A Constructive Proof of Existence and Mass Gap for Pure SU(3) Yang-Mills in Four-Dimensional Space-Time

In this work we present a constructive proof that pure SU (3) Yang-Mills theory on R⁴ exists as a nontrivial Wightman quantum field theory and exhibits a strictly positive mass gap. Our approach embeds the four-dimensional gauge theory as the zero-mode sector of a five-dimensional orbifold regulator that preserves gauge invariance and reflection positivity. A convergent joint polymer expansion on the Wilson lattice provides uniform control of the continuum (a->0) and infinite-volume (L->inf) limits; an Osterwalder-Schrader reconstruction yields a unique-vacuum Wightman theory; a nonperturbative BRST/Nielsen argument ensures gauge-parameter independence; a rigorously controlled operator-product expansion matches one-loop beta-function data; and a Sturm-Liouville analysis of five-dimensional fluctuations, combined with transfer-matrix spectral projections, isolates a strictly positive glueball mass m0. All steps rest on explicit epsilon-delta estimates and combinatorial bounds for SU (3), leaving no remaining gap between heuristic physics and mathematical proof.