Quote: "The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward. But in the process of capturing the reward, Z usually introduces new concepts and new open problems, reinjecting latent value into the Overhang.
LLMs can be trained on the entirety of the mathematical corpus. Thanks to their phenomenal memorization and pattern-matching abilities (without always being able to map out their associative logic and attribute due credits), they are in a unique position to harvest the Overhang. By contrast, professional mathematicians have typically read a few hundred articles in their career, out of millions of existing references, less than 0.1% of the total.
This will lead to great discoveries, which is unambiguously exciting. But it could also lead to a sad new deal, where human slaves painfully curate the Overhang while AIs systematically beat them at the finish line."
I've made an entire career out of being 'jack of all trades, master of none'. Being able to synthesize connections from relatively trivial knowledge in a bunch of domains is SOP for many humans as well. I think AI just has deeper knowledge and better pattern matching to make up for it's (at least now) lack of strength in cognition and 'ex nihilo' creativity.
(Which probably isn't 'ex nihilo' at all, and has more to do with the plethora of modalities that humans live in vs. large language models. For example, why do we pick the color red for notating important things and why do we say a schedule 'slips'...these are informed by a shared human experience borne of distinct physical sensation deep in our wiring that LLMs can only infer from what we write.)
In favor of the generalist, I think AI is also quite limited in its scope of how it generalizes. I'm mowing through hundreds of mythos-generated security findings right now for work and while it's amazing that it can build an exploit chain 20 steps deep, it's completely lacking in all of the external layers that render it's speculation moot.
I've been trending "quiet" lately, because I don't like the "friction"/convincing aspect of it all. It's hard to get people to see things from a different angle, or even convincing them there's a problem to begin with!
The last project required a complete redesign from a problem I pointed out during the first review, and second, and third, but now I'm seeing even more friction.
Maybe this is just corporate life, after a group gets large.
Any tricks/advice?
That overhang seems like a precious resource for AI companies. They can exploit that overhang to inflate the impression of AI's capabilities, and hopefully that exploitation will discourage the next generation of mathematicians from pursuing math. If they play their cards right, OpenAI and Anthropic can dominate the field even if they ultimately can't replicate the creativity of human mathematicians, because they'll have driven their competition out.
What we should be trying to achieve is a ladder-breaking maneuver: knock out the lower rungs so no person can reasonably climb to the top-reaches of mathematical skill anymore. That may ultimately result in stagnation, but it's what's best for AI, so it's what should be done now.
We need to do everything we can to create the greatest-possible dependence on AI tools.
When I was a software library developer, I came to resent application developers. I noticed a pattern. Libraries solved hard problems and did so carefully, thoughtfully, in a way that others could reuse. Apps would come along and carelessly, recklessly glue together several high quality libraries into a piece of software targeting a general audience. The apps would then harvest all the credit.
What's happening in mathematics right now feels similar. Applications (theorems) were always how one built objective reputation, but libraries (concepts, definitions, boring lemmas) were also rewarded socially within the mathematics community. And individual mathematicians often managed to both build their own libraries, and use them to prove an important result. And then those libraries were sometimes of use in other results.
Bessis asks whether AI Lean proofs will land in Mathlib or Mathslop. Or in my framing: will they be libraries, or applications?
At present they're mostly Mathslop. The proven result is perhaps useful, but the methods employed aren't novel or reusable. I worry that this trend will only worsen, because applications make headlines, and the libraries they used do not. We are not properly incentivizing library development in OSS, or in math, or in infrastructure writ large. There's a serious credit assignment problem here.
What might change this? Once the low hanging fruit is picked, will citation count rise in relative status again? Will we get result fatigue and start to reward legibility — no one cares unless the paper has an accompanying ELI5 tiktok video? A labeling regime that certifies the proof was produced sustainably, organically, by local artisans with no AI additives?
Many problems are solvable, but require months of work, and thousands of pages of proof. So people do not even try to create or verify the proof. AI changes that, it can verify and perhaps even simplify it, to more digestible form.
(I mean actually randomly, not asking an LLM to do the randomness.)
Most of the output would be incoherent (like many dreams), but occasionally you would get a gem.
The other part is, humans don’t really want to fund other humans doing this.
Very few want to be a math major; and of those that do, fewer complete a grad degree; and for those that do get grad degrees, there’s scant few research jobs; and for those who do get jobs there’s hardly any research funding to go around.
There does seem to be unlimited money for ai researchers to use ai to solve these problems though.
We’ve turned education into job training, so because there’s no jobs in solving math problems, few aspire to do it. If there were more opportunities for people, more people would do it, and more low hanging fruit would be plucked.
Once OpenAI heard that Navier-Stokes was solved, this caused them to immediately revisit the problem and throw a ton of compute at it, apparently using a more (very) recent model than what they had tried before. What we don't know is just how recent this model was, and therefore what it may have been trained on. Buckmaster/Levant had apparently been working towards this for at least a year, and made their "forced" blow-up breakthrough on August 15th.
Presumably any anonymized prompts that are being trained on are part of pre-training, so older, but once OpenAI had heard that Navier-Stokes had been solved and wanted to revisit it, it seems possible they may have done a few weeks of incremental RL training on anything Navier-Stokes adjacent they could come up with, in addition to then throwing unlimited compute at it, now confident that there was something to find.
>The Wednesday evening statement from OpenAI was more emphatic: “We can say categorically that it is impossible for Dr. Buckmaster’s Codex prompts over the last two months to have influenced the system in any way, including training.”
>The statement added, “After investigating, we can say with full confidence that no user inputs past July 3rd could have influenced this system in any way.”
https://www.nytimes.com/2026/09/10/science/tristan-buckmaste...
two people worked on this for a year before the breakthrough. Perhaps that earlier work reduced the search space sufficiently to brute force the problem with 10,000 agents?
It's comparable to Magnus Carlson saying that if he wanted to cheat, all he would need would be for someone to tell him to spend more time thinking about a specific move (just a wink would be enough) as an indication that a computer had found something interesting.
It's as-if after OpenAI first failing on Navier-Stokes (which OpenAI had just tweeted about 2 days earlier!), someone winked at them and said "you might want to try a little harder ...".
Obviously the result of OpenAI's investigation was that no usage data has interacted with the system after that date.
What else do you expect them to investigate?
If Buckmaster and co. provide their chats, OpenAI could potentially search for them in the anonymized opted-in usage data. Then they could say if any data has been used.
By all accounts individual usage data does not have the direct impact on the model most here fantasize about. To prove this, OpenAI would need to do new training runs to replicate the system used minus the particular usage data in question, if it exists, and then benchmark this on the problem again.
Potentially multiple times, in order to reach a conclusion.
The cost might be in the hundreds of millions.
(TIL: paltering: exact and technically correct statement usage to create misleading impression)
1) OpenAI by their own admission, only re-tackled Navier-Stokes because they heard it had already been solved (but not yet published). This isn't advancing science or helping the mathematical community, this is just being a dick.
2) OpenAI, specifically Sebastien Brubeck, then threaten to "not be nice" and "ruin the career" of one of the mathematicians whose work they had succeeded in duplicating, unless he agreed (which he refused to do) that his collaborator, an Anthropic employee, was not named. This is not only against mathematical norms of credit assignment, it is also being a pathetic human being.
OpenAI would have you believe this result shows how powerful their mystery better-than-Astra model is, but the reality here is that this model needed 10,000 agents, $20M of compute, and the assistance of a whole team of people at OpenAI, to replicate (then exceed) the work that just took two people, with some academic grants as an AI spending budget to achieve (a few $100K - listed below).
https://cims.nyu.edu/~tristanb/
I'd say advantage humans this time. Better luck next time OpenAI - and if you don't want unfavorable comparisons then maybe choose to work on problems that have not been solved yet, and that humans are NOT making nice progress on.
I think you have to work pretty hard to minimize what OpenAI achieved here like this.
The Navier-Stokes equations have been around since 1850. The smoothness problem has been well known for over a hundred years and has only gained importance. It's been a Millennium Problem since 2000.
Levent Alpöge and Tristan Buckmaster did great work to solve the related Euler problem, but didn't solve the Navier-Stokes smoothness problem.
The Navier-Stokes smoothness problem has previously had significant resources working on it. Computational fluid dynamics is one of the most important tools in modern engineering and is closely related.
You speak of 10,000 agents as though it is somehow extreme, and yet within the past month I've had a single task that used over 100 agents on a mere Anthropic team plan. I think two orders of magnitude more compute to solve one of the greatest unsolved physics problems[1] is nothing.
I don't excuse Brubeck behavior because of this, but that doesn't minimize the achievement here.
[1] Wikipedia quote: In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...
2. Yes Brubeck's comments were weird at face value. That said, Open AI's proof isn't a duplication of anything. Not only is Tristan's work a sub problem but the methods are different. And what OpenAI didn't want was Levant on the paper OpenAI authored not whatever they were working on (Euler). It's petty sure but it's fair enough. Tristan and Levant didn't have anything to do with the Navier Stokes solution, so it's really their call if they didn't want to collaborate on their own paper with the Anthropic employee.
>OpenAI would have you believe this result shows how powerful their mystery better-than-Astra model is, but the reality here is that this model needed 10,000 agents, $20M of compute,
$20M in approximated API prices doesn't mean they spent $20M worth of compute. The real number would obviously be substantially less.
>and the assistance of a whole team of people at OpenAI
You can't eat your cake and have it. What sort of guidance do you think is happening in a 10k agent, 320b token, 88 hour run ? AI did this one.
>I'd say advantage humans this time....to work on problems that have not been solved yet, and that humans are NOT making nice progress on.
Interesting way to frame progress that didn't move along till an LLM generated proof.
If you read the PDF release by Buckmaster, apparently the initial claim from Brubeck was that there as very little human input involved, then as the call progressed more and more people popped up that has been involved with it.
Does this aspect really matter? Not really, other than OpenAI wanting to present this as all the work of their model.
**
https://cims.nyu.edu/~tristanb/statement.pdf
I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used.
I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI.
As it seems and as they tell it, they started the run modestly and diverted more resources towards it as it looked more and more promising. The run didn't start with 10k agents for instance. The point is there isn't anything humans are doing in this timeframe against all this text that would count more than "little human output". It's still a fair assessment I would say.
This part of your argument is totally wrong. The OpenAI approach begins with the B/L work. The belief / knowledge that their approach would pan out is worth a lot - it means essentially “depth-first” search in this direction will be more fruitful than a general search.
Unless you are counting the B/L work as LLM generated. Is that your argument? Even if you do consider it that way, to me racing in for a scoop isn’t a good look.
This is an odd way to gloss over threats.
Given Buckmaster's telling, this seems beyond "poor choice of words"... It was a veiled threat, that he then doubled down on with his "If you don’t want me to be nice, then I don’t have to be nice." follow-up.
**
I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”
**
FWIW there are also other people on Twitter, such as this DeepMind researcher, saying this is a pattern for Brubeck.
https://x.com/dheeraj_nagaraj/status/2097266146445774924?s=2...
I see none, and consequently Brubeck's explanation makes more sense to me. I understand he meant these words, which he supposedly retracted on the spot, in a "why would you ruin your career with this behavior / turning down the opportunity I am offering" way.
You don't think that is the likely explanation?
Well it looks like they will announce at least one other millenium solution soon. In the same link they say they have "made substantial progress" on another millenium problem. The rumor mill before that statement was Hodge is done and Birch and Swinnerton-Dyer is on its way out.
They've pretty much said their own work was heavily agent driven. Levent is in a particularly bad place here because while he probably had a lot of background in the Jacobian Conjecture problem, he made the solution to that one sound like someone asked the question and he just fed it to Fable during the world cup. Whether that nonchalantness was to just seem hip or was to promote Anthropic, which he has stock in, or was just the truth I don't know though. But it makes this one seem similar, when they might have had really had nearly a year of very valuable feedback to the models.
Terrance Tao has lamented this practice as being unhelpful for mathematics, and likely to lead to humans working in private to avoid this.
Tao has also noted that many of these AI math proofs don't really help mathematics (nor does it seem they are intended to), since for many of them the proof was never the point, it was the math expected to be needed to be developed along the way, which the AI solutions don't provide.
A related point is that the actual solution approach is never revealed. What was the role of humans guiding the agents ? was it fully autonomous ? etc. It is in the incentive of the AI labs to trump the powers of the LLM, but in practice it is humans guiding the agents on the overall approach, This is never admitted. For example, in the announcement on NS there was only an output artifact given but no indication of how it was arrived at, and not even a writeup. This is what disappointed many folks as it was done purely for one-upmanship. As other have noted, the benefit is in the journey or process and not in arriving magically at a destination.
You can train on a sequence of outputs. In the end, OpenAI outputs are OpenAI's property.
You can learn a lot from a single side of a conversation.
Why should we trust them?
The only way the chat could have been used would be for Open AI to baldly violate their policies.
That said, sometimes it take very little information to point someone in a given direction, "I'm working on Navier-Stokes" said by someone with a given specialization might itself be very useful information.
OpenAI's statement says that they began training their new model on August 28.
edit: ffsm8 makes a great point below, it doesn't matter. I'm not great with dates, sorry.
There was much excitement, then, as now, for this kind of approach and there were several systems that followed along the same lines, e.g. Automated Mathematician by Doug Lenat.
Eventually it became clear that this approach is limited by what it can generate: you may have a sound and complete verifier, but if the generator, i.e. the first step in the generate-and-test pipeline, is incomplete, then the entire thing will run out of steam sooner or later.
The difference with LLMs is that they are... well, large. They are the most powerful generators ever created. That means their limits are not in sight and it will probably take us a very long time to find them.
Which is all to say that, yes of course, automatic verification is indispensable. But without an LLM generating an unprecedentedly large number of plausible theorems, there would be no AI mathematics, or in any case AI mathematics wouldn't have gone as far as it has.
But I find it interesting that Lean, a validator/compiler made by humans, is what enables those discoveries. But somehow all the praise goes to the models
Of course another way to look at this is, the people that wrote the validator got praise for that years ago. Now and up and coming actor is solving problems that took us 100s of years to create in insanely short time periods so of course it's going to get a lot of attention as it well should.
1. OpenAI couldn't have solved the problem without the researchers' private data for training.
2. OpenAI models can solve math problems
These mathematicians’ prompts are not like “hey chat, please solve Navier-Stokes for me”. They add real expertise and intuition from the cutting edge of their field.
1. Anthropic employee working on monumental problem but didnt receive/ask for the full backing of the company's resources
2. May or may not be mixing unreleased Claude output with Codex without zero data retention agreement
3. Victory lap on Twitter and giggling around the city before they finished the job, sparking rumors for competitors
Recklessly prompting OpenAI without a care to the safety of their knowledge.
And after that trying to cast aspersions at OpenAI?
Hopefully we get some better facts, because OpenAI are disliked enough that a smear campaign could work against them.
Edit: also the narritive is getting framed as OpenAI versus Anthropic. A highly political extremely capitalist fight is going on, and facts are victims.
Everyone in this thread seems to have made up their mind about OpenAI's guilt though.
Especially if there really is a long list of them.
"Here are a few hundred proofs" is far more convincing than "We really Navier Stokes and coincidentally someone else did too but we don't know the details or anything, who us, definitely not."
It's a PR fiasco, and a cynic might wonder if it's entirely about the IPO.
I'm consistently entertained by how these companies, with the most advanced models on the planet, consistently do the most idiotic things.
An article post that wouldn't even amount to a white paper + the LEAN proof is not evidence of how they got to produce it.
Of course it’s not an endless source. They had to burn millions of dollars to solve a single problem.
I'd like to adjust that to "They had to burn a lot of energy (create a lot of entropy) to solve a single problem. As we go into the super-intelligence age the current paradigm of money as humans understand it may break at some point. For example to a paperclip-maximizer money at best is a short term instrumental goal, hard power of matter conversion machines is what it wants and once it has those money no longer has purpose.
If you were in 1999 you'd be saying pets.com = internet.
Right now, a lot of money is going to train new models. And we need to train new models because they get gated by their training data. And models are only as useful as their training data.
So let's say the money stops.
Do we stop training models? Do we train them slowly? Do we accept the then current models as the limit?
Look at how much we spend on single bombers, how many training runs can you do for that much?
Well, the money will stop when the value of problems the LLM can solve is not increased by adding compute. Since current LLMs are getting quite good at solving problems, that might be a while.
Spinning it negatively like that doesn't do anybody good.
Were mathematicians the "victims" of calculators? of Matlab?
Were writers the ""vIcTiMs"" of word processors?? (apparently yes, according to old TV shows about computers during the 1980s, that you can see on YouTube)
> "tHiS iS nOt ThE sAmE" — Everyone every time.
No, just look it up. Look into old magazines and TV shows or newspaper articles from whenever a disruptive new technology came out.
I'm a professional mathematician and all the better mathematicians I know are in crisis mode. Most of us hadn't taken this sufficiently seriously and don't know how to use these models effectively but we play with them and immediately see that the entire way we've worked all our professional lives has to change. We worry less about ourselves than about the younger folks. I've got good ideas ai still doesn't know about ... Younger folks may not get the chance.
This is the same problem for software engineers too. I am now asked: what can you do that AI cant ? The answer to this could be intangibles like taste, aesthetics, and insights which collectively fall under creativity, and often accompanies experience. And there are no shortcuts to accumulate experience and perversely the more AI is used the harder it becomes. Soon, there will be a closure of all AI generated solutions, ie all low-hanging fruits are taken. Then, experts will again become needed to guide beyond the AI knowledge closure.
So fucking make it so that people don't -need- "jobs"
It's about fucking time already.
Don't fucking try to hold back electricity just so people still have to manually light street lamps to earn food and shelter: https://en.wikipedia.org/wiki/Lamplighter
They are posting here to try to convince their super intelligent AI overlord that the people will be less likely to revolt / better sheep if the overlord provides universal basic income.
Obviously these are unbiased and trustworthy sources.
As far as I understand it, users can opt out from the training aspect, but they cannot stop their conversations (“User Content”) being used “[t]o improve and develop our Services and conduct research, for example to develop new features”.
The answer is almost certainly yes, and this is a problem for most users.
I mean, we’ll know as soon as they decide they want to provide verifiable proof. Really dragging their feet on this front so far.
I’m inclined to believe this is false.