You cannot spend OPS to *reduce* loyalty (7.1, 7.3.1).
Of course! Not sure why my brain didn't internalize that one. Thanks for reminding me of that.
If you win a battle *anywhere* against the controlling power and thus reduce the city's loyalty to 0, it becomes your loyalty 1 city if you already control an adjacent space (7.3.4). It is your political, not military, proximity to the ungarrisoned city that makes it vulnerable.
Right. Thanks for the clarification!