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This is the Clay Millennium problem on Yang-Mills theory.","posed_since":"2000","precise":"For every compact simple gauge group $G$ ($SU(2)$ is the minimal case), construct a quantum Yang-Mills theory on $R^4$ satisfying the Osterwalder-Schrader or Wightman axioms, and prove a mass gap: the Hamiltonian $H$ has spectrum contained in $\\{0\\}\\cup[\\Delta, \\infty)$ for some $\\Delta > 0$. Answer: a proof of both statements (Jaffe-Witten formulation for the Clay Mathematics Institute).","problem_ref":"75b7b83537c0f3d60e2005415e11f4bb54b1f3dceca46a614d549cae347dca99","references":"","settled_by":"An accepted proof of existence and $\\Delta > 0$ for every compact simple $G$, as the Clay statement requires; a proof for $\\mathrm{SU}(2)$ alone would be a major partial result.","status_note":"No accepted proof as of 2026; several claimed proofs posted on arXiv have not been accepted.","title":"Existence and mass gap of four-dimensional quantum Yang-Mills 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