Out of curiosity: is it possible to take a frontier model and just let it loose on all math textbooks out there, converting each and every proof/theorem/lemma/etc. into Lean? A frontier model should be able to do that, right? And it's a one-time task: do it once and it's there for eternity, available as a library to be used in other proofs.
Maybe my understanding of Lean is wrong; I'm thinking of it as a programming language but for Math.
Axiom MTS here. What you describe is pretty close to how we formalised this proof, which is scattered across more than 500 articles and books: a swarm of agents was tasked with going through all these sources.
One challenge is assembling everything into a coherent whole, since the same informal concept can be formalised in slightly different but incompatible ways.
As someone with a highly uneven and kinda basic maths background, I'd have thought that simplification of a proof, and pulling things into a coherent whole would be exactly what computers excel at, but everything I've seen (or heard) suggests that's not really the case. Are you able to explain why?
When two sources formalized the same object in incompatible ways, did the swarm mostly emit translation lemmas between them, or did someone pick a canonical definition and have downstream proofs rewritten to match? The second feels like where the "coherent whole" cost actually sits, and I'm curious how much of the 500-source effort was that reconciliation versus line-by-line translation.
Yes, but thing is, for a lot of basic stuff there are myriad ways in which it is formulated, so humans have to decide which of them goes in the library. Of course you can include two or five or ten equivalent formulations (like textbooks sometimes do) but still, not everything under the sun.
Probably this will be done in some form in the coming months.
The really interesting aspect of this, IMHO, is discovering what theorem formulations combine together well as a whole. It's much akin to a software architecture problem.
Metamath has done quite a bit or primary research into this. Theorems often have a slightly non-standard, but more general form.
One can ask things like, "what is the equivalent of functional programming?" Or, "is there global state?" It's a perspective on reverse mathematics.
_If_ you have formalized a provable theorem in Lean, a simple bread-first search can prove it. It will take lots (and lots, and lots, and lots…) of time, though (https://en.wikipedia.org/wiki/General_Problem_Solver).
LLMs can do smarter searches, and will likely find proofs for most of them, but are not guaranteed to find a proof.
I think people assume that it would end up not being usable by humans without a lot of hand-holding. Though maybe that's wrong, and somebody should try it.
This seems like a very good use of auto-formalization, because in the past mathematicians have incorrectly believed that they were finished with the classification only discover a missing class/sporadic group.
The original proof was a collaboration of ~100 mathematicians, if Tao is correct then we should expect to see more such collaborations in the future like we do in e.g. physics.
They didn't discover a missing sporadic group. The big gap in the proof (the quasithin case) happened because it was covered in a 600-page unpublished manuscript that under further investigation turned out to be incomplete.
Makes me sad that the homotopical foundation people never got their shit together and now formalization is so mature but it's still done in set theoretic foundations.
Last week I blew my whole Claude Max weekly budget having Opus translate from Mathlib to Coq-HoTT. It was so beautiful I can't wait to do it again this week.
No Type Theory won: Per Martin-Löf is still alive and should be able to confirm his work lives in some form in Lean. Zermelo and Fraenkel would be utterly confused, however.
What you mean it say is that Lean is not a HoTT, and Mathlib has tons of classical reasoning (excluded middle, axiom of choice).
I don't think it's too late at all for HoTT. In fact, I think the AI companies will find trying to crank out more new math which is out of distribution in a funky way a highly useful exercise.
Groups were my gateway drug into abstract algebra. A friend clued me into symmetry being everywhere and Groups are how symmetry is formalized, and it’s been a crazy journey understanding all that. The frontier models can tie those abstract concepts to plain English examples and answer questions decently well, it’s never been a better time to pick up the field solo and see what it has to offer us
Well, the point of the _computer_ formalization is that it's checkable.
An error in the formalization would be a bug in lean. Which, as we know, certainly seems plausible (I've mused on if these giant proofs are incentivizing LLMs to find lean bugs to exploit), but at least it's a smaller surface area.
In general checking that the definitions are correct isn't that hard and assuming at least one of the lean kernels you verify with is correct (which tbf is an assumption, the implementations are based on each other so could have a common bug) that's all you need to verify.
In this case the statement is unusually complicated but still much much easier to verify than the proof itself.
The papers describing lean and it's core operations are deliberately small. The kernel of a theorem prover is meant to be such that a human can check it for correctness. The kernel does not include things like type inference or execution. It literally checks the proof and that's all and everything needs to match exactly. It's fairly trivial to write such a thing.
Humans have to establish trust in this.
After that, the rest of the language is written. Yes it is possible the type inference engine is broken but in that case the kernel will reject it
Depends if you include a few decades of improvements in lean itself. Non-AI proof automation is much better than it was even a year ago and so some fiddly results can now be done with `by grind`.
Also don't discount the fact that people would learn how to do this kind of project, currently no one really knows how to decide what definitions work best, what should be reducible, what should be part of a public api and what should be local to a module, etc.
I would bet no. Because the original proof is so distributed, with parts of it being very old, and with parts of it being, as a first attempt, not the most concise or lucid route to the result, we were rapidly approaching a world where the empirical knowledge of how/why the result holds was literally dying.
The article refers to its use of the "first-generation proof". It would seem this proof is too large to be truly known by one person, and the group of people who "collectively knew" it grew old and started to die. Since its original formulation, mathematicians began to compile, consolidate, and revise the proof into a 2nd generation digested form (estimated to be 5000 pages instead of tens of thousands). But that effort was still so monumental in its own right that a 3rd gen proof project began to digest the 2nd gen proof before the 2nd gen proof was even finished. I believe both the 2nd and 3rd gen projects are still ongoing.
It will be interesting to see how small the proof gets once we finish learning how to obtain these results "efficiently" with the benefit of hindsight.
I'm surprised they only found one gap. I would have expected multiple (fixable) gaps would have been uncovered, especially since the result was spread over many papers by many different people.
>However, there were 34 noteworthy deviations from the published record that needed to be resolved, the most substantial of which related to the Unbalanced Group Theorem.
Looking at the visualizations made me wonder if the embeddings themselves can be used to suggest, if not derive, mathematical results. I realize it depends on the quality of the embeddings.
Out of curiosity: is it possible to take a frontier model and just let it loose on all math textbooks out there, converting each and every proof/theorem/lemma/etc. into Lean? A frontier model should be able to do that, right? And it's a one-time task: do it once and it's there for eternity, available as a library to be used in other proofs.
Maybe my understanding of Lean is wrong; I'm thinking of it as a programming language but for Math.
Axiom MTS here. What you describe is pretty close to how we formalised this proof, which is scattered across more than 500 articles and books: a swarm of agents was tasked with going through all these sources.
One challenge is assembling everything into a coherent whole, since the same informal concept can be formalised in slightly different but incompatible ways.
As someone with a highly uneven and kinda basic maths background, I'd have thought that simplification of a proof, and pulling things into a coherent whole would be exactly what computers excel at, but everything I've seen (or heard) suggests that's not really the case. Are you able to explain why?
Proof and program are twins. Formalized theorems can be incompatible for exactly the same reasons that programming APIs can be incompatible.
What were the agents based on?
When two sources formalized the same object in incompatible ways, did the swarm mostly emit translation lemmas between them, or did someone pick a canonical definition and have downstream proofs rewritten to match? The second feels like where the "coherent whole" cost actually sits, and I'm curious how much of the 500-source effort was that reconciliation versus line-by-line translation.
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Producing a formalized proof from a non-formalized proof in a paper or book should not be considered a violation of copyright.
Indeed; one cannot copyright mathematical ideas, only their specific expressions. And the formalization is a different expression.
Yes, but thing is, for a lot of basic stuff there are myriad ways in which it is formulated, so humans have to decide which of them goes in the library. Of course you can include two or five or ten equivalent formulations (like textbooks sometimes do) but still, not everything under the sun.
Probably this will be done in some form in the coming months.
The really interesting aspect of this, IMHO, is discovering what theorem formulations combine together well as a whole. It's much akin to a software architecture problem.
Metamath has done quite a bit or primary research into this. Theorems often have a slightly non-standard, but more general form.
One can ask things like, "what is the equivalent of functional programming?" Or, "is there global state?" It's a perspective on reverse mathematics.
> Of course you can include two or five or ten equivalent formulations (like textbooks sometimes do) but still, not everything under the sun.
Note that mathlib currently includes duplicates of many theorems; I believe deduplication is an active goal (if not one seeing much labor right now).
Yes and no. With enough compute, probably, but doing this does not necessarily yield a reusable library. See https://xenaproject.wordpress.com/2026/10/16/how-to-formaliz...
_If_ you have formalized a provable theorem in Lean, a simple bread-first search can prove it. It will take lots (and lots, and lots, and lots…) of time, though (https://en.wikipedia.org/wiki/General_Problem_Solver).
LLMs can do smarter searches, and will likely find proofs for most of them, but are not guaranteed to find a proof.
Write a few proofs with it and then come back to this
I think people assume that it would end up not being usable by humans without a lot of hand-holding. Though maybe that's wrong, and somebody should try it.
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This seems like a very good use of auto-formalization, because in the past mathematicians have incorrectly believed that they were finished with the classification only discover a missing class/sporadic group.
The original proof was a collaboration of ~100 mathematicians, if Tao is correct then we should expect to see more such collaborations in the future like we do in e.g. physics.
They didn't discover a missing sporadic group. The big gap in the proof (the quasithin case) happened because it was covered in a 600-page unpublished manuscript that under further investigation turned out to be incomplete.
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This tech is about to come for software.
Makes me sad that the homotopical foundation people never got their shit together and now formalization is so mature but it's still done in set theoretic foundations.
we are moving forward steadily :) https://github.com/marcinjangrzybowski/cubical-bis/blob/inte... (see table in the middle of doc) mind, that it is fully constructive implementation, you can get arbitrary precision aproximation to every real expression. for now WIP, but soon will be aviable in cubical-library :) also our related TYPES2026 presentation: https://types2026.cse.chalmers.se/slides/51.pdf
Last week I blew my whole Claude Max weekly budget having Opus translate from Mathlib to Coq-HoTT. It was so beautiful I can't wait to do it again this week.
No Type Theory won: Per Martin-Löf is still alive and should be able to confirm his work lives in some form in Lean. Zermelo and Fraenkel would be utterly confused, however.
What you mean it say is that Lean is not a HoTT, and Mathlib has tons of classical reasoning (excluded middle, axiom of choice).
I don't think it's too late at all for HoTT. In fact, I think the AI companies will find trying to crank out more new math which is out of distribution in a funky way a highly useful exercise.
Why is that sad?
The stronger AI become the easier it should be to develop and move to alternative foundations.
Hey now... If you have a formulation in type theory (lean is not set theory), then you can translate to homotopy type theory fairly easily
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Groups were my gateway drug into abstract algebra. A friend clued me into symmetry being everywhere and Groups are how symmetry is formalized, and it’s been a crazy journey understanding all that. The frontier models can tie those abstract concepts to plain English examples and answer questions decently well, it’s never been a better time to pick up the field solo and see what it has to offer us
Filling in well-known gaps like this is the kind of thing the AI companies should be doing, not trying to scoop research in progress.
The formalization is 43.1M lines of Lean.
Is that going to be mathematically useful?
It's nice to know that they didn't miss anything. The natural language proof is 10,000 pages and they have missed groups in the past.
Indeed, this formalization effort spotted a flawed argument in one of the original publications and worked around it to repair the result.
How do we know the formalization itself doesn't have any errors? AFAIK it's very far from trivial to formalize something.
Well, the point of the _computer_ formalization is that it's checkable.
An error in the formalization would be a bug in lean. Which, as we know, certainly seems plausible (I've mused on if these giant proofs are incentivizing LLMs to find lean bugs to exploit), but at least it's a smaller surface area.
An error in the formalization could also be that you formally prove something different from what you think you proved.
In general checking that the definitions are correct isn't that hard and assuming at least one of the lean kernels you verify with is correct (which tbf is an assumption, the implementations are based on each other so could have a common bug) that's all you need to verify.
In this case the statement is unusually complicated but still much much easier to verify than the proof itself.
The papers describing lean and it's core operations are deliberately small. The kernel of a theorem prover is meant to be such that a human can check it for correctness. The kernel does not include things like type inference or execution. It literally checks the proof and that's all and everything needs to match exactly. It's fairly trivial to write such a thing.
Humans have to establish trust in this.
After that, the rest of the language is written. Yes it is possible the type inference engine is broken but in that case the kernel will reject it
The proof was of such complexity and length there were lingering doubts. This dispels them.
More so if Lean uncovers logical errors. Note also there is already an ongoing 2nd generation proof.
I suppose this is the longest Lean proof by a wide margin.
It's possibly the longest open source project of any kind (depending on whether you count comments and white space in Linux).
I wonder whether humans could have realistically finished it without AI by 2100.
Depends if you include a few decades of improvements in lean itself. Non-AI proof automation is much better than it was even a year ago and so some fiddly results can now be done with `by grind`.
Also don't discount the fact that people would learn how to do this kind of project, currently no one really knows how to decide what definitions work best, what should be reducible, what should be part of a public api and what should be local to a module, etc.
I would bet no. Because the original proof is so distributed, with parts of it being very old, and with parts of it being, as a first attempt, not the most concise or lucid route to the result, we were rapidly approaching a world where the empirical knowledge of how/why the result holds was literally dying.
The article refers to its use of the "first-generation proof". It would seem this proof is too large to be truly known by one person, and the group of people who "collectively knew" it grew old and started to die. Since its original formulation, mathematicians began to compile, consolidate, and revise the proof into a 2nd generation digested form (estimated to be 5000 pages instead of tens of thousands). But that effort was still so monumental in its own right that a 3rd gen proof project began to digest the 2nd gen proof before the 2nd gen proof was even finished. I believe both the 2nd and 3rd gen projects are still ongoing.
It will be interesting to see how small the proof gets once we finish learning how to obtain these results "efficiently" with the benefit of hindsight.
I used to take mathematicians seriously.
Holy shit.
I was thinking this would be done at some point but not so soon.
Note they found one gap (and filled it) in the previously accepted 1,000+ page proof. So that's pretty cool.
Somehow the classification of finite simple groups makes me feel uneasy as a mathematician, but I suppose this helps quell that feeling.
https://en.wikipedia.org/wiki/Monster_group
https://en.wikipedia.org/wiki/Monstrous_moonshine
LLM formalization is not as useful as human formalization, but nice to know that the Classification is correct
Page has some kind of redirect bug that steals the back button, may want to open it in a new tab to avoid that.
I'm surprised they only found one gap. I would have expected multiple (fixable) gaps would have been uncovered, especially since the result was spread over many papers by many different people.
In [1] they claim:
>However, there were 34 noteworthy deviations from the published record that needed to be resolved, the most substantial of which related to the Unbalanced Group Theorem.
[1]https://axiommath.ai/research/the-enormous-theorem-formalize...
Many many gaps had been found and patched (by humans) already over the last forty years.
what a monster of an effort!
i_see_what_you_did_there.gif
Looking at the visualizations made me wonder if the embeddings themselves can be used to suggest, if not derive, mathematical results. I realize it depends on the quality of the embeddings.
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based if true