Mathematical Foundations of an Emergent Constant: An Axiom-Theorem System for Constraint Network Dynamics

We construct a deterministic discrete dynamical system—the Constraint NetworkModel—defined by three axioms: energy units move at constant speed, uponencounter they undergo symmetric collision, and in asymmetric encounters aresidual unit may merge if directional alignment occurs. The system is fullyformalizable in ZF set theory. We prove eight theorems: the evolution iswell-posed and total energy is conserved; merging is irreversible and themaximum aggregate number is non-decreasing; the system must converge to asteady state in finite time; the aggregate number of a saturated sealed nodeis bounded by a sphere-covering problem; the aggregate number of a sealednode must be even, and this emergent constant is uniquely determined by thesystem parameters; an odd neighbor inevitably appears adjacent to a sealednode, with conditional stability depending on the existence of a chain; thesealed node is a global attractor; multiple sealed nodes can coexist, eachlocked at the same emergent constant, connected by chains into a stablenetwork. The specific numerical value of the emergent constant is not givenin this paper. Its determination is deferred to subsequent work.