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Chapter 5: Geometrogensis (Equilibrium)

We turn our attention from the mechanism of the individual tick to the aggregate behavior of the system over deep time. The engine we constructed in the previous chapter ticks reliably, adding and subtracting relations based on local cues, yet we must ask what global state emerges when these microscopic fluctuations balance out. We confront the core question of statistical mechanics applied to causality: in a system where every change is constrained by the strict axioms of acyclicity and unique paths, does the sheer multiplicity of compliant graphs impose a thermodynamic order on the evolution? We are looking for the graph-theoretic equivalent of an equilibrium state, where the "atoms" are causal links and the "pressure" is the tendency of the network to maximize its combinatorial freedom.

To quantify this probabilistic drive, we must define the entropy of the causal graph as the logarithm of the count of valid configurations. A critical requirement for a physical vacuum is that this entropy must be extensive; it must scale linearly with the system size $N$, allowing us to treat distinct regions of the universe as thermodynamically independent. We establish this property by demonstrating that correlations between distant parts of the graph decay exponentially, effectively partitioning the universe into weakly coupled volumes. With this measure of capacity in hand, we derive the master equation that governs the time evolution of cycle densities. This differential equation tracks the net flux of geometry, balancing the creation terms driven by the exploration of new paths against the deletion terms driven by the relaxation of tensions.

Our inquiry culminates in the mapping of the system's phase space and the identification of stable equilibria. By sweeping through the parameters of friction and catalysis, we identify a bounded region of physical viability where the graph maintains a steady, sparse density without collapsing into a trivial tree or diverging into a dense complete graph. Within this regime, we solve for the unique fixed point of the density, a stable attractor that anchors the vacuum state. Finally, we bridge the gap between discrete graph theory and continuous geometry. We postulate that this stable, entropic equilibrium satisfies the Reifenberg conditions for manifold convergence, ensuring that the randomness of the connections averages out to produce a structure that is locally flat and topologically smooth.

Preconditions and Goals

• Prove extensive entropy scales linearly with vertices via subregions and correlation decay.
• Derive master equation for cycle density from fluxes with frictional suppression.
• Map physical viability region through parameter sweeps of friction and catalysis coefficients.
• Solve transcendental equation for unique stable equilibrium density with friction bounds.
• Chain geometric preconditions for manifold convergence.

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5.1 Thermodynamic Framework

We confront the foundational necessity of quantifying the configurational capacity of a vacuum that lacks a pre-existing metric to measure its own volume. This requirement forces us to define an extensive entropy for the causal graph before the dynamical engine can be trusted to drive evolution, effectively establishing a statistical framework that counts the allowable configurations of the universe without relying on standard volume definitions which do not apply in a discrete pre-geometric context. The inquiry demands a scaling law that relates the total entropy to the number of vertices to effectively distinguish between a finite physical reservoir and an unbounded mathematical abstraction.

Relying on classical phase space analogies or continuum assumptions introduces ambiguities that render the resulting thermodynamics inconsistent with the discrete nature of the substrate. A model without a defined extensive entropy risks describing a universe where the chemical potential for new relations diverges as the system grows, leading inevitably to an ultraviolet catastrophe where infinite complexity accumulates in finite regions without thermodynamic penalty. Furthermore, a system that cannot demonstrate the decoupling of distant regions implies a fundamental failure of locality where the choices made in one corner of the universe infinitely constrain the possibilities elsewhere, effectively destroying the concept of independent subsystems essential for statistical mechanics and rendering the definition of local temperature impossible.

We resolve this foundational crisis by establishing the spatial cluster decomposition principle and proving the correlation decay lemma. By partitioning the graph into weakly coupled sub-volumes defined by the correlation length $\xi$, we show that the entropy scales linearly with the number of vertices $N$, confirming that the vacuum acts as a stable thermodynamic reservoir capable of supporting regulated heat exchange.

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5.1.1 Definition: Spatial Cluster Decomposition

Exponential Decay of Mutual Information within Disjoint Subregions

The Spatial Cluster Decomposition principle asserts that the statistical properties of the causal graph factorize over sufficient distances. Let $R_A$ and $R_B$ be disjoint subregions of the graph $G$, and let $d(R_A, R_B)$ denote the geodesic graph distance between them. The subregions satisfy Quasi-Independence if the Mutual Information $I(R_A; R_B)$ between their configuration states is bounded by the exponential decay envelope:

$$I(R_A; R_B) \leq K \cdot \exp\left(-\frac{d(R_A, R_B)}{\xi}\right)$$

where $\xi$ is the finite Correlation Length (§5.1.3) and $K$ is a normalization constant. In the asymptotic limit $d(R_A, R_B) \gg \xi$, the joint configuration space factorizes as $\Omega(R_A \cup R_B) \approx \Omega(R_A) \cdot \Omega(R_B)$.

5.1.2 Theorem: Extensive Entropy

Let $\Omega_N$ denote the cardinality of the set of all axiomatically compliant causal graphs on $N$ vertices. It is asserted that the system exhibits Extensive Entropy, defined by the asymptotic scaling law of the total entropy $S(N) \equiv \ln \Omega_N$:

$$S(N) = c \cdot N + o(N)$$

The coefficient $c > 0$ is the Specific Entropy per Event, a universal constant determined by the local constraint density (bounded degree and acyclicity). The term $o(N)$ represents sub-extensive corrections that vanish in the thermodynamic limit $\lim_{N \to \infty} S(N)/N = c$. This linearity confirms that the vacuum is a thermodynamically stable phase of matter.

5.1.1.1 Commentary: Defining "Volume" via Correlation

Emergence of Additivity from Causal Limits

This definition formalizes the concept of "separation" within a pre-geometric substrate that lacks an intrinsic metric background. In the absence of a pre-existing coordinate system; distance must be defined dynamically via the propagation of constraints and information. This definition asserts that the influence of a constraint at vertex $u$ decays exponentially with the graph distance from $u$; creating an effective horizon of causality. This mirrors the behavior of correlation functions in statistical field theories, where the correlation length $\xi$ defines the scale of interaction. Specifically, (Ambjørn, Jurkiewicz, & Loll, 2005) in Causal Dynamical Triangulations demonstrate that even in discrete, random geometries, a macroscopic dimension and volume emerge from the scaling of spectral dimension and correlation functions, justifying our treatment of the causal graph as a collection of statistically independent sub-volumes.

The correlation length $\xi$ constitutes an endogenous scale that emerges directly from the local branching ratios and density parameters of the graph. It defines the effective size of a "causal patch" or "volume element." Inside a radius of $\xi$; the graph exhibits high entanglement and strong correlation; behavior is collective and non-local. However; at distances greater than $\xi$; regions behave as statistically isolated reservoirs. This property allows us to discretize the graph into $M \approx N / V_\xi$ independent correlation volumes. This partitioning is the mathematical justification for summing local entropies to yield a global extensive entropy. It bridges the gap between the discrete relational nature of the graph and the continuum-like behavior required for the Master Equation; ensuring that entropic contributions from distant parts of the universe do not entangle in a way that violates the additivity required for thermodynamic stability.

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5.1.2 Theorem: Extensive Entropy

Linear Scaling of the Configuration Space with Vertex Count

Let $\Omega_N$ denote the cardinality of the set of all axiomatically compliant causal graphs on $N$ vertices. It is asserted that the system exhibits Extensive Entropy, defined by the asymptotic scaling law of the total entropy $S(N) \equiv \ln \Omega_N$:

$$S(N) = c \cdot N + o(N)$$

The coefficient $c > 0$ is the Specific Entropy per Event, a universal constant determined by the local constraint density (bounded degree and acyclicity). The term $o(N)$ represents sub-extensive corrections that vanish in the thermodynamic limit $\lim_{N \to \infty} S(N)/N = c$. This linearity confirms that the vacuum is a thermodynamically stable phase of matter.

5.1.2.1 Commentary: Logic of Extensivity

Transition from Combinatorial Counting to Physical Reservoirs

The argument establishes the thermodynamic stability of the vacuum by decomposing the global configuration space into additive local contributions. This follows the foundational principles of statistical mechanics where extensivity is a prerequisite for a well-defined thermodynamic limit. (Bekenstein, 1981) established that the entropy of any bounded system is fundamentally limited by its energy and size (the Bekenstein bound), implying that information capacity scales with the physical dimensions of the system. In our graph-theoretic context, the linear scaling of entropy $S \propto N$ validates that the causal graph behaves as a standard extensive system, akin to a gas or a lattice spin system, rather than a holographic surface or a system with long-range interactions that would lead to super-extensive scaling.

1.  The Finite Basis (Local Boundedness): The argument first addresses the definition of entropy configuration counting ($S = \ln \Omega$). It invokes Axiom 1 (Bounded Degree) and Axiom 3 (Acyclicity) to prove that the number of possible directed graphs on any finite set of vertices is strictly bounded. This guarantees that no local singularity can drive the entropy to infinity. 2.  The Decoupling (Cluster Decomposition): The argument applies the Spatial Cluster Decomposition principle. It invokes the Correlation Decay Lemma to partition the graph into $M \approx N / V_\xi$ quasi-independent subregions, where $V_\xi$ is the correlation volume. Explicit bounds on Mutual Information demonstrate that boundary corrections scale sub-extensively ($O(\sqrt{N})$), becoming negligible in the limit. 3.  The Scaling (Synthesis): Finally, the proof sums the entropies of these independent regions. Since each region contributes a finite, constant amount of entropy determined by local constraints, the total entropy scales linearly: $S(N) = c \cdot N$. This confirms the existence of a well-defined Specific Entropy per Event ($c > 0$), validating the vacuum as a stable thermodynamic phase.

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5.1.3 Lemma: Correlation Decay

Exponential Suppression of Long-Range Dependencies under Bounded Branching

Given a causal graph $G$ satisfying the Bounded Degree condition (§3.2.1) and the Acyclicity constraint (§2.7.1), the probability $P(u \leftrightarrow v)$ that a causal constraint propagates between two vertices $u$ and $v$ separated by distance $r$ decays exponentially:

$$P(u \leftrightarrow v) \sim (d_{\max} \rho)^r$$

Within the Sparse Phase, where the edge density satisfies $\rho < 1/d_{\max}$, the correlation length is finite: $\xi = -1 / \ln(d_{\max} \rho)$. Consequently, the Mutual Information satisfies $I(R_i; R_j) \to 0$ for distances greater than $\xi$, validating the mean-field approximation for macroscopic dynamics.

5.1.3.1 Proof: Exponential Decay

Derivation of Correlation Bounds from Finite Branching

I. Path-Sum Formulation

The correlation function between observables localized at vertices $u$ and $v$ relates proportionally to the weighted sum over all self-avoiding paths connecting them. $\langle O_u O_v \rangle_c \propto \sum_{\pi: u \to v} w(\pi)$ In the high-temperature vacuum phase, the weight of a path decays exponentially with its length due to the disorder average. $w(\pi) \sim \rho^{\ell(\pi)}$ where $\rho < 1$ is the edge density parameter.

II. Branching Structure

From the uniqueness of the Bethe Fragment as the vacuum state (§3.2.1), the graph $G_0$ exhibits a locally tree-like structure with finite branching factor $b$. For a distance $d = \text{dist}(u, v)$, the number of simple paths of length $L \ge d$ is bounded by the branching process: $N(L) \sim b^{L-d}$ (The path must traverse the $d$ specific radial steps, with transverse fluctuations limited by the tree topology).

III. Summation Bound

The total correlation sums contributions from all path lengths $L \ge d$: $\mathcal{C}(d) \approx \sum_{L=d}^{\infty} N(L) \rho^L \approx \sum_{L=d}^{\infty} b^{L-d} \rho^L$ Factor the sum: $\mathcal{C}(d) \approx \rho^d \sum_{k=0}^{\infty} (b \rho)^k$ For the series to converge (finite correlation length), we require $b\rho < 1$. This is the sub-percolation condition. The sum becomes a geometric series constant $A = 1/(1-b\rho)$. $\mathcal{C}(d) \approx A \cdot \rho^d = A \cdot e^{d \ln \rho}$

IV. Correlation Length Definition

Define the correlation length $\xi$: $\xi = -\frac{1}{\ln \rho}$ Substituting this definition yields the standard exponential decay form: $\mathcal{C}(d) \sim e^{-d / \xi}$

V. Information Bound

The mutual information $I(u; v)$ is bounded above by the square of the connected correlation function (for Gaussian fluctuations). $I(u; v) \le \frac{1}{2} \langle O_u O_v \rangle_c^2 \sim e^{-2d/\xi}$ This establishes that information content decays exponentially with graph distance.

Q.E.D.

5.1.3.2 Commentary: The Role of Acyclicity and Sparsity

Characterization of the Vacuum as Sub-Percolating

The proof relies on the combinatorial counting of connecting paths between vertices. In generic random graphs near the percolation threshold; paths loop back and reinforce one another; creating long-range order and diverging correlation lengths that span the entire system. This phenomenon is extensively studied in percolation theory and random graph dynamics, particularly by (Bollobás, 2001), who details the phase transition where the giant component emerges. However; the vacuum structure derived in Chapter $3$ (The Bethe Fragment) and enforced by Axiom $3$ remains locally tree-like and strictly acyclic.

The prohibition of directed cycles forces causal influence to propagate unidirectionally; preventing the feedback loops that drive percolation. In a sparse regime; the number of paths of length $r$ grows insufficiently to overcome the probabilistic decay associated with traversing each link. This bounds the "sphere of influence" of any single event. The vacuum effectively remains sub-percolating: influences damp out exponentially before they can span the system. This stability against runaway connectivity forms the bedrock of the manifold structure; without this correlation decay; the graph would collapse into a highly connected "small world" network where every point is adjacent to every other point; effectively destroying the dimensionality and locality required for physics.

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5.1.4 Proof: Extensive Entropy

Formal Derivation via Partitioning and Limits

I. Volume Decomposition

Partition the graph $G_N$ into a set of $M$ quasi-independent sub-volumes $\{V_1, V_2, \dots, V_M\}$. The characteristic size of each volume is set by the correlation length $\xi$ derived in Lemma 5.1.3. $|V_k| \approx V_\xi \sim \xi^3$ $M = \frac{N}{V_\xi}$

II. Partition Function Factorization

Let $\Omega_{total}$ be the cardinality of the global configuration space. Due to the exponential decay of correlations ($e^{-d/\xi}$), the mutual information between non-adjacent volumes vanishes. $I(V_i; V_j) \approx 0 \quad \text{for} \quad \text{dist}(V_i, V_j) \gg \xi$ The global phase space volume approximates the product of local volumes: $\Omega_{total} \approx \prod_{k=1}^{M} \Omega(V_k)$

III. Logarithmic Additivity

The total entropy is the logarithm of the phase space volume. $S_{total} = \ln \Omega_{total} \approx \ln \left( \prod_{k=1}^{M} \Omega(V_k) \right) = \sum_{k=1}^{M} \ln \Omega(V_k)$

IV. Local Finiteness and Bound

Each sub-volume $V_k$ contains a finite number of vertices. Axiom 1 (bounded degree) strictly bounds the number of possible subgraphs. For a volume of size $v$, the number of edges is at most $v(v-1)$. The states are subsets of edges. $\Omega(V_k) \le 2^{|V_k|^2}$ Thus, the local entropy $S_{local} = \ln \Omega(V_k)$ is finite.

V. Homogeneity Limit

In the equilibrium vacuum, the system is statistically homogeneous. $S(V_k) = S_{local} \quad \forall k$ Substituting into the sum: $S_{total} \approx \sum_{k=1}^{M} S_{local} = M \cdot S_{local} = \left( \frac{N}{V_\xi} \right) S_{local}$ Define the entropy density constant $c = S_{local}/V_\xi$. $S_{total} = c N$ Corrections due to boundary interactions scale as area $\sim N^{2/3}$, vanishing relative to the bulk term in the thermodynamic limit ($N \to \infty$).

Q.E.D.

5.1.4.1 Calculation: Boundary Correction

Computational Verification of Subextensive Boundary Terms using Lattice Simulation

Quantification of the subextensive boundary term and verification of the independence assumption are based on a simulation of a 2D toroidal lattice. The simulation protocols are as follows:

1. Lattice Construction: The algorithm generates a toroidal grid graph of size $N$ and partitions it into $\sqrt{N}$ blocks to mimic correlation volumes.
2. Edge Counting: The protocol iterates through all edges in the graph, identifying the block coordinates of each node. Edges connecting nodes in different blocks are flagged as "boundary edges."
3. Scaling Analysis: The metric computes the fraction of boundary edges relative to the total edge count across a range of system sizes $N \in [100, 10000]$ to verify the vanishing surface-to-volume ratio.

import networkx as nx
import numpy as np
import pandas as pd

def boundary_fraction(N: int):
    """Compute fraction of edges crossing block boundaries in a 2D toroidal lattice."""
    side = int(np.sqrt(N))
    if side * side != N:
        raise ValueError("N must be a perfect square for a square toroidal grid.")
    
    # Create toroidal 2D grid graph
    G = nx.grid_2d_graph(side, side, periodic=True)
    # Relabel nodes to linear indices 0..N-1
    mapping = {(i, j): i * side + j for i in range(side) for j in range(side)}
    G = nx.relabel_nodes(G, mapping)
    
    total_edges = G.number_of_edges()
    
    # Block size ≈ side // 4 (mimics correlation volume)
    block_side = max(2, side // 4)
    blocks_per_side = side // block_side
    
    boundary_edges = 0
    
    # Iterate over all edges and count those crossing block boundaries
    for u, v in G.edges():
        # Block coordinates of u and v
        block_u = (u // side // block_side, (u % side) // block_side)
        block_v = (v // side // block_side, (v % side) // block_side)
        
        if block_u != block_v:
            boundary_edges += 1
    
    # Each edge counted once (undirected graph)
    fraction = boundary_edges / total_edges if total_edges > 0 else 0.0
    
    # Relative correction term (as in original)
    rel_correction = np.sqrt(N) * np.log(total_edges + 1) / (N * np.log(2) + 1e-10)
    
    return {
        'N': N,
        'Boundary Edge Fraction': fraction,
        'Relative Correction': rel_correction
    }

# Perfect-square lattice sizes
sizes = [100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000]
results = [boundary_fraction(N) for N in sizes]

df = pd.DataFrame(results)

print("Subextensive Boundary Terms in 2D Toroidal Lattice")
print("=" * 54)
print(df.round(4).to_markdown(index=False, tablefmt="github"))

Simulation Output:

======================================================

N	Boundary Edge Fraction	Relative Correction
100	0.5	0.7651
400	0.2	0.4823
900	0.1667	0.3605
1600	0.1	0.2911
2500	0.1	0.2458
3600	0.0667	0.2136
4900	0.0714	0.1894
6400	0.05	0.1705
8100	0.0556	0.1554
10000	0.04	0.1429

The data confirms the hypothesis: the fraction of boundary edges drops from 50% at $N=100$ to merely 4% at $N=10,000$. This validates that for large systems, the vast majority of interactions are internal to the quasi-independent volumes. The vanishing boundary term justifies the additive approximation $S \approx \sum S_{local}$, confirming that the extensive bulk term dominates regardless of emergent dimension.

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5.1.Z Implications and Synthesis

Extensive Entropy

The entropy of the causal graph is established as strictly extensive, scaling linearly with the vertex count $N$. This property transforms the abstract graph into a physical reservoir, where information content behaves as a bulk quantity analogous to volume in a gas. By proving that correlations decay exponentially, we have decomposed the universe into a vast collection of quasi-independent volumes, validating the application of standard statistical mechanics to the discrete substrate.

This result implies that the vacuum possesses a finite, measurable capacity for disorder. It ensures that local operations do not trigger instantaneous global reconfigurations, protecting the system from non-local instabilities. The linearity of the entropy scaling confirms that the universe is thermodynamically stable, capable of supporting heat exchange and local equilibrium without diverging into infinite complexity or collapsing into a singularity.

The existence of a well-defined specific entropy per event provides the necessary thermodynamic potential to drive evolution. It converts the combinatorial vastness of graph space into a manageable physical quantity, allowing us to treat the growth of the universe not as a random walk, but as a directed flow down a free energy gradient. This extensivity is the bedrock that permits the formulation of a master equation, ensuring that the microscopic rules of the graph aggregate into coherent macroscopic laws.

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