The Scretching–Planck Vacuum Density and the Yang–Mills Mass Gap

This paper expresses the finite-lattice Kogut–Susskind spectral gap in terms of the Scretching–Planck (SP) vacuum density. In the unit-sphere metric, the ground-state transformation gives \(\Delta=(g^2/8a)\lambda_{\mathrm{unit}}(P_0)\), where \(\lambda_{\mathrm{unit}}(P_0)\) is the optimal Poincaré spectral constant. The paper checks the identity in a double-well model and an exactly solvable one-plaquette \(SU(2)\) model, with relative agreement of \(3\times10^{-5}\) in the latter. For the four-dimensional \(SU(2)\) Wilson measure, a plaquette Hessian estimate gives the curvature bound \(\operatorname{Ric}+\operatorname{Hess}U\ge(2-24\beta)I\). This yields volume-uniform Poincaré and log-Sobolev bounds for \(\beta<1/12\) that pass to the transfer-matrix vacuum slice. The range extends to \(\beta<0.0908\) under the stated Otto–Reznikoff assumption. Explicit configurations rule out a uniformly positive single-scale curvature bound beyond \(\beta=1/3\), and a separate numerical calculation places the obstruction near \(\beta=0.238\). An iterated two-scale estimate for Gaussian fields shows how powers of the correlation length are lost unless the free flow is treated exactly. Lattice calculations report positive scalar-channel gaps in \(SU(2)\), \(SU(3)\), and \(SU(2)\)–Higgs theory. The \(SU(2)\) data are consistent with two-loop scaling and give a continuum extrapolation of \(\Delta/\sqrt{\sigma}=3.96(27)\). In the sampled ensembles, large-field regions cease to percolate near \(\varepsilon_*\approx0.48g^2\). The paper states the remaining multiscale step as Conjecture C, a uniform lower bound in physical units across the crossover. The lattice results provide numerical evidence for a positive continuum gap, and the rigorous four-dimensional Yang–Mills construction and gap proof remain open.