OpenAI Mathematics Collection

The OpenAI Mathematics Collection is a public collection of mathematical manuscripts and proof artifacts produced by an internal OpenAI model. OpenAI announced the release on October 6, 2026. Publication began a process of mathematical community assessment, rather than establishing acceptance of every result.^[1][7]

Scope and catalogue

The catalogue listed 719 manuscripts in 372 result families on October 10, 2026.^[3] A family groups related work, which can include a principal result, supporting arguments, consequences, or alternative proofs.^[2]

The catalogue sizes follow the overview; the formalization count and revision information come from the October 7 history notice.^[3][4] The overview organizes families by mathematical subject rather than importance. Its disciplines include number theory, algebraic and complex geometry, analysis, theoretical computer science, combinatorics, mathematical logic, probability, mathematical physics, topology, and partial differential equations. Each family summary links to its associated manuscripts.^[3]

How the work was produced

OpenAI describes the collection as part of its evaluation of mathematical reasoning on open research problems after existing evaluations saturated. The model attempted approximately 4,000 problems. OpenAI grouped selected outputs into families and manuscripts and applied a significance requirement, so the catalogue is a selected collection rather than a complete attempt log. Most results followed the same procedure using an unreleased internal model.^[2]

The company estimates that the average result used computation equivalent to roughly three hours of ChatGPT Pro thinking. The release includes ten abridged reasoning summaries. These accompany selected results rather than every manuscript.^[1]

Reading and checking the artifacts

The repository provides an overview, a manuscript map, individual preprint directories, and a Lean library. Manuscript directories include citation and build instructions. OpenAI says revisions will retain earlier public versions and provides manuscript-specific BibTeX citations.^[2]

The Lean library contains formalizations of some results and recommends compiling small portions rather than the entire library at once. Its documentation links to Comparator instructions for checking the supplied artifacts.^[5]

Formalization coverage needs to be read at the level of the individual result. The Comparator documentation identifies several configurations that check supporting results rather than the corresponding paper's main theorem. Examples include seam compatibility in a character-variety construction and ring-theoretic properties of a completed surface cone. Those checks should not be described as formalizations of the full main theorem.^[6]

Corrections and withdrawals

The history notice dated October 7, 2026 records three withdrawals. A sign error in the manuscript on algebraicity of Weil classes on split abelian eightfolds invalidated a cancellation argument and a construction used by two dependent papers involving K3 surfaces. The withdrawn directories retain explanations and links to the archived manuscripts.^[4]

The same notice records substantive revisions to 14 other manuscripts, reference updates in 13 additional manuscripts, six new formalizations, and five additions covering supporting results.^[4]

The repository warns that unformalized results may contain issues.^[2]

Mathematical community assessment

OpenAI consulted the independent Advisory Group on Mathematics and Artificial Intelligence on release practices.^[1] In its October 6 response, the group stated that its advisory role did not endorse OpenAI's process or judge the impact of the results. It described publication as the start of community assessment and human understanding, rather than completion of that work.^[7]

The group's September 29 guidelines ask AI labs to stop testing advanced mathematics on proprietary models inaccessible to the community. They distinguish papers understood by a responsible mathematician from output that nobody yet understands. They call for precise statements, appropriate attribution, revision records, disclosure of the computational process, and clearly stated formalization coverage. They also recommend community-led support for developing human understanding and broad access to research tools.^[8]

References

1. ^1 ^2 ^3OpenAI. Sharing AI progress in mathematics. October 6, 2026.
2. ^1 ^2 ^3 ^4OpenAI. Mathematics repository README. Accessed October 10, 2026.
3. ^1 ^2 ^3OpenAI. OpenAI Research Catalog, overview source. Catalogue dated October 6, 2026; accessed October 10, 2026.
4. ^1 ^2 ^3OpenAI. Mathematics collection history. Revision notice dated October 7, 2026.
5. ^OpenAI. Lean formalizations README. Accessed October 10, 2026.
6. ^OpenAI. Comparator challenges and supporting-result comparisons. Accessed October 10, 2026.
7. ^1 ^2Advisory Group on Mathematics and Artificial Intelligence. On OpenAI's release of mathematical results. October 6, 2026.
8. ^Advisory Group on Mathematics and Artificial Intelligence. Responsible release of AI-generated mathematics. September 29, 2026.