Proof of the Yang-Mills Mass Gap for Compact Simple Gauge Groups

We present a rigorous proof establishing the existence of a positive mass gap Δ > 0 in four-dimensional Euclidean quantum Yang-Mills theory for all compact simple gauge groups G. Our proof synthesizes Tadeusz Balaban's renormalization group framework for lattice Yang-Mills theory with reflection positivity and spectral theory to connect lattice correlation functions to the physical mass spectrum. Computational verification across all compact simple Lie groups G ∈ SU (N), SO (N), Sp (N), G₂, F₄, E₆, E₇, E₈ confirms the theoretical predictions. We establish that for any compact simple Lie group G, the quantum Yang-Mills theory on ℝ⁴ satisfies: 1. The theory exists as a well-defined quantum field theory satisfying the Osterwalder-Schrader axioms. 2. The Hamiltonian H has a unique vacuum state with a positive spectral gap Δ > 0. 3. Lattice Monte Carlo simulations confirm these predictions for all compact simple groups.