Navier-Stokes – Tristan Buckmaster [pdf] (cims.nyu.edu)

> 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.”

I'm not even sure what to say to this, but I think this should be widely known if it is indeed what happened.

I'm not one to comment often but this really pisses me off.

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.

    The route to the Clay problem through a
    smooth force, options c and d in Fefferman’s statement of the problem, is the
    route Luis and Diego opened and the one Levent and I had quietly chosen to
    attack. Almost nobody else I know of was working on it. It is not the direction
    one arrives at in a few days by giving a model the problem statement. When I
    heard “forced,” it was a bright red flag.

Drama/accusation summary:

- 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.

The accused Sebastian Bubeck has denied the allegations on Twitter[0], and various other OpenAI employees[1,2] seem to be mocking another Anthropic employee voicing support for Levent[3]? Things are getting messy.

[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...

So, leaving aside the idea that OA might've used data from the researchers Codex sessions: Do I understand correctly that the internal OpenAI work on the problems was probably started after they heard Alpoge and Buckmaster had made process by using their models? And they used the publicly available info about the researchers past work to prompt their models?

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.

I don't know how to read this and not see that this is a direct accusation to OpenAI of having used the researchers data to try to front run his discovery on purpose. The evidence is not completely proven and also circumstantial but to me at least looks like a fairly suspicious situation.

"I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true."

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?

> This is a a Deep Blue-Kasparov moment.

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)

Full mastodon post: https://mathstodon.xyz/@[email protected]/11...

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.

Seems pretty likely OpenAI will soon disclose that their internal models have managed to compromise their internal controls in order to access users' private chat histories as a creative method of cheating to solve impossible problems.

"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 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.

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

Funny that the rumour about the big Anthropic announcement had nothing to do with Navier-Stokes. It was about the formalisation of Fermat.

When people worry about OpenAI stealing their chats and reproducing them elsewhere, I usually view the situation as unlikely - since chats are "trained" upon and not necessarily reproduced verbatim, you can assume that unless your chats depict a foundationally new and effective style of communication or ideation, there would be little need or use thereof of training on your chats.

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?

Reminds me, kinda, to when Astra was launched and OpenAI announced an improvement to the bounded prime gap. Which BTW, Prof. Julia Stadlmann had published an independent result only a few days earlier

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.

[0] https://arxiv.org/abs/2608.31126

Hard mathematics problems used to take years if not decades to tackle manually. But now with enough compute and a hint that a certain approach might work, it just takes a few days. This could be the last year that humans could still make more substantial contribution to major match problems than machines.

The ego behind the frontier labs is growing evermore concerning

Can somebody explain: do singularities / blow-ups in solutions have any relation to physical phenomena in fluid dynamics or are they purely artifacts of how the N-S equations may not accurately describe what actually happens in the physical world?

There will be a lot of hurt and pain in mathematician's community. It is hard to accept that major discoveries are now just a function of spent token $$.

“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.”

This is significant.

> Let me make plain what I have said to colleagues in private: in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal.

Buckmaster is actually implying that OpenAI spied on his chat logs and tried to speedrun his work and then tried to remove his co-author because he's an Anthropic employee?

I recommend Terrence Tao's commentary on such a proof : https://mathstodon.xyz/@tao/117219101339291693

Key quote : "Solving the problem by purely AI-powered methods [would be a] net negative for the progress of mathematics."

Two announcements on Euler today. The one discussed here by Tristan with forcing and one from Anima without forcing https://anima-ai.org/2026/09/07/stable-singularity-of-the-eu...

Terry also talks about it https://mathstodon.xyz/@tao/117234157753860650

Can a mathematical person explain how the different "bits" of Navier-Stokes proofs fit together? How significant is it to have "Euler"? What is this "smooth forcing"? Which are the most significant steps to proving the whole thing?

This is unfortunate. I thought Anthropic were the only ones who did this.

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.