As you may have heard, Open AI claimed to have solved the Navri-Stokes problem https://openai.com/index/navier-stokes-solution/

However, there are multiple discussions around that claim. For now, the full solution has not been presented so there are a few uncertainties regarding the validity of the claim:

Two weeks ago OpenAI claimed a solution to one of the biggest open problems in math—the Navier-Stokes problem—an achievement worth a $1-million prize from the Clay Mathematics Institute. The proof ignited a powder keg of concern over artificial intelligence companies’ race to disrupt the subject.

But with the dust still far from settled, a different controversy is emerging: Did OpenAI even solve the right Navier-Stokes problem?

Generated by an internal large language model (LLM), OpenAI’s proof relies on an approach that many experts find unnatural. It solves a variant of the problem that mathematicians say is disconnected from reality and thus less interesting. In a sense, the LLM found and exploited a loophole in the framing of the question.

“The most important problem is unsolved,” says Luis Silvestre, a mathematician at the University of Chicago. “The Clay problem is settled, but the main problem for the Navier-Stokes equations is not.”

There is also another controversy:

Over the course of August, Dr. Buckmaster and another mathematician had made significant progress on a longstanding puzzle involving the Navier-Stokes equations, which describe the flow of fluids like water and air.

But then OpenAI, the artificial intelligence behemoth, jumped into the fray.

Rumors that their archrival, Anthropic, had solved one or two of math’s “Millennium Prize problems” spurred OpenAI leaders to tackle the same list of problems, which offers $1 million for the first proof that passes a rigorous review.

The list includes the Navier-Stokes puzzle, and in less than a week, OpenAI beat Dr. Buckmaster and all of the world’s other human mathematicians, announcing on Tuesday that it had found the answer.

Over the past couple of days, a firestorm has ignited across the social media universe: perceived threats, dangled prize money and suggestions that OpenAI’s A.I. agents had somehow found and incorporated Dr. Buckmaster’s recent work into the company’s proof.

On Wednesday evening, in response to questions from The Times, OpenAI said in a statement that it was “categorically” impossible for its A.I. system to have been influenced by anything Dr. Buckmaster had done in the past two months.

https://www.nytimes.com/2026/09/10/science/tristan-buckmaster-openai-math-navier-stokes.html?unlocked_article_code=1.AFE.CA12.DVpcZwQn9Yl6&smid=url-share

That opened up a completely different question. Let's assume that a mathematical problem is solved without human input but at high cost (the estimated computation time was valued at around 15 mio) what is ultimately worth?

In truth, it was precisely the opposite: the clearest evidence yet that generative A.I., rather than aiding scientific progress, may be thwarting it.

For starters, two mathematicians who had been trying for months to reach their own solution to the famous problem say the new proof appears to have ripped off their unpublished work, which also used an OpenAI model. The company, after first equivocating, quickly changed its tune: It said the model it used for the solution had not looked at any of the prompts that the lead mathematician had entered into the OpenAI model within the last two months. Whatever the case, to get to the finish line in so little time, OpenAI had deployed an advanced model not available to the public, using computing power that would have cost an outsider an estimated $15 million. How could regular researchers compete with a secret tool that has the power to steal your work and mint its own money?

Flashy finishes like this don’t necessarily contribute to mathematical knowledge. Terence Tao, perhaps the most prominent mathematician of his generation, is generally positive about A.I. and math — so positive that he was featured in one of OpenAI’s ads. But contemplating the possibility of Navier-Stokes being solved by A.I., he wrote a long post explaining that “in most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves.” Instead, mathematicians want to see all the work that goes into achieving the solution — all the not-quite-right hypotheses that got adjusted this way or that, all the seemingly dead ends that “in fact end up being highly instructive in the nature of their failure.”

I also like the final point made in the last paragraph here:

Many scientific fields are already drowning in plausible-seeming but unverified papers generated with significant help from a large language model. What’s slop and what’s not? Is there an actual gem amid the gazillion new hypotheses? It’s becoming increasingly impossible to tell. Just last week, the editor in chief of Arxiv, a site where researchers can share papers that have not yet been peer reviewed, proposed implementing oral exams for authors who submit work, to screen out A.I.-generated junk. The scientific conversation cannot survive the deluge.

If OpenAI and Anthropic really want to help advance science, there is one obvious way they can do so: They can spend a tiny fraction of their vast fortunes to fund the kind of humble but essential research that’s increasingly starved for resources.

Often, AI compute time is seen as free, but it ultimately is expensive. And I wonder whether 15 mio in form of grants to further mathematical research group would have yielded more insights, for example. I also see parallels in the use of AI in biological research where (as also mentioned in the article) the actual work starts after hypothesis generation.

https://www.nytimes.com/2026/09/22/opinion/artificial-intelligence-ai-danger.html?unlocked_article_code=1.DVE.uGAZ._2SuQ4ojyQES&smid=url-share

1 hour ago, CharonY said:

“in most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves.” Instead, mathematicians want to see all the work that goes into achieving the solution — all the not-quite-right hypotheses that got adjusted this way or that, all the seemingly dead ends that “in fact end up being highly instructive in the nature of their failure.”

I've said it before, but according to a quote I've seen, a neural network has the second-best solution to any problem.

1 hour ago, CharonY said:

As you may have heard, Open AI claimed to have solved the Navri-Stokes problem https://openai.com/index/navier-stokes-solution/

Okay, I'm going to bite here. Somebody has offered a possible solution to (AN IDEALISED CASE OF ONE OF THE) Navier-Stokes (SMOOTHNESS) problems. Please forgive the capitalisations but without the extra words, some innocent fellow might be lead into believing that we were referring to a general solution to the Navier-Stokes equations.

In context, the smoothness problem is akin to the fractal coastline problem: prove that triangulation of the mean HWM length of the British coatline tends to infinity for a sufficiently short measuring stick. Solving this doesn't mean you've solved Earth Science in general.

Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly?

In which universe are we supposed to find an incompressible fluid?

The maths might be interesting... useful even. But of any use to those for whom Navier-Stokes are a tool for use in the real world? Doubt it.

Edited 3 hours ago3 hr by sethoflagos

Seems to me, that if Dr. Buckmaster had published any of his work, any AI with internet access would be able to make use of it.
And I really don't think OpenAI included crediting Dr. Buckmaster's work in its 'footnotes'.

1 hour ago, sethoflagos said:

Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,”

Sorry not you, but your link.
Anyway, whenever 'singularities' arise, we know that our model is no longer effective under those conditions.
Does OpenAI know this ???

1 hour ago, sethoflagos said:

Okay, I'm going to bite here. Somebody has offered a possible solution to (AN IDEALISED CASE OF ONE OF THE) Navier-Stokes (SMOOTHNESS) problems. Please forgive the capitalisations but without the extra words, some innocent fellow might be lead into believing that we were referring to a general solution to the Navier-Stokes equations.

Absolutely fair.

2 hours ago, sethoflagos said:

The maths might be interesting... useful even. But of any use to those for whom Navier-Stokes are a tool for use in the real world? Doubt it.

I think that goes towards the argument made in the article. From a mathematics rather then physics perspective, developing and approaching questions, even if they do not have any real-world counterpart can be relevant for the field itself, but only if the work is being demonstrated and can be read and understood by humans.

28 minutes ago, MigL said:

And I really don't think OpenAI included crediting Dr. Buckmaster's work in its 'footnotes'.

I think OpenAI offered to publish with Buckmaster as author. However, there was a second mathematician (Alpoege) who worked with Buckmaster which they didn't want to have on the paper. The reason was because Alpoege works for Anthropic. In addition to the maths and the issues of potential negative AI influence on science we also can add conflict of interest and the perhaps the negative impact of companies on scientific integrity....

1 hour ago, MigL said:

Anyway, whenever 'singularities' arise, we know that our model is no longer effective under those conditions.
Does OpenAI know this ???

Every 1st year Chem Eng student learns the known scope of applicability of the N-S equations: I'm sure it's written down formally somewhere (therefore accessible to LLMs), but from the dusty recesses of my gut recollectiom:

When system characteristic dimension approaches the order of magnitude of particle mean free path, the continuum approximation of N-S becomes invalid, and the system is better analysed via the domain of statistical mechanics (yuk).

Okay, found one version of it from MIT at The Limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics

We focus on the modeling challenges arising from the breakdown of the Navier-Stokes description, observed when characteristic lengthscales become of the order of, or smaller than, the molecular mean free path.

Near enough.