The candidate I would push forward as the proper reification would be Mark Burgess' Promise Theory. Mark's starting point isn't, "what's the most general algebra I can form for encoding mathematical objects", but "what happens if I generalize points in spacetime to be a special case of any kind of graph". When you do this, you make each point in space responsible for either promising information to another point, or accepting the outcome of a promise, as bidirectionality cannot be directly inferred.
So if every point in spacetime cannot be assumed to be connected both ways, then you have to explicitly define the communication channels that exist between points. This idea unifies both physical and digital "spaces", as everything is now modeled as a communications network from atoms up. Thus, Promise Theory makes the connection between compositionality and information and the rest of reality more clear than categories alone.
Mark acknowledges that Category Theory and Promise Theory are both similar and related. For instance, they both deal with notions of interiority. And Mark encodes his axioms with categories. He's written several books on the topic, like "In Search of Certainty" and "Smart Spacetime", and has published papers on arxiv. I recommend "Smart Spacetime" to see how powerful this kind of semantics can get.
Rodney Brook's articulation of these laws of robotics is the first thing I have come across that seems to point the way to formulating something for permatech. That is, if we generalize "robots" to any kind of automation that may have to interface with humans.
The first principle, that the form of automation makes a promise to humans, reminds me a lot of both Christopher Alexander's ideas on architecture, as well as Mark Burgess's Promise Theory. Alexander has a lot of ideas on design that goes beyond buildings, and is very influential to the Human-Computer Interface design and OOP. Burgess's ideas speaks a lot about how systems of agents (both human and machine) can voluntarily cooperate and coordinate together.
The second principle about agency connects to a lot of things. Among them, I can draw connection to the Living Systems world view, in contrast to the Machine World view. The former retains meaning and purpose for humans, and the latter can be summed up as "beat to fit and paint to match".
The third principle struck me as practical yet narrow in scope. On a second pass, I think there's something more general that can be teased out of it. I'm not exactly sure what it is. In some ways, it reminds me of how software can be improved iteratively, or what Alexander had wrote about "unfolding", where designs can change to suit inhabitants of a building.
Interesting cross-over with Mark Burgess's Promise Theory, where a promise serves somewhat as a single discrete expectation.
CFEngine being based on Promise Theory is circular, since Promise Theory means whatever it's convenient to mean, or possibly maybe "whatever CFEngine does"
2014 introductory article: https://www.linuxjournal.com/content/promise-theory—what-it
Basic book on the subject: https://www.amazon.com/Thinking-Promises-Designing-Systems-C...
It's not "maths" like arithmentic but it's "maths" like graph theory:
Promise Theory, in the context of information science, is a model of voluntary cooperation between individual, autonomous actors or agents who publish their intentions to one another in the form of promises. It is a form of labelled graph theory, describing discrete networks of agents joined by the unilateral promises they make.
The book I mentioned (Thinking in Promises) is the introductory-level public book.
https://www.amazon.com/Promise-Theory-Principles-Application... is the heavy-duty scientific stuff.
"In Search of Certainty" (https://www.amazon.com/Search-Certainty-Science-Information-...) is somewhere in-between.
I would say promise theory is its own kind of logic and notation. Thanks for the correction @dozzie.
