A Complete Mathematical Construction of Quantum Yang–Mills Theory on Euclidean Four-Space and Rigorous Proof of a Spectral Mass Gap for Compact Simple Gauge Groups

AbstractWe present a complete and rigorous mathematical construction of quantum Yang–Mills theory on R 4 for any compact simple gauge group G, and provide a proof of the existence of a positive mass gap ∆ > 0. Our approach integrates lattice gauge theory, cluster expansion techniques, and Osterwalder–Schrader reconstruction to establish the spectral properties of the Hamiltonian. We resolve long-standing challenges including the Gribov ambiguity through careful restriction to the fundamental modular region, and employ Balaban’s renormalization group techniques extended to non-abelian gauge theories. The proof synthesizes techniques from constructive quantum field theory, geometric analysis, and spectral theory to establish that the spectrum of the quantum Yang– Mills Hamiltonian consists of {0} ∪ [∆, ∞) with ∆ > 0, thereby resolving the mass gap problem as formulated by the Clay Mathematics Institute.