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.
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.
_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 The big gap in the proof (the quasithin case) happened because it was covered in a 600-page unpublished manuscript that under further investigation turned out to be incomplete.
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
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.
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.
Sounds like you are admitting to copyright theft, Jonathan.
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 expression. 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.
> 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.
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.
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
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.
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 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.
Holy shit.
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.
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.
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.
LLM formalization is not as useful as human formalization, but nice to know that the Classification is correct
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.
Page has some kind of redirect bug that steals the back button, may want to open it in a new tab to avoid that.
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.
based if true