# P36: Yang-Mills existence and mass gap

- ID: `P36`
- Reference: `yang-mills-existence-and-mass-gap`
- Page: https://theoremdb.org/statements/P36
- Record maturity: Reviewed problem with recorded work

## 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 \(\Delta>0\).

### Context

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

### Problem setup

- **Definition (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.
- **Definition (The required axioms, nontriviality condition, and relationship to classical Yang-Mills theory are those in the official problem formulation).** The required axioms, nontriviality condition, and relationship to classical Yang-Mills theory are those in the official problem formulation.
- **Remark.** The problem asks for a rigorous mathematical construction underlying a central quantum field theory and for a proof of its spectral gap.

### What 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.

## Status

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](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: 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.
- Status and formulation were checked against the Clay Mathematics Institute page on 2026-07-22.
- Consult the cited official problem description for the exact axioms, admissible gauge groups, and meaning of the mass gap.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `yang-mills-existence-and-mass-gap`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://www.claymath.org/millennium/yang-mills-the-maths-gap/
   - Also cited at Unsolved status and overview
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>Arthur Jaffe and Edward Witten, Quantum Yang-Mills Theory, official Clay Mathematics Institute problem description. claymath.org checked 2026-08-01. Official problem statement https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf
   - website; primary source; checked 2026-08-01
   - Source use: original_summary
   - Defines the exact acceptance condition.