How An AI math breakthrough ignited a controversy
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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
> "Of course we don’t know whether that is true"
Yep. Who is verifying these claims? We all know how trustworthy Altman & Co are.
Separate from all the allegations of more nefarious actions and ethical issues, that’s the most charitable version of what happened here.
And according to them at some point they saw that one was close to being solved, so they pointed all the agents at it.
The same thing happens to humans - at this time there are no simple problems left, so solving the hard ones requires using prior knowledge and attempts at solving things.
A better example would be playing chess against a player slightly stronger than me and using a chess computer to suggest some good moves. I could win, but it certianly wouldn't be just my brain that wins. It would be an amalgamation of my brain with a machine that suggests good moves.
One cannot simply reason by analogy.
My iPhone keyboard does
This likely doesn't happen exactly on an analog keyboard, but then many text-processing environments that do the same thing in post. My keyboard just edited 'yuor' to 'your' even though I successfully input the prior string.
Professor says the assistants - why don't you dig in this direction, I have a hunch it might produce something valuable. And AI assistant does just that, proving or disproving a hunch. This would take the professor a lot of time if doing by themselves.
If I write a book and pass it through a spelling and polish checker, I still wrote the book and its core IP. I didn’t “rely on” the tool to create the IP.
The OpenAI effort was a pure brute force attempt. I'm not even sure an LLM was actually involved. I think they just used their hardware to run the matrix multiplies required by the search for a counter example. Perhaps some clever approach guided the search but that seems to be about it.
Work on something novel -> llm is kinda useless and low value-add -> Keep at it and in the process feed it more information -> keep doing this periodically -> a few months go by and you realise the model outputs are almost like-for-like regurgitations of what was inputted in some prior period.
Once it’s accumulated new info can it produce something automated that is somewhat useful? Sure.
But by itself - absolutely not.
I clearly see humans will be needed - the best ones that is. For ‘rote work’ and stuff that is not IP sensitive firms will be ok with employees putting that as inputs into models.
But I’d wary about trusting the labs. They will push the letter of the law to the max.
Personally I’ve stopped doing anything novel with these models. If I do use a model on something adjacent but not totally novel I have to craft the inputs in a strategic way not to give much away.
I’d wager firms will soon realise this and that growth rate of revenues of the frontier labs will become questionable. The economic cost that firms have brought out thus far is only financial. There’s a whole bunch of other costs people aren’t talking about.
I guess I'm having trouble unraveling your experience and personal usage vs. what you're concluding about the labs.
To avert that dynamic, the frontier labs must deflect and otherwise act to prevent this controversy from breaking through. Both to the general public, but also more specifically to the firms' decision-makers. All of whom are generally aware of the IP issues, and some of whom are aware of what happened with Cursor and Figma, but with few exceptions have not yet themselves acted to protect their property.
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.
Also, OpenAI wanted the actual mathematician taken off the resulting paper. I'm not sure I would describe what OpenAI did as research. What the other team was doing does seem to be more like research but the hardware was still in those cases mostly brute forcing things and then doing something like a genetic algorithm to compose an actual proof based upon the results of a large set of brute force attempts.
Do you think it is possible that better math will lead to better physics models?
Not all.
Ehrhart’s volume conjecture
Quantum parallel repetition for general two-player quantum games
Erdős Problem #183 on multicolor Ramsey numbers
Erdős–Sárközy Problem #12(i)/(ii)
Erdős Problem #125
Log-concavity of codimension-3, type-2 pure O-sequences
Optimal O(1/t) last-iterate convergence for Anchored Gradient Descent-Ascent
Prior updated :)
This proof is just checking the boxes for mathematicians.
> 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.
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.
> This proof is just checking the boxes for mathematicians
There are already quite a lot of summaries of the story that lead to the solution, and the impact that the intermediate results have had.
It certainly seems like any problem that is amenable to reinforcement learning will be solved.
you're not actually spending that money. it's sunk cost, as you already bought the hardware. at least for the big pharmaceutical companies for drug development. then you run your own local model, trained on special data, with special etc, etc... to the end of buying GPUs for what, 3.5-6.5M/rack or so (GB300 NVL72, Google AI summary pricing quote) becomes a bargain (vs the double digit billions you need to spend on a new drug R&D).
Solve for how to implement and synthesize physically? Not likely.
Humans solved for launching rockets to the Moon on paper decades before it happened.
Pareto type thing; the logical work is the easy 80%. The last 20% is fighting physics.
There is no beating physics but there is still plenty of room for us to improve our understanding of it.
Which we weren't focused on at all sitting millions primates at well understood physical computers searching for Shakespeare Python and Ruby code yet merely getting same old contemporary software outputs.
It would be like saying proving ergodicity more generally for physical systems would unlock condensed matter physics, ignoring how well stat mech has served us regardless.
I am not anti-AI and I don't think we should stop throwing them at conjectures. I'm against this fundamentally misleading type framing that's become prominent. Millennium prize problems are important. Treating this specific aspect of NS as the one missing piece is just harmful. If we just throw compute at formal conjectures voila cancer and fusion.
I think the better example of "AI" usefulness toward solving problems is AlphaFold, and immensely powerful tool. But also suffering from a false framing/marketing problem as "solving protein folding". It feels like the right use of compute. Considering many factors that we can't hold in our head at once. "Solving" something that was already "solved" via computation (simulation) but now much more efficiently. The output is a valuable tool itself, it was not about "solving the protein folding problem", which it didn't do. It is a tool to solve problems requiring a sequence->ground state calculation. Which is a very broad set.
Formal verification of a conjecture we set up as a benchmark we set to test human understanding is not valuable in the same way.
I'm failing to make multiple points and gotta run, but i think that final point is important. The millennium prizes are not about technological/practical value, at least not intentionally. They're about shit that seems fundamental to us, things that feel[1] to us based on our understanding are important AND feel like they should be solvable in a human-comprehensible way. So formally resolving them with pure compute is not really the point. It seems closer to that story about one of those prime conjectures where some guy just ran brute force enumerations to find a counterexample. Valuable for sure, time-saving. And knowing the answer makes it a lot easier to solve a problem.
TL:DR science and math are more than formally resolving conjectures, they're about building up understanding and tooling that you can then build more on. AI should be an increasingly big part of it, but declaring "AI will solve fusion because it's smart" is like the rest of the fucking owl meme. I have no doubt it will help, most likely via simulations/quicker testing/calculations and verification. Maybe partly via reactor designs. Maybe partly being fed conjectures about bounds/limits that would be useful as inputs for the next iteration. And maybe even in the form of resolving some formally stated conjectures (I don't know enough plasma physics to name any).
[1] obviously to the mathematicians it's more than a feeling..
> 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?
This is explained on the page[1] linked to from the privacy section of the pricing page.
[1]: https://help.openai.com/en/articles/5722486-how-your-data-is...
Unless OpenAI can show that training was permitted, this will erode the limited trust that many users have had in such toggles and may lead to further, uncomfortable inquiries.
Where are you seeing that? He only says "We used several LLMs throughout: Anthropic’s Claude, OpenAI’s Codex, especially with GPT-5.6 Sol and, more recently, Astra. The latter was only used for writeups and auditing our arguments."
I might be missing something, but he doesn't seem to confirm that he opted out, at least in the written writeup, maybe he has on social media? He also likely had early access to Astra given the timing, and I thought early access customers couldn't opt out? (Am I wrong about that?)
"I pay for the tools my group uses out of my own research funds, including footing a large bill to OpenAI."
I have reason to believe OpenAI doesn't respect training opt outs, but there is a good chance they let something get through without an opt-out too.
It's not given in the statement, so I don't know
> And did he share drafts of the work with anyone who fed it back in on a consumer subscription without disabling training?
To this, the statement does have the following:
"... the one Levent and I had quietly chosen to attack"
Note his collaborator (Levent) works at Anthropic, so presumably he knows about opt ins/outs, etc.
> I have reason to believe OpenAI doesn't respect training opt outs, but there is a good chance they let something get through without an opt-out too.
I believe that, at this point in time, OpenAI's reputation speaks for itself. The stakes for success are already enormous, and the CEO has been repeatedly described by those around him as a pathological liar.
People who choose to expose anything remotely sensitive can surely be expected to know the risks of doing so.
> I pay for the tools my group uses out of my own research funds, including footing a large bill to OpenAI.
This is blatant scientific misconduct.
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".
Later on the author claims that the OpenAI rep threatened to ruin his career if he didn't go along with them.
Worth noting that there were two versions of the problem:
- the proof in the equations with viscosity (which OpenAI claims to have solved), and
- the proof with no viscosity (which Tristian and Levent solved)
What is confusing is that if OpenAI can prove their independence from Levent and Tristian, they could take full credit for proving the viscous version of the problem. Offering to give one author credit for something they proved seems like a strange choice: if nothing else it seems obvious that it would drive a very deep wedge between the two authors of the non-viscous version.
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.
OpenAI’s behavior here — even if you only consider [their] side of the story — was (at best) in bad taste.
It is mean spirited but nevertheless sold their model
They just used that occurrence as a PR stunt but it isn't worse than the others done at scale every second.
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.
2. Even their pursuit of this problem was itself unprofitable - $15M in compute to solve a problem with a $1M prize. Not that that was the point, but still.
The frontier labs are likely betting on lay observers (read: investors) confusing headline-grabbing results in abstract mathematics with phenomenal profitability in more grounded endeavours. There is an implicit fallacy that "If our models can solve mathematics they can do everything else."
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.
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.
I think you cannot assume so, because pattern matching is not reasoning.
Quanta Magazine article that also discusses some of the controversy: https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-...
> 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.
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.
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.
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.
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.
Aristocrats were a dime a dozen.
> 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?
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.
Or they can simply turn off the setting that allows OpenAI to use their chats to improve the model. Or they can use the API where it is off by default.. If the advantage of using OpenAI models will be significant enough for the business in question, these are the options.
You do realize that businesses that have contracts with them can specify if they want their data used for training or not right?
Pick a better analogy.
(Not that I think the comparison to pi digits make sense)
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.
Or is it just capitalism doing capitalism stuff?
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 think going forward, any researcher should consider anything submitted to an LLM to be copied/stolen.
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'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)
This doesn't guarantee that the statement is correct (Lean cannot do that), but makes it highly likely.
> The main branch of formal-conjectures does not contain the path `FormalConjectures/Millenium/NavierStokes.lean.`
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.
This isn't the way you build trust.
This is just lawyer speak. Maybe there was a small reward from a possible thumbs up on any of the chat sessions. Open AI have no way of knowing if that happened or not and the chances that, if this did happen, that it had anything to do with their solution of Navier Stokes is extremely unlikely.
>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.
There was never a threat to not publish their own work with whatever credit to whoever. This was about the offer to be a lead author on the paper that Open AI authored, an invitation that was not extended to Levant.
You're right - it is exactly "lawyer speak", rather like Bill Clinton's "I did not have sex with that woman".
Navier Stokes is a test of how high the intelligence is.
Even if these two mathematicians were using the AI services under a clause that nominally allows OpenAI to train on their data, it is still another level above for there to exist some pathway within OpenAI to know that someone is doing something very valuable and important with their AI and to swoop in to try to steal the value of that work. It doesn't matter exactly what that pathway is, just that it exists is a big deal. Today it's a Millennium Prize, but tomorrow, is it the next hot new product category? Is it the patent someone is working on? Does OpenAI constitutionally believe that you may be using their product but whatever you produce really belongs to them and you are just borrowing it? Can we trust contracts that promise that they don't believe that if this is how they act?
Even people who are consciously aware that their sessions are being used to train the AI don't expect the AI companies to be in some way scanning the stream of what AI is working on for high-value propositions to snipe. That changes the relationship between these companies and enterprises, and anyone who thinks they may be doing this sort of work, deeply and fundamentally.
It isn't just that OpenAI is now something we should be suspicious about for math. It's all high-value science. Is there anyone trying to figure out how to develop a next-generation AI architecture using OpenAI? Better stop, OpenAI could see it and scoop you out of who knows how much value.
There is an old story attached to many historical figures that goes:
Churchill: Madam, would you sleep with me for a million pounds?
Woman: My goodness… well, I suppose I would.
Churchill: Would you sleep with me for a pound?
Woman: Certainly not! What kind of woman do you think I am?
Churchill: Madam, we’ve already established that. Now we are
just haggling over the price.
Honestly the human drama, while the compelling story, is in monetary terms orders of magnitude less important than the question of whether OpenAI has a mechanism to steal ideas from its customers. Has this happened before and they just successfully covered it up? Or maybe the people stolen from thought it was innocent and just a coincidence that they were scooped? If it established that there is a price, I have a lot of questions about that price.
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.
What does the company they work for (only 1 of them, mind you) have anything to do with the topic at hand? They were using both Anthropic & OAI models during their research, and they were doing it separate to their work as independent researchers rather than as employees of any specific company. In fact, this makes OAIs actions even worse because they tried a bribe in order to slice the Anthropic employee out of the deal.
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.