{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"30f09e4774ef3aa1177b799294a2b0b5b59b6794696231894e74ff42282185ff","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["4910c35b91df40c4b4ae6c3b2c07380b680f81b93f76b71f30889c5407bdc3c1","5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"77ca780e3272fa95b1b9eb1718ffb9676eee40b5d615d0f03d086aa720703a74","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.yang-mills.mass-gap","kind":"well-posed","literature_status":"open","n":"1","parents":[{"note":"","parent_revision":"4910c35b91df40c4b4ae6c3b2c07380b680f81b93f76b71f30889c5407bdc3c1","relation":"special_case"}],"plain":"Prove that the quantum theory of gluons exists as a well-defined mathematical object and that its lightest particle has a strictly positive mass. 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 $\\{0\\}\\operatorname{union}[\\Delta, \\infty)$ for some $\\Delta > 0$. Answer: a proof of both statements (Jaffe-Witten formulation for the Clay Mathematics Institute).","problem_ref":null,"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 theory","topic_ref":"c1a7c5336cf6e262c1ec8a96010ceec0183209142167af7077f763d0eba2c1d0"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"75b7b83537c0f3d60e2005415e11f4bb54b1f3dceca46a614d549cae347dca99","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"3b144ea531f2d2e86c1c0f0094693b1ee226a74e3fe8d7bf356e381a45f402d5","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"OTAPOHL1cvswO9FrZQyHPPZnVsT1U5nPQP1fprLz9SkLZxIZ7pHGWFlpJvvGgfi0MidsnrzEPYHW-8ITZs3LCg"},"schema":"pubphys.envelope/1"},"record_hash":"75b7b83537c0f3d60e2005415e11f4bb54b1f3dceca46a614d549cae347dca99","leaf_index":1662}