1. It shows what even this wave of AI can actually do.

2. I wish it were done by different folks, ideally under some kind of public control like NASA research or the NPR model.

3. Keep in mind: natural science is different. It's not always a matter of computation. Computer science folks often struggle with this -- but this virtual world here does not actually exist. Everything is physical, including information. Any natural science PhD or otherwise knows just how complicated nature actually is -- e.g. mention any research topic and try to encapsulate all the relevant phenomena present there. Pure mathematics is different because we define the problem, rarher than explore nature. We are in my view far away from removing humans in natural science R&D. Advancements in AI however can greatly assist us in all natural sciences, which is already beginning to happen.

What a landmine sentence to bury in this report, you can't rule out your models were spying on other researchers?

We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.

WOW?

Unlike the vanilla read of the OpenAI press release, it is much more unfiltered and outlines some particularly aggressive behavior by specific OpenAI employees

The dark forest awaits..

- There are at least two versions of a model more powerful than Astra at OpenAI at the moment.

- The less capable version was used to solve the unforced Euler problem (while the one solved by Levent Alpöge and Tristan Buckmaster was forced Euler) with 100 agents.

- The more improved version was used to solve Navier-Stokes, given the results of the unforced Euler problem from their earlier attempt, with 10000 agents.

- OpenAI initially tried a shotgun approach against the 6 Millennium Prize Problems until it emerged that Navier-Stokes was the most likely to succeed.

So the timeline was:

Shotgunning 6 open Millennium Prize Problems -> solved unforced Euler problem with 100 agents -> concentrating on Navier-Stokes with 10000 agents -> solution.

If so, that is fantastic development and a huge success (despite all the drama surrounding it)! Congratulations!

It's like that story about George Dantzig solving open problems as a student because he thought they were simply homework: https://en.wikipedia.org/wiki/George_Dantzig

It's also unfortunate that such a potentially momentous occasion is overshadowed by so much drama. Which I suppose is expected given the technology and the people involved are so polarizing.

This specific problem having had a $1 million bounty on its head and still remaining unsolved for 26 years after the bounty was placed is pretty clear evidence that many of the world's best human mathematicians would have solved this problem if they could have, and none were able to until LLMs came along.

Hard to claim at this point that LLMs aren't capable of novel STEM creativity and genius to a degree that will soon far surpass that of humans.

If anyone has counterpoints to this I'd love to hear them!

https://mathstodon.xyz/@tao/117219101339291693

A major goal of our work is to empower scientists to advance research and technology that benefits all of humanity.

And what's a better way of empowering people than robbing them.

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.

But there is 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 does not mean what they imply it means.

https://smackernews.com/item/49605915 HN

https://bsky.app/profile/quantian.bsky.social/post/3muyhwbcd...

https://cims.nyu.edu/~tristanb/statement.pdf

IPO+rumour driven research.

I appreciate the achievement, but it doesn't feel right.

While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .

Which seems a bit irresponsible/rash?

https://x.com/rynorhn/status/2097223532438487463

At all times we maintained the same strict safeguards that we apply to all our frontier model evaluations, including monitoring and isolation.

Looks like they're shifting away from the "unprecedented hacking ability" backroom-PR strategy into more benevolent messaging.

the first "Country of geniuses in a datacenter" moment.

OpenAI got wind that a millenium problem was being solved. And that feels a bit like the critical move in chess. That is - it was a signal that AI advanced far enough that it would be worth spending a lot of time and resources solving a millenium problem.

At a conservative estimate of GPT 6 Astra pricing, this would have cost upwards of 15 Million dollars for anyone using the OpenAI API!

To me this is the one silver lining. Yes, they can solve millennium prize problems, but it still costs a fortune.

Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens

Don't even try to do the math on how much that would cost at normal API prices. And we don't even know how much more expensive this internal-only model would be!

Or, in this case, stealing prompts from competitors.

Do not use stealing chatbots for research even if you think you have data agreements. The people running these companies have worked on hookup apps for Christ's sake. Get real.

   At all times we maintained the same strict safeguards that we apply to all our frontier model evaluations, including monitoring and isolation.

Most (all?) of the big discoveries have been counterexamples, which is just sort of a systematic tearing down human ingenuity. I know that counterexamples are an important part of progress and discovery, but it just feels bad to me.

But I'm not a mathematician, maybe I'm totally misreading the vibe.

Cure all illnesses Utopia or Robot Wars Dystopia, both are pretty exciting.

[0] Struggle relative to its ability to disprove, not struggle relative to people's ability to prove theorems.

On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved

What's the other one?

Aug 28: OpenAI starts training a new model.

Sep 1: OpenAI sees a rumor on Twitter that two Millenium Prize problems were solved and starts their own effort to attack all the prize problems using the new (4 day old!) model.

Sep 3: The new model makes some progress toward Navier-Stokes. Based on this progress, OpenAI focuses on Navier-Stokes over the other Millenium Prize problems, using several approaches in parallel.

Sep 5: Navier-Stokes is solved. Assuming Astra API prices, $15m in output tokens were used by the whole effort.

In this account of the story, no specific information about Tristan and Levent's work is used to inform OpenAI's approach. The focus on Navier-Stokes and the choice of approaches to pursue came from OpenAI's own progress, not specific knowledge of Tristan's concurrent work.

There is a caveat that they "can't rule out" the possibility that Tristan's Codex data could have been part of the training set of the new model, though it is described as "unlikely" and the proofs are substantially different.

This timeline is insane. Navier-Stokes was solved start-to-finish in 5 days? A model in training for at most eight days dramatically outperforms Astra and Fable, and not just in mathematics?

When your hosting provider has unlimited resources to throw at any problem, all they need to know are the good problems, and they can learn that from your logs, how can you trust them?

They could easily have looked at the logs. We don't know. We'll never know!

You can't trust places like OpenAI or Anthropic with your IP if you're a business. They can easily review all of your logs for interesting discoveries. For example, if your drug discovery pipeline fails to find something that they think might work with 1000x the compute, they can do it. And now suddently they have a new business and you don't.

I wonder whether a team of 60 mathematicians working solely on this for a year would have cracked this. (Assuming $250k total compensation.)

To what extent should one trust a statement that a program is free of Trojan horses? Perhaps it is more important to trust the people who wrote the software.

https://www.cs.cmu.edu/~rdriley/487/papers/Thompson_1984_Ref...

OpenAI already has a model that is at the very least twice as smart as Astra.

Oh god.

https://cims.nyu.edu/~tristanb/statement.pdf

Why would OpenAI volunteer to give credit / share authorship if they believe they indeed solved the problem independently and in a different way?

Based on how they seem to operate in general, this smells more like CYA than it does being honorable, but that's my bias.

Also it sounds like the human research effort spanned weeks if not years from Tristan’s statement so it is extremely likely the work and prompts of these human researchers was used in the OpenAI knock-off.

While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .

Once again, I'm no closer to understanding what https://openai.com/policies/how-your-data-is-used-to-improve... actually means.

If I run Codex against a project that includes a private API key, is there a chance a future user of ChatGPT could ask for an API key and get back mine?

I've actually asked someone at OpenAI this question and they said that was the "regurgitation" problem and is something which they actively work to prevent happening.

That's reassuring, but I want to know more. I still don't have an intuitive understanding of what kind of data I should avoid sharing with a model if I'm worried about that data causing me problems when it's used for future training.

Is it safe for me to brainstorm future directions for my company with a model, or might that risk someone getting that information in response to a prompt like "What potential directions could company X consider in the future?" in six months time?

so far the proof looks more along the lines of another euler blowup proof we had, off of whose ansatz naming we were making really stupid puns like “smooth criminale”, unlike the much better “ideal fluids explode”, Tristan

There are too many ambiguities around OpenAI. Unanswered questions making this ambiguity more.

Why they didn't properly explain to Tristan about usage of their data.

If the singularity is in the physical space?

Is this just a result of ignoring things like friction and energy dissipation via heat, etc?

For those of you who don't care about the drama and want to see this distilled to 3 lines:

https://x.com/nadermx/status/2097414953225310280

Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize for this result.

Does OpenAI have a policy of not claiming math prizes like this, or is this them trying to avoid any concerns (right or wrong, I'm sure we will hear more in the future) about how they got there?

Everyone uses the classification. Nobody has great confidence in the proof. Nobody understands it. There are attempts to reprove it.

If it can be formalized, that would demonstrate that AI is ready to formmalize all of mathematics.

The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents… The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched.

The Millenium Prize is $1M, what is the ROI? (Edit: since I was not clear, and confused some - I mean for a hypothetical of a third party paying commercial rates to use AI to solve mathematical challenges and claim prize money, not for scientific value alone or as a promotion of an AI lab’s capabilities)

My napkin math - If you get 33 output tok/s each agent will burn 10.5M tokens over 88 days. At $50/MTok (Astra cost), that is $525 per agent. With 10,000 agents, you’d spend $5,250,000 to get back a million.

(We also know that they were running more groups that varied in size and this model is a generation ahead of astra)

The answer to this will obviously shape the near future of mathematics, but there's also something even bigger than that at play: It has always been the case that the questions in math were stronger than the answers; you have stuff like Fermat's great theorem that is easy to state but monstrous to prove. This seems to be a property of mathematics, not of humans... but is it true?

A question by Scott Aaronson from 2011 (3) about P vs. NP seems relevant here: "Will humans manage to prove P≠NP before they either kill themselves out or are transcended by superintelligent cyborgs? And if the latter, will the cyborgs be able to prove P≠NP?" Later, he notes that if P≠NP, "once the robots do overtake us, they won’t have a general-purpose way to automate mathematical discovery any more than we do today".

---

(1) https://mathstodon.xyz/@tao/117207849921390904

(2) I'm not sure whether this is a hard distinction -- e.g. Tao also has some partial results towards Collatz (https://terrytao.wordpress.com/2019/09/10/almost-all-collatz...).

(3) https://scottaaronson.blog/?p=690

[T]he group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.

While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .

Wow. That sounds like an admission of guilt.

I think the market for local models/private datacenters (for bigger businesses) is going to be big. Even if you don't have unique tech/idea/implementation sharing your business secrets with Altman/Dario/Elon/Zuck doesn't look very appealing going forward.

How we found the proof

Easy, we stole it from Levent and Tristan

https://x.com/kyanyang_/status/2097211154669998337

The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched.

If this actually holds up, solving a Millennium Prize problem in 88 hours is mind-boggling.

I'm only half joking. Should researchers perhaps put hashes of their attempts on a public blockchain tied to their own public keys, verify their claims asynchronously, and then whoever reveals the first believable attempt gets the credit?

I know some people started doing this years ago but now it might need to become standard practice.

The sequence for Navier-Stokes was that these researcher spent a year working on it, then they published a possible breakthrough, OpenAI then spends $15M within a couple days to finish it.

This was incredibly opportunistic.

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PLEASE DO NOT TRAIN ON OUR PAID ACCOUNTS. There is a fundamental trust violation at stake here, no wonder mathematicians are mad. Using our data should be opt - IN!

>“we cannot rule out that de-identified data derived from their usage of our products helped improve our models”

Other simpler words for this sort of thing are “IP leak.”

There’s some quite concerning issues burried in this rah rah PR post that seems like potentially the real story here.

Much more clarity is needed on what happened here beyond this eh, some strange stuff could have happened comment.

Another way of reading this is never give these models anything that’s not already public knowledge as otherwise OpenAI is admitting it could, potentially, steal your IP or idea. Thats quite scary for anyone in the business of IP generation and explains why the maths community seems quite upset today.

Feeding it your paper and asking for help (even just editing and grammar) now looks like a terrible idea.

Now, keep in mind, I'm only asking strictly pure mathematical questions - nothing at all related to cyber or protein creation or biohacking or anything like that... And, like I said, only the OpenAI models are doing this. To be fair, all of the prompts have always eventually returned a satisfactory answer, as far as I can tell, and haven't used a weaker model to answer them. Maybe? I dunno, it has just struck me as odd every time it has given me that message to pure math prompts.

Or will access to internal frontier models provide a big boost?

I think they should be able to unravel whether or not any sessions by Tristan or Levent went into the training data for this model.

First of all, there has been published work from Diego Cordoba and Luis Martinez-Zoroa that will be in every training set. It was suggestive of the pathway to solve Navier-Stokes.

Then Tristan Buckmaster and Levent Alpoge built on this work using LLMs from OpenAI and Anthropic. Possibly internal models were used from Anthropic. And of course Anthropic wants to credit for solving the first Millennium Problem just as bad as OpenAI. It seems they were getting close and were aware that they might get to Navier-Stokes.

OpenAI swoops in. At a minimum they are aware that Anthropic has either solved a Millennium problem or is close to it. At a maximum they may have Tristan and Levant’s unpublished proofs of related problems.

They then throw a truly staggering amount of compute at Navier-Stokes. They seem to be aware it is the best candidate problem. And they crack it. They are the first with a verified proof.

So the outcome here is that we have a solved Millennium Problem. It’s not the extremely simple narrative that would be easy to understand, “solve Navier-Stokes make no mistakes.” It was a messy race to finish against two unpublished frontier models, a whole bunch of brilliant mathematicians and enough compute to drain a lake. It’s kind of irrelevant which company got there first. They were both within a few months of being capable. I think the thing to remember here is that without LLMs, I don’t think we would have a proof to Navier-Stokes in hand today.

We've seen in the past they will go to any means to satisfy the desired outcome

a cached version of the internet

Interesting detail. A heavily pruned version, I assume?

There you go, the suspicion of the "concurrent work" (https://cims.nyu.edu/%7Etristanb/statement.pdf) mathematicians might not be that unfounded after all...

YT playlist on Millennium Prize Problems By Harvard math department in March 2026

https://www.youtube.com/watch?v=3j1VW9REm7s&list=PL0NRmB0fnL...

On Navier-stokes problem definition:

https://www.youtube.com/watch?v=XoefjJdFq6k

https://www.youtube.com/watch?v=ERBVFcutl3M

https://www.youtube.com/watch?v=Ra7aQlenTb8

this is surely the line which confirms they plaigiarised the solution.

Running agents and prompting excessively to produce 'slopcode' to solve mathematical problems and generate a solution.

If this is what anyone calls 'slop' then slop has no meaning.

I'm all for it on the use case of solving mathematical breakthroughs!

The reality is clear though. The chances of AI models overtaking majority of mathematics within next 10 years is becoming very high. Especially if it becomes cheaper to run these models.

As math formalizations improve, AI can have faster progress in math, compared to even computer science or software engineering.

It is simultaneously the best and the worst time to be a mathematician right now.

Not in a happy-go-lucky "if we just ignore the problem of politics and resource allocation for a bit" world, but in ours. Do y'all really think this will make the world a better place?

Maybe stop building the Torment Nexus, you numbskulls.

I guess solution had not yet appeared in training set.

"The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.

Diagram of a swirling vortex illustrating inward spiral and axial stretching. A snapshot of local incompressible motion. Orange marks faster angular rotation; teal marks slower rotation. Circulating speed also depends on radius. The trajectories show inward spiraling and axial stretching."

Hmmm, isn't that interesting!

(Side note: Apparently we can't "compress" something, but apparently we can move more units of that thing over a specific space in a specific time... hmmm, I wonder what the difference between those two concepts could be...)

Main Observation: The image on OpenAI's web page above, looks sort of like the one for the Hopf Fibration:

https://en.wikipedia.org/wiki/Hopf_fibration

https://www.google.com/search?q=hopf+fibration&udm=2

1. This used an awful lot of compute.

2. The solution to the issues regarding whether or not OpenAI stole the result, would normally be to move to a self hosted solution, however those researchers are unlikely to be funded for 1.

Which means they can learn from whatever you discuss with ChatGPT unless you are going through a clean API (perhaps Bedrock? Anyone know?)

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).

While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .

Is the biggest fuck you to the mathematics community.

Credit? Nah if we think you’re close we’ll use your data and swamp you with our improved model. Then we’ll threaten you.

Millennium Prize Problems were used as examples of something the current approach to AI just wasn't capable of, discussions that would result in "we'll need a totally new architecture".

Cause OpenAI will hear about it and beat you to publishing.

Who made me the genius I am today The mathematician that others all quote? Who's the professor that made me that way The greatest that ever got chalk on his coat?

[Chorus] One man deserves the credit One man deserves the blame And Nicolai Ivanovich Lobachevsky is his name Oy, Nicolai Ivanovich Lobach—

[Interlude] I am never forget the day I first meet the great Lobachevsky In one word he told me secret of success in mathematics: Plagiarize

[Verse 1] Plagiarize Let no one else's work evade your eyes Remember why the good Lord made your eyes So don't shade your eyes But plagiarize, plagiarize, plagiarize Only be sure always to call it please, "research"

[Chorus] And ever since I meet this man My life is not the same And Nicolai Ivanovich Lobachevsky is his name Oy, Nicolai Ivanovich Lobach—

[Interlude] I am never forget the day I am given first original paper to write It was on analytic and algebraic topology Of locally Euclidean metrizations Of infinitely differentiable Riemannian manifolds

Боже мой

This I know, from nothing What I'm going to do I think of great Lobachevsky and get idea, haha

[Verse 2] I have a friend in Minsk Who has a friend in Pinsk Whose friend in Omsk Has friend in Tomsk With friend in Akmolinsk His friend in Alexandrovsk Has friend in Petropavlovsk Whose friend somehow is solving now The problem in Dnepropetrovsk And when his work is done Haha, begins the fun From Dnepropetrovsk to Petropavlovsk By way of Iliysk and over Novorossiysk To Alexandrovsk to Akmolinsk To Tomsk to Omsk To Pinsk to Minsk To me the news will run Yes, to me the news will run

[Verse 3] And then I write by morning, night And afternoon, and pretty soon My name in Dnepropetrovsk is cursed When he finds out I published first

[Chorus] And who made me a big success And brought me wealth and fame? Nicolai Ivanovich Lobachevsky is his name Oy, Nicolai Ivanovich Lobachev—

[Interlude] I am never forget the day my first book is published Every chapter I stole from somewhere else Index I copy from old Vladivostok telephone directory This book was sensational! Pravda—well, Pravda—Pravda said: "Жил-был король когда-то, при нём блоха жила”…it stinks But Izvestia! Izvestia said: "Я иду туда, куда сам царь идёт пешком”…it stinks Metro-Goldwyn-Moskva buys the movie rights for six million rubles Changing title to 'The Eternal Triangle' With Ingrid Bergman playing part of hypotenuse

[Chorus] And who deserves the credit? And who deserves the blame? Nicolai Ivanovich Lobachevsky is his name Oy

(Tom Lehrer put all of his work in the public domain prior to his passing. Find versions of his performances on YouTube.)

https://tomlehrersongs.com/disclaimer/

https://mathstodon.xyz/@tao/117237320796901560

"We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential. The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science and do serious long-term damage to the future of the field."

So we can be suspicious that there is some truth, one way or another that they could have reused prompt/data generated by the user session.

Just saw this a few mins ago.