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

We construct a four-dimensional Euclidean Yang–Mills theory in background gaugesand derive a strictly positive Hamiltonian mass gap through coercivity, step scaling,thermodynamic control, and exponential clustering. The construction starts from arenormalized DeTurck regularization on finite four-tori and a local cylinder algebra ofgauge-invariant observables. Quantitative lower bounds for the mixed Faddeev–Popovoperator on quantitative Gribov regions control the background margin and providethe slice estimates used in the determinant block. A multiscale step-scaling schemepropagates an ultraviolet Poincar´e seed inequality to a fixed physical scale. A boundedoverlapargument on the limiting local core yields an L-independent Poincar´e constantfor uniformly localized observables; this local spectral input gives exponential Euclideanclustering. Osterwalder–Schrader reconstruction then gives a Hamiltonian with a strictlypositive spectral gap above the vacuum.Canonicality and nontriviality are proved by finite-dimensional estimates. The determinantcontribution is organized by a quasi-local primitive polymer expansion withexponential diameter decay and connected one-step bounds. The coupling window iscontrolled in a normal-form coordinate by finite shell arithmetic and an explicit sparsecofinal schedule. Nontriviality is proved by a concrete gauge-invariant plaquette observablewhose Wick-free fourth interaction cumulant survives in the canonical limit. Backgroundand regulator universality are established on gauge-invariant local observables. The finalshort-distance package includes the local curvature-field algebra, asymptotic freedom,finite-order operator-product asymptotics, and the renormalized stress tensor.