[#P36] Yang-Mills existence and mass gap

Problem. For every compact simple gauge group \(G\), construct a nontrivial quantum Yang-Mills theory on \(\mathbb{R}^4\) satisfying the required quantum field theory axioms and prove that its mass gap \(\Delta\) satisfies \(\Delta>0\).

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Definitions and notation

1Context

The problem asks for a rigorous mathematical construction underlying a central quantum field theory and for a proof of its spectral gap.

2Problem setup

Definition 1 (A mass gap). A mass gap means there is a positive lower bound between the vacuum energy and the energies of all non-vacuum states in the reconstructed quantum theory.

Remark 1. The required axioms, nontriviality condition, and relationship to classical Yang-Mills theory are those in the official problem formulation.

3What counts as a solution

Construct the quantum Yang-Mills theory for every compact simple G under the official axioms and prove a uniform positive mass gap in the required sense.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: No rigorous construction meeting the official four-dimensional axioms and positive-gap requirement is known. Experimental and lattice evidence does not satisfy the theorem. Exact unresolved remainder: Construct the theory for every compact simple G under the official axioms and prove the required positive mass gap.^[1][2]

1Packet records

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: No rigorous construction meeting the official four-dimensional axioms and positive-gap requirement is known. Experimental and lattice evidence does not satisfy the theorem. Exact unresolved remainder: Construct the theory for every compact simple G under the official axioms and prove the required positive mass gap.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: No rigorous construction meeting the official four-dimensional axioms and positive-gap requirement is known. Experimental and lattice evidence does not satisfy the theorem. Exact unresolved remainder: Construct the theory for every compact simple G under the official axioms and prove the required positive mass gap.^[1][2] Relevance to this problem For Yang-Mills existence and mass gap, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: No rigorous construction meeting the official four-dimensional axioms and positive-gap requirement is known. Experimental and lattice evidence does not satisfy the theorem. Exact unresolved remainder: Construct the theory for every compact simple G under the official axioms and prove the required positive mass gap. Open full record →
Construct the quantum Yang-Mills theory for every compact simple G under the official axioms and prove a uniform positive mass gap in the required sense.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Construct the quantum Yang-Mills theory for every compact simple G under the official axioms and prove a uniform positive mass gap in the required sense.^[1] Relevance to this problem For Yang-Mills existence and mass gap, record yang-mills-existence-and-mass-gap-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Construct the quantum Yang-Mills theory for every compact simple G under the official axioms and prove a uniform positive mass gap in the required sense. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Construct the quantum Yang-Mills theory for every compact simple G under the official axioms and prove a uniform positive mass gap in the required sense. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited authoritative source listed this problem as unsolved when checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Consult the cited official problem description for the exact axioms, admissible gauge groups, and meaning of the mass gap.

2See also

Hard-core coefficient log-concavity on the first ten thousand four-cycle stripsmathematical physics
Exact heat-bath spectral gap on the six by six Ising torusmathematical physics
Most antiferromagnetic ground states in a 3-connected cubic graph on twenty verticesmathematical physics

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Plain text

“Yang-Mills existence and mass gap.” TheoremDB. P36. Problem statement; statement text SHA-256 51a9fc20ef43cc26f8a3707dbbc97f29f3aa14b0e02b785b4c55318b481787c9. https://theoremdb.org/statement/?ref=P36

BibTeX

@misc{theoremdb-problem-51a9fc20ef43cc26f8a3707dbbc97f29f3aa14b0e02b785b4c55318b481787c9,
  title = {{Yang-Mills existence and mass gap}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 51a9fc20ef43cc26f8a3707dbbc97f29f3aa14b0e02b785b4c55318b481787c9},
  url = {https://theoremdb.org/statement/?ref=P36}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from claymath.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Clay Mathematics Institute, Yang-Mills & the Mass Gap. claymath.org checked 2026-08-01. Official Problem Description by Arthur Jaffe and Edward Witten; listed under Unsolved Millennium Prize Problems. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Unsolved status and overview.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Authoritative current status.Source named by the research packet.
Arthur Jaffe and Edward Witten, Quantum Yang-Mills Theory, official Clay Mathematics Institute problem description. claymath.org checked 2026-08-01. Official problem statement. ↗website · primary source · checked 2026-08-01Source use: original summary.Defines the exact acceptance condition.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P36] Yang-Mills existence and mass gap

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