But it's for the first time now I think verified AI is to open up collaboration.
Either it's human-AI collaboration.
Well, before Blueprint, that's human-human collaboration.
And Lean was a grounding, was a verification, formal language.
And then human-AI collaboration like we're seeing now, future AI agent-agent-agent like collaboration.
Like I think verified AI is for openness.
It's not for meeting the requirements of closed industries.
And I think, just like I think verification should not be about.
Oh, I remember, like you know, there's this article like chatbots mixed up.
Oh, there's math solution to hallucination.
Verification to me is not about lousiness.
Verification to me is about scaling brilliance, compounding brilliance.
It's like just kind of going back to the collaboration point.
It's about Ramanujan being a much stronger mathematician.
He was already a really strong one, but verification helps him extend the brilliance, like both kind of like scale up and scale out.
Welcome to the LatentSpace AR for Science podcast.
I'm Brandon Anderson.
I build RNA therapeutics at Atomic AI and I'm joined by RJ Haneke, the CTO of Miraomics, working on spatial transcriptomics.
It's a pleasure to have Karina Hong, CEO and founder of Axiom Math.
Axiom has made a splash in several different areas.
First, they got a perfect score in the Putnam last December, I think.
They also had the claim of the first AI to prove research conjectures using formal verification.
And very exciting, they just yesterday announced quite a large Series A. Welcome to the show.
Thank you for having me.
You just raised 200 million which, as one of your colleagues said, this is like basically the entire like US math budget for math research each year.
Is that true, actually?
According to his LinkedIn post, yeah.
Okay, wow.
$250 million is our apparently annual math budget.
I think we should spend more on math research.
Yeah, it's kind of sad.
Yeah, I know.
But anyway, like, you know, as a, you know, as a nerd who loves math, that's like really cool.
But I mean, I'm just like, that kind of blew my mind.
Like, what?
Okay, so like, yeah, how is it 200 million, I guess, 1.6 billion valuation?
Yeah, I don't know.
Yeah, well, super, super excited to be here.
Also, I think like, You know, this is a series A, so it's a very, very interesting, timely podcast.
We're like a seven, eight-month-old company, so it definitely means a lot to us.
It's a really cool milestone.
We're currently about like 30 people now, right?
I think this amount of funding will give us a feel that it needs to accelerate the strong execution momentum that we have so far.
I think people think of us, there are many ways to think about Axiom.
People think of us as a mass startup.
Yeah.
Matt's startup, Lynn's startup.
The other obviously, things that we do that are formal verification.
We think verification is a really good, best first market format.
And so I think this fundraiser is going to let us explore some of the applied domains.
As my colleague CTO Shubo said in the launch video of the Series A, we had it.
Lets us broaden our dreams.
So, yeah.
But still like $200 million and I guess a $1.6 billion valuation.
How is there a market for that?
I mean, obviously you're not doing this just for the fun of proving things, although I'm sure there's a lot of that.
So let's just bring us back to 2024.
So when you know 01 reasoning models like just came out, what was Anthropic kind of like secretly working on back then?
It was coding.
And everyone knows they're working on coding, like OpenAI, Meta, Accent.
Everyone has full knowledge that Anthropic was working on coding.
And they just like overlooked it.
They thought, oh, they are at B2B plays.
They just want one vertical.
People think of coding as one vertical.
And now look at where we are today.
Coding, kind of like strong transfer learning from coding to reasoning to basically, you know, a monopoly in the future of reasoning.
And I think that's really, really shocking.
The people who are working on coding, I think back then believed in something that we believe you know similarly with math and lean now, which is that if you have more structured and formal data, it's going to be a lot more horizontal than the specific vertical we are tackling.
So you know, if today we are doing you know math in an informal way, like the standard chain of thought, data train, a math model based on human preference, then I would say well, perhaps we're just a math startup, right?
But you know, while we are pursuing math, we're also doing things that do have transfer learning to other domains.
So I think that's kind of like.
The broader picture is that, while the DNA of the company remains math and all of us are math nerds and this is a very strong cultural statement everyone has a great mission of having AI be a superhuman mathematician, like we're seeing on a batch of research conjectures.
In fact, we have another batch coming.
We're also thinking that this is going to be fundamental to verified reasoning.
And we kind of talk a little bit about verified AI.
I want to talk a little bit about verified AI next, because I think you have another.
Yeah.
I have several things I want to hear about the verified AI.
I do want to dig in a little bit.
Do we know that Anthropic and OpenAI and everyone they're not doing formal verification and using that for their rollouts and whatever?
I think I have a lot of rumor mill that I probably shouldn't put it on the record.
I think, you know, researchers talk, they play card games.
Yes, yes.
But there are really interesting reasons if they are or are not doing it.
I think that's like kind of the takeaway I have, which is that if you're like at a frontier lab and the direction actually does change a lot for lots of reasons beyond your control,
So I want to kind of like bring us back to the alpha proof moment, right?
Like alpha proof was such an amazing that really the 2024 28 out of 42 performance was the IMO moment for me.
It was not gold in 2025, because across 2024 and 2025, AI models could solve all the problems that are not combinatorics.
The only difference is that you know if you get all the problems that are not competent Rx, you get 28 in 2024 and 35 in 2025, because there's only one competent Rx question in 2025.
After AlphaProof kind of like, we didn't see a lot of the formal math you know, results or kind of progress from Google DeepMind.
That's actually because of reasons that are not necessarily technical.
But if you're at a startup and you have very singular focus, that is, formal math and verified AI, then you know you get to work on a really cool problem for a long time.
And you have like a lot, a lot higher likelihood to get to where you want to be in terms of like progress and breakthrough unlock.
So yeah, just define that for us.
Yeah, like a lot of people think about formal verification as an ancient you know subject.
Um, it existed like as long as you know, way before like like deep learning, and it existed in the time of rule-based computer science.
Uh, there's this really strong push of like formal verification around like since ever, since 1980s.
Uh, Really interesting historic anecdotes, such as, I think the Paris trade union demanded that the automatic switching of the subway system needs to be formally verified for safety purpose.
So quite interesting trade union for technology.
And I think around the time of Challenger, both before and after, European Space Agency was using formal verification for the Ariane spacecraft.
It's also interesting.
Boeing, Airbus, formal verification.
And then more recent years.
Right, I think there's a lot of push about automated reasoning at AWS because they have a lot of enterprise customers that really require things to be 100 verified and there's no edge cases missed and just general testing doesn't satisfy the need.
So a lot of people think about verification as something that's like annoying because it's like tax and compliance.
Like it's making sure that we are good to go.
Right.
And like that's really not the way.
And so we talked about like verification.
I think our competitor, when they launched, they talk about formal verification, pre-reasoning.
They talked about it in the time of hallucination.
And maybe for them, formal verification is about the lousiness, the hallucination.
For us, no.
For us, verified AI is about the brilliance.
It's about scaling and compounding superintelligence.
So this is quite a deep point, and sometimes it takes a little bit of explanation.
So, if you think about the place of brilliance, for example, Ramanujan he's a brilliant mathematician.
He was able to find a lot of interesting formulas just by intuition before he knew how to do proofs.
So he went to Cambridge. you know, work with Hardy and Littlewood.
And you know, in the famous movie The Man Who Knew Infinity, there's this like storyline of how hard it was for Hardy to force him to no longer rely on intuitions and do proofs.
After he learned proofreading, he came out as a much more powerful mathematician whose results, like intuitions, turn into theorems.
And future generations of mathematicians build on those theorems.
So it is a way to kind of scale and compound the intelligence that we already have.
Another example mathematicians kind of have been writing code in English or their respective country's natural language for thousands of years.
And why do I call it writing code?
Because there's this sort of community standard of rigorous logical deduction.
Everything has to be step by step correct.
Otherwise, you will get outcasted by your math community.
So it's a law.
Well.
Or rules in the community.
So it's interesting, right?
Because that is kind of human mathematician enforced, right?
And so it's a peer review process.
Peer review of a paper currently takes two years.
Okay so, but proof assistants and you know formal proof checkers like Lean still found its place right.
And why?
Like if you, if I'm a mathematician and you know any like my work can be peer reviewed by other humans.
Like why do we even, why do mathematicians even play with Lean right?
And why do we even like talk about kind of like you know, you know lean-based assisted, like serum, proving it's because like it handles a low level.
For example, we're not even talking about AI.
We're talking about, for example, the grind tactic in lean.
It can currently handle a lot of mass proofs. like at a very low level.
And this is pretty shocking because I have seen you know, actually another company working in the same space, like you know some of their demo.
And I look at the demo like it can actually completely be handled by Grind, which is a tactic in Lean.
Can you explain what Lean is to non-experts?
Oh, okay.
Yeah, yeah.
I think our order is like a little wrong.
Yeah.
So Lean is a computer program, a bit like for math proofs.
It is a formal language, just like its cousin, Isabelle, COQ or ROC and some other further cousins like Daphne, Agda, like these formal languages, whole sector.
And what does it do?
Basically, if you have a proof written in the programming link and then, assuming there's not any weird things happening, like unintended use of sorry, which is a tactic that lets you take things for granted, assuming everything is safe.
Hence people have tools like Comparator, SaveVerify and Axiom recently released rollout verify proof.
That's like 100 times faster than comparator.
Then you know, once you kind of execute that program, like once it compiles, and it tells you that it's correct, then it is, the proof is actually correct.
So it's like a type checker?
Yeah, that is based on this result called Carl Howard correspondence, which turns proofs into programs.
So I want to talk about the magic of Lean.
Why I think it's a really good programming language is because, on one hand, if you don't care about the formal part at all, if you don't care about the logic part, you just want to use Lean to write code.
You can.
Like we have had candidates actually currently, you know, the person is working at the Lean FRO.
He wrote AutoGrad in Lean in our interview process.
So it's a train-complete language.
That's right.
So you can write, you can do a lot of things with Lean.
It's a functional programming language, right?
And then you can also use it to, so you use it to do coding, you can use it to do math, two-in-one.
And kind of going back to what I was kind of getting at, if mathematicians are already enforcing that, most proofs you know say maybe not all mathematicians but the ivory tower people in academia, all proofs are correct.
Why do we even need linked model checker?
It's because Lean has tactics that help them handle the low-level calculation or proof or deduction, not calculation then for them to be able to navigate in a high-level intuition space.
So this is my point that it is not about like formal verification or verified AI to us.
It's not just about handling or like kicking out the lousiness, the hallucinations, the mistakes.
It's about scaling brilliance.
It's about super intelligence.
I actually, Terrence Tao has a great video also about using lean to as a way you can collaborate.
Exactly.
That's another point I want to I want to talk about.
A lot of people think about, you know, what is our what is our market?
It has to be some like really niche industrial societies area that is mission critical, safety critical.
No, that's not the TAM.
The TAM is all code.
The TAM is a right of first refusal on all AI generated code.
Like right of first refusal, meaning, you know, you get to choose whether you want to verify it.
So this is the important part I want to kind of come across, which is that people talk about formal verification as almost like painful because it has all these like stringent requirements has been.
Yes, yes.
And to us, it's actually verified generation means performance gain.
It means higher sample efficiency.
It means a startup like us with, like you know, still we raise some money, but lesser compute budget, lesser data budget, then Frontier Lab will be able to match and even exceed, you know performance on superhuman tasks.
In fact, for the pandemic exam that we just competed, December 2025, which we did in real time, Mass Arena, which is this organization that evaluates a lot of LLMs, found the best LLM.
DeepSeek got 103 points out of a 120-point exam.
The best human, obviously, we now know is a student from either MIT or Chicago.
We don't know which one because they don't announce the top five.
Winner score got 110 and we got 120.
So it's the first time, actually.
I remember when we were starting this, people were like is it even possible that a formal mass system with so much orders of magnitude last data can match or beat an informal LLM?
And pun them, it's the first time it beat LLM.
Right.
And so we're not thinking about it just about the painfulness, the challenges it posed.
We are thinking about the verified generation performance gain, the improvement, the fact that you can.
You know, just like you would expect RL for Lean to have improvement because of seeing evidence of RL encoding.
So this is the second point I want to make about like how to think about verification, verified AI.
So maybe we can talk a little bit about why.
Can you describe what is different about what you do versus what the Frontier Labs you know, at least when they're building their standard RL enhanced LLMs?
What's different about what you do?
Yeah.
So we heavily rely on a kind of data called lean data.
And we kind of talked about lean is all the data that we have.
That's lean proofs.
You know it's correct.
So you know it's correct or not.
And that's quite important.
So, you know, we have a system of models.
These models are post-trained and using RL or SFT.
So LLM, like some sort of foundation model that you get off the shelf and you post-train it, or continuous
Yeah.
And there's obviously an inclination for open source, you know, based models.
So it does speak English.
Yeah.
Probably knows how to code.
Yeah, but also you fine-tune it or continue?
Yeah, and the base model might be similar to what everyone else is using as well, right?
If they're not kind of pre-training their model.
Right.
And then we basically do this, you know, RIL for formal masks, kind of.
There's, I think, a standard... pipeline or like, you know, tricks of the trade that people use.
We try to innovate really on top of it as much as we can.
I think that we found scaling inference to have almost no wall, recursively decomposing you know, a proof goal into many sub goals and learning to backtrack as well.
Is there a risk that, like you start out with this, you know what you know and a certain domain of data sets and so on, and then you start rolling out recursively in a space but now all of your training data is localized in some domain that you It still is only so like, maybe logarithmically in some large space, growing from your initial training data?
So you could get trapped essentially in that you know you could be really good at this, but you just created a big, jagged frontier where some other domains are just far from them.
Distribution shift, are we talking about?
So you know, it is an open question whether a system that can do really well in number theory can do well in give me, you know, another field of math.
Yeah, exactly.
Well, actually, I think the way we think about it is it depends.
It depends on whether topology has a lot of the existing definitions as, almost like you know, the math infrastructure existing.
Because what people have found in the past is when people were building out math lib, like you know for the algebra, you know.
So mathlet being the lean like undergraduate library.
That's right.
So it's like all the proofs that you learn in undergraduate math.
Yeah.
And they're all sort of in lean.
Yeah.
Yeah.
So, for example, some of my friends who currently are at Axiom is, you know, crazy like full circle back moment.
Kenny, we're like friends for like, you know, five, six years.
And he was the first one to tell me about Lean.
He was working with Kevin Buzzard to build out MathLib.
It's a lot easier to codify algebra in MathLib than for analysis.
So that's interesting because for analysis a lot of the definitions around convergence limits et cetera becomes tricky.
And so I don't think there's a lot of like topology in math lib today in terms of like differential topology, differential geometry kind of stuff.
So you know, our system likely will not do very well on those domains because it doesn't even have the definitions to build off on top of.
For the places where the definitions are in.
We actually are doing quite okay in terms of distribution diversity.
We have good performance, you know, having solved open research questions and questions.
Number theory, commutative algebra, algebraic geometry, some discrete math that come into our exam probability.
So earlier you said that, like with the Putnam exam, the 2024 version, when all of the questions were that were not that alpha proof, did not get right.
The IMO, International Rhapsody.
Yes.
For the IMO, all of the ones that got wrong were in combinatorics.
Is there is there like a weakness there in that specific domain?
I would say so.
For Olympia in math, people are seeing combinatorics being a little bit more tricky.
Seems like the steps are quite creative.
So I'm a human and you know, when I have friends who are really good at combinatorics, which I never consider myself really the top of combinatorics
I'm kind of better at number theory, but I know some people who are just their IMO gold perfect score Putnam, fellow perfect score, and like all the way.
And then when they do like tricks and combinatorics, I'm like I don't know how you thought of that.
But, you know, after you give me that construction, it actually becomes a lot more trackable.
I think a lean-based system will struggle in those very creative ways places, which is why we at Axiom actually also invest on something called mathematical discovery.
It is not useless.
And we have some major news in the coming weeks basically open sourcing, entire code bases of mathematical discovery coming up.
You want to tell us a little bit?
Yeah, yeah, sure.
So we are currently having two code bases being open source.
So the goal is for if you're a mathematician or you're a theoretical physicist and you have a problem that you would like to solve.
For example, you want to find a construction that is a very complicated graph construction, then we would suggest you follow the very detailed manual intended for mathematicians to run the code that we write.
It's a tool for mathematicians to make mathematical discoveries.
Mismetal discoveries is this idea that proof is not enough for math.
In fact, before you kind of start proving something, you don't know where you want to start.
So you will try to construct some interesting examples.
This can be usually, say, sequences, right?
If you want to understand the property of a sequence, you will write out a few of the first terms.
This can also be graphs.
So if you want to, you know, figure out what the graph that you're looking for should have say, a certain property, then you will start by doing some simpler version of the graph.
Now, constructions cannot be done by lean.
So we believe in having AI for mass discovery.
And we have, you know, one of the OGs in that field, Franรงois Chartrand, a member of technical staff at, And he previously have done pattern boost and end-to-end, you know, set out this proof a 30-year-old conjecture.
By finding a counterexample found the solution to a 130-year-old problem the global lampenov function.
That is a kind of mathematical object showing up in the three-body problem.
So we are thinking that you know, mathematical discovery tools should be open to the mass community.
So we are open sourcing entire code bases for that.
So discovery, meaning it makes new conjectures?
Yeah, it's a pre-conjecturing step, actually.
Okay, oh, I see.
Yeah, so you start to form intuitions, right?
If you're a mathematician and your goal is to solve a really hard, hard conjecture, Action Prover can't just solve it for you.
You might want to try to formulate some sort of lemmas, conjectures that you want to say then give to Action Prover.
If you're a human mathematician, you will start by wanting to formulate that conjecture.
You don't know where to go.
You want to find constructions.
Now, the code bases that we're going to open source is going to help you hopefully significantly.
So one thing that maybe there's a lot of computer scientists listening and one of the things that will immediately kind of come up, especially when you're talking about formal verification and so forth, is Rice's theorem and decidability and incompleteness theorem and maybe some arguments about computational complexity in LLMs.
So I'm curious to hear.
Rice's theorem says you cannot prove nontrivial things about programs for all programs.
Right.
So how are you navigating this space?
Obviously, formal verification, you know, does is able to do some things.
Yeah.
So yeah, I think like it's very clear that you just like there's theoretical result telling you you cannot formally verify all programs, right?
But I think it's good to formally verify a majority of the useful programs, right?
So you know, like I remember, there's this MIT like little, like documentary or another documentary, like an advertisement for, you know, people who are admitted students.
And then there's this famous line by Tame the beaver, the mascot of MIT, saying that what does theory give you?
Which is kind of like, it doesn't stop us from trying to push it as much as possible.
So the goal that we have for the future is suppose you are, you know, doing the coding.
You want to wipe code, a really complex task.
So you know, currently it's front end websites, but in the future we might want to wipe, code much more complicated things, whole distributed systems.
Even then we want to be able to say decompose it and there's maybe a high level kind of like sketch plan this we can make, other people can make.
But say, you know, you have Claude give you like, you know, kind of break it down into 10 things.
And then at one point, it will decide to call Axiom.
And Axiom will give you a computer program that you know is formally verified.
Or it will say, this is still too hard for us.
So you write the program.
Yeah.
You give it to Axiom.
Yeah.
It makes changes to it, maybe.
So we're talking about kind of two sort of phases.
It is possible that we are the verification partner.
So you already have a computer program and you want us to verify it.
In fact, like, you know, GPT found a proof to an unsolved Erdos problem and our competitor Harmonic, you know, Aristotle, you know, verified it.
But we can do, we want to do verified generation, right?
We might want to say hey, you know this little component, everything that we generate and provide for you is formally verified.
I see so.
So the idea would be you, you generate, you co-generate both, and so that and i can imagine this fitting into um, you know the idea of a promise or a sorry sorry, and then Sorry.
Sorry, meaning it's a lemma that is unproven, but you're just taking it as given until you can take.
Have the time to prove it.
Right.
Is that a good way to think about a sorry?
That is a good way to think about a story, but not necessarily in the coding context.
So I can imagine you can say, assuming that this module is verified, then this module is correct.
And so that you can decompose a problem small enough that you can verify?
So let's say we want to, you know, like, wipe code control flows.
Right.
That's quite hard.
You will likely, you know, break that down into multiple steps.
And then it will continue to break down these steps into more fine-grained steps.
And at one point, you want something that is absolutely correct.
And then this is also something that is likely within reach.
Then we want to generate, you know, both.
We want to generate a piece of computer program and underlying.
Lying is a guarantee that there is also the proof that has been generated which tells you that the thing that you specify, this program, can solve for you.
Yeah.
So the vision we have is anything that can be, which anything is.
You know, it's a little bit marketing because, as you said, it's a radical bound, but mostly well, almost surely, hopefully anything that can be defined can be executed.
Anything that can be specified can be proven.
So the way I think about it is if you have a program times a, you know a program times a statement or problem, it maps to verifiability conditions times a proof.
So, while the program verification community has given you say, the verifiability conditions, and we're trying to kind of recruit a really strong team to help us do that ActionProver is going to give you the proof.
So just help me map from...
The program to the proof because, like I could say, you know, this two line lean program verifies, you know, sort of like whatever, whatever I claim it solves.
How do I know that it actually verifies the thing that I think it verifies?
Yeah.
So for example, there is this, currently there's this benchmark called CodeVarina.
It's a code verification benchmark that's supposed to be limb friendly.
And so, you know, every problem is a coding problem.
And the goal is to generate there's a code part and there's a proof part two different computer programs.
Yeah.
And then the goal is to generate code with proof.
So you know the code that supposedly solved this problem, and then the proof that this program indeed does solve the problem.
I see.
Now, how do people do on this benchmark?
I kind of want to like talk about this a little bit because it's interesting.
It was wrote out, I think, by Berkeley and Meta researchers in 2025.
And they found, I think whatever version of GPT they evaluated does like pass one like 36 iterative, something like 22.
Now, you know, how does the formal math systems models do?
COBRA, which is a system, because in a system you iterate and define, so pass.
One doesn't quite work, but still they evaluate it.
Pass one of the system about like, I think 11, 12%.
And then also DeepSeq Prover and Godot Prover, very strong.
Prover model, 11%, 12%.
And I think our competitor has released last year the only proof part, 96%.
And we actually recently, with no modification to the pandemic system, we saw a 99 out of the 189 problems we solved, 187.
We missed only two.
Code was proof.
So if you want to train something, to do co-with proof and you want to do reinforcement learning, it's actually quite annoying because look, it's mixed.
If you want proof to be informal math, It's very annoying because then that just makes objective function.
Your code is something like Python.
Your proof is, say, natural language math proof.
You will not have a strong RL kind of performance, right?
But if you have proof as lean and you have you know code you can choose Rust, which is a strongly typed language.
It's more, it's more conversion.
So you're going to have much better performance.
I can't wrap my head around.
How do i tie?
So i like i can say that this proof solves fermat's last theorem right yeah, i don't know that like yeah, but it's two lines in lean.
Obviously it doesn't.
So how do i know that the program that i wrote matches the proof that i generated?
You will basically look at the coding problem and you look at the program and then you try to see if it satisfies the verifiability conditions.
But like, how do I know?
Right.
Like if I read it.
Right.
You know, like I can I can do is like eyeball it and I can say, and then like traditionally, how mathematicians have done this is they they.
You know, they take the paper and they read it and they say I agree that this proof solves the problem.
And this other person says, no, wait, it doesn't for, you know, like, look at this.
And then people disagree.
And eventually there's consensus that that like this proof solves this problem.
Yeah.
So like, how do how are you?
But you check it step by step.
Yeah.
So you basically will look at the verifiability conditions and see if it does actually satisfy that.
So suppose, suppose like we're looking at like, you know, a piece of computer program.
Yeah.
Right.
And then whether it does actually solve the coding problem.
You will have a judgment about that right.
Yeah.
So you will not solely rely on testing, even though that is the way.
So somebody looks at the proof and says yeah, that actually solves the problem that we think it's supposed to solve.
But then now you're basically producing a you know, formal verification program that satisfies the verifiability conditions about this program and this statement.
So again, the function is taking you from the program and the statement to verifiability conditions and proof.
Okay, so I can see how this works in a benchmark.
Then if I have, let's say I have a flight control system that is like very, Then the problem becomes very annoyingly.
You know the specification.
I think the word is gonna.
You know, even if we say successful, like anything that you know, we will have a specification problem.
Yeah.
So, like, here comes a bank saying that, like, please, do I have a really safe financial audit?
Sorry, like, prove the financial audit for me, right?
Yeah.
Like, What does that mean?
Like we can't specify.
Humans are bad at specifying everything that we want.
There's always like some sort of saying that we are not specified.
And if it's not specified, it's not proven.
Okay.
So what do you do about that?
Yeah, so we're not there yet.
Okay.
Currently, you know like again, the vision as of currently is anything that can be specified can be proven.
Okay, yeah.
Now, obviously there are people who have been really good at you know maybe where that's the informal kind of reason they come in.
The informal reasoner can and this is I want to kind of you know, call the literature of testing like testing are great, because testing is like hey, have you thought about that?
Right.
Like I want to highlight the work mutation based, you know, LM unit test generation by XMCTO Shubo, and he was that a director of Facebook.
AI research, like the way you kind of think about it, is like the AI will be like hey, have you thought about this case?
And so this is a little bit like conjecture.
So the conjecture is going to help with the specification.
I see.
And then the prover does the proof.
And so this is an interactive process.
Maybe that the person, so that when we're actually giving good specifications,
I think this is a future of coding.
Yes.
I think this is a future of coding.
And I think this is where you know, this is where I think, even if we are supposed like, given the assumption that everything can be formally verified, you know, like studying sort of like.
You know, automatic task generation is still interesting because it is basically giving you the specification proposal, right?
And then another thing is, let's talk about auto-formalization, right?
Which is the ability to define it.
It is kind of converting something that is more informal into something that is more formal auto-formalization.
So suppose I have a coding problem that is written for ICPC and this problem is written in English, like Alice and Bob blah, blah blah.
Okay, now I want to convert that into a formal statement, like a formal spec.
How do I do the auto-formalization step? right?
Now, this is going to be, because I have not solved the problem yet, so I don't have any signal, I don't have any grounding, the test cases in put-out-prepare is going to ground my yeah so I know I have to know I'm going to give this input I'm going to give this output it has to have these characteristics and so and so I write test cases and I write So is there an equivalent in Lean of this, right, where the specification where you just know the sort of like outcomes that you are expecting so that you like the statement of the result and then but the proof is completely unproven?
So Lean is actually quite annoying because it's like a lot of the times it's proof.
So you don't actually have the numerical answers to ground it.
Okay.
So auto-formalization is quite a hard thing to do.
Yeah.
Because you know what's generally happened is you, can't you?
Just it's hard to ground the auto-formalization of a statement.
You can obviously ground the auto-formalization of a proof, but because you can then just run it.
But you need human to eyeball it.
How big is a lean proof of like a formalized, you know, of a formalized program of significant size?
I mean, do they grow with the size of the program or do they grow super linearly?
Yeah.
Currently, actually, you know, for each line of code written, there could be like 20 lines of proof.
Okay.
It's not looking that great.
But is that like a linear relationship, or is it as the complexity of the program gets greater than it like it, you know, sort of also grows?
I don't have a good answer to the scaling law.
Okay.
Yeah.
Because I know that that's a problem in formal verification.
That's right.
Right.
Where you have these huge like, you have to have these very, very long proofs for even simple programs.
Yeah.
So then, are you going to run into sort of like limitations in in the capabilities of LLMs when you start to get to large, larger?
What we believe fundamentally is we are building a reasoning engine.
Mm-hmm.
And we have seen Axiom Prover deal with really huge trees that are like, you know, tree of a proof.
We have seen it scale from 40 nodes to 4,000 nodes.
So what you're saying, Axiom Prover is the LLM?
Action Prover is an ensemble system of multiple models that we do post-training.
I see.
Okay.
And it also includes, obviously, the tools that we have open-released.
Sorry.
Yeah.
So we have seen it being able to deal with more and more complex tasks.
I see.
We don't think it's perfectly bound.
You could ask, you know, is it bounded at one point on the pre-trained base model?
I think that's a good question.
I think, you know, mid-training could be very interesting because it does actually...
You know, a lot of the sort of capability gain does come from that part, right?
You could argue that even if you try to reinforcement learn some person who is not very talented, that person might perform a lot less well than an un-post-trained Ramanujan.
You can argue that, very sad reality of things.
So at one point, we might consider doing that.
But we think there's so much to push.
So you just feel like there's so much overhead right now, or so much space to glow, that you're not running into theoretical constraints at this point.
I just wonder because you know there's been recent results in the computational complexity of the problems that LLMs can solve fundamentally.
And I don't think that they're really a concern, for you know, when I'm writing code with Cloud Code.
But I can imagine problems becoming big enough in a system like this where you have a gazillion lines of lean.
You can't keep them... get them into the context window.
So you have to like be smart about that.
And then you have to summarize, and then you're summarizing and summarizing and pretty soon you're like kind of losing track of what's going on.
And it just seems like with a very large system like that, you might run into it.
Yeah, I think this is interesting.
It's always a problem of abundance.
So simply you just like keep really the mathematical discovery.
Renaissance has come.
Action prover does try to prove everything.
You end up with like tens of thousands of lines of limb proof.
So, first of all, auto-informalization is a lot easier than auto-formalization, minus the problem of no grounding right.
So, you know, every model has seen a lot of text and a lot of lean.
So you can always, you know, convert that lean back into informal.
And then there's the problem of, well, how do you know if you're correct or not?
You can rely on cyclic like, So you then formalize it again and like prove like program equivalence, something like that
So that's... Oh, so you, like, informalize and then formalize.
Yeah, you can use it to ground.
You can use it, yeah, yeah.
Like, and although informalization is, you know, obviously less hard a problem.
So you can always do that.
So for a lot of the you know the link code that we output, we can have an informal summarizer of like big chunks of links.
It's actually doing okay.
So, you know, that's the thing.
And there's another question of like which I think is very interesting is I think there's a panel at ICML Vancouver last year at the AI for Maths workshop.
There's like Leo DeMora, Jeremy Avigad, and Shubo, and CTO was there.
And they were talking about like will humans or mathematicians at some point stop trying to understand what's going on there, right?
Because like, suppose you're a really ambitious mathematician.
You're like, I want to proofread my hypothesis.
And bang, here's the lean proof.
And like, it's actually correct.
And it's just like, you know, problem, one million lines.
Yeah, isn't that like a big negative for the community?
Because I mean usually, when someone comes up with a big proof of something, oftentimes the process is massive.
Yeah, I was about to get there, right?
It's like, well, will that negative outcome happen was the question the panel was discussing.
It's completely hypothetical.
No one's, like, you know, model system can prove remote hypothesis, right?
So disclaimer, please don't cut that part.
Just stand alone.
But, like, you know, were people still trying to understand what's going on?
And I think the answer is usually, is always yes.
I think curiosity and the desire to understand what is going on you know, mathematically or in other domains as well, is a basic human need.
And I think that is like, I think, a dose of optimism in an era of, I think, verified superintelligence.
Suppose we get there is that even if all the outputs are going to be produced and at a much faster pace and much more exponential volume compared to what humans could possibly consume, they're still going to try to consume it.
And they're still going to try to consume the ones that they deem important.
So then basically attention is the bottleneck.
And if attention is the bottleneck, then really intuition and taste.
You know of which statement is probably worth the consumption of human
And also maybe in a finite computer resource worth the consumption, worth the consumption, the sort of spending of compute resources.
That's where human mathematicians' taste will always guide us.
And I think that's incredibly beautiful.
Is it worth like internally, taking like results so you can prove one way, and then trying to send your system at many different routes to get like orthogonal, conceptually orthogonal proofs?
And so you kind of get a diverse set of different proofs ways of reasoning about the same thing.
Because, you know, I think it could be very valuable if you give it a problem to say, oh well, like here's kind of the brute force natural way that, like maybe some humans would do it.
And then there's, like, a really much shorter elegant way of doing it.
So have you essentially thought about training your models to be elegant in some way?
Yeah, at one point we're going to get to there because, you know, I think the conjecture will probably depend on what you know, will probably depend on what we mean by taste.
Elegance Feels like an alignment problem to me, you know, like you know who gets to say what is elegant.
Humans get to say what is elegant.
That's what makes it fundamental, right?
There's something about hard work, right?
What you work on hard is what you're going to be good at.
Yeah, yeah.
And we're going to have a problem about that, I think, like pretty much in a lot of the domains as well right,
Not just math.
Like, how do you be that senior programmer with, you know, really good high level understanding?
Well, I guess full stack understanding high level and low level if you haven't spent the year of training.
I mean, I would argue that you don't.
This is very philosophical.
But, like, you know, I don't need to be good at assembly language programming, right?
Like, not many people are good at that.
A few people are because it's important for their job.
It's not experience, but curiosity.
But it feels to me a little different because not being good at like proving things, for example right,
That seems like a fundamental gap in like that.
Maybe my mind doesn't develop in the same way if I am not doing that.
Whereas if I'm just not good at assembly language program well, but I'm good at like higher level programming.
So maybe that doesn't matter.
I think that's probably because how the, Maybe how the education system, the pipeline works, which is that if you do not show early signs of brilliance, you don't sometimes go through the process of pre-training in math.
Yeah, yeah.
So that maybe you can argue that you don't need to say you know, learn everything to develop a sense of taste.
But there's like a threshold you kind of need to meet.
Yeah.
So, for example, you probably need to be able to code even if you don't need to understand assembly language.
And that thing might transfer my intuition or, you know, my intuition might transfer from the Olympiad math problems into some other research areas.
And I tried to pursue.
And Kamalantara's transfer is more direct, very similar, and number theory could be further, but still okay.
And then when it gets to like something that's a lot more different than olympian math, it transfers that strong, but kind of like you know you need to diligent, as you said right, like you need to diligently go through some amount of training yeah, and if people over rely on strong ai, that doesn't happen.
I want to switch gears.
Yeah.
You mentioned software verification.
What are the domains?
How are you going to make enough money to justify the valuation?
And congratulations.
Thank you.
Yeah.
So what's the?
Give us the high level summary of like.
What is the?
What is the vision that you show, you put in front of investors about?
Why does this actually make a lot of money?
Yeah.
So first of all, this round is kind of preemptive.
So I think a lot of the investors have pretty high interest about Axiom.
In terms of kind of what we believe in.
We believe the future of coding is going to be somewhat constrained by verification capability.
Mm-hmm.
And we believe in solving formal math is a very natural starting point.
And then by extension, you can increase the verification capability across hardware and software.
And for hardware, for example, that's quite revolutionary.
I mean, there is no, as we know, there's no partial credit for a mostly verified GPU.
No.
It's all or nothing.
It is all or nothing.
And you do need a perfect prover.
I want to stress this point, which is that suppose I am someone who loves solving math.
I think there are a lot of Twitter users who enjoy Pokemon hunting or those problems.
And then I just try to, you know, use a non-deterministic LLM, like TPT say, to try to get the full proof for that.
Yeah.
Now I can... do that many, many times.
And I might succeed and I might not.
And I might not have a problem with whether I actually succeed or not.
This absolutely does not work for hardware verification.
So For those kind of domains, which I call like hardcore verification needed, it is a pain point.
It is a current pain point.
There are hundreds of humans and thousands of licenses being dedicated to solve one local grid problem.
Verification
Just as an aside.
My understanding is that the industry standard for design to verification in ASIC project is like one to three, one to four.
One to three, one to four, correct.
Both in, say, team size and in duration.
Yeah.
Right.
So if you... apply that yeah square and then I think so that's that's a I would say like you know it's a it's a must cover and now for the software verification it is it is interesting right because you know as probably we all realize like my nephew vibe codes a lovable website and There is absolutely no need to formally verify that piece of code.
Like why would you?
Now I heard a story from K-Math actually that New York Times reporter who told me the story which is like however, if you think about, like you know, in the time of agents like my open claw can probably do all sorts of things and probably can do some bad things.
Like, my open claw can decide to, like, tag something bad to my professor, right?
Yeah.
And you can say that perhaps is that a problem of formal verification?
Probably still not, right?
You can change something about the action space and make it more limited, so you don't need to rely on formal verification.
So you can have a lot of cases.
But you can think about, you know, maybe an enterprise that is dealing with a lot of regulatory kind of stuff using agents.
They might want to do something like It's their choice.
But I will argue that the improvement of verification capability, both in latency, you know, and in accuracy, all these stuff, the performance holistically, is going to determine whether people rely on formal verification or not.
Sure.
So in a way, we want to make it so good that basically we can make that a choice.
So why did the investors think that you could do this?
Right.
Because, I mean, people have been working on verification for so long.
And I think everyone agrees it's an important problem.
And I think certainly if I can just have a verification proof for every program that I write like hey Claude, like give me the proof also.
And then it just... produces it and it looks good to me, I would absolutely do that.
But so why is it?
What was it that the investors saw in your opinion that persuaded them that okay, this is the moment I'm going to put in my 200 million, or whatever?
I think when it comes to faith, you either have it or you don't.
So you either dream the dream without it or you don't.
And that's okay.
Because when we realize the dream, the company is going to be worth $10 billion.
Yeah.
So I think that's kind of the feeling that I have, which is that we believe verification is the critical, critical part to superintelligence.
Our version of superintelligence is absolutely verified.
We don't think there's any other possible future.
We do not believe that I'm going to say on the record.
We do not believe that an informal mass system is going to be the mass AGI solution.
Why not?
We just don't believe that.
I mean, the counter argument is oh, you know like we just do a lot of good RL and you know we've seen GPT, you know solving, you know, I think, some of those problem and like whatever.
So why do you think that that runs out of gas?
Yeah, so you can say that if you're a frontier lab and you have infinite resources, there's by definition no running out of gas, right?
If you think infinite means there's no running out of gas.
I don't think it's going to scale to superintelligence.
So you think that you run out, like you run out of money, basically, or run out of power.
So we as a startup, first of all, cannot do that.
We, first of all, as a startup, cannot do that.
But we generally think that formal math and by sort of converting math proofs to programs to code give us much better performance.
So it's just, it's your sample efficiency argument and so forth, that you just, and maybe you just can't, that you can't bend the curve enough if you don't use formal solutions.
The thing is, the informal stuff is also available to us in a way.
If you really, you can have both informal and formal system.
And that is going to be very strong.
The thing that I kind of like I think my suspicion about, like you know, whether we can scale to mass AGI just by the informal approaches is you're going to keep having, you know, the LMS judges solution or you have human experts who grade and It's just human experts.
Like doesn't scale that well.
And if you really argue infinite infinity, then sure, then you also have infinite money and you can pay infinite.
There's so many.
Is there really infinite number of people who can understand and prove at say, like about, like?
You know, a result, a natural result in Langland's program.
I think, you know, good luck finding those people.
And in fact, I think how Frontier Maths came together is because they couldn't assemble a benchmark.
They are expert pool, so they have to, you know, collaborate with Epoch to do it, right?
And I think that's kind of what I worry about, about having the human part.
So they have LMS judges and then now stochastic judging.
The problem is like whether something is impossible to achieve versus something is incredibly expensive and like really incredibly expensive and incredibly expensive to achieve.
Get kind of like mixed in the end.
And then, of course, investors always want to know why you.
Right.
So I've read a little bit about your background and I think we would do a disservice to the audience if you didn't hear a little bit just about your personal story.
I see.
Do you want to talk just a little bit about like you've done some really interesting stuff.
So I'd love to hear like you and then your team.
What makes Axiom special?
Yeah.
I think Axiom is very special because there are really expert mathematicians.
Basically, they are users of the system we are developing.
And that iteration loop is very fast.
It is extremely fast.
You have some of the strongest mathematicians.
And both in research and Olympia contest.
And you also have people who are, you know, mass lib contributors maintainers developers, link gurus, really.
And combine them with people who come from like applied ML, really strong organizations like MetaFair and Golden Age Affair, as well as people who have co-gen expertise, who work with like compilers, like Kernogen.
Have kind of these backgrounds of people together.
I think that sort of interdisciplinary way of thinking about things is quite helpful.
We think AI for math has traditionally been quite interdisciplinary.
People are borrowing techniques from even AI for science.
People are borrowing techniques from physics, from the co-gen literature, and people are borrowing techniques from obviously the broader, like you know frontier, like applied ML, to try to apply on the niche problem of AI formats.
So we also think having this sort of very special, special team is a differentiation.
We also think that you know, as you say, there's no permanent mode.
The proprietary data that we generate and a little bit of a flywheel we are seeing is a time mode.
On me personally, I love math.
I think You know I kind of have been doing math since I was very young and, like math, sometimes gets really hard when the problems you're solving are just a little bit out of reach and it gets a bit depressing.
And times to times I wonder if I can just have an AI help me.
And yeah, I think I figured why not build such a thing.
You did a master's at Oxford in neuroscience.
Yeah.
Has that informed your thinking here?
That's a great question.
I think like my experience with neuroscience is you learn very well about what's hard.
What's impossible?
I mean, it's very interesting.
I think that year of neuroscience gives me some feelings about what's hard and almost no feeling about what might work.
Okay.
But I think I was kind of under the pretense of neuroscience like hanging out at the UCL Gatsby Institute and was fortunate to do AI research with some really cool faculties.
And so I think that was a very productive year of AI study.
So it was mostly for you studying AI.
That's right.
That's right.
I think in the UK if you back in the 20th century, if you call something AI, you will not get the donation.
But if you call something brain science, you might have a chance.
So the UCL Gatsby, which is a premier AI hub where a lot of people actually go, you know, from there to DeepMind, including Demis himself.
It's a very wonderful research environment.
I remember those kind of like tea time talks were very amazing.
And people were basically just doing AI.
It's called the Gatsby Computational Neuroscience Institute.
Yeah, I think how that kind of you know happened was because so I was in the Master of Neuroscience program and then quickly realized that you need to like kill rats and kind of don't want to do that.
And computational neuroscience sounds more appealing.
And when you look at the project and you see, like Transformer, you're like you absolutely want to do that.
Yeah.
We're all excited about that.
So after the Gatsby, you started a math PhD program at Stanford?
I started actually one year full-time at the law school.
Oh.
Because the JD-PhD program is structured in a way where you have to spend one full residency year.
So that was also a very fun year of learning things that are just quite fascinating, like criminal law, looking at homicide cases.
Exciting.
Do you ever feel like the legal system is under or over-specified in some way that maybe you could acclimatize and improve?
That's a great question.
I think for a lot of things, it's definitely under-specified.
For some other things.
I was actually quite excited about sort of transfer learning from mathematical reasoning to those specific things.
I think appellate litigation, the legal gymnastics.
You see some really good appellate scholars and lawyers that just come from mass training.
Not many, but like Lawrence Tribe for one, you know.
Harvard law professor um one of the you know strongest like um you know appellate litigation and SCOTUS briefs like brains on the left uh, democratic party.
Um and uh, And I think there's a lot of other domains such as antitrust.
That's incredibly flowcharty.
Contract law, sometimes also flowcharty.
Bankruptcy tax, more on the corporate side.
I just love litigation side.
Yeah.
Yeah.
So actually, just because we're talking about litigation, it's not the same thing.
But there was a Erdos problem that Axiom saw.
I don't know if it was Axiom Prover or whatever.
Is that right?
There was a controversy about it because it had represented that it had solved the problem, when in fact the proof had been.
It had discovered the proof and then just formalized it.
Yeah, so actually what happened was our competitor harmonic, decided to publicize that they have solved unsolved problems um erdos, number uh, 124 and 481 and then we trusted their literature review, believing that these problems are really, truly unsolved, and We were a really young company at the time.
We wanted to test if our system can attempt to try the problems that our competitor can.
We fully did not expect that it actually solved them.
But it turns out that we were both wrong, that in fact, the problem has been solved before.
I see.
It's not the only time that we relied on others' literature research.
And, you know, we wish on it.
The other time was this paper called Dead Ends in Square Free Walks.
You know, Professor...
Professor Miller have this problem that actually turns out to have been solved.
But we, I mean, we really should have done our part.
That is.
You know, The point I'm trying to maybe elicit is not like you guys did something wrong, but rather You know, there's this like Japanese, like advertisement of, like a whole company, like hundreds and thousands of people, like apologizing in the advertisement.
It's like, you know, sorry, we raised our price by like five cents and That's the advertisement.
I was like thinking that maybe I should just do that.
It's so embarrassing.
No, but I think that the question of provenance of information and sort of like, how do you?
It goes back to the question I was asking before about like, how am I connecting the answer to the question?
Yeah, this is a great question.
I think after the Erdos thing, we're like extremely careful.
And so we kind of like, you know, we didn't really look at the other Erdos problems.
I believe that Harmonix still continue to claim they have solved Erdos problems that might exist.
Might not.
I don't know.
You know there's a.
I think Terence Tao and a lot of other people have a database about all the Erdos problems and the status.
I think you know like it is really, by the way, like it's a really easy mistake to make because there are so many Erdos problems that actually have been solved.
Right.
And I think that that's kind of interesting.
Indeed, I think like, you know, search and retrieval is a hard problem.
Like you don't know if that argument or an equivalent version of that.
In fact, I think the most interesting part about that entire database is there are a lot of problems that are not directly self-solved but can be just An very easy extension, almost a trivial extension, of another result that has been solved, or sometimes not even resolved.
Sometimes I think in this dead-end square-free walks case, which is nothing to do with harmonic complete axioms, fault that we actually didn't realize.
And Professor Kondansandararajan actually pointed to us and to Professor Miller is that it was actually from a stack mass overflow or stack overflow post.
Like a user pointed out that there is a 1936 results.
It's fascinating.
I think it's hard to find out.
Why search is a hard problem.
I guess that means that you do.
Does the conjecture engine or whatever does that, use search as part of its process?
Or is that something that you kind of, the human does and then feeds it?
I think knowledge graph or knowledge base is a very, you know, important component of any company.
Yeah.
And I don't think it's talked about enough.
And you guys?
It sounds like you don't want to give us too many details, but so you guys have a knowledge graph.
I mean that brings up also.
I read somewhere that you guys have a really massive database of lean proofs that you've generated.
So synthetic data in some sense, but this maybe is a competitive advantage for you.
I think everyone is trying to accumulate like a data, which is not a mode.
It's just time and time mode.
Yeah.
It's all about, like you know, whether you can execute fast enough to make sure that you have like, a certain buffer because of say, your data set.
You know accumulation.
But that is only just a buffer.
Have you ever thought about doing something like an alpha zero for math, where you start from nothing and let it just make up axioms and see what happens?
This is a wonderful question.
I think that's a very interesting approach, actually.
Yeah, I think we believe in something which is that, like you know, suppose action prover can be a really strong mathematician and then really the thing that it is proving every day should hopefully help it improve right.
I think this sort of self-improvement design is extremely valuable.
And I think there are other people in the AI for Math community.
I think Professor Gabriel Praja's work is very interesting.
I think there are some of the kind of more conjecturing type of exploration.
Suppose we just kind of change.
You know a lot of the.
There are specific things you can do in certain ways that can try to see if your system can learn to conjecture and build theories.
I think that the topic is really interesting and important because it really You're claiming that to get to superintelligence, there's sort of this like
It's just not going to be possible.
Maybe if you had infinite resources, you could just RL and it would work, maybe.
But the reality is is that you just can't be sample efficient enough or whatever it is to do that, so that you need some sort of verifier in the loop with the inference process, rather than Because you do have verifiers sort of during the training process and you just don't have them during the inference process.
Yeah, I think a lot of them are just secretly like...
Trying to use this to ground their reasoning.
Yes.
I mean, I would.
I was surprised that, like when O1 was, you know, everyone knew O1 was coming, but it hadn't come out.
I was sure they're going to announce that they're using Lean to do like formal verification of proofs and actually generate proofs and then verify them so that they're grounding in reasoning.
I mean, that was my intention.
When Ilya was there, there was GPTF.
That was a great piece of work.
There's also MiniF2F.
These are all formal math work at OpenAI.
Okay.
So presumably those guys are doing something.
No, no, they all left.
Oh, they all left.
So that's my point, which is that if you are, like you know, an intern, I guess you can't be an intern forever.
So let's say you're like a junior, you know, like member of technical staff, and you want to work on something for like as long as it takes to solve it.
Weirdly, people think about startup as this sort of your runway can just run out and they can just like all fall apart thing.
You might have a better chance of staying focused on the same problem for as long as it takes at say, a startup like Axiom or one of the other new labs.
Yeah.
If you're aligned to the mission of the company rather than, like somebody, decided that what you're doing is no longer.
Yeah.
Yeah.
It can be your VP lost some political fight.
And so, yeah.
Yeah, absolutely.
No, obviously, if we succeed, then they're all going to, you know, start doing that again.
Yes.
And then, like I guess, as a talent, then there are more, like you know, potential places to choose from as well.
Yeah.
So then your job is to go fast so that they're struggling.
So actually we haven't talked about it, but you actually also just released an API for doing lean verification.
Yeah.
And I actually tried it with Cloud Code because it's easier than setting up.
You know, your own lean tool chain.
Yeah.
And, you know, like tried to get lean to prove some stuff.
Yeah.
And the infrastructure is maybe non-trivial, especially at scale.
So you want to talk a little bit?
Yeah, yeah.
So we just released Axle, A-X-L-E, stands for Axiom Lean Engine.
And it's really a set of kind of proof, validation and manipulation tools that are built for Lean, in the language of Lean.
So it's a bunch of metaprogramming tools.
Now, metaprogramming talents are extremely, I think, like you know, hard to find and We're so grateful to have a really crack team working on that.
We want to release it to the community to use for free, because we think that there are probably other people doing also large-scale lean operations and these tools are going to make their stuff go a lot more robust and faster, and do so at scale.
And Excel is currently, I think, 14 like such tools, starting from verify proof, which is the sort of to make sure that there's nothing weird, you know, going on like no sort of cheating by link code.
You don't axiom something out.
You know, you don't assume weird things.
If you axiom n plus n equals n, you can prove 2 plus 2 equals 2, which you are, you know for sure not.
That's not the right answer.
There are also like, you know, a lot of other kind of generation tools.
For example, you can try like different repair attempts.
So, you know, broken lean in and then good lean out.
And, you know, there are currently other repair methods by ILM.
So hopefully this what we provide can be just a lot cheaper and more kind of straightforward and it's just, you know, straightforward.
I think strong and better engineering can get you to a place that's quite far.
A lot of the people from the Ling community has been using Excel even if it's just been a week to do all sorts of different interesting things.
We have seen People from the kind of blockchain community use it to do interesting things in sector.
And we have seen also we have heard from a lot of the people that Cloud plus Axel is kind of their go-to setup for now.
We think that these are really interesting tools.
I think famously.
I think today there is this mathematician who said he formalized the download news system, you know, using cloud to prove, I think, a result, a Ramsey result, and then to formalize the lean proof.
And then that is also using XO tool.
So we're really glad to see people kind of already using it.
I mean, I feel like this is a great opportunity for the collaborations that Terrence Tao was talking about as well where, once people have access to the common tools and it becomes easy to do.
And I mean...
Like, if you have an intuition even not a strong mathematician like myself you might be able to participate in the, you know, sort of like an effort to prove a larger theorem or something like that.
Yeah, I think that's a very interesting view, which is that, if you think about, mathematics has been not as collaborative as software engineering.
You don't have hundreds and thousands of people working on something together.
I think polymath was an instance when that happened.
That was fantastic.
So if you have a lot of really good sort of setup, indeed like commoditized kind of access, then people can all participate in.
In fact, that's how I think some of the large formalization projects have been done.
Things are divided into subtasks.
But really the blueprint writing process by say, Terence Tao and Alex Kondorowicz, of assigning the tasks to different people and how things kind of fit together.
That blueprint writing part is extremely important.
And there has been, I think, a result about sphere packing, I think by one of the other companies out there.
And The blueprint part for the eight dimensions is still pretty much built on what the sphere packing community, the link community, the humans blueprint.
And similar with some of their other results as well.
The blueprint part has still been human generated.
And I think auto-generated blueprint is going to be a technical bottleneck that many people are trying to solve around the same time.
So is there value in me, as a you know cloud code user, trying to attempt like some small lemma or whatever, where I don't have a great understanding of the math?
Maybe I have a high level understanding.
Depends.
Are you trying to formalize or are you trying to prove?
To prove new things.
That's a good point.
Yeah.
So maybe you would obviously probably start with formalization, right?
Yeah.
You already know the proof and you just can't get it.
Yeah.
Nobody has been able to get the formalization correct.
I do actually have seen people use lean and formalization and they try to do it by hand, you know, not using any AI as a way to learn mathematics.
No, it's, you know, it's all the formalization.
You don't have that.
Well, it's interesting because I think a lot of my friends who started working on Lean and Math Lib was because they are in PhD and the problem is really hard.
We get stuck all the time and we want to kind of review some of the undergrad classes, a time where we still understand what the math was about, and we do so by hand, you know, doing lean.
And I think that's very beautiful.
The materials.
Yeah.
But if you have, for example, like you know, access to action provers that also can formalize other formalized things and you don't have, you lose that part of the learning process.
Yeah.
Yeah.
But I do think that you know like for you and I we can set up like Axel and try to like see you know what results we might be able to prove.
And I think that's quite interesting.
And thanks to Axel sort of making the speed a lot faster, you don't have to wait very long.
I remember the pandemic exam day.
We were all, like, in the war room.
It was a Saturday.
We're all really excited.
And we just got the exam paper from the official, like, organizing the proctor of the pandemic exam.
We just like were looking at like how much workout Axel was like getting and without it we couldn't have solved it with, I think, the eight problems within the time limit, definitely not within the time limit.
And I think one thing about these tools is like It's very interesting in that potentially you can have interesting reward for RL as well.
What do you mean by that?
So, for example, verify proof can be a reward for just basically a proof is completely correct and validated.
I see.
I think formal verification tooling can be interesting direction to pursue with RL.
Yeah, so you mean, for example, you should formalize auto-formalize, the informal proof and then verify it and then use that as a reward.
Or do you mean...
No, as in like you pass like lean programs in these formal tools, right?
Like, and you will have some sort of score.
Okay, yeah.
I think if I were to build a one or something I would have in my mind, I would have use what I just described.
But you're saying just to learn how to do lean.
Sure, sure.
The valid proposition, which is interesting about Frontier Lab, is that suppose you are a 2C business, then sure you can just not do what we are doing.
And We have seen, for example, DeepSeek originally having a formal team and then later dissolved that team because of strategic direction change.
That's all completely reasonable.
Now, suppose you are focused on coding, right?
And you have talent, who want to work on what we're doing, it makes a lot more sense for you to do code generation.
Further your strength and moot.
You can partner with Axiom, just like how, for example, Frontier Labs partner with startups that work on search, such as Excel and Parallel right.
Just call Excel API for search and Potentially, you know if you're a frontier lab, I think you should call XMAPI for verification.
Yes.
It doesn't make sense.
I mean, it's just you know potentially, I think, the talent, the finickiness of lean, the sort of data code.
Like you know, there's no reason to.
Yeah, I mean it took me five minutes to set up.
Why did you decide to start Haxiom?
Right.
Why did I decide to?
You were a grad student at Stanford.
Yeah.
And, you know, in math.
Yeah.
So what made you decide?
I wasn't in math for very long.
I was, I was, I think like almost as soon as I started the PhD, I just started fundraising.
So it wasn't like.
Oh, really?
Yeah.
Yeah.
Was that the plan, or did you start there and you like, almost immediately realized that this is the direction?
Right.
So the year of law school right was very, very interesting to me, like on an intellectual level.
But it's also the first year where I had no science, technology, math whatsoever in my life.
It's a weird year, right?
Like, I'm reading a lot.
I'm practicing.
Well, I'm learning how to write.
I'm learning how to read.
Like...
But, like, I'm just kind of... I want to, like, be obsessed about something in technology.
Like, that was also what's going on that year.
So, yeah, the year of law school, right?
And it was very, very interesting to me because it's like okay, like I just I need to be obsessed with like a technical thing, because otherwise I get too.
I don't think I'm bored, because I really love like everything about law.
I really, really loved it.
It was something that's incredibly interesting to study.
But I just, I mean, I've been basically, like you know, very excited about like, the progress of reasoning.
I was looking at a lot of the post-training kind of papers.
I was learning all of these, like, just by myself.
And then at one point, it got to a point where I'm like, I think this is for sure happening.
And like I think, talking to Shubo right at Verve, like every weekend also, like it didn't help, like soothing this thought.
So I got more and more obsessed.
And at a point I'm like okay, if I'm doing this like literally every minute and I can't think about something else, like you know, I need to do something about it.
I mean, it's like I fall madly in love with the idea that AI is going to do math.
And like, OK, now do I do I do math?
It's really, really crazy.
Like at a time where I remember the obsession was quite I just couldn't get out of it.
And then I went to this Knight Hennessy event.
Knight Hennessy scholar Denninghaus hosts all sorts of like free lunch events.
And those are great because you get free food and you get interesting intellectual exposure to things.
And I remember Julie Zhao, who was, I think, a Facebook, first Facebook PM came to speak.
And then after that I just like basically walked up to her and I said like, like what do you do if you want to do a startup?
And you really wanted to do academia because you kind of love math.
And then she's like, well, you know, what's your time spent on these two different things?
And I'm like, 100%, 0%.
And then she's like, well, you kind of have to follow your energy.
Yeah.
I mean, if you're, if you are completely obsessed with it.
Yeah, I was completely obsessed with it.
I thought it was going to be big.
And I thought it just has to be a for-profit startup because it's so much broader than making mathematical breakthroughs.
If you think about recursive self-improvement And like really the kind of more high level like concept of like you really want to have just AI scientists.
Like the math reason is going to be, it's going to be a pretty big part of it.
And now I think the sort of belief by Cursor and Claude and other folks is like okay like, just like math, transfer to coding coding, transfer to math as well.
I think that's true.
It's just that like, you know, why not push it directly?
I don't get it.
You need to push that directly.
And then there's this other, like you know thought, which is that, and maybe kind of going back to the collaboration point right,
Verification has traditionally been thought of as okay.
Well, there are some industry where there's a lot of guardrails.
So if you're working in defense, military use.
OK, you need to like basically satisfy a lot of barriers to entry to meet those stringent requirements.
So it's something that's verification is for the industries that are closed, but it's For the first time now, I think verified AI is to open up collaboration.
Either it's human-AI collaboration.
Well, before Blueprint, that's human-human collaboration.
And Lean was a grounding, was a verification, formal language.
And then human-AI collaboration, like we're seeing now, future AI agent-agent-agent-like collaboration.
So like I think verified AI is for openness.
It's not for meeting the requirements of closed industries.
And I think, just like I think verification should not be about.
Oh, I remember, like you know, there's this article like chatbots make stuff Up, is AI the solution to sorry, it's math solution to hallucination.
Verification to me is not about lousiness.
Verification to me is about scaling brilliance, compounding brilliance.
It's like just kind of going back to the collaboration point.
It's about Ramanujan being a much stronger mathematician.
He was already a really strong one, but verification helps him extend the brilliance, like both kind of like scale up and scale out.
So verification... It seems more rigorous.
Verification to me is not about, you know, like, erasing the mistakes.
It's about scaling brilliance.
And the third point is that, like verification to me is not about like the sort of you know, just talking about rigor.
It's actually about performance gain.
It's not just about the stringent requirements, the hurdles that you need to overcome.
It is about like actual verified generation is going to make it so much better, and I think like kind of these three points.
I think the last point is that a lot of the people think that you work on verification because of your distrust for technology.
Like it sounds really well to, I think, the general public including, like my parents, like oh, why we're doing verification because, like you know, technology make mistakes.
It's No, we don't think verification is because of the distrust for technology.
It's because that's what like expected rapid, exponential scale up and the deployment and the creation of technology and technological progress is what that compels and demands.
It's a very mathematical perspective, right?
Yeah.
Because you're saying proofs drive math, right?
A lot of math is about proofs.
Yeah.
And math drives a lot of science and innovation in the world.
And the innovations in math drive innovation in the world.
Yeah.
So that...
But it doesn't need to even go through like in terms of you know the math solve everything thing, like obviously stands.
Like.
My point is like transfer learning doesn't like.
Transfer learning is about like pushing math reasoning.
It just...
So there are kind of, I guess, like there are a couple of narratives here.
Like for some people, it's like you solve math and then math are the you know fundamentals of science.
So that's actually the from AI for math, like take this radical layer of AI for science.
It's that narrative.
We actually believe in just like general transfer learning.
Like I think Axiom is on the infrastructure stack.
And you think that this is just a first step to you know, basically unlocking capabilities in many domains, in science and law, for example.
Yes, I think it's, So again, there are like, you know, multiple, multiple kind of like beliefs.
One belief is that there is math and there is, like you know formal, the power of formal verification.
Suppose we actually, you know, solve and have a really strong informal math reasoning engine, we do not expect that temp to be as large as solving math through the formal way.
Why?
I mean, code as it is language, but it is indeed on the more structured end.
Yes.
It bridges informal and formal.
Yes.
What we are doing is it's not informal versus formal.
We're not taking this sort of like completely formal alpha proof approach.
Like it's bridging between informal and formal.
It is bridging between high level and low level.
It is a direct, it's sort of like a direct improvement to reasoning.
So transfer, like transfer learning.
And it's also indirect in that like okay well like, math is going to unlock a little science and sure.
And that is really what we're seeing.
So you think that it enables transfer learning?
Yeah.
I see.
I think that is pretty much a consensus.
I think it is a consensus and this is a bet that has been pretty much kind of overlooked by others, because math sounds pure and it doesn't sound like there's any commercial value.
Well, I do obviously understand the opportunity cost if you're like a really like a frontier lab of solving this problem.
But I definitely think this is a problem that, if you're like a well-resourced startup, you should be doing.
That's an interesting perspective.
Did you get everything out that you wanted to say?
Yeah, I think it's like.
You know, like the question of like is Axiom math or is Axiom verification?
The DNA of the company is math.
We think that verification is the best first market.
And we think that sort of like solving math, and especially like formal math, is going to like help us like tackle the really ambitious quest of verified AI.
Now, when we're done with that, we might have other, that second market, including effort science we just talked about, but on the theoretical layer right.
I think real-world testing is important and potentially we can stay in the digital world and software stuff and for other things to be getting real-world, like physical world signals.
But do you think that the sort of The capability of doing really powerful reasoning.
Once you have that powerful verified reasoning engine, that's the moment when okay, now we've unlocked that for you know software verification and hardware or whatever.
Yeah.
Okay, so now what about biology?
What about chemistry?
So that could be one.
The other one is then like really how far are you to recursive self-improvement?
Okay, so just AGI.
Yeah, I think there is this sort of question and different people, because of their probably different backgrounds, have different
It's really where your energy and your passion leads you.
Like for some people actually I have heard this actually, you know, with my friends they want to work on AGI because they believe AGI solves death.
There are other people who come from a more like medicine background.
They really believe they can solve death and they don't solve AGI and then solve death.
They just solve like AI for science.
Now, which way is correct?
I don't know.
And so the recursive self-improvement angle.
It sounds to me like you're saying that the combination of verification plus the sort of like language which is informal, it's that combination that enables really good recursive self-improvement.
I think recursive self-improvement is going to happen anyways.
We're trying to have like formal verification earn its place.
So again, whether formal verification can be welcomed and deployed and become a consensus depends on how well we execute.
And I think when you boil down that problem into an execution problem, you should just go for it.
Looking forward, what's the biggest bottleneck that you see in the field for both Axiom and maybe just the field abroad, in terms of?
Fragmentation.
So I think we're in a market where people like to start, like you know, a thousand people.
They don't join force, they start a thousand things.
I think that's actually the biggest like kind of bubble indicator.
I think there are categorical bubble and there are like other categories where there are moonshots.
It's not bubble, it just looks a little bubbly.
In the field.
If people who are like, really like, really legit, you know backgrounds, decide to join force and work in a team for the mission rather than for ego, for kind of the status of Neolab founder, I think that category is really bullish, and vice versa.
So I think the bottleneck actually is about Potentially.
I think it's annoying because it's like we are in a, if you believe we are in an age of research, if you believe in like deep techs are the interesting directions to go after.
The market sort of conditions currently is good and bad.
And that good, it enables these sort of long-term, long-horizon bets to be funded.
Bad, because there's too much noise in the market and some other like irrational players.
You know, we try to work with really incredible venture firms like they are.
The partners.
They are our intellectual partners and there's a lot of alignment and we really bounce like very cool ideas technical and non-technical, each other like for long, long hours.
And you know we spend like a lot of time off work and weekend together to really intensely build the company.
But there are also other people who just want to like park, like capital somewhere and you know, while we don't work with them, these encourage, these are market conditions that encourage fragmentation, and when things get fragmented like no one gets there.
Like I think every category, regardless of how right the idea is, is pretty much in a sort of earning the right to exist stage.
And if that is the case then, for example, great deep tech company SpaceX and people do actually join force to work on that dream and potentially in that case also a very charismatic founder.
I think a really kind of concerning thing for me personally is that for other, probably some categories that I'm personally quite bullish about their action about and just like looking at things generally, fragmentation is a problem.
Like the sort of you know.
We see stock polling professors from university to work on something when It really is a really interesting kind of situation.
Maybe this is a naive question, but like right now, when you're talking about players in, let's say, AI for Math, where you know you Harmonic and then you know the big labs right.
Am I missing someone?
Is that actually fragmented, really?
I guess fragmentation, I think, is a bottleneck for the entire, like...
AI landscape.
I think AI for math is a category that is actually not a bubble because it is not fragmented.
Because people who are really amazing talents do like to join force.
So for example, the fact to get Ken Ohno and Francois Chartand on one team.
Like, this is fantastic.
Like you have someone who's a core contributor frontier math tier four, really great benchmark setter.
Francois, who's on the AI for math discovery, proving and discovery, they work together.
Then you are suddenly a player with both proving capability and construction capability.
And that's fantastic.
And I believe you know, as you said, like harmonic probably also have some really great talents like joining force together.
I think Afro-mass is a good category because of the absence of fragmentation.
But even, you know, from our perspective.
In our perspective, the sort of, for example, you know RL right being.
I don't think that's like a category per se, but you know RL talents.
Currently it's quite hard to attract and retain right for literally everyone.
And there are a lot of companies being started and then sold like three months later and just...
Each month where you could have worked on a technical problem and you're instead working on deals.
It's a month that is wasted.
And I say that, like you know, also with some amount of pain and suffering, because I'm having gone through two fundraisers.
Yes, yes, yes.
Yeah.
So what's the biggest bottleneck in AI for Math?
For Axiom, for AI for Math.
Not Axiom, but just the community.
But the community of AI for Math.
Yeah.
Where is it going?
What is the thing that everyone just really wants to break?
I expect fragmentation to start to happen as Axiom and Harmonic establish category leadership.
So I expect people kind of, you know, that's one thing.
But I also think that another bottleneck could be the pressure of short-term versus long-term.
I think that we are... doing things in a very sort of fast-paced manner.
But that does not mean we can always, or it does not mean it is always correct to do things in the most fast-paced manner.
Like we did things in a fast-paced manner because well, we were founded on the day of the International Mass Olympiad.
So we couldn't have competed in that anyway.
The next Mass Olympiad is Putnam.
And we're quite excited because it's, I mean, it's undergraduate exam.
And this year's IMO, 2025 IMO was easy on the MOHS scale and putting them could be hard.
And in fact it was harder than the IMO and the MOHS scale.
If you look at AI, you know how much, how many scores the AI has retained on average and on the max scale.
You know difficulty with the problem.
Pandemic is harder in both axes.
So we want to try.
And so there's only a gap of four months, but it doesn't mean I'm always going to set four months goals.
If I build a company only setting four months goals, I might build a really short-sighted company.
So they're like, I think, longer horizon problem.
I think, for example, market forces could force other players into chip verification.
Well, it is possible that co-verification is a holy grail.
It's possible that if you solve that, then you also naturally solve chip verification with some amount of like epsilon.
Caveat of like distribution shift.
But I strongly believe that like a bottleneck like could be the pressure.
But I think that Axiom is fortunate that one, we are early enough.
Two, we are like a team of just incredibly like high agency people that our execution generally surpasses expectation.
But I think like...
What I think could be a bottleneck for the entire AFR math field is that potentially trying to prove commercial value is going to distract significantly from the core capability improvement.
Yeah, that makes sense.
Cool.
Thank you for driving up and coming to see us.
Thank you so much.
I know the traffic was horrible.
Yeah, thank you.
And it's been really a pleasure speaking with you and we look forward to seeing how things develop.
Yeah, thank you so much.
Thank you.
Awesome, thank you.
Yeah.
Okay.
