We prove that four-dimensional SU (N) Yang–Mills theory, N ≥ 2, has a mass gap Δ > 0. The proof proceeds in four stages. At strong coupling, a center-vortex percolation seed is established via a local center-event mixing argument (MIX route), with an explicit finite-energy lower bound that renders the Liggett–Schonmann–Stacey domination hypothesis checkable. Supercritical percolation is converted to an area law for Wilson loops by a coarse-plaquette dephasing argument. The lattice theory is transported to a continuum Euclidean Schwinger family via a Balaban renormalization group sequence, with a certified suppression margin ΩG ≥ 2. 01 × 10⁻⁴ controlling the KP mixing entry. Osterwalder–Schrader reconstruction applied to the limit theory yields a Hilbert space with a self-adjoint Hamiltonian satisfying H ≥ m_ξ/ξ > 0. All principal steps are proved in this paper; the five imported facts are standard and explicitly stated in Appendix B.
Zoltan B Feher (2026) studied this question.