Grant Sanderson (‪@3blue1brown‬) – AI disproved a famous math conjecture. Now what?

It's an interesting question because it's hard to answer without knowing what the solution looks like ahead of time. If we take the IMO, the spirit of your question three years ago was in looking at how some of the solutions to these problems really seem to require creativity. The designers of these problems try to come up with things that you can't train for as easily. The dirty secret with the IMO is that you really can train for a lot of them. With the whole AI and math project underway, as you point out, one of the reasons it's interesting at all is that there's a spiky frontier to AI, and math is just right there in one of the spikes. But there's a fractal nature to that spikiness, because when you zoom into the specific progress within math, you have some things that are a lot easier than others. If we just think about IMO, which is old news at this point. It's been two years since they're really doing quite well. They would have gotten a gold in 2024 if not for the following reason. They're very good. They just cold-solved geometry basically. The IMO has these four categories of problems: geometry, number theory, algebra, and combinatorics. Geometry, it just solves it in nineteen seconds since 2024 because it's a brute force solver. The dirty secret is that for students, there's also a brute force way you can go at it. Combinatorics is the wild card: much more playful, puzzly-seeming problems. There were two combinatorics problems on that year's test, and there's not always. There are four categories and six different problems, so it's a toss-up which one is going to have two questions. Had it been more geometry questions, they would have gotten a gold that year. But it struggles on those combinatorics ones. Someone who's trying to keep that torch of the last holdout of math for humanity might say those are the ones that require more creativity. Even then, the spirit of your question—if they're solving a Millennium Prize problem, does that also service a lot of white-collar work?—suggests that whatever the rate limiter is between where we are now and that is the same as the rate limiter for making things better at white-collar work. We could paint a couple of different ways. If we focus on the Riemann hypothesis, what would it look like to solve that? These things are extremely good at a specific domain of knowledge, knowing it very deeply, and then knowing another domain, and another. You've pointed this out. It's bizarre to have something with this superhuman breadth that knows all the fields so well, and yet isn't finding those lightning bolts that connect them. I think we're starting to see sparks of it actually finding connections between the things it's an expert at. I'm sure we'll talk about it. If the nature of the solution to the Riemann hypothesis was something like that, that feels pretty distinct to me from what's necessary to get good at white-collar work. And there's a reason to believe that might be the nature of the solution. I don't know if you know the story of Hugh Montgomery and Freeman Dyson at the IAS. This is a side tangent, but it's a fun story. I don't know if it was over lunch or something like that, but you have this number theorist who is just trying to understand the statistical correlation between pairs of zeros of the Riemann zeta function. The Riemann hypothesis is all about whether all these zeros sit on a straight line. He finds this quantitative question you could ask, and he writes down a formula. It looks like one over sine squared or something like that. Freeman Dyson, a physicist, is like, "I know that expression. That expression comes up in studying the eigenvalues for random Hermitian matrices," which was something that comes up in studying the energy levels of a nucleus. The idea that the statistics of those two seemingly different things were the same prompted an exploration of whether there are aspects of random matrix theory that might be relevant to the Riemann zeta function. I think it's a little bit of an open question whether there is fruit to be had there. But that bridging together of two different fields—if it turned out that the solution to the Riemann hypothesis was exploring an idea like that even further—has the character of how you expect LLMs to be good at math. They're experts at quantum physics. They're experts at analytic number theory. They should be able to see that similarity in a way that doesn't require Montgomery and Dyson to be having lunch and happen to talk about it. That's totally different from white-collar work. To the extent that you have a hard time using an AI as an editor, it's not because they know everything and you just need them to find that lightning bolt in between. A different possibility would be… What's the right analogy? Maybe if we think of Fermat's Last Theorem, between the moment of Fermat phrasing the question and what the solution itself looks like, where the solution ultimately involves such heavy machinery in math. The beauty of that problem is you can phrase it so simply. You ask about xⁿ + yⁿ = zⁿ. Do you have integer solutions for this when n is bigger than three? It's something you might expect there to be an elementary number theory approach to, but as far as we can tell, there's just not. Whereas the actual solution, maybe there is something simpler, but this might be what it has to be. There's such a complicated set of ideas that build on centuries of work centered around elliptic curves. Then there's this other mountain of ideas centered around these things called modular forms. Both of those mountains have to be built before you can ask the right question that connects them. If the solution to the Riemann hypothesis involved building a new mountain, that's a kind of skill—the ability to come up with the right new ideas—that feels sufficiently different from the character of how they're intelligent right now. It's not like that's what you need from your hired video editor per se. But if it's capable of building mountains that are the correct new theory crystallizing how we should be thinking about a subject, that's just such a level of intelligence that it would be surprising if it didn't permeate into other aspects of the economy besides just the mountain-building for math itself. Or at the very least, even if it couldn't literally do every single thing white-collar humans can do, it would just have transformative effects in the way that getting gold in the IMO did not have transformative effects on the world.

You highlight two pretty good examples there. For anyone curious about the unit distance conjecture, there's this really nice video by a math channel called Polylog where they talk about it. All of these discussions cause people to reflect on the process of doing math. They're like, "Oh, this thing can do this impressive stuff. What does that mean for us?" One of the people in that video highlights this quote: "Good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions." That's more or less exactly your framing here. We need the conjecture generator and then the definition generator. That's the premium-tier mathematician. I don't understand how exactly you'd make that a benchmark. Usually, when I think of the word benchmark, I'm thinking of something that is a goalpost. The ball is through the goal or it's not. You can clearly say, "Yes, this is done." Partly that's to be able to do things like RLVR, but also partly just to know that you haven't moved the goalpost in answering. OpenAI can have their headline on disproving the unit distance conjecture because it's a clear, distinct thing. It did it. Whereas imagine trying to have a headline on GPT-5.4 coming up with a really good conjecture. "We promise, everyone thinks it's a good conjecture." It just doesn't land the same way. But maybe that doesn't negate the fact that it's the right thing to be thinking about. I would be surprised if it ever took the form of looking like a benchmark, where we have a score saying it's passed because we can quantify how good a conjecture is. The nature of what it would take is probably that you'd feel a tone shift in conversations with mathematicians about the way it's useful to work with. This series you referenced, which is not at all produced yet and probably won't be for a couple of months, takes the form of us interviewing a lot of mathematicians. What's interesting is that we started doing this over a year ago, and it's fun to see a little bit of a tone shift in the way they talk about AI between mid-2025 and where we are now in 2026. In the real world, that's a very short amount of time. In the AI world, that's eons. We're able to see this tone shift over those eons. I think the way you'd measure conjecture-generating ability is going to be more subjective, based on that tone shift. It will be mathematicians saying they're not just using it to solve their problems, but that as they step back and decide what their research field should even be, a conversation with such-and-such model was genuinely helpful for that. I don't think it's likely you'd see it in the form of a headline saying this was yet another benchmark knocked down.

You struck a nerve, because I had this project about Galois I was going to do in 2022 that I put on the shelf, but I spent a year of my life thinking a lot about what he did. There's a risk of me accidentally talking too long on the specifics, which you can hold me back on. It's a perfect example for your case, because describing why it was a valuable insight does not come from immediate utility. Certainly, if you're thinking about RLVR environments, this is going to be really hard to do. But it's interesting to note that even with human verifiers at the time, it took a really long time to recognize it as being useful. With Einstein and GR, people could feel this was a good theory right away. What makes Galois theory such an interesting example is that you literally have this hundred-year segment of an idea that flows through many different people's heads before it settles into something the math community agrees is good. To back up a little bit… Do you want the background on the problem at all? We all learn about the quadratic formula in school.

We all learn about group theory… No, the quadratic formula. This was known. In some sense, the Greeks could solve quadratics, but they didn't really write things in algebra. It's really the Arabs who wrote down that formula. There's this delightful story about dueling Italian mathematicians—not real duels, just intellectual challenges—who secretly found a formula for the cubic, and then very shortly thereafter found a formula for degree-four polynomials. So a natural open question for mathematicians is, can you find a formula that solves degree-five equations? The degree-four formula is a monster. It would be wild to write it down. You usually don't write it down in full. You break it up as a procedural thing. You might believe these things have this exponentially increasing complexity. So for many hundreds of years, nobody was really answering that question. Usually, we say Abel was the first to prove it. He was this young, precocious Norwegian mathematician. He showed it's simply impossible. It's not that you can find a quintic formula. He thought he found one initially, but he showed it's impossible. I think the real credit though, you have to back up a bit and talk about Lagrange. He found the right kind of question to ask about this. I'll give it at a very high level. He was studying the question and recognized that being able to solve these polynomials is very related to understanding the way certain algebraic expressions are symmetric. If I write down a + b + c + d, just adding four variables, and I permute those, it doesn't change the value of the expression. Whereas if I write a + b * c + d, some of the permutations don't change it, but some of them do. He had this really nice insight about how if you can find expressions that have four free variables, but all the permutations take on three distinct values, that has this unexpected relationship with being able to reduce degree four into degree three. He started approaching the question of whether we can find a quintic polynomial by wondering if he could extend that method. To extend that method, you would have to have an expression that has five free variables such that as you permute them over all the five factorial permutations, it takes on only four values or fewer. You could put that in a puzzle book. You could put that in a brain teaser that a twelve-year-old could engage with. It's not too hard to find yourself feeling like that's an impossible task. Lagrange is sitting there saying, "Here is a strategy to solve this problem of finding a quintic polynomial. It seems like it might be impossible, at least from this strategy." But that was the first time in history that people had the instinct that some kind of question about symmetry was the right way to study these polynomials. In his mind, it was just a way. It had yet to be discovered that there was actually a tighter connection. Also maybe rather than searching for the formula, we should be asking the opposite question: can you prove that it's impossible? He sort of planted that seed. Around fifty years later, Abel definitely read Lagrange and was influenced by it. We know that Galois loved Lagrange when he was falling in love with math. It's very hard to imagine that these two young geniuses coming up with pretty similar insights around that problem wasn't born from Lagrange. But to your question on whether you are able to verify that this was a good idea, there wasn't any result that Lagrange came to. He didn't solve the problem, so it wasn't a case of knowing it was the right question to ask based on a solution. He just asked it. There's some intrinsically interesting thing about it. It also wasn't very important for math at the time. Most people were more interested in the applications to physics. This was almost a side, recreational, hobbyist-type thing. Abel started working on quintic stuff, but then he was advised to spend more of his efforts studying elliptic functions, so more of his work was on that before he died young. He died at twenty-six from tuberculosis. And then Galois pushed both of those ideas in the right direction, where he really understood the nature of abstraction. He had this really nice piece that he wrote while he was in prison. We could talk all about his life story. It's pretty wild. But he's this teenager, he's in prison, and he had tried to submit his math papers and they had been rejected. So again, thinking about verifiable reward, the verifier function that is the academy at that time is rejecting what he wrote. Frankly, it was not very coherent. It wasn't a complete proof. He wasn't giving a clear thought of what the theory actually was. He was just a young fledgling mathematician getting his bearings. The verified reward there is, "No good." But he has some instinct that there's something there. So he's writing this diatribe on the nature of math being something that undergoes these shifts over time. He talks about the advent of algebra itself and going from just thinking in terms of numbers to having a certain fluency with pure algebraic expressions, where you're not tied to interpreting those expressions. He has this instinct that there seems to be another layer of abstraction that we should be doing, where rather than thinking about the formulas themselves, we're thinking about what symmetries underlie those formulas. But it was still a pretty ill-defined theory. If you're trying to say the verified reward is that he solved a problem that other people haven't, well, Abel already proved that quintics are unsolvable. So what was Galois doing? In principle, Galois theory lets you take a specific polynomial, and it gives you the rules to say whether that specific polynomial has roots that you could write down. For example, with x⁵ - 1, you know that a solution is 1. Or x⁵ - 2, you can write down the fifth root of two. So it's not that you can't write down the solution for every quintic polynomial, but could you find a specific one where you prove you can't write the solution using radicals? He also didn't even solve that exactly. He didn't show for a specific example that he couldn't. Even describing what problem he solved is very tricky. He then dies. It's this very romantic story of him having this duel. There's a lot of myth around how he supposedly writes up all his ideas the night before the duel, but really, he tried to get them published five times before. Working on the quintic doesn't seem to be good for your health. It's very bad. If you're a young genius, don't work on the quintic. He asks his brother and his close friend to get his notes to Gauss, to get these notes to the important mathematicians of the day, because he thinks there's something there. Even then, it didn't really take. His brother and his friend tried to get them out, but it was another twenty years until Liouville sees these notes, sees that maybe there's something in them, and tries to clean them up and understand what Galois was getting at. Even then, it was another twenty years or so until Jordan actually puts together something like a modern treatment of group theory that they attributed to Galois. You could easily imagine history turning differently, where these ideas were coming about from other points in math, and Galois could have been forgotten in history if he was a less florid character. But between the time of Lagrange having this inkling that maybe symmetries of roots is the right way to go, to where it all looks like modern group theory, you've got this long span. A lot of the time, it's not even passing the verified reward of human reviewers. It gets on someone's desk and they say, "I don't really know if there's anything here." You have to have this one person recognize it. Even then, it's not really solving practical problems at that point. You pointed out cryptography and physics and things like that. You have to get into the twentieth century before you have Gell-Mann thinking that maybe understanding the nature of how certain groups break down has a relationship with what particles are made out of. He anticipates quarks based on a purely group-theoretic question. That's one of the more interesting applications of group theory: to even predict the existence of quarks is a group-theoretic question. That's so long after Lagrange before you have anything like that. So you have to ask, what is the way of measuring progress that's not based on solving a problem, but that is somehow capturing the instinct inside Galois's mind when he says, "I think there's something here"? What's the instinct inside Lagrange's mind when he says, "I think this is the right way to think about it"? What's the instinct inside Liouville's mind when he says, "These scattered notes from this long-dead youngster might have something to them"? It's so hard to put a finger on that. A different series of videos I'm making right now is about the whole "compression is intelligence" idea. Even though this isn't really the angle I'm taking, there is something to the idea that the smaller expression that's more predictive feels more intelligent. So I wonder the extent to which you can give some kind of verifiable reward around not just whether you solved it or what it is solving, but around the smallness of the concepts required to do it.

I think a third way it could happen is it just straight-up works harder. In the same way, you could maybe have an elementary proof of Fermat's Last Theorem that's just spelled out over thousands of pages that would be incoherent. But the cleaner way to view it is with elliptic curves and all that. Maybe there's some thousand-page proof of the Riemann hypothesis that no one's really getting anything out of, and what you actually want are the succinct, compressed versions of those ideas that would then lend themselves to human understanding. Maybe you throw Kolmogorov complexity into your attempt to quantify what you mean by elegance. I don't think it's easy, but I do think it's something you would have to do in order to reward the Galois-like instinct, rather than just rewarding whether you solved a problem. It's very hard to come up with the heuristic for science. But it's clear humans have been doing this somehow, and obviously, AIs will do it at some point. It's relevant also not just in terms of verified reward, but presumably, the end goal is understanding, human understanding. Even if you do have some thousand-page proof of some math thing or some grand new physical theory, the goal is understanding. Maybe if the goal is predictiveness, you can just have automated engineers go off and build rocket ships where we have no idea how they work, but we can get between stars. But there are going to be a lot of people who want to understand. You're still going to want whatever the concision function is that distills down this complicated way of thinking into the right one, like the equivalent of the universal law of gravitation for Newton. You would still want to train AIs to be able to do that and find the compressed representation.

I think it depends on the nature… If we break down the three possible ways of solving the Riemann hypothesis… The other big one from this year was a certain Erdős problem numbered 1196, about these things called primitive sets. It had that character of bringing an idea from a seemingly different field. As soon as you present the basic idea to a mathematician… You say, "What if we try the Markov chain process where we show that this thing is one from the bottom up probabilistically rather than the top down, and use the von Mangoldt function?" If you say that to someone in the know, they'd know how to run with it. You have this very small idea that has the form of expertise in one field and expertise in another, drawing a little lightning bolt between them. Those are going to be very human-parsable, because all you have to do is show the start and end point of what those connections are. If the character of it is mountain building, you have to put in a lot more time to understand that new mountain that was built, because it's a new thread, not just a lightning bolt between them. And if the nature of the progress was just raw hustle—a super long chain of reasoning with no new theories—then you would have that worry of this whole digestion process. So I don't think there's one clear answer. It depends on what the solution would look like. On the mountain building side, that would actually be really interesting to see. Is it by default very human-understandable, the way we see new theories from great mathematicians? Or is it an alien, different kind of mountain being built where we have to reprocess the kinds of abstractions we engage with? The closest example here would be the attempted solution of the abc conjecture. We maybe shouldn't get into that one, but it probably is not a correct solution. Basically it's this whole new way of thinking that this otherwise reputable mathematician in Japan had come up with. It took mathematicians a long time to even parse what he was saying, but it had the feeling of an alien bit of mathematics that's theory building, not just a long chain of reasoning. He called it inter-universal geometry. The biggest fear would be that an AI does that, and then much like the abc conjecture, people work for years to go up the mountain, and they're like, "Dang it. This just isn't right." If it turns out to be wrong, but it really looked right. Even if it was right, there's just a lot of effort to hike up a new mountain. If we end up in that situation, David Bessis had a really great blog post called "The Fall of the Theorem Economy". He's talking about how historically, as you were saying, mathematics is coming up with these definitions and problems, and it's about proving theorems about them. The theorem-proving stuff is what gets all the credit, but it's really a parasite on the coming-up-with-the-definition stuff. Historically, this has not been a problem in terms of credit apportionment, because if you come up with a definition, you're probably going to be the guy who comes up with a theorem. But now we're in a situation where if the valuable work is coming up with the insight and AI automates the latter part… Imagine a scenario where an AI comes up with Abel-like direct arguments about a bunch of important conjectures in the world, and then we just have these proofs. Now it's up to humans or future AIs to consolidate.

I think it would be hugely helpful. So much of trying to discover new math is mostly being wrong. You're trying to solve a problem, and it doesn't feel like constantly taking the correct step up the mountain. Mostly it feels like a random drunken walk, where you're doing a thing and then you're wrong and constantly discovering that. If at the very least you know that trying to digest what you have is ultimately leading to a correct solution, that feels like progress, simply because of the sense of knowing it leads to a solution. There are plenty of instances in the recent history of math where it feels like the reach has exceeded the grasp, where things are proven long before they're understood. One of my favorite openings to a paper—it's not even a research paper, it's more like an expository one—is from a mathematician named Timothy Chow, who was trying to understand a concept called forcing. There's this problem called the continuum hypothesis that more or less asks: you have a size of infinity for the natural numbers, and a size of infinity for the real numbers. Is there something in between? The answer is both yes and no. It depends on your axioms. It's outside the scope of our usual axiom systems, which is an interesting answer. But the method to describe it is really hard to understand. It's this thing called forcing. In the beginning of this paper, he writes, everyone knows the idea of an unsolved research problem. I want to propose the idea of an unsolved expository problem. Sure, we've proven it, but we don't really know why it's true. Then he proposes a partial solution to that expository problem. You can imagine why I loved that framing, because this is my whole life. I don't do research math. It's wholly about what's the most clear way to understand this, even if it's proven. There is a difference between proof and explanation, and I think you're getting at the importance of that distinction.

There's a couple of interesting things there. One is that there seems to be a really strong correlation between the people who come up with genuinely novel insights and the people who are actually quite clear in their communication of it. You might imagine the opposite, given that the experience of a university student is often that the expert teaching them is not necessarily the best explainer of that topic, because they're so spoiled by their expertise. But what seems, at least in some cases, to be the case is that the people who are really coming up with something quite novel—you've got Einstein or Claude Shannon or someone—you read their papers, and they're really lucid. It doesn't feel like this is just for the experts and you have to chop through it with a machete. They're very good expositors. Feynman has this characteristic too, he's a very good expositor. Maybe the same part of the brain that comes up with the correct new way of thinking about it at a research level also has this knack for good explanation. I think this is pertinent to AI. I used to think that AIs would become these automated theorem provers, but the role of mathematicians was going to shift towards my job, explaining these things. Now I suspect that actually they'll also be quite good at doing that, probably better than most humans are at explaining and distilling. So digesting and explaining what was going on is probably actually not what's left for mathematicians, by the nature of how these things are going. We can talk about ways this might not be it, but probably the same thing that comes up with the really good new idea that solves some new problem is also just good at explaining it. That's a way my beliefs have changed.

I will probably be doing something like what I am until I die. If the doomers are right, maybe it'll be for the same reason. Yeah. You build a man a fire, and he's warm for one night. But set a man on fire, and he's warm for the rest of his life. So that's where I am with AI. Some of the function of an explainer or a teacher is to add clarity to a thing that someone's curious about. That's one thing. But some of it is more relational, providing motivation and a sense of curation. One interesting take that I've heard about what mathematicians will end up being is that it's actually more analogous to art museum curators than anything else. The AI solved the thing, so the art exists. They even know how to explain it really well. But you still want someone to help you navigate this nearly infinite space of what ideas are worth engaging with. Even if AIs were in some sense better at that, I think we would always still prefer a human that we had a relationship with, because the way we get motivated to be interested in things is a social phenomenon. If you have some specific technology you're trying to build, that might be different. But the people listening to this podcast trust your curation on what's an interesting topic in the first place. It's not that they're landing here because whatever your next topic is, that's what they wanted to understand in a prior sense. They're trusting you as a curator. So my role, and arguably that of other mathematicians, might actually just shift subtly into that curation direction of what ideas are worth pursuing. That's a lot of my job right now. I think people assume a lot of the time for a video goes into the visuals. Sure, it does. It's not immediate. But actually a lot of it is just deciding what's worth saying in the first place, what's worth putting there. I want to engage with that, and I think I have a trust with certain people, and they're curious what I would choose to put forward even if the AIs are better than that. It's the same reason human musicians are always going to have a role: that social function of the story behind them, even if the objective quality of the MP3 file coming out of some model is better. That's what I see happening to my job.

There's a lot more to do on that one, by the way. Just because a couple lightning bolts have been thrown… I think there's this flourishing future over the next couple years of really connecting.

A lot of the conversation has focused on problem-solving and that nature of math, ticking off Erdős problems or something. But I'd say it's not even a majority of mathematicians who would characterize their work as really targeting the next problem to tick down. Are you familiar with the Langlands program?

It's not even a field of math so much as it is a research ethos. Fermat's Last Theorem is one inkling of this. You had these two seemingly disparate things, and a connection between them led to a solution. Langlands was a mathematician. He has this famous letter essentially spelling out how it seems likely that there's a lot more connections like that. He even got a little bit more specific about the nature of the connections, such that you might imagine this large map, and you've got this valley over here and this mountain over here and this set of plains over there. There's a lot of mathematicians who would characterize their work as being part of trying to understand the threads on this map. The progress there, it's not even "Here's this one specific problem that we know will be solved by that connection." It's more that time and time again, there have been cases where big problems were knocked down by finding connections, such that it's almost preemptively finding the connections. It's actually very interesting. Anytime you run into a mathematician, ask them whether the character of their work is more akin to the Langlands program or to targeting one particular problem. You get a certain bifurcated split there. The possibility of AIs being supercharged connectors feels like it might be an amplifying tool in that pursuit. It's hard to measure, though. This cuts to what we were saying earlier: How do you assign a score to say, "Yes, you've done it"? If it's knocking down a problem, you have a clear way of saying, "Yes, you've done it." You can write the headline. You can have your PR move as the AI company to say, "We did it." Whereas if it feels like that was the right connection to draw, you can write theorems around it. That's the nature of what the papers in that field look like. But I think it will require a lot more "human in the loop" to say, "What was the kind of connection that we were going for?" That's my guess on what most of the useful progress from these models will look like in the next five years. It's just really filling in that landscape of connections that you can draw if you're an expert in multiple fields. As you've pointed out, it's kind of surprising we haven't already had this. I would be curious to know at a technical level what causes the unlock there. On the one hand, you can paint an explanation in your head for why you could be an expert in all of these things and not be drawing those connections. When the method of reasoning is this autoregressive chain-of-thought phenomenon… Autoregression is actually a really weird way to produce stuff, if you think about it. You're an intelligent person. Imagine I've locked you in a box, and the only way you have of interacting with the world is that you receive a slip of paper, and someone says, "Can you predict what will come next?" You predict what will come next, and then your memory's wiped. You get another slip of paper. Imagine that was done a whole bunch of times, and then what comes out on the other end. They say, "Look at this essay that you wrote." You might look at that and say, "This is awful. That's not the essay that I would've written." The process of repeatedly predicting something is just pretty different from how you would think as a writer to compose it and think it through. In particular, what would probably happen is that you're a slave to your context. You might be answering some question about a particular field, so you draw on all the context around that. But the connection where all the substance is going to come from is, by its nature, a very unlikely one. You can do all the RL that you want to try to get better in some way, but what's the thing that's specifically upweighting and incentivizing making these unlikely connections when the vast majority of them aren't the predictable next token that would come in there? So it might be the case that you just have this intelligence locked inside that box, but it's a weird way of interacting with it. The thing I'm curious about is: do you ever get any fruit by questioning the premise of how tokens are generated? I don't think it would be as simple as manipulating the temperature, but are there any things that you can do that take the existing level of intelligence but find the right ways of sparking those connections that unlock these sorts of things that we've seen? Or do you just need a little bit more intelligence, such that at the level of prediction, it's predicting that it should be making that lightning bolt to another field? I think it's more productive to reason, instead of architecture or even loss function, about data. We have diffusion models that do text, and the kinds of things they produce are not of a wholly different character. They've just not been explored as much. I think the more relevant thing is: what is the data on which whatever architecture or loss function you have is incentivizing you to produce? It does seem like they're getting better. Forget about math. We did have a couple of examples of this kind of thing, but if you just look at why they're getting better at being autonomous agents… They're in an environment where they're autoregressively producing the step that says "Let's step back and do a search over the whole codebase," and then "Let's step back and assess my mistake," is the thing that works. I assume what happened in the case of progress in science or maybe in math is you have frontier math-like problems. Mathematicians have specifically designed them because they require connecting together two different fields. I'm guessing there's all kinds of clever, partially synthetic ways to make harder and harder problems like that that require these kinds of connections—for example, by eliminating assumptions and still requiring the AI to get to the answer—and then it doesn't really end up mattering what the loss function is. It's really about, can you come up with an environment that incentivizes this ability? It feels like you should be able to. I certainly can't speak to the correct ways of doing that to unlock all this, but it would just be pretty surprising. Don't you think it would be surprising if, over the next three years, there weren't a lot more of those lightning bolts?

In the history of mathematics, who makes a few connections and then dies in a duel. It's universally applying that waterline across all problems that are accessible at that level of capability. This is among the many advantages that digital minds inherently have that we don't think enough about. The other ones being that they can merge all their knowledge together—or at least that there will be techniques that allow this to happen—and that you can spawn off copies with identical levels of knowledge. This parallelization is quite an important property. I'd be curious about your predictions.