Reflection Positivity and a Finite- a Strong-Coupling Gap in Lattice SU( N ) Yang-Mills: Part(1)

For Formula: see text with Formula: see text, we develop a reflection-positive transfer-matrix framework for four-dimensional lattice Yang-Mills which, on a nontrivial strong-coupling window Formula: see text, yields a strictly positive spectral gap at fixed lattice spacing Formula: see text, with bounds uniform in the spatial volume. The construction is compatible with OS reflection: on each Euclidean time slice we select a gauge-invariant transverse representative Formula: see text by Landau functional minimization within the fundamental modular region, and we insert a smooth “horizon” spectral projector as a slice-local positive weight that preserves reflection positivity. In the same regime Formula: see text, a Kotecký-Preiss cluster expansion reorganizes the partition function and gauge-invariant correlators; it converges uniformly in the volume and implies exponential clustering for connected gauge-invariant observables with a decay rate bounded away from zero uniformly in the volume. OS reconstruction then promotes clustering to a nonzero lower bound for the spectral gap of the positive, self-adjoint transfer operator Formula: see text (equivalently, of the transfer Hamiltonian Formula: see text) at fixed Formula: see text. We also establish a Wilson-loop area law throughout this window. The conclusions are stable under admissible variations of the slice-wise selector and of the smooth projector profile, and they quantify the existence of a finite-Formula: see text mass gap for Formula: see text Yang-Mills at strong coupling.