The Scretching–Planck Vacuum Density and the Yang–Mills Mass Gap Part 2

This paper uses Scretching–Planck density–phase variables to express the finite-lattice \(SU(2)\) Kogut–Susskind gap as \(\Delta_{\mathrm{KS}}=(g^2/8a)\lambda_{\mathrm{unit}}(P_0)\). Here \(P_0\) is the vacuum density, and \(\lambda_{\mathrm{unit}}(P_0)\) is the infimum of the weighted Poincaré quotient over nonconstant, gauge-invariant test functions in the product unit-sphere metric. Double-well and one-plaquette calculations check the ground-state transformation. For the four-dimensional \(SU(2)\) Euclidean Wilson measure, a plaquette Hessian bound gives \(\operatorname{Ric}+\operatorname{Hess}U\ge(2-24\beta)I\). For \(\beta<1/12\), this yields volume-uniform Poincaré and log-Sobolev inequalities that pass to the transfer-matrix vacuum slice. A conditional Otto–Reznikoff argument extends the range to \(\beta<0.0908\). Explicit configurations rule out a globally positive curvature bound beyond \(\beta=1/3\) analytically, and numerical calculations locate an obstruction near \(\beta=0.233\). A Gaussian two-scale example shows how the bound deteriorates as the mass tends to zero. Lattice calculations estimate scalar-channel masses in pure \(SU(2)\) and \(SU(3)\), including an exploratory \(SU(2)\) extrapolation of \(m_{0^{++}}/\sqrt{\sigma}=3.96(27)\). A separate \(SU(2)\)–Higgs scan reports \(W\)-channel effective masses, and a finite-volume large-field scan finds a threshold near \(\varepsilon_*\approx0.48g^2\). The uniform gap bound in physical units (Conjecture C), the Hamiltonian limit, and the four-dimensional continuum construction remain open.