OpenAI's Navier-Stokes claim hinges on a term most mathematicians leave out

The dispute turns on forcing. Over the past few years Diego Cordoba and Luis Martinez-Zoroa worked out a way to break the fluid equations using a piece of them that mathematicians had deemed inconsequential ^[8]. Scientific American reports that this piece sits in the equations as written in the Clay problem, and that most experts write the problem without it anyway, on the assumption that a blowup would look the same with or without it ^[7]. Whether that assumption holds is the whole question, and the accounts diverge. AFP reported OpenAI as saying its work answers the question as set out by the Millennium Prize ^[6]. Forkast reports that the institute's criteria are defined on the unforced equations, so the forced result leaves the prize unclaimed ^[5]. Terence Tao, writing on Mastodon, called the Alpoge-Buckmaster work a remarkable achievement and said he sees no fundamental obstacle to extending it to the full equations ^[12]. Seeing no fundamental obstacle is different from having closed the gap.

What the result would mean, physically, is that the model misbehaves rather than the fluid. OpenAI's Ven Chandrasekaran said the proof shows there exist fluids that start out perfectly normal and reach infinite speed in finite time under the equations, which real fluids cannot do, so the equations stop being a reliable mirror of reality in those cases ^[4].

The two clocks here are not close. One of the seven Millennium problems has been solved since the list was published in 2000, and each carries $1 million ^[28]. Clay's rules require publication in a peer-reviewed journal, then two years of acceptance in the mathematical community, before the institute will even convene a committee ^[19]. Two years is 17,520 hours; the agent run reached its solution in 88, a ratio close to 200 to 1 ^[24]. Martin Bridson, the institute's president, told both New Scientist and AFP that evaluation is deliberately unhurried and will be absolutely rigorous ^[20]. That caution serves a prize committee's purpose, though it leaves a field trying to keep pace without a matching timeline.

The 10,000 agents measure what a lab was willing to spend, not how hard the last step was. Mark Chen put the bill emphatically in the millions, and Sebastien Bubeck said it was roughly a thousand times OpenAI's previous spend on mathematical results ^[3]. The method was already in the literature ^[8], the adjacent Euler result had been proved 24 days before the announcement ^[30], and in the same week Anima Anandkumar's group at Caltech released a zero-viscosity solution built on a physics-informed neural network rather than a general-purpose language model ^[25]. Scale is one route to these results, alongside other methods currently producing them.

Bubeck has called Buckmaster's account false and inflammatory ^[14]. Both accounts can be given in good faith, because the mechanism at issue leaves no trace either party could inspect: nothing in a training run, and nothing in a formalisation pipeline, records where a proof strategy came from. Tao's complaint about pace, that "now there's no speed limit, and suddenly things are breaking down" ^[26], is the same problem from the other side. The tools for checking whether a proof holds got much faster this year, while the tools for establishing whose proof it is have not kept pace.