterize conservation and limits. This paper establishes theorem family T1–T26 under minimal axioms, with detailed proofs and reproducible experimental protocols.

2. Notation and Preliminaries

2.1 Space, Alphabet, and Configuration

• Space $L={\mathbb{Z}}^d$, spacetime $L\times \mathbb{Z}={\mathbb{Z}}^{d+1}$; finite alphabet $\Sigma$.
• Spacetime configuration $x\in {\Sigma }^{{\mathbb{Z}}^{d+1}}$. Restriction to window $W\subset {\mathbb{Z}}^{d+1}$ denoted $x{|}_W$.
• Convention: $|\cdot |$ denotes both string length and set cardinality (disambiguated by context).

2.2 Neighborhood and Global Evolution

• Finite neighborhood $N\subset {\mathbb{Z}}^d$, local rule $f:{\Sigma }^{|N|}\to \Sigma$: $$x(\mathbf{r}+N,t):=(x(\mathbf{r}+\mathbf{n},t){)}_{\mathbf{n}\in N}\in {\Sigma }^{|N|},x(\mathbf{r},t)=f(x(\mathbf{r}+N,t-1)).$$
• Global map $$F:{\Sigma }^{{\mathbb{Z}}^d}\to {\Sigma }^{{\mathbb{Z}}^d},(F(x))(\mathbf{r})=f(x(\mathbf{r}+N)).$$

2.3 SFT and Eternal Graph

• Space-time SFT $$X_f:=\{x\in {\Sigma }^{{\mathbb{Z}}^{d+1}}:\forall (\mathbf{r},t),x(\mathbf{r},t)=f(x(\mathbf{r}+N,t-1)\left)\right\}.$$
• Eternal graph $G=(V,E)$: vertices $V$ encode local patterns (events), edges $E$ encode causal/consistency relations.
• Edge shift $$Y_G=\{(e_t{)}_{t\in \mathbb{Z}}\in {\mathcal{E}}^{\mathbb{Z}}:\forall t,\mathrm{tail}(e_{t+1})=\mathrm{head}(e_t)\}.$$

2.4 Foliation Decomposition and Leaf-by-Leaf Reading Protocol

• Unimodular transformation: $U\in {\mathrm{GL}}_{d+1}(\mathbb{Z})$ (integrally invertible, $\det U=\pm 1$), time direction $\tau =U{e}_{d+1}$.
• Acceptable leaf: There exists a primitive integral covector ${\tau }^{\star }\in ({\mathbb{Z}}^{d+1}{)}^{\vee }$ and constant $c$, with leaves being level sets $$\{\xi \in {\mathbb{Z}}^{d+1}:⟨{\tau }^{\star },\xi ⟩=c\},$$ satisfying $$⟨{\tau }^{\star },\tau ⟩=b\ge 1,$$ to ensure monotonic cross-leaf advancement.
• Leaf-by-leaf reading: Using block code $\pi :{\Sigma }^B\to \Gamma$, advancing leaf-by-leaf from layer $c\mapsto c+b$, applying $\pi$ to corresponding windows to produce visible sequences.
• Leaf counting and time-slice cuboid families: For time-slice cuboid family windows $W=R\times [t_0,t_0+T-1]$ ($R\subset {\mathbb{Z}}^d$), define $L(W)=T$ as the temporal thickness (number of leaf layers traversed). Under step-size $b$ protocol, the observation step count is $\lfloor L(W)/b\rfloor$ (ignoring $O(1)$ boundary effects). Such window families are compatible with the one-dimensional Følner theory of the time subaction ${\sigma }_{\mathrm{time}}$. The complexity upper bounds in this paper can absorb this $O(1)$ boundary effect into the $O(\log |W|)$ term (see T4).
• Time subaction notation: Denote ${\sigma }_{\mathrm{time}}$ as the one-dimensional subaction of $X_f$ along the time coordinate, ${\sigma }_{\Omega }$ as the time shift of $\Omega (F)$.

2.5 Complexity and Measure

• We use prefix Kolmogorov complexity $K(\cdot )$ and conditional complexity $K(u|v)$.
• $\mu$: invariant and ergodic for time subaction ${\sigma }_{\mathrm{time}}$ (unless otherwise noted); $\nu$: universal semimeasure (algorithmic probability).
• Window description complexity: $K(W)$ is the length of the shortest program generating $W$; Følner family $\{W_k\}$ satisfies $|\partial W_k|/|W_k|\to 0$.
• Entropy notation convention: This paper distinguishes two types of entropy: $h_{\mu }({\sigma }_{\mathrm{time}})$ (compatible with leaf count $L(W)$ normalization) and $h_{\mu }^{(d+1)}$ (compatible with voxel count $|W|$ normalization). They generally have no fixed conversion relation; results throughout are stated and proven under their respective compatible normalizations, without cross-normalization conversion. Furthermore, for any finite spatial section $R\subset {\mathbb{Z}}^d$, denote the observation partition for the time subaction as $${\alpha }_R:=\{[p{]}_{R\times \{0\}}:p\in {\Sigma }^R\},$$ and define the relative entropy $h_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R)$; by Kolmogorov–Sinai definition $$h_{\mu }({\sigma }_{\mathrm{time}}):=\underset{R\text{ finite}}{\sup }{h}_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R).$$ For observation factor $\pi$, the corresponding observation partition is denoted $${\alpha }_R^{\pi }:=\{[q{]}_{R\times \{0\}}^{\pi }:q\in \pi ({\Sigma }^B{)}^R\},$$ where $[q{]}_{R\times \{0\}}^{\pi }$ denotes the cylinder set of configurations yielding visible pattern $q$ when $\pi$ block-reading is applied at $R\times \{0\}$.

2.6 Causal Thick Boundary (for T4)

• Explicitly use $\infty$-norm: $$r:=\underset{\mathbf{n}\in N}{\max }|\mathbf{n}{|}_{\infty }.$$
• Define $$t_{-}=\min \{t:(\mathbf{r},t)\in W\},t_{+}=\max \{t:(\mathbf{r},t)\in W\},T=t_{+}-t_{-}+1.$$
• Bottom layer $\mathrm{base}(W)=\{(\mathbf{r},t_{-})\in W\}$.
• Past causal input boundary at bottom (standard coordinates) $${\partial }_{\downarrow }^{(r,T)}{W}^{-}:=\{(\mathbf{r}+\mathbf{n},t_{-}-1):(\mathbf{r},t_{-})\in \mathrm{base}(W),\mathbf{n}\in [-rT,rT{]}^d\cap {\mathbb{Z}}^d\}\setminus W.$$ Convention: $T$ in this section and in subsequent T4 always refers to the number of time layers traversed (consistent with $L(W)$ in §2.4). For non-standard leaf cases, first map back to standard coordinates via $U^{-1}$ then take image.

2.7 Coordinate Relativization of Eternal Graph (Anchored Chart)

$G$ carries no global coordinates. Choose anchor $v_0$, relative embedding ${\phi }_{v_0}:{\mathrm{Ball}}_G(v_0,R_0)\to {\mathbb{Z}}^{d+1}$ with ${\phi }_{v_0}(v_0)=(0,0)$, path layer function along $\tau$ monotonically non-decreasing, spatial adjacency as finite shift. Fix in advance the minimal presentation radius $R_0$ for “local pattern $\to$ vertex” encoding; all related definitions and constructions below use this fixed $R_0$. Layer function $$\ell (w):=⟨{\tau }^{\star },{\phi }_{v_0}(w)⟩,{\mathrm{Cone}}_{\ell }^{+}(v):=\{w\in \mathrm{Dom}({\phi }_{v_0}):\exists v⇝w\text{with }\ell \text{monotonically non-decreasing along path}\}.$$ SBU (Static Block Unfolding) $$X_f^{(v,\tau )}:=\{x\in X_f:x{|}_{{\phi }_{v_0}({\mathrm{Ball}}_G(v,R_0))}=\mathrm{pat}(v)\text{and}x\text{is a consistent extension on}{\phi }_{v_0}({\mathrm{Cone}}_{\ell }^{+}(v)\left)\right\},$$ where “consistent extension” means: all cells within this cone forced by $v$ and the local rule match $x$.

2.8 Eternal Graph–SFT Dual Representation (Working Principle)

• Dual representation: All conclusions stated in terms of static block $X_f$ in this paper have an equivalent version in terms of eternal graph edge shift $(Y_G,\sigma )$; the two are given by standard finite-type presentation/encoding. For conciseness, the main text uses $X_f$, with path version statements bracketed as “(EG)” where necessary.
• Correspondence: Window $W$ and thick boundary correspond to finite path segment and finite adjacency radius; leaf-by-leaf reading corresponds to time-shift reading along paths; observation factor $\pi$ defined on $X_f$ can be equivalently realized on $Y_G$ via path block code ${\pi }_G$.

Below we discuss SBU only on the graph domain where this relative embedding exists.

Definition: Realizable Event. Let eternal graph $G=(V,E)$. Call $v\in V$ realizable if there exists $x\in X_f$ with some relative embedding ${\phi }_{v_0}$ and radius $R_0$ such that $x{|}_{{\phi }_{v_0}({\mathrm{Ball}}_G(v,R_0))}$ is consistent with the local pattern of $v$ (according to the “local pattern→vertex” encoding convention in the text).

3. Minimal Axioms (A0–A3)

A0 (Static Block) $X_f$ is the set of locally constrained models. A1 (Causal-Local) $f$ has finite neighborhood; reading uses acceptable leaves. A2 (Observation = Factor Decoding) Leaf-by-leaf reading with $\pi$ applied yields ${\mathcal{O}}_{\pi ,\varsigma }(x)$. A3 (Information Non-increase) For any window $W$, $K(\pi (x{|}_W))\le K(x{|}_W)+K(\pi )+O(1)$.

4. Leaf-Language and Observational Equivalence

Fix $(\pi ,\varsigma )$ and leaf family $\mathcal{L}$, $${\mathsf{Lang}}_{\pi ,\varsigma }(X_f):=\{{\mathcal{O}}_{\pi ,\varsigma }(x)\in {\Gamma }^{\mathbb{N}}:x\in X_f,\text{leaf-by-leaf reading per }\mathcal{L}\},x{\sim }_{\pi ,\varsigma }y⟺{\mathcal{O}}_{\pi ,\varsigma }(x)={\mathcal{O}}_{\pi ,\varsigma }(y).$$

5. Preliminary Lemmas

Lemma 5.1 (Complexity Conservation for Computable Transformation). If $\Phi$ is computable, then $$K(\Phi (u)|v)\le K(u|v)+K(\Phi )+O(1).$$ $\square$

Lemma 5.2 (Describable Window Families). For $(d+1)$-dimensional axis-aligned parallelotopes or regular windows described by $O(\log |W|)$ parameters, $K(W)=O(\log |W|)$. $\square$

Lemma 5.3 (Thick Boundary Coverage). For radius $r={\max }_{\mathbf{n}\in N}|\mathbf{n}{|}_{\infty }$ and time span $T$, computing $x{|}_W$ requires only the previous layer of $\mathrm{base}(W)$ within $[-rT,rT{]}^d$ past input; i.e., ${\partial }_{\downarrow }^{(r,T)}{W}^{-}$ covers all dependencies (propagation radius measured in $|\cdot {|}_{\infty }$). $\square$

Lemma 5.4 (Factor Entropy Non-increase). If $\varphi :(X,T)\to (Y,S)$ is a factor map, then $h_{\mu }(T)\ge h_{{\varphi }_{\ast }\mu }(S)$. $\square$

Lemma 5.5 (Time Subaction Version SMB/Brudno). For ${\sigma }_{\mathrm{time}}$-invariant and ergodic $\mu$ and time-slice cuboid family $W_k=R\times [t_k,t_k+T_k-1]$, where $T_k\to \infty$, and $R\subset {\mathbb{Z}}^d$ is a fixed finite set. Denote $L(W_k)=T_k$ (temporal thickness), we have $$-\frac{1}{T_k}\log \mu ([x{|}_{W_k}])\to h_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R)(\mu \text{-a.e.}),\underset{k\to \infty }{\mathrm{limsup}}\frac{K(x{|}_{W_k})}{T_k}=h_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R)(\mu \text{-a.e.}).$$ This is the one-dimensional SMB/Brudno theorem for time subaction ${\sigma }_{\mathrm{time}}$ or equivalently $(\Omega (F),{\sigma }_{\Omega })$. The window shape must be time-slice cuboid family (or satisfy equivalent “time-uniform slicing” condition) to ensure cylinder sets $[x{|}_{W_k}]$ are generated by iteration of the one-dimensional generating partition of the time subaction, making normalization match the action. If general $W_k$ is used, one can only guarantee that the limit with $|W_k|$ normalization is the ${\mathbb{Z}}^{d+1}$ action entropy $h_{\mu }^{(d+1)}$, or recover the time entropy limit under additional uniform slicing/density assumptions. $\square$ Furthermore, for fixed finite $R$, the above limit equals $h_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R)$; taking ${\sup }_R$ recovers $h_{\mu }({\sigma }_{\mathrm{time}})$. If instead using $|R_k|\to \infty$ with generating section families (hence corresponding generating partitions), the limit directly equals the complete $h_{\mu }({\sigma }_{\mathrm{time}})$ (this case is a supplementary remark, not within the fixed $R$ premise of this lemma). 【Note】Only when ${\alpha }_R$ is a generating partition for the time subaction (e.g., using generating section family $R_k\uparrow$), does the limit equal the complete $h_{\mu }({\sigma }_{\mathrm{time}})$; for fixed finite $R$ the limit is $h_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R)$.

6. Main Theorems with Detailed Proofs (T1–T26)

T1 (Block–Natural Extension Conjugacy)

Proposition. If $X_f\ne \varnothing$, then $$\boxed{(X_f,{\sigma }_{\mathrm{time}})\cong (\Omega (F),{\sigma }_{\Omega })},\Omega (F)=\{(\dots ,x_{-1},x_0,x_1,\dots ):F(x_t)=x_{t+1}\}.$$

Proof. Define $\Psi :X_f\to \Omega (F)$, $(\Psi (x){)}_t(\mathbf{r})=x(\mathbf{r},t)$. By SFT constraint we have $F((\Psi (x){)}_t)=(\Psi (x){)}_{t+1}$. Define inverse $\Phi :\Omega (F)\to X_f$, $\Phi ((x_t))(\mathbf{r},t)=x_t(\mathbf{r})$. Clearly $\Phi \circ \Psi =\mathrm{id}$, $\Psi \circ \Phi =\mathrm{id}$, and $\Psi \circ {\sigma }_{\mathrm{time}}={\sigma }_{\Omega }\circ \Psi$. Continuity and Borel measurability follow from product topology and cylinder structure. $\square$

T2 (Unimodular Covariance; Complexity Density Invariance)

Proposition. For any shift-invariant ergodic measure $\mu$ and two sets of acceptable leaves (given by $U_1,U_2\in {\mathrm{GL}}_{d+1}(\mathbb{Z})$), let $W_k=U_2{U}_1^{-1}(W_k)$. Assumptions: (i) Both $\{W_k\}$ and $\{W_k\}$ are time-slice cuboid Følner families, with spatial section $R\subset {\mathbb{Z}}^d$ being a fixed finite set independent of $k$; (ii) the two leaf families are given by primitive integral covector–time vector pairs $({\tau }_i^{\star },{\tau }_i)$ with pairing constants $b_i=⟨{\tau }_i^{\star },{\tau }_i⟩\ge 1$ being constants independent of $k$. Then for $\mu$-a.e. $x$, $$\underset{k\to \infty }{\mathrm{limsup}}\frac{K(\pi (x{|}_{W_k}))}{L(W_k)}=h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},{\alpha }_R^{\pi }),\underset{k\to \infty }{\mathrm{limsup}}\frac{K(\pi (x{|}_{W_k}))}{L(W_k)}=h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},{\tilde{\alpha }}_R^{\pi }).$$

Proof. The integer isomorphism $U=U_2{U}_1^{-1}$ preserves Følner property: if $\{W_k\}$ is Følner then so is $\{W_k\}$, with $|W_k|=|W_k|$ (integer determinant $\pm 1$; here $W_k$ denotes the lattice image set $U(W_k)$, even if shape is non-axis-aligned, lattice point count remains equal). Under the time-slice cuboid family assumption, there exist constants $c_{-},c_{+}>0$ depending only on $(U,{\tau }_1^{\star },{\tau }_1,{\tau }_2^{\star },{\tau }_2)$ (independent of $k$) such that leaf counts (temporal thickness) satisfy $c_{-}L(W_k)\le L(W_k)\le c_{+}L(W_k)$. Applying Lemma 5.5 to the factor system for each window family yields time entropy limits relative to respective observation partitions: $h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},{\alpha }_R^{\pi })$ and $h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},{\tilde{\alpha }}_R^{\pi })$. If the two observation schemes are related by a finite-memory invertible block code isomorphism along the time subaction, they are equivalent and yield the same entropy rate; under this condition coordinate choice does not change the limit value. $\square$

T3 (Observation = Decoding as Semantic Collapse)

Proposition. ${\mathcal{O}}_{\pi ,\varsigma }:X_f\to {\Gamma }^{\mathbb{N}}$ is a factor map of the time subaction, inducing equivalence class $x{\sim }_{\pi ,\varsigma }y$. One observation selects a representative within the equivalence class; the underlying $x$ remains unchanged. $\square$

T4 (Information Upper Bound: Conditional Complexity Version)

$$\boxed{K(\pi (x{|}_W)|x{|}_{{\partial }_{\downarrow }^{(r,T)}{W}^{-}})\le K(f)+K(W)+K(\pi )+O(\log |W|)}.$$ where $T$ is the number of time layers traversed (consistent with $L(W)$ in §2.4). 【Premise Note】The following upper bound holds under the single-step time dependence premise of §2.2; if the rule depends on past $m>1$ layers, change the past causal input boundary to the corresponding thick boundary at $\{t_{-}-1,\dots ,t_{-}-m\}$, replace $rT$ with $r(T+m-1)$, and the rest of the reasoning remains unchanged.

Proof. Construct universal program $\mathsf{Dec}$:

1. Input: Encoding of $f$, encoding of window $W$ (including $(t_{-},T)$ and geometric parameters), encoding of $\pi$, and conditional string $x{|}_{{\partial }_{\downarrow }^{(r,T)}{W}^{-}}$.
2. Recursion: Generate layer-by-layer from layer $t_{-}$ according to time subaction. For any $(\mathbf{r},s)\in W$, compute via $$x(\mathbf{r},s)=f(x(\mathbf{r}+N,s-1))$$ with required right-hand side either already generated from previous layer or from conditional boundary (Lemma 5.3). Generate by layers $s=t_{-},t_{-}+1,\dots ,t_{+}$, avoiding dependency cycles. For each layer $s$, first generate all cells needed for the forward closure of $W$ within the propagation cone (allowing temporary generation of values outside $W$ but within $[-r(s-t_{-})$, $r(s-t_{-}){]}^d\times \{s\}$), finally restrict to $W$.
3. Decoding: Apply $\pi$ within $W$ according to protocol to obtain $\pi (x{|}_W)$.

Program body is constant size, input length is $K(f)+K(W)+K(\pi )+O(\log |W|)$ ($\log |W|$ for layer depth/alignment cost). By definition of prefix complexity, the upper bound follows. $\square$

T5 (Brudno Alignment and Factor Entropy)

Proposition. For fixed finite spatial section $R$ and time-slice cuboid family $\{W_k=R\times [t_k,t_k+T_k-1]\}$ (where $T_k\to \infty$ and satisfying window premise of Lemma 5.5), normalizing by temporal thickness $L(W_k)=T_k$ (【Note】Only when ${\alpha }_R$ is a generating partition, does the limit equal the complete $h_{\mu }({\sigma }_{\mathrm{time}})$; for fixed finite $R$ it is $h_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R)$): $$\underset{k\to \infty }{\mathrm{limsup}}\frac{K(x{|}_{W_k})}{T_k}=h_{\mu }({\sigma }_{\mathrm{time}},{\alpha }_R),\underset{k\to \infty }{\mathrm{limsup}}\frac{K(\pi (x{|}_{W_k}))}{T_k}=h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},{\alpha }_R^{\pi })\le h_{\mu }({\sigma }_{\mathrm{time}}).$$

Proof. By Lemma 5.5 (time subaction version SMB/Brudno, window family shape matching normalization), the first limsup equality holds. For factor image, $\pi$ is computable transformation and factor entropy is non-increasing (Lemma 5.4), so the second limsup equality holds and does not exceed $h_{\mu }({\sigma }_{\mathrm{time}})$. Moreover, applying Lemma 5.5 to the factor system $(\pi (X_f),{\sigma }_{\mathrm{time}},{\pi }_{\ast }\mu )$ (same window premise), we get the limsup value equals $h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},{\alpha }_R^{\pi })$ for $({\pi }_{\ast }\mu )$-a.e.; by $\mu ({\pi }^{-1}A)={\pi }_{\ast }\mu (A)$ the limsup equality also holds for $\mu$-a.e. $x$. If using generating partition (or growing section family $|R_k|\to \infty$), the right-hand side can be directly written as $h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}})$. $\square$

T6 (Program Emergence: Macro-block Forcing; SB-CA $\Rightarrow$ TM)

Proposition. (Allowing finite higher-block representation/alphabet extension) There exists a macro-block forcing embedding scheme $\mathsf{Emb}(M)$ such that if the finite-type constraint family of this scheme is non-empty (extended SFT non-empty), then there exists $x^{\mathrm{ext}}\in X_f^{\mathrm{ext}}$ (if using only higher blocks without alphabet extension, write $x^{[k]}\in X_f^{[k]}\cong X_f$) that under decoder $\pi$ can be decoded to yield some (expected) run trace of Turing machine $M$. If further assuming the embedding constraints are complete and without spurious solutions, we obtain “if and only if”.

Proof (Construction). Take macro-block size $k$. Extend alphabet to ${\Sigma }^{\prime }=\Sigma \times Q\times D\times S$ (machine state, tape symbol, head movement, synchronization phase). At macro-block scale, implement transition $(q,a)\mapsto (q^{\prime },a^{\prime },\delta )$ via finite-type local constraints, and implement cross-macro-block synchronization via phase signals. Denote the extended SFT obtained from above finite-type constraints as $X_f^{\mathrm{ext}}$, and denote the forgetting projection $\rho :{\Sigma }^{\prime }\to \Sigma$ (if needing to relate extended configuration back to base). Decoder $\pi$ reads macro-block centerline to output tape content. If these finite-type constraint families are globally compatible (extended SFT non-empty), then by compactness we can take limit to obtain global configuration $x^{\mathrm{ext}}$; non-emptiness thus depends on compatibility premise, not automatically derived from compactness. $\square$

T7 (Program Weight Universal Semimeasure Bound)

Proposition. Let program codes be prefix-unambiguous, then for any decodable program $p$, $\nu (p)\le C\cdot 2^{-|p|}$.

Proof. By Kraft inequality ${\sum }_p{2}^{-|p|}\le 1$, universal semimeasure $\nu$ as weighted sum satisfies upper bound, constant $C$ depends only on chosen machine. $\square$

T8 (Section–Natural Extension Duality; Entropy Preservation)

Proposition. $X_f$ and $\Omega (F)$ are mutually section/natural extension dual, with equal time entropy.

Proof. By conjugacy $(X_f,{\sigma }_{\mathrm{time}})\cong (\Omega (F),{\sigma }_{\Omega })$ of T1. Natural extension does not change entropy; conjugacy preserves entropy, so the conclusion holds. $\square$

T9 (Halting Witness Staticization)

Proposition. Under compatible embedding scheme of T6 (i.e., corresponding extended SFT non-empty), $M$ halts if and only if there exists $x^{\mathrm{ext}}\in X_f^{\mathrm{ext}}$ and finite window $W$ such that visible pattern $\pi \left(x^{\mathrm{ext}}{|}_W\right)$ contains “termination marker” $\square$; conversely likewise.

Proof. “If” direction: If $M$ halts at step $\hat{t}$, then macro-block center decoding output shows $\square$, forming finite visible pattern. “Only if” direction: If $\square$ appears in visible layer, by local consistency backtrack to halting transition; construction ensures $\square$ is not produced by other causes. Above equivalence all premise on global compatibility of embedding scheme (extended SFT non-empty). $\square$

T10 (Unimodular Covariance Information Stability)

Proposition. If window families satisfy $K(W_k)=O(\log |W_k|)$, then under any integer transformation $U\in {\mathrm{GL}}_{d+1}(\mathbb{Z})$, the T4 upper bound and T3 semantics are preserved; window description complexity difference of transformed windows is $O(\log |W_k|)$, not involving data complexity $K(x{|}_{W_k})$ or $K(\pi (x{|}_{W_k}))$ pointwise upper bounds.

Proof. By Lemmas 5.1–5.2, window description complexity $K(W)$ is the shortest program length for generating window geometric parameters. Encoding of integer transformation $U$ and translation adds only $O(1)$ constant; bounded distortion of thick boundary ${\partial }_{\downarrow }^{(r,T)}{W}^{-}$ under $U$ is absorbed into $O(\log |W_k|)$. By measure-theoretic version of T2, data complexity density after normalization is coordinate-independent. $\square$

T11 (Model Set Semantics)

Proposition. $$X_f={\mathcal{T}}_f(\mathsf{Conf})=\{x\in {\Sigma }^{{\mathbb{Z}}^{d+1}}:\forall (\mathbf{r},t)x(\mathbf{r},t)=f(x(\mathbf{r}+N,t-1))\}.$$ Proof. By definition. $\square$

T12 (Computational Model Correspondence)

Proposition. (i) SB-CA and TM mutually simulate; (ii) Various CSP/Horn/$\mu$-safe formulas $\Phi$ can be equivalently embedded in EG-CA.

Proof. (i) By T6 and standard “TM simulates CA” gives bidirectional simulation. (ii) Convert each clause of radius $\le r$ to forbidden pattern set ${\mathcal{F}}_{\Phi }$, obtain $X_{f_{\Phi }}$. Solution models are equivalent to models of $\Phi$ (finite-type + compactness). $\square$

T13 (Leaf-Language $\omega$-Automaton Characterization; Sofic Sufficient Condition)

Proposition. If (i) using path version $(Y_G,\sigma )$ or there exists $k$ such that $X_f$ under time subaction can via higher-block representation $X_f^{[k]}$ make cross-leaf consistency depend only on adjacent $k$ layers (time direction can be Markovianized); and (ii) decoder $\pi :{\Sigma }^B\to \Gamma$ has kernel window $B$ with finite cross-leaf thickness, then ${\mathsf{Lang}}_{\pi ,\varsigma }(X_f)$ is sofic (hence $\omega$-regular), accepted by some Büchi automaton $\mathcal{A}$: $${\mathsf{Lang}}_{\pi ,\varsigma }(X_f)=L_{\omega }(\mathcal{A}).$$

Proof (Construction). Under time direction finite-memory condition, take higher-block representation $X_f^{[k]}$, encode finite state set $Q$ into extended alphabet and implement transition $\delta$ via finite-type constraints. One cross-leaf reading corresponds to one automaton step. Acceptance condition is realized via local safety/regularity constraints (e.g., “infinitely often visit $F$” via loop memory bit). Thus obtain equivalent $\omega$-language. $\square$

T14 (SBU Existence for Any Realizable Event)

Proposition. For realizable $v$ and acceptable $\tau$, $(X_f^{(v,\tau )},{\rho }_{\tau })$ is non-empty.

Proof. Take finite window family consistent with $v$, forming directed set by inclusion; finite consistency given by “realizable” and local constraints. By compactness (product topology) and Kőnig lemma, there exists limit configuration $x\in X_f$ consistent with $v$, hence non-empty. $\square$

T15 (Causal Consistent Extension and Paradox Exclusion)

Proposition. $X_f^{(v,\tau )}$ contains only restrictions of global solutions consistent with anchor $v$; contradictory events do not coexist.

Proof. If some $x\in X_f^{(v,\tau )}$ simultaneously contains event contradictory to $v$, then on ${\phi }_{v_0}({\mathrm{Cone}}_{\ell }^{+}(v))$ it is both consistent and contradictory, violating consistency definition. $\square$

T16 (Time = Deterministic Advancement (Apparent Choice))

Proposition. Under deterministic $f$ and thick boundary conditions, each minimal positive increment advancement of $\ell$ is equivalent to deterministic advancement on the future consistent extension family; unique under deterministic CA.

Proof. By construction of T4, given previous layer and thick boundary, next layer values uniquely defined by $f$; if two different advancements exist at same layer both acceptable and mutually conflicting, then at some common cell produces inconsistent assignments, contradiction. $\square$

T17 (Multi-anchor Observers and Subjective Time Rate)

Proposition. Effective step size $b=⟨{\tau }^{\star },\tau ⟩\ge 1$ reflects chapter rhythm; different $b$ only changes reading rhythm, under time-slice cuboid Følner family premise with fixed or uniformly bounded spatial section, entropy rate normalized by temporal thickness $L(W_k)=T_k$ is consistent.

Proof. Changing time subaction to ${\sigma }_{\mathrm{time}}^{(b)}$ is equivalent to “sampling” the $\mathbb{Z}$ subaction (${\sigma }_{\Omega }^b$). Measure entropy satisfies (version relative to chosen partition also holds) $$h_{\mu }({\sigma }_{\Omega }^b)=b\cdot h_{\mu }({\sigma }_{\Omega }).$$ For time-slice cuboid $W$, observation step count is $\lfloor L(W)/b\rfloor$ (see §2.4), while normalization uses temporal thickness $L(W)=T$. Hence $$\frac{K(\pi (x{|}_W))}{L(W)}\sim \frac{\lfloor L(W)/b\rfloor }{L(W)}\cdot h_{{\pi }_{\ast }\mu }\left({\sigma }_{\mathrm{time}}^b\right)=\frac{1}{b}\cdot b{h}_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}})=h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}}),$$ where $h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}}^b)=b{h}_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}})$. If using fixed finite section $R$, should write $$h_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}}^b,{\alpha }_R^{\pi })=b{h}_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},\underset{i=0}{\overset{b-1}{\bigvee }}{\sigma }_{\mathrm{time}}^{-i}{\alpha }_R^{\pi }),$$ and only when the above joined partition generates (or there exists finite-memory invertible block code isomorphism along time) can we write $b{h}_{{\pi }_{\ast }\mu }({\sigma }_{\mathrm{time}},{\alpha }_R^{\pi })$. Observation step count $\lfloor L(W)/b\rfloor$ factor $1/b$ exactly cancels $b$, entropy rate after normalization independent of $b$ choice. By Lemma 5.5, for $\mu$-a.e. $x$ the two family density limits agree. $\square$

T18 (Anchored Graph Coordinate Relativization Invariance)

Proposition. Two embeddings $({\phi }_{v_0},{\phi }_{v_1})$ if sourced from restrictions of same integer affine embedding $\Phi$, then on intersection domain after removing constant-radius boundary strip, differ only by ${\mathrm{GL}}_{d+1}(\mathbb{Z})$ integer affine and finite-radius relabeling; additional encoding/description overhead between two embedding protocols $\le O(K(W))$ (for describable window families is $O(\log |W|)$), not involving data complexity or entropy difference pointwise upper bound for observation sequences themselves.

Proof. There exist $U\in {\mathrm{GL}}_{d+1}(\mathbb{Z})$ and translation $t$ such that ${\phi }_{v_1}=U\circ {\phi }_{v_0}+t$ holds on intersection domain. Finite radius difference corresponds to removing boundary strips. Window encoding in two coordinates adds only finite description of $U,t$; this is additional description cost for protocol conversion, absorbed into $O(K(W))$ (Lemmas 5.1–5.2). Data complexity of observation sequences given coordinate-independence after normalization by measure-theoretic version of T2. $\square$

T19 ($\ell$-Successor Determinism and Same-layer Exclusivity)

Proposition. Under deterministic $f$, radius $r$, if context of $u$ covers information needed to generate next layer, then unique ${\mathrm{succ}}_{\ell }(u)$ exists; edge $u\to {\mathrm{succ}}_{\ell }(u)$ has exclusivity against same-layer alternatives.

Proof. Next layer values uniquely determined by local function of $f$; if two different same-layer alternatives can both continue and mutually conflict, then at some common cell produce inconsistent assignment, contradiction. $\square$

T20 (Compatibility Principle: Apparent Choice and Determinism Unified)

Proposition. Leaf-by-leaf advancement at operational level manifests as “representative selection”, while holistic static encoding is “unique consistent extension”; determinism holds, compatible with A3/T4.

Proof. By T14 global consistent extension exists; T15 excludes contradictory branches; T3 shows “observation = select representative in equivalence class”; T4/A3 ensure choice does not increase information. Hence apparent freedom and ontological determinism are consistent. $\square$

T21 (Information Non-increase Law: General CA and Observation Factor)

Proposition. Let $F$ be radius $r$ CA in $d$ dimensions, take any Følner window family $\{W_k\}$ (axis-aligned parallelotopes satisfy $K(W_k)=O(\log |W_k|)$). Define spatial information density (per cell) $$I(x):=\underset{k\to \infty }{\mathrm{limsup}}\frac{K\left(x{|}_{W_k}\right)}{|W_k|},I_{\pi }(x):=\underset{k\to \infty }{\mathrm{limsup}}\frac{K(\pi (x{|}_{W_k}))}{|W_k|}.$$ Then for each fixed $T\in \mathbb{N}$ and configuration $x$, $$I\left(F^{T}x\right)\le I(x),I_{\pi }\left(F^{T}x\right)\le I\left(F^{T}x\right)\le I(x).$$ Note: This theorem is for $d$-dimensional spatial configurations, normalized by voxel count. When applied to time subaction of spacetime SFT, should use leaf count normalization (see T5).

Proof. Thick boundary and propagation cone give dependency domain: there exists $W_k^{+rT}$ such that $\left(F^{T}x\right){|}_{W_k}$ can be computably recovered from $x{|}_{W_k^{+rT}}$ (see Lemma 5.3’s $r,T$ control). By computable transformation complexity upper bound, $$K((F^{T}x){|}_{W_k})\le K\left(x{|}_{W_k^{+rT}}\right)+O(\log |W_k|).$$ Følner property gives $|W_k^{+rT}|/|W_k|\to 1$, taking $\mathrm{limsup}$ yields $I(F^{T}x)\le I(x)$. Factor decoding non-increase of information (A3, or computable transformation of Lemma 5.1), yields $I_{\pi }(F^{T}x)\le I(F^{T}x)$. Combining gives conclusion. $\square$

T22 (Information Conservation Law: Reversible CA)

Proposition. If $F$ is reversible and $F^{-1}$ is also CA (reversible cellular automaton), then for each fixed $T\in \mathbb{N}$ and configuration $x$, spatial information density (per cell) is conserved: $$I\left(F^{T}x\right)=I(x),I_{\pi }\left(F^{T}x\right)\le I(x),$$ with equality when $\pi =\mathrm{id}$ or lossless factor conjugate to $F$. If $\mu$ is shift-invariant ergodic measure, applied to time subaction of spacetime SFT $(X_f,{\sigma }_{\mathrm{time}})$, by conjugacy of T1 and entropy preservation under reversibility we know $h_{\mu }({\sigma }_{\mathrm{time}})$ is invariant; spatial marginal distributions at each time $({\nu }_t)$ satisfy stationarity ${\nu }_{t+1}=F_{\ast }{\nu }_t={\nu }_t$; this notation has no direct mixing with computation of $h_{\mu }({\sigma }_{\mathrm{time}})$. Hence $\mu$-almost everywhere time direction information density is conserved.

Proof. By T21 applying to $F$ and $F^{-1}$ respectively yields $I(F^{T}x)\le I(x)$ and $I(x)\le I(F^{T}x)$, combining gives $I(F^{T}x)=I(x)$. Non-increase for $\pi$ given by A3. Measure-theoretic version by conjugacy $(X_f,{\sigma }_{\mathrm{time}})\cong (\Omega (F),{\sigma }_{\Omega })$ and entropy preservation under reversibility (T8), combined with Brudno theorem of leaf count normalization (Lemma 5.5). $\square$

T23 (Observation Pressure Function and Information Geometry)

【Source Mapping】For each visible category $j$, let $a_j$ be the a priori weight (or count weight) for decoded unit time slice, ${\beta }_j$ be the fixed vector of corresponding leaf-by-leaf statistical features (e.g., frequency vector/energy cost); when taking limit over window family $W_k$, $\{p_j(\eta )\}$ are these observation statistics reweighted by exponential family.

Definition. To avoid confusion with leaf family notation $\rho$ in §2, below we use $\eta$ for parameters. Take a set of visible categories (given by decoding/counting rules) indexed $j=1,\dots ,J$ (here take $J<\infty$), with weights $a_j>0$ and vectors ${\beta }_j\in {\mathbb{R}}^n$. Define $$Z(\eta )=\sum _{j=1}^J{a}_j{e}^{⟨{\beta }_j,\Re \eta ⟩},P(\eta )=\log Z(\eta ),p_j(\eta )=\frac{a_j{e}^{⟨{\beta }_j,\Re \eta ⟩}}{Z(\eta )}.$$ In the domain where $Z(\eta )$ converges, and satisfying standard conditions of local uniform convergence allowing sum/derivative interchange, we have $${\nabla }_{\eta }P(\eta )={\mathbb{E}}_p[\beta ]=\sum _j{p}_j{\beta }_j,{\nabla }_{\eta }^{2}P(\eta )={\mathrm{Cov}}_p(\beta )⪰0.$$ Hence $P$ is convex, its Hessian being the Fisher information. Along direction $\eta (s)={\eta }_{\perp }+s\mathbf{v}$, $$\frac{d^2}{d{s}^2}P(\eta (s))={\mathrm{Var}}_p(⟨\beta ,\mathbf{v}⟩)\ge 0.$$ If we further denote Shannon entropy $H(\eta )=-{\sum }_j{p}_j\log p_j$, then $$H(\eta )=P(\eta )-\sum _j{p}_j\log a_j-⟨\Re \eta ,{\mathbb{E}}_p[\beta ]⟩.$$

Proof Sketch. By standard differentiation of log-sum-exp (under aforementioned local uniform convergence condition allowing interchange of sum and differentiation), obtain gradient and Hessian expressions; directional second derivative is variance. Entropy identity obtained by substituting $p_j\propto a_j{e}^{⟨{\beta }_j,\Re \eta ⟩}$ into $H=-\sum p_j\log p_j$ and expanding. $\square$

T24 (Phase Transition/Dominance Switching Criterion; Finite $J$ Version)

Proposition. Let amplitude $A_j(\eta ):=a_j{e}^{⟨{\beta }_j,\eta ⟩}$, and define $$H_{jk}=\{\eta :⟨{\beta }_j-{\beta }_k,\eta ⟩=\log \frac{a_k}{a_j}\},\delta (\eta ):=\underset{j<k}{\min }|⟨{\beta }_j-{\beta }_k,\eta ⟩-\log \frac{a_k}{a_j}|.$$ If $\delta (\eta )>\log (J-1)$, then unique index $j_{\star }$ exists such that $A_{j_{\star }}(\eta )={\max }_j{A}_j(\eta )$ and $$\sum _{k\ne j_{\star }}{A}_k(\eta )<A_{j_{\star }}(\eta ),$$ hence no dominance switching; dominance switching can only occur in thin band $\{\eta :\delta (\eta )\le \log (J-1)\}$, whose skeleton is hyperplane family $\{H_{jk}\}$.

Proof. By definition of $\delta (\eta )$, $\log A_{j_{\star }}-\log A_k\ge \delta (\eta )$, hence $A_k\le e^{-\delta (\eta )}{A}_{j_{\star }}$, summing yields conclusion. $\square$

T25 (Directional Pole = Growth Exponent; Countable Infinite Version)

Proposition. Fix direction $\mathbf{v}$ and decomposition $\eta ={\eta }_{\perp }+s\mathbf{v}$. Let index family $\{(a_j,{\beta }_j){\}}_{j\ge 1}$ be countably infinite, and assume there exists ${\eta }_0$ such that ${\sum }_{j\ge 1}{a}_j{e}^{⟨{\beta }_j,\Re {\eta }_0⟩}<\infty$. Let weighted cumulative distribution along $\mathbf{v}$ $$M_{\mathbf{v}}(t)=\sum _{t_j\le t}{w}_j,t_j:=⟨-{\beta }_j,\mathbf{v}⟩,w_j:=a_j{e}^{⟨{\beta }_j,{\eta }_{\perp }⟩},$$ have exponential–polynomial asymptotics as $t\to +\infty$ (and assume $M_{\mathbf{v}}$ has bounded variation and moderate growth) $$M_{\mathbf{v}}(t)=\sum _{\ell =0}^L{Q}_{\ell }(t)e^{{\gamma }_{\ell }t}+O\left(e^{({\gamma }_L-\delta )t}\right),{\gamma }_0>\cdots >{\gamma }_L,$$ and $M_{\mathbf{v}}$ has bounded variation and satisfies moderate growth. Then its Laplace–Stieltjes transform $${\mathcal{L}}_{\mathbf{v}}(s):={\int }_{(-\infty ,+\infty )}{e}^{-st}d{M}_{\mathbf{v}}(t)=\sum _j{w}_j{e}^{-s{t}_j}$$ converges for $\Re s>{\gamma }_0$, and can be meromorphically continued to $\Re s>{\gamma }_L-\delta$, with poles of order at most $\mathrm{deg}{Q}_{\ell }+1$ at $s={\gamma }_{\ell }$. In particular, the real part of the rightmost convergence line equals the maximal growth exponent ${\gamma }_0$. If $J<\infty$, the above sum is finite and pole case does not occur.

Proof Sketch. Belongs to classical Laplace–Stieltjes Tauberian dictionary: integrate segment-by-segment for exponential–polynomial asymptotics using integration by parts/residue control to obtain pole locations and orders; absolute convergence domain critical value given by ${\gamma }_0$. $\square$

T26 (Reversible vs Non-reversible: Criterion and Consequences)

Proposition (Criterion). Global map $F:{\Sigma }^{{\mathbb{Z}}^d}\to {\Sigma }^{{\mathbb{Z}}^d}$ is CA-reversible $⟺$ it is bijective and $F^{-1}$ is also CA (there exists inverse local rule of finite radius). On ${\mathbb{Z}}^d$, Garden-of-Eden theorem gives: $F$ surjective $⟺$ $F$ pre-injective; reversibility equivalent to simultaneously surjective and injective.

Proposition (Consequences). If $F$ is reversible, then:

1. Information density conserved: $I(F^{T}x)=I(x)$ (see T22);
2. Non-increase under observation factor: $I_{\pi }(F^{T}x)\le I(x)$;
3. No true attractors: no unidirectional attraction compressing open sets into proper subsets (each point has bidirectional orbit, may have periods but no information dissipation into irreversible collapse to single fixed point).

Proof. Criterion is standard result; consequences 1–2 immediately follow from T21–T22; consequence 3 given by reversibility and bidirectional reachability (if true attractor exists contradicts bijection). $\square$

7. Constructions and Algorithms

7.1 From Rule to SFT: From local consistency of $f$ derive forbidden pattern set $\mathcal{F}$, obtain $X_f$.

7.2 From SFT to Eternal Graph: Use allowed patterns as vertices, legal tilings as edges to construct $G_f$; use level sets of $\ell$ to give leaf ordering.

7.3 Decoder Design: Choose kernel window $B$, block code $\pi :{\Sigma }^B\to \Gamma$; define leaf-by-leaf reading protocol $\varsigma$ stratified by $\ell$.

7.4 Macro-block Forcing Program Box: Self-similar tiling embeds “state-control-tape” and can be decoded (see T6).

7.5 Compression-Entropy Experiment (Reproducible) $$y_k=\pi \left(x{|}_{W_k}\right),c_k=\mathrm{compress}(y_k),r_k=\frac{|c_k|}{|W_k|},\mathrm{plot}(r_k)(k=1,2,\dots ).$$ Note: Here normalizing by $|W_k|$ for experimental convenience; theoretically for time subaction entropy should use temporal thickness $L(W_k)=T_k$ normalization (see T5, requires time-slice cuboid family). To align with time entropy of T5, should keep $R$ fixed and simultaneously report $|c_k|/T_k$; if $|R_k|$ varies or using general Følner windows, then $r_k$ reflects $h_{\mu }^{(d+1)}$ scale rather than time entropy.

7.6 Constructing SBU from Event Node (Forcing Domain Propagation) Input: Realizable $v$, orientation $\tau$, tolerance $ϵ$. Steps: Using $v$ and local consistency as constraints, perform bidirectional constraint propagation/consistency checking, compute the set of cells forced by $v$ on growing $W_k$, and leaf-by-leaf extend according to $\ell$ until locally stable. Output: Forcing domain approximation of $\left(X_f^{(v,\tau )}\right)$ on $W_k$ and information density curve.

7.7 Anchored Graph Relative Coordinate Construction: BFS stratification (by $\ell$/spatial adjacency) → relative embedding ${\phi }_{v_0}$ → radius $r$ consistency verification and equivalence class merging.

7.8 From CSP / $\mu$-Safe Formula to CA: Given CSP or Horn/$\mu$-safe formula $\Phi$, for each clause of radius $\le r$ generate forbidden pattern ${\mathcal{F}}_{\Phi }$, define $f_{\Phi }$: $$X_{f_{\Phi }}={\mathcal{T}}_{f_{\Phi }}(\mathsf{Conf})=\{x:\forall (\mathbf{r},t),x(\mathbf{r},t)=f_{\Phi }(x(\mathbf{r}+N,t-1)\left)\right\},$$ if necessary use finite control layer to maintain synchronization (without changing equivalence class).

7.9 From $\omega$-Automaton to Leaf-Language: Given Büchi automaton $\mathcal{A}=(Q,\Gamma ,\delta ,q_0,F)$, choose $(\pi ,\varsigma )$ and construct extended SFT $X_{f,\mathcal{A}}$ such that:

1. $\pi$ encodes cross-leaf observations as $\Gamma$-words;
2. Implement $\delta$ via finite-type synchronization conditions: overlay finite-type “control/synchronization layer” on $X_f$ (or equivalently first take $X_f^{[k]}$ encode $Q$ into alphabet and simulate transition via local constraints), obtain extended SFT $X_{f,\mathcal{A}}$;
3. Acceptance condition expressed via safety/regularity constraints. Thus $${\mathsf{Lang}}_{\pi ,\varsigma }\left(X_{f,\mathcal{A}}\right)=L_{\omega }(\mathcal{A}).$$

8. Typical Examples and Toy Models

Rule-90 (Linear): Three perspectives consistent; SBU of any anchor uniquely recursed by linear relations; complexity density after Følner normalization consistent; leaf-language is $\omega$-regular.

Rule-110 (Universal): Macro-block forcing embeds TM (T6); halting witness corresponds to local termination marker (T9); leaf-by-leaf advancement excludes same-layer alternatives (T19–T20).

2-Coloring CSP (Model Perspective): Graph 2-coloring local constraints → forbidden patterns; anchor certain node color and leaf-by-leaf unfold, forming causal consistent event cone; leaf-language under appropriate conditions is $\omega$-regular.

2×2 Toy Block (Anchor–SBU–Decoding–Apparent Choice) $\Sigma =\{0,1\},d=1,N=\{-1,0,1\},f(a,b,c)=a\oplus b\oplus c$ (XOR, periodic boundary). Anchor $v_0$ fixes local pattern at $(t=0,\mathbf{r}=0)$. By T4’s causal thick boundary and leaf-by-leaf advancement recurse $t=1$ layer, obtain unique consistent extension; same-layer points contradicting anchor excluded (T19). Take $$B=\{(\mathbf{r},t):\mathbf{r}\in \{0,1\},t=1\},$$ $\pi$ reads out 2D block as visible binary string—“next step” only reads, does not increase information (A3).

9. Extension Directions

• Continuous Extension (cEBOC): Generalize via Markov symbolization/compact alphabet SFT; restate complexity/entropy and clarify discrete→continuous limit.
• Quantum Inspiration: Simultaneously describe multiple compatible SBUs of same static block $X_f$, measurement corresponds to anchor switching and locking + one $\pi$-semantic collapse; provides constructive foundation for information and computation based quantum interpretation (no state-vector assumption).
• Categorical/Coalgebra Perspective: $(X_f,\mathrm{shift})$ as coalgebra; anchored SBU as coalgebra subsolution injecting initial values; leaf-language as automaton coalgebra homomorphism image.
• Robustness: Fault-tolerant decoding and robust windows under small perturbations/missing data, ensuring observable semantic stability.

10. Observer, Apparent Choice, and Time Experience (RPG Metaphor)

Level Separation: Operational level (observation/decoding/leaf-by-leaf advancement/representative selection) and Ontological level (static geometry/unique consistent extension). Compatibility Principle: View $X_f$ as RPG’s complete data and rules; leaf-by-leaf advancement like unlocking story according to preset chapter rhythm $b$. Player “choice” is to select representative among same-layer compatible branches and exclude other branches; story ontology (static block) already written, choice does not generate new information (A3), compatible with determinism (T20). Subjective Time Rate: Effective step size $b=⟨{\tau }^{\star },\tau ⟩$ embodies “chapter rhythm”; entropy rate after Følner normalization consistent (T2/T5/T17).

11. Conclusion

EBOC under minimal axioms unifies timeless geometry (eternal graph/SFT), static block consistent body, and leaf-by-leaf decoding observation-computation semantics, forming complete chain from model/automaton to visible language. This paper provides detailed proofs of T1–T26, establishing information non-increase law (T4/A3), Brudno alignment (T5), unimodular covariance (T2/T10), event cone/static block unfolding (T14–T16), multi-anchor observers and coordinate relativization (T17–T18), and other core results, with reproducible experimental and construction protocols (§7).

Appendix A: Terminology and Notation

• Semantic Collapse: Information factorization of $x\mapsto {\mathcal{O}}_{\pi ,\varsigma }(x)$.
• Apparent Choice: Advancement by minimal positive increment of $\ell$, representative selection among same-layer alternatives; only changes semantic representative, does not create information.
• Time-Slice Cuboid Family: Windows of form $W_k=R_k\times [t_k,t_k+T_k-1]$, where $R_k\subset {\mathbb{Z}}^d$ is spatial section, $T_k\to \infty$ is temporal thickness; compatible with one-dimensional Følner theory of time subaction ${\sigma }_{\mathrm{time}}$.
• Leaf Count (Temporal Thickness): For time-slice cuboid $W=R\times [t_0,t_0+T-1]$, define $L(W)=T$ as number of leaf layers traversed, corresponding to observation step count.
• Primitive Integral Covector: ${\tau }^{\star }\in ({\mathbb{Z}}^{d+1}{)}^{\vee }$, $\gcd ({\tau }_0^{\star },\dots ,{\tau }_d^{\star })=1$; its pairing with actual time direction $\tau$ as $⟨{\tau }^{\star },\tau ⟩=b\ge 1$ defines leaf-by-leaf advancement step size.
• ${\mathrm{GL}}_{d+1}(\mathbb{Z})$: Integer invertible matrix group (determinant $\pm 1$).
• Følner Family: Window family $|\partial W_k|/|W_k|\to 0$.
• Cylinder Set: $[p{]}_W=\{x\in X_f:x{|}_W=p\}$.
• Entropy Normalization Correspondence: $h_{\mu }({\sigma }_{\mathrm{time}})$ compatible with temporal thickness $L(W)=T$ normalization (time-slice cuboid family); $h_{\mu }^{(d+1)}$ compatible with voxel count $|W|$ normalization (general Følner family).

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