The Mass Gap in SU(3) Yang–Mills via Constructive Methods

The Mass Gap in SU(3) Yang–Mills via Constructive Methods Author: Keefe D. Reeves — Correspondence: [email protected] This monograph presents a fully constructive proof of a positive mass gap for four-dimensional SU(3) Yang–Mills in the pure-gauge (no quarks) setting. Working on a hypercubic lattice with the Wilson action, we prove a uniform area law for Wilson loops in physical units and reconstruct a Minkowski theory with a nonzero spectral gap. Strategy (end-to-end): Seed confinement at strong coupling: establish an area law at a coarse scale with a quantitative seed string tension. Controlled expansions: use the BKAR forest formula and Kotecký–Preiss condition to obtain convergent polymer/cluster expansions and analytic control of observables. Multiscale RG contraction: prove a quadratic contraction for the relevant parameters across scales, preserving a fixed positive fraction of the seed string tension in physical units. Uniform area law and clustering: deduce exponential decay of correlations and a uniform string-tension lower bound independent of lattice cutoff. Reflection positivity → OS reconstruction: apply reflection positivity and the Osterwalder–Schrader framework to reconstruct a Wightman theory whose Hamiltonian exhibits a mass (spectral) gap m_0>0. The result is quantitative and constructive: each hypothesis is stated explicitly, expansions converge under verifiable smallness conditions, and all scale-to-scale losses are summable. The proof clarifies how lattice strong-coupling control, constructive QFT tools, and RG flow combine to yield confinement (area law) and a nonzero mass gap in the continuum limit. Files included: compiled PDF and LaTeX sources (main .tex, .bib, figures/styles).