OpenAI withdraws three AI-generated math papers over a sign error

OpenAI has withdrawn three papers from the AI-generated mathematics collection it released on October 6, 2026. According to the October 7 entry in history.md, the file that tracks updates to the repository, a sign error was found in the paper "Algebraicity of Weil classes on split abelian eightfolds". The error invalidates a simplification argument in that paper and the construction used by two dependent papers. The other two withdrawn papers are "Algebraicity of Kuga-Satake Correspondences for K3 Surfaces" and "The rational Hodge conjecture for products of K3 surfaces". GlobalFeed reported the original release the day before.

The withdrawn papers claimed results connected to the Hodge conjecture, one of the best-known open problems in mathematics. Part of the collection has not yet been checked by computer, and this entry is the first update showing how OpenAI handles results found to be flawed. According to the withdrawal notices, the decision concerns the proofs, and OpenAI does not claim that the mathematical statements are false. With the change, the catalogue went from 722 to 719 papers, while the number of result families stays at 372.

A sign error kept a count away from zero

A sign error means counting a value as positive when it should be negative, or the reverse. According to the withdrawal notice, the proof counts the points where a surface crosses itself (double points) with plus and minus signs, and it needed that signed total to be zero. When the total is zero, a known cancellation theorem by Eliashberg and Murphy can be applied to remove those crossings. The paper counted each piece it added to balance the total as plus one. With the orientations accounted for correctly, each piece counts as minus one, so the added pieces bring the total to twice its starting value. Because the theorem's condition is not met, the later steps of the proof are left unsupported.

The error carried over to two papers built on the same construction

In mathematics, a proof often rests on a construction or result established earlier. When the underlying step fails, proofs built on top of it lose their support as well. According to the withdrawal notices, the two papers on K3 surfaces used an adaptation of the flawed construction, so their main theorems are now considered unproved. The pages of the withdrawn papers now carry a notice explaining the gap and a link to an archived copy of the pre-withdrawal text. The notices give October 6, 2026 as the withdrawal date. The entry does not say who found the error or how.

The same update fixed 14 papers

According to OpenAI's update log, the same update revised 14 papers with proof repairs, corrected statements and clearer hypotheses. These include four papers on Lipschitz heights and Ashkin-Teller currents, six papers on Kähler minimal model programs, and a corrected equality claim in the paper "Taming implies compatibility". As a result of these fixes, 13 more papers were updated to cite the revised editions of companion papers. The repository keeps earlier editions of revised papers accessible.

More proofs can now be checked in Lean

Lean is a programming language that lets a computer check a mathematical proof step by step. Because every step of a completed Lean proof is checked by the computer, calculation errors of this kind can be caught. The update added 6 new Lean formalizations and 5 other additions covering supporting results. According to the entry, as of this update the top-line results of 300 of the 719 papers are formalized in Lean. The repository states that not all papers have accompanying Lean formalizations and that some unformalized results could have issues. The repository's formalization catalogue did not list a Lean counterpart for any of the three withdrawn papers.

The results in the collection are the company's claims and await independent review. A Lean proof checks the formal statement given to the computer; whether that statement fully matches the claim in the paper is left to mathematicians to judge. Since the model that produced the papers is not available, the process cannot yet be reproduced from outside. For a fuller picture, follow evaluations from independent researchers as well.

Sources: GitHub: openai/math history.md, GitHub: withdrawal notice for Algebraicity of Weil classes, GitHub: openai/math, OpenAI: Sharing AI progress in mathematics.