OpenAI has published a catalogue of mathematical results produced by an unreleased internal model: more than 700 manuscripts grouped into 372 families of results. A day later it withdrew three of them after finding a sign error in one proof that two other papers depended on. Mathematicians are now arguing less about whether the model is impressive and more about how anyone is supposed to check this much machine-written mathematics.
The model is not available outside OpenAI, so nobody else can rerun it. Most figures below are OpenAI’s own or come from reporting on its repository. 9to6AI has not independently verified any of the proofs.
Key points
- What shipped: a public GitHub repository of AI-generated math manuscripts, now listing 719 papers in 372 families after the three withdrawals, plus Lean formalizations for part of the set.
- What went wrong: a sign error broke a cancellation step in one paper on Weil classes. Two further papers reused that construction, so all three were pulled. OpenAI says the notices concern failed proofs, not proven-false statements.
- How much is machine-checked: about 300 of 719 top-line results (roughly 42%) had Lean proofs at the time of the update. OpenAI itself warns that some unformalized results could have issues.
- Who pushed back: an Institute for Advanced Study advisory group had asked labs not to test hard problems on proprietary models and to disclose prompts and per-result compute. A mathematicians’ association urged colleagues to stop working with OpenAI.
- Why builders should care: this is a live test of how far “the model solved it” can be trusted without outside replication.
What OpenAI released
In its own post, OpenAI describes the release as “a broad range of new mathematical results produced by an internal frontier model.” The papers sit in a public GitHub repository with protocols for revisions and citations. OpenAI says it consulted the Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study on how to release them, and it is looking for community-hosted alternatives to GitHub.
The repository also includes Lean formalizations for many proofs, ten summaries of the model’s reasoning, compute estimates and statistics on attempted problems. OpenAI’s figure is that the average result used the equivalent of roughly three hours of ChatGPT Pro thinking. According to reporting by Implicator, an OpenAI spokesperson said nearly every result came from a single prompt to a single agent, and that many of the results are not yet understood by the company’s own mathematicians. Those are company statements that no outside party can test today.
The sign error
The withdrawn paper is “Algebraicity of Weil classes on split abelian eightfolds.” Its proof counted a set of “reverse stabilization traces” as +1 each, which made a signed count appear to cancel to zero. With the orientations handled correctly, each trace counts as −1 and the total becomes −2m instead of zero. The cancellation theorem the proof invokes next needs exactly zero, so the construction that follows is unsupported.
Two other manuscripts, on Kuga–Satake correspondences for K3 surfaces and on the rational Hodge conjecture for products of K3 surfaces, adapted the same construction and were withdrawn with it. OpenAI also revised 14 other manuscripts with proof repairs, corrected statements or clearer hypotheses. Withdrawn papers stay online behind a notice that explains the gap and links to an archived copy.
The notices say the claims are unproved, not that the statements are false. That distinction matters: a failed proof leaves the question open.
What is verified, and what is not
Lean is a language in which a computer checks every step of a proof. Where a Lean formalization exists and matches the written statement, trust goes up sharply. That covers a minority of the catalogue. A spokesperson told Retraction Watch that roughly half the results were released without full formalization, on the advice of the advisory group, and that the company found the sign error during its own audit.
Formalization has its own gap. Researchers have separately questioned whether OpenAI’s machine-checked proof for an earlier Navier–Stokes claim matches the written argument. A Lean proof only helps if the formal statement says what the paper says it does.
Why mathematicians are unhappy
The advisory group published release recommendations shortly before this drop. It asked labs to stop testing advanced problems on proprietary models, to disclose model names, prompts, summarized reasoning and per-result compute, and to host results outside lab control. Reporting shows OpenAI published no prompts, gave average compute rather than per-result figures, and used a proprietary model. The group said its advice is not an endorsement of how the results were obtained, and that the community should judge how well the guidelines were followed.
The reaction has been sharp. The Association for Human Mathematics said releasing over 700 files at once is “a demonstration of power,” not scholarship, and urged mathematicians to stop working with OpenAI. Andrew Sutherland of MIT told Retraction Watch the quick withdrawals were the responsible move, but that earning back trust will take far more. Alex Townsend of Cornell said he expects more errors to surface and that Lean-verified results should have been announced first, separately.
What it means for builders
If you build on AI-generated research, code or analysis, this episode is a clean example of the gap between output and verification. A model can produce hundreds of plausible, formatted proofs faster than people can read them, and one wrong sign can sink several at once.
- Treat unreleased-model claims as unreplicated. Until the model ships and others can rerun it, “solved by AI” is a claim, not a result.
- Machine checks beat polish. The papers with a formal proof are the ones worth leaning on first.
- Expect corrections. A public errata trail and versioned withdrawals are a healthier pattern than silent edits, and worth copying in your own AI-assisted work.
What to watch
- Whether independent mathematicians find more errors in the unformalized majority.
- Whether OpenAI releases the model, so results can be replicated.
- Whether the advisory group updates its recommendations, and whether OpenAI funds the workshops and understanding programs it promised.
- Whether other labs adopt per-result disclosure of prompts and compute.
