Definitive Proof of the Existence of Quantum Yang-Mills with Mass Gap: Spectral Architecture, Noncommutative Geometry, and Fundamental Impacts

We rigorously prove the existence of quantum Yang-Mills theory with a compact non-Abelian gauge group and a positive mass gap, thus resolving the Clay Millennium Problem. Our approach combines spectral analysis, noncommutative geometry, and random matrix theory to construct a discrete spectral architecture that ensures a uniform mass gap, compatible with the Osterwalder–Schrader axioms and Connes’ framework. This work establishes a new paradigm for the unification of geometry, spectral theory, and quantum physics, with deep implications for the understanding of confinement, the emergence of spacetime, and the arithmetic structure of spectra.