Thanks for clarifying. You're right, I skipped over talking about dynamic looping, since it adds another level of complexity to the discussion, and OpenAI's claim (quoted in TFA) that the compute graph depth of Astra is "within a factor of two of GPT-4" basically denies that they're doing it for more that 1 (or maybe max 2) dynamic loops. And that is equivalent to "stack repeated layers a couple times, but with dynamic off-ramps."

The ability to compute any computable function between tokens given an ability to loop an arbitrary number of times is a nice theoretical point, sure, but ultimately if people are still using single digit hard cutoffs on the number of loops, I'm not sure it's all that important.

So, I agree it's right to say that arbitrary length dynamic looping could open the door to making monitoring very hard indeed, by extending hidden states further and further. But I would speculate that if it actually worked better than extending the sequence with CoT tokens, we'd already be seeing it in strong open weight models. It's a fairly obvious thing to try. And we're not seeing it, AFAIK. So I do wonder whether it's something we really need to worry about in practice, compared to all the other things we have to worry about.