On the use of grant funds for extracurricular research
I've also been funded by NSF's Division of Mathematical Sciences, so I can respond personally.  To first-order approximation, the only resources that research grants in mathematics (and similar fields, like theoretical computer science) provide is money to buy food to supply brain cells.

If my brain cells (or my students' brain cells) produce research results during the active period of a grant, I always acknowledge the grant in the resulting paper and I always list that publication in my reports to NSF.  It doesn't matter whether the work is directly or even tangentially related to the research outlined in the proposal.

If that seems inappropriate, consider the alternatives: (1) Shut off any part of my brain that is not thinking specifically about the proposed work.  (2) Allow myself to think about other stuff, but don't acknowledge the grant if I actually make progress.  The first alternative requires irreversible  surgery that I am unwilling to perform on myself, and the second is completely unethical.