Chapter 3: Object Model

3.3 Only Maximal Parallelism Preserves Vacuum Symmetry

The perfect symmetry of the Bethe vacuum imposes an unavoidable constraint on the mechanism of time evolution and forces us to design a scheduler that advances the state of the universe without breaking the indistinguishability of its components. We face the operational challenge of processing the graph without introducing artificial distinctions between topologically identical locations which would effectively determine the outcome of physics through the arbitrary choices of the update order. The clock of the universe must function as a global operator that respects the inherent equality of the vacuum and ensures that the relativity of the system is preserved.

Any update strategy that relies on sequential execution or arbitrary subset selection introduces a persistent historical scar into the vacuum and distinguishes otherwise identical sites based solely on the moment they were processed. This introduction of extrinsic information destroys the covariance of the theory as the physical state becomes dependent on the hidden variable of the scheduler's queue rather than the intrinsic geometry of the causal graph. A universe driven by a serial processor exhibits preferred frames of reference defined by the processing sequence and creates a reality where the laws of physics are not invariant under translation or rotation.

We establish maximal parallelism as the protocol for time evolution by mandating that the rewrite rule acts simultaneously on every compliant site in the universe during a single logical tick of the clock. This global synchronization ensures that the automorphism group of the vacuum is strictly preserved through the update and guarantees that the symmetry of space is not violated by the passage of time. By processing all potential events in a single unified wave, we ensure that the universe evolves as a coherent whole and prevents the scheduler from imprinting its own arbitrary patterns onto the vacuum.

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3.3.1 Definition: The Annotated State Space

Formal Specification of Graph States and Rewrite Sites as Annotated Structures

The physical state of the universe at Logical Time $t$ (§1.2.1) is defined as the Annotated Directed Graph $G_t = (V, E, \mathcal{A})$.

1. Annotation Structure: The annotation $\mathcal{A}$ is defined as the ordered pair of functions $(a_V, a_E)$, where $a_V: V \to \mathcal{X}_V$ maps vertices to a finite set of vertex labels, and $a_E: E \to \mathcal{X}_E$ maps edges to a finite set of edge labels. The codomains $\mathcal{X}_V$ and $\mathcal{X}_E$ include the History Mapping (§1.3.1) and local Syndrome values (§3.5.5).
2. Annotated Automorphism: An automorphism $\varphi$ of $G_t$ is defined as a bijection $\varphi: V \to V$ satisfying the conjunction of the following conditions:
   • Structural Isomorphism: $\forall u, v \in V, (u, v) \in E \iff (\varphi(u), \varphi(v)) \in E$.
   • Vertex Annotation Invariance: $\forall u \in V, a_V(u) = a_V(\varphi(u))$.
   • Edge Annotation Invariance: $\forall (u, v) \in E, a_E((u, v)) = a_E((\varphi(u), \varphi(v)))$.
3. Candidate Rewrite Site: A candidate rewrite site $s$ is defined as the ordered tuple $s = (F_s, p_s)$, where $F_s \subseteq G_t$ constitutes the finite footprint subgraph required by the rewrite rule, and $p_s$ constitutes the deterministic local transformation rule defined on the domain of $F_s$.

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3.3.2 Definition: The Formal Symmetry Framework

Axiomatic Constraints on the Update Mechanism regarding Equivariance and Determinism

A graph rewrite system satisfies the Symmetry Preservation Constraints if and only if the Update Map $\mathcal{U}$ and the Site Identification Function $\mathcal{S}$ satisfy the following four axiomatic conditions with respect to the automorphism group $\text{Aut}(G)$:

1. Assumption A1 (Locality and Equivariance): For every automorphism $\varphi \in \text{Aut}(G)$, the induced action on the set of candidate sites $\mathcal{S}(G)$ is a bijection that preserves the isomorphism class of the site footprints and their associated local proposals.
2. Assumption A2 (Universality of Eligibility): The eligibility function determining membership in $\mathcal{S}(G)$ depends exclusively on local structural invariants preserved under the action of $\text{Aut}(G)$.
3. Assumption A3 (Deterministic Acceptance): The acceptance function $\mathcal{A}$ governing the update is strictly deterministic, conditioned solely on the state $G$ and the specific set of selected sites.
4. Assumption A4 (Joint-Update Equivariance): The simultaneous application of a selected set of site updates commutes with the action of the automorphism group, such that $\varphi(\text{Update}(S, G)) = \text{Update}(\varphi(S), \varphi(G))$.

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3.3.3 Theorem: Preservation of Automorphisms

Necessity and Sufficiency of Maximal Parallelism for Symmetry Maintenance established by Biconditional Proof

It is asserted that an update map $\mathcal{U}: G_0 \to G_1$ preserves the full automorphism group of the vacuum state, such that $\text{Aut}(G_1) \supseteq \text{Aut}(G_0)$, if and only if $\mathcal{U}$ constitutes a Maximally Parallel Scheduler. A Maximally Parallel Scheduler is defined as the operator that applies the rewrite rule simultaneously to the complete set of compliant sites $\mathcal{S}_{sites}(G_0)$ permitted by the axiomatic constraints. (Wolfram, 2002)

3.3.3.1 Argument Outline: Biconditional Symmetry

Structure of the Preservation Proof via Sufficiency and Necessity Arguments

The proof establishes that maximal parallelism is the only update strategy compatible with the principle of background independence.

1. The Sufficiency (Lemma 3.3.4): We first demonstrate that if the scheduler is maximally parallel, symmetry is necessarily preserved. This relies on the Equivariance of the site definition: if the inputs are symmetric, the simultaneous output must be symmetric.
2. The Resolution (Lemma 3.3.5): We verify that this preservation holds even with overlapping sites, provided the Conflict Resolution logic itself satisfies the same equivariance constraints.
3. The Necessity (Proof 3.3.7): We demonstrate that if the scheduler is not maximally parallel, symmetry is necessarily broken. Any selection of a proper subset of sites introduces a distinguishing property ("selected") that partitions the vertex orbits, collapsing the automorphism group.

3.3.3.2 Diagram: Scheduler Symmetry Outcomes

Visual Comparison of Symmetry Outcomes under Sequential vs Parallel Schedulers

SCHEDULER SYMMETRY OUTCOMES
---------------------------
      [ SEQUENTIAL ] ---------> Breaks Symmetry (|Aut| ~ 1)
            |                   (Introduces arbitrary order)
            v
      [ SUBSET / RANDOM ] ----> Breaks Symmetry (|Aut| ~ 2)
            |                   (Distinguishes chosen vs unchosen)
            v
      [ MAXIMAL PARALLEL ] ---> PRESERVES Symmetry (|Aut| = Max)
                                (Treats identical sites identically)

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3.3.4 Lemma: Equivariance of Site Definition

Commutativity of Rewrite Site Identification with Graph Automorphisms

Let $\mathcal{S}_{sites}(G)$ denote the set of candidate rewrite sites for a graph $G$. Then the identity $\varphi(\mathcal{S}_{sites}(G)) = \mathcal{S}_{sites}(\varphi(G)) = \mathcal{S}_{sites}(G)$ holds for any automorphism $\varphi \in \text{Aut}(G)$.

3.3.4.1 Proof: Equivariance of Site Definition

Verification of Invariance via Isomorphic Mapping

I. Site Definition

Let the set of candidate rewrite sites $\mathcal{S}_{\text{sites}}(G)$ be defined by a predicate function $P(s, G)$ that evaluates the local structural eligibility of a subgraph $s$:

$$s \in \mathcal{S}_{\text{sites}}(G) \iff P(s, G) \text{ is True}$$

The predicate $P$ depends exclusively on:

1. Topological Isomorphism: The subgraph $F_s$ matches the required template.
2. Causal Constraints: The site satisfies the Principle of Unique Causality (§2.3.3).
3. Timestamp Ordering: The site satisfies the strict monotonicity requirements (§2.6.3).

II. Automorphic Mapping

Let $\varphi \in \text{Aut}(G)$ be an arbitrary automorphism of the graph. The mapping $\varphi$ acts on a site $s = (F_s, \tau_s)$ by mapping vertices, edges, and timestamps:

$$\varphi(s) = (\varphi(F_s), \varphi(\tau_s))$$

III. Preservation of Structural Properties

Since $\varphi$ constitutes an isomorphism, it preserves all relational properties defined on the graph:

1. Topology: $F_s \cong \varphi(F_s)$.
2. Causality: If $s$ satisfies Unique Causality in $G$, then $\varphi(s)$ satisfies Unique Causality in $\varphi(G) = G$.
3. Order: If $\tau_u < \tau_v$, then the preservation of structure implies that the mapped timestamps satisfy the corresponding order in the mapped site.

IV. Predicate Invariance

The evaluation of the eligibility predicate is invariant under the automorphism:

$$P(s, G) \iff P(\varphi(s), \varphi(G))$$

Since $\varphi(G) = G$ for an automorphism, this yields:

$$P(s, G) \iff P(\varphi(s), G)$$

It follows that if $s \in \mathcal{S}_{\text{sites}}(G)$, then $\varphi(s) \in \mathcal{S}_{\text{sites}}(G)$.

V. Bijective Conclusion

The map $\varphi$ restricts to a bijection on the set of sites:

$$\varphi(\mathcal{S}_{\text{sites}}(G)) = \mathcal{S}_{\text{sites}}(G)$$

Furthermore, the local update operation $\mathcal{U}_{loc}$ commutes with the automorphism:

$$\mathcal{U}_{loc}(\varphi(s)) = \varphi(\mathcal{U}_{loc}(s))$$

This establishes complete equivariance.

Q.E.D.

3.3.4.2 Commentary: Physical Justification

Derivation of Formal Assumptions from Principles of Background Independence

The four formal assumptions $(A1)$ through $(A4)$ do not constitute arbitrary mathematical conveniences; they are the encoding of the fundamental physical principles required to establish background independence; relational uniformity; and the absence of privileged reference frames within the quantum vacuum.

Assumption $(A1)$ (Locality and Equivariance) embodies the principle that physical laws remain local and identical everywhere in the universe. It asserts that no hidden global coordinates; external clocks; or absolute labels may influence where or how the rewrite rule applies. The dynamics must depend exclusively on the intrinsic relational structure that automorphisms preserve; ensuring that if two regions of the graph are topologically identical; the laws of physics act upon them identically.

Assumption $(A2)$ (Universality of Eligibility) enforces the Generalized Copernican Principle: the criteria for "where geometry can emerge" must remain the same at every structurally identical location. Any deviation would introduce preferred directions or privileged positions in the vacuum; violating the cosmological principle of homogeneity at the foundational level. The vacuum must be a perfect isotrope; offering equal potential for creation at every valid site.

Assumption $(A3)$ (Deterministic Acceptance) implements strict determinism at the level of the selection mechanism itself. While the outcome of the universe may be probabilistic due to thermodynamic weighting; the procedure for accepting a valid candidate must be purely a function of the state. No additional randomness or hidden variables may influence acceptance beyond the explicit state configuration and the thermodynamic selection criteria.

Assumption $(A4)$ (Joint-Update Equivariance) guarantees that the physical outcome of simultaneous local modifications remains consistent under symmetry transformations. This requirement is critical to avoid the "updating artifacts" identified by (Wolfram, 2002) in his analysis of cellular automata and network systems. Wolfram demonstrated that sequential or partial updates inevitably introduce arbitrary, history-dependent asymmetries (breaking the graph's automorphism group), whereas maximally parallel updates preserve the underlying rule invariance. By enforcing joint-update equivariance, we ensure the scheduler does not imprint a spurious "preferred frame" onto the vacuum, maintaining the discrete precursor to General Covariance.

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3.3.5 Lemma: Conflict Resolution

Preservation of Automorphism Group in Overlapping Site Resolution

For any overlapping rewrite sites, the resolution mechanism preserves the automorphism group $\text{Aut}(G)$ if and only if the logic satisfies the Symmetry Preservation Constraints (§3.3.2). In particular, for any automorphism $\varphi$ mapping site $s_1$ to site $s_2$, the resolution outcome for $s_1$ maps to the resolution outcome for $s_2$ under $\varphi$.

3.3.5.1 Proof: Conflict Resolution

Demonstration of Equivalence between Symmetry Preservation and Maximal Parallelism

I. Sufficiency ($\implies$)

Let $\mathcal{U}_{max}$ denote the maximally parallel update map acting on $G_0$, and let $\phi \in \text{Aut}(G_0)$. Equivariance of Site Definition (§3.3.4) implies $\phi(\mathcal{S}_{sites}) = \mathcal{S}_{sites}$. The map $\mathcal{U}_{max}$ applies the rewrite rule $\mathcal{R}$ to every element in $\mathcal{S}_{sites}$:

$$E_{new} = \bigcup_{s \in \mathcal{S}_{sites}} \mathcal{R}(s)$$

The automorphism $\phi$ acts on the new edge set:

$$\phi(E_{new}) = \bigcup_{s \in \mathcal{S}_{sites}} \phi(\mathcal{R}(s))$$

The equivariance of the rule $\mathcal{R}$ (Assumption A1) implies:

$$\phi(E_{new}) = \bigcup_{s \in \mathcal{S}_{sites}} \mathcal{R}(\phi(s))$$

Since $\phi$ permutes $\mathcal{S}_{sites}$, the union over $\phi(s)$ is identical to the union over $s$:

$$\phi(E_{new}) = E_{new}$$

The map $\phi$ preserves $E_0$ and $E_{new}$. It follows that $\phi \in \text{Aut}(G_1)$.

II. Necessity ($\Longleftarrow$)

Let $\mathcal{U}_{partial}$ denote an update map that selects a proper subset $S' \subset \mathcal{S}_{sites}$:

$$S' \neq \emptyset \land S' \neq \mathcal{S}_{sites}$$

Consider $s_a \in S'$ and $s_b \in \mathcal{S}_{sites} \setminus S'$. The vacuum state $G_0$ is a vertex-transitive and site-transitive graph (§3.2.1). There exists $\sigma \in \text{Aut}(G_0)$ such that $\sigma(s_a) = s_b$.

In the successor state $G_1$, the neighborhood of $s_a$ contains new structure $\mathcal{R}(s_a)$, while the neighborhood of $s_b$ remains unmodified. An extension of $\sigma$ to $G_1$ implies mapping the modified neighborhood of $s_a$ to the unmodified neighborhood of $s_b$:

$$\sigma(\mathcal{R}(s_a)) \neq \emptyset \quad \text{but} \quad \mathcal{R}(s_b) = \emptyset$$

This contradiction establishes that $\sigma \notin \text{Aut}(G_1)$ and symmetry is broken.

III. Conclusion

Only the map where $S' = \mathcal{S}_{sites}$ avoids this contradiction. We conclude that symmetry preservation necessitates maximal parallelism.

Q.E.D.

3.3.5.2 Calculation: Cycle Resolution

Algorithmic Resolution of Symmetric Overlaps in a 6-Cycle Graph using Parallel Operations

Initial state with timestamps: A → B (H=1), B → C (H=2), C → D (H=3), D → E (H=4), E → F (H=5), F → A (H=6). Initial syndromes: For triplet A-B-C, $\sigma_{\text{geom}} = +1$ (vacuum), similar for all triplets.

Step 1: Addition of Chords Add C → A (H=7), D → B (H=8), E → C (H=9), F → D (H=10), A → E (H=11), B → F (H=12). Post-addition syndromes: For A-B-C-A, $\sigma_{\text{geom}} = -1$ (excitation), similar for all new 3-cycles. with all chords: C→A, D→B, E→C, F→D, A→E, B→F

ASCII Before/After Addition

    C→A E→C A→E
     ↑ ↑ ↑
A → B → C → D → E → F → A
     ↑ ↑ ↑
    D→B F→D B→F

Step 2: Parallel Deletion on Overlaps Delete B → C, D → E, F → A (flagged -1 overlaps). These shared edges undergo removal, which breaks the original 6-cycle while resolving the overlaps. Each 3-cycle retains two original edges and one chord, and the residual edges preserve geometric identity with resolved flux.

ASCII Post-Deletion

    C→A E→C A→E
     | | |
A → B C → D E → F A
     | | |
    D→B F→D B→F

(deleted: B→C, D→E, F→A; original cycle broken, 3-cycles remain via chords and residual edges)

This expanded 6-cycle example demonstrates overlap resolution in a smaller symmetric graph and now progresses to the 8-cycle example, which introduces greater complexity through a larger dihedral group and more overlapping sites.

For an 8-cycle with vertices A-H, the dihedral $D_8$ group governs symmetries (rotations/reflections). This graph contains 8 overlapping 2-paths: $s_1$: A → B → C, s_2: B → C → D, ..., $s_8$: H → A → B.

1. Add all 8 chords (C→A, D→B, E→C, F→D, G→E, H→F, A→G, B→H), which forms 8 3-cycles (A-B-C-A, B-C-D-B, etc.), with shared edges like B-C flagged -1.
2. Parallel deletion on -1 overlaps (e.g., B→C, D→E, F→G, H→A).

It is confirmed that $D_8$ receives preservation: Rotations/reflections map remaining structures equivalently.

3.3.5.3 Calculation: Symmetry Metrics Pre/Post-Update

Computational Verification of Automorphism Preservation

Analysis of the scheduler's impact on vacuum symmetry established in the Overlap Determinism Proof (§3.3.5.1) is based on the following protocols:

1. State Initialization: A balanced $N=7$ Bethe fragment is constructed. The graph topology possesses an initial $S_3$ symmetry group due to the structural indistinguishability of its three primary branches.
2. Sequential Perturbation: The algorithm simulates a sequential scheduler by selecting a single compliant chord $(1,2)$ arbitrarily from the set of valid moves and modifying the graph topology.
3. Parallel Perturbation: The algorithm simulates a maximally parallel scheduler by simultaneously instantiating all compliant chords $\{(1,2), (2,3), (1,3)\}$.
4. Group Analysis: The size of the automorphism group is re-evaluated post-update for both scenarios to determine if the scheduler operation broke the initial symmetry state.

import networkx as nx
import math

def get_automorphism_count(G):
    """Calculates the size of the automorphism group."""
    matcher = nx.isomorphism.GraphMatcher(G, G)
    try:
        return len(list(matcher.isomorphisms_iter()))
    except:
        return 0

# 1. Setup: Balanced Bethe Fragment (N=7)
# Structure: Root(0) -> Level1{1,2,3} -> Level2{4,5,6}
# Symmetries: Permutation of branches {1,4}, {2,5}, {3,6} => S3 Group
G0 = nx.Graph()
G0.add_edges_from([(0,1), (0,2), (0,3), (1,4), (2,5), (3,6)])

print(f"{'State':<20} | {'|Aut|':<10} | {'Symmetry Status'}")
print("-" * 65)

aut_0 = get_automorphism_count(G0)
print(f"{'Initial Vacuum':<20} | {aut_0:<10} | Perfect Symmetry (S3)")

# 2. Define Compliant Sites (Chords between Level 1 siblings)
# Potential edges: (1,2), (2,3), (1,3)
sites = [(1,2), (2,3), (1,3)]

# 3. Scenario A: Sequential Update (Random Choice)
# Scheduler picks site (1,2) arbitrarily.
G_seq = G0.copy()
G_seq.add_edge(*sites[0])
aut_seq = get_automorphism_count(G_seq)
status_seq = "BROKEN" if aut_seq < aut_0 else "PRESERVED"
print(f"{'Sequential Update':<20} | {aut_seq:<10} | {status_seq} (Distinguishes Branch 3)")

# 4. Scenario B: Parallel Update (All Sites)
# Scheduler executes all compliant updates simultaneously.
G_par = G0.copy()
G_par.add_edges_from(sites)
aut_par = get_automorphism_count(G_par)
status_par = "BROKEN" if aut_par < aut_0 else "PRESERVED"
print(f"{'Parallel Update':<20} | {aut_par:<10} | {status_par} (Equivariant)")

Simulation Output:

State                | |Aut|      | Symmetry Status
-----------------------------------------------------------------
Initial Vacuum       | 6          | Perfect Symmetry (S3)
Sequential Update    | 2          | BROKEN (Distinguishes Branch 3)
Parallel Update      | 6          | PRESERVED (Equivariant)

The computational verification provides empirical evidence for the necessity of Maximal Parallelism:

1. Initial State ($G_0$): The vacuum fragment exhibits $S_3$ symmetry ($|Aut|=6$), reflecting the indistinguishability of the three branches.
2. Sequential Update ($G_{seq}$): The application of a sequential scheduler, picking exactly one of three equivalent sites, fractures the symmetry group down to $|Aut|=2$. The "choice" of the scheduler injects information into the system, creating a preferred direction (the updated branch vs. the non-updated branches).
3. Parallel Update ($G_{par}$): The simultaneous application of all valid updates preserves the full $S_3$ symmetry ($|Aut|=6$). The transformation is equivariant; it commutes with the automorphism group of the state.

This confirms that any update rule other than Maximal Parallelism introduces a "scheduler artifact," breaking the isotropy of the vacuum and violating the principle of background independence.

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3.3.6 Theorem: Scalability of the Scheduler

Logarithmic Time Complexity via Quasi-Local Checks

Assume the graph remains in the sparse regime (§3.1.2) subject to quasi-local constraints (§2.3.3) with a bounded check radius $R \propto \log N$. Then the time complexity of the maximally parallel update operation is bounded by $O(\log N)$. Moreover, the probability of conflict chains spanning the system decays exponentially.

3.3.6.1 Proof: Scalability of the Scheduler

Derivation of Time Complexity via Radius Bounding

I. The Interaction Radius

Let $R$ denote the graph distance required to verify all local constraints for a given site $s$. In the sparse vacuum graph $G_0$, the edge density is minimal.

1. Footprint: The rewrite site possesses radius $r \approx 1$.
2. Constraint Check: Verification requires traversing paths of length up to a constant $k$ (cycle detection limit).
3. Interaction Zone: The radius $R$ is bounded by a small constant in the vacuum topology.

II. Propagation Complexity

The time $T_{step}$ required to resolve overlaps and verify consistency scales with the diameter of the interference patch:

$$T_{step} \propto R$$

While $R$ scales with $N$ in a generic graph, the Axiom of Geometric Constructibility (§2.3.1) enforces a tree-like regular structure (Bethe lattice) for $G_0$.

III. Error Suppression Limit

Consistency requires that the probability of an undetected long-range conflict vanishes. Let $P_{err}(R)$ denote the probability of a conflict chain extending beyond radius $R$. In a sub-critical sparse graph, this probability decays exponentially:

$$P_{err}(R) \propto e^{-\lambda R}$$

Global consistency with high probability ($1 - \epsilon$) as $N \to \infty$ requires:

$$N \cdot P_{err}(R) < \epsilon$$ $$N \cdot e^{-\lambda R} < \epsilon \implies R > \frac{1}{\lambda} \ln \left( \frac{N}{\epsilon} \right)$$

IV. Complexity Bound

Substitution of the bound for $R$ into the time complexity yields:

$$T_{step} \sim O(R) \sim O(\log N)$$

This logarithmic scaling establishes computational feasibility for cosmological $N$.

Q.E.D.

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3.3.7 Proof: Demonstration of Mandatory Parallelism

Formal Proof of the Inevitability of Maximal Parallelism for Symmetry Preservation through Contradiction

I. The Indistinguishability Premise

The vacuum state $G_0$ is defined by maximal symmetry (§3.2.1). For any two compliant sites $s_i, s_j \in \mathcal{S}_{sites}(G_0)$, there exists an automorphism $\sigma$ such that $\sigma(s_i) = s_j$. This renders $s_i$ and $s_j$ informationally indistinguishable within the state $G_0$.

II. The Selection Function

Let $\mathcal{U}$ be an update function defined by a selection vector $\mathbf{v} \in \{0, 1\}^{|\mathcal{S}|}$, where $v_k=1$ implies site $s_k$ updates. If $\mathcal{U}$ is not maximally parallel, $\exists i, j$ such that $v_i = 1$ and $v_j = 0$.

III. Information Generation

The application of $\mathcal{U}$ generates a bit of distinguishing information $I_{diff} = 1$ bit (distinguishing $s_i$ from $s_j$). The source of this information cannot be $G_0$ (where $I(s_i, s_j) = 0$). Therefore, the information must be extrinsic (arbitrary or random).

IV. Covariance Violation

The physical laws must be covariant; the update rule must depend only on intrinsic state information.

$$\text{Output}(G) = F(\text{State}(G))$$

An update depending on extrinsic selection violates covariance. To eliminate extrinsic variables, the selection must be uniform.

1. Null Selection: $v_k = 0 \quad \forall k$ (Trivial identity map).
2. Full Selection: $v_k = 1 \quad \forall k$ (Maximal parallelism).

V. Conclusion

Since evolution requires non-trivial change, the Null Selection is rejected. The Full Selection (Maximal Parallelism) is the unique non-trivial update mode preserving the information-theoretic symmetries of the vacuum.

Q.E.D.

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3.3.Z Implications and Synthesis

Only Maximal Parallelism Preserves Vacuum Symmetry

The requirement to preserve the automorphism group of the vacuum during time evolution mandates that the scheduler must be maximally parallel, executing all possible rewrites simultaneously. Any sequential or partial update strategy introduces arbitrary distinctions between identical sites, effectively "measuring" the vacuum and collapsing its symmetry into a particular historical trajectory. Maximal parallelism acts as the guardian of covariance, ensuring that the passage of time respects the indistinguishability of spatial locations.

This establishes the universe as a massively parallel computer rather than a serial Turing machine. The "clock" of the cosmos ticks everywhere at once, advancing the global state in a unified wavefront of computation. This mechanism prevents the scheduler from imprinting a preferred frame or sequence onto physical reality, maintaining the discrete precursor to general covariance where no observer's clock is privileged over another's.

The imposition of maximal parallelism resolves the conflict between discrete time and relativistic covariance at the fundamental level. By forcing the universe to update as a synchronous wavefront, we prevent the arbitrary serialization of events that would otherwise imprint a preferred reference frame onto the vacuum. This ensures that the causal structure remains invariant under observation, defining time not as a local variable but as a global computational heartbeat that drives the collective evolution of the graph without privileging any specific observer or location.

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