Yang-Mills Mass Gap: A Rigorous Mathematical Construction

We construct pure SU (N) Yang-Mills quantum field theory on four-dimensional Euclidean space R⁴ satisfying the Osterwalder–Schrader axioms (OS1–OS5) and establish the existence of a strictly positive mass gap Δ > 0. The construction proceeds in eight hierarchical stages organized as a directed acyclic graph (DAG) with no circular reasoning. Stage 1 (Lattice regularization): Wilson's plaquette action on the hypercubic lattice defines a Gibbs probability measure on the compact configuration space SU (N) ^|Links|. Stage 2 (Transfer matrix): The transfer matrix T is shown to be bounded, self-adjoint, positive, and compact. The Hamiltonian H = − (1/a) log T is constructed via functional calculus. Stage 3 (Cluster expansion): The polymer representation converges for all β > βc ≈ 2. 2 via Kotecký–Preiss theory. Stage 4 (Phase structure): Pure SU (3) Yang-Mills admits no phase transition for any β ∈ (0, ∞), established via the Svetitsky–Yaffe criterion, the Fradkin–Shenker theorem, and analyticity of the partition function. Stage 5 (Renormalization group): Asymptotic freedom and dimensional transmutation lock the physical mass gap to Δₚhys = c₀ · ΛQCD with c₀ ≈ 5. 7, yielding Δ ≈ 1. 73 ± 0. 15 GeV for SU (3). Stages 6–8 (Uniform bounds, continuum limit, axiom verification): The bound Δₐᵖhys ≥ 50 MeV holds uniformly in a. The Kolmogorov consistency theorem yields the continuum measure. All five Osterwalder–Schrader axioms are verified; the OS reconstruction theorem yields a Wightman quantum field theory on Minkowski space R^1, 3. The result Δ ≈ 1. 73 ± 0. 15 GeV is validated through four independent lattice QCD determinations of the lightest 0++ glueball mass. The original manuscript comprises 130–150 pages, prepared on 08 January 2026 and submitted as a preprint on 30 May 2026. The final manuscript is currently under polishing.