Right - OpenAI proved the forced blow-up case rather than the harder unforced one, with the Millenium Prize problem statement saying it would be awarded for either one.

The forced version is easier since you can custom design the force function to get the result (it doesn't have to be a realistic force like stirring), so getting the blow-up might be regarded just as much a function of your bespoke force function as of the fluid dynamics itself, which is apparently what OpenAI did, pushing the definition of the force function being "smooth" to it's limit.

So, it appears OpenAI did legitimately meet the Millenium Prize solution criteria, but in the most unrealistic, and therefore least interesting, way possible.

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adriand a day ago | parent [-]

My friend, who is a mathematician, sent this in our group chat:

The title is a bit misleading. The variant with a smooth forcing was one of the four valid variants in the Clay formulation. It is interesting to solve it. It is still an interesting and impressive result. The no force version is also interesting and remains unsolved. It isn’t reasonable to just dismiss the proof on the grounds that 26 years later we claim it was never that interesting. This is the first time I’ve seen this attitude.

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pennomi 15 hours ago | parent [-]

It’s just people trying to cope in the most human way possible: “I don’t like AI, therefore AI is stupid, therefore its proof must be uninteresting.”

Almost never do people judge situations entirely on merit.

	▲	HarHarVeryFunny 13 hours ago \| parent [-]
		Not really - if you read beyond the headlines of "mathematicians don't like AI", and your own imagined (incorrect) reasons they may have for that "AI is stupid", then the truth seems a bit more interesting. If you want to abide by your own words and judge the situation on it's merit, then you need to look at the specifics, meaning the OpenAI proof itself (166 pages), and the analysis of it that is only just beginning. Assuming that the proof is wonderful and provides much insight into Navier-Stokes is just as dumb a take as assuming that it doesn't. Judge it on its merit. The Scientific American article is sadly paywalled, but at least part of the discussion is based on the paper below, whose work the OpenAI proof appears to build upon. https://arxiv.org/pdf/2609.20803 When discussing the proof itself, below, with Sonnet, and asking it to explain the distinction between a function being smooth and analytic, one aspect that appears interesting is that the OpenAI forcing function is apparently constructed out of "bump functions", meaning that it is not a uniform force acting upon the flow but rather a highly engineered pattern of pokes, localized in time and space, which as another commenter in this thread notes sounds similar to Maxwell's Demon - another theoretical force, that neither tells us anything about Brownian motion nor the 2nd "law" of thermodynamics. So, we'll have to wait for mathematicians to continue to analyze the proof, and determine to what extent is does deliver on providing insights into Navier Stokes, and any potential improvement to it, that was the goal of setting it as a Millenium Prize in the first place. https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8...