OpenAI is reportedly close to a proof of the Hodge conjecture, which would make it the second Millennium Prize problem the company has taken down. The first, Navier–Stokes, was handled with a variant of OpenAI's next pretrained model, codenamed "Doug," at an estimated cost of millions of dollars. What the company wants out of either result is unclear — apparently including to people who work there. Some employees reportedly believe the work feeds recursive self-improvement.
That belief sits awkwardly against what OpenAI said at the launch of Astra. Jakub Pachocki, the company's chief researcher, said then that OpenAI would concentrate first on recursive self-improvement, an AI's ability to optimize itself, and not on mathematics. One solved problem and a second within reach later, the internal account is that the mathematics was the self-improvement work all along. That is a tidy reframing of what otherwise looks like a detour.
Millions of dollars is the number attached to the Navier–Stokes run, and it reads large until you set it beside what it costs to pretrain the model that did the work. The ratio is the more interesting figure here: a Millennium Prize problem priced at a fraction of a single training budget. If that holds for the Hodge conjecture too, the constraint on this kind of result stops being compute and starts being whether anyone can check the output.
It is worth keeping the hedges in view. Close to solving is not solving, and reportedly is not announced. This is a secondhand account of an unfinished result produced by a variant of a model that has not shipped and that nobody outside the company can run.
The mathematical community's response has been closer to irritation than admiration, with some mathematicians treating the work as a threat to their field. Irritation strikes me as the rational reaction, though not mainly for the reason it will be read as. A system that can generate candidate proofs of the hardest open problems faster than humans can verify them does not take work away from mathematics. It relocates the work to the least rewarded part of the discipline. Someone still has to read the thing line by line, and that someone is not getting the Millennium Prize.
The question the reporting leaves open is what OpenAI plans to do with a proof, and the employees' answer runs backwards from how I would read it. A Millennium Prize problem is an unusually clean instrument: hard, unsolved, unmemorizable, and with an answer that is either right or wrong. If you are trying to measure whether a model can meaningfully improve itself, that is exactly the kind of task you need. Which suggests the mathematics is not serving the self-improvement program. It is the scoreboard for it.
If that is right, the Hodge conjecture is not the product. It is the readout on a model that has not shipped — and OpenAI is letting the readout reach the press well before the model reaches anyone else.