alt Hacker NewsOn the Navier–Stokes Millennium Prize Problem
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>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)
I'm not an expert in fluid dynamics, but does this result have any positive implications for nuclear fusion research?
Seriously starting to think we are not going to make it out alive of the near-future.
My takeaway:
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.
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...).
> 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?
This is a great day to re-read Ken Thompson's "Reflections on Trusting Trust":
>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...
Astounding. Would be interesting if one day the archive of those prompts / messages / tool calls would be released publicly.
> 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.
Well, a few weeks ago I made some snarky comment on another LLM maths proof article like “I’ll be impressed when they point it at a Millenium Prize problem and make progress”. Guess I have no excuse not to be impressed now.
Elsewhere in the thread, others have calculated $15mm at API rates for just the output token. (So I’ll assume this cost about that much, taking input and human researcher time.)
I wonder whether a team of 60 mathematicians working solely on this for a year would have cracked this. (Assuming $250k total compensation.)
It does make you think about the old question "are we discovering or inventing mathematics?"
> [T]he group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.
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.
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!
>>“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 nor 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 what the maths community seems quite upset today.
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.
This is the problem Yu Deng got this year's Fields Medal for I think?
The actual solution link https://t.co/tz1shoCZZo
They should release the entire session trace if they really have nothing to hide
Damn how long before the simulation stops if all the unanswered problems get solved .
If OpenAI doesn’t claim the millennium prize for this, who gets it? No one?
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.
Questions: can new research like this be done using publicly available models?
Or will access to internal frontier models provide a big boost?
Fuck OpenAI. Fuck everyone who works there. Like seriously, to all the people who gift their life's work to this monstrosity, do you actually think something good will come of any of this?
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.
Too many unverified claims from OpenAI at this point.. why are we still talking about these people anyway?
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
How can they "not rule out" that Tristan and Levent's data was used for training?
"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.
It is so disappointing that we can't have such a monumental moment in history without the controversy. OpenAI leadership clearly doesn't seem to care too much about ethics. Is it a requirement to completely lack integrity to have a ground breaking company?
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.
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?
I think it's over guys
The named OAI employee has released a statement: https://xcancel.com/SebastienBubeck/status/20973794116915163...
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 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.
45 pages only. God damn that internal model is crazy
"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...
"While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models"
this is surely the line which confirms they plaigiarised the solution.
Why almighty openAi doesn't solve PvNP problem :(
I guess solution had not yet appeared in training set.
Well, if that's actually true, I think America needs to start talking about the nationalization of both OpenAI and Anthropic, maybe even merge both under a new federal bureau.
I hope everyone is as Navier-Stoked about this as I am.
This kind of thing is one of the reasons I really hate how AI is coming to fruition. These companies get a whiff of something valuable and they use their vast resources to take it for themselves. For everyone else, the only recourse is extreme secrecy.
Has this been verified by the Clay Institute?
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?