I’m having difficulty following your comments about adversarial choice as related to the axiom of choice. What you’re saying doesn’t sound well-defined (in the mathematical sense).
It's actually very well defined (or which part are you confused by). I'm stating these things informally as a formal treatment is right now beyond me and it also doesn't really exist. Look up Chu spaces esp their relationship to linear logic. This paper is pretty informative
It's a path that unifies several very formally related. The unification itself isn't fully formalized, you need to look at the symmetries in the well formalized subparts.
