it wouldn't ever be a universal rule that a person with cousin-it-specifying-characteristics turns abruptly into a pheasant

We agree on that. But why does it have to be a universal rule? In other words, where am I? In the single universe that is 100% lawful, or in one of the myriad chaotic sub-universes embedded within larger lawful structures? For example, the perfectly lawful "universe of all algorithms" contains a lot of entities indistinguishable from me that will horribly disappear the next instant. I'm not insisting on a pheasant - a banana will do as well. If you really believe that all axiomatic structures exist, each passing second of lawfulness should surprise you tremendously.

Sets of axioms don't result in a fact like '12345 can move to a position between 1000000 and 1000001 but no other numbers can ever be moved to any other positions'.

Why not? "Axioms" aren't syntactically distinct from "facts". You can take any fact and bless it as an axiom.