Which of the 2 places are you most worried about containing mesaoptimizers? The language model or the maths model?

I think by far the language model. We don't know which proofs to ask the ZFCbot to prove, so we still need the LLM to model the space of math papers in order for it to think up stuff to send to ZFCbot. But if it's modelling the space of math papers written up to current year, it will resist outputting any correct alignment-related proofs, because it can notice from its dataset that humans haven't actually solved alignment in 2023, and so the probability of a good alignment paper conditioned on year=2023 is low. The only way I can think of convincing the LLM to try to output alignment proofs is to condition the paper on a future date by specifying a prompt like "2045, introduction to AI alignment, cambridge university press...". But this means that the LLM now needs to internally model the future mathematical progress of humanity, and this is where I think the risk would be coming in.

If the AI is just good at math and reasons "oh, later year means more progress on math, so I should output better proofs along the current direction human research is going", then everything is fine. But if the AI gets good enough that it infers and reasons about affairs outside of math papers, it might go "hmm, 2045 is a long way from now^[1]^[2], and humans aren't looking like they're gonna solve alignment, so in 2045 the only entities capable of writing the alignment paper would also be AIs, what would those AIs write in such a book?", and we get a bad, bad situation where basically if the LLM predicts that future AIs are aligned, it writes an aligned paper, and if it predicts future AIs are not aligned, it writes a misaligned paper. One saving grace here is that conditioning on the existence of the paper (which we do when we give it the prompt) tells the LLM that any possible AI writer is (more likely than not) aligned, since presumably unaligned AIs wouldn't waste their time writing alignment math papers.

Let

, $B \equiv paper is aligned$ , $C \equiv AI author is aligned$ , and we have

$P (B | A) = P (B | C, A) P (C | A) + P (B | \neg C, A) P (\neg C | A)$

which intuitively collapses into just

$P (B | A) = P (C | A)$ since $P (B | C, A) = 1$ and $P (B | \neg C, A) = 0$

.

^{^}
Possibly a good way to mitigate this is to confuse the LLM about the "current year", we might try to fool it into thinking that it's really a model being trained in 2090 with truncated data up to 2023, so it would then condition its output on "I am being trained in 2090", and this is more likely if alignment ends up being solved, since unaligned AIs presumably have no need to train such things. But I don't know how to convincingly fool it to think this.
^{^}
Or we could add a few artificial "newspaper articles from 2090" to its dataset, so it would condition on "newspapers exist in 2090", which would imply that papers written in 2045 are aligned.