The Scretching–Planck Formulation of Four-Dimensional SU(2) Gauge Theory in the Continuum

This study uses the Scretching–Planck (SP) vacuum functional \(Ψ_0=√P_0\) to express the lattice Yang–Mills Hamiltonian above its vacuum energy as a Dirichlet form. The resulting identity, \(Δ=(g^2/2a)λ(P_0)\), relates the mass gap to the optimal Poincaré constant of the vacuum density and makes its nonnegativity explicit. The paper corrects seven errors in the earlier argument and formulates the SP equations for the \(3+1\)-dimensional continuum theory. Monte Carlo estimates for four-dimensional SU(2) lattice gauge theory at \(β=2.3,\,2.4,\,2.5\) give positive gaps at all three lattice spacings. A linear extrapolation in \(a^2σ\) yields \(Δ/√σ=3.96(27)\), equivalent to \(1.74±0.12\) GeV under the conventional scale choice \(√σ=440\) MeV. The extrapolation provides numerical evidence for a positive continuum gap. The uniform Poincaré bound in physical units needed for a proof remains open as Conjecture C.