2) that she wouldn't be so illogical as to assert a circular argument where F would merely be a premiss, instead of being equivalent to the proper (valid) conclusion ~T.

Oh, now I see what you mean. I interpreted F as a new promiss, a new axiom, not a whole argument about the (mistaken) reasoning that proved T. For example, (wikipedia tells me that) the axiom of determinacy is inconsistent with the axiom of choice. If I had proved T in ZFC, and Ms. Math asserted the Axiom of Determinacy and proved ~T in ZFC+AD, and I didn't know beforehand that AD is inconsistent with AC, I would still need to find out what was the problem.

I still think this is more consistent with the text of the original post, but now I understand what you meant by " I was being charitable with the puzzles".

Thank you for you attention.