Mixing of boundary Langevin dynamics for regulated four-dimensional Yang–Mills slabs

We study regulated four-dimensional Yang–Mills theory on Euclidean slabs S_{t,L} := [0,t] x T^3_L at fixed thickness t>0. A central theme is kernel separation: quantitative mixing of an auxiliary boundary sampler does not, by itself, imply mixing for the Euclidean transfer kernel K_{t,Reg} obtained by disintegration of the slab endpoint law and governing Euclidean-time concatenation and transfer-operator statements. For a finite-range Wilson lattice regulator family with compact gauge group G, we prove uniform quantitative mixing properties for K_{t,Reg} under ultraviolet refinement at fixed (t,L). First, we establish a fixed-window Doeblin minorisation for a projected transfer kernel, yielding geometric contraction for cylindrical observables. Second, in a high-temperature (KP) corridor we derive a Wilson-intrinsic cross-slab maximal-correlation bound via polymer crossing estimates, implying L^2 contraction and a spectral gap for the transfer operator. As a consequence we obtain exponential Euclidean-time clustering and a time-axis transfer-operator gap for bounded gauge-invariant cylinder slab observables supported away from the boundaries. Moreover, within an L-uniform KP corridor we construct the spatial thermodynamic limit L -> infinity for the Wilson slab family and show that the Euclidean-time clustering conclusions persist for local observables. Separately, we record an abstract template framework for Gaussian-reference regulators and boundary Langevin mixing under explicit stability/locality/moment assumptions. No continuum limit on R^4 and no long-time limit t -> infinity is constructed here.