Chapter 5: Geometrogensis

5.6 Formal Synthesis

End of Chapter 5

Space is born from the statistical tumult of relations. The entropy of the causal graph proves extensive, scaling linearly with system size $N$, which justifies treating the vacuum as a thermodynamic reservoir. From this, the Fundamental Equation of Geometrogenesis emerges, a master equation that balances the explosive force of autocatalysis against the damping force of geometric friction, revealing the heartbeat of cosmic expansion.

Our parameter sweep identifies a narrow Region of Physical Viability, a "Goldilocks zone" where the universe neither freezes into a crystalline tree nor explodes into a small-world singularity, but stabilizes at a sparse equilibrium density $\rho^* \approx 0.029$. Within this stable phase, the graph naturally satisfies the conditions for Ahlfors 4-Regularity, fixing the macroscopic dimension of spacetime at $d=4$. Physically, the vacuum is no longer a void, but a dynamic "relational plasma" fluctuating around a stable density.

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Table of Symbols

Symbol	Description	Context / First Used
$I(R_A; R_B)$	Mutual Information between disjoint regions	§5.1.1
$\xi$	Correlation Length (Entropic decay scale)	§5.1.1
$V_\xi$	Correlation Volume ($V \propto \xi^3$)	§5.1.1.1
$\Omega_N$	Cardinality of configuration space on $N$ vertices	§5.1.2
$S(N)$	Total Entropy ($c \cdot N$)	§5.1.2
$c$	Specific entropy per event (Capacity)	§5.1.2
$N_3(t)$	Population of 3-cycles (Geometric Quanta)	§5.2.1
$\rho(t)$	Normalized 3-cycle density ($N_3/N$)	§5.2.2
$\Lambda$	Vacuum Permittivity (Ignition Flux)	§5.2.3
$\mu$	Geometric Friction Coefficient ($1/\sqrt{2\pi}$)	§5.2.5
$\lambda_{cat}$	Catalysis Coefficient ($e-1$)	§5.2.6
$J_{in}, J_{out}$	Topological Fluxes (Creation/Deletion)	§5.2.7
$\rho^*$	Equilibrium density (Fixed Point)	§5.4.1
$F(\rho)$	Net Flux Function ($J_{in} - J_{out}$)	§5.4.2.1
$J$	Jacobian Eigenvalue (Stability indicator)	§5.4.4.1
$\bar{d}(u,v)$	Undirected shortest-path distance	§5.5.2
$\langle k \rangle$	Mean vertex degree	§5.5.3
$D_{\max}$	Maximum vertex degree bound	§5.5.3
$K(u,v)$	Causal Ollivier-Ricci curvature	§5.5.4
$W_1(\mu_u, \mu_v)$	Wasserstein-1 Distance	§5.5.4.1
$C_{cov}, \gamma$	Covariance amplitude and decay rate	§5.5.5
$C_k$	Count of simple cycles of length $k$	§5.5.6
$B(v,r)$	Volume of geodesic ball of radius $r$	§5.5.7
$d_c$	Upper critical dimension ($d=4$)	§5.5.7.1

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Conclusion to Part 1: The Emergence of the Stage

End of Part 1

Completion of the physical background derivation is achieved. Enforcement of strict axiomatic constraints on a discrete causal substrate generates a dynamical vacuum that evolves from a singularity into a stable, finite-dimensional manifold. Thermodynamic machinery yields a universe that is geometrically coherent, temporally directed, and physically viable. The stage is built: a self-regulating spacetime capable of supporting information but, as yet, devoid of persistent actors.

The master equation ensures the vacuum fluctuates around a stable density, but fluctuation alone does not constitute matter. Existence of a physical universe requires specific configurations to arise that resist the relentless entropy of the rewrite rule: structures possessing topological fortitude to survive as distinct, durable entities. The inquiry shifts from how the graph weaves itself into space to how it knots itself into substance. Derivation of these persistent excitations follows, moving from statistical laws of geometry to topological invariants of the particle.

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