Inside OpenAI’s Breakthroughs in Mathematical Reasoning

Key Takeaway

Use AI as a persistent thinking partner for problems you have already started—not merely as a search engine. Today, take one stalled question or project, provide the context and your current approach, ask for a rough strategy, then explicitly ask it to challenge assumptions, explore an alternative path, and “push this further.” Treat its output as a set of testable leads: verify key claims, retain what works, and use the saved time to focus on judgment, synthesis, and the next important question.

Episode Overview

Leisha Li speaks with OpenAI mathematicians Mark Selke and Metavswani about AI systems making new progress in sphere packing, coding theory, and group theory. They argue that advanced models do more than brute-force search: they execute delicate arguments, backtrack from errors, connect fields, and help humans both generate and absorb mathematical knowledge faster.

Full Transcript

Transcript of Inside OpenAI’s Breakthroughs in Mathematical Reasoning from A16Z. Auto-generated from episode audio; may contain minor errors.

Often as a practicing mathematician, you have an idea, and then you kind of think it might work. Then you try for a few hours, a few weeks, and at some point you give up. Whereas for GPT, like, okay, a human told me to do this, let's just do this. And so that's why we're sort of in this renaissance of reachable results. This is the best part about this problem, which is really nobody had any idea. Is the model just guessing in some insane way? It doesn't seem like there's a limit so far, but it doesn't have that context yet.

It'd be nice for the world if applied mathematics went a lot faster. The ceiling for difficulty of a math problem is pretty high. Even if AI continues getting exponentially better at math, plausible will never solve something like P versus N. What's the ideal way that this is being taken up by the math community? Probably most at this point are like, okay, AI is obviously doing some non-trivial stuff. So AI isn't just getting better at math benchmarks. It's beginning to make progress on mathematical problems that have resisted humans for decades.

In this episode, A16Z Infra-partner, Leisha Li, sits down with OpenAI mathematicians Metavswani and Mark Selke to understand what's actually changing. They walk through recent results in sphere packing, coding theory, and group theory, and explain why these advances can't be reduced to brute force. The models try different approaches, abandon dead ends, connect ideas across fields, and in some cases, produce reasoning that reads surprisingly like the notes of a human mathematician. They also tackled the bigger question, what happens to mathematics when proving a result becomes less of a bottleneck?

From mathematical taste and human judgment to understanding an explosion of new results, Leisha, Metavswani, and Mark explore how AI could change not just what problems get solved, but what it means to practice mathematics. Well, thank you guys for coming. This is really exciting because I think math has been moving so fast. I love working with AI. I just love to get to both practicing mathematicians and who work at OpenAI to chat on some of these results. So we have with us Mark Selke and Metavswani. We're connected actually because Yufei was actually your advisor.

So both of you guys have worked much more deeply in math since I have quit many, many like over a decade ago. So this is very exciting to kind of hear a download of your thoughts on how OpenAI has been sort of approaching this and also just like where you think math is going with the incredibly rapid advance of how AI has been helping. We can start off with some very basic questions. What do you do to the extent that you can, of course, share? And how did you come from being a practicing mathematician to working at OpenAI?

Yeah, I mean, I guess we both broadly got excited last year when the model started to really take off in math. So I joined a little bit before Metavswani. I saw the IMO gold medal last summer, basically. I thought, this is amazing. I want to see what the heck they did. Yeah, let me go see. And then, yeah, I guess in the fall, Mark gave me a GPT-5 account and then I started playing with the models and very quickly became convinced that, yeah, it was extremely exciting to play with them.

And you two were collaborating before that. Yeah, we've known each other for a while. We have one paper we actually wrote jointly. Yeah. So GPT-5 was your conversion? Yeah. What was the magic that sort of, what question do you throw at it and what process? Yeah. So I think actually, yeah, so I think how this started was, at least for me, the starting moment was something like, there's a collection of problems called, so Paul Erdos is a very famous mathematician. He posed a bunch of problems and so they've now all been collected on this site.

And so I specifically work in combinatorics and a lot of these questions are among the most important. So it's always fun to flick through the slide. One thing that often happened to me that was extremely frustrating was I would look at a question, see that it's marked as open and then not actually know if it's correct, not actually know if it was still unsolved because the literature is often quite hard to search. And one instance, I just plugged it into GPT-5 and five minutes later, it found a reference.

And this was a case where a few of my friends actually started thinking about the problem on the site. I was talking with them and I mean, we'd spent a few hours, it wasn't clear if the problem was within reach. And it was just very nice to be told, yes, this is in reach, here's how you do it. And yeah, GPT-5 told me this and I told Mark about this and yeah, this is sort of, yeah, this for me was quite a surprising moment. Yeah. And then we looked more into it and we found 10 more cases sort of like this.

At the time, I feel like being better at maybe making connections between, as you're saying, like the search for whether there's been a result or a related thing earlier is just kind of humanely hard, but maybe better for machine. But I imagine as the progress has happened in the last year, what has been impressive has kind of reached beyond that and maybe through talking about it more abstractly or if it's more natural to talk about it through one of the problems that has been recently announced through, you know, Astra and kind of enlighten me as to like how the recent progress has been a lot more than just searching through more areas, making these connections between the field and perhaps just actually deeper, more mathematical reasoning that's similar to a working mathematician.

Yeah. I mean, I think this like search point of being familiar with everything is still definitely like a relative strength that maybe informs like the types of problems that AI is solving now. I think there are some other relative strengths and weaknesses. Another relative strength that's pretty noticeable is just like it's very good at executing on some like idea once it has it. Whenever you have an idea, there's usually some amount of getting everything lined up as epsilon, like smaller than delta, this kind of thing.

You have to get everything correct. And like for a human, it's easy to get lost in these kinds of details and the AIs just kind of always nail these kinds of arguments, I find. Yeah. I feel like you guys will know more detail on this, but like for the unit distance problem, it was just like the approach. There was definitely contributions from OpenAI, but like the approach perhaps was suggested even originally by Erdos. And then it's just that the actual reasoning was a very, very like momentous feat.

And so for a human, you're like, well, I only have a limited amount of time. And if after so many steps, it is still not clear, I mean, maybe you're like Andrew Wiles and you actually spent 10 years alone and do something, but like it's not clear that the risk reward is not good enough. Whereas for GPT, okay, like a human told me to do this, let's just do this. And so that's why we're sort of in this renaissance of like reachable results. Does that track?

And did you feel like with the ASTRA results, is that sort of like where the strengths have been primarily, or there's an extra ingredient or magic here? I feel, I think the unit distance example is about, it's quite telling. In the sense of maybe the exact construction, you can make it look very similar to what people have tried before. But I think, I mean, often as a practicing mathematician, you have an idea and then you kind of think it might work, then you try for a few hours, a few days, a few weeks, and at some point you give up.

And then a not so uncommon experience is that you find out a year or two later that somebody else got the idea to work that you thought that didn't work. So somehow getting an idea to work can even be a large portion of the battle. And I think, especially in the case of the unit distance conjecture, there's just a lot of extraordinarily finicky details. And very often when you're doing mathematics, you're kind of gambling against the problem. You're like, maybe I should try this approach, but it seems really unlikely and just not worth my time.

And the model, I think, in several of these cases, both by combining what it knew and sort of having good taste, kind of made the correct bet. And you can kind of see this in the summarized chain of thought we released. You can sort of look at it. It's reasoning like a mathematician, and because it knows a few very correct bits, it makes the right decisions and eventually able to prune the search tree. It's not really trying everything. It tries a lot of different things. It's extremely dogged.

I mean, it can't try every idea. It has to try a limited set of ideas. And it's able to kind of use its knowledge plus good mathematical judgment and find the right path to go along. So I mean, that for me was like, because this was a problem which a lot of people have thought about. And the fact that the idea is not so foreign probably indicates that a lot of people have tried it, or at least a few very serious mathematicians have tried. And I think that's what made it really interesting to see.

I think something else that I feel when I see these proofs is like, if I have an idea and I'm trying to execute, it might be that I have some wrong plan for like how to get things to work. And as a human, if you have some wrong path you go down for a while, it can be hard to like rewire your brain to start over and try a different path. The initial idea is kind of linked in your brain with these other things that ended up not working.

It's sort of like your context window is like a little polluted and you can't just make another clone of yourself from last week and say, don't do this, try something else, build your intuition in another direction. But it's very easy to do this with an AI. So I think this is another reason that it's like getting the details right once you have some good general direction is much less of a barrier all of a sudden. And when you say it's much easier to do with AI, it's like it's not actually being directed with human interference to it.

As you're saying, the reasoning traces, it's like making these choices. Maybe it backtracks, but then it's able to not be distracted by maybe the context in which it's thinking about the problem, like by these machinery. And it's like, do you see it kind of go back as well or is it just making good choices? Like is it a lucky sample or is it actually reasoning like a mathematician or, okay, it doesn't do well in this path, but it goes back, but then it doesn't let that pollute.

No, I mean, it definitely makes mistakes and then it goes back and thinks about it. I think it's somehow very calculating, very correct. I mean, as human mathematicians, you're not always perfect at making these decisions. The first time something doesn't work, you automatically kind of downgrade how likely this approach is to work and you keep doing this a few times. The model somehow is much better able to like, it seems for several of the solutions we've seen, it seems much better able to update how likely the path is to work versus rejecting a path versus a human doing it.

But I think even if it weren't, the fact that you could just start another model session over means that it's always going to be the case that it's a disadvantage here. So in some sense, it is still leveraging the fact that you could run kind of parallel agents on the problem. But if it were kind of backtracking, then it does make it seem much more like a human mathematician. And perhaps it is kind of doing some of that stuff too, because obviously we have to make mistakes in order to even gain intuition for why that solution space is not in the set of paths that it could be in.

I mean, I think this kind of thing happens with humans too, where if you get stuck on some approach, you might tell another human your kind of general idea, and then they'll come back and figure out how to get it to work. It's just, it takes more time to do this with humans. I wonder, I mean, maybe this gets to the extent that you can actually talk about sort of like, obviously don't talk about the training recipes or whatever, but it's interesting that if you're just studying, for instance, for math papers, it's like a very poor training set, like a priori for math, because I mean, maybe math textbooks are even a pure example of this.

It's like really bad at actually reconstructing the motivation for why things were, you know, like don't, I mean, maybe some people like it, but don't learn real analysis from Rudin. It's just like, it's very clean already and crisp. And I think that that's bad because it doesn't show the struggle that made us formulate definitions in a certain way. Like, why do we even need to have real numbers be defined in this like super abstract way, et cetera. And so, you know, I think papers also, I mean, unless you're, most people don't write papers with the context of I need to educate somebody to be a mathematician.

And so like the actual maybe curriculum of like learning math is not inherent in like a lot of our artifacts as mathematicians. So maybe another way of asking this question is if the reasoning tracers are actually producing things that's like, okay, this is actually more close to mathematical thought, like how does that arise? I mean, I, yeah, I guess OpenAI has been like the pioneer of reasoning models and teaching AI to reason in this way. So, you know, we're doing a lot of work at kind of all possible directions on, you know, teaching models to reason better and for longer and, you know, in all kinds of different domains.

I mean, I think we're training general purpose reasoning models and kind of one, a lot of these behaviors that we're describing mathematically, like backtracking or kind of starting again, I mean, these are not really specific to mathematics. I mean, we're seeing them specifically in mathematics in these examples, but kind of they're general purpose tools for reasoning. And I think if you work hard at reasoning, you should see these patterns eventually. So it's just emergent because it's, I mean, I do think that's why the OpenAI approach was so, I mean, it's like, it doesn't rely on, you know, doing auto-formalization in order to like guide the reasoning.

I think that's like obviously more like us. But it's just like, so not obvious that if you're just like training on say a corpus of like math proofs, maybe auto-formalize and lean that you get, get the sort of like projection of like how to think well, like put another way, like with maybe, maybe if we think about it with code, like code is such a good corpus to train on because it's one of the few data sets that has such large context. You just like, I mean, maybe you see this kind of with books, but they're less structurally interconnected.

There's just like less structure there. I think it's safe to say like on average in a book compared to like a piece of code. And so like with math papers, I feel like maybe what we're so bad at with coding models is stuff that that data set doesn't contain, which is like kind of the semantics, like the syntax is there, but there's a little bit of like the higher level semantics of what produced, like, why do I have to write it this way is not, I'm kind of getting too much of the philosophical, but it is just like really interesting how it's still emergent that it's doing good mathematics.

And we'll probably get into this in more detail if you guys wanted to talk in more detail about some of the problems, which is just like, it's not just doing like the expected, like we'll push the brute force thing. Like you clearly are impressed with some of the reasoning traces and it's just not obvious that's gleaned from, you know, what we would imagine would be the easy training set here. Yeah, absolutely. I mean, I think this kind of thing is one reason we decided it was important to release these summarized chains of thought for these kinds of results, because if you've never seen these and you just see all these proofs coming out, you're kind of, you're not sure what it means.

Like, is the model just guessing in some insane way? Is it thinking in some totally foreign, like, what's going on? But actually, it's reasoning kind of shockingly like an expert human would. Yeah, yeah. Yeah, it's very much like reading a colleague's notes. I mean, it's a little more disorganized in some ways, but kind of, especially if you work close enough with a collaborator, sometimes you'll just see them spill out their thoughts in an email to you. And it kind of, it feels like reading a lot of those chained together.

So it's, yeah, it's very, it's quite surprising the first few times. Were you two sort of very involved in choosing the problems to release in this like last 10 problem set that Astra has applied to? Which was your favorite? Yeah, we are definitely involved. Do you want to start? Yeah, I mean, yeah, I guess. Yeah, so I guess my personal favorite among these problems is the following. It's extremely simple question, which is just like, it's just about how efficiently can you put a bunch, my circles are not very good and they're not all the same size, but.

But we're assuming they are. Yeah, so the question is just like, how dense can you place a bunch of, so you have a bunch of spheres, you have a bunch of spheres of radius one and D dimensions. So the question is, how densely can they pack? And so, yeah, so in two dimensions, it's kind of like the, so D equals one, this is not an interesting question, kind of, it's just the real line. And yeah, you can cut it up and a sphere in dimension one is just a unit segment.

So, okay, you can cover everything. So in D equals two, it's kind of the picture that you know, that everybody loves. It's just like, it's just a bunch of spheres which sort of form like a hexagonal lattice. Hopefully I've drawn it well enough that I can draw the hexagon. I'm kind of betraying my naivete on this problem. Is that like obvious? Is it like a very elegant proof that it's a hexagonal lattice? Yeah, it's not so obvious that this should work. It was only proven in the 60s, I think.

There's a short argument, but it's not so easy. Yeah. Where's the intuition? Like, what is kind of like the machinery of the argument? I mean, it kind of like, I mean, it definitely looks like it should work. That's why I'm really shocked. But so did you. Yeah, I think this is the best part about this problem, which is really nobody had any idea what's up with this problem. So, yeah, so I mean, honestly, the best intuition that I have for this is that like bees do this.

And if there was a more efficient way, then probably bees would pack honeycombs some other way. Evolution is efficient. Yeah, I think beyond that, like I don't have a great argument. I mean, and I think how little we know is demonstrated by the fact, so, okay, D equals three. The answer is just like, it's how you pack like oranges in a grocery store. And this was only, this was proved by Hales sometime in the 2000s. And like, and we don't have a short proof of this.

Like, I think the shortest proof is like a few hundred pages. What area does it like draw from? So it's a lot of linear programming arguments and it's very delicate, like geometry. It's quite ugly, actually. Yeah, it's like. This is like a famously ugly argument. Oh, no. And then the two most famous results are D equals eight and 24. Eight and 24. It must be some like weird subspace. Yes, yeah, exactly. So this was done in some. Like gluing some. Yeah, so this was done in 2017.

Sounds slightly prettier though. Yeah, so the reason it works out in these two very special dimensions is that, so this is called a lattice packing. So it's like kind of very regular. And it turns out in these two dimensions, there are two very special lattices. They're called the E8 and leach lattice. And they're very nice and they're like unusually dense. Like kind of, they're just very, very pretty structures coming from other areas of math. And it turns out that they're the optimal structures. But they're still like regular.

Yeah, they're very regular. But I mean, beyond this, so we don't know any more exact dimensions. We know these five dimensions and we kind of don't know anything else. And I mean, like to give an indication of how little we know. So there are two very surprising things about this. So you can define like delta D to be like the densest sphere packing in D dimensions. So there's kind of an easy lower bound of like two to the minus D. Basic, yeah, this is not so hard to show.

Basically, any packing where you can't put in another sphere has to have this density. So, okay, it's not ridiculously small. And we know that it has to decay exponentially. So it has to decay, like it grows like one minus C for some, at least for some constant. So in large dimensions, you can only cover like a vanishingly small portion. But we know like basically nothing else. And so- And that's just because like the high dimensional sphere like thing where it occupies, it just like, yeah, the volume behavior is weird.

Yeah, so basically, I mean, basically they don't want to touch next to each other. I don't know if, I don't think there's a particularly short way to see that it's like exponentially small, but it's known to be exponentially small. And for a long, long time, the best bound was something like this funny number, like two to the minus 0.599 D. And this was proved by two mathematicians in the 70s. Kapitansky. Okay. That's a weird number. Where's that? Spit out. It's not- It's like combinatorial or? It's the answer to some extremely ugly optimization problem.

There's like a nice underlying strategy. Okay, but let's- Yeah, yeah. What's the music of it? Yeah, yeah. I'll say one last thing about this, yeah. So, yeah, I mean, these were these two Russian mathematicians in the 70s. They're thinking very hard to find their paper. Like one page, it's like two pages long. Yeah, they don't write very many details because paper was fine, but yeah, and so- So two negative Ds, they're just like the square lattice, like the dumb one or? So, yeah, it's actually not so easy to- So the argument for this is as follows.

Basically, imagine that you have a set of spheres and I tell- You can construct a set of spheres so that like you can't put down another sphere because if you could put down an extra sphere, you just keep putting it down. So you have a set of spheres so that there's no other sphere which you can put down. That's your, it's like almost like a- Just, yeah, just take any such packing. Okay, yeah, yeah. And I claim that this has to cover at least two to the minus D fraction.

The reason is that like, if you blew up each of these spheres by a factor of two, then like they have to cover every point in space. And the reason is otherwise you could put down, if there was any empty space, you could put down a sphere there. Yeah, I didn't. So I guess if you take the usual lattice, there are actually like more places you can put things kind of diagonally. Yeah. Okay, yeah, yeah. So that's actually not, it's like a worse bound. Yeah, you can just keep plopping things in.

Yeah, this is- So this is like more, okay. Yeah, this is related to this like really funny fact where you like put a sphere on every point. If you take a cube in high dimensions, you put a sphere on every point. It's like vanishingly small. It's so small that you can put another sphere in the middle. Yeah, yeah, yeah. And it fits. High dimensional sphere behavior. Yeah, it's very weird. And so, okay. So the great part is that, so the model shows the following. So I'll write two things.

So this is Astra, I guess. That's probably the right way to refer to this. So, okay. I'm gonna write something called the LP bound. I'll explain this in a second. And it shows that it's smaller than this very nice number. Do you wanna say equals? Yeah, it's equals actually. D to the two pi plus little one to the D. And if you can work out what this number is, it's like roughly something like two to the minus 0.6. Close enough. Yeah, it's surprising. Wait, but this one, you're like, oh, maybe there's some nicer kind of structure there that fell out.

But it's at most, this is roughly something like two to the minus 0.601, dot, dot, dot, D. That's the numerics? I thought it was six or four. Great. Yeah, this shows mine. Yeah. Okay. So there are a couple of things. So first, what is this LP of D? So Vyazatsky's work actually builds on some earlier work. It turns out that there's a way to attack sphere packing by what's called a linear programming bound. So LP just stands for linear programming. So Kohn and Elkies gave an approach for sphere packing based on linear programming.

So it's like a linear optimization problem over a convex set, but it's all kind of infinite dimensional here. And basically what this reduces down to is you try to understand the following. So what you try to show is you, basically you construct a function F. So this is in D dimensions and it's mapping to R and it has the following properties. So first, F of X. So this is a function in D dimensions. So it's always less than zero if like the size of X is bigger than one.

And you second have that the Fourier transform of X, this is always non-negative. So this is just, this is a linear program because the Fourier transform is a linear operator. And Knight Race. So you're taking just some arbitrary F that satisfies this property? Yeah, so you can take any F that satisfies these properties. And what they prove is that delta D is bounded by the ratio of the Fourier transform at zero to the F of zero over the Fourier transform of zero times the volume of the ball of radius one half in D dimensions.

And so, okay, this proof is not so short for experience about partition. It's like, it's a half a paragraph to prove it, but a little bit tricky. And the point is, so it turns out, so this is a relaxation problem. There's no guarantee that taking the optimal F will give you a good bound on delta D. But, so what Vyazoska did, and this was sort of the key, I mean, a large part of the reason she won a Fields Medal in 2020 was that, or in 2022, was that she constructed a function in eight and 24 dimensions, such that this upper bound matches exactly these two very special lattices.

And these are kind of miracles of nature that both you can construct this function and that it gives you the optimal bound. But you can just, this is a very, very natural problem. I mean, it's a function with two very simple properties, and you just want to understand how this behaves in, for large dimensions, D. And that was a big mystery. There was a numerics paper by Kohn and several others, which conjectured that, just based on doing numerics, that this was the answer, but they had no idea why this would be the answer.

And what the model shows is that actually the linear programming bound in large dimensions has this extremely nice asymptotic behavior. And the proof kind of explains where this is coming from. And because you understand this LP bound perfectly, this actually just gives a better bound on delta D. It turns out that this old bound can be kind of reinterpreted in this framework. And what the model does is it shows you the best possible bound you can get by this framework. So the model sort of made the connection.

And what is the sort of like, yeah, what did the model do? I think, so the model gives a function F, which, so first it constructs a function F, which gives you this bound. And then it shows that there's no function F which does any better. So it's inequality, which is quite strong. So in like sort of, we now understand this problem in high dimensions very well. And that's pretty remarkable. And the model was just kind of told like, analyze this linear program in high dimensions.

You know, go have fun. Got it. And to give an indication of like how it was known, I think this conjecture was based basically only by doing numerics. Extremely clever numerics, but numeric. And so, yeah, you have to kind of figure out why this is the right thing to aim for. And it does. And that was pretty remarkable. Yeah, I mean, I had actually thought about this problem for about six months at some point when I was a graduate student. And yeah, I just, I remember making like absolutely zero progress on it.

So it was very nice to be like explained why it was, yeah, why it was true. So that was a pleasant experience. I think also in general, it was one of these solutions which I knew several people had tried the problem. It's pretty remarkable because like the model solution, especially for this being like, the LP can't do better than this was like quite short. It's a few pages of like complex analysis, but it's kind of exactly the right approach. Like once you see it, it's kind of, it's like unbelievable.

Like why, why hadn't somebody done this before? It was, it was like, there are many types of good mathematics, but I think one of them is just like, you see it and you're like, oh man, why didn't I think of this? And it was really fun. And I mean, I sort of knew why I didn't think of it, but it was quite nice to see it. And it was fun to see. Yeah, that's why I like this problem a lot. Yeah, so this is the first of the 10 problems that Astra solved.

But the second is actually closely related. So this was sphere packing. The second one is a spherical and binary codes. So what's like your draw code? Did you draw a packing? It's going to be the same picture. Okay, sure. Otherwise we're going to have his picture. He's got a little packing in our minds. Yeah, a spherical code is literally just a sphere packing, but on another sphere. Yeah. Yeah, so I mean, a spherical code is basically just a sphere packing on the surface of another sphere.

So. Yeah, it looks like a sphere. Yeah. Okay, so same picture as before, except you're kind of on a curved surface. Okay, okay. Okay, so why is it called a code? Well, you can, I guess the reason is because of binary codes, which is, again, the same sort of thing, but now it's on a cube. Okay, yeah, fine. Let me draw a picture of a cube and some like simplest possible code on it. So like when you're like sending, so this is really like about error correcting codes.

So what are error-correcting codes? So, you know, it's like, I send you some string of bits, right? And maybe I'm worried that some of the bits I send you get corrupted, right? So maybe just because of some errors in my system, this one gets changed. And we want some communication protocol so that, like, you can decode this, like, small amount of error and, like, recover what I was trying to tell you. And, you know, like, normal English language kind of has this sort of property, right?

If I make a few typos, you're going to be able to understand what I'm saying. But if we have some, like, really brittle communication scheme, it's not going to work. So codes are kind of the way you solve this. And mathematically, it just means, like, what's a binary string like this of a fixed length? It's like a point on some hypercube. And we want a dictionary of allowable code words that are, like, separated from each other. So, like, in this case, if I don't want any two to be adjacent, I would kind of take these four vertices, kind of the, like, even ones, if you sum up the digits, right?

And, like, okay, I guess, okay, in this case, I guess if I have an error, you can't tell which one it's from, but at least you can tell it's, like, not, at least you can tell there was an error. Oh, I see, I see. Yeah, because it's, like, kind of sparse in the, yeah, it's, like, it's not true adjacent. So that, like, is this, like, when the hamming distance? Yeah, yeah, yeah, right, right. You want, yeah, so you want, like, a large hamming distance between any distinct points in your dictionary.

And, yeah, I guess if you take two opposite corners, then if I have, like, a single bit error, I can always, like, recover which point it was coming from. One that it's definitely closest to. Yeah, so there's, there's kind of the, you know, same question in both of these cases, like, in a very high dimensional setting, what kind of rate can you get? And, like, for binary codes, it's really, like, you know, an extremely practical question. It's sort of, like, if I send you, like, an n bit string, and there's, like, you know, 1% error rate, like, how much longer does my message have to become to tolerate that amount of errors?

There's, like, some fundamental, fundamental information theoretic limit of, like, you know, communication. And, you know, but you can see, like, certainly this spherical case is, like, it looks very much like sphere packing. For example, if you, like, if you make all these little spheres really small, then, like, the curvature of the big sphere is kind of not going to matter so much, and it looks like just packing spheres in full space. And, in fact, yeah, like, these problems turned out to be very related. So, for these problems, there was, like, there were similar bounds coming from the KL authors, and, like, there's, like, something for the sphere and something for the cube, but it's all kind of the same stuff.

And our models found better bounds for these cases as well. And, like, I mean, the techniques look pretty different, actually, if you, like, write them out. So, this full space analysis of this linear programming was using, like, just complex analysis. But if you, like, the method for these cases, we're using representation theory. Like, both the sphere and the cube have a lot of symmetry. And basically, the idea of the proof was to really leverage this symmetry. Like, there's some amount of this in the previous, like, existing method, and really the improvement is to, like, lean into the representation theory, like, really hard and kind of make the algebraic symmetry, like, enter in a more sophisticated way.

And then it, like, turns out that from the representation theory formulas, if you kind of take this, like, small sphere limit in the spherical code case, you recover, like, part of this result, and you recover this value. So, like, this result isn't a special case. You kind of only went one direction of the bound from looking at it from the code's point of view, but, like, there's, like, a very close connection. Okay. Yeah. You guys let this, like, run in parallel. So, it's, like, kind of discovering, or because you're not sort of, like, feeding it.

So, actually, this was the one case where there was some interactivity. Oh, interesting. So, for all of, so, except for this pair, it was just, you know, we had some problems. We fed them in, and we, you know, the model came back with some solutions. What happened here is actually pretty interesting. So, we first asked it to improve the bounds for the codes, and it came back with an improvement that, like, used some amount of representation theory, and then we kind of asked it, hey, can you, like, push this further?

Like, you know, what happens? And then it came back with some, like, much more sophisticated representation theory, and, like, it turned out that you got this conjectured value for full space sphere packing, like, out of that method by pushing it as far as it can go. So, then we kind of asked to directly analyze the sky and try to complete the picture. Okay. Yeah. So, the relationship, like, isn't a coincidence. Yeah, yeah. It's, like, interesting when you're saying the first prompt, which is, you know, maybe so basic, which is, like, can you push this further?

It does require some judgment from mathematicians, but, like, eventually, you would imagine by scaling the models, you don't need to do that. Or there's another view that the harness actually does matter, and this is kind of part of the harness apparatus. Do you guys have any views on that with your working with Astra, especially generations of models, and how much you have to kind of input or how much the harness matters versus not? I mean, yeah, I guess there have been some, like, funny quirks like this that just come from, like, exactly what you asked the model to do, basically.

Like, in this case, what the model was asked to do originally for codes was to improve the bounds by, like, some exponential factor. So, it really, like, shows up in this, like, leading constant up here. And, you know, it improved the bounds, and it didn't try to push things too much further. Like, sometimes you see it do, but sometimes it just doesn't bother. But yeah, you know, you just ask it again, and it goes further. So, it wasn't, like, a capabilities issue. It just didn't feel like it at the time.

Do you call that judgment, or, like, what is the... Because, like, there is a... Yeah, what do you call that? Models tend to be pretty task-oriented. If you tell it to do a task, it accomplishes the task. That's good. I'm training for that. It's pretty happy. So, yeah, the task-orientedness, it's like, but do we expect that level to kind of ascend up to... It's not that they will be less good at being task-oriented. It's just, like, they'll ascend to the level of, like, okay, no, let's go in this direction.

You'll have the judgment, too, because you guys have the judgment, too. Like, okay, this is pretty promising. Looks like you're using a lot of representation theory. It doesn't seem like there's a limit so far, but it doesn't have that context yet. But, like, I guess what I'm trying to say is, like, this one, it's hard to, maybe, harder to extrapolate. But from, like, previous generations, when you had to give it more, maybe prompting, more of that harness work, but eventually, you probably had to give it less.

So, it probably gives you some confidence that there's this, like, really fast ascension. And do you see... Yeah, like... Somehow, solving a harder math problem is, like, you have to solve many smaller, like, somewhat less hard math problems. And the fact that the math problems are getting harder is kind of an indication that the model is able to take on more and more work in, like, a single continuous unit. And I think that's the thing that looks very promising somehow. Like, any of these solutions, it's not, like, one idea, and then you're kind of home free.

You need several pieces to kind of interact and talk to each other. And the model doesn't come up with all the ideas at once, right? It doesn't pull everything out in an instance. So, kind of the fact that it needs to sort of see how this piece interacts with another piece, that's kind of like solving a problem in itself, or piecing together many problems in itself. It could just be that, okay, when you're telling it, okay, push this even further, that was of the same order of, like, magnitude as, like, all the smaller things it's solving as well in between.

And so, you don't think that this is kind of, like, a privileged direction. It's just sort of like, hey, let's give it, like, one more help. Or you actually think that there's, I guess what I'm trying to get at a bigger question is, like, is there a good sense of, like, you know, taste? Because, like, when people talk about, for instance, how well the models are getting at, like, doing research, for instance, and that's what we need, we want a little bit of RSI. And sort of, like, there's surprising things about how that improves.

And then there's, like, oh, you know, maybe right now it's at a level of still, like, a junior researcher. It's, like, not really asking, like, the right problems. And so, I'm just trying to get, like, maybe a sense of, like, where you're seeing that progress through the model advancements each generation. I mean, what is taste even? Yeah, I think somehow I tend to be pretty utilitarian in my view of taste. And, like, if you're able to solve problems faster by making better judgments, like, I think that's, like, the best, like, general proxy I have for taste.

And somehow the fact that it's solving harder problems means it has, kind of by definition, means it has better taste. I think there are these, yeah, I think occasionally, because they are task-oriented, you do occasionally get these these symptoms of, like, oh, it clearly has made a breakthrough. It kind of understands it's made a breakthrough, and then it doesn't kind of push all the way to the limit because that's not what you asked. But that seems, yeah, that seems rather minor compared to, like, the state of progress we've seen so far.

Okay, yeah, no, I think that's pretty clear. Like, I think it's, like, maybe you're liable to get confused if you're trying to, like, do a concrete long-horizon task and show taste kind of at the same time. But, like, you know, if you have, like, one model that's responsible for taste and one model that's responsible for going out and, like, you know, working for a long time at solving a hard problem, kind of as the, like, you know, underling of the supervising AI, I feel like that's kind of going to be fine currently.

Oh, interesting, because that is, like, saying that these two things are somewhat separate. If not separate, at least they shouldn't kind of pollute each other's context, which is a little bit, I mean, it could be potentially, like, a stronger statement than, I guess, yeah, no, it's just kind of interesting because it might just be, like, to your point, it's, you know, let's take the utilitarian answer. It's solving harder and harder problems. It's doing a lot more than just, like, you know, brute-forcing something. It's making choices.

It's, like, pruning, you know, a vastly large space of possible paths into something that's, like, really, you know, it's both tractable but then ends up being, like, it's a diminishingly small path within that space. But, like, having, like, why would we be, like, a separate model, a separate generation of something that's a different version of the model that would contribute to taste? Or maybe that's totally, like, it's too abstract, doesn't make any sense, you know, we should just let the actual, this might, like, a related question be, like, you know, what is the thing that gets us to a better version of intelligence, the harness and the model or is it just the model?

And it's, like, we see this in, you know, at least in applied AI or, you know, startups where it's, like, it's a continual battle of, like, you need the harness but then the harness adapts very poorly to a new model because sometimes, like, a very, very minimal harness is still the best way to expose to the raw power of the model. But then now we also have these, like, training regimes where we require the harness to be, you know, trained with, I mean, part of this is to keep things more proprietary and harder to, harder for other people to use it.

But I think partially it's maybe actually that it helps have more control on, like, the reasoning traces you care about. It's a long rambling way of saying it's, like, yeah, I don't actually, like, this is so interesting to see how the models have gotten better at math and maybe something that's, like, very abstract and hard to describe, like, taste is a way to tease out, like, what is actually necessary here. I think my only, like, non-trivial thought here is that, like, when you're working, I mean, just when you're doing any task, occasionally you get pigeonholed and you, like, work really hard and just having a friend look over your shoulder and be, like, what are you doing?

And then just, like, just having that one bit of, like, step back for 10 seconds, like, this is often very useful. Yeah, yeah. I mean, I see no reason why humans would be so different than models somehow. Or models would be so different than humans. Having a few humans working together is often more powerful than just having one. Yeah, it's, like, this kind of collaborative thing. You actually, you kind of, yeah, artificially created it, but it's very similar and dynamic. But I think a lot of taste is also, like, having a sense of what problems you or, like, some method you have in mind are going to be good at solving.

Like, it's, I mean, certainly there's some amount of, like, absolute aesthetic point, right? But there's also just, like, you know, having a nose for what you might want to pursue because you'll be able to make progress. And, you know, I think for that, like, there's, you know, you would expect that as a side product of being good at completing tasks, you would get there, sort of, right? Let me know if we still want to do, like, a section on Soffit groups because I think, you know, up to you guys, it's definitely super interesting.

So maybe the first question is, what is a group? Let's remind ourselves. So a group is a set of elements with some multiplication operation. And basically, this is how mathematicians think about symmetry. So you're, like, basically, like, if G and H are elements of your group, then G, H and H is some other well-defined element of your group. And you have, like, associativity. And you have an inverse. So for every G, there's some inverse. And there's some, like, specific element in the group that is kind of the identity.

Okay, so it's some abstraction of composing operations. So these could be numbers, they could be multiplying matrices, they could be rotating something, which is a special case of multiplying matrices. And a group is Sophic. Well, there's some precise definition, but roughly it means it... So I should say groups that can be finite or infinite. So if you have a square, all the rotations of it form a group with four elements. If you have a circle, then the rotations form a group with uncountably many elements. And so Sophic groups are either finite or countable.

You should think of them as being countably infinite, so there's the same number of elements as the integers. And it's Sophic if in some sense can be approximated by finite groups. So we didn't know if there was a non-Sophic group. So the result that Aster proved is simply that there exists a non-Sophic group. Yeah, and without... Before going to that proof, it is like... I feel like a lot of the programs in math is like, okay, we are such finite creatures, let's see how well our finite approximations do.

And in this case, especially for the countable case, maybe you'll be relating it to the Aldous-Leon thing. It helps anchor the picture of it seems like such a... I mean, it's a nice result if it were true, but it's not. And it seems almost reasonable. And so, yeah, I actually didn't go... I would love to hear the explanation of how it found a counterexample. Yeah, I mean, I would say that the hope that there was no non-Sophic group, so every group has this kind of approximation, maybe this is sort of like people hoping that there's a miracle.

Because it turns out that groups like this have a lot of nice properties, because you can run certain proofs for finite groups and then kind of approximate them in whatever way the definition of being Sophic lets you approximate them and get the result. So there's this notion of being a surjunctive group. So there's some fact that any group which is Sophic is also surjunctive. Surjunctive is some property of dynamical systems on the group. And I guess the original question was whether every group is surjunctive. This is some question of Gottschalk from the 70s.

And this fact that follows this pattern of prove it for finite groups and then do this approximation is what motivated the question about if there's a non-Sophic group. Yeah, maybe I'll say a little bit about this Aldous Lyons conjecture. So I guess I had heard of this a little bit beforehand, because there's a related stronger conjecture in probability that was made popular by Aldous and Lyons. This conjecture, roughly what it says is any infinite graph with some nice property called unimodularity, a modular random graph can be approximated by large finite graphs.

So maybe the way to explain what these kinds of things are trying to say without getting into technical weeds is to say what they mean about the integers. So how would I draw the integers as a graph? So this is called the Cayley graph. You're just going to connect nearest neighbors. So there's some kind of canonical way in which this is like the graph that represents the integers. And there's some sense in which you can approximate this by finite graphs. Why? Well, if you look at integers mod n, then you kind of get the same picture, but you have a big circle instead of an infinite line.

And the point is, if you look at any point here and any point here, in nearby, things look the same. You have to go very far away to see this global geometric structure that you have a circle and not a line. And in fact, the integers and integers mod n are both groups just by adding numbers or adding numbers mod n. So these integers mod n are like SOFIC approximations for the full integers. So this approximation is why the integers are a SOFIC group. So the statement that every group is SOFIC is sort of a generalization of the fact that you can do this approximation with groups.

And this Aldous-Lyons conjecture is kind of a broader conjecture that, like any network, you can do this. And you don't require as much algebraic structure, roughly. So it's kind of a broader conjecture. So this conjecture was disproved earlier, like two years ago, and it was kind of a really tour-de-force work. It was like 250 pages building on another 200 pages. It uses quantum complexity theory. So it really builds this very complicated bridge. And I think not many people could understand this. So since this is a stronger conjecture, the disproof is weaker than disproving this statement that all groups are SOFIC.

But it turns out that the direct proof that there's a non-SOFIC group was much shorter and easier than this really amazing disproof of the Aldous-Lyons conjecture. It's like 15 pages, maybe. And it doesn't have any of this very complicated connection with quantum complexity. It just kind of stays in group theory land. I mean, it uses some important existing results by other mathematicians, like Kuhn and Tom, but it's like a very reasonable, normal kind of proof. Yeah. And it's kind of spelling maybe out the obvious, but the connection between the SOFIC group statement is just you take the Cayley graph, and that's the one that is what they use for the Aldous-Lyons.

And so that's why it's like a subset of... Right. So basically what happens is for a group, you can take exactly a Cayley graph. So you take some elements that generate the group, and you kind of connect elements that are adjacent. So in this case, this is a Cayley graph of the integers. So when you do that from a group, you get a deterministic graph. You just get a single graph. You fix some set of generators. So this conjecture is stronger basically because it allows a broader set of graphs that aren't deterministic.

It allows them to be random, but have some extra modularity property that constrains exactly how it can be random. But yeah, basically that's the difference. Like here, you kind of have to give a deterministic network instead of a random one. Yeah. Anything kind of interesting, surprising about the results? I mean, you mentioned some things, which is like it stayed within group theory, the techniques. I mean, I think maybe it's like a nice example of this general pattern that theorems produced by AI have generally been like...

The proofs are pretty short, generally. They're like... Like the counter examples so far. Yeah. But this one, it's like, okay, it's sort of a counter example, but there's some stuff you have to do to analyze. The difficult part here is that the property of being a Sofit group is not so easy to get your hands on. So you have to find a concrete way of producing a way of saying this group cannot be Sofit. And the proof is actually very short. It's a combinatorics argument, but it's a very delicate combinatorics.

The model... Somehow you need to both have the right statement, and know what piece of literature, and then execute it correctly. And that's very nice. The difficulty of this problem is that it's just a really, really hard... It's very hard to get your hands on being approximated by any possible finite group. Yeah. It's like, what is happening at that countable infinity that's resisting this approximation? Do you guys kind of... Do you give a sense of... Do you do post-mortem when you're like, okay, asterisk, explain to me, what is the...

Oh, yeah, we did that. Yeah, yeah, yeah. What was a good explanation you got out of it? I think there's some concrete combinatorial instruction. Basically, it's hard to explain, but there's some concrete combinatorial structure, which if you read the previous papers, you realize that that's what they couldn't rule out. And Aster found a way to say, okay, no, no. If you add this one extra algebraic fact, this weird conspiracy can't happen. It's very clearly trying to rule out a conspiracy in the sort of previous authors had implicitly written about.

Those were the actual suspects, it turned out. So they were sort of on the right track, and then this did the last mile of, well, whatever, however you quantify that. But I think it's... A year ago, I would have been very surprised to learn that all of these AI proofs are very short and elegant. You're kind of afraid that they're going to generate all these thousand-page things. Yeah, like nobody's verified that, yeah. But it's been kind of the opposite. Only humans can generate 200-page proofs right now.

Yeah. Well, and also I was asking, if you do that post-mortem, it ends up usually engendering more mathematics. Because when you do that with humans, that's what breeds new mathematics. So maybe if you kind of alter the prompt a little bit and be like, how would you generalize this or something? Yeah, I don't know if that's been a technique for you guys to have it explore and exploit what it has already developed. Well, there has been some follow-up on this already, actually, by Kun and Tom, who this was always built on.

So the math community is coming on. Yeah, which is kind of what we're hoping. We don't want to be writing lots of follow-up papers ourselves, but if there's some interesting follow-up, we're very, very happy that there's some follow-up building out these ideas more and giving more examples of non-Sophic groups. Yeah. Well, actually, maybe that's a great segue into what's the ideal way that this is being taken up by the math community? Because I feel like there's a spectrum of answers from working mathematicians. Probably most at this point are like, okay, AI is obviously doing some non-trivial stuff.

It would be a disadvantage not to admit that in my workflow. I've definitely heard some stories where people would find it hard to either take AI as a co-author, or how do you even do attribution this way? But I don't know. Maybe to paint the more optimistic picture, you're saying you want the mathematicians to be building on these results. It definitely generates a lot more results to be verified, so it puts pressure on the community and the profession. How do you expect the evolution of uptake and collaboration with mathematicians?

Well, I mean, given that the fact that the models can produce sophisticated mathematics means that they can help you understand sophisticated mathematics. I mean, I don't know. Occasionally, I enjoy looking at the archive, and I want to understand some proof. I could read the introduction, but in practice, it's just much faster to take the PDF, put it into my favorite model, and then get an output of what is the rough proof strategy. I mean, of course, models are going to help us produce exponentially more mathematics, but they also make it much easier to absorb it.

And right now, okay, it's still a bit of a challenge back and forth, but I think it's, for me at least, much, much faster at understanding. It's much, much faster to understand a piece of mathematics with a model than without it. It's helping solve the problem it creates anyways. Yeah, I feel like that at least. I don't view it as creating much more problem, but again, I don't have such high stakes in like, okay, I'm not going to get tenure, et cetera. So I agree, making it more accessible.

If I'm not spending so much time absorbing an area, I can put it into GPT and then expect to, I mean, you guys have an even more powerful model, hopefully releasing, for other people to enjoy as well. I think the positive version of that is actually more people can participate in mathematics. It's like people might be coming with other intuitions and they could actually maybe generate good mathematics. Is that sort of closer to the vision of what you're hoping this is pushing towards? Or what things do you think mathematicians should be wary of to kind of adapt fast enough to take advantage of AI?

Yeah, I mean, I think certainly there will be a lot of changes, right? I guess in math, there are a lot of things that are kind of important for a given result, right? You need someone to come up with it, but you also need people to understand and absorb it and internalize it enough to do more with it and figure out where it fits into humanity's understanding, right? And a couple of years ago, proving the result was so hard that the other stuff was just kind of coming along for the ride, right?

You know, if you manage to prove this thing yourself, you're automatically going to understand it quite well. You're kind of responsible for maintaining it in some sense and explaining it to other people. And yeah, now this kind of what was the main bottleneck before is kind of much less of a bottleneck. And these other kind of constraints come into play. So yeah, the optimal structuring for organizing the knowledge could look rather different. Yeah. How does that look? I mean, does this make the field a lot more kind of empirical?

Will people do sort of the hard, like the first thing that was scarce, which is like all the reasoning and then more? I mean, not that it's like a bad thing to make it empirical, but it's almost like it functions as a very different discipline. Like a lot of the fun stuff is understanding, you know? And so understanding, communicating, maybe assembling, having still the human taste, does that sort of remain rarefied? And that's how, you know, current mathematicians need to adapt and reward, you know, contributions or is this too much of a caricature?

It's like something else. I think certainly understanding how to put, as we get more and more mathematics, put in like a proper framework and sort of how sort of like being able to explain it to other humans so that they can also appreciate it. I mean, so implicitly we valued this, but it was usually because you were the person proving the results that gave everybody else the understanding. But I think increasingly it would be a function of like you're sort of helping, you're the human who can sort of give this understanding to other people and sort of help them with it.

I think that more sort of that communal understanding will, I think, become, it was much more implicit in how we viewed math and gene, but I think it'll be an increasingly more explicit and valuable part of the subject. I mean, a nice thing about math is that the ceiling for difficulty of a math problem is pretty high. So even if, you know, kind of even if AI get, you know, continues getting like exponentially better at math, like it might, you know, plausible will never solve something like P versus NP.

And it could be that like the field kind of becomes more, you know, attached to like these big mysteries and less to like smaller mysteries that are more like routine now. Yeah, yeah. I think that's a positive vision of that. I mean, also like, I don't know, there are things I spent like months or years of my life wondering about, not getting to know. And now we get that. Yeah, some portion of them will get to know the answer to it. I'm pretty happy about that.

No, exactly. No, I'm excited about this like renaissance of results and understanding. And I feel like, I mean, this is such an infinite, you know, field, like, no pun intended, but like, it's just like, it's just, there's so much that you can actually create here. So, I mean, especially for somebody like me, who's not going to have the time to actually like practice mathematics. Now there's like a lot more that you can actually do in the activity of math. So, yeah. Yeah, I think the ability of someone who's not working on math is like their literal job all the time to like understand what's going on and like, you know, learn about some of the mysteries they might have wondered about will go up quite a lot.

Also, you know, if you're like, if you're working on something that requires some math, you know, suddenly you don't need to like find a world expert on this topic to be able to, you know, use it in your own work. Sorry, mathematicians. No, it's true. I mean, I think there was just like a dearth of actual like people who could do that. And so I think this is helpful. Maybe it's helpful for theoretical physics, like we'll see. But a lot of other applied areas as well.

It'd be nice for the world if applied mathematics went a lot faster. Yes. I mean, I'm of that opinion. Well, thank you guys for joining. This is a lot of fun. And I'm, you know, just so excited for how much the models are advancing. So maybe we'll have you guys back soon. Thanks so much for having us. Yeah. Thanks for having us. Thanks for listening to this episode of the A16Z podcast. If you like this episode, be sure to like, comment, subscribe, leave us a rating or review, and share it with your friends and family.

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