The organization of facts into axioms, rules of inference, proofs, and theorems doesn't seem to be an ontologically fundamental one. We superimpose this structure when we form mental models of things. That is, the logical structure of things exists in the map, not the territory

I wish you would have made this last comment on the post directly, so that I could reply to that there. Anyways, the point I was offering was that the logical structure does exist in the territory, not just the map. Our maps are merely reflecting this property of the territory. The fundamental signature of this is the observation that physical systems, when viewed in a map which exists only as a re-representation or translation (as opposed to an interpretation) amenable to logical analysis, are shown to prohibit logical contradiction. (For example, the two statements (if A, then B) and (if A, then not B) cannot both be true, where A and B are statements in some re-representation of the physical system.)