A Holonomy–Anchoring Approach to the Yang–Mills Mass Gap

We present a purely mathematical resolution of the four-dimensional Yang–Mills existence and mass gap problem in the sense of the Clay Mathematics Institute.The approach is based on a geometric anchoring functional, defined via local holonomy (Wilson loop observables), which obstructs arbitrarily small-scale curvature concentration for finite-energy non-flat connections. Using Uhlenbeck gauge fixing, Sobolev control, and trace estimates, we prove that finite Yang–Mills action enforces a strictly positive anchoring radius. This anchoring yields a global Hardy–Poincaré type coercivity inequality for the quadratic Yang–Mills form, implying a strictly positive spectral gap above the vacuum sector. Beyond the spectral argument, we address the existence of the quantum theory. We show that anchoring provides non-perturbative ultraviolet control, ensuring tightness of regularized Euclidean Yang–Mills measures. The continuum limit exists in the space of distribution-valued gauge fields, satisfies the Osterwalder–Schrader axioms, and admits a self-adjoint Hamiltonian via OS reconstruction. The spectral gap implies exponential clustering of correlation functions. All technical issues concerning measurability, support of the measure, uniqueness of the vacuum, stability in the infinite-volume limit, and non-triviality of the continuum theory are resolved in an appendix.The result establishes, within a single geometric–analytic framework, both existence and mass gap for four-dimensional Yang–Mills theory.