A Compositional Equivalence Framework for Closed Systems: Topology, Graph Theory, and the Reconstruction of Dispersed Configurations

We introduce a mathematical framework for studying the equivalence and reconstruction of physical configurations in closed systems. A hierarchy of three equivalence relations — compositional, graph-theoretic, and topological — is established, and topological equivalence is identified as the precise condition under which reconstruction is theoretically possible. Main results include obstruction theorems, a proximity bound relating structural similarity to bond edit distance, an NP-hardness result for computing bond distance, a topological diversity classification via knot types and linking numbers, and a dimension dependence theorem showing that topological obstructions vanish in ambient dimensions n ≥ 4, making n = 3 the unique critical case. Sufficient conditions for convergence of bond dynamics and a formal continuum limit are also established.