An explicit area law for SU(2) lattice Yang-Mills theory up to beta_W = 2/(d-1), and a machine-checked area law for SU(N)

For SU(2) lattice Yang-Mills theory with the Wilson action in d >= 2 dimensions we prove Wilson's area law for every Gibbs measure and every torus measure at |beta_W| < 2/(d-1) (2/3 in d = 4, 1 in d = 3), with string tension at least arccosh(2/|beta_W| - (d-2)) and exponential decay of Polyakov loop correlators at every temporal extent. In d = 4 the range is four times that of Cao, Nissim and Sheffield at N = 2 and passes the deconfinement points of Z_2 gauge theory. The proof combines the slab decomposition of Durhuus and Frohlich with a sharp Wasserstein bound for von Mises-Fisher laws on S^3. Second, for every gauge group with a nontrivial central character, in particular SU(N) for all N >= 2, a short positive-expansion and centre-twist argument gives the area law at 4(d-1)|beta_W| < 1; this theorem is machine-checked in Lean 4 with Mathlib using only the standard axioms (github.com/bharathsrivats/fisher-lorentz-formalization, tag v2.0.0). All statements are at fixed lattice spacing. Nothing here concerns the continuum limit or the Clay Millennium problem. The deposit contains the note (PDF) and its LaTeX source with all numerical checks.