A single minus sign just killed three mathematical proofs.
On October 7, OpenAI published a large batch of mathematical results produced by its internal models on GitHub. The first version of the catalog contained 722 manuscripts, grouped into 372 result families.
A day later, three manuscripts were gone.
One of them, a geometry paper dated September 18 about the algebraicity of Weil classes, carried a short withdrawal note: "The proof contains a sign error."
The proof needed the positive and negative counts of geometric intersections to cancel to zero before a later theorem could be invoked. An operation expected to add a positive term had actually added a negative one. In simplified arithmetic, the authors intended "−2 + 2 = 0" and instead produced "−2 − 2 = −4." The theorem's conditions were never met, and two other manuscripts borrowing the same construction lost their support.
This is not a story about failure alone.
On the same day the repository went live, combinatorialist Gil Kalai called the release "an amazing milestone for mathematics." Then he added the necessary caution: the proofs still have to be verified and digested by human mathematicians.
The debate over AI mathematics had already begun weeks earlier. In September, a claimed proof about critical percolation from Claude triggered widespread discussion. On September 11, Terence Tao and 24 other Fields Medalists signed an open statement acknowledging the progress of AI in mathematics while criticizing companies that treat problem-solving as a benchmark race and neglect understanding and the transmission of knowledge.
One side points to results that could reshape a field. The other side points to withdrawals that have already happened. So what, exactly, did these AI-generated math manuscripts deliver?
What OpenAI Handed to Mathematicians
The public repository contains paper PDFs together with their typesetting source files, so peers can inspect the arguments line by line. Some manuscripts include Lean proof artifacts — mathematical statements and reasoning written as strict code that a program can check.
Ten papers also ship with abridged reasoning traces. They show part of the exploration process, including attempts that did not work.
These manuscripts are effectively preprints: research texts released openly for reading, discussion, and checking. A preprint can contain major results, but its appearance says nothing about external peer review.
OpenAI says the overwhelming majority of results came from a single pipeline and a single internal model. If the major claims hold, that means one model system participated in research across many branches of mathematics.
The announcement also converted the average compute per result into a familiar unit: roughly three hours of thinking time on ChatGPT Pro.
What 722, 719, and 372 Actually Count
The first two numbers count manuscripts. The third counts groups. OpenAI calls each group a result family; the repository itself uses the plainer phrase "related papers."
Think of a family as a research folder. Inside it, papers divide the labor: one proves the main theorem, one fills a critical gap, another extends the conclusion to different conditions. Each is a separate manuscript, while the folder shares one family number.
A single paper can contain several conclusions, and a single folder can answer several different questions.
The largest folder, family 034, contains fourteen manuscripts in algebra and complex geometry, including both a general argument and a version specialized to four-dimensional objects.
A smaller example makes the logic clearer. Family 088 has only two papers, both studying a geometric indicator loosely connected to the "shadows" a shape casts in every direction. One paper asks which shapes minimize that indicator. The other supplies a counterexample showing that the presumed maximizer is not maximal at all. Same indicator, different questions, one family.
Grouping like this is common. Of the 372 families, 205 contain a single paper and 167 contain multiple papers. The multi-paper families are less than half the total, yet they hold 514 manuscripts — about 71 percent of the current collection.
Seventeen Fields, Seven Dominant Clusters
Across OpenAI's own subject classification, the 372 families span seventeen fields. The seven largest groups account for 227 families, about 61 percent. Theoretical computer science leads with 40 families, combinatorics follows with 37, and algebra with complex geometry contributes 36.
The fields ask different kinds of questions. Theoretical computer science measures how much computation an algorithm requires. Combinatorics studies how discrete objects can be arranged and counted. Algebra and complex geometry examine the properties of shapes defined by equations.
Headcounts tell two stories. Theoretical computer science contributes 40 families and 73 manuscripts, while probability and statistical mechanics contributes 29 families and 105 manuscripts. The first field has more folders; the second has more pages.
Five Shapes a "Breakthrough" Can Take
Based on a sampled reading of the math manuscripts, the contributions fall into five forms. This is not OpenAI's grading system, and one paper can combine several of them.
An answer to an open question. Pi can be approximated endlessly by fractions. The research asks how fast the error can drop, infinitely often, relative to the denominator. The new paper claims the "irrationality exponent" measuring that speed equals 2.
Wider applicability for an existing conclusion. Studies of particles interacting with electromagnetic fields often require the initial state to be small or specially symmetric. The new paper claims both restrictions can be removed, under the stated model and initial conditions, while long-term smooth solutions still exist.
Crossing a line presumed impassable. Longer integers usually demand more multiplication steps. The new paper claims that, under one fixed machine model, the number of steps can grow more slowly than the long-conjectured limit.
A premise shared by many conclusions. Numerous studies assume the Unique Games Conjecture, then prove limits on how well certain algorithmic tasks can be approximated. If the new paper holds, those studies no longer need to list it as an unproven assumption.
A new route to a known answer. One geometry counterexample paper openly states that "their result already gives a negative answer," then offers a direct construction. The added value is the proof path itself — whether it yields a clearer explanation or a reusable method remains to be seen.
The Three Withdrawals and a Twenty-Seven-Year-Old "If"
The withdrawn papers are the algebraicity of Weil classes on split abelian eightfolds, the algebraicity of Kuga–Satake correspondences for K3 surfaces, and the rational Hodge conjecture for products of K3 surfaces.
The latter two both involve K3 surfaces, and both use rewritten versions of the faulty construction from the first.
Crucially, a withdrawal is not a verdict that the statement is false. A gap in one argument does not disprove the proposition, nor does it rule out a different, valid proof.
