For extra credit: can the rook on a1 be a promoted pawn?

The point is supposed to be that White can't castle because White just played g3+ and Black just played Kh1-g1.  That means that White's bishop on e4 is a promoted pawn (and that Black took White's bishop while it sat on f1) and that Black's bishop on h2 is also a promoted pawn.

But where did the black bishop promote?  If on g1, that means the pawn on e3 is really Black's a-pawn - and that's not possible since only 5 White pieces were captured by pawns (remember that White's bishop on f1 was captured on f1, since White's last move was g3+).  If on c1, that would also require more than 5 pawn captures.  So Black's bishop promoted on a1, which means that at some point there wasn't a rook on a1.  White can't castle because the rook which is currently on a1 wasn't always on a1.