This repository collects pure Yang–Mills submissions, the group's independent checks, and its assessments. The first package concerns OpenAI's submission of 28 October 2026. The assessment is dated 10 November 2026. In our opinion, this submission proves that a relativistic quantum field theory satisfying the Wightman axioms can be constructed as a scaling limit of four-dimensional Wilson lattice gauge theory for any simple compact gauge group. It does not prove that the infinite-volume limit is unique or that there is a mass gap.
Each submission has a complete package containing the original source, our checks, reproduction instructions, and dated evidence. This lets readers inspect a submission on its own and compare it with later submissions. The original material and the group's additions remain distinct.
| Submission | Assessment | Formal checks and evidence |
|---|---|---|
| OpenAI, 28 October 2026 | Assessment 1 (PDF) | Check scope ·dated results ·rerun instructions |
| OpenAI, 4 November 2026 | work in progress | — |
The October package includes the original Lean sources, the combined acceptance theorem, bridges to previous relevant formalizations, and a separate OPE check. Its historical evidence records what was built and which Comparator checks were run, including their limitations.
The commented Lean examples provide a reading route through the assessment’s A1–A8 checks: lattice law and scale, the construction chain, operator content, OS axioms, translation Ward identity, the leading trace law, and OPE bounds. Each places explanations beside compilable statements and identifies the scope of the result. The documentation index links the assessment to the specifications, proofs and verification records.
docs/ Assessments, reading guides and indexes
assessment-1.pdf Stable location of the first assessment
submissions/
openai-20261028/
submission/ Original, unchanged submitted source
checks/ The group's specifications, adapters and proofs
verification/ Comparator references, configurations and tool pins
scripts/ Reproduction runner
evidence/ Dated build and verification records
provenance/ Import hashes and upstream correspondences
docs/ This submission's explanations and examples
New submissions receive their own directories using an author and date, as described in the package convention. Each package retains its own pinned tools and checks. Assessment PDFs have stable top-level paths such as docs/assessment-1.pdf. Submission-specific explanations and examples live inside their packages; indexes link to those documents. Comparisons spanning submissions will also live in docs/.
The separate original-submission repository preserves the October distribution. This results repository contains its own full copy of that source and the inputs for the group's checks, so reproduction does not require access to that other repository. Lean, Mathlib and Comparator are pinned external dependencies.
To check the October package's integrity from a fresh clone:
cd submissions/openai-20261028
python3 scripts/verify.py integrity
For the full build and Comparator replay, read the prerequisites and reproduction instructions, then run python3 scripts/verify.py all from the same directory. The recorded macOS verification mode does not provide Linux/Landrun isolation. Generated files stay within the package and are excluded from version control.
To propose a submission, email yangmills4d@gmail.com with a short summary and a link to a fixed revision or dated copy of the materials. Identify the authors and the pure Yang–Mills results you claim. Proposals concerning QCD or chiral SU(5) should identify that category so they can be considered for the corresponding results repository. Include an explanatory document and any formal source code, together with the license or terms under which the group may host the materials.
Submissions must meet a minimum standard of relevance, clarity, and mathematical substance. State the problem, definitions, assumptions, and claimed results precisely enough to assess them, and provide substantive arguments supporting those claims. Distinguish formalized results from informal arguments and identify known gaps or unproved steps. For formal results, include the toolchain and dependency versions, instructions for reproducing the checks, and a description of any additional axioms or proof placeholders.
The group decides which submissions meet this standard and which to include. The group does not promise to post or review submissions, or to reply to submitters, even when these requirements are met. Posting a submission does not constitute endorsement or certification of its claims.
Code and documentation are licensed under Apache-2.0, subject to the preserved upstream notices. See NOTICE and the notices within each submission package.