I agree this is a factor in problem-solving. I've found it to be important too. However, I suspect this can't be the main reason behind the "magic genie" phenomenon because if it were, you'd expect that mind-quieting meditations between bursts of mathematical effort would be vastly more productive than spending hours on problems and taking breaks. E.g., spending fifteen minutes on a tough problem and then spending five minutes meditating, cycled three times, would produce vastly better results than thinking about the problem for an hour. I'm not aware that this is the case, nor that the 20% most prodigious mathematicians are above average in their interest in mindfulness.

I agree that it's not the main thing, but not with your analysis. For one, this "mindfulness" thing is never really unpacked well. It could be that the habit of not focusing and bouncing between ideas (that's considered "unmindful", right?) is what it takes to not get stuck in ruts, and that the helpful mindfulness related bit is meta-awareness that "I'm noticing that I'm stuck" - and then fixing it instead of freaking out about it.

Instead of practicing mindfulness by itself, I'd hold mindfulness as an ideal and attack the specific blocks more directly.

Oddly, it turns out that most mathematicians do not relate to the supposed experience of a solution presenting itself in a dream. Many, including Hadamard, have mentioned waking up to have the answer to a problem they've been working on trumpeted into mind, but not clearly as a result of dreamed experiences

This actually doesn't change my hypothesis much. I'm hypothesizing something that happens without requiring awareness. I occasionally notice myself making strange metaphorical connections that were there outside my awareness for some time and finally got bumped in. I very much expect this to happen completely outside awareness a lot. Heck, Milton Erickson was famous for doing this on purpose as a technique in therapy!

By and large, no. But in this domain, yes! [...]what I was talking about seemed to be the division between the sympathetic and parasympathetic nervous systems.

Which domain exactly?

Interesting. I'll have to look into that distinction more, and generally spend more time in that perspective. I have gotten similar benefits, just on a fairly small scale - it'd just allow me to make sense of things there were a bit elusive and import it back to the individual thoughts/feelings level model.