No, they were not proof of anything. They were nearly worthless, and people discarded them all the time. You could collect a fistful of them on and around the bus.

Anyway, that is not how the fare system worked with paper transfers. There were no caps at all. You either paid full fare, or you showed a valid transfer. If you didn't have the transfer, then you paid again and again on any given day. The fare caps only came into use when the system was capable of tracking them, for example a unique preloaded or prepaid card.

Now I have limited experience with American public transit, so perhaps there were some "capped fare" systems in the 1980s or before, but I never saw one that worked that way.