Chapter 2: Constraints

2.8 Formal Synthesis

End of Chapter 2

The three axioms forge the substrate's unyielding frame, erecting a rigid skeleton upon which the fabric of reality can braided. The Causal Primitive acts as a ratchet, directing influence without reversal and sharpening the arrow of time. Geometric Constructibility mandates the tiling of the vacuum with 3-cycle quanta, ensuring space is woven from fundamental areas. Finally, Acyclic Effective Causality projects these local rules into a global order, preventing the universe from trapping itself in the paradox of closed loops.

This triad delimits the boundaries of the possible. Our countermodels prove that each axiom serves as a unique load-bearing pillar of the theory, independent and necessary. Furthermore, the mechanism of Decomposition ensures that complex tangles dissolve into simplices, enforcing an inexorable drive toward geometric simplicity. Physically, the graph now accretes as a directed lattice, where every cycle resolves to a quantum of area and every edge preserves the integrity of history.

But a set of rules is not a universe; laws require a jurisdiction. Possessing the constraints but lacking the initial state, the investigation must now determine the specific configuration of the graph at $t=0$ that satisfies these strictures while maximizing potential. This leads us to Chapter 3, where the unique topology of the vacuum is derived.

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Table of Symbols

Symbol	Description	Context / First Used
$(u, v)$	The Directed Causal Link (Atomic relation $u \to v$)	§2.1.1
$E$	The set of edges within the graph	§2.1.1
$\implies$	Logical implication	§2.2.1
$\forall$	Universal quantifier ("for all")	§2.2.1
$\mathcal{T}_{self}$	Self-Loop Addition Operation	§2.2.3
$\Omega(G)$	Cardinality of the set of Simple Paths	§2.2.3
$\Delta S$	Change in Entropy	§2.2.3
$k_B$	Boltzmann Constant	§2.2.3
$\mathfrak{T}_{add}$	Edge Addition Operation	§2.3.1
$\Pi_{\ell \le 2}(u, v)$	Set of Simple Directed Paths from $u$ to $v$ with length $\le 2$	§2.3.1
$L$	Length of a cycle or path	§2.3.1
$\gamma$	Geometric Quantum (Directed 3-Cycle)	§2.3.2
$\Phi(G)$	Lexicographic Potential $(L_{\max}, N_{L_{\max}})$	§2.3.4
$L_{\max}$	Length of the longest simple cycle in $G$	§2.3.4
$N_{L_{\max}}$	Count of distinct cycles of length $L_{\max}$	§2.3.4
$\mathcal{R}$	The Rewrite Rule (Edge addition mechanism)	§2.4.2
$C$	A Simple Directed Cycle	§2.4.3
$\text{dist}_C(u, v)$	Distance between vertices along a cycle $C$	§2.4.3.1
$\mathcal{O}_{add}$	Composite Addition Phase (Chord insertion)	§2.4.5
$\mathcal{O}_{del}$	Composite Deletion Phase (Entropic breakage)	§2.4.5
$\mathcal{S}_{step}$	Composite Update Step ($\mathcal{O}_{del} \circ \mathcal{O}_{add}$)	§2.4.5
$\le$	Effective Influence Relation (Strict Partial Order)	§2.6.1
$H(e)$	History Timestamp of edge $e$	§2.6.1
$\pi_{uv}$	A specific Simple Directed Path instance from $u$ to $v$	§2.6.1
$\neg$	Logical negation	§2.7.1
$N$	Total number of vertices in the graph	§2.7.2
$R$	Radius of local computational patch	§2.7.3
$\rho$	Edge density of the graph	§2.7.3
$t_{crit}$	Critical time where cycle diameter exceeds horizon	§2.7.3
$P_{err}(R)$	Probability of paradox evasion at radius $R$	§2.7.4
$E_{sync}$	Energy required for global synchronization	§2.7.5
$D$	Graph Diameter	§2.7.5

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