OpenAI published 722 mathematics manuscripts written by an unreleased internal model on October 6, 2026, grouped into 372 results across 17 fields, with Lean proofs for the main result of 162 papers. The claims include a zero-free half-plane for every Dirichlet L-function, the Mahler conjectures, a counterexample to Kaplansky’s zero-divisor conjecture and the isomorphism of the free group factors. OpenAI’s announcement opens: “We’re releasing a broad range of new mathematical results produced by an internal frontier model.” The papers sit in a public GitHub repository under Apache 2.0. This page reads the announcement, the repository’s README, its 41-page overview catalogue and its Lean formalization list, as of the evening of October 6, 2026.
Update, October 11: OpenAI withdrew three manuscripts on October 7 after a sign error and revised 14 others, and the first week brought sharp responses from mathematicians, including NYU’s Tristan Buckmaster and Fields medallist Hugo Duminil-Copin. Both are in the first week below.
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OpenAI published 722 mathematics manuscripts from an unreleased internal model on October 6, 2026, grouped into 372 result families across 17 fields, in a public GitHub repository under Apache 2.0.
The model was posed about 4,000 problems after OpenAI’s existing math evaluations saturated; OpenAI kept the results it judged significant and grouped related papers into families.
Lean formalizations cover the main result of 162 papers, including the zero-free half-plane Re(s) > 7/8 for Dirichlet L-functions, the symmetric Mahler conjecture and a Kaplansky zero-divisor counterexample.
Each result used about three hours of ChatGPT Pro thinking on average, OpenAI says, and the repository adds 10 abridged summaries of the model’s reasoning.
Claims without a Lean proof yet include the isomorphism of the free group factors, the irrationality exponent of pi being 2 and Hilbert’s tenth problem over the rationals; the README warns unformalized results could have issues.
The model is not public and the papers sit on OpenAI’s own GitHub, two points the Advisory Group on Mathematics and AI asked labs to change; OpenAI says it is working on both.
§ 01What OpenAI released
The repository holds manuscripts, a catalogue and proof files rather than one paper. Its README says how the model got there: “We expanded these evaluations after performance on our existing mathematical evaluations saturated.” Over the evaluation the model was posed about 4,000 problems; OpenAI kept the results it judged significant and grouped related papers into families.
| Item | Count or detail |
|---|---|
| Released | October 6, 2026, OpenAI announcement and GitHub repository |
| Model | Unreleased internal OpenAI model |
| Manuscripts | 722 |
| Result families | 372, across 17 fields |
| Problems posed | About 4,000 |
| Compute per result | About three hours of ChatGPT Pro thinking, on average |
| Papers with a Lean-formalized main result | 162, per the repository’s formalization catalogue |
| Reasoning summaries | 10 abridged summaries of the model’s reasoning |
| License | Apache 2.0 |
On cost, the announcement says: “The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking.” The README names two exceptions to the fixed procedure, work on a zero-free region for the Riemann zeta function and a proof of the Hodge conjecture for CM abelian varieties, and says the write-up for the Re(s) > 11/12 zero-free region “was human edited for readability”.
§ 02Where the results fall
We counted the families under each field heading in the overview catalogue. Theoretical computer science, combinatorics and algebraic and complex geometry lead; mathematical logic has the fewest.
| Field | Result families |
|---|---|
| Theoretical computer science | 40 |
| Combinatorics | 37 |
| Algebraic and complex geometry | 36 |
| Number theory | 31 |
| Probability and statistical mechanics | 29 |
| Differential geometry | 29 |
| Mathematical physics | 25 |
| Operator algebras | 19 |
| Algebra | 18 |
| Topology | 18 |
| Real and complex analysis | 16 |
| Partial differential equations | 16 |
| Convex and metric geometry | 15 |
| Group theory | 14 |
| Dynamical systems and ergodic theory | 12 |
| Functional analysis | 11 |
| Mathematical logic | 6 |
§ 03The headline claims, and what is checked
The catalogue states each result as proved. A Lean formalization means a computer has checked the formal statement against the formal proof; it does not by itself show that the formal statement says exactly what the paper says, or that the result is new. The README is direct about the rest: “Some of the unformalized results could have issues.”
| Family | Claim as stated in the catalogue | Lean main result in the catalogue |
|---|---|---|
| 003 | Every Dirichlet L-function, including the Riemann zeta function, is zero-free in Re(s) > 7/8 | Yes |
| 087 | The symmetric and nonsymmetric Mahler conjectures, in every dimension | Yes, for the symmetric case |
| 196 | A counterexample to Kaplansky’s zero-divisor conjecture | Yes |
| 197 | Kaplansky’s direct-finiteness conjecture fails, even without torsion | Yes, characteristic two and odd characteristic |
| 287 | All nonabelian free group factors are isomorphic | Not yet |
| 017 | The irrationality exponent of pi is exactly 2 | Not yet |
| 004 | Hilbert’s tenth problem over the rationals has a negative answer | Not yet |
The remaining 560 papers without a formalized main result sit in the same repository; OpenAI says it “will continue to update this repository with Lean formalizations as we obtain them”. Readers checking a single claim should open its family in the manuscript map, where each paper links to its PDF and citation file.
§ 04How the release answers the advisory group
Two weeks earlier, the independent Advisory Group on Mathematics and Artificial Intelligence, hosted at the Institute for Advanced Study, published its guidelines for releasing AI-generated mathematics (September 29). They open with a request to labs: “we ask them to stop testing advanced mathematical problems on proprietary models”. OpenAI says it consulted the group and drew on those recommendations. The release follows several of them: problem counts, compute estimates, reasoning summaries and Lean files are published. Two points are still open by OpenAI’s own account. The model is not public; OpenAI says it is “working to responsibly release the model that produced these results”. And the papers sit on OpenAI’s own GitHub; OpenAI writes that it is “continuing to explore other community-hosted alternatives for this release which meet the committee’s guidelines.”
OpenAI also commits money to the reading: “We will be funding a series of workshops, conferences, and special programs around the understanding of major results produced by AI”. This is its second public batch: an earlier repository, openai/ten-proofs, held ten results, and our record of its Navier-Stokes claim covers the September episode.
§ 05The first week: withdrawals and reactions
| Date | What happened | Source |
|---|---|---|
| October 7 | OpenAI withdrew three manuscripts after a sign error broke one argument and two dependent constructions; 14 others revised | Retraction Watch, October 8 |
| October 8 | Tristan Buckmaster (NYU) on CNBC’s Squawk Box: early-career research programs wiped out | Business Insider, October 10 |
| October 8 | The Association for Human Mathematics urged mathematicians to stop working with OpenAI | AHM statement, relayed October 8 |
| October 8 | Fields medallist Hugo Duminil-Copin, at a Société mathématique de France video call: 16 questions he worked on are covered | Le Monde, October 10 |
The errata came first. A day after the release, OpenAI withdrew three papers: one sign error invalidated an argument in one manuscript and the construction two others relied on, Retraction Watch reported, and the withdrawn papers carry notices explaining the gap. An OpenAI spokesperson told the outlet the errors surfaced in an audit, that the advisory group had recommended releasing without waiting for full formalization, and that roughly 50% of the results went out unconfirmed: “We welcome scrutiny and feedback from the mathematical community.” That fits the line the README had already printed, that some unformalized results could have issues. Le Monde now counts 719 manuscripts, which matches 722 less the three withdrawn.
The reactions were about the field, not only the proofs. On CNBC, Buckmaster said: “So we’ve had whole research programs wiped out on Tuesday by their dump,” per Business Insider, which reported that OpenAI did not respond to its request for comment. The Association for Human Mathematics wrote: “Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power.” And Le Monde quoted Duminil-Copin, the 2022 Fields medallist, telling a video call that 16 questions he cared about were among the claims: “Ils ont annihilé mon domaine de recherche” (they wiped out my research field). He said he cried on the Tuesday night of the release.
None of this changes which claims are checked. The Lean column in the table above is still the line between a computer-verified main result and a manuscript waiting for readers, and the first week has shown the second group is where errors turn up.
§ 06October 15: a $15 million fund to read the results
Nine days after the release, the Simons Foundation announced a $15 million effort with OpenAI and Anthropic, initiatives that will “help the mathematics community navigate the rapid rise of artificial intelligence in the field.” The foundation manages and awards the money, including through an open call for proposals, for workshops and programs that help researchers evaluate AI-assisted results, learn the tools and set shared standards. The $15 million is “combined funding and resources”, not a cash pool.
| Party | Commitment | In its own words |
|---|---|---|
| OpenAI | $5 million, plus model access and compute for Simons-supported researchers | Funding for workshops, conferences and special programs, including on the results in this release |
| Anthropic | $5 million of funding, compute and Claude access | Workshops, conferences and lectures by domain experts |
| Simons Foundation | Manages and awards the $15 million total | An open call, community-driven awards |
OpenAI added the commitment as an update at the top of its own release post: “we are contributing $5 million to a new Simons Foundation initiative to help mathematicians examine, understand, and build on results produced by AI, including those shared in this post.” That turns the workshop pledge quoted above into a number. Anthropic’s post on X says it has “committed $5 million of funding, compute, and Claude access to help the math community navigate the massive changes that AI brings.” Neither company breaks out cash versus compute, and the foundation does not say how much of the total is its own.
The foundation’s president, David Spergel, put the governance point first: “The community should be driving discussions of how these tools are evaluated, adopted and used, not any single company or institution.” After a first week of withdrawals and protest, that is the answer on offer: the labs pay, a foundation the field already trusts decides who gets the money. The Simons Foundation also notes it funded the development of the Lean proof assistant, the tool behind the 162 checked papers on this page.
§ 07What we are watching
- Independent checks. The first mathematician write-ups on individual families, starting with the zero-free half-plane and the free group factors, and any further withdrawals after the three on October 7.
- The advisory group’s answer. OpenAI told Retraction Watch it will ask the group whether its release recommendations change after this one.
- More Lean files. How fast the 162 formalized papers grow toward 722.
- A community-hosted copy. Whether the papers move to a repository OpenAI does not control, as it says it is exploring.
- The model itself. OpenAI’s release plan for the model that produced the results.
- The Simons open call. Who wins the first awards, and whether any fund independent checks of the 560 unformalized papers.
§ 08Update log
- October 6, 2026: page opened, the evening of the release.
- October 11, 2026: added the first week, the October 7 withdrawals and the reactions from Buckmaster, the Association for Human Mathematics and Duminil-Copin.
- October 16, 2026: added the $15 million Simons Foundation initiative with OpenAI and Anthropic, announced October 15.
§ 09Sources
- OpenAI, “Sharing AI progress in mathematics”, October 6, 2026: openai.com
- OpenAI mathematics repository (README, overview catalogue, manuscript map, Lean catalogue): github.com/openai/math
- Lean formalization catalogue: lean/formalization.yaml
- Advisory Group on Mathematics and Artificial Intelligence, “Responsible Release of AI-Generated Mathematics”, September 29, 2026: agmai.org
- OpenAI ten-proofs repository: github.com/openai/ten-proofs
- Retraction Watch, “OpenAI withdraws three preprints a day after releasing 722 manuscripts”, October 8, 2026: retractionwatch.com
- Business Insider on Tristan Buckmaster’s CNBC interview, October 10, 2026: businessinsider.com; CNBC video, October 8: cnbc.com
- Simons Foundation, “Simons Foundation Launches $15M Effort To Help Mathematicians Navigate AI Era”, October 15, 2026: simonsfoundation.org
- Anthropic, post on X, October 15, 2026: x.com/AnthropicAI
- Le Monde, October 10, 2026 (free portion): lemonde.fr
Q1What did OpenAI release on October 6, 2026?
OpenAI published 722 mathematics manuscripts produced by an unreleased internal model, grouped into 372 result families across 17 fields. They sit in the public GitHub repository openai/math under the Apache 2.0 license, with an overview catalogue, Lean formalizations and 10 reasoning summaries.
Q2Are OpenAI's math results verified?
Partly. Lean formalizations cover the main result of 162 papers, which means a computer checked the formal proof of the formal statement. The other 560 papers have no formalized main result yet, and the README warns that some unformalized results could have issues.
Q3Did OpenAI prove the Riemann hypothesis?
No. The catalogue claims a zero-free half-plane Re(s) > 7/8 for every Dirichlet L-function, including the Riemann zeta function, which it calls the quasi-Riemann hypothesis. The Riemann hypothesis itself concerns Re(s) > 1/2. The 7/8 result has a Lean formalization in the catalogue.
Q4Which model produced the results?
An unreleased internal OpenAI model. OpenAI says the average result used about three hours of ChatGPT Pro thinking and that it is working to responsibly release the model.
Q5Has OpenAI withdrawn any of the math papers?
Yes. On October 7, 2026 OpenAI withdrew three manuscripts after a sign error invalidated an argument in one and the construction in two that depended on it, and revised 14 others, Retraction Watch reported. An OpenAI spokesperson said roughly half the results were released before formal confirmation.
Q6What is the Advisory Group on Mathematics and AI?
An independent group hosted at the Institute for Advanced Study that published guidelines for releasing AI-generated mathematics on September 29, 2026. OpenAI says it consulted the group; the guidelines ask labs to stop testing advanced problems on proprietary models.