Mathematical formulae (illustrative). Image: Wallpoper / Wikimedia Commons, Public domain, cropped OpenAI has put 722 maths manuscripts written by a model nobody outside the company can use into a public GitHub repository, and within a day it had withdrawn three of them. The company announced the release on Tuesday, October 6, saying the papers come from an “internal frontier model”.
The independent group of mathematicians OpenAI consulted said the same day that its advice should not be read as approval, and a second group of mathematicians called the release “a demonstration of power”.
What is in the repository #
The collection is grouped into 372 “families” of related papers, each with a principal result and companion arguments, according to its README. OpenAI says the model was set about 4,000 problems during its internal testing, and that each result used, on average, the equivalent of about three hours of ChatGPT Pro thinking.
About 42% of the headline results (300 of 719) come with formal proofs in Lean, a programming language that lets a computer check every step. The rest have not been machine-checked, and the README warns that “some of the unformalized results could have issues”. OpenAI has also published abridged summaries of the model’s reasoning for 10 results, including the irrationality exponent of π, the Mahler conjectures and Kaplansky’s direct-finiteness conjecture in characteristic two.
Two pieces of work fell outside the fixed procedure used for the rest: a zero-free region for the Riemann zeta function, whose write-up was edited by people for readability, and a claimed proof of the Hodge conjecture for CM abelian varieties. None of the papers has been through peer review, so every claim about what they solve is OpenAI’s own for now.
Three papers pulled in a day #
On October 7 the repository’s change log recorded its first withdrawals. A sign error in “Algebraicity of Weil classes on split abelian eightfolds” invalidated a key argument, OpenAI wrote, along with the construction two other papers relied on. All three, including “The rational Hodge conjecture for products of K3 surfaces”, were withdrawn, which took the count from 722 to 719.
OpenAI also revised 14 other manuscripts with “proof repairs, corrected statements, clearer hypotheses”, and updated 13 more to cite the fixed versions. Withdrawn papers stay online with a notice explaining the gap, and OpenAI says every revision will be kept as a new version rather than overwriting the old one.
“Not an endorsement” #
OpenAI said it shaped the release on advice from the Advisory Group on Mathematics and Artificial Intelligence (AGMAI) at the Institute for Advanced Study in Princeton, whose members include the Fields medallists Tim Gowers and Martin Hairer and the physicist Edward Witten. The group published guidelines for AI labs on September 29. In a statement on October 6 it called the release “an important event for mathematics” and its talks with OpenAI “constructive”, but added:
AGMAI’s advisory role should not be interpreted as a judgment of the impact of these results or an endorsement of the process by which OpenAI obtained them.
Advisory Group on Mathematics and Artificial Intelligence
It said only the wider mathematical community can judge the work, and whether its recommendations were followed. “This release is the beginning, not the completion,” the group wrote, warning that mathematicians must stay free to choose their own questions.
“A demonstration of power” #
The Association for Human Mathematics, a group that campaigns to keep maths a human endeavour, went further. In a statement from its communications working group, it said “mathematicians did not ask for this work to be done” and argued that OpenAI had ignored the advisory group’s central advice that labs should not test advanced problems on internal models. It rejected OpenAI’s claim that the release advances the subject:
Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power.
Association for Human Mathematics
It urged mathematicians to stop working with OpenAI. OpenAI hasn’t commented on either statement. It says it will fund workshops and conferences on results produced by AI, and that it is “working to responsibly release the model” itself, without giving a date or a name.
Checking hundreds of papers is a big job for a field already worried about being swamped: arXiv limited submissions to two a month last week after a record surge, and mathematicians have been arguing for weeks over OpenAI’s Navier-Stokes claim.
Why it matters #
This is the largest public batch of AI-generated maths so far, and putting it on GitHub with formal proofs and a change log means anyone can check it. The three withdrawals in a day show why that checking matters: until mathematicians have been through the papers, the solutions it reports to hundreds of open problems remain OpenAI’s claim, not established results.
Sources: OpenAI, openai/math on GitHub, AGMAI, Association for Human Mathematics