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 (wₔₕ 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 = 1---the optimal balance point where cooperative capacity is maximized---is equivalent to the closure of the spectral gap of the constraint graph: = 1 ₁ 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 t_, 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 G₏₇ₘₒ 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 -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