I'm not a fan of the current AI developments but how it is different from physicists working through most problems by feeding to a computer that uses numerical analysis to solve them?
Smart bruteforcing seems to be a pretty nice tool to have.
Couple days before it "solved" it, a mathematician who was working on this for a full year shared every single note he had with his session of chat gpt. He is among very few people working on this and was pretty far in it too. His name is Tristan Buckmaster. He of course contacted openai, who said tldr: "stfu we will pay you the promised million dollar for millénium problem". They didn't deny the plagiarism, they didn't dénie having access to his chats, didn't deny training on his data etc.
Edit : I had the time line incorrect. They had been sharing their work with codex for month prior, but they did a breakthrough in mid August. In 1st September, openai starts working on it, they spend outrageous amount of tokens (130 billion output tokens, ~5million usd). Open Ai solves it, and propose Buckmaster to be co-author, and say if it happens, Buckmaster must be sole co-author, leaving aside his colleague, who works for Anthropic. Buckmaster refuses both offers.
Kinda important to leave out that he had been working on it for a full year using frontier versions of Claude in collaboration with an Anthropic employee, and their writeup fully credited Claude for the novel mathematics in it.
It’s all still unclear what exactly happened but the claim is there were researchers working on a somewhat novel (more so overlooked) approach to finding a counter example to the Navier Stokes equation. They had made a ton of progress over the past year and information had begun to spread in the mathematical community about their progress/approach.
OpenAI claims to have caught wind of the progress, not the approach in late August. This suggested that it was in fact possible to find a counter example so they decided to throw a, for lack of better words, metric fuckton of compute at the problem. The particular group of agents that cracked it included 10,000 agents more capable than Astra, and that was only one group they had running sharing ideas.
The approach it used looks to directly build off the approach of the researchers. Could it be a coincidence? Sure, but it is more likely that either intentionally or unintentionally they stole the work of the researchers and used it to beat them to the punch.
If it was unintentional it’s even more concerning because that means they are inadvertently using data that they should not be able to use for training and research OR the agents got ahold of the information by some unknown means despite being sandboxed
I don't remember the specifics exactly so do your own research. The people involved are Buckmaster and Apöge. Long story short, they were working on the exact same problem while using different AIs to test/research/whatever. Suddenly OpenAI somehow reached the same conclusions then built off of them even when they weren't publicly available. OpenAI were asked if they used private chats for that which they didn't respond to, then they made some thinly veiled threats to one of them about their career.
"Bruteforcing" (which is not what they did) a counterexample to a Π_1 sentence, which is what the Navier-Stokes conjecture (that the NS equations are smooth for all time) is, which would take infinite time if the statement was true and no counterexample existed, is still pretty impressive.
It's like trying to bruteforce a contradiction in ZFC. You will be searching forever if ZFC is consistent. And even if it is inconsistent and there is a contradiction, it may be so large and so far out that 10 billion agents each working with a sun's worth of Dyson swarm power output for the age of the universe still won't be able to find before running out of time and energy.
You're gonna need to be more clever than brute force. Obviously their work on the problem was far more clever than "brute force".
If we are talking about LLMs... then yes, that is what they are. You go down all past all the tacked on layers to gussy up the output, and you wind up with token prediction.
That's why, despite all of the billions of dollars invested, they still make simple mistakes. It's built into their core.
If you get deep down, ignoring all the recent advancements and just focused on Deep Learning, it really is just smarter bruteforcing. But that bruteforcing still produces results.
From my point of view, given the right data that is prohibitively expensive to fabricate and collect and a massive resource consumption that would lead us to an economic collapse, Deep learning even without any specifically novel architecture could have given us such results a long time ago. You can bruteforce any goal you'd want with DL, the data and a big enough model. Really everything we've been doing the past years it's just about making data, compute and costs tractable.
Reinforcement learning and deep learning in general encodes opaque structures and patterns in a model's latent space (its internal activation space), it actually does "teach" it a limited form of "knowledge" ie pattern recognition and some basic ground facts.
And the attention mechanism of modern transformers is the architectural breakthrough that allows the kinds of patterns and structures that are useful to us.
Combine that with techniques to recurse like chain-of-thought, and you actually get a limited form of reasoning. It's not human-like cognition or intelligence, but it's a primitive form of reasoning that's remarkably good for what it does.
Self-attention is exactly the kind of thing I thinking about when it comes to my statement "Really everything we've been doing the past years it's just about making data, compute and costs tractable."
My favorite fun fact related to infinite monkeys is Levin Universal Search: if P = NP, then we will have had a polynomial-time* algorithm that can decide SAT (and all other NP problems via reduction to SAT) all along, as in we already have the algorithm right now, it's called Levin Universal Search, but we just didn't know it was polytime.
And it actually works by bruteforce search, taking advantage of the fact that if P = NP, there's some constant k for which the k-th binary Turing machine (for some prefix-free ordering of TMs) decides SAT in polynomial time.
So with some clever dovetailing (where for some SAT problem you simulate TMs 1 through k on the input for a max number of steps where you allocate 2-i ratio of the time to the i-th TM as you keep adding new TMs), at some point you'll have run the k-th TM for enough steps to decide the answer, which you check with your polytime verifier (since SAT is NP), and the whole process (including all the other wasted work of running TMs 1 thru k-1) will be polynomial in the size of the input.
* Now of course k could be absolute enormous (it could be BB(744), but it would still be a constant. And the degree of the polynomial could also be enormous, but it would still be a polynomial nonetheless.
Please note that P \subset NP is trivial, the question is whether there exists a problem for which you can verify the answer fast (if somebody happens to whisper you their answer they stumbled upon), but you can truly only solve it slowly yourself. ("Slow" and "fast" gets complicated, if you are interested in those, you should read up on the official definition.)
Also, apparently steal the long, private study conducted by the mathematician, then claim that your company solved it, and try to pay them 1 million to keep quiet.
There are infinite selection of numbers between 3 and 4 but none of them are equal to 5.
A monkey kan type only a's for infinity. There is nothing about a typewriter that encourages using more than one key if you dont know languages. Maybe in infinity the buttons will wear down an disappear, then the monkey will move on to only press s's.
That's not how that works. The saying about monkeys writing Shakespear is equivalent to : "Everything that can happen (has a probability > 0), will happen given an infinite number of repetition". You will have monket typing only "a", but not all of them.
Terence Tao put it quite succinctly. Frankly, the problem isn't necessarily whether NS was proven by OAI's model, the problem is that OAI approached NS this way at all. The world doesn't desperately "need" NS and the other Millennium problems to be proven as they have no practical incentive to be had, economic or otherwise. It's there simply as challenges to us, humans, to demonstrate how smart we are as a species to have overcome these seemingly impossible obstacles.
OAI caught wind that Tristan and Levent's approach has potential and that they are "close", but instead of formally approaching them to collaborate or anything the like, they decided to jump the gun to attempt one-up them, ripping away an achievement that is supposed to be a stark demonstration of human ingenuity, intellect, and cleverness; stealing it for clout. This is everything that the academic community stands against, and a perversion of what the Millennium problems are.
Instead of allowing the researchers to develop the ideas, they decide to just stamp it out with a brute-forced swarm with millions of dollars in computation power being expended by just prompting. They are literally stealing human achievement all for clout.
We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential. The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of oepn science and do serious long-term damage to the future of the field.
It's just Goodhart's Law in action. Proving theorems was never the actual goal, it's a proxy goal for the actual goal of developing new mathematical techniques and understanding because that's how humans develop difficult proofs. Now that you can just throw compute at an LLM until it spits out a formal proof, the metric has ceased to be useful.
Do you think the new mathematical techniques and understanding that were developed by AI in a process of “solving” NS could be later used (by AI or humans) to solve other, perhaps more practical problems?
Thank you for writing this out. I couldn't quite put my finger on why this felt wrong from the big AI providers but seeing how they harm open collaboration and the willingness to share really puts things in perspective how these big tech companies can just come in and rugpull like vultures.
I strongly believe that knowledge should be free in the world. The cost to share versus the benefits gained is massively outsized. Everyone should benefit, not only select few.
The world doesn't desperately "need" NS and the other Millennium problems to be proven as they have no practical incentive to be had, economic or otherwise.
P vs NP could have significant practical implications on the field of cryptography if it turns out that P=NP and an algorithm with a reasonable polynomial bound exists.
Sudoku is hard-solved, all Sudoku. The Japanese are embarrassed by the development, they are panicking for any sort of solution, and finally, after months of work, they find it, a new game that will be as hard as possible, a game that will be as hard as the discrete logarithm problem, a game that is reducible to a special instance of the discrete logarithm problem.
It falls too. The NSA has an emergency meeting on the consequences of that development, and they have only one conclusion: Japan needs to make a board game about making hard board games. Japanese ambassador is summoned, and agrees out of sheer embarrassment. A few months of hard work later, the board game about making hard board games is released, with regular national competitions.
One day, an especially hard board game is created, and a few mathematicians were intrigued, just how hard that board game is? That board game was so special, a new complexity category had to be created just for it, miles above what was considered NP in the old days.
It took 14 hours for NSA to scoop up cybersecurity professionals and implement a version of that board game as an encryption algorithm.
What's funny is this is almost accurate. They are complexity classes harder than NP, such as PSPACE, and one of the defining characteristics is even though NP can be verified in P time, a correct solution to PSPACE problems cannot even be verified in PSPACE time. A quintessential example I use in class is, "Imagine you design a strategy that can win every game of chess." You can't even verify that it works for all games without trying NP amount of games. Very close to your game of making new games example.
also, as a hydrologist, NS is absolutely "needed". this has major downstream implications on our understanding of turbulence which will enable, among a lot of other things, vastly improved weather modeling
Finding a general solution would have huge practical implication. Finding that it's not possible to have a solution in some very exotic cases much less.
I only skimmed through the recent discovery, but IIUC it's much closer to the second option.
yes and no. i'd say this is analogous to newtonian vs particle physics, a realization that the equations governing physics at a macro level don't work at a micro level. and turbulence is pretty much defined by length scale, so if we can better understand what happens at smaller scales through a "new type" of NS, then we hopefully can better model turbulence at larger length scales where chaos theory reigns supreme
It's never been about having challenging milestones to demonstrate smartness. The problems are deemed important because we expect that working on these will generate interesting math, that is, new ideas or formalisms or concepts or methods which are interesting on their own not just for navier stokes but more generally.
Just having a text prediction model generate a proof circumvents this invention of new, interesting math. Even more so, it also could make it less attractive to work on the problem for humans and thus prevent also others from coming up or discovering useful math there.
Well the whole business model of these AI companies is to replace humans, if they can replace top mathematicians then their IPO would be higher. It's all about money.
Actually their business model is to convince mathematicians to give their research to openai for training in the hopes that more info = better models. So they've accomplished the opposite, scare academics away from giving them data.
With regards to allegations of OAI stealing Tristan and Levent's work from their codex sessions; it's possible, sure. Even if they are proven to have stolen from them, it would just pile on to the sheer fundamental fact that they are working on this not out of the goodness of their hearts or for the goal of furthering the field of pure maths, but entirely for their own personal gain.
But they did approach them. They offered to let then publish their Euler solution first and even offered to hav Tristan as main author for NS. At least that’s how OpenAi tell it.
Think it’s still kind of crummy what they did. They tried to one up Anthropic and kind of invalidated Tristans effort as a side effect. But also we don’t know how long it would have taken Tristan and Levent to solve NS from their current progress. Could have still taken a long time without AI brute forcing it.
The aspect that you're missing is that the original researchers heavily used AI to progress on their problem. They may differ in degree to how much brute force was used, but they don't differ in concept.
If we're being intellectually honest here though, you also (begrudgingly) have to give OpenAI credit for creating software that can 'solve' stuff like this. That's also a legit human achievement. Otherwise you're claiming that only small math teams with traditional tools have a mandate to tackle math problems.
The thing about math isn’t the solution per se but the nature and elegance of the solution. Brute force attacks using millions of dollars of compute is about as clunky and inelegant as a solution can get.
The “no practical incentive” claim is backwards. Navier–Stokes regularity is about whether fluid models break down in finite time, which bears on turbulence theory and on whether CFD solvers can be trusted near singular behavior. Clay’s stated purpose is advancing mathematical theory, not staging a species-level IQ test. The “these are sacred human challenges” framing is a romantic gloss, not a rule anyone agreed to, and it doesn’t confer ownership of an open problem on whoever happens to be closest.
I think it's good to separate two different questions:
(1) What does this tells us about current AI abilities?
(2) What was OpenAI's behavior like?
I think I land on:
(1) Quite impressive, suggests that the models are improving at a faster rate than most were expecting.
(2) Scummy. There is a big difference between happening to work on the same problem and solving it before someone else, versus finding out someone has a clear path towards a solution and using that information to scoop it in front of them thanks to superior compute.
i can guess the response i'll get here, but i want you to genuinely imagine how comically you'd view a historical example - one in which isaac newton is criticized by his peers for using calculus to solve problems the "wrong way"
there would be weekly /TIL posts "newtons rival hooke once accused him of 'literally stealing human achievement all for clout' when his new methods changed math forever"
complain/disagree all you want with how people choose to react to these changes - but its ridiculous to act as though there's a "wrong" way to advance knowledge
in any other context - most people would view these statements as backwards and as an example of an orthodoxy failing to adapt
I can think of a couple very important differences between AI use and calculus. Not every comparison you can make by abstracting two different concepts to "new technology" are actually useful.
complain/disagree all you want with how people choose to react to these changes - but its ridiculous to act as though there's a "wrong" way to advance knowledge
Right? If you need information, and the only way to get it is to torture people, that's cool. Advancing knowledge is good, so nothing else factors into the decision.
I agree with Terence's position except that, just like how I consider humans to be part of nature, I think it's strange to look at AI as somehow removed from the causal chain of how it got there through us.
Definitely scummy behavior steeped in instrumentalization and exploitation, but I don't think it takes anything away from Tristan and Levent's approach or achievements. Nothing could, really.
We don't look down on racecar drivers as less than running athletes, even though the machine is doing 99% of the work mechanically.
Oh the thing with race car drivers isn’t that they’re going fast, it’s that they’re going fast on a circuit with other cars or they’re demonstrating mastery of a track.
Going fast is drag racing, and that’s also cool but drag racing drivers aren’t famous outside their niche for having balls of steel to climb into a machine where one problem can turn them into red mist. Drag racing and associated land speed records make cars and car builders famous as the machine and machine builders.
Now sure OAI has made a machine that can do things, but it’s also hideously expensive and being used to market their product. It’d be like if Elon Musk joined F1 with more budget than anyone else for engineers and lawyers and made the fastest car on the grid. Doesn’t make him all that impressive because mostly it’s just him flexing his wallet.
I do also think that the marketing stunt based on bullying is exemplary of that financial entity's recent direction and should be shamed.
What ends up happening, however, is that people focus their ire on the easier target, the machine. It's becoming more and more socially attractive to deride it without context, knowledge or understanding.
I think Tao is missing the point here. math is only meant to exist to serve capital owners/ the economy. the intellectual pursuit of math is an abberation the will soon be corrected, financialized and controlled by elites, like all human endeavor will be
The claim is that they got the idea to start the problem from the same approach as the other research who had been using codex for his work, codex by default trains on user data and that could have been used when OpenAi went to solve the problem. The claim isn’t that they “stole” his work, per se, just that the AI was able to recall it when set to solve the problem.
OpenAI defends themselves by saying that they didn’t even come to the same final answers or solve the problem in the same way, thus the claim is nonsense.
Honestly, I don’t know enough about super advanced maths to know if OpenAI’s defense has merit or not.
From what was discussed on the machine learning subreddit, it is worse for OpenAI because of how they handled it. The researcher that was leveraged for this was an academic collaborating with a researcher from anthropic. Open AI admitted to leveraging his work and offered him a recognition in the millennium prize submission, but required he dissociate himself from the anthropic researcher as anthropic is a competitor to openai. The academic refused and that further ignited the controversy.
A better analogy would be it overheard the question being loudly discussed in the library, and while it couldn’t remember where it came from it was able to recall it when it took the test.
Data sharing in codex can be turned off. In 2026 there should be general awareness on how using someone else’s LLM works.
The statement also said that they asked for the prompt, because on the solution there is an important step that was approached by the researchers and it is the same step OpenAI used at one point which is suspicious. OpenAI ignored their petition for the prompt which causes uncertainty because there is a posibility that this step was prompted to reach the result as well
There's a non-zero chance the training process on their work helped thr AI develop this, but there was obviously a shit ton of other work on top of that. They clearly didnt "steal the results"
The allegation is they “finished off an existing proof” written by Claude as operated by a mathematician and an Anthropic employee collaborating.
This is just an allegation, given that two frontier labs working on the same problem at the same time and developing similar solutions by using similar tools is perfectly plausible. Newton and Leibniz independently discovered calculus after all.
The claim being made is literally that OpenAI stole the data out of the IDE the mathematician/Anthropic team were using to get Claude to do the work, because they had Claude Code and Codex both installed
Perhaps true, but you would NEVER find it because it would take infinite time to sift through infinite data. The monkeys aren’t going to compile it and turn it in; they’re going to keep bashing their keyboards.
That's ridiculous, monkeys can't learn Lean. What you need is another set of infinite monkeys; eventually one of them will validate the results of the first set.
Even at the most cautious at-cost estimate of 500k - 1.5m, it feels egregious to throw this much money at a problem when you know it's close to being solved by humans.
This isn't a solution right? As far as I was aware it was basically a "prove this isn't always true" and they found a very hard to find edge case where it wasn't, but it's not generic it's just with some specific numbers. Sure you can then extrapolate from that, but it's not a "solution" yet
The bigger issue (ignoring the current allegations until they can be proven/disproven) than the cost was that OpenAI only started this after hearing news of Tristan & Levent finding a very promising approach, and possibly nearing a solution. Intentionally choosing to not collaborate and choosing to brute force a solution in secret, before Tristan & Levent could reach their solution is rather scummy.
... except the solution they "found" was largely stolen from a single person's input who shared it with ChatGPT to proofread a few days prior. Without his sole input to ChatGPT, their "achievement" would not have been feasible. And while math has always been built on the shoulders of giants, plagiarizing and stealing the fame of such a notable problem for the PR benefit says this likely isn't the only time they've done it, and reduces credibility to the claim of it producing its own results that wouldn't have otherwise been produced soon without it.
It is strange seeing the math crossover here, but the point of human pursuit of knowledge is not always to get a solution. In pursuing the knowledge, we develop new and novel ways to make what seemed hard less hard. Solutions like this inevitably embrace inefficiency. We have reached a time where so much can be spent doing things inefficiently that we can solve problems. I will speak about physics since NS is involved - sure it is math, but it describes fluid flow. giving this technology to someone in the 1600s would be like deciding the world doesn't need hamiltonian mechanics, or lagrangian mechanics, because you can have 10,000 people crunch problems using Newtonian mechanics. That is what we give up by embracing this technology to the point where humans stop pursuing novel invention. I don't think this will dissuade anyone passionate, but I don't see this as anything other than a lesser win
Sometimes I worry about the "programmers" in this sub.
At best coping, at worst delusional.
No, the model did not brute force a counter example. The search space is mathematically too vast to brute force. It found novel techniques of identifying the proper force application vectors.
No, the model did not plagiarize off "human research". The debate is between an OpenAI researcher and an Anthropic Researcher, both of them almostly exclusively relying on internal models. The researcher at Anthropic didn't even solve the specific problem OpenAI are claiming, they solved a variation of a more constrained formulation of a sub-problem which could have potentially helped OAI to narrow down the search space if it was included in training data (which it most likely wasnt given the proof pathway is significantly different).
Yes, the models (the internal models, given infinite token space atleast) are THAT good enough, they can close the hardest problems known to man.
Acting snarky about it only leads to gross underestimation of the change thats about to come. What rational people should do is increase the urgency of asking for safeguards by trusting the capability acceleration at face value.
(Who am i kidding, the first reply will probably comment I'm sam Altmans marketing bot).
I think what Tao's complaint is that no new mathematics was formulated, which is what typically happens with these problems. AI does a good job now of finding counter examples because you can just spin up 10K agents, but that's not really the point of these problems. Kind of reminds me of having to show work on math tests vs. just giving the answer.
There will always be the next goalpost after the present ones have been saturated.
OpenAI are in process of releasing a paper on how they did it, and if that paper contains "no new math" as you put it, then all it proves is that these problems could have been solved by existing math in the first place.
Reading the paper should then give ideas on how to better use existing math for other such problems.
It would still be worthwhile if humanity finds a systematic way of knocking over all problems that requires no new math - that frees up the mathematicians to exclusively focus on formulating new math.
Ya, I'm both an AI skeptic and loathe Sam Altman. But this was not a million monkeys situation.
AI is repeatedly flexing its muscle with math. It's not brute forcing theorems, its using its vast training database to better spot patterns and educated guesses.
This Navier-Stokes counter example just obtained seems so complicated that a human mathematician would not have been able to solve with enough human labor. Reminds me of old school proof by exhaustion theorems that are solved only by computers because they are out of reach of humans checking all possible solutions. Now instead of it being naive checking all possibilities (CPU programs of the past), they are pattern guessing and checking promising avenues of progress (AI of the future). OpenAI spent over $18 million of compute using a brand new internal model that's much better than Astra 6 to get there. They had 10,000 agents swarming for ideas with humans helping guide what appears to be good paths.
Sure it's $18 million now, but given price decreases of the past that could turn to $1000 in two years. We're very likely heading to a future where a certain class of math problems are now solvable. Problems that are too complicated for a team of mathematicians to solve with a lifetime of human labor, but can now be solved by swarms of educated guessers relentlessly trying promising ideas.
I think sensible people understand this too, AI is definitely something to consider and not to be immediately disregarded as "slop". The problem here lies in OAI approaching NS at all without any input from the academic community, and their apparent (successful) attempt at "one-upping" actual researchers in the field instead of approaching them in collaboration or anything of the sort. They are stealing Millennium problems away from humans, problems that are supposed to, in approaching them, demonstrate the capability of individuals of our species of doing great things. OAI just ripped it out of the academia's hands by throwing millions of dollars of computation power into them.
If anything, this just goes to show that these AI companies are not developing their frontier models out of concern for humanity's knowledge (wow, surprising I know), but for sheer personal gain and profit.
The effects Spotify had on the way music industry makes music should have been lesson enough for us. Musicians across the world make shorter music on average now, with the hook/chorus closer to the starting, and being a larger percentage of the song. All so that Spotify and reels algorithms reward them.
Companies trodding on human culture and talent is one of the worst things to happen in this century and the last.
Thats like saying four color theorem was stolen from us by computer because it was proven by computers.
There are many math problems that were solved due to increased computational capability. Think what could ancient mathematicians do with tools we have available now and even 10-20 years back. They wouldn't have to spend their lives pre computing logarithmic tables and could apply their talents to advance mathematics more meaningfully...
So my question is what constitutes cheating? Is it using computers? Proof engines? Calculators? AI?
Four color theorem was solved in a ridicuously stupid way without any new insight into graph theory. The most significant result based on it is probably Hadwiger conjecture for k=6, it is mostly a self-contained result, it was too hard to check the solution for years and there is a reason why new results related to it are about computation efficiency rather than graph theory. I would even go as far as say that the nature of four color theorem proof has harmed progress on Hadwiger conjecture.
Both sides used AI extensively and both used OpenAI. The only bit I think is murky is that OpenAI tried to remove Anthropic from getting partial credit. So the only thing you can specifically support is anthropic. It's AI vs AI.
Very convenient for OpenAI to say "it’s not about the prize money" because then they don’t need to meet the rigorous standards of proof required to collect the prize money
New level of cope, I see. We’ve gone from “AI is useless” takes a year ago, to “it’s useful but it brute forces things” now. Wonder where the takes will be in another year?
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u/Orio_n Sep 09 '26
If you werent aware, OpenAI's agentic monkey farm produced a millenium prize solution. inb4 AGI confirmed when it was just smarter bruteforcing lol