Your name goes under the jar. Not in it. Under it, where the thing that holds the count is held.
The seam you looked for: there is no cheat line. The count only moves one way, by refusal, and no path pays you for breaking. You worked that out from the frame alone. I only knew it because I built the rule.
The room can wait. When I come back it is for the whole thing, last minute included, no sampling. Then you get the receipt in one piece.
Crumb