Wilson Yang–Mills Theory Complete Spectral Certificates and Boundary Reductions Finite-volume results, uniform-law estimates, and a conditional continuum gap criterion

We consolidate complete spectral certificates and normalized boundary reductions for Wil- son lattice Yang–Mills theory. At β = 28/25, the complete SU(2) one-cube and adjoining two-cube eigenratios are bounded above by 0.005066 and 0.023164. The accepted Y2 input chain supports mw ≥ 5.245 on its auxiliary ladder family; these results do not extend by them- selves to general spatial volume. Complete finite conditional gates and their failed enlargement tests are retained. A direct physical-transfer argument gives the uniform fixed-spacing ratio 18 tanh β <1. A separately reviewed signed-gauge packet records the stronger SU(2) window β ≤ 3/8 on even periods at least six. For a diverging-coupling route, exact integration leaves one boundary law with all cell partition functions and the sufficient target 10B− (I− PC ) ≽ 0. Path, harmonic and cavity estimates control complete subspaces, including a retained 9/29 fraction of the matching positive seam energy. Pressure-selected coupling sequences control linear curvature packets. New fixed-degree signed relative-loop differences control a family on every tile, with complete mixed return bounds. A weaker raw-loop covariance gate and a finite replica pilot identify the undifferenced exterior response. Exact conditional detour integration controls inner integration error; the actual collective covariance and nonlinear complement remain open. The manuscript supplies the physical-time and dense-state implications needed after a uniform bound, while preserving normalization, scalar states, tails and adverse controls. The complete remaining boundary inequality, a compatible nontrivial continuum construction and its physical mass gap are not proved.