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.