Mass Gap for Yang–Mills Theory in Four Dimensions via Geometric Depletion

This manuscript develops a fully non-perturbative, geometric and functional framework for four-dimensional Euclidean Yang-Mills theory with compact gauge group SU(N), establishing a strictly positive spectral gap in the reconstructed Wightman theory. The central innovation is the Geometric Depletion Functional, which measures local concentration of curvature and its covariant derivative. Small depletion implies quantitative Uhlenbeck regularity, explicit Coulomb gauges, and stability of renormalization counterterms, converting ultraviolet divergence control into a deterministic geometric patching problem. Main results: 1. Global Coercivity: A multi-scale patching argument based on Besicovitch coverings and Azuma-Hoeffding concentration establishes uniform Poincare and Log-Sobolev inequalities with constants independent of the lattice spacing. 2. Continuum Stability: The spectral gap persists in the limits of vanishing lattice spacing and infinite volume, ruling out logarithmic drift of functional constants in dimension four. 3. Boundary Extinction: The Yang-Mills measure vanishes structurally at the Gribov horizon, ensuring essential self-adjointness of the Hamiltonian. The limiting measure satisfies all Osterwalder-Schrader axioms. Reconstruction yields a Hilbert space, a unique vacuum, Wightman functions, and exponential clustering of gauge-invariant observables. The spectrum of the Hamiltonian exhibits a mass gap with explicit lower bound. This work addresses the Yang-Mills existence and mass gap problem, one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute.