Existence and Positive Mass Gap for Four-Dimensional Yang–Mills via OS-Constructive Methods and Gauge-Covariant Recursive Operators

This work presents a rigorous, non-perturbative proof of the Yang–Mills Existence and Mass Gap problem, one of the Clay Mathematics Institute’s Millennium Prize Problems. Using a constructive Osterwalder–Schrader framework, gauge-covariant recursive operators, and explicit cluster-expansion bounds, the paper demonstrates: The existence of a mathematically well-defined four-dimensional quantum Yang–Mills theory for compact simple gauge groups. Verification of the Osterwalder–Schrader axioms, followed by reconstruction to a Wightman QFT with a self-adjoint Hamiltonian. Exponential decay of gauge-invariant two-point correlation functions, establishing a strictly positive spectral gap above the vacuum. The approach unifies heat-kernel regularization, fixed-point recursion, and BRST-invariant gauge control into a constructive continuum theory. The result provides both the existence of Yang–Mills in four dimensions and a positive lower bound on the mass spectrum, satisfying the Clay Institute’s stated requirements. This paper offers a full mathematical proof that the quantum fields underlying the strong nuclear force are stable and have a minimum ‘mass gap’—a result foundational to particle physics and essential for potential applications like warp field engineering and biological energy stabilization.