Constraint Network Dynamics: An Axiom-Theorem Foundation for Emergent Constants

We construct a deterministic discrete dynamical system—the Constraint Network Model—defined by three axioms: energy units move at constant speed, upon encounter they undergo symmetric collision, and in asymmetric encounters a residual unit may merge if directional alignment occurs. The system is fully formalizable in ZF set theory with the Axiom of Dependent Choice. We prove nine theorems establishing: well-posedness and conservation; irreversibility of merging and monotonicity of the maximum aggregate number; finite-time convergence to a steady state; a geometric upper bound on sealed node capacity from sphere packing; the parity constraint (the aggregate number of a sealed node must be even); the rigorous reduction of the emergent constant determination problem to a constrained optimal covering-packing problem on S^2, whose unique solution, if it exists, uniquely determines the constant; the conditional stability of an odd neighbor; the global attractivity of the sealed node; the coexistence of multiple sealed nodes in a chain network; and nine comprehensive properties of the emergent constant. The specific numerical value of the constant is not given in this paper; its determination via the solution of the problem formulated herein is the subject of companion work.