Wilson–Haar Renormalization and a Positive Mass Gap for Four-Dimensional SU(3) Yang–Mills Theory

This preprint develops an end-to-end constructive framework for pure four-dimensional SU(3) Yang–Mills theory, starting from exact finite-volume Wilson lattice measures and culminating in a claimed positive spectral gap for the reconstructed physical Hamiltonian. Its principal contribution is the integration, within a single regulator-uniform argument, of exact Wilson–Haar disintegration, complete local coefficient control, absolutely convergent BKAR/Kotecký–Preiss polymer expansions, a physical Wilson renormalization-group map, fixed-shell two-loop matching and dimensional transmutation, transported gauge-invariant sources and contact terms, reflection positivity, complete physical clustering, and Osterwalder–Schrader reconstruction. A central technical contribution is an owner-anchored aggregation argument that upgrades bounds for individual connected polymers to the complete marginal, relevant, and irrelevant renormalization-group coordinates, while controlling position sums, connected-support entropy, source marks, combinatorial multiplicities, and finite-volume normalization uniformly. Within the Wilson–Haar/BKAR framework developed in the manuscript, the work constructs a nontrivial continuum Schwinger family and presents a claimed strictly positive spectral threshold above the vacuum. The result is specific to SU(3). This deposit is an author preprint presenting a complete claimed proof within its stated framework; independent line-by-line specialist verification and ordinary peer review remain pending.