Navier-Stokes – Tristan Buckmaster [pdf]
1705 points - yesterday at 5:42 AM
Further discussion: https://simonwillison.net/2026/Sep/8/on-navier-stokes/, https://news.ycombinator.com/item?id=49621697
Comments
- Aug 15th: Tristan Buckmaster & Levent Alpöge make progress on a few important math problems, "finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler."
- they do NOT have a proof for the $1,000,000 Millenium Prize problem. BUT, they do claim to have a proof for a similar (non-Millenium) Navier Stokes problem that could help lead the way there
- Levent works at Anthropic, but this research was independent of his work there, with a mix of GPT and Claude models. Tristan is not related to Anthropic.
- Early Sep: Rumor spreads to OpenAI that Anthropic solved a major problem. Tristan emails OpenAI to clarify, without revealing the problem they solved or how they did it.
- After hearing of the rumor, OpenAI started researching Navier Stokes with a new internal model.
- Sep 6th: OpenAI's Sebastien Bubeck tells Tristan that they solved the $1,000,000 Millenium Prize Navier Stokes problem. The approach is very similar to Tristan & Levent's approach to the non-Millenium problem.
- Tristan is suspicious of the timing, as only few others were trying this approach. OpenAI says the model didn't access his user data directly, but leaves unanswered whether Tristan's chat conversations were part of the training.
- OpenAI says they would partially credit Tristan for the $1,000,000 discovery (even though Tristan did not solve the $1,000,000 problem) — but only if they remove Levent as an author, as he works for Anthropic.
- Sep 8th: Tristan refuses to remove Levent, and rushes to publish their results independently.
I'm not even sure what to say to this, but I think this should be widely known if it is indeed what happened.
> While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models
This is the crux of it. If Tristan's work and insights were not used to train OpenAI models, then this just looks like a case of hyper-competitive academic sniping that has been going on for decades (check out Watson and Crick!) accelerated by AI as a tool.
The fact that this is ambiguous even to OpenAI leaves one huge question: did Tristan opt out of model training for his ChatGPT and Codex sessions? If the answer is no, then this seems fair game. If the answer is yes, then OpenAI's ambiguity is strongly suggestive that opting out of model improvement does not mean what they imply it means.
OpenAI looked at user data, stole world class researchers' work, and then tried to threaten those researchers to do what would make their corporation profit (which they would anyways!).
Imagine you have been working on a terribly difficult math problem for a decade. This is a result you have spent years on, and what you will likely be remembered for. And to have some punk from OpenAI lie to you, threaten you, and tell you that they are willing to go on the record that you "deserved" it? What is this, the Godfather?
If OpenAI solved Navier-Stokes, that is an astounding result! - yet they'll still be remembered as those who thought credit was more important than results. That winning was more important than collaboration. If this is true, they're burning any trust left with academia.
[0] https://xcancel.com/SebastienBubeck/status/20972141224714323...
[1] https://xcancel.com/polynoamial/status/2097215233119211902
[2] https://xcancel.com/danintheory/status/2097214838003138603
[3] https://xcancel.com/_sholtodouglas/status/209721833169057800...
If compute is cheap, and the difficult thing with scientific discovery is now mostly in steering agents into promising areas, there's an obvious incentive for OA mathematicians to simply monitor closely which researchers are close to releasing exciting results, make some assumptions about their prompts based on their past work, and quickly prompt their own (stronger) model to look into the same areas.
If you think these companies are not training on your prompts you are incredibly naive. These models were built by stealing and pirating literally everything they can get their hands on no matter the legality. AI companies are always very specific about what they're not doing - in a way that you can drive a truck through the loopholes
"Oops! We really did mean it when we said we wouldn't train on your data. Our models are just so good they decided to anyway."
I don't know how the math community handles this but normally I would think if X mathematician comes up with an idea and Y mathematician uses it to solve some problem, Y would get credit. But does that change if Y heavily relied on LLMs? I suppose we're going to find out.
I guess this is true in more ways than one. Kasparov famously accused IBM of cheating during the match, by spying on his preparation (edit: though the main cheating accusation was live human intervention during the games, on top of IBM downplaying the heavy human involvement behind the AI, which also mirrors this situation)
The money in nerdy frontier math is very little. The money in Big AI is very very much.
So the deal is this: We will pay an army of you guys very well and you will get to work on your favorite problems. The only thing is if you find something you will have to credit the Machine God.
Do you think you can handle that?
These are the kinds of people in charge of the reins, folks.
Mathematical explanation by Terrance Tao: https://mathstodon.xyz/@tao/117233527638291447
It seems there is much background drama behind this, and this is what I've pieced together of what happened:
Over the past year, Buckmaster and Alpöge have been using AI to work on fluid dynamics maths problems. Alpöge works at Anthropic, which will cause future issues.
In mid-August, they found a counterexample for a simpler version of the Navier-Stokes problem. They spend the next few weeks preparing their paper.
In early September, rumors start spreading on X that Anthropic has solved a Millennium prize problem (and that it's Navier-Stokes). Buckmaster reaches out to OpenAI to explain this is their own personal research, not an Anthropic project.
A few days later, OpenAI gets back to him, and tells him an internal model found has a counterexample for Navier–Stokes, potentially worth the $1 million Millennium prize. The proof uses the same method that Buckmaster and Alpöge chose to work on. They don't show him the proof.
Buckmaster pressed them for more details. OpenAI reveals they had an entire team had been working on the problem, and that they started work in the past few days, after the rumors that Anthropic had solved a Millennium prize problem.
Buckmaster says OpenAI talked about a shared publication timeline. They want to Buckmaster to publish first, then give Buckmaster shared credit for the Millennium Prize when they publish the full result. But they want to exclude Alpöge as an author because he works at Anthropic. An agreement is not reached. Buckmaster had been using OpenAI Codex to draft/check his work, and asks if his private AI chats were used to accelerate OpenAI's result.
Buckmaster and Alpöge think they have found a counterexample for Navier-Stokes, but the paper is not yet presentable. It's unclear what date they found this result.
Because of the situation with OpenAI, they published their existing papers earlier than planned (today), alongside this statement announcing they have a tentative result on Navier-Stokes and revealing the OpenAI drama.
The post is missing context from both sides, and this isn't my field, so hopefully someone else can unpack what's happening here.
We congratulate Levent Alpöge and Tristan Buckmaster on their remarkable mathematical work.
We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem.
While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.
However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs. unforced).
For eg: "Hey ChatGPT my name is X and I am 6 and a half feet tall. Am I anaemic?" This is a query, and while it might suggest to an AI model that tall people may worry about iron deficiencies, it's not really necessary to include in training. The user may be tall or short, but the idea that one may randomly ask about anaemia is not exclusive to this dataset. At best, this chat is an example of linguistics, not anything else, and the models figured out how to write and answer such questions years ago. It is ignored in training.
But when your work involves solid complex and unique mathematical proofs, the data is suddenly worth training upon. If I understand it correctly, the LLM may view your approach as a brand new path to take to solve an otherwise intractable problem. Its reinforcement training emphasises that it should do this in order to improve. And since it leads to results - large internal teams likely flag the model that reached this stage, the model is rewarded and given compute and attention - it is a desireable outcome both for the model and for OpenAI.
OFC, OpenAI becoming an advertising company will suddenly have incentive to treat all data as valuable. But while they are a "we need to make headlines" company, it's more rational that they view these examples of data as more valuable than others.
I don't doubt that they trained on his chats. This seems like the ideal usecase for "mass surveillance but using training" as a sort of filter.
But even so, one wonders how the model differentiates. If the researcher entered proofs into ChatGPT every day that mentioned "strawberries", while no other math paper on the topic did so, does that mean their chats would be audited?
IMO the parsimonious answer seems to be OpenAI has a pretty good model (because it did finish) and stole someones work... and threatened them over it. TBH all OpenAI need to do is solve another millennial problem and none of it would matter - people expect them to behave heinously regardless - but if they have generalized superhuman math model... well I guess they're allowed io.
For those who know nothing about the context - the Diego mentioned was a student of Fefferman and Luis was a student of Diego's - these people have all worked hard on these problems for a long time and are genuine experts. The mathematicians at OpenAI are strong mathematicians, but not expert on these particular problems. The particular approach is claimed to be the key to the whole thing.
The allegation is not different in spirit to alleging that a particular group of astronomical researchers "discovered" a new planet because they had access to the logs of another group that had already pointed its telescope at the planet.
This post is not intended to assess the correctness of the allegation.
"We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models . However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced)." https://openai.com/index/navier-stokes-solution/
NOTE: there are a couple duped threads around this. i replied on a different one first before seeing this one
"Strong agree. I know that there is rivalry between the labs but it's important that we learn to work together given what's coming. <quote tweet [1] above>" -- Noam Brown, an OpenAI researcher [2]
We all should heed the implied warnings of these top researchers about what's coming. The world is far from ready and everyone who can should pitch in.
[1] https://x.com/_sholtodouglas/status/2097224624274911368 [2] https://x.com/polynoamial/status/2097225279366414541
Key quote : "Solving the problem by purely AI-powered methods [would be a] net negative for the progress of mathematics."
This is significant.
https://mastodon.social/@tristanbuckmaster/11723341370570119...
And here are Terence Tao’s comments on the results: https://mathstodon.xyz/@tao/117233527638291447
This sounds like a very big coincidence and it looks really bad for OpenAI but there is an alternative explanation that I can only state as a conjecture.
Suppose that the ability of LLMs to generate mathematical proofs is like a quiver full of arrows: each arrow, one proof. The same quiver is shared between all instances of one model and substantially similar models share substantial subsets of the arrows in the same quiver.
That would allow two independent teams to converge on the same LLM-aided solutions to the same problems. Even more likely so if the quivers were small and finite and their arrows were specific to a distinct class of problems (without being able to suggest a particular class from what we've seen so far).
This would explain the kind of LLM-mediated results we've seen so far that tend to be ... sparse. By which I mean that every time there's a new model release we get some new results and then they seem to dry out, until the next release.
It would also explain how OpenAI was about to prove the same result as Buckmaster and Alpoge, while absolving OpenAI of any misconduct. And this is one reason to prefer this explanation: one should not favour accusations of misconduct as long as there are conceivable alternatives.
But, that's just a conjecture that I can't prove.
Given Tristan doesn't explicitly say he was using the API, and given he doesn't mention anything about the API TOS (which disallows training on chats) in his call with OAI, it's highly likely Tristan was using the consumer OAI product (whose TOS allows training on chats).
This is unethical behavior from OAI. And it is 100% consistent with their long and public history of unethical behavior, so nobody should be surprised.
The only thing interesting I see here is OAI PR dilemma. If they claim the prize they get the blowback we're seeing in this thread and all over the web right now. But most people don't follow AI closely and shut off their brains when they see "Navier-Stokes", so 90% potential investors (the only people OAI really care about) probably only see the headline "OAI solves famous hard math problem" and think "OAI models are really smart, better invest before they take all the jobs." If they don't claim the prize, then maybe they let Anthropic their mortal enemy claim it. Anthropic is already IPOing first. Can't let that happen.
Yeah as I write this there it's clear there is no dilemma. For a company whose secret motto is "do be evil" this is a super easy discussion.
> I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI.
> I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.
> Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic. It was also said that if OpenAI posted after us, they would say that we deserved the Clay Prize, and that we were the “closest humans to the problem”. I declined both offers.
> I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”
Wow, that's some VERY friendly communication. Besides, will the career of the person be ruined because of “Why would you ruin your career?” came out of his or her own mouth?
Like, to me this looks like academic slap-fighting from Bubeck and Levent. People working at OpenAI are saying, "hey, we don't have that particular data in our models," others are saying, "we used a different approach to do it with Navier-Stokes" this feels like much ado about nothing.
Then in these comments I see some wild accusations.
If OpenAI is telling the truth (I don't really see a reason to lie here, if anything that sounds kind of like a dumb idea given the context), then they heard, "oh, shit, someone might be able to solve Navier-Stokes, don't we have some guys working on that? Give them 10,000 agents!" Then 88 hours later, out pops a similar solution. It's not like there's probably an infinity of ways to do this, the proof is probably similar.
Read this:
> Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic. It was also said that if OpenAI posted after us, they would say that we deserved the Clay Prize, and that we were the “closest humans to the problem”. I declined both offers.
So, really, it sounds like academic slap-fighting nonsense and corporate bureaucracy. Literally, OpenAI's best move would have been to say, "ok, we're going to not say anything, do your thing" and let it happen. Ego and vanity got in the way.
Still, the stupid drama of this doesn't really do the results justice. There are maybe 1000 people on planet earth who are qualified to solve a problem like this. Even if the human "loosened the jar" a bit, that's astounding that their model was able to figure it the rest of the way out. Why are people dialed in to the human interest story here and not looking at the bigger picture!
Thats how most people use LLMs? If I was back in my student days working in Navier-Stokes, I guess I would also punch at blow ups. The number of students doing this at the same time, posting open efforts to GitHub then retraining of the models. If there is solutions to the problem, it’s a real possibility that it was not a result of this effort?
Using a large amount of tokens is not a good augment that it’s not likely others have done the same. Good questions is the difference between $10 and $10M in token usage to solve a problem.
- OpenAI did related research around similar timeframe.
- Tristan claimed OpenAI offered a proposal that included dropping the Anthropic-affiliated co-author.
- Sebastian (a prominent OpenAI researcher involved) denied these claims.
- Tristan have no concrete evidence that OpenAI accessed their session.
- OpenAI's theory may hold up, but it will require long-term validation to confirm.
Stadlmann improved it from 246 to 240, OpenAI later claimed 186 I think?
Maybe someone can help clarify? I am no expert at all, but I can't help but see similarities.
Terry also talks about it https://mathstodon.xyz/@tao/117234157753860650
- the OpenAI researchers claimed that they had "just told it to work on the problem" with little human input
- in fact, they had a whole team working on it
- and used, among other things, the work of third party human researchers to drive the work
- then threatened? a researcher who tried to go against theit planned narrative
Just from this document (which is of course only one side of the story) it really sounds like OpenAI was hoping to publish and say "we just told the model to try harder and it solved a Millennium problem!". Not great if true.
This part in particular was especially egregious:
> I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”
"I don't want to live in a world where someone makes the world a better place, better than we do."
It's amazing how transparently OpenAI is running the standard silicon valley playbook.
but they probably won't and if it's happened on some obscure math research it's happening everyday everywhere else.
Fully local AI compute can't come fast enough, these guys have IP theft baked into their bones.
https://www.quantamagazine.org/computer-helps-prove-long-sou...
If I publish something, and disclose that I used AI for assistance, do I have to credit everyone who previously used the same AI to try the same problem? Because their prompts inevitably made it to the training data for my prompts?
Time will almost certainly reveal a lot more about the drama and the related ethics, but let's get excited about the actual breakthrough as well!
OpenAI's board has fired Sam Altman. https://news.ycombinator.com/item?id=38309611
Apple sues OpenAI, accuses ex-employees of stealing trade secrets. https://news.ycombinator.com/item?id=48865019
Ah yes, taking code from a person/company is fine if its deidentified!
And if we think this only applies to academic fields then we're doubly fooling ourselves. They do not have the ethics or incentives to be good stewards of the technology.
But this (if it is true) is really abominable and a show of force from the techno-feudalists.
This should stop. What is possible does not mean it should be implemented.
What I'm curious to know is whether this was a manual snooping, or automated farming that occurs for anything of value that happens in chats.
Telling OpenAI that Anthropic has apparently solved an important problem but most likely that refers to him and he is using OpenAI models (not Anthropic's)?
And he wants to clarify that with OpenAI in advance? And get a pardon for Anthropic's likely but false press statements?
I dont get it.
[edited] needless to say, the behavior of the OpenAI employee is really despicable
$15m in tokens; but what about labor?
What about compressible fluids?
I'm just wondering how much real input Buckmaster gave here that he thinks the proof is his. I guess at the end of the day OAI still wins if ChatGPT was used to prove this successfully.
Who still wants to use AI to solve cancer and other major problems?
Literally who cares who solved the problem just publish the results.
Academia was always politics first results second and I AM GLAD that LLMs are becoming superhuman at math. I like better theorems, not better politics.