Mass Gap of Four-Dimensional Lattice Yang–Mills Theory for Every Compact Simple Group

We prove that four-dimensional lattice Yang–Mills theory with any compact simple gauge group has a unique infinite-volume state, exponential clustering and an explicit transfer-matrix gap at couplings past the curvature limit of the ungauged measure, up to $\beta_W=0.345$ for \SU(2) and $\beta_W=0.76$ for \SU(3), in a gauge that fixes one link of every plaquette. We prove that a Dobrushin–Shlosman estimate in one cube of fixed physical size, or any exponential decay of renormalised correlations in physical units, gives a mass gap for every continuum limit. The cube estimate is open along the continuum tuning; for \SU(N) it needs cubes of side at least of order $(\beta_W/\log\beta_W)^{1/4}$. For block averages, a cut-off of Bałaban's average composes across all scales on fields of small curvature for every compact group, and under the \SU(2) Wilson measure the large fields form clusters of vanishing physical size.