Graph-Theoretic Dynamics of the Energy-Efficiency Cycle: From Edge-Weight Fluctuations to Topological Phase Transitions

The Energy-Efficiency Cycle (YEC)—Disturbance, Response, Stabilization, Constraint, Transition—has been formulated as a universal phenomenological grammarfor non-equilibrium steady-state systems. This paper establishes its ontological foundation in the graph-theoretic framework of Energy-Efficiency Theory (EET).We show that the YEC is the geometric dynamics of a weighted constraint graph G(V, E), where vertices represent constrained-state energy (particles, bound struc-tures) and edges represent free-state energy flux (gauge bosons, interactions). The energy ratio η = ˙Eresp/ ˙Emain controls three fundamental graph operations: edge-weight fluctuations (η > 1), edge stretching and condensation (η → 1), and edge freezing with subgraph clustering (η < 1). The five YEC stages are mapped ontograph rewiring dynamics: Disturbance triggers local η spikes and edge excitation; Response drives edge proliferation and enhanced connectivity; Stabilization prunesredundant edges via spectral gap relaxation; Constraint freezes effective edges into stable subgraphs; Transition marks topological phase transitions—percolation, sub-graph isolation, and symmetry breaking. We derive the degenerate limits to mainstream physics: η = 1 with action quantization recovers quantum mechanics; η → 0with large action yields classical mechanics; the N → ∞ continuum limit of Ollivier-Ricci curvature gives general relativity; and irreversible edge dissipation implies thesecond law of thermodynamics. The framework unifies gauge fields (edge coloring, Wilson loops), the Higgs mechanism (isotropic edge stretching), and parity viola-tion (VL subgraph isolation). We propose three framework-level falsifiable predictions: edge-fluctuation spectra in ultracold atoms, macroscopic superposition cutofffrom graph percolation, and spectral gap scaling in complex networks. This paper supersedes the earlier phenomenological YEC formulation, providing its definitivegraph-theoretic ontology and establishing the YEC as the universal meta-grammar of constraint graph dynamics. Keywords: Energy-Efficiency Cycle; graph theory; edge-weight dynamics; topo-logical phase transitions; Ollivier-Ricci curvature; spectral gap; constraint graph