[kirkules]

Is derived information not distinct from the input? I mean, tagged information tells you about the tagging system in addition to the tagged information.

Specifically, it tells you how the tagging system interacts with the tagged information.

In this case, from a physicalist perspective, seeing red tells Mary how her specific brain (or whatever physical system explains color sight experiences) interacts with the experienceable subject "red", be it red light or whatever choice you make for the "true" subject for the sake of the thought experiment.

The ordinary discussion of the thought experiment simply assumes she can somehow already know this without experiencing it, and hand-waves away concerns about whether she could obtain that knowledge without having the experience (she is a brilliant neuroscientist, after all).

My position is that this is precisely the new information she obtains from seeing the apple.

The minimum value from a finite list of integers is derived from the list, but until you compute it (the analog of "seeing the apple"), you don't know the minimum. Even if you "know all the integers in the list" otherwise.

Edit: typo