OpenAI Yang-Mills Lean Proof: Gross Group Validation and Public Release

On November 9, 2026, the Yang-Mills Working Group chaired by David Gross announced it would publish OpenAI''s Lean formalization of pure Yang-Mills existence the next day. By November 11, the OpenAI-YM-20261028 repository on GitHub is public, and the group''s assessment PDF is linked from its third committee statement.

This is the first machine-checkable release from the physics shepherding process OpenAI described on October 26. It does not close the Clay mass gap question that RoundSquare and social posts keep arguing about.

TL;DR

What the committee actually claims

The November 9 PDF names two artifacts:

1. Full Lean code OpenAI submitted October 28, 2026 — now at OpenAI-YM-20261028.
2. A validation document summarizing checks on the October 28 submission. The group's YM repository README states that, in its opinion, the formalization constructs a Wightman QFT as a scaling limit of Wilson lattice gauge theory — existence machinery, not a mass-gap proof.

The wording is careful. It validates existence under the definitions the committee accepts. It does not claim a public proof of Δ greater than zero on R⁴ in the Clay Millennium sense.

The group''s first statement already said confinement and the mass gap remain very important open challenges. The second statement described definition alignment work before release.

Green verification seal evoking machine-checked Lean review of OpenAI Yang-Mills formalization

Timeline: from model output to GitHub

OpenAI''s October 26 post gave these dates for Yang-Mills:

• September 9, 2026 — natural-language existence proof produced inside OpenAI.
• October 16, 2026 — Lean formalization finished.
• October 28, 2026 — Lean package delivered to the Gross group (per the November 9 statement).

Between November 3 and November 9, the committee said it was checking that definitions match standard statements and that the formalization aligns with informal treatment. The November 10 release date was announced one day ahead, matching the procedure in the second statement.

For ETH and Kakeya, OpenAI shipped public Lean earlier. Yang-Mills stayed withheld until this shepherding finished — the pattern we tracked in our launch explainer.

What is in the GitHub repository

The published repo description reads "OpenAI construction of pure Yang-Mills obtained 10/28/2026". It is Lean 4 source intended for community inspection, not a polished paper.

If you maintain scientific agent pipelines, treat it like any large third-party formal artifact:

• Pin a Lean toolchain version before you Mathlib-pin.
• Read #print axioms on modules you depend on.
• Compare theorem statements against the informal problem sheet in the assessment PDF.

The committee credits Ron Nissim and Fred Rajasekaran on the assessment work. That mirrors how human peer review often splits authorship (OpenAI) from verification reporting (the group).

Chalkboard-style diagram for Yang-Mills existence proof and Clay mass-gap debate context

Mass gap, RoundSquare, and headline confusion

Two parallel stories collide in headlines:

Social posts that say "Peano disproved the $100B proof" usually mean RoundSquare''s infrared argument, not a refutation of every line in OpenAI''s withheld package. Now that existence Lean is public, readers can ask sharper questions: do the definitions in OpenAI-YM-20261028 match the assessment''s reading, and do they match the Clay statement Jaffe and Witten published?

We are not physicists signing off on QFT measure theory. The actionable read for builders is definition alignment: automated provers are only as meaningful as the formal statement they close.

How this compares to other lab Lean collaborations

Anthropic and other labs have shipped formal math with outside audits — see Aaronson–Ambainis conjecture coverage and Opus formalizing CPython. The Yang-Mills arc adds months of withholding and a Nobel-laureate committee as gatekeeper.

That matters for agent evals and research harnesses:

• Time-to-public can exceed time-to-generate by an order of magnitude.
• Custody may sit with domain committees, not the lab blog.
• Partial releases (existence first, mass gap later, natural-language write-up later still) are plausible even when Lean exists internally.

What the assessment says versus what Lean checks

Lean does not read like a physics paper. It is a dependency graph of definitions, each theorem pinned to axioms Mathlib already exports or to definitions the submission introduces. The committee''s assessment PDF is the human-readable map between informal Yang-Mills existence language and those formal statements.

When you audit, copy this checklist:

1. Locate the main existence theorem in the repo README or top-level module index.
2. Run lake build with the toolchain file the repo pins; do not mix Mathlib versions blindly.
3. For any claim you care about, run #print axioms on that theorem — unexpected axioms are a red flag.
4. Compare the formal gauge group, measure, and Hilbert space wording against Jaffe–Witten and against the committee PDF, not against Twitter summaries.
5. File GitHub issues on definition mismatches if you find them; that is how public formal math improves.

The committee thanked Ron Nissim and Fred Rajasekaran on the assessment — typical for a large formalization where statement alignment is as much work as proof length.

AGMAI, withholding, and why release took weeks

The Advisory Group on Mathematics and Artificial Intelligence asked OpenAI in late October to release Yang-Mills outputs without delay once Lean existed. The Gross group chose a slower path: validate definitions, then publish Lean first while the 1,147-page natural-language package waits for another quality gate.

That split matters for research agents:

If your pipeline ingests arxiv + GitHub only, you may now index existence Lean but still miss the full narrative proof — plan crawlers accordingly.

Reading list for builders who are not QFT experts

You do not need to master constructive QFT to extract value:

• Wilson lattice intuition — why discretization appears before continuum limits.
• Balaban RG program — the historical line OpenAI says its proof extends.
• OS/Wightman axioms — the package RoundSquare argues is over-constrained on R⁴.
• Clay problem sheet — the exact conjunction journalists conflate with existence alone.

Our Millennium prize AI scorecard and OpenAI 722-manuscript repo guide help separate headline counts from checkable artifacts.

What people building with formal methods should do next

Clone and compile. Start from the committee''s repo URL, not a fork you cannot trace.

Read the assessment before the lore. The PDF is the committee''s summary of checks, analogous to a referee report''s "main theorem verified" paragraph.

Separate existence from prize language. If your product marketing says "solved Yang-Mills," you are ahead of the committee''s own mass-gap disclaimer.

Watch for the natural-language 1,147-page package. The first statement said the group may release full LLM proofs once satisfied with quality. That is a second drop; today''s release is Lean-first.

Pair with complexity-news hygiene. The same October batch included ETH and Kakeya Lean with different public timelines. Do not assume one release policy fits all three.

Implications for AI evals and science agents

Teams building ** theorem-proving agents** should log three metrics on this release:

Time from generation to public Lean — roughly seven weeks from OpenAI''s stated Lean completion to GitHub, dominated by human committee time, not compute.

Custody model — artifacts live under a third-party org, not the lab that generated them. Your agent must follow redirects and mirror policies.

Partial claim scope — automated headline tests should flag posts that say "Clay solved" when sources say existence only.

Pair with AI peer review planted-errors benchmarks if you score models on reading referee-style PDFs, not just compiling Lean.

Questions readers search after this headline

Did OpenAI disprove RoundSquare? Not automatically. RoundSquare targets a different conjunction on R⁴. Both can be formally valid under different statements.

Can I cite this in production? Cite the committee assessment and theorem statements you verified, not uncopied social threads.

Will API prices change? No signal in KITP or GitHub releases. This is science process, not product pricing.

Does this help coding agents? Indirectly — it is another datapoint that long-horizon formal tasks can produce gigabyte-scale artifacts needing expert gates before trust.

What is still unverified or open

• Independent mathematicians beyond the committee may publish critiques of definitions or analytic steps inside the Lean.
• Clay Mathematics Institute has not, in sources we checked, issued a prize decision based on this release.
• Mass gap and confinement remain open in the committee''s own framing.
• RoundSquare and OpenAI existence may both be formally consistent if they formalize different problem statements — that is a job for experts, not a headline.

Related reading

• OpenAI''s October triple-result hub
• RoundSquare vs mass-gap claims
• DeepMind AlphaProof and formal verification trend
• Opus 5.5 on CPython in Lean
• AI peer review planted-errors benchmark
• METR–Anthropic acceleration research

Repository URLs, committee statements, and OpenAI announcement text are accurate as of November 11, 2026.