Yang–Mills Existence and the Mass Gap

Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang–Mills theory exists on $\mathbb{R}^4$ and has a mass gap $\Delta > 0$.

The classical Yang–Mills action
$$S[A] = \frac{1}{4}\int d^4x\, F_{\mu\nu}^a F^{a\,\mu\nu}$$
describes massless gauge bosons, yet the quantum theory is expected to confine and produce only massive excitations. A rigorous construction satisfying the Wightman axioms, together with a proof that
$$\inf \big(\operatorname{spec}(H) \setminus \{0\}\big) = \Delta > 0,$$
remains open. This is one of the Clay Millennium Prize Problems.