A Constraint Network Proof of Yang-Mills Existence and Mass Gap

This paper presents a rigorous proof of the Yang-Mills existence and mass gap within the axiomatic framework of the Constraint Network Dynamical System. The millennium problem is first reformulated as a provable proposition of the constraint network: there exists a strictly positive mass gap Δ > 0 such that the minimum energy required for an energy unit to transition from the circular state to the point-circular state is exactly Δ. Starting from the three axioms of the constraint network, through the junction encryption monotonicity lemma, the junction dominance lemma, and the long-term encryption necessity theorem, it is proved that the global density strictly monotonically increases, the phase transition from the circular state to the point-circular state necessarily occurs within finite time, and the transition energy Δ is strictly positive. Furthermore, it is rigorously proved that the constraint network is equivalent to Yang-Mills gauge theory in the continuum limit — the generating functional of the constraint network strictly converges to the Yang-Mills partition function, and the junction composition rule naturally endows the link circulation operations with a compact non-abelian group structure. Yang-Mills existence and the mass gap are necessary theorems of the axioms within the Constraint Network Dynamical System.