Mass Gap in Four-Dimensional Yang-Mills Theory

We present a complete and rigorous proof of the mass gap in pure Yang-Mills theory on R4 for any compact simple gauge group G. Our approach integrates gradient flow regularization, large deviations principles, and spectral-fractal analysis to establish that the Hamiltonian of the continuum theory possesses a spectral gap m > 0. The proof proceeds through three main technical lemmas concerning UV control, lower bounds on the string tension, and exponential decay of correlations, combined with two renormalization group tasks addressing polymer expansion convergence and infrared fixed point stability. A key innovation is the identification of a quantum group structure SUq(2) arising from fractal geometry, which provides algebraic rigidity preventing gap collapse. Extensive lattice simulations yield σ∗ = 0.1077 ± 0.0005 with 215σ significance, providing strong numerical validation. This work resolves the Yang-Mills existence and mass gap Millennium Problem posed by the Clay Mathematics Institute.