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Those problems can’t be formally verified with an automated theorem prover. We have a lot of physics based simulation tools, but they tend to focus on small subsets of the full design problem and they make limiting approximations because otherwise they’d be too computationally expensive, or we just don’t have the right data to parameterize them beyond describing qualitative behavior. Agents are helping accelerate research in these fields but I think it’s mostly a different class of problem that’s a lot harder to specify and verify
Separate from all the allegations of more nefarious actions and ethical issues, that’s the most charitable version of what happened here.
This proof is just checking the boxes for mathematicians.
OpenAI (claim to) show the existence of *a* finite time singularity. It could stimulate more research in PDE solving, and maybe physics, but it has zero impact on practical applications, that I can see. The Millenium problems were chosen based on hardness not practical relevance.
> For any practical application, numerical solvers for Navier-Stokes already exist and do a good job.
or
> This proof is just checking the boxes for mathematicians.
Do you think it is possible that better math will lead to better physics models?
So yes, the models do seem to be "solving" the problems themselves, but not necessarily in the way we think of mathematical discoveries happening. Academic mathematics has historically been resource constrained: There are a limited number of top-level mathematicians, and they only have so much time and brain power to spend. So when approaching a problem, they are essentially forced to be as efficient as possible, not just searching for a solution, but for one that can be achieved within their cognitive budget. This induces them to develop novel techniques and abstractions, and it is actually those techniques and abstractions that tend to be the valuable part for further research, not the proof itself.
An agentic swarm is like getting a single skilled mathematician, cloning them a hundred times, then locking them in a room with the single objective of solving a problem. No longer constrained by time or brain power, they can approach it differently, using pre-existing techniques to gradually build their way to a solution. This process might not require a single intuitive leap or new discovery, and the solution will not be simple or elegant, but they will probably get there. It is more like a process of intelligently guided search than invention.
It certainly seems like any problem that is amenable to reinforcement learning will be solved.
> However, communications quickly became contentious. According to Buckmaster, OpenAI offered to give him sole authorship on the Navier-Stokes solution—but only if Alpöge’s name was removed from the work and if the write-up would acknowledge the problem had been resolved by an internal OpenAI model. Buckmaster refused, in part because he was troubled by the question of what OpenAI's system had actually seen. For example, Buckmaster said the company did not initially give him a clear answer about whether its agents had access to the pair's logs on Codex (which is an OpenAI product).
> OpenAI executives have denied that any employee or AI agent saw the pair’s work before the researchers released it publicly on 7 September. But there still remains a separate question: Could the pair's work have reached OpenAI's models through its training data?
> OpenAI’s blog announcing the Navier-Stokes solution does not dismiss the possibility: “While unlikely, we cannot rule out that de-identified data derived from [Buckmaster and Alpöge’s] usage of our products helped improve our models .”
Real solid business model there, how could it ever fail?
Add "We might take credit for things you figure out, if we can infer it from your prompts" and it might actually affect people's usage of these tools.
Well yeah, if you use the free product they train on your data, ... I thought this was widely understood?
> I pay for the tools my group uses out of my own research funds, including footing a large bill to OpenAI.
[0] https://cims.nyu.edu/~tristanb/statement.pdf
This is blatant scientific misconduct.
For comparison, if you offer me to collaborate in a paper about Algebra I may agree to go alone, but if the paper is about Quantum Chemistry I have to piggyback a few coworkers because we are collaborating in that topic for a long time and I already discussed may of the topics and I may even discuss the new paper too.
However, if they had published first, it's hard to imagine Anthropic not taking the opportunity to claim "our employee solved this Millennium prize problem using our AI".
But I think what’s being overlooked in the race to claim absolute credit is that both sides ultimately relied on a LLM (and one of OpenAI’s at that). Either a human researcher made a breakthrough discovery with the help of Codex, or the latest GPT model made a breakthrough with the help of human training data, or a little of both… either way it is undeniable that LLMs have quickly become an integral part of R&D workflows and are accelerating research.
This would be a major win for any normal company. You could even build a bigger collaboration with this guy, give him a big budget and push for extensions to this preliminary result, and in return do a write up on how he uses your model in his workflow. Huge PR win. What this says to me is that their valuation is so astronomical that they feel the only way to justify it is to demonstrate a fully autonomous discovery bot… which it simply is not.
OpenAI’s behavior here — even if you only consider [their] side of the story — was (at best) in bad taste.
They just used that occurrence as a PR stunt but it isn't worse than the others done at scale every second.
It is mean spirited but nevertheless sold their model
Quanta Magazine article that also discusses some of the controversy: https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-...
In short: The problem is about whether a solution (of the NS Equations with external driving force) can be found that blows up in finite time. i.e. exhibits infinite velocity at a point even for a viscous flow.
The solution: Take a circular curl ansatz which shrinks in xy-direction and elongates in z-direction and see whether you can find linearized waves so that these waves show a blow up when propagated on the curl. Then prove that the higher orders of the perturbation are regular before the T0 singularity time and you have solved the problem. The external force is just the remainder of the NS-Equation right hand side.
It is a lot of tedious formula juggling of all the higher orders and some singular perturbation expansions. Perfectly suited for algebra systems. OpenAI was using probably python sympy for the formula work and the researchers had to guide the LLM what to do in higher mathematical language.
Here is the paper:
https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8...
when you upload it to chatgpt astra can explain what they do and why it works, have fun.
LLMs seem particularly suited toward these existence-proof problems. Working mathematicians seem absolutely essential for universally quantified results, still. I strongly doubt, for example, that if Fermat's Last Theorem hadn't been proven three decades ago, that an LLM would be able to do work equivalent to inventing the mathematics as Andrew Wiles did to solve the problem. I have similar doubts about P vs NP, the twin prime conjecture, even the Riemann Hypothesis (unless the latter has at least one counterexample).
And I want to be clear: I'm not downplaying the achievements of these models. This is remarkable! I simply think that the pattern of success is in existence proofs or finding counterexamples, which makes sense based on how LLMs function and are trained.
I think you cannot assume so, because pattern matching is not reasoning.
Reductively, math can be said to be either problem solving or theory building - it seems the latter is a much harder thing to do right now.
Maybe those things are true, so maybe they should be convinced?
I don't see why mathematicians think they should be an exception here.
Seven, not six. One is solved already, but is still a millennium problem.
> Navier-Stokes is one of six [open] “Millennium Problems” on a list compiled by the Clay Mathematics Institute in 2000.
> According to Buckmaster, OpenAI offered to give him sole authorship on the Navier-Stokes solution—but only if Alpöge’s name was removed from the work and if the write-up would acknowledge the problem had been resolved by an internal OpenAI model.
I wonder if an appropriate response from the mathematical community would be a good old-fashioned shunning. Mathematicians are allowed to use OpenAI's tools as much as they want, but no one with any current or prior OpenAI affiliation gets published in a reputable journal, ever.
But given the cost of a college textbook this is a pretty silly complaint to lobby against a subscription that's $200 a month, in the context of the cost of a variety of other materials and tools out there. (If you think that's expensive you've clearly been lucky enough to never have to deal with commercial software costs) Also not sure how quickly this stuff uses up limits; $100 or even $20 subs might be enough for students. And if a student is scrappy and figures out that Luna can meet their needs then I'd imagine Luna is effectively unlimited on some of these subs. Luna Max scores pretty high.
taxes.
In any case mathematics is humanity's oldest open source project going on for millenia, it never belonged to a single country, institution or class.
Pick a better analogy.
> OpenAI, meanwhile, says its experience with Navier-Stokes could open the door to solving puzzles with more practical relevance. “We are now able to spend millions of dollars on a problem that we really care about and that really matters: developing new materials, finding cures to diseases,” Bubeck said. “All of those things that we have been talking about for a long time—now they seem to be at our fingertips.”
Their AIs?
That said... most of us are not working on problems as famous as Navier-Stokes. Even if OpenAI could scoop me based on my back-and-forth with ChatGPT, which I presume they could if they threw $15 million worth of compute at it, I highly doubt they'd bother.
I am going all in on sovereign ai even if its worse, these companies have shown they not only dont deserve trust but are actively stealing past and present intellectual property from humanity and users.
I think going forward, any researcher should consider anything submitted to an LLM to be copied/stolen.
Or is it just capitalism doing capitalism stuff?
You cannot risk companies like Anthropic, OpenAI or their business partners like Microsoft having unfettered access to proprietary data on your company/businesses.
It it likely that they or rogue employees will use the information to make a profit? It's pure speculation, but I'd say more than likely, and we will never hear about it or read it on the news unless there's whistleblowers in high enough positions to know about it.
Assuming you and your employees aren't careful with what data you share, they will have intimate knowledge about your company from files and conversations logs. Likely personal user data too which they'll gladly create databases to link to and create extensive profiles on you, your employees and your businesses.
It's not far-fetched to see them leveraging insider information shared with LLMs to play the stock market, leveraging data against competing businesses in other markets they might want to explore, and likely a bunch of other things that are escaping me right now as I write this.
At the end of the day it's on those people for sharing such sensitive data, but it's not like these AI companies are innocent and won't gladly exploit every little byte of data without telling you, we know it happens.
And the AI company for making the search program that searched through the data and found the solution.
Snark aside, the researchers working on this, who built the foundation, should get credit, and they are.
This doesn't guarantee that the statement is correct (Lean cannot do that), but makes it highly likely.
I'm not claiming to be an expert on Lean4 (although I have contributed tactics) but this is one of the most direct formalisations I've seen of a serious result (second only to FLT of course, which has a horrible proof but it is trivial to verify the statement)
Eh? There's no connection at all between the Navier-Stokes work and those things.
Why would anyone use OpenAI models for anything commercially valuable, or where secrecy is important, when it appears that if OpenAI "becomes aware" that you are doing so they may try to compete with you?
Not only did OpenAI, by their own admission, rush to re-solve Navier-Stokes once they heard the rumor that it has been solved (the rumor being that it was Anthropic that had done it), but they are leaving the door open ("we cannot rule out that") as to whether the model they used to do it had been trained on the anonymized date from the researchers who's approach they ended up copying.
Terrance Tao has recently said as much for mathematics - that there appears to be a trend (not just this Navier-Stokes incident) of the AI companies going after math problems wherever there is an "rumor" of progress, and that he thinks this may sadly result in breakthrough mathematics being conducted in secret to avoid this.
The rush to steal another researcher's thunder is bad enough, but it also appears that one of OpenAI's employees acted in a very thuggish manner to try to threaten the professor who had been working on this not to publish and to co-operate with their telling of the story.
Navier Stokes is a test of how high the intelligence is.
Sorry the sarcasm, but really your point makes absolutely no sense. It is completely different to design something with AI and then execute, validate, evolve vs just prompt it machine-g-brrrr style and get a result. This brings an important question. Nowadays I don't write code, I review code, I review systems behavior and get paid for it. Will that be the same for math researchers? Their prompt/problem is already well-posed out there. Ours, in the day-to-day, are not. Will the first one to verify AI work get the credit? or is it going to be the dumdum that types a simple prompt and has the compute to run it for 21321 hours? I absolutely don't get your point here. Or you are just rage baiting
By contrast, what the world-class 2 mathematicians did was sat down and started working on their proof for over a year, using AI along the way to help with their research. A much more grounded and realistic use of these tools, but one that doesn't generate nearly as much hype as the alternative.
The cracks in OAI's story has been immediately disproven, and they seemingly plagiarized the work of the 2 and then threw the team of researchers and the 15 million dollars at the problem after the fact. It doesn't exactly bode well for their hype machine when you consider the chain of events here, which is why people care about this, as OAI's constant and incessant lies they spew every minute of every day is finally hopefully catching up to them, and right before their big IPO too.
Editing to add: And I think it's all such a shame. We live in a time with genuinely insanely cool technology that is doing some truly incredible, ground-breaking stuff, but it's all tainted by a gaggle of greedy sociopaths and reprobates whose only goal in life is to have the largest number in their bank accounts. LLMs could've been such an amazingly neutral and cool and useful tool had more level-headed people been at the wheel, but instead we're stuck with this childish bullshit and giving the likes of Sam Altman real power to enact societal collapse.
If they were not related to anthropic I'd probably agree with you. OpenAI is much more for science than they are imo. Anthropic culture is all about "machine go brrrr" more than all of the other labs. If they had access to better models they'd probably would've one-shotted the solution. When the creator of bun was just "vibe-sciencing" it was ok. There is little to no evidence that they've been working using AI in this problem for over a year. Maybe they've been working on the problem for decades. So many other scientist have. Are they better because they threw a prompt and let it go brr??
When we put this in the perspective of how agents are changing the landscape of math/science, true it is shitty and weird. When folks are saying these scientists by anthropic that vibe-science'd the solution are victims, just because they did it with a smaller model, it is not defensible imo. And credit loses meaning here. The credit is shared with all the scientists that contributed somehow with the data in the AI pre-pos/training and not the prompter.
https://arxiv.org/abs/2506.06941
Ergo, they can't prove any theorem whatsoever. How do people at OpenAI expect that we believe in claims like that? This is yet another before-the-IPO stunt in my opinion..
Personally, I won't believe any of these claims until the community of mathematicians says otherwise.