alt Hacker NewsOn the Navier–Stokes Millennium Prize Problem
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What happens to real fluid in this particular cases?
If the singularity is in the physical space?
Is this just a result of ignoring things like friction and energy dissipation via heat, etc?
I have a dumb feeling that the proof will be wrong with serious flaws but that will be found out only after the ipo
>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)
> 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?
For now I think more or less the same thing as with all recent math announcements: This is in a range where human work still exists (see Terry Tao, (1)). I wonder whether the trend will extend into the problems that (as far as I can tell) are considered complete brick walls right now -- P vs. NP, Collatz, Goldbach, odd perfect numbers, problems that aren't part of any research program. (2) In other words, is the progress coming from putting together vast amounts of existing work and computational power, or is it more from RLVR and self-play and autonomous effort?
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".
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(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...).
> [T]he group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.
Sebastien Bubeck’s (OAI project lead) response: https://x.com/sebastienbubeck/status/2097379411691516310?s=4...
Is blockchain going to finally be the solution to something?
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.
Seriously starting to think we are not going to make it out alive of the near-future.
> 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.
The problem is the precedent this creates. For non-famous people using public APIs like this it could mean AI companies sucking up the information and throwing millions in compute at it.
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.
I really hope OpenAI doesn't take the bad press some people are giving them too seriously here. They should throw their whole weight behind the rest of the Millennium Prize Problems. To think – if everyone lets their egos calm down we could have the Riemann Hypothesis solved by the end of the year...
Astounding. Would be interesting if one day the archive of those prompts / messages / tool calls would be released publicly.
I'm not an expert in fluid dynamics, but does this result have any positive implications for nuclear fusion research?
There's a loophole in the terms of service at least for Anthropic which allows the use of dark patterns to "borrow" your (even paid) data.
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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!
It's probably important that some humans verify these proofs "by hand".
created a simulation of the solution to describe what's happening and why it's important for engineers, climate modeling, etc : https://navier-stokes-singularity-simulator.netlify.app/ (updated so that it works better on mobile)
In CS speak very roughly this would mean something like disproving an algorithm by giving it a case that fails it. Right?
Just so everyone knows, although openAI pretends that the model generated solution and wrote the paper by itself ""with very little human input"" as Buckmaster himself mentioned in his statement. In reality they have team of researchers guiding the system, along with, probably training on user data, probably Buckmaster in this case, in order to come up with the proof.
OpenAI cribbing from other researchers. We just have to assume OpenAI is actively adversarial in future. Accidental cyber intrusion is also well within model capability.
>>“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.
Can't wait for the BobbyBroccoli series on this in a couple years.
This is the problem Yu Deng got this year's Fields Medal for I think?
So, is that basically the Taj Mahal of counter examples?
It’s a pity they had Astra do the writeup. I was curious to see how “GPT7” writes.
I think it would be better for the proof to go through the peer-review process.
If OpenAI doesn’t claim the millennium prize for this, who gets it? No one?
Can't help but shake an unsettling feeling about all this, frankly. I engage in some limited mathematical research and will often use any one of the latest frontier models to check some ideas. Lately, only the OpenAI models have been giving me a temporary message that says something like (paraphrasing from memory), "We're thinking extra hard about your request before we answer. You can choose another model to answer now or click here to learn more about why." When I click to read why it's doing this "extra thinking", the help page says that for cybersecurity and biosecurity-related information, it will review the answer and could refuse.
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.
It does make you think about the old question "are we discovering or inventing mathematics?"
Questions: can new research like this be done using publicly available models?
Or will access to internal frontier models provide a big boost?
I feel like something is being lost in the drama here.
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.
"While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."
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.
Damn how long before the simulation stops if all the unanswered problems get solved .
They should release the entire session trace if they really have nothing to hide
How can they "not rule out" that Tristan and Levent's data was used for training?
With these massive Lean proofs how do we know the model didn't just find some bug in Lean and exploit it?
We've seen in the past they will go to any means to satisfy the desired outcome
What is the other clay prize that's might be solved now/soon?
>a cached version of the internet
Interesting detail. A heavily pruned version, I assume?
We are living in the future.
I think it's over guys
The named OAI employee has released a statement: https://xcancel.com/SebastienBubeck/status/20973794116915163...
The real story here: the priority dispute and its implications on AI.
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.
"While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."
There you go, the suspicion of the "concurrent work" (https://cims.nyu.edu/%7Etristanb/statement.pdf) mathematicians might not be that unfounded after all...
Resources:
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
The problem that I want to see them tackle is formalizing the classification of finite simple groups.
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.