A Constructive Five–Dimensional Gradient–Flow Proof of the SU(3) Yang–Mills Mass Gap

In this work, we present a constructive proof that pure \(\mathrm{SU}(3)\) Yang–Mills theory on \(\mathbb{R}^4\) exists as a nontrivial Wightman quantum field theory and exhibits a strictly positive mass gap.  Our approach embeds the four-dimensional gauge theory in a five-dimensional gradient-flow slab that preserves gauge invariance and reflection positivity, and supplies a geometric smoothing parameter that replaces weak coupling.  Finite-range decomposition, a \(\beta\)-independent polymer/cluster expansion, a reflection-positive functional renormalisation-group flow, and a five-dimensional transfer-matrix gap combine to yield the bound \(\operatorname{Spec}H=\{0\}\cup[m_0,\infty)\) with \(m_0>0\).  All constructive constants are shown to be independent of both the lattice spacing and the bare coupling once a universal small-field threshold is fixed, leaving no remaining gap between heuristic physics and mathematical proof.