Should the proof of a theorem (taken in isolation) allow us to reconstitute precisely its statement? That seems like an interesting question, and I guess my personal answer would be that it should, more or less, given maybe enough context information (and with some restrictions on the length of the proof).
However, there might be other opinions. For instance, it is clear that if the proof does not make use, explicitly, of one of the assumptions, by hiding it in a computation or check left to the reader, then the reconstruction from the proof might miss it (I mentioned earlier reading a proof of the Banach-Alaoglu theorem where the important fact that one works with the weak-* topology is hidden from view).
In this spirit, today’s challenge is to find the theorem for which this short sentence is supposedly a proof:
The heat kernel defines a renormalization-group invariant plaquette action.
