Rhyme had never gone into Mistfall's Academy, and he didn't know anyone there. However, he had a gut feeling that something was wrong, so he entered the Academy grounds anyway. In any case, if people from the Academy saw him running around despite not being a student, he could just show them a pass from the Emperor to prove his identity—that he was not a spy from another Empire and meant no harm.
Technically, he would still get punished for trespassing, but hey, at least he won't get killed on the spot.
The people in the frontlines were expecting the weapons to be delivered at 5:00 AM the next day so they could commence their attack immediately afterward. If they get the weapons later than that, even if it's just an hour late, they would have to readjust their strategy, and doing that isn't possible in a few minutes. It would take them a few more hours.
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