OpenAI's Yang–Mills claim, the Gross committee and a public Claude attempt

On 28 October 2026, the Kavli Institute for Theoretical Physics (KITP) posted a two-page statement from the five-member committee shepherding OpenAI's claimed Yang–Mills proof. The group, chaired by David Gross, says the AI-produced proofs run to 1,147 pages. It will release OpenAI's Lean formalization only after checking that its definitions match the standard ones, and it will release the full proofs only together with its own assessment. It also says what the work leaves open: "confinement and the mass gap remain very important open challenges."

A second AI-assisted attempt on the same problem is public. Since 6 October, Ethan Sussman and Levent Alpöge have been posting manuscripts generated with Claude to a GitHub repository named Yang--Mills-LLM-experiment. Its README opens with a warning: "Its content has not been satisfactorily vetted. All claims should be understood as tentative, pending verification."

One claim on a Millennium Prize problem is held by a committee; the other sits in the open with its authors' warning attached. Below: what each says it has shown, what anyone can read today, why both lean on Balaban's papers from the 1980s, and why neither addresses the mass gap. Our 27 October explainer covered OpenAI's announcement. Every source was checked on 30 October 2026. Neither claim has been independently confirmed.

What did the Yang–Mills shepherding committee say?#

The statement is titled "Proof by OpenAI of the Existence of Quantum Yang-Mills theory". It is signed by Michael R. Douglas, David J. Gross (chair), Arthur Jaffe, Igor R. Klebanov and Nikita Nekrasov, the group OpenAI named on 26 October. It explains how the group formed: "On Oct. 16, David Gross was asked by OpenAI to assume custody of the proof, decide on the best way to release it to the community, and shepherd the release process. David Gross then chose the rest of the committee."

The committee says it works "on behalf of the academic community" and adds: "We have no responsibility to OpenAI." It names the risks it wants to avoid, "such as putting out wrong results or carelessly omitting attributions", and calls the arrangement "an experiment in how the academic community can deal with the rapid development of new knowledge by AI."

On the result itself, the statement is brief: "The new proof by OpenAI appears to be an important step towards putting QFT on a rigorous mathematical footing." Its one sentence on what was proved credits the renormalization-group papers Tadeusz Balaban published in the 1980s: "Balaban's ideas were among the important elements that recently enabled OpenAI to follow the theory all the way to infinite distances and prove the existence of its proper continuum limit." The next sentence is the one quoted above: "But confinement and the mass gap remain very important open challenges."

The statement does not give the theorem's exact wording, the gauge groups it covers, or what "existence" includes. Checking the definitions and the problem statement is the committee's first step.

When it was published. The PDF is dated only "October 2026". Its file metadata gives a creation time of 27 October, 20:25 UTC. KITP's post on X linking to it followed at 00:16 UTC on 28 October.

When will the Lean proof be released?#

The statement gives no date. It sets out steps in order, each with a condition:

1. Check the definitions. "Our first objective will be to check that the definitions and problem statement correspond with those which are standard in physics and mathematics, and that the formalization aligns with the informal treatment."
2. Release the Lean proof. "Once this validation is done, we will release the Lean proof."
3. Work to understand it, with help. The committee will then work on "a deeper understanding" of the proof and intends "to draw upon the talent and expertise of other members of the community to help us with this difficult task."
4. Release the full text. "When we are satisfied with the quality and correctness of the proofs, we will provide our assessment together with the full 1147-page LLM proofs."

Its reason for waiting is one sentence: "Before we release it, we would like to be convinced that it is correct."

The first step matters most for a Lean proof. A proof assistant checks that a formal statement follows from formal definitions. It cannot tell whether those definitions describe what physicists and mathematicians mean by a quantum Yang–Mills theory on four-dimensional space. People make that comparison, and for now only the committee and OpenAI have the files.

OpenAI's announcement says: "We finished our formalization of Yang–Mills’ existence on October 16th, 2026". The committee writes that "OpenAI is providing a formalization". OpenAI mentions a finished formalization only for Yang–Mills existence, not for its claims about quantum chromodynamics (QCD) and a completion of the Standard Model.

Why did AGMAI call for immediate release?#

On 26 October, the day of OpenAI's announcement, the Advisory Group on Mathematics and Artificial Intelligence (AGMAI) posted a "Recommendation for Immediate Release of Yang–Mills AI Output":

We call on OpenAI to make public without delay the natural-language and Lean outputs behind its announced existence proofs for Yang–Mills theory, quantum chromodynamics, and a completion of the Standard Model.

Its argument rests on OpenAI's own timeline: "OpenAI have said they delayed release until the Lean formalization was complete. By their own account this is now complete, so there is no scientific reason for delaying the release any further." Behind that is its general position that "understanding should develop in the open, with everyone given the same access." The statement also says: "We have great respect for the scientists asked by OpenAI to lead this work."

AGMAI is a group of nine mathematicians that advises AI companies, including on releasing mathematical results. It says it "operates independently of any AI company", and that it formed "after OpenAI approached some of its members about establishing an external advisory board." One member is Edward Witten, who wrote the Clay Mathematics Institute's problem description with Arthur Jaffe, now on the shepherding committee.

The committee's statement does not mention AGMAI; its answer on timing is that it wants to be convinced first. It is the trade-off our earlier post described: open access for everyone at once, or a first check by a few experts. As of 30 October we found no further statement from either group.

What is the Sussman–Alpöge Yang–Mills repository?#

It is a public GitHub repository described as "an attempt with Levent Alpoge to get Claude to finish Balaban's program". Scientific American describes Sussman as a mathematician at Northwestern University and Alpöge as an Anthropic employee. The README gives no affiliations.

The plan is to work up to four dimensions from easier cases: "Following Dimock and others, we are beginning with superrenormalizable theories in three spacetime dimensions, in which the ultraviolet singularities are less severe."

The README is frank about how the manuscripts are made. "The manuscripts cited above are the result of cutting material from literally thousands of pages of slop generated by Claude." That output "is then fed to Codex to extract a minimal manuscript", and "Occasionally it introduces minor errors not found in the original slop." One shorter manuscript, RPM1, is "half-written by one of us [E] and completed by Astra". The README names the models only as Claude, Codex and Astra, with no versions.

On credit, the disclaimer ends: "Novel mathematical ideas should be attributed to Claude (under the guidance of LA)." LA is Alpöge; notes marked ES are Sussman's.

What does the repository claim about QED3, YM3 and YM4?#

Three things, with three levels of confidence:

What the Osterwalder–Schrader axioms are. They are properties of a theory's correlation functions in Euclidean space. Konrad Osterwalder and Robert Schrader showed in 1973 and 1975 that functions with these properties determine a relativistic quantum field theory, with a vacuum and an energy operator. The Clay problem counts this as part of existence: "Existence includes establishing axiomatic properties at least as strong as those cited in" its references to Osterwalder and Schrader and to Streater and Wightman.

Three dimensions is a step, not the prize. The Clay problem is set on ℝ⁴. The README says: "We believe that the ideas developed in the QED3 and YM3 manuscripts can be applied to YM4." YMM2's commit message says it was posted "for the record" and "has not been edited to even our standards".

The YM4 file is the only public document in either effort that states a four-dimensional theorem. Titled "SU(2) Yang–Mills theory in four dimensions via Bałaban's renormalisation group: construction, axioms and non-triviality", it is dated 6 October 2026 and has no author line or AI statement. Its abstract says its results hold "Assuming the proof inputs declared clause by clause in Section 9", and full rotation invariance follows only from "a displayed hypothesis" that it does not derive. Beside it sit three raw TeX files, about 27 MB in all, titled "a complete account of Theorem 0", the draft's theorem. The account labels each statement "proved", "stated; proof incomplete", "not proved" and so on.

Outside readers have started. On 27 October, a GitHub user formalizing parts of RPM1 in Lean opened an issue: "Nothing I found breaks a result, but a few things might be worth a look". One of them: a QED3 manuscript takes "628 of its 636 citations" from companion papers that are not in the repository. Sussman replied on 30 October: "That's referring to Levent's original slop." The issue's author says AI (Opus 5.5 and GPT-6 Astra) helped with the work. It is a reading, not a verification.

On 30 October the repository added a 125-page manuscript, "Massive SU(2) gauge theory in three dimensions", in a new QCD3 folder.

What is public for each claim?#

Side by side, as of 30 October:

The dates rest on different kinds of evidence:

OpenAI's September and October dates are its own statements; no file from those days is public yet. The repository's dates come from its commit history, and the pushes also appear in GitHub's public event log. Alpöge's note on YM4, for example, was pushed at 06:33 UTC on 16 October.

The README itself carries no date, and its wording has changed. Earlier on 26 October, before an edit at 16:18 UTC, it named Astra rather than Codex for the condensing step and said: "We take no mathematical credit for any results proven in this repository." Anyone citing it should cite a commit, not just the page. Peter Woit cited the repository in an update to his 26 October blog post, "Secret Unified Physics Slop": "Unlike OpenAI, they’ve been posting things publicly".

Why do both rely on Balaban?#

Both build on Tadeusz Balaban's renormalization-group papers rather than starting again. The committee credits "Balaban's ideas" by name, and the repository's stated goal is to "finish Balaban's program".

Scientific American, on 27 October, described the papers as "a series of massive, dense papers that sketched out a path to prove the “existence” part of the problem." It quoted Roland Bauerschmidt, a mathematician at New York University: "He basically did it," and "But I think there’s no one who understands these papers and can tell whether they’re correct or not."

The YM4 draft says most of its statements are "complete deductions from Bałaban’s theorems as cited" and from its companion account. That account gives statements Balaban printed without proof a status of their own, "printed without proof; cited as printed", and does not count them as proved. A Lean formalization checks only what is formalized. If a proof takes some of Balaban's results as given, the open question moves to those papers. The committee's statement does not say whether OpenAI's formalization proves Balaban's results or assumes them.

Why is Yang–Mills existence not the mass gap?#

Because the Clay problem asks two questions, and both efforts answer at most the first. Arthur Jaffe and Edward Witten's problem description states it: "Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0."

Existence means a mathematically well-defined theory on ℝ⁴ that satisfies the axioms above. The mass gap is about energy: "every excitation of the vacuum has energy at least Δ", so the lightest particle has a positive mass. Jaffe and Witten say it is "necessary to explain why the nuclear force is strong but short-ranged". An important consequence, they write, is clustering: correlations between distant points die off exponentially.

A construction can exist and still say nothing about the gap. The committee describes asymptotic freedom: "forces weaken at short distances and grow stronger at long distances". Balaban's work treated the short-distance, or ultraviolet, side; in the YM4 draft's words, "the ultraviolet problem was treated by Bałaban". The gap concerns long distances, where the interaction is strong. Asymptotic freedom, the committee writes, "is believed to lead to the confinement of quarks within hadrons and to a mass gap". Believed, not proved.

The YM4 draft says so directly: "the present paper does not address the mass gap", and "a gap is an infrared property of the infinite-volume theory, on which the construction gives no information." It adds that "Theorem 0 asserts no decay at all."

When Jaffe and Witten wrote the description, they saw no method for either the gap or the passage from a finite box to ℝ⁴: "no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume. Nor do present methods suggest how to obtain the existence of the infinite volume limit T⁴ → ℝ⁴." They suggested that a uniform gap "may play a fundamental role in the proof of existence of the infinite-volume limit". Both new efforts describe a way to reach ℝ⁴ without one. Whether that meets the description's meaning of existence is part of what the committee says it will check first for OpenAI's version.

Two further conditions apply to the existence half:

• Every gauge group. The problem asks about "any compact simple gauge group G". The YM3 and YM4 work in the repository is for SU(2). On 21 October Sussman wrote in a GitHub issue: "Re. YM, we're focusing on SU(2) for now." OpenAI has not published the statement of its theorem.
• The right observables. The description asks for local operators built from the curvature, "such as Tr F_ij F_kl(x)". In three dimensions, the YM3 manuscript covers tr[F²]; the README says a sequel that handles tr[F_jk F_ℓm] is being processed.

So even if every page of both efforts holds up, the Millennium Prize problem as Clay states it would remain open, because the mass gap is part of the question. And under Clay's rules, a proposed solution is considered only after publication in a qualifying outlet, at least two years, and "general acceptance in the global mathematics community".

What happens next?#

For OpenAI's claim, the next public step is the Lean proof, after the committee's check of the definitions. Until the formal statement is public, no one outside the committee and OpenAI can rebuild the proof or compare its definitions with the Clay description. The 1,147 pages come later. No dates have been given.

For the repository, the files are public and the authors ask readers to report mistakes. They hope to pair each submanuscript with "a human-written equivalent"; so far the one example, RPM1, is half human-written.

What we changed on Papex#

The Yang–Mills existence and mass gap page lists OpenAI's claim as a partial result, as its source states it. With this post, we add two responses to that claim: the shepherding committee's statement and AGMAI's call for immediate release. We also add the Sussman–Alpöge repository as a separate claim, marked tentative, in its authors' own word. Both claims are listed on AI-assisted claims.

Open questions#

• Should the formal statement of a claimed proof be public before the proof is understood, so that others can check the definitions in parallel?
• When the only dated evidence for a discovery is a company's announcement, what counts as the date of the claim?
• When a README changes after people cite it, should the citation point to a commit, an archived copy or a signed version?
• Where should an attribution like "under the guidance of LA" live, so that it survives when thousands of pages are cut down to a paper?

Sources#

• Proof by OpenAI of the Existence of Quantum Yang-Mills theory: Statement by the shepherding committee (Douglas, Gross, Jaffe, Klebanov and Nekrasov, dated October 2026; PDF created 27 October 2026), checked 30 October 2026
• Statement by the Yang-Mills Working Group (KITP), checked 30 October 2026
• KITP post on X linking the statement (28 October 2026, 00:16 UTC), checked 30 October 2026
• Recommendation for Immediate Release of Yang–Mills AI Output (AGMAI, 26 October 2026), checked 30 October 2026
• About AGMAI (AGMAI, 22 September 2026), checked 30 October 2026
• Working with academic communities to build scientific understanding (OpenAI, 26 October 2026), checked 30 October 2026 (Internet Archive copy of 28 October 2026; the live page blocked automated access)
• ethanwsussman/Yang--Mills-LLM-experiment: README, commit history and event log (GitHub), checked 30 October 2026
• The field-strength Osterwalder–Schrader axioms for QED in three dimensions (QED3/qed3.pdf), checked 30 October 2026
• Scalar fields from three-dimensional lattice Yang–Mills theory (YM3/YMM1.pdf), checked 30 October 2026
• Curvature observables, dynamics, and conditional confinement in three-dimensional lattice Yang–Mills theory (YM3/YMM2.pdf), checked 30 October 2026
• SU(2) Yang–Mills theory in four dimensions via Bałaban's renormalisation group: construction, axioms and non-triviality (ym4_manuscript_zeroth_draft.pdf, dated 6 October 2026), checked 30 October 2026
• The Yang–Mills continuum limit along Bałaban's renormalization group: a complete account of Theorem 0 (raw TeX, part I of III), checked 30 October 2026
• Massive SU(2) gauge theory in three dimensions (QCD3/QCDM1.pdf), checked 30 October 2026
• Issue 5: Lean formalization of parts of RPM1, and a few notes from reading QED3 and YM3 (GitHub, 27 October 2026), checked 30 October 2026
• Issue 1: QED and QCD in 4D (GitHub, 20 October 2026), checked 30 October 2026
• Peter Woit, Secret Unified Physics Slop (Not Even Wrong, 26 October 2026, with updates), checked 30 October 2026
• Joseph Howlett, Dueling proofs from OpenAI and Anthropic astound scientists (Scientific American, 27 October 2026), checked 30 October 2026
• Jaffe and Witten, Quantum Yang–Mills Theory (Clay Mathematics Institute), checked 30 October 2026
• Rules for the Millennium Prize Problems (Clay Mathematics Institute), checked 30 October 2026