Dear Chris,
Having myself first learned to think more like a physicist and only in the last 1.5 years or so more like a mathematician (though I find thinking like a physicist still more intuitive, or, I should say, easy to come by), I think I can well appreciate where you are coming from.
Perhaps it would be worthwhile to step back for a moment and think about exactly what role you want this to play in the connection between physics and mathematics. The spectrum ranges on one end from something like a philosophical or physical principle to the other as a rigorously formulated mathematical axiom. Where you consider your idea to fall on this spectrum determines how precisely you need to state it. Your choice of calling it an "axiom" led me to believe that you are in fact considering it as something much mores imilar to the latter. In that case, expressions like "maps to statements", which, I agree, are perfectly acceptable in physics, require some work to be intelligible to mathematicians. Your expression reminded me that set theory is not the only way you can try to incorporate your idea into the foundations of mathematics. Although I know at this time still very little about category theory, from what I do know I have the impression that "maps to statements" might be more easily accommodated there than in set theory.
As a final note, part of my interest in this probably reflects the fact that I am myself struggling with incorporating a general philosophical principle as a rigorous axiom into the foundations of mathematics.
Best,
Armin