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Why OpenAI withdrew three Hodge conjecture papers: the sign error explained

OpenAI withdrew three AI-produced papers on the Hodge conjecture within two days. A new arXiv note by Lekili and Sheridan explains where the sign error came from.

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On this page
  1. What does the Hodge conjecture ask?
  2. What does mirror symmetry have to do with it?
  3. What did OpenAI's three Hodge papers claim?
  4. How and when did OpenAI withdraw the papers?
  5. Who found the error?
  6. What do Lekili and Sheridan say went wrong?
  7. Can the proofs be repaired?
  8. What does the note not analyze?
  9. A claim's life cycle, step by step
  10. Where to follow the claims on Papex
  11. Sources

On 22 October 2026, two mathematicians posted a nine-page note to arXiv explaining where three of OpenAI's papers on the Hodge conjecture went wrong. In "Rigidity, flexibility, and a sign error", Yankı Lekili of Imperial College London and Nick Sheridan of the University of Edinburgh trace the failure to a sign convention. The papers counted the self-crossings of a geometric object one way; the 2013 theorem they relied on counts them the other way. OpenAI had released the three papers on 6 October, among hundreds of manuscripts produced by an internal model, and withdrew them within two days.

The three papers have now been through a full life cycle in public. A company released them and then withdrew them with notes of its own. An independent reader checked the signs and reported the problem, and two specialists diagnosed it in the open literature. Each step was recorded in a different place, by different people, with its own date.

Below: what the Hodge conjecture asks, how mirror symmetry was meant to help, what the papers claimed, how they were withdrawn and diagnosed, and where to follow them on Papex. Every source was checked on 30 October 2026. Papex does not assess the mathematics; we report what each source says.

What does the Hodge conjecture ask?#

It asks when the shape of a solution set can be described by further equations. The Clay Mathematics Institute, which lists it among its seven Millennium Prize Problems, puts it this way: "The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations."

Take a shape cut out by polynomial equations, which mathematicians call a projective variety. Topology describes it through its cohomology classes, a kind of catalog of its holes and cycles in each dimension. Some of those classes come from smaller shapes inside it that are also cut out by polynomial equations. These are algebraic cycles. Every class that comes from an algebraic cycle passes a test from Hodge theory: it is of "type (p, p)". The rational classes that pass this test are called Hodge classes.

The conjecture says the test is enough. In Pierre Deligne's official problem description for Clay: "On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles."

W. V. D. Hodge posed the question in an address to the 1950 International Congress of Mathematicians. Clay's page says the conjecture "is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown." Deligne names the obstacle: "Progress is blocked by a lack of methods to construct interesting algebraic cycles."

That last sentence explains why the OpenAI papers drew attention. To prove a case of the conjecture, you have to build an algebraic cycle that represents a given Hodge class. The three papers proposed a new way to build one.

What does mirror symmetry have to do with it?#

Mirror symmetry is a phenomenon first observed by physicists studying string theory. In his talk at the 1994 International Congress of Mathematicians, Maxim Kontsevich proposed "a refined version of the Mirror Conjecture formulated in terms of homological algebra", now called homological mirror symmetry.

It pairs a space X with a "mirror" space Y. X is studied with algebraic geometry, through equations. Y is studied with symplectic geometry, a different branch built around areas. The conjecture says that two collections of objects match exactly:

  • on X, algebraic objects: subvarieties, vector bundles and their generalizations, called coherent sheaves;
  • on Y, Lagrangian submanifolds, surfaces of exactly half the dimension of Y with a special property, which make up what is called the Fukaya category.

Lekili and Sheridan's note sets out how the OpenAI papers meant to use this. First, show that the variety X satisfies homological mirror symmetry; the note credits the key results for the relevant cases to Mohammed Abouzaid and Paul Seidel. Then, to represent a Hodge class on X by an algebraic cycle, "it then becomes sufficient to represent a corresponding homology class" on the mirror Y "by an unobstructed Lagrangian immersion". Build the right Lagrangian on the mirror side, carry it across, and you have an algebraic object on X that carries the class.

The catch is the word "unobstructed". Only unobstructed Lagrangians count as objects on the mirror side. A Lagrangian immersion is allowed to cross itself, and each crossing point has a degree. Holomorphic discs with boundary on the Lagrangian contribute to an "obstruction", and the Lagrangian is unobstructed if a correction term can cancel it. Crossing points of degree 2 can feed the obstruction; crossings of odd degree cannot.

Think of mirror symmetry as a dictionary between two languages. A hard sentence in one (an algebraic cycle) can be written in the other (a Lagrangian) and translated back. But the dictionary only accepts well-formed sentences, and showing that a Lagrangian is well-formed is itself hard. Lekili and Sheridan: "This is a valid approach to representing Hodge classes by algebraic cycles. The problem is that constructing unobstructed Lagrangians is also a hard problem."

What did OpenAI's three Hodge papers claim?#

On 6 October, OpenAI opened the GitHub repository openai/math. Its first README described the contents as "mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model". OpenAI's announcement that day: "We're releasing a broad range of new mathematical results produced by an internal frontier model."

Three of the manuscripts claimed cases of the Hodge conjecture through homological mirror symmetry:

Paper (all credited to "OpenAI")Date on the paperPagesMain claim, from the abstract
Algebraicity of Weil classes on split abelian eightfolds18 September 202681"We prove that every rational Weil class on a split abelian eightfold of Weil type is algebraic."
Algebraicity of Kuga–Satake Correspondences for K3 Surfaces3 October 202669"We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface."
The rational Hodge conjecture for products of K3 surfaces4 October 202665"We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic."

Each is a special case, not the full conjecture. The eightfold paper says so itself: it "makes no assertion about all Hodge classes on arbitrary abelian eightfolds or about the Hodge conjecture for arbitrary smooth projective varieties." It presents the eightfold case as a step beyond Eyal Markman's work on fourfolds and sixfolds. The Kuga–Satake paper describes its question as "a particular case of the Hodge conjecture, with a specified tensor to be realized."

None of the three had a Lean formalization. The catalog of formal proofs in the repository's first version does not list them, and that version's README warned: "Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly."

The same release contained other Hodge-related manuscripts, among them one claiming the conjecture for abelian varieties with complex multiplication. OpenAI has not withdrawn those. They are separate claims, and this post is not about them.

How and when did OpenAI withdraw the papers?#

Each paper's README in the repository was replaced by a withdrawal notice. The notice for the eightfold paper explains the error in a few lines. In the paper's convention, the proof starts from a signed count of crossing points, I(f₁) = −m. It then inserts m local pieces, "reverse stabilization traces", that it counts as +1 each, so the total should come to zero. The notice:

Accounting for the opposite source orientations of the two branches of the standard cusp gives sign −1 for each reverse trace in this convention.

So the new count is −2m, not zero. "The Eliashberg-Murphy cancellation theorem invoked at this step requires zero signed double-point count, so its hypothesis is not met. The subsequent oriented-surgery and embedded-brane construction is therefore unsupported, and the paper does not establish its claimed algebraicity theorem."

The other two notices are shorter. Each says the proof "relies on an adaptation of the flawed stabilization-trace construction" in the eightfold paper, and that "the resulting gap" leaves the claimed theorem unproved. All three end with the same sentence: "This withdrawal concerns the proof; it does not assert that the mathematical statement is false."

The repository's change log records the decision in one line: "In 'Algebraicity of Weil classes on split abelian eightfolds' a sign error invalidates a stabilization-trace cancellation argument and the construction used by two dependent papers." The same update revised 14 other manuscripts with proof repairs and corrected statements.

OpenAI did not delete the withdrawn work. Each notice links to the pre-withdrawal PDF at a fixed revision of the repository, so anyone can still read exactly what was claimed. That turned out to matter: Lekili and Sheridan cite the papers by those fixed links.

The withdrawal carries three dates, depending on where you look:

  • the notices say "Withdrawn on October 6, 2026", the day the repository opened;
  • the change log lists the withdrawals under 7 October;
  • the commit that added the notices to the public repository is timestamped 05:03 UTC on 8 October.

The sources do not explain the difference.

Who found the error?#

OpenAI's notices and change log do not say how the error came to light. Lekili and Sheridan do: the sign problem "was apparently first pointed out and reported to OpenAI by Nathan Humphrey".

Humphrey published his check on 7 October in two places: a GitHub repository, created at 18:17 UTC, and a Substack post a few minutes later, titled "A sign slip in OpenAI's split-eightfold proof, and why the natural repair looks blocked". He had wanted to use the paper, not refute it. He had been working on Weil classes himself: "I tried to use that route for my case and ran into a peculiar problem in Proposition 3.1."

His post gives a check that can be done by hand. Take a sphere immersed with a single crossing point, in dimension 8. The paper's self-intersection formula gives a count of −1; the formula of Ekholm, Eliashberg, Murphy and Smith gives +1. "Same object, opposite sign." Switching to the other convention throughout does not help either, he wrote: then the total is +2m instead of −2m. He also wrote: "I'm not saying the theorem is false", and "I would be happy to retract or correct if someone finds an error."

The repository's README says the work was "reported to OpenAI the same afternoon" and "Worked out by Nathan Humphrey with Claude and a GPT model". An error in a model-produced paper was found, by his own account, by a reader working with AI tools.

What do Lekili and Sheridan say went wrong?#

Their abstract is two sentences: "The documents [OAI26a, OAI26b, OAI26c] claimed to prove cases of the Hodge conjecture via homological mirror symmetry, before being withdrawn by OpenAI. We discuss the source of the error (but do not analyse the rest of the argument)."

According to the note, all three papers "broke down in the same way, at the point where they attempted to construct an unobstructed Lagrangian immersion". They first built a Lagrangian immersion using the Gromov–Lees h-principle, a general method for building such objects. They then claimed, citing Theorem 1.1 of a 2013 paper by Tobias Ekholm, Yakov Eliashberg, Emmy Murphy and Ivan Smith, that it could be modified so that every crossing point had odd degree. That step was crucial, because only crossings of even degree, "in fact only those of degree 2", contribute to the obstruction.

But the 2013 construction, the note says, "always produces an immersion with a distinguished self-intersection point p of degree 2; in particular, it can never eliminate all even self-intersection points." With that one crossing left, the proof that the Lagrangian is unobstructed breaks down.

Where did the mistake come from? "The error arose because [OAI26a, OAI26b, OAI26c] computed signs of self-intersection points with the standard convention, whereas [EEMS13] implicitly use the opposite convention." In the note's notation, the OpenAI papers assumed a number I(f) equals J(f), when it equals −J(f). Had the signs been the other way round, the note says, the hypothesis of the unobstructedness theorem "would have been satisfied".

The note does not blame the 2013 paper. Its authors "do not say explicitly that their results are stated using the opposite of the standard convention; on the other hand, they use this convention consistently, the convention can be deduced from formulae in the paper and from reading the proof, and the results are correct if their choice of convention is understood."

The eightfold paper took a slightly different route, and the note treats it separately in a remark. That paper resolved all the crossings to get an embedded Lagrangian. The note explains that this needs every crossing to be odd, so "the even self-intersection point provided by the correct statement of Theorem 3.2 will destroy the grading of the Lagrangian."

Can the proofs be repaired?#

The note goes beyond the broken step. "Not only does the proof of unobstructedness break down, the Lagrangian immersions produced by [EEMS13, Theorem 1.1] are typically obstructed." The leftover degree-2 crossing has a rigid holomorphic disc ending on it, and that disc produces a leading term in the obstruction. The note explains why this makes the Lagrangian obstructed in a certain regime, and adds: "Even outside this regime, there is no reason to expect that this term in the obstruction would be cancelled."

The note's title refers to a distinction in symplectic topology. Flexible problems can be solved with the general tools of differential topology, such as Gromov's h-principle. Rigid problems cannot. In the note's words, "unobstructed Lagrangians are generally rigid. Typically one can get from something rigid to something flexible, but not the other way round." The leftover degree-2 crossing "is the cost of flexibility". This, they write, "caused several researchers to be sceptical" of the papers' construction before they were withdrawn. They add that "such intuitions can of course be mistaken".

The note does not use the word "irreparable". On 24 October the Proofs & Prompts newsfeed summarized it as explaining the error "and why it is probably irreparable without new inputs". That is the newsfeed's reading of the note. Humphrey's post makes a narrower argument about one route: that "the natural repair" through a different kind of filling "looks blocked". He calls that conclusion conditional on an analytic result that, he writes, has been proved only in a more specific setting.

OpenAI's own position is in its notices: the withdrawal "does not assert that the mathematical statement is false." The note does not say the theorems are false either.

What does the note not analyze?#

Most of the three papers. The abstract says so, and the section on strategy repeats it: the approach is described "modulo important technical details (which, however, are not the subject of this note and which we do not evaluate)". In particular:

  • The rest of each argument. The note examines one construction, the unobstructed Lagrangian, and nothing after it.
  • A side argument. The spaces in the OpenAI papers are not simply connected, so the papers supply their own version of the 2013 theorem. The note says this "is not relevant for the rest of this discussion" and does not check it.
  • Whether the statements are true. The note explains why the proofs fail. It does not claim the theorems are false.
  • OpenAI's other Hodge manuscripts. The note cites only the three withdrawn papers.

It also finds something to keep. The authors list three "positive outputs" of the release. The first: "The approach to cases of the Hodge conjecture via homological mirror symmetry which they propose is interesting and worth pursuing; to our knowledge it has not appeared in print as such, although it has been tried without success." If the key difficulty, building unobstructed Lagrangians, is overcome, the papers "may well provide a useful guide to other technical aspects of the problem." The other two outputs: the sign convention of the 2013 paper, and whether its Lagrangian immersions can be unobstructed, "has been clarified".

The note itself is a preprint; arXiv lists no journal reference for it.

A claim's life cycle, step by step#

Put the sources side by side and the three papers have a complete public history:

When (UTC)What happenedWhere it is recorded
18 September to 4 OctoberDates printed on the three manuscriptsThe PDFs
6 October, 21:58First commit of openai/math, with the three papersGitHub
6 OctoberOpenAI announces the releaseopenai.com
7 October, 18:17Humphrey publishes his sign check; he says he reported it to OpenAI that afternoonGitHub and Substack
8 October, 05:03A commit adds withdrawal notices dated 6 October and a change log entry under 7 OctoberGitHub
22 October, 09:18Lekili and Sheridan submit their notearXiv
24 OctoberProofs & Prompts summarizes the noteNewsfeed
31 OctoberPapex adds the note as a response to the three claimsPapex

Three things stand out.

Each step has different authors. The claim is credited to "OpenAI", for work its README attributes to an internal model. The withdrawal is signed "OpenAI". The catch comes from an individual working with AI tools, and the diagnosis from two academic specialists. A full record of the episode has to name all of them.

Fixed links made the diagnosis citable. Because OpenAI kept the pre-withdrawal PDFs at a fixed revision, Lekili and Sheridan could cite exactly the text they were analyzing.

Dates depend on the source. The notices, the change log and the repository history give three different dates for one withdrawal. A record meant to last should keep each date with its source rather than pick one.

Where to follow the claims on Papex#

The Hodge conjecture page on Papex lists the three papers as withdrawn claims, each with OpenAI's withdrawal note. With this post, we have added Lekili and Sheridan's paper as a response on all three. The claims also appear on /ai-claims, under "Revised or withdrawn".

Papex lists each claim as its source states it. We do not check proofs or decide whether a claim holds. The problem's status comes from its source: Clay lists the Hodge conjecture as unsolved, so it stays Open on Papex. For the other Millennium Prize Problems and how to read an AI claim about them, see Has AI solved a Millennium Prize Problem?.

Sources#

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