Mass Gap of Lattice Yang–Mills Theory up to the Dobrushin Ceiling

We prove that the Hessian of the Wilson action on $d$-dimensional lattice gauge fields is bounded by $4d$, sharply, so lattice \SU(N) Yang–Mills theory has a unique infinite-volume state with a positive mass gap for $|\beta|<1/(8d)$. Every von Mises–Fisher law on the three-sphere moves by at most a quarter of its parameter in Kantorovich distance; consequently four-dimensional \SU(2) theory has a mass gap for $\beta_W<2/9$, the ceiling of all single-link Dobrushin arguments. A gauge-invariant finite-box criterion for decay and the area law is also given. The continuum mass gap remains open.