THE SIGN SLIP THAT SANK THREE PROOFS

The Hodge conjecture is one of the deepest open questions in mathematics. Its statement, in this paper, is taken from a chapter by Pierre Deligne in the Clay Mathematics Institute’s book The Millennium Prize Problems. Very roughly, it says that certain “shapes” hidden in the topology of a geometric space defined by polynomial equations can always be built out of smaller pieces that are themselves defined by equations.

In September and October 2026, OpenAI released three documents, signed “OAI”, claiming to prove the conjecture in special cases: for certain surfaces called K3 surfaces, their products, and certain eight-dimensional spaces. OpenAI later withdrew all three. Yankı Lekili, of Imperial College London, and Nick Sheridan, of the University of Edinburgh, now explain exactly where they went wrong.

A valid route through the mirror

The documents took an elegant path. A powerful idea called homological mirror symmetry, conjectured by Maxim Kontsevich, links a geometric space X to a “mirror” space Y of a quite different kind, studied in symplectic geometry — a geometry inherited from the physics of motion, where areas rather than lengths are measured. Once the link is established, a question about X can be traded for a question about Y.

Here, to settle the Hodge question for X, it is enough to build a particular object in the mirror Y: a surface-like shape called a Lagrangian immersion, which must also be unobstructed — a technical condition that makes it a legitimate member of the mirror’s toolkit. Lekili and Sheridan agree this is a valid approach. The catch: building unobstructed Lagrangians is itself a hard problem. That is where all three documents broke down.

Where the crossing points come in

An immersion is allowed to cross itself, and each crossing point carries a number, its degree. Crossings of odd degree are harmless here. Crossings of degree 2 can create an obstruction.

The documents first built a rough immersion using the h-principle of Gromov and Lees, a general-purpose tool from differential topology, of the kind mathematicians call flexible. Then they invoked a 2013 theorem by Tobias Ekholm, Yakov Eliashberg, Emmy Murphy and Ivan Smith to tidy it up, so that every crossing would have odd degree. With no even crossings left, unobstructedness would follow.

A plus that should have been a minus

The 2013 theorem is correct. But it counts crossings with the opposite sign convention from the standard one. Its authors never say so explicitly; they use their convention consistently, and it can be deduced from their formulas and proof. The OpenAI documents used the standard convention instead — one sign flipped. This matters because the theorem’s outcome depends on whether a certain tally of the crossings, counted with pluses and minuses, comes out negative or not. The documents had built their immersion so that the tally was negative — the case in which they believed every crossing could be made odd. With the theorem’s own sign, the same tally is positive: the other case. The issue was apparently first spotted and reported to OpenAI by Nathan Humphrey.

With the sign read correctly, the theorem tells a different story. Its construction always leaves one crossing point of degree 2, which Lekili and Sheridan call p. Every other crossing can be made odd; p stays. And p is enough to break the proof.

It is worse than a gap. The authors show that the immersions produced this way are typically obstructed: a small “rigid” disc — an isolated surface, impossible to deform, whose edge lies on the immersion and passes through p — creates an obstruction that, in a certain regime, cannot vanish — and outside that regime there is no reason it would cancel. In their words, this crossing is “the cost of flexibility”.

Rigid versus flexible

Mathematicians in this field distinguish flexible problems, which yield to general-purpose tools from differential topology, from rigid ones, which do not. Unobstructed Lagrangians are generally rigid, and one can usually go from rigid to flexible but not back. That intuition made several researchers sceptical of the documents before they were withdrawn. Yet the authors add a note of humility: intuition can fail, and the 2013 theorem itself once contradicted expectations.

What survives

Lekili and Sheridan do not analyse the rest of the arguments, and they list three positive outcomes. The overall strategy — attacking the Hodge conjecture through mirror symmetry — is interesting and worth pursuing; if someone overcomes the Lagrangian obstacle, the documents may guide the remaining technical steps. The hidden sign convention of the 2013 paper is now clarified. And so is the question of whether its immersions can ever be unobstructed.