I think there's a tautology hidden in there someplace. If two ice cubes plus two ice cubes equal one puddle of water, and two guinea pigs plus two guinea pigs equal an unspecified number of gunea pigs, you can say that isn't what you meant by addition, but I think that what you mean is that the physical isomorphism has to be arranged so that you get the answer you were expecting.

As long as the person using math and asserting its relevance (compressive power) in a situation can specify, in advance, what must be true in order to for the isomorphism to hold, there's no tautology. This situation exists for all theories: the theory's validity, plus the validity of your observation, plus several other factors, jointly determine what you will observe. If it contradicts your expectations, that reduces your confidence in all of the factors, to some extent. Mathematical predicates, depending on how easy they are to verify, can be more or less resilient than other factors in the face of such evidence.

Is it true that there's always a physical isomorphism for math? My impression is that some math sits around just being math for quite a while, and then someone finds physics where that math is useful. It's at least plausible that no one will ever find a physical isomorphism for some math, even if the math is logically sound.

Right, that's the caveat I was referring to here:

(Note that this discussion avoids the more narrowly-constructed class of mathematical claims that take the form of saying that some admittedly arbitrary set of assumptions entails a certain implication, which decompose into only 2) above.  This discussion instead focuses instead on the status of the more common belief that “2+2=4”, that is, without specifying some precondition or assumption set.)

In other words, some mathematical claims are about arbitrary axiom sets, not necessarily related to physical law, and simply assert that some implication follows therefrom. This article isn't about those cases. Rather, it's about bare claims like "2+2=4", not "under this axiom set, with these definitions, 2+2=4". Therefore, their truth will hinge partially on the meaning given to the terms, and claims without an explicit axiom set have an assumed one, and necessarily hinge on the presence of an isomorphism to physical law.