Brand New AI Solves a Millennium Prize

The first Millennium Prize, Navier-Stokes, has fallen to AI.

A deeply unfortunate situation has arisen involving what should have been some combination of a positive story about new progress in AI-assisted mathematical research and yet another opportunity to freak out about rapid AI progress.

Or, as we call it around here, Tuesday.

The Real Story Is The New Model That Is Better Than Astra

Keep your eyes on the prize. There are three stories here.

The first story is much more important than the second story, which in turn is much more important than the third story.

1. OpenAI’s next model took a week to get a generation ahead of Astra, and they are telling us this because everyone is totally freaked out about what is happening.

   1. Or, in official language: ‘We believe it is important to inform the world about the pace of AI progress and what to expect from upcoming models’ and that we ‘may require more deliberate choices about the pace of progress.’

2. This new AI has, eight days after it started training, solved Navier-Stokes.

3. A bunch of drama over who gets the credit for math involving Navier-Stokes.

Jeffrey Ladish: I haven’t looked into the human drama around the Navier–Stokes problem but sorry give me a minute because HOLY SHIT AI JUST SOLVED A MILLENNIUM PROBLEM.

I cannot emphasize the top story enough.

I am still going to tell all three stories, but again: Eyes on the prize.

Setting the Stage

The story on this particular Tuesday begins in the morning.

Tristan Buckmaster and Levent Alpoge had worked for a year and offer us a series of remarkable results: Finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3D incompressible Euler. They also believe they have a blowup for hypo-dissipative Navier-Stokes, but the Lean verification of that result is not finished.

Tristan Buckmaster: The program this fits into was not started by us nor was it proposed by a Large Language Model. The credit for the basic idea of this program goes to Diego Cordoba and Luis Martınez-Zoroa, who for several years have been
exploring the construction of forced blow ups. We took their work as a starting
point, using Large Language Models to push their program to completion.

Concretely, what Levent and I did was to take the Cordoba and Martınez-Zoroa program, which achieved blowup results with rough forcing, and, with a
great deal of help from LLMs, push it to smooth forcing and to the incompressible Euler equations. The ideas making this line of attack possible are due to
Cordoba and Martınez-Zoroa.

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

So far this is great. As he notes requires rethinking about how all of advanced math will function going forward. Terence Tao offers commentary on the underlying results.

The part that is not so great is where they felt forced to publish early, without the opportunity to spend the weeks necessary to make the proofs what passes among mathematicians as readable. Buckmaster outright apologizes for the way the reports look, comparing the Euler writeup in particular to AI slop.

What happened?

I Heard a Rumor

On September 1, OpenAI heard a (false) rumor that two Millennium Problems had been resolved. In response to the possibility that someone else might make the world a better place, OpenAI spent millions of dollars in inference to explore all the unsolved Millennium Problems, using an internal model stronger than Astra, and cracked the whole of Navier-Stokes in 88 hours, plus another 17 for Astra to do the Lean formalization and verification.

You can view this as ‘OpenAI got curious to see what their new baby could do’ or you can view this as ‘they spent millions trying to scoop what they thought was an Anthropic project.’ My money is on a little from column A, a little from column B.

All it took was the rumor.

Our Price Cheap

OpenAI: Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.

Depending on what costs you count, this would have cost a regular customer on the order of $22 million, or several million for internal marginal costs. A small price if it works, but also it is crazy how many people don’t think two steps ahead.

roon (OpenAI): like all other technologies your equivalent agent swarm will cost a buck fifty in like a year. all this stuff will mean an unprecedented Enlightenment no matter the costs today

I mean, no, maybe $150k or if you’re lucky $15k, but the point stands.

Sam Altman (CEO OpenAI): ugh AI is such a bubble, i heard they are selling tokens at a loss, did they know this was only worth $1 million?

A Millennium Prize result is worth vastly more than a million dollars. OpenAI does not intend to claim the prize money. This was never about prize money, for anyone. It was always about the credit.

An Accusation Is Made

After hearing internet rumors, Buckmaster reached out to OpenAI, and according to Buckmaster OpenAI’s Sebastien Bubeck said that an internal OpenAI model had produced a proof of finite time blowup for the forced Navier-Stokes equations over the past few days. Buckmaster claims that over the course of two calls, it became clear he had initially been misled and that an entire team had been working on the problem, using an ‘insane’ amount of compute.

Which we now know was 130 or 300 billion output tokens, depending on what counts.

If this part is true, it would be quite bad.

Tristan Buckmaster: 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.”

… I would like to be clear about what I am not claiming. I have not seen
OpenAI’s proof. I do not know what their model did, or how. I do not know
whether our data was used. I am not accusing anyone of anything. I am stating
what I was told, when, and what was proposed to me. I am stating it because
the alternative is to let a sequence of announcements say something I know to
be false.

Or here’s Levent Alpoge telling his side of the story, and Sebastien responding:

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

i mean props to them for straight coming clean.

(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)

so i’ll now give a bit on my thinking here. i actually woulda been pumped to collaborate on this, there are a lot of people at oai i like (ok, clearly some were indirectly dicks to me because of being part of the whole situation, but im a big boy, i still like them), idgaf about authorship on that step anyway, coulda been me Tristan and every fte at oai for all i care (on that Tristan would disagree:p). but on hearing the loud convo in the hallway, especially the part where a millennium prize was offered if i’d just be removed from the paper, it was kinda clear the die had been cast and things were locked. pretty wacky, unstrategic, and unnecessary, since on my side things were mostly me and claude having a good time yoloing random stuff in the corner rather than anything institutional. i also like the idea of the labs cooperating, and even better on scientific progress. it’s a shame!

Sebastien Bubeck: nothing at all was locked we were just willing to talk but you didn’t talk to us ... sorry but that’s just untrue, we were willing to go above and beyond and have as many discussions as you would have liked to reach resolution.

Sebastien strongly denies that he did anything wrong, as does Noam Brown, and Sebastien offers screenshots he says demonstrate good faith.

Also a counter-accusation:

Alexey Guzey: I have many friends at Anthropic. Almost all of them stopped talking to me once I joined OpenAI.

Completely unsurprising that Levent refused to talk to Sebastian to discuss the announcement plans and then started blaming OpenAI for it.

Sam Altman also strongly claims his team acted ethically, and tried to collaborate in good faith.

How Did You Get That Idea?

The implication of all this, that many drew, was that there was a soft accusation that OpenAI had stolen part of their result from Codex chat logs, or that the chat logs had been used in training data and that had contributed to the result.

Charles🔸: Why big corporations want ZDR, exhibit A

OpenAI: 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).

We now have explicit confirmation that the Codex data was not used in any way in training runs:

roon (OpenAI): i was on a call during the first announcement where people were trying very hard to say the precise lawyerly truth having not looked into where codex (opt in) data was used and whether it made it into training runs. it's now clear it would have been impossible, the chances are 0

I agree with Sholto Douglas that it is extremely unlikely that user data had any influence here via either method:

will depue: there is literally no chance that OAI researchers spied on Tristan’s private codex chats. i don’t think y’all realize that openai like any other large regulated online platform has highly locked down user data access permissions. it’s not 2010 guys

Sholto Douglas (Anthropic): fwiw I think it is _extremely_ unlikely that user data had any influence here - there is no way OAI would pull user transcripts for this, or knowingly train on it in a way that would’ve influenced this.
I think its pretty important people don’t run away with ‘your user data isn’t safe in codex’ - because it surely is (based on everything I can assume from the outside).

OpenAI Almost Certainly Did Not Misappropriate User Data

It would be an earthquake, perhaps the worst possible story for their business, if we found out OpenAI or Anthropic had raided someone’s user data to gain competitive advantage, and the facts make more sense without this. The funniest hypothesis, from Jerry Tworek, which I also find highly unlikely but which we sadly cannot rule out, is that OpenAI did not intend to use any such data, but the agentic swarm assigned to the problem hacked them and got it anyway, and OpenAI has no idea.

My strong presumption is that the user data was not intentionally accessed, that they would never (or, rather, that they at least recognize they can only do it once and this is not that once), and also that user data would have been of little help.

Thane Ruthenis points out that there was previously a school-shooting incident, where (highly reasonably) the logs were examined and there was internal debate over notifying law enforcement. Even that sent chills into some users. Where will they draw the line? And yes you should understand your data and logs might not stay private, if you give them to a lab. It is reasonable, as Andreas Thorn also is doing, to wonder about whether your proprietary research or math data has stayed private.

I still reiterate that, in practice, I would put a very, very low probability on OpenAI intentionally looking at such data. The risk-reward is just so, so bad. And I believe Roon’s later report that they have now confirmed the logs could not have reached the training data.

Our Top Labs Cannot Get Along Even On A Feel-Good Math Story

No matter who is and is not at fault, it is rather alarming that the labs cannot cooperate on something like assigning credit for a mathematical proof. This is a very bad sign and also a wake-up call.

Sholto Douglas (Anthropic): It is extremely sad that this didn’t end up as an example of how the labs could cooperate/coordinate, because the stakes will be so much higher in the future.

Noam Brown (OpenAI): Strong agree. I know there is rivalry between the labs but it’s important that we learn to work together given what’s coming.

Sholto Douglas (Anthropic): 🤝

Kevin Roose: These labs (especially OpenAI/Anthropic and OpenAI/DeepMind) are fueled by spite for each other to a degree that isn’t obvious from the outside. Partly because of stuff like “who gets credit for this famous math problem,” and partly just personal animosity between leaders.

As a guy writing a book about the AGI race, all of this is great dramatic fodder. But not great if the eventual goal is to have the labs cooperate on safety and pacing!

The continued inability to get along is a big deal, and will only get bigger.

What Next?

If it worked on Navier-Stokes, what else will it work on? Time to find out.

Nat McAleese (Anthropic): Navier Stokes is an incredible achievement, and reflects awesome progress. Huge oai swarms should be applied to other areas of science; ideally those with near term applications, including alignment.

This is indeed reported to be happening, including for fun questions like P=NP. Which, if it happened to be constructively proven true, would break a lot of things. You should also be nonzero worried about what such agent swarms might do as an incremental step, given OpenAI’s history.

The Mathematicians Are Not Happy

Solving math problems is their entire jam. Yet they are very not happy.

Andrew Curran: Twenty-five Fields Medal winners have published a joint declaration warning about what they see as a severe misalignment between AI companies and the mathematics community.

Math and AI (signed by 25 Fields Medal winners including Terence Tao): Over the last few months, the mathematical capabilities of LLMs have improved dramatically, to the point that they can solve major outstanding problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI companies and the goals of the mathematical community are severely misaligned. We see these as part of broader alignment issues impacting other scientific and creative professions, as well as the whole of society.

… Famous problems have often served as landmarks and lighthouses against which one can measure an improved understanding of this landscape.

… In recent months, the success of AI in solving major mathematical problems has made headlines even outside mathematical circles. But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight.

… Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions.

This is the general complaint, on top of particular complaints about lab conduct.

One could respond that no, the point of math problems is to solve math problems, and the mathematicians are mad that AI is disrupting their status games. That is certainly part of what is happening, but like Tyler Cowen I am not so cynical, and unlike him I do not think that ‘a bit of patience is needed’ or to wait until the damage is done before once can object. People can extrapolate. Did you know mathematicians are good at drawing straight lines on graphs?

There is a real complaint here. Getting people with talent to do real math for long periods of time, for very little money, and develop real math intuitions and concept mastery, when they could do other things for far more money, is no easy feat. If you ‘take away their status games’ and also far more importantly take away their fun and feelings of accomplishment and elegance and beauty, what then?

Robin Hanson suggests that then we should just fix the incentives and reward mathematicians better.

Robin Hanson: "there is nothing stopping the mathematics community from awarding status, pay, and promotions to people who 'fill in the important blanks in math understanding,' even if an AI already has proven or disproven the underlying theorems."

He doesn’t use the word ‘just’ but this is definitely a just, as in if only people would just [X], and as we all know humans have never justed and are not about to start now. This is not an actual thing humans might do. As even he then realized hours later:

Robin Hanson: Actually it seems to me quite a bit harder for the math academic community to coordinate to judge who "filled in blanks" in math, in ways other than proving theorems. Math has coordinated heavily around evaluating proofs, and hasn't developed many ways to agree on other stuff.

We can and will get some rewards for those who fill in blanks and improve elegance, but it is not going to be the same and will be a Herculean effort, at best.

Scott Armstrong argues that this is a change in order, but not in result. Human mathematicians will take a few weeks to understand the proof, but then they will gain the understanding. In the past, the understanding came before the proof, but either way you get the understanding. That’s what matters. He thinks what people are mad about is that the machines are smarter than them.

As always, AI is the best tool both for learning and not learning. The mathematicians are saying that they are being forced into ‘not learning’ mode, to letting the AI do their homework, and they don’t like it, because actually the homework teaches you.

You can say who cares, that it’s fine to have AI do the frontier math. And yes, I do think a lot of people will still want to do math all day, and look for elegant versions of ugly AI things and such. But it won’t be the same.

OpenAI’s New Model Was A Step Change Above Astra Four Days Into Training

Now for the most important part of the story.

RIP various OpenAI training pauses, June 22, 2026 to August 28, 2026.

roon (OpenAI): by my mark, the period from June 22nd (hugging face) to August 28th is upwards of a month of frontier pause.

On August 18, OpenAI said they had paused frontier RL for two weeks, and their largest planned training run remained on hold. Then on August 28, OpenAI started training another model that quickly became more capable than Astra across the board (by their own reports), now that they’ve solved all those pesky supervision, infrastructure and alignment issues, and it’s good, with a ‘step change’ being observed after only four days:

Terrified? You should be. That model is still training, had been training for less than two weeks, and is not especially aimed at mathematics.

That is all we know about the model. That is enough, when we combine it with the repeated stories of OpenAI and people at OpenAI virtuously freaking out over its alignment and other problems, and over the pace of AI progress, in a preference cascade.

And we also combine it with OpenAI dropping so many pieces of alarming information, that it could have avoided dropping, because they very much need us to understand the situation. The Astra model card counts here, so does An Alien Mind, so does Paul Christiano’s statement, so do the reactions to the HuggingFace attack including Pacing the Frontier.

And so does the OpenAI report on recursive self-improvement (RSI), which is the next section.

OpenAI and Anthropic have entered the RSI era.

OpenAI Research Progress Is Accelerating Due To OpenAI Research Progress

OpenAI has been opening up lately about many things.

The most important issue of all is that of the acceleration and automation of AI R&D. This is the thing that is likely to kill us, the thing that is most scary, the thing all the lab employees are warning about, and also the thing both OpenAI and Anthropic are driving towards as quickly as possible.

They gave us a report about that, too. I agree with Tejal Patwardhan that this is excellent transparency.

OpenAI (September 6, in Research Acceleration: The View Inside OpenAI): We aim to safely build an automated AI researcher that can work under human supervision to further progress on deep learning and alignment, enabling iterative improvements.

According to our measurements, we have now reached the goal, announced last fall, of having an automated research intern by September of this year.

… Whenever we find that proceeding would pose an unacceptable safety risk, we will respond appropriately including by slowing or stopping our development or deployment.

… We believe that we and other companies should be required to publicly track our progress towards RSI.

Line this up with the timeline revealed in their announcement of the proof of Navier-Stokes. They started training their new model on August 28.

Meanwhile, they believe they have a research intern that the timeline suggests is a different distinct internal model. What will they have next week? Next month? What does Anthropic have?

OpenAI explains that highly capable AI offers great upside, but of course it does. That is not why everyone is so determined to go this fast. They are in a race.

This is one more call, among the many from OpenAI and its employees lately, that are a combination of ‘we need to work together to end this madness’ with a large amount of ‘somebody stop me.’

OpenAI: These are reasons to develop useful automated research capabilities, but they do not mean that rapid RSI is necessarily an outcome we should pursue. Whether and how to proceed must depend on our ability to preserve human control and on informed democratic choices about the benefits and risks.

We do not yet know how to safely get all the way to aligned, full RSI. We are working to scale alignment and safety measures alongside capabilities. But we cannot assume that progress in alignment and safety will keep pace, and more capable systems can become harder to monitor.

Careful alignment and safety work is at the center of this effort, and it starts with measuring and mitigating the safety problems we see today in agentic coding systems.

Once again I object to the view that the prosaic or current problems are the main issue, but yes, they are trying to warn us, over and over that (translating out of corporate speak a bit):

1. OpenAI and Anthropic have a path to recursive self-improvement (RSI).

2. If either goes for it now, it will probably go terribly and everyone will die.

3. OpenAI is prepared to exercise some amount of expensive caution, but there are limits, and without a deal or law someone is going to try it soon.

The bulk of the post is a detailed snapshot of how much agentic systems have contributed to RSI progress at OpenAI in recent months, as in since Astra came online internally. The answer is quite a lot.

If anything, the surprise is that use remained this low for so long:

OpenAI: At the start of this year, the median researcher ranked by agent usage at OpenAI was using coding agents only in modest amounts. By mid-August, the median researcher was integrating agents daily into their work, using more than $600 per day of inference at API prices. The 90th percentile user in our research organization now uses more than $7,000 of tokens per day.

OpenAI: Before June 2026, total agent runtime across the research organization was still below that of total human labor. That has since changed. In terms of a standard 8 hour workday, as of mid-August, in total, the research organization uses 3.1 agent-workdays of effort for every workday of human labor.

They are not not bragging, and they are not not confessing. They are warning.

The binary measurement here likely makes this look less of an acceleration than it is:

I do wonder how much humans can properly handle 4+ concurrent workflows, says the man with dozens of open tabs and a dozen active post drafts.

It turns out this is not as impressive as it looks, because they’re counting subagents as workflows. That makes it more surprising that, while the median researcher spends $600 a day, there are still the 30% that aren’t doing intense usage.

Code shipments and experiments are accelerating fast starting around the point Claude Code and then Codex started being a big deal.

Until recently, most of what AI got used for was building research and infrastructure code, and the humans mostly did the rest. Now the AI does a lot of work on launching, monitoring and debugging runs, and on technical help and review, as well, and is branching out into analyzing results and other places.

They took our jobs, ‘human troubleshooters’ edition:

Here’s a different variation of the famous METR graph, with production tasks, a bit hard to read as a static screenshot but the progress is rapid. These are not years. These are months:

Quantifying OpenAI’s Pause

They have a graph of RL compute spending over time, I am surprised how little Astra has been used before July 20, but there you go. I notice they do not extend this into the present and presumably the Astra use went back up over time:

This Is the Way the World Ends

Was using the swarm to prove Navier-Stokes, only days into the training of the new model, itself risking a serious loss of control incident? June Jimenez argues yes.

If you unleash a 10,000 agent swarm of a new model you cannot possibly have tested, on a potentially impossible and definitely extremely difficult task, and collectively give it 300 billion tokens, what else might it have done? I definitely considered ‘maybe it hacked into OpenAI in some way to get the logs’ but there are so many possibilities.

Quickly, There’s No Time

When a former lab employee, here Andrew Ho, says ‘I notice that over a >3 month timescale I don’t think my productivity has increased over 100% or perhaps even over 50% because I get distracted’ as a reason to be bearish, that’s not all that reassuring even if true. Imagine only being 50% more productive every three months and thinking that this is bearish.

He is saying ‘even if AGI is eventually achievable, this implies a significantly longer timeline’ on the same day Greg Brockman said ‘welcome to the AGI era.’

On top of that, this level of math progress was not expected a few weeks ago.

Every time one of these problems falls, people try to retcon it into being not so surprising. They are often wrong. No, most people, even most forecasters, did not presume that this would happen. Perhaps you did. After all, you are very intelligent.

Henry Shevlin: As recently as April this year, prediction markets gave AI less than a 40% chance of solving any Millennium Prize Problem before 2030