It's great to see someone working on this subject. I'd like to point you to Jim Crutchfield's work, in case you aren't familiar with it, where he proposes a "calculii of emergence" wherein you start with a dynamical system and via a procedure of teasing out the equivalence classes of how the past constrains the future, can show that you get the "computational structure" or "causal structure" or "abstract structure" (all loaded terms, I know, but there's math behind it), of the system. It's a compressed symbolic representation of what the dynamical system is "computing" and furthermore you can show that it is optimal in that this representation preserves exactly the information-theory metrics associated with the dynamical system, e.g. metric entropy. Ultimately, the work describes a heirarchy of systems of increasing computational power (a kind of generalization of the Chomsky heirarchy, where a source of entropy is included), wherein more compressed and more abstract representations of the computational structure of the original dynamical system can be found (up to a point, very much depending on the system). https://www.sciencedirect.com/science/article/pii/0167278994902739
The reason I think you might be interested in this is because it gives a natural notion of just how compressible (read: abstractable) a continous dynamical system is, and has the mathematical machinery to describe in what ways exactly the system is abstractable. There are some important differences to the approach taken here, but I think sufficient overlap that you might find it interesting/inspiring.
There's also potentially much of interest to you in Cosma Shalizi's thesis (Crutchfield was his advisor): http://bactra.org/thesis/
The general topic is one of my favorites, so hopefully I will find some time later to say more! Thanks for your interesting and though provoking work.
This post really helped me make concrete some of the admittedly gut reaction type concerns/questions/misunderstandings I had about alignment research, thank you. I have a few thoughts after reading:
(1) I wonder how different some of these epistemic strategies are from everyday normal scientific research in practice. I do experimental neuroscience and I would argue that we also are not even really sure what the "right" questions are (in a local sense, as in, what experiment should I do next), and so we are in a state where we kinda fumble around using whatever inspiration we can. The inspiration can take many forms - philosophical, theoretical, emperical, a very simple model, thought experiments of various kinds, ideas or experimental results with an aesthetic quality. It is true that at the end of the day brain's already exist, so we have that to probe, but I'd argue that we don't have a great handle on what exactly is the important thing to look at in brains, nor in what experimental contexts we should be looking at them, so it's not immediately obvious what type of models, experiments, or observations we should be doing. What ends up happening is, I think, a lot of the types of arguments you mention. For instance, trying to make a story using the types of tasks we can run in the lab but applying to more complicated real world scenarios (or vice versa), and these arguments often take a less-than-totally-formal form. There is an analagous conversation occuring within neuroscience that takes the form of "does any of this work even say anything about how the brain works?!"
(2) You used theoretical computer science as your main example but it sounds to me like the epistemic strategies one might want in alignment research are more generally found in pure mathematics. I am not a mathematician but I know a few, and I'm always really intrigued by the difference in how they go about problem solving compared to us scientists.
Thanks!