An explicit strong-coupling gap for the four-dimensional Wilson transfer Proof and a conditional route to the continuum Yang–Mills gap

This proof note establishes a volume-uniform spectral gap for the gauge-invariant Wilson transfer matrix in four-dimensional lattice Yang–Mills theory at strong coupling. On periodic three-dimensional spatial tori at fixed lattice spacing, the ratio of the largest excited eigenvalue to the leading eigenvalue is at most 18 tanh(β) when 18 tanh(β) is below one. The proof uses the original Wilson Gibbs law and its exact normalization. A projection-angle inequality gives a route to a one-cube block Poincaré bound. For growing SU(2) tori, the straight-path decomposition isolates a conditional variance estimate under the full Wilson law. A periodic transfer-trace inequality shows how a long-time bound would pass to fixed physical times in a compatible continuum limit. The uniform conditional variance estimate and the nontrivial continuum construction remain open.