Dear Christi,
I'm glad to see you enter this contest; your essays always bring an interesting point of view or novel argumentation to the table. This year's does not disappoint.
Intriguingly, there seems to be some degree of confluence of thought between your particular neutral monist stance, and the one I defended in a recent publication (https://doi.org/10.1007/s11023-020-09522-x). Basically, your view seems to be close to Strawson's physicalist panpsychism, which I characterize by the following two main theses:
PP1: In experience we are immediately presented with the intrinsic nature of physical stuff, which is inherently experiential.
PP2: Theories are purely structural or relational in form.
The second thesis, which goes back to Russell's causal theory of perception, and has roots as well in Eddington's structuralism (who talks memorably about the 'inner un-get-atable nature' of matter), I think both of us share, and likewise, the thesis that this relational structure does not suffice for conscious experience.
I think you may want to accept the first thesis---your S essentially being the intrinsic properties grounding the relational mathematical structure of P---whereas I ultimately reject it, substituting instead a different model of how 'structure-transcending properties' of the world can come to attention in the mind, thus grounding experience without themselves necessarily being experiential (or elements of sentience).
So I do agree that the relational structure as present in our theories needs grounding by some structure-transcending properties, and that it is those which are, properly understood, ultimately what ground subjective experience; but I don't think that these properties simply are experiential in themselves, essentially because I think the resulting panpsychism or pan-experientialism faces some difficult challenges, such as the notorious combination problem. Hence, I propose that the intrinsic must be used in the right way---essentially, as grounding our models of the outside world---in order to yield conscious experience.
We also seem to look to similar sources for inspiration---the quote from the Tao you use to preface section 6, I used as the hook for my entry in the previous FQXi contest, 'Four Verses from the Daodejing', which contained precursors to many of the notions in the 'Minds and Machines'-article. Indeed, some of it even turned up in my very first FQXi-entry, 2013's 'Informational Ontologies and 'Hard' Problems', which pointed out the underdetermination of the character of physical objects by relational structure: "[I]nformation is only concerned with differences. Wherever there is a difference, wherever you can tell apart one thing from another, you have information; and only this difference structure is straightforwardly encoded in the information state. [...] But this difference structure is not enough to recover the physical state of an object: the informational underdetermines the physical."
Furthermore, later, it goes on to argue that "information has an intrinsically relational character"---hence arriving at the insufficiency of a purely relational characterization. And of course, the basic point of that essay was that such underdetermination is at the root of the 'Hard Problem' (which I still think is the case, even if I no longer agree with the solution outlined back then). There's also already a tentative connection to quantum mechanics being made, which I've since gone on to flesh out some more, see this year's essay and my paper in 'Foundations of Physics' (https://doi.org/10.1007/s10701-018-0221-9).
Anyway, I'm sorry about the extensive self-quoting; I don't mean to be vain, but I wanted to underscore that there indeed seems to be some amount of convergence between our views. This, to me, always indicates that we're not merely barking at shadows---if two people are led to similar views along independent routes, one just might hope that there's something worthwhile to find at that destination.
Hence, thanks for this eminently readable and thought-provoking essay. I'll have to take some time to digest your larger piece, and perhaps return with some further comments and questions. Good luck in the contest!
Cheers
Jochen
