There's a claim going around that OpenAI has constructed a finite-time blowup for the Navier-Stokes equations, potentially resolving one of the million-dollar Millennium Prize Problems. There's also speculation that OpenAI used Tristan Buckmaster's private research to do it. The claimed breakthrough and the dispute over credit have now reached the news.
I haven't seen OpenAI's proof, and I don't know what happened inside the company. But my current guess is that we're looking at a case of multiple discovery: different groups reaching similar results independently, at nearly the same time. My own experience using Paratelligent for mathematical research makes this seem considerably more plausible than much of the discussion suggests.
Tristan Buckmaster's mathematical claims so far are a blowup for Euler, or for hypodissipative Navier-Stokes with weaker dissipation. These are excellent, but do not by itself solve the full Navier-Stokes problem for the Millenium Problem. For now, we don't even know that the Millennium Problem has been solved, as it's just a rumor until OpenAI announces otherwise.
Three explanations that seem unlikely to me
-
Someone at OpenAI read Tristan's private chats and used his work for research. That is a serious allegation, and I haven't seen evidence establishing it. I would expect a company handling sensitive customer data to restrict and audit employee access, making this a firable offense for a rogue employee if discovered. That expectation isn't proof that misuse could not happen, but neither is the existence of chat logs evidence that it did.
-
The AI went rogue and hacked internal logs to obtain the chats. This seems extraordinarily unlikely to me. It adds an entire unobserved security breach to the story. I see little reason to believe it.
-
The chats entered training data, and a later model used what it learned. This is a more plausible mechanism in principle, but unlikely due to training timelines and the specifics of what was claimed solved. Since training and deploying new models takes time, it is unlikely that recent chat logs would have made it into the model used today. And a proof for a related fluid equation would still leave substantial mathematics to do before reaching ordinary Navier-Stokes. Seeing such a proof could certainly help but a lot of work remains.
Occam's razor suggests looking first for a simpler explanation. A rumor that Navier-Stokes was close to being solved could have encouraged OpenAI to devote more effort to it. Given enough compute, OpenAI might then have made independent progress. If its claimed proof holds up, that would be a fairly ordinary research dynamic operating at an extraordinary speed.
That is my current hypothesis for how this happened. Hearing that a problem is tractable can influence the choice to work on it without revealing how to solve it. As AI speeds up research, I expect many more discoveries to follow one another this closely.
My agents were exploring the same strategy
About a month ago, before these rumors, I worked on Navier-Stokes with Paratelligent. I had roughly ten agents investigating it, most of them looking for blowups rather than trying to prove that blowup is impossible.
Last night, I went back through their work. Agent #2 had investigated the Córdoba–Martínez-Zoroa strategy. One claim I've seen on social media is that this approach is so nonobvious, and so rarely pursued, that two groups choosing it would be suspicious. My own experiment is a counterexample to the idea that practically nobody else was exploring it. I also asked Grok for promising blowup strategies last night, and the Córdoba–Martínez-Zoroa program appeared on its shortlist.
This is a public research program. For example, Córdoba, Martínez-Zoroa, and Fan Zheng posted a paper on forced hypodissipative Navier-Stokes blowup in 2024. That paper does not resolve the Millennium Problem, but it gives anyone surveying the literature a concrete direction to investigate.
I suspect many people have tried to solve Navier-Stokes with AI in recent months, and many of their systems would have explored this program. Choosing the same published starting point is weak evidence of access to someone else's private work. The difficult part is getting from that starting point to a correct proof.
Paratelligent did not finish that job. Looking back, I wonder whether I gave up too easily by allocating too little compute, which ultimately means money. More compute would likely achieve a solution. However, I did not know this was the correct path, nor dedicate a ton of compute to Navier-Stokes. Finding the strategy and successfully extending it are very different achievements.
Why multiple discovery feels familiar
Navier-Stokes isn't the only reason I've been thinking about this. I've recently encountered several situations that make overlapping AI discoveries feel less surprising.
The first involves the θ(pc) = 0 conjecture for Bernoulli percolation. Last week, I read a mathematician's blog post suggesting that an AI might solve it. I put a single Paratelligent agent on the problem, without necessarily expecting success. A day later, I heard that Anthropic had already solved it but hadn't yet published the result.
When I checked my agent's work, I found what appeared to be a solution too. I got excited and considered posting it. But the approach looked substantially similar to what I heard from Anthropic. It wasn't identical, yet it seemed close enough that I worried people would assume copying, even though their proof hadn't been made public when my agent did the work. I decided against posting it.
My guess is that Anthropic's result generated a rumor, that rumor reached the mathematician, and the resulting blog post prompted me to investigate. I don't know that this was the actual chain of events. What I do know is that the post prompted my experiment. It is easy to see how news that a problem might be within reach could lead several groups to attack it at once, without any of them receiving another group's proof.
A second example comes from another subfield of mathematics. Paratelligent recently produced a solution to a significant problem, and I emailed a mathematician who works in that space about it. He told me his group had just solved it with AI too and was preparing the write-up. We're now working together, with plans to share the results in the near future.
That experience makes me wish the Navier-Stokes story could end with a collaborative write-up as well. I don't know what has prevented cooperation among OpenAI, Buckmaster, and the researchers at Anthropic. Whatever the commercial tensions, I'd like the outcome to be a clear proof and fair credit for the people who contributed to it.
Finally, I've had Paratelligent try to recreate solutions to recently solved problems, as a way to improve the system. Difficult problems that have just become tractable are useful places to study what an AI research system can do. Sometimes Paratelligent produced a methodologically distinct solution from the public one. I expect other groups are running similar experiments and getting multiple discoveries too.
My bet is that multiple discovery will become much more common as AI research accelerates. A rumor, a blog post, or a newly published technique can send many teams toward the same problem on the same day. To establish copying, we need evidence beyond that overlap.