Why? The constraint that a system output a "string" is too strict; it suffices that they output something interpretable as a string.

I agree now that "string" is too strict. I should have said "symbol", where a symbol is anything with physical tokens. My proposed label is now

Physical system S outputs the symbol A whenever it is fed the symbol B

where

• "S" is the name of a specific concrete physical system, and

• "A" and "B" are the names of specific physically-manifested symbols,

such that a token of the symbol A physically makes S output a token of the symbol B in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4.

I think that the work that you want to do by adding the word "interpretable" to the label is done by my conditions on what S, A, and B are.

An isomorphism M is a one-to-one mapping between two phenomena X and Y. In this context, then, the physical referent of M is whatever physically encodes how to identify what in Y it is that the aspects of X map to.

Then you should be able to make the label refer directly to that physical encoding of M. That is, instead of mentioning the isomorphism M, you ought to be able to refer just to some specific physical system T that "encodes" M in the same way that my physical system S above encodes the operation of adding 2 to 2.

However, if you're still unhappy with my label, then you would probably be unhappy with this unpacking of your reference to M. But I can think of no other way to make good your claim to refer only to physical things.

(A strict platonist would say that even my label refers to nonphysical things, because it refers to symbols, only the tokens of which are physical. I'm happy to ignore this.)