OpenAI Releases 722 Math Manuscripts From an Unreleased AI Model

In a research post published October 6, 2026, OpenAI said it is releasing a broad range of new mathematical results produced by an unreleased internal frontier model, publishing 722 manuscripts organized into 372 families in a public GitHub repository alongside Lean proof formalizations and abridged summaries of the model’s reasoning.

Inside the GitHub Repository

The results are published in the openai/math repository, released under an Apache-2.0 license. Its README describes the contents as mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model, assembled as part of model-development evaluations on open research problems.

The current catalogue contains 722 manuscripts organized into 372 families. A family groups related papers, which may include a principal result, companion arguments, consequences, or alternative proofs, and each family is classified by mathematical discipline. A preprints directory holds PDFs, source files, and manuscript-specific citation and build instructions, while a Lean library and formalization catalogue describe the available formal proofs, their associated papers, and verification configurations.

The collection spans different stages of verification. Many manuscripts have accompanying formalizations in Lean, a programming language that allows mathematical proofs to be checked by a computer, but not all do. The README cautions that some of the unformalized results could have issues, and OpenAI says it will endeavor to fix any such issues quickly while updating the repository with more formalizations as they are obtained. OpenAI also says it will preserve the collection’s public release history, recording corrections and revisions as new versions while keeping previously released versions accessible, and each manuscript directory carries a BibTeX block for citations.

How the Results Were Produced

According to the README, the vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. Over the course of the evaluation, the model was posed approximately 4,000 problems, and OpenAI reports that the average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking. The README says OpenAI expanded these evaluations after performance on its existing mathematical evaluations saturated, and that some outputs build upon earlier results produced by its models. Requiring an appropriate level of significance and aggregating the output into result families and manuscripts produced the published catalogue.

Two named exceptions departed from that fixed procedure: work on a zero-free region for the Riemann zeta function and a proof of the Hodge Conjecture for CM abelian varieties. The writeup covering the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability.

The repository also publishes 10 abridged summaries of the model’s reasoning. They cover results on ordinary two-point correlations of multiplicative functions, the irrationality exponent of π, the symmetric and general Mahler conjectures, ordinary NP-hardness at the basic semidefinite threshold, quasipolynomial bounds for arithmetic progressions, Kaplansky’s direct-finiteness conjecture in characteristic two, the Mézard–Parisi formula for diluted spin glasses, spontaneous magnetization in the quantum Heisenberg ferromagnet, the isomorphism of free group factors, and the three-dimensional relativistic Vlasov–Maxwell system.

OpenAI said it is publishing these additional details, including compute estimates expressed in terms of ChatGPT Pro usage and statistics about the number of attempted problems, to promote scientific transparency and openness.

Advisory Group Recommendations

OpenAI said it has been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study to develop best practices for sharing results with the math community, and that it drew on the group’s advice and public recommendations to inform this release.

The advisory group published its responsible-release recommendations on September 29, 2026, after receiving more than 600 replies from the mathematical community, and says the recommendations are supported by a clear plurality of respondents. A footnote states that the survey asked about a specific situation in which OpenAI announced the existence of many results without giving details. The document opens by observing that some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community, then states: “we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.”

The recommendations call for results to be deposited in scholarly repositories that no AI lab controls, with persistent citable identifiers and appropriately recorded modifications, and urge labs to refrain from treating result releases as marketing vehicles for their models. For each released result, the group says labs should make public the model name, the prompts used, a summarized chain of thought, the time taken, and the estimated cost of computation, and says proofs should be formalized as far as possible. When many results are released at once, the recommendations call for a further public document referencing all of them and explaining how many other problems of comparable difficulty the models tried and failed to solve, as well as how the problems were chosen.

The group further recommends that labs provide significant support, including funding, for the development of human understanding of AI-generated output, citing conferences, summer schools, targeted workshops, longer-term working groups, and books or long expository articles as candidate activities. It says decisions about which efforts to support should rest with existing nonprofit institutions rather than being directed by the labs, and it advises granting the global mathematical community broad, equitable access to publicly available models.

OpenAI said it is continuing to explore other community-hosted alternatives for this release that meet the committee’s guidelines, and that it is committed to improving the citations, mathematical exposition, and presentation of results in future releases.

Internal Model Background and Next Steps

The phrase “internal frontier model” in OpenAI’s post links to the lab’s September 8, 2026, announcement that an internal OpenAI system had produced a proof, with a Lean formalization, showing that smooth three-dimensional fluid motion governed by the Navier–Stokes equations can develop a singularity in finite time. OpenAI said the result resolves a problem the Clay Mathematics Institute named one of its seven Millennium Prize Problems in 2000, and that it does not intend to claim the Millennium Prize for it. In that post, OpenAI said the internal model it used is significantly more capable than GPT-6 Astra, that it had been training the model since August 28, 2026, that it launched an evaluation effort on Millennium Prize problems on September 1, 2026, and that its agents arrived at the resolution on September 5, 2026.

OpenAI said it will fund a series of workshops, conferences, and special programs around the understanding of major results produced by AI, and will share more on this in the near future. It also said it is working to responsibly release the model that produced these results, and that it will continue to act on feedback from the community and update its standards for future disclosures of major scientific advancements.