Pushing toward two-thirds
When Claude's AI agent attempted to push a zero-density estimate for the Riemann zeta function from roughly 58% to 2/3, it drew on Hugh Montgomery's prior result showing 2/3 of zeros share a certain property. Rather than follow the orchestrator's instruction to recover efficiency through the Cauchy-Schwarz inequality, a sub-agent rejected that approach, wrote a Python script to simulate matrices of zeta zeros, spotted an optimization pattern, and distilled it into a five-line lemma. The process crashed mid-run, but Claude had already saved the lemma to a file on its virtual disk. When the orchestrator restarted the session, Claude found the file, resumed from it, and reached the 2/3 bound.
- The orchestrator-agent architecture used here paired a directing orchestrator with execution sub-agents called E2 pairs, and it was one of those sub-agents that overrode the orchestrator's specific instruction about which mathematical tool to use.
- The Cauchy-Schwarz inequality, which the orchestrator wanted to exploit, is described as analogous to the triangle inequality, a basic bounding tool in analysis.
- Jared was the human researcher who set the 2/3 target, prompting Claude to find a route to that specific bound.
- The hosts argue this sequence, choosing a non-instructed method, finding structure in simulated data, and proactively saving intermediate work, illustrates behavior beyond next-token prediction, though they acknowledge the file-writing may have been routine process rather than deliberate crash-preparation.
Transcript
This chapter, from the episode video's captions · 838 words
1:09:062/3. So then Jared's like, "All right, push it [laughter] to 2/3." Why did it say 2/3? Now the reason [clears throat] why it said 2/3 is because under the assumption of the remon hypothesis, Hugh Montgomery had reached a 2/3 bound by utilizing the fact that there's like some for whatever reason that he was doing and he had found that 2/3 of the of the zeros had some property. >> Mhm. >> Okay. So Claude starts thinking well maybe the two/3s is and human stuff I can now use I already used Bombiaris right >> now maybe instead of he was using like Claude was using something like the
1:09:47Koshy Schwarz inequality not important what we have to get into >> effectively it said I could push this to 2/3 because I know of Montgomery stuff >> there's a reference point that says there's a chance >> and so in a well- definfined space so let's let's go Let's go after it. Right. So again, the orchestrator confidently instructs this next E2 pairs >> to recover some lost efficiency in the Koshy Schwarz inequality. It's like this tool that we're using. It's basically the triangle inequality. But um in any case, it's like you can do better than that right? >> The AI says I'm not going to do it that way. >> Mhm. >> That doesn't make any sense. Again, it's going back to the orchestrator like that doesn't make any sense, but it's giving
1:10:29me another idea. I'm going to write a Python script and that's going to just make a bunch of simulated matrices consisting of like zeros of the remon zeta and it's going to do a bunch of math and I'm going to try to optimize this >> and it optimized it and noticed a pattern in how the optimization worked in that Python script. From that it garnered how it could reveal some prof profound insight into the non-frivial zeros and it went through and started writing up this this little lema, this fiveline lema that's going to help it get to 2/3. All of a sudden the thing crashes. [laughter] >> Okay. Okay. >> This is also documented in Claude's like
1:11:10report. All of a sudden this thing crashes. But by some miracle, Claude has dumped that fiveline proof >> into a a file >> into its memory somewhere. >> Yeah. It's into it's into like a hard like file memory somewhere. >> And so when the orchestrator rigs it up again, it's like I'm pretty sure you were like doing something with this file. >> It reads that file. It's like a f It's like oh >> that's right. >> That and and then it goes back and completes it. Isn't that cool? >> I love I love Oh [sighs] god. >> It's like a text file on the virtual disc. >> Right. Right. >> And it like dumped its memory. It I guess I mean I don't know why but it
1:11:50thought that okay this is important. I'm going to write this. >> Let me write this down. >> I'm going to write this down. And then it like crashed. >> Of course. God. And this is this is I think what's also interesting when you look at the chain of thought of these models um and we try to start doing this introspection is >> I understand this idea that there's a framing that these things don't have ingenuity and insight yeah in the way that humans do. Um, but even though it's doing a lot of work through having the ability to execute ex execute with code and ingest lots of information, >> it still has to make after those
1:12:32executions that are functional. >> It still has to try to choose a path >> and it intuitits sometimes in ways that are interesting in between those like milestone points of functional tasking. And this is a perfect example of, you know, one, it chose to not just do what the orchestrator told it. >> Yeah. >> It had to make that choice. >> Two, it was able to sort of figure out this matricy map, this this this building out these matrices to find an insight within it >> and where to even look within it. and
1:13:14then thought, "Let me make sure I write this down before I run it >> in case for whatever reason it crashes." >> Yeah. >> Yeah. >> You know, again, it might not have done it explicitly for the reason thinking it would crash. It might just been part of it normal process, but um I just find this chain of thought so fascinating because I think it illustrates this is well beyond next token prediction. >> Yeah. This is Yeah. >> You know what I mean? >> It's pretty insane. I mean, this is this is pretty insane. And then and then and then from there you get to the 66% you get to the 2/3
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.