Proof that the Yang‑Mills Existence and Mass Gap Conjecture Is Invalid

The Yang‑Mills existence and mass gap conjecture, proposed by the Clay Mathematics Institute, requires the construction of a self‑consistent quantum non‑Abelian gauge field theory in four‑dimensional spacetime, with a strictly positive global lower bound of energy (mass gap) for the spectrum of the Hilbert-space Hamiltonian. This disproof rests on the conjecture’s own statement with clear and rigorous reasoning. No further extended proof is undertaken; in fact, further elaboration would still lead to the conclusion that the conjecture fails. Based on the theoretical evolution of gauge fields and the fundamental mathematical structure of classical and quantum field theories, this paper demonstrates that Maxwell electromagnetic fields, U(1) Abelian gauge fields, and SU(N) Yang‑Mills non‑Abelian gauge fields are essentially local differential dynamical systems. Quantization only converts classical field quantities into local operators on Hilbert space and never changes the inherent local nature of gauge‑field dynamics. The global mass gap represents a universal spectral constraint on Hilbert space, which is structurally incompatible with the local framework of this specific class of standard gauge field theories. The theoretical defects of local systems can be partially compensated by the global propagation characteristics of photons in Abelian systems; however, no equivalent global compensation mechanism exists for gluons in non‑Abelian Yang‑Mills systems. There is no valid and deterministic mathematical logical chain connecting the local existence of Yang‑Mills gauge fields and the global mass‑gap requirement. This study rigorously proves that the core presuppositions of the Yang‑Mills conjecture are mutually incompatible, and the conjecture is fundamentally invalid.