Can the proof be machine-verified?
Claude's work toward a higher bound on the Riemann hypothesis was checked two ways: by mathematician Al Pogate, known for work on the Jacobian conjecture, and by running it through Lean 4, a strictly typed machine-checkable proof assistant hosted on GitHub. Lean 4 cannot catch every flaw a human reviewer might, but it can eliminate logical hallucinations entirely, making it a useful complement to traditional peer review. The result has not yet cleared formal peer review, but the hosts note that even a partial advance on a high-priority open problem can hand other mathematicians an unexpected angle to work from.
- Lean 4 is described as strictly typed, meaning every step must conform to formal logical rules the software can verify mechanically.
- The hosts draw a distinction between open problems that have sat unattended and those like the Riemann hypothesis that have had sustained, focused effort from leading mathematicians, arguing the latter category makes any AI-assisted progress more significant.
Transcript
This chapter, from the episode video's captions · 443 words
1:13:49>> and which can be formalized this this lean this lean software which can you can actually then have people look at it. >> Exactly. Yeah. So then so then I mean they actually had like the actual people look at it. So I mean um I think they got um Pog um Al Pogate the the same guy who did >> the uh Jacobian conjecture. He took a look at it because he's a proper mathematician in his own right. They also actually like put it into the lean software to take a look and see if it actually worked. Um lean is you know human peer review is infallible but um you can completely illumin eliminate the possibility of logical hallucination by
1:14:30using this software. It's called Lean 4. That's the latest one. It's strictly typed machine checkable proof assistant. >> Mhm. effectively like truth statements and things like that. >> Yep. Yep. >> Um it's hosted on GitHub and you can go take a look. It's it's fairly cool that I think they've done it. I mean it still has to go through traditional peer review >> for sure, >> right? Um but it seems that they have gotten a higher bound of the Remon hypothesis. And again, when this goes back to what we were just talking about earlier, when when folks who are mathematicians start to try to look at what did it do in these arenas because a lot of the argument has been, oh, these
1:15:11models are attacking these there there are all these um sort of open problems and like there are sort of maybe two categories. will say one category is open problems that are open but have not had concerted attention put towards them, you know, and so they're just open and gathering dust on the shelf uh but not necessarily either for whatever reason meaningfully being trying to be solved by leaders in the field. And then there are ones like this that are have been a high priority and a focal point. Um, and so it's not solving the hypothesis, but some of the insights that are
1:15:52gleaned might trigger somebody to have >> to have a completely different Oh, that's interesting. >> Out of left field insight. >> I could take that >> and go this way with it. >> We'll see. Who knows? We'll see. >> I just find a lot of this it's just it's happening. >> Yeah, it's happening as we speak. I mean, it's it's pretty crazy. a few interesting things that I came away with. >> Um, is the fact that first, as you said,
From What Claude Actually Did to the Riemann Hypothesis
Claude takes a real run at the Riemann Hypothesis, forcing us to ask what agentic AI can now do in mathematics, before we open the summer transfer window for America’s scientists.