OpenAI's AI proof of Yang-Mills theory passes committee review

Overview

Updated 42 minutes ago

On September 9, OpenAI's models produced a written proof that quantum Yang-Mills theory, the mathematics of the strong nuclear force, can be built from first principles. The company announced the claim on October 26. It handed a machine-checked version, written in the Lean proof language, to a five-member panel. On November 10 the panel said the formal proof passes.

The result covers only the existence half of the Clay Mathematics Institute's Yang-Mills problem. The harder mass-gap half, which would explain why particles have mass, remains open, and the $1 million prize is unclaimed. Critics say a review run by a committee OpenAI chose is not the same as open peer review.

Why it matters

If AI can generate and machine-check a foundational physics proof, the scarce skill in mathematics becomes understanding proofs, not finding them.

Questions about this story

Free account needed to ask — your question is kept and asked for you right after sign-up. Answers are public.

No questions yet — be the first to ask.

Key Indicators

250,319

Lean declarations audited

The committee checked every declaration the construction imports for added axioms or placeholders.

9,470

Lean modules rebuilt from scratch

The full build reported zero errors and took about 184 minutes.

100,168

Declarations in the dependency chain

Only part of the audited code is actually used by the main theorem.

Committee members

David Gross chose four colleagues to join him on the review panel.

$1,000,000

Unclaimed Millennium Prize

The Clay award requires the mass-gap half of the problem as well.

Voices

Curated perspectives — historical figures and your fellow readers.

People Involved

David Gross

Theoretical physicist; Nobel laureate

Chaired the five-member review committee

Michael R. Douglas

Theoretical physicist, Harvard University

Committee member

Scott Armstrong

Mathematician, New York University

Public critic of the review process

Levent Alpöge

AI researcher, Anthropic

Pursued a competing Yang-Mills result

Organizations Involved

OpenAI

Artificial intelligence company

Produced the proof and convened the review committee

The AI lab produced the Yang-Mills existence proof and appointed the panel that reviewed it.

Yang-Mills Shepherding Committee

Independent review panel

Reviewed and affirmed the Lean proof

A five-member panel chaired by David Gross that reviewed OpenAI's Lean proof.

Clay Mathematics Institute

Nonprofit mathematics organization

Offers the $1 million Millennium Prize that includes Yang-Mills

The institute set the seven Millennium Prize Problems in 2000, including Yang-Mills.

Anthropic

AI company

Its researchers pursued a competing Yang-Mills result

The rival AI lab whose researchers pursued a competing Yang-Mills result.

Timeline

September 2026 November 2026

10 events Latest: 3 days ago

Tap a bar to jump to that date

November 2026
Affirmation reaches the wider research community
Latest Publication

The committee's document and Lean code circulate for independent checking.
3 days ago
Committee affirms the proof and releases code
Review

Panel posts its assessment and makes the Lean proof public.
5 days ago
Second OpenAI submission listed
Research

OpenAI files a revised proof that the committee marks as work in progress.
November 4th, 2026
Committee's combined theorem passes
Verification

The panel issues a second statement as its combined theorem checks out.
November 3rd, 2026
October 2026
Comparator passes the main theorem
Verification

An independent tool checks the proof's central theorem.
October 30th, 2026
Lean code handed to the committee
Review

OpenAI gives the formal Lean version to the review panel.
October 28th, 2026
OpenAI announces the Yang-Mills claim
Announcement

OpenAI says its model solved the existence half of the problem.
October 26th, 2026
Anthropic pair claims a 3D result
Research

Alpöge and Sussman claim a three-dimensional version using Claude agents.
October 25th, 2026
Gross takes custody of the proof
Appointment

OpenAI asks David Gross to decide how to release the proof.
October 16th, 2026
September 2026
OpenAI models finish a written proof
Research

OpenAI says its models complete a written Yang-Mills existence proof.
September 9th, 2026

Scenarios

Independent review confirms the Lean proof is sound

Likely Resolves by Q2 2027

Discussed by: Scientific American; the committee's November 10 assessment

The committee released the Lean code and a short document. Mathematicians can now rebuild the 9,470 modules and check the definitions against standard physics. If no one finds a mismatch, the existence result stands.

A team builds on the proof to prove the mass gap

Possible Resolves by End of 2029

Discussed by: Committee members; Clay Mathematics Institute problem statement

Existence was the easier half. The mass gap needs a quantitative bound. With a rigorous construction now available, researchers could try to derive the gap from it.

Close reading finds a gap between the formal statement and the physics claim

Possible Resolves by Q2 2027

Discussed by: Critics including Scott Armstrong; mathematics commentators

The committee checked that definitions match standard ones, but admitted it does not fully understand the proof. A subtly wrong definition, or a theorem that formalizes a weaker statement than physicists mean, would undercut the claim.

A rival lab publishes a competing, open proof

Possible Resolves by Q2 2027

Discussed by: Scientific American reporting on Anthropic's Alpöge and Sussman

Anthropic researchers claimed a three-dimensional version the day before OpenAI's announcement. They or another group could release a four-dimensional proof developed in the open.

Historical Context

3 moments from history that rhyme with this story — and how they unfolded.

June 1976

Four-color theorem (1976)

Kenneth Appel and Wolfgang Haken proved that four colors suffice to color any map, relying on 1,936 cases checked by computer. The proof ran to hundreds of pages, and no human checked every case by hand.

Then

Mathematicians accepted the result but debated whether a computer-assisted proof counted as a proof.

Now

It opened the field of computer-assisted mathematics and set a precedent for trusting machine-checked cases.

Why this matters now

The Yang-Mills proof is far larger, and its acceptance depends on software that checks every step rather than on humans reading it.

1998-2014

Kepler conjecture and Flyspeck (1998-2014)

Thomas Hales proved how spheres pack most densely. Reviewers said they were 99% certain but could not certify the proof, so Hales spent over a decade formalizing it in proof assistants.

Then

The formal verification finished in 2014, giving the result machine-checked status.

Now

It showed formalization can settle proofs too long for human referees.

Why this matters now

OpenAI skipped the long human review and went straight to Lean, the same strategy Flyspeck pioneered.

1993-1995

Wiles and Fermat's Last Theorem (1993-1995)

Andrew Wiles announced a proof in June 1993. Reviewers found a gap that took him and Richard Taylor nearly a year to repair.

Then

The corrected proof was published in 1995.

Now

It became the standard example of peer review catching a flaw in a celebrated proof.

Why this matters now

The Yang-Mills review was closed and fast, and the committee says it does not fully understand the proof, raising the question of whether a gap could slip through.