Open problemChecked July 24, 2026

Yang-Mills existence and mass gap

A rigorous construction problem at the foundations of quantum field theory.

Mathematical physics Quantum field theory Analysis

Mathematical statement

For every compact simple gauge group $G$ , construct a nontrivial quantum Yang-Mills theory on $\mathbb{R}^4$ whose spectrum has a positive mass gap $\Delta>0$ .

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

MillenniumYangMills.clay_prize_yang_mills

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem clay_prize_yang_mills :    ClayYangMills := by  sorry

Statement source: Lean Millennium Prize Problems
Lean version: v4.31.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Clay Mathematics Institute problem pageofficial Yang-Mills existence and mass gap overviewbackground