Comment by caughtinthought

The material is interesting, but unless the statement that is proved in lean is not blowup for Navier-Stokes, then it's still proven.

What the examples seem to show is that the proof method is different between the natural language proof and the lean proof. Which, if the lean proof actually proves blowup, would suggest that the natural language proof is subtly wrong, but the strategy was close enough to be used to create a real lean proof.

A little worrying, but part of the purpose of formalizing things in Lean, it forces you to be more accurate than natural language does. It's surprisingly common for major theorems to have slight inaccuracies early on that can be repaired. Famously, the initial proof of Fermat's Last Theorem had a flaw that took a year to repair (though I think that's unusually difficult).

So the most fundamental question is: does the Lean theorem faithfully state the right theorem?

That assumes the natural language paper came first and then was formalized in lean. I haven't looked too deeply into how these labs solve these problems (or if they even specify this publicly) but you could also start with lean and then write the natural language proof based on it.

For what it's worth the initial lean specifications for the top-level theorems generally come from human written formalizations such as in https://github.com/leanprover-community/mathlib4/blob/021ce6... so we can be reasonably confident about their correctness.

Not quite — the formal statement of the problem in Lean may be correct, and therefore the Lean proof gives quite a lot of confidence that the statement is true. It's just that the proof given in natural language doesn't necessarily match up with the Lean proof, so the natural language proof might be unsound even though the statement it's proving is true.

> If the lean proof doesn't match the natural language one (which is the one the AI generated to solve the problem), it sounds like the lean proof isn't verifying the intended claim?

No, the other way around. The natural language proof was derived from the lean code, badly. This is my experience with using claude and lean to prove things. Its natural language explanations drift a lot from the lean, both before and after. But the lean code is the lean code.