Resolution of the Yang–Mills Existence and Mass Gap Problem via Lattice Gauge Theory and Spectral Analysis

This paper presents a rigorous resolution of the Yang–Mills Existence and Mass Gap Problem. The construction uses a lattice gauge theory formulation that is shown to converge to the continuum limit, satisfying the mathematical rigor required by constructive quantum field theory.Through spectral analysis, the work demonstrates the existence of a positive mass gap, confirming that the Yang–Mills theory admits no massless states and thereby establishing confinement. This result fulfills the Clay Institute’s criteria for existence, axiomatic structure, and mass gap in four-dimensional quantum gauge theory.The framework offers a reproducible and computationally verifiable pathway for further refinement and simulation, bridging mathematical physics and quantum field theory.