A Renormalized DeTurck Construction of Four-Dimensional Yang–Mills Theory with Mass Gap

We construct a four-dimensional Euclidean Yang–Mills theory in background gaugesand derive a positive Hamiltonian mass gap by an analytic route through coercivity,step scaling, and exponential clustering. The starting point is a renormalized DeTurckregularization on finite four-tori, together with a local cylinder algebra of gauge-invariantobservables. Quantitative lower bounds for the Faddeev–Popov operator on quantitativeGribov regions yield control of the Gribov margin. A multiscale step scalingscheme propagates an ultraviolet Poincare seed inequality to large scales withregulator-stable constants and produces a uniform spectral gap for the limiting generator.This spectral gap implies exponential clustering of Euclidean correlations.Osterwalder–Schrader reconstruction then gives a self-adjoint Hamiltonian with astrictly positive spectral gap. Nontriviality and universality with respect to the backgroundconnection and the ultraviolet regulator are proved within the same framework.