The Graph-Theoretic Ontology of EET: Constraint Network as the Discrete Syntax of Existence

What is the fundamental mathematical structure of the constraint network? Is the graph merely a convenient modeling tool, or is it the necessary discrete syntax that any constraint network must obey under Finite Distinguishability? Energy-Efficiency Theory (EET) provides a first-principles answer: the graph is the constitutional mathematical grammar of the constraint network---not an optional representation, but the exact discrete form that the network must take when expressed in the language that finite distinguishability demands. This paper develops the complete graph-theoretic ontology from the generative foundations of EET Core Rules v5.4 and the companion ontologies of Constraint (v2.0) and Difference (v2.0). Version 2.4 upgrades this ontology from an L2 structural realization to an L0/L1 constitutional mother text, establishing three foundational theorems derived from first principles: 1. The Discreteness Theorem. Under Finite Distinguishability (L-1), the state space of any constraint network admits a countable discrete basis. Continuity is an emergent property of coarse-graining in the macroscopic limit, not an ontological primitive. The constraint network is irreducibly discrete at L1. 2. The Graphic Sufficiency Theorem. A weighted graph $G = (V, E)$ with vertices V and edges E is the minimal mathematical structure capable of representing: distinct existence (vertices as Maintained Differences), relational influence (edges as free-state channels), and weighted capacity (wᵤᵥ as transmission capacity). No structure simpler than a graph can simultaneously capture these three irreducible features. 3. The Optimal Balance Spectral Theorem. The energy ratio η = 1---the optimal balance point where cooperative capacity is maximized---is equivalent to the closure of the spectral gap of the constraint graph: η = 1 λ₁ → 0. This theorem provides the precise graph-theoretic signature of universal criticality across all scales of EET. We establish the constitutional definitions of the two fundamental edge types---Type I (maintained connection, inertial) and Type II (free-state channel, non-inertial)---and their interconversion dynamics. Transient edges are introduced as a new constitutional edge type, representing temporary couplings with finite persistence time τ ~ Δ tₘᵢₙ, providing the graph-theoretic expression of quantum fluctuations, neural spikes, and momentary market transactions. Constitutional Dictionary of graph operations is established: every graph operation (vertex creation, edge weight adjustment, subgraph contraction, spectral truncation) corresponds to a precise EET physical or cognitive operation (constraint formation, sliding, encapsulation, projection), each with exact energetic conditions. We establish complete interfaces to all 24 EET mother texts---from the physical constitution (Constraint, Difference, Inertia, Space, Time, Phase Transition, Statistical Mechanics) to the cognitive constitution (Observer, Information, Generative Grammar, Modelology, Xu-Shi, Complexity). The graph is revealed as the universal syntax of the Dual Helix: the physical constraint graph Gphys and the cognitive constraint graph GM (MEER Audit Graph) share the identical graph grammar, differing only in their carrier semantics. External validation from nine independent research clusters (2024--2026)---including Combinatorial Quantum Gravity, Discrete Gravity Ontology, the revival of the Page-Wootters mechanism, Ollivier-Ricci curvature as the network Einstein-Hilbert action, and physical information graph neural networks---confirms that the constitutional role of the graph is not an isolated EET construct but part of a broader paradigm shift occurring across theoretical physics. Falsifiable predictions include spectral gap scaling with hierarchical depth, a percolation threshold for macroscopic superposition, and η-dependent edge-weight relaxation. The graph is not a model of the constraint network---it is the constraint network, expressed in the discrete language that finite distinguishability demands. {Keywords}: Graph theory; constraint network; Laplacian; spectral gap; Ollivier-Ricci curvature; Type I/II edges; transient edges; constitutional syntax; Energy-Efficiency Theory; Dual Helix