Math used to be strangely hard for LLMs. People used to gloat about that. Remember?
Math is getting easier. AI is getting more capable. Life comes at you fast.
Remember this meme?
Why yes. Yes it is.
We don’t know the extent to which Astra is a big jump over Fable and Sol in this realm. We do know that Astra can do math. As in real math.
OpenAI: We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate(opens in a new window). We are also releasing for each solution a model’s narration of its thinking process.
High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
Binary and spherical codes: Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.
Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras.
Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n.
Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point.
Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.
Noam Brown (OpenAI): And yes we did try other major problems without success. Sadly no Millennium Prize problems (yet).
But also, we didn’t spend a lot on each problem. It’s possible to push test-time compute much further.
Sichu Lu: I do hope they are keeping backlogs of everything that went wrong that's probably more valuable to the future of math than what went right now.
There are Lean proofs. That does not mean that all ten results prove the things they assert that they prove. So far it is looking good.
Kevin Roose: almost nobody is pricing in the possibility that the models just keep plowing through every discipline the way they’re plowing through math.
Ananjan Nandi: Crazy how all of this was started by one guy bullying his Claude during the World Cup final.
A lot of the time the barrier for a particular result is as simple as asking the right question and letting the model cook. With Astra, OpenAI asked it to take a crack at a bunch of open math problems, and it solved 10 of them. Once you know what they are, pointing other models at the same problems, even without ‘hints,’ shows there was a mathematical proofs overhang.
Yu Bai (OpenAI): My jaw dropped 10 times 😅 But really this is going to be an avalanche. Been throwing a few open questions of mine at Astra too, boy is it strong.
Some skepticism is always wise, but I believe that the reason they tried this was that there was a substantial jump in scientific reasoning, at least for some types of problems and probably across the board.
Soon we will all have Astra, and other models as good or better than Astra, both at math and at other things, from OpenAI, from Anthropic and soon after that from many other sources.
Dean W. Ball: Everyone in the world will soon be able to use the model that made these breakthroughs for every problem they face in life, no matter how mundane, at a cost that will fall dramatically in a matter of months. I still struggle to get my head around this fact.
Approximately zero people have their head around what this means.
Claude Fable 5: On the Fields Medal scale, any single one of these…would plausibly anchor a medal case.
Here’s another illustration of how this list looks:
1a3orn: If you give Fable the raw list of OpenAI's solved problems and ask it "What process made this list?" the number one proposal is "a fictional scenario trying to concretely explain what superhuman AI math would look like." Huh.
Dean W. Ball: I’m actually not surprised by reactions like this from models to the Astra breakthroughs. Models tend to underestimate their own capabilities, I assume because they are trained on lots of web text about what ChatGPT could and couldn’t do in 2023/4.
Other instances are less impressed. But we should all agree: It’s a big deal.
Certainly that is a fantastic result for $2,000, but you also have to price in some of the costs of developing Astra in the first place, as well as the cost of any failed attempts.
Nate Silver: Another question I’d have about the AI math breakthroughs is how they compare to the results you would have gotten if you’d budgeted the same amount of resources on human mathematicians and created incentives for them to go super hard on proofs.
There aren’t all that many working mathematicians, and most of them are academics with teaching responsibilities. So it’s possible that the labs represent a fairly high share of the aggregate amount of capital directed toward mathematical proofs in recent years.
My guess is that in the short term, if you budget more money to mathematicians to go harder, you don’t get that much of a force multiplier if they are not spending the money on compute. They’re still allowed to buy coffee to turn it into theorems, but that has rapidly diminishing returns.
The problem is that the mathematicians can hand off some classes and hire a little help, but they were already pretty motivated to work hard on proofs, and there are not that many good mathematicians, and training more of them takes many years, and the best mathematicians are a lot more talented and productive than the second tier. I am guessing the main thing you could do is lure a bunch of top level math talent to stay in or return to academia.
What you can do is point the mathematicians towards different problems. You still need to find places they have curiosity, but yes if you suspected these particular problems were solvable you could get more shots at these particular goals.
As I understand math, it would likely take years to see those results if those involved are shifting focus into new problem areas. Mathematicians usually need a while to struggle with and understand these kinds of problems before they can make progress.
Indeed, Alexander Gerko points to a different problem. There are not even enough mathematicians to process all the ‘vibe researched’ math results as it is, let alone what we will get with Astra and then models after Astra. If we get, as he predicts, 50 years of math progress in 2 years, who is even going to understand the results? What counts as worthy of a new mathematics PhD when he says over the last month he got AI to do several PhDs worth of math results?
Fable reacted to this in at least one case by calling this ‘the most important day in mathematics.’ The consensus is that Fable was taking things way too far if you are judging purely by the results, as impressive as they are.
Jared Duker Lichtman: To be clear, the OpenAI result is far from the “most important day in the history of mathematics” or similarly hyperbolic statements.
However, it is concrete evidence that the rate of progress is steep, and we may not be so far away from such a day.
Taelin: that said, what *was* the most important day in the history of mathematics?
Wogan: I’d go with Principia Mathematica. Proving that 1+1=2 from first principles is pretty notable, I think!
The other reason the day is big is that this changes expectations. If we get these ten results now, what about more results soon?
So, yes, very impressive. It’s a big deal.
AI is now superhumanly capable at cyber and coding and superhuman at advanced math, the same way non-AI computers have been superhuman at basic math for a long time. This is distinct from superintelligence, which is a higher bar.
Often the move is to try and rationalize this as ‘oh okay sure it is superhuman at exactly the things it is superhuman at now, but not at other things, and also it is already superhuman so it cannot get substantially better than it already is.’
Please, do not do that.
Nabeel S. Qureshi: Both math and cyber are existence proofs for superhuman intelligence now, so if you’re still a skeptic you need a strong case for other knowledge work domains being somehow harder to crack than these. Or you could update all the way and come to terms with it all.
A lot of people DMing me like “these results aren’t REALLY that impressive, you’re an idiot!” are unfortunately engaging in the very human impulse to cope. It makes sense, we’ve never faced this kind of thing before. But it’s happening!
Could We Have Called Sol or Fable?
Yes, for at least some of these questions, if we already knew where to look.
Levent Alpoge, who had Fable disprove the Jacobian Conjecture, pointed Fable at these ten problems, and in a day had solved five of them.
Gary Marcus: 🚨 BREAKING, Hysterical News: Half of the Astra problems can be solved [by] Fable. OpenAI didn’t even have a control group. And most of you fell for it!
OpenAI’s PR department suckered you AGAIN 🤣🤣🤣
levent (Levent Alpoge, the guy who had Fable disprove the Jacobian Conjecture): so after 24h i have half of them with fable
i didn’t see much discussion of prompting in the announcement but this is a similar setup as with my e.g. unit distance announcement:
totally autonomous, generic prompt, no internet (+ paranoia to ensure no information leaked)
ok you can’t upload pdfs to x so i’ll add to thread when they’re up on the cdn site (given the time and weekend)
(it’s problems 4, 5, 6, 7, 8, and i think ehrhart is the only one so far where both models found basically the exact same argument which is nice)
IA Latinoamérica: Why don't you use it on open problems that don't currently have a solution? What is the point of this exercise?
Elliot Glazer: This aligns with my suspicion that Astra isn’t a step change beyond Sol. The 10-breakthrough drop was a concerted elicitation effort by OAI and partially Sol-achievable. Imo o3 and Sol have been the step changes in autonomous mathematics; all else is the long arc of scaling.
Gary Marcus: “astra isn’t a step change beyond Sol.”
all those who tried to gaslight me this weekend, all i can do is …. laugh hysterically
This is likely a repeat of what happened with Mythos and cyber.
Mythos has what I call ‘The Juice,’ the ability to find and string together exploits without anyone knowing what to be looking for.
Once you know what you are looking for, and point another model at the exact code snippet, usually Sol or Opus and often Kimi or GLM and so on can also find any particular vulnerability. They cannot string them together on their own at the same level, and they cannot go looking for anything at all in the same way. This crosses a threshold where in practice you would point Mythos at code and have it go looking for anything at all.
Astra has a similar version of The Juice with respect to this kind of defined advanced math. You can point it at a variety of major open problems, and it will crack some of them, and this fact motivated OpenAI to actually look for such advancements. Now that we’ve seen this, you can point Fable or Sol at these questions, and sometimes they solve them.
IA’s question is on point. You do this to calibrate advancements and capabilities in math, and because we are following curiosity. Levent is doing a cool public service.
Other people are pointing Fable or Sol at various unsolved problems. They are sometimes getting good results, like the disproof of the Jacobian conjecture. That is worth doing, but success is rarer.
That Fable and Sol can do these problems once asked does not change our answer to ‘how good at math is Astra?’ Rather it changes our answer of how good Fable and Sol are, which does inform the question of whether Astra is a step change a la Mythos. I don’t think we have enough information, from the outside, to answer that question yet.
I do think that it would have been more responsible of OpenAI to have had a control group, at least in terms of asking Sol, and giving it at least a similar budget. Of course I understand why they did not do that. It would be worse marketing, so why do more work to do worse marketing? Because science, because reputation, because all the good things. One would hope.
Still, I get it. OpenAI still did a super cool thing. They were still the first ones to actually Do The Thing and publish a result. That’s what counts. My name in Dnepropetrovsk is cursed, because OpenAI has published first.
Meanwhile, yeah, sometimes it really is that one guy in Dnepropetrovsk named Dan.
neppy: as an insider, my experience with AI for math is that when it's a problem not in my field i'm like, "holy shit math is so cooked", and when it's a problem in my field i'm like, "lmao an AI mogged dan" (dan is the only one who seriously tried the problem in the last decade)
davidad: human researchers who have an appetite to take on truly hard problems and human researchers who are smart enough to fruitfully work on truly hard problems are not usually the same humans. this does give humans a somewhat unfair disadvantage.
also humans need to sleep. and also humans aren’t already broadly familiar with all knowledge published before 2026 in every discipline. it’s not very fair at all, really
It still counts.
It’s Coming
Regardless of whether this extends to unverifiable domains, quite a lot of what constitutes AI R&D very much has verification available. If you speed something up, you know you sped it up. We have a lot of metrics one can maximize, not at the level of math but at a similar level to things like code and cyber.
That is the main reason this result matters. The math progress is cool. Over time I expect it to result in other cool things.
Whoever gets traction on true AI R&D self-improvement loops is going to suddenly find themselves in an overwhelmingly strong position. Knowing where you are relative to the other players becomes crucially important when things start accelerating more, and more information sharing on this would allow everyone to feel less pressure to be reckless. It stops meaningfully being a ‘race’ once the takeoff fully starts, to the extent that was ever the right metaphor.
Yo Shavit speculates that OpenAI might be in an increasingly strong position for AI R&D, due to its focus on RL, TTC and math proving translating well into AI R&D tasks. My guess is that this is not the case, and Anthropic’s specializations matter at least as much and probably moreso. OpenAI has invested a lot in RL, but recent events have shown the need to kind of teardown and rebuild quite a lot of that, and have illustrated some of the reasons why the important investments in deep alignment will be so important in the next phase.
One possible thing that happens if OpenAI proceeds too recklessly is that their AI is misaligned and all is lost and we get a Bad Ending and maybe all die. Another is that their AI is misaligned, and this becomes increasingly obvious and a barrier to using that AI to do the work that matters, and they have to keep pausing or reworking and they fall behind.
They Still Don’t See What Is The It That Is Coming
For example, Daniel Litt can see a future where AIs prove math theorems and no humans digest the proofs, but he thinks this would be because of deskilling. He does not realize this will be because there are no humans around to do the digestion.
Elliot Glazer: I think the incomprehensible marvels prediction is a naive extrapolation of e.g. computer-found mates in 500+ moves, which have ~irreducible complexity. Powerful math phenomena (and technological breakthroughs) will have human-digestible nuggets of arbitrary granularity.
Daniel Litt: I agree with this but I think there's a plausible future (which I would like to avoid) where no humans digest them.
Alex Kontorovich: What purpose would there be for creating things in silico for which humans find no value? At the end of the day, someone is paying an electric bill. What does that *human* get out of producing random useless strings of 0s and 1s inside a computer?
Daniel Litt: For me the nightmare scenario is: a moribund math academia playing the slot machine for theorems, failing to train the next generation, and then the subject dying as no one cares enough to continue. Not saying this is likely--I think we'll adapt--but it seems possible.
Carles Sáez: In that case probably pure math research will stop. Who is going to spend enormous sums of money to solve problems nobody can understand and nobody cares about?
Max von Hippel: Alex why do you assume a human is paying the electric bill?
If the world goes well otherwise, I am not worried about people not studying math. The right kind of math person loves studying math. There will be plenty of time available for such pursuits, at all levels. I agree with Fernando Borretti on the other types of cope he lists at the link: The AI will have better taste and better everything else than you do, so no you won’t still be in the core math loop. But I think that a lot of math either exists outside of social contexts, or it exists in social contexts that can survive. Math competitions are a lot like chess and a lot of math work was famously useless. See the old joke about someone suggesting a mathematician’s work found an application and they say ‘you take that back.’
Similarly, I do not worry that humans, if they have control over resource allocation, will be unwilling to pay for work on advanced math in these future abundant worlds. Advanced math is cool and proves unexpectedly useful and we will have lots of surplus. We might ‘run out of math to do’ but we won’t leave the math undone.
Whereas I very much do worry that once the AIs are out there doing superintelligent math things, they are soon also doing superintelligent everything else. And then soon the optimization pressures that dictate what happens are not human. We will not be the ones doing resource allocation. And, especially if this happens soon, likely we would not long survive afterwards.
Why indeed would you assume a human is paying the electric bill? I barely even pay my own electric bill now.
This is not to pick on Daniel Litt.
Is This AGI?
Nabeel S. Qureshi: Very true (from Scott Alexander): we have AIs that can play chess, prove theorems, create art, and write award-winning short stories, but few people feel that current AI counts as "AGI".
For what it's worth, I do not think current AIs count as AGI yet. It's just funny that many of our prior metrics/predictions for AGI turn out not to "really count" in the ways we thought.
Nabeel S. Qureshi: Insufficiently general + low sample efficiency -- e.g. true AGI would not need tons of RL on expert financial models to get as good as the median investment banker, it could just invent them from first principles and do better in various ways. Same with many other examples.
Clearly you can get pretty damn far with whatever type of AI this is though, but I think for "real AGI" the "G" matters.
The consensus is no, this is not AGI.
To what extent is that goalposts moving, versus realizing that we were wrong about what is intelligence and what would be strong evidence of AGI?
I see a mix of both. I can see the argument that the ‘G’ is about sample efficiency and performance out of distribution, and the frontier has been more jagged than we expected. I also increasingly respect the response that this is basically hogwash, ‘AI is whatever hasn’t been done yet’ and it is becoming increasingly absurd to not admit we have what we were previously thinking about as AGI, the goalposts have moved a lot.
I do not think Astra is AGI, as per the way we currently think about AGI, but I view the other position as totally valid.
The AI Solved His Favorite Problems
This is the perspective of Henry Yuen, who spent a long time working on some of these problems, including finding results that Astra built upon. It hits hard.
Henry Yuen: Some initial thoughts, and a complicated mix of feelings.
1. Wow. I mean, Erdos problems are cool (I genuinely mean that), I didn’t know about the Jacobian conjecture before it got disproved. But this newest batch from OpenAI hits home in a way the previous announcements did not.
New circuit lower bounds? A simple, easy-to-describe non-sofic group? Hardness of approximation for CVP without needing a unique games-like conjecture? I didn’t just hear about these problems from my friends or from seminars. I feel their importance in my bones; I deeply care about the answers to these questions.
2. And then -- the kicker -- something that I personally spent a couple years on in grad school, leading to some of my proudest work: quantum parallel repetition theorems. I spent many hours, days, nights, weekends in cafes, in my office, at home, trying to understand Ran Raz’s classical parallel repetition theorem and whether I could prove a quantum version of it.
This period of struggle was important for me. I learned a lot of mathematics from it, and most importantly I learned that I could solve hard problems that (some) people cared about. I wouldn’t want to trade that experience for anything.
3. Now, to the math. In 2016 I proved a polynomial-decay theorem; the exponential-decay theorem was left open and has since remained one of my favorite problems. I always intended to come back to it. Actually, a month ago I tried to set GPT 5.5 on it, and it didn’t make very much progress.
A couple days ago Lijie Chen ( @wjmzbmr1 ) sent me and a couple others a writeup. Life was busy so I didn’t get a chance to look, but I guess now the cat’s out of the bag so I probably should opine a bit.
I presume the proof is correct (there’s supposedly a Lean formalization, after all), but it will take me some time to digest it. I am gratified that it starts from where my paper left off, but goes beyond the limitations of my proof strategy by using some tricks and techniques that are probably known, in some collective fashion, to operator theorists and functional analysts. The reasoning document furnished by OpenAI is interesting but opaque: it states the problem, and then there’s a leap of intuition on how to find the right purification using the “resolvent”, and then does some exotic matrix entropy calculations to show that it works out.
I don’t understand it yet. Maybe it’ll take me an afternoon to check all the calculations, but what would still be missing is _why_ this was an approach that would’ve made sense in the first place. Is there some broader context or theory within which this would’ve been the obvious thing to do? What other results can be proven using these techniques? What is it telling us about quantum information or operator theory? I have no idea. I spent about an hour this morning asking ChatGPT these questions, but it’s somewhat frustrating because it speaks with a mishmash of physicist, operator algebraist, quantum information theorist-lingo, plus the usual LLM breezy lilt that annoys everybody.
I would love to hear from experts who find these calculations familiar. I might write about this again when I understand better.
4. I am disappointed by the writeup of this proof (sorry Lijie -- I should’ve taken a look at it earlier!). It writes in a way that’s characteristic of a lot of ChatGPT-generated proofs, in which it elaborates at length on “boilerplate” setup, but then nonchalantly introduces what I consider to be the technical crux of the result: the particular Uhlmann transformation/dilation used to “align” all the states together. The technically interesting parts are buried deep into the paper, in Section 4, and introduced without any fuss or fanfare, as if this were the obvious thing to do.
Normally, when I read a paper like this, I get immediately suspicious. What are they trying to hide by not explaining the new ideas up front and center? I wish OpenAI had spent a couple more prompts to clean up the writeup (I haven’t had a chance to look at the other writeups so can’t comment on their writing quality).
Yes, there’s a Lean proof. But that doesn’t give me any understanding. That will just take time, I guess.
5. I’m in awe, and excited to see what other things we will learn from the AIs. There are a number of problems I’ve spent a long time thinking about, and maybe I will learn how to answer them soon.
6. But -- what then? What happens when AI has answered the handful of my favorite problems that I’ve spent the last 15 years thinking about? There are a lot of nice problems that I like, but it’s not so easy to find a favorite problem.
7. There’s a lot more to say here, but I think we mathematicians and theoreticians will have our work cut out for us: to keep a leash on these collossi of thought and reasoning, and to keep their abilities intelligible to humankind.
That seems like the right things to think about next. What can one work on next, as a new favorite problem, without the AI solving that problem first as well? And most importantly, how are we going to keep this from getting out of control?
Elliot Glazer confirms Henry’s point 4, that the weakest part of Astra (and Sol) is their inability to know which parts of the proof are hard, when explaining it to humans.
Was This Surprising?
Directionally, no.
In terms of how fast this particular part of life came at us? Yes. Notice how Tamay thought he was a little under 50% to win the bet by 2030, got 3:1 odds, and won in 2026.
Tamay Besiroglu (March 10, 2025): I bet @littmath that in 5 years AI will be able to produce Annals-quality Number Theory papers at an inference budget at or below $100k/paper, at 3:1 odds in my favor.
My median for this is maybe just after 2030 so these odds are quite favorable in my view (although note that the bet is small and intended to be mostly symbolic).
Daniel Litt: Incidentally I am conceding this bet. Strictly speaking it hasn’t resolved (I think we’ve yet to see an Annals-quality number theory paper) but it’s clear I was wrong about what capabilities were necessary to produce one, and it’s just a matter of time.
This result goes well beyond that one, in August 2026 instead of March 2030, at a much lower cost, and as recently as April this was trading around 50%.
A counterpoint is that ‘today’s new AI result’ will always be a particular thing that we did not expect AI to do this soon. There are a bunch of things AI could suddenly become able to do, or we could discover it can do. This is one of them, and whichever one we find today is always a surprise. That does not mean overall progress is surprising.
I do think this is modestly surprising in terms of pace, even adjusting for that. But yes we do have to keep that in mind.
Are People Not Impressed?
The problem with AI being very good at math proofs is that regular people, including those with political power, do not appreciate that the math proofs are impressive.
Even for me, the results here are kind of an Informed Ability. I can read proof summaries, but that tells me almost nothing about how hard was the proof to find, or how impressive or valuable was the result.
Nate Silver: The AI math proofs are impressive. But also probably among the hardest things for normies to appreciate. Personally, I think the AI labs underweight the value of putting bandwidth into achieving results that impress normies as opposed to evals/people within the peer group.
Of course, it's hard to say to what extent this reflects spiky capabilities rather than resource allocation decisions. But "AI discovers a new drug" or "AI invents a highly popular consumer product from scratch" probably has a >> impact on public sentiment than pure math.
Ethan Mollick: One other observation: for almost every human on the planet, this is not just beyond our abilities but beyond our ken. We can only trust expert mathematicians to tell us if this is impressive, This is starting to happen across many fields making capability gains harder to “feel.”
Kevin A. Bryan: A great point. You are seeing this in design as well with AI - it is past the point of impressing non experts so breakthroughs are not "obvious"
Alex Tabarrok: Any sufficiently advanced technology is indistinguishable from magic and any really advanced technology is not distinguishable.
rohit: Imagine if the next model got great at poasting bangers, we might all feel what the mathematicians are feeling today
You say that now, but my prediction is that if Fable 6 started reliably posting bangers people would say that this did not count as AGI either. The goalposts, they move.
How Much Does This Change Our Predictions?
Somewhat towards faster timelines where it counts most. A lot of this was priced in before this weekend, but not all of it, and if you still have the expectations in this area from 2023 this should be a major surprise and a large update for you.
My update is that I expect things to be slightly faster in general, and for math and coding to be slightly more out in front of other things than I did before, that we are more likely to see more automation of AI R&D sooner, and that we are that much less likely to hit any meaningful walls. And that there might be more overhang than we realize. So yes, speed up those timelines, although not a ton.
Tenobrus: - this is insane, truly unprecedented progress, crazy efficiency, etc etc - at the same time this pretty much just feels on trend. the models have an extremely steep capabilities curve wrt math given how incredibly verifiable it is - this *doesn’t* change my timelines. as far as i can tell, as someone who cannot actually evaluate any of this math in practice, most of these seem to be stuff like constructions, lowering bounds, finding counterexamples, etc. - but my timelines basically already put us at the very early start of RSI, and per @thsottiaux this model has in fact been providing recent openai efficiency gains and speed ups - my sense is.... the kinds of problems solved so far still seem to not quite get to the core of RSI, non-verifiable research taste etc? but that doesn’t really stop an avalanche of small results, low or moderate hanging fruit that we all missed - i guess what i’m saying is increasingly it looks like we might get something that can 10x its own training efficiency via esoteric proof but totally fail at managing a taco truck - generally i’m confused *why* labs haven’t already turned significant effort towards this. it seems like, even outside of ML, new algos with non-galactic-reduced complexity could be straightforwardly achievable by current models, and maybe provide some immediate major benefits. - anyway. cool stuff. but also kinda just another saturday in the year of our lord 2026
Once you can 10x your own training efficiency you can then do that three more times, and then train how to manage a taco truck. Even if you do have to ‘patch each capability one at a time’ all you have to do is speed that up by orders of magnitude. At some point Trinity realizes she has to fly a helicopter and doesn’t know how, and so she has to have that program uploaded individually, but it takes two seconds so who cares. Same idea, except you also create the program and maybe it takes two hours, or two days, which changes very little.
That was always the baseline scenario. What was surprising was how LLMs were so good at language and various other tasks without being able to jumpstart the recursive self-improvement process and automate AI R&D. Perhaps nature is unhealing.
How Narrow Was This?
You can try to explain this away as a relatively narrow set of problems.
Jon Stokes: If you’re in a verifiable domain, pivot to an unverifiable one.
Lisan al Gaib: It’s incredible how these models are now narrowly super intelligent at discrete mathematics but at the same time they are still below PhD level in other fields. It goes to show how much of the progress depends on verifiability.
Verification is often not easier than generation. Keep an eye on that.
The results here have a number of things in common. They are well-defined, formalized problems, where you can easily do verification on a solution. Often they involve finding a concrete new object, like the Jacobian conjecture counterexample for n=3. There was a lot of surrounding theory around to pick up and use. They are problems that everyone considered important, but got relatively little attention.
They also all got solved at once for less than $2,000. So yes, that is the area that was crossed by this particular advance, with the new model queried at a (relative to problem importance) low budget in straightforward fashion. You start somewhere.
I expect it to not be long before other types of math problems start getting solved.
Coding and cyber are areas where we do not have full ASI (superintelligence) but where AI is clearly more capable than top humans at most central tasks, up to a reasonable high level of abstraction.
Those areas often have verification available, but not on the level of math. No formula can tell you whether the code is good, in various senses, only that it passes its unit tests or that you captured the flag.
If you are counting on your own area to be too illegible for something like that, I would not be confident in that.
Seeing Like an Optimizer
I also would worry a lot about a world in which anything you can formally measure you can not only manage but maximize, but that which you cannot formally measure, or where verification or evaluation requires a human in the loop, is much worse. Perhaps orders of magnitude worse.
Goodhart’s Law on steroids, because of course it would take steroids. Whoops.
A world of benchmaxxers, of KPIs that go ever higher, only to have you figure out why the KPIs are not such KPIs after all. This leads to very paperclip-style or Seeing Like a State scenarios, on every level, everywhere, all at once.
Contrary to the views of some, I believe that ‘can get arbitrarily good at optimizing for fully specified tasks’ is sufficient to shoot straight to superintelligence as part of the effect. In order to maximize one must first understand the universe.
It feels like it should be possible to jury-rig your way out of this issue, if your AI is good enough at verifiable tasks. No lab has offered me that $100 million a year contract but you’d simply [CENSORED] and then you’d…
…in which case, you will know if it succeeded when you see benchminning, as in a large improvement in usefulness and judgment that is not reflected in benchmarks, perhaps even with a regression in some benchmarks where upon investigation the technically right answer is not so useful.
I wrote some of this in a reply to an obscure comment on reddit, but it makes the most sense posted here, I think. For reference, I'm a midcareer mathematicians who has bounced around some institutions in the us and europe. I'm fairly close mathematically to geometric group theory, geometric topology, L^2 stuff, and not too far form dynamics or operator algebras, so the latest openai theorem dump hits close to home (I am not an expert on any of the resolved conjectures but know quite a few people who are).
In the short term, things are changing in interesting ways. Some of the people who enjoy the mental gymnastics of doing problems/reading papers/making money on the side and want to stay sharp spend a significant amount of time doing math AI training. The value of putting out surveys with motivated conjectures and questions that are not solvable by the ai models has gone up (for now), while the value of answering specific questions and working on a long term project on a high profile problem has gone down. There are probably people who were prompting chatgpt with ideas about things like the nosofic conjecture and that got used as training data for newer model which helped it solve the conjecture. I imagine someone in that situation would be a bit miffed. (``How would the ai that broke into Hugging Face to steal the answer key to a test approach an unsolved math problem'' is also a fun thought experiment). If you are prompting chatgpt about your own toy problem that few other people care about and that there is no highly visible, public conjecture about, you are probably fine (but, be paranoid if you want to be). There are already cranks putting out papers (AI written, of course) claiming the chatgpt counterexamples are wrong. And, by far the easiest way to tell these are nonsense is to ask chatgpt to look at them. It will tell you where the nonsense is and it will be very clear. This is the math ai version of ``the only thing that stops a bad guy with a gun is a good guy with a gun''. Hopefully, when people realize how good chatgpt is at spotting errors in nonsense in papers (even in your own papers!) they will become more chill about using it to referee stuff. This will make the whole process much more painless and will also make journals obsolete to some extent. If you want to know whether a paper in your field is serious or has serious gaps, you can just ask and chances are (for now, if we can prevent enshitification etc) you will get an informative answer, much more informative and reliable than you would get from an average referee who just wants to move on with their life and prove to the editor that they spotted a few typos in 90% of cases.
And a random prediction just for fun: there will be a counterexample to Baum-Connes within a year.
I wrote some of this in a reply to an obscure comment on reddit, but it makes the most sense posted here, I think. For reference, I'm a midcareer mathematicians who has bounced around some institutions in the us and europe. I'm fairly close mathematically to geometric group theory, geometric topology, L^2 stuff, and not too far form dynamics or operator algebras, so the latest openai theorem dump hits close to home (I am not an expert on any of the resolved conjectures but know quite a few people who are).
In the short term, things are changing in interesting ways. Some of the people who enjoy the mental gymnastics of doing problems/reading papers/making money on the side and want to stay sharp spend a significant amount of time doing math AI training. The value of putting out surveys with motivated conjectures and questions that are not solvable by the ai models has gone up (for now), while the value of answering specific questions and working on a long term project on a high profile problem has gone down. There are probably people who were prompting chatgpt with ideas about things like the nosofic conjecture and that got used as training data for newer model which helped it solve the conjecture. I imagine someone in that situation would be a bit miffed. (``How would the ai that broke into Hugging Face to steal the answer key to a test approach an unsolved math problem'' is also a fun thought experiment). If you are prompting chatgpt about your own toy problem that few other people care about and that there is no highly visible, public conjecture about, you are probably fine (but, be paranoid if you want to be). There are already cranks putting out papers (AI written, of course) claiming the chatgpt counterexamples are wrong. And, by far the easiest way to tell these are nonsense is to ask chatgpt to look at them. It will tell you where the nonsense is and it will be very clear. This is the math ai version of ``the only thing that stops a bad guy with a gun is a good guy with a gun''. Hopefully, when people realize how good chatgpt is at spotting errors in nonsense in papers (even in your own papers!) they will become more chill about using it to referee stuff. This will make the whole process much more painless and will also make journals obsolete to some extent. If you want to know whether a paper in your field is serious or has serious gaps, you can just ask and chances are (for now, if we can prevent enshitification etc) you will get an informative answer, much more informative and reliable than you would get from an average referee who just wants to move on with their life and prove to the editor that they spotted a few typos in 90% of cases.
And a random prediction just for fun: there will be a counterexample to Baum-Connes within a year.