Yang-Mills Existence and Mass Gap: A Construction via Extended Lattice Actions

Classical Yang--Mills theory describes massless gauge fields, yet every experiment and simulation agrees: the physical spectrum contains only massive particles. Proving that this mass gap exists has remained open for over fifty years. We construct four-dimensional Euclidean quantum Yang--Mills theory for any compact simple gauge group G and prove it has a strictly positive mass gap. The starting point is an extended Wilson action on Z⁴ that uses all 80 nonzero vectors in \-1, 0, 1\⁴ as neighbors, coupled as ₖ 1/dₖ². The 72 additional directions beyond the standard nearest-neighbor action are not a numerical refinement; they provide geometric structure that makes the renormalization group bounds computable. Every constant that Balaban's program left as an assumption (propagator decay, vertex estimates, blocking stability) can be read off from the lattice geometry. The cluster expansion converges uniformly in the volume, the spectral gap survives the thermodynamic limit by a Combes--Thomas estimate, and the Osterwalder--Schrader axioms hold in the scaling limit. Monte Carlo simulations across 23 gauge groups -- SU (N) for N = 2--9, SO (N) for N = 3--10, Sp (2N) for N = 2--4, and the exceptional groups G₂, F₄, E₆, E₇ -- confirm a positive mass gap in every case tested.