(2026-09-14) Zvim Brand New Ai Solves A Millennium Prize

Zvi Mowshowitz: Brand New AI Solves a Millennium Prize. 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. The first Millennium Prize, Navier-Stokes, has fallen to AI.*

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

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.

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.

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

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

I cannot emphasize the top story enough.

Setting the Stage

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.

Our Price Cheap

*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*

An Accusation Is Made

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.

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.”

Sebastien Bubeck: nothing at all was locked we were just willing to talk but you didn’t talk to us

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.

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.

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

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

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.

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.

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

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.

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.*

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.

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.

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."

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

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.

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

Now for the most important part of the story.
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.

*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.

*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.’*

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

*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.*

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

*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*

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.

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?*

Quickly, There’s No Time

*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.*

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