A Mathematical Proof of the Yang-Mills Mass Gap in the Constraint Network Framework

The Yang-Mills existence and mass gap problem is one of the Millennium PrizeProblems of the Clay Mathematics Institute. The problem requires provingthat for any compact gauge group, quantum Yang-Mills theory exists and itsexcited states have a positive mass gap. This paper presents a mathematicalproof of this problem within the axiom-theorem system of Constraint NetworkDynamics. The Constraint Network Model is a deterministic discrete dynamicalsystem defined by three axioms, fully formalizable in ZF set theory, whoseeight theorems have been rigorously proved in prior work. We prove thatwithin this framework, the chain—a bi-directional flow channel of N = 1units between sealed nodes—is the origin of the strong interaction. Theformation and maintenance of a chain requires a specific energy thresholdthat is strictly positive and uniquely determined by the system parameters.This threshold is precisely the Yang-Mills mass gap. The existence of themass gap is guaranteed by the chain network stability established in Theorem8, and its positivity is jointly derived from the geometric constraint ofTheorem 4 and the irreversibility of accretion of Theorem 2. This proofintroduces no assumptions beyond the axiom system of the Constraint NetworkModel, and relies on neither numerical simulation nor perturbative expansion.