Local, Volume-Uniform Covariance Coercivity for Plaquette Sources in Finite SU(2) Wilson Gauge Theory

We establish explicit, volume-uniform lower bounds for the covariance matrix of all plaquette observables in finite SU(2) Wilson lattice gauge theory with independently varying, signed, and spatially inhomogeneous couplings. For periodic four-dimensional hypercubic lattices with side lengths at least four, we prove a positive diagonal lower bound for the full coupling-response Hessian of the logarithmic partition function. The coefficient associated with each plaquette depends only on the absolute couplings of plaquettes sharing its boundary links and is expressed through scalar covariance functions involving modified Bessel functions. In particular, when the absolute values of all plaquette couplings are bounded by M, the covariance Hessian satisfies H(λ) ⪰ [1800(1 + 6M)³]⁻¹ I, with an improved coefficient [1200(1 + 6M)³]⁻¹ when all lattice periods are even. We also obtain stronger quadratic-coupling lower bounds for separated plaquette sources, quantitative response and relative-entropy inequalities for distinct coupling profiles, and extensions to freely integrated finite plaquette subcomplexes, general dimensions, and certain short-period lattices. The proof combines exact tilted SU(2) Haar covariance on the three-sphere, two successive conditional-variance estimates, opposite-link incidence geometry, and finite link coloring. No spatial mixing assumption or gauge fixing is required. These results concern finite-volume static susceptibilities and quantitative convexity of the coupling-response functional. They do not establish a continuum Yang–Mills construction, exponential clustering, or a positive Yang–Mills mass gap. Version: v1.1r2 (1 November 2026). Independent research preprint; not peer reviewed. This revision updates the manuscript's presentation and formatting without changing its mathematical results.