Uniqueness and universality of the continuum limit in 4D SU(N) Yang-Mills: Part (4)

Abstract

A sound theory must not depend on the scaffolding by which we reach it; only the invariant content is real. Under standard constructive hypotheses-reflection positivity, locality, clustering, and spectral regularity-we show that four-dimensional SU(N) Yang-Mills has a Euclidean continuum limit that is both unique and universal within a natural class of regulators. Within the Osterwalder-Schrader scheme, an explicit disintegration of a single time slab yields the one-step transfer kernel which, together with a common one-slice marginal, fixes all Schwinger functions by time-slicing and positivity. The limit is independent of the regulating lens: for gauge-covariant, reflection-symmetric schemes built from completely monotone spectral projectors and finite-range decomposition (FRD) blockings, single-scale Lipschitz control, telescoping in Euclidean time, and BKAR polymer bounds transmit stability to connected cumulants and hence to the continuum. A measurable, reflection-covariant Landau selector keeps the slice construction compatible with positivity. The bridge to weak coupling is modest and precise: a one-dimensional implicit-function/continuity tuning brings the flow into a contracting domain of the FRD map; along this trajectory the renormalized coupling diminishes-an operational sign of asymptotic freedom. No step relies on perturbation theory; a one-loop check is recorded only as a signpost, and all estimates are uniform in volume.