For instance, it looks like you get a piece of paper with your ballot that has a bunch of numbers, one for each candidate? Can you say with certainty that the poll worker didn't slip you a different piece of paper with the numbers in different order.

The numbers are not written on ballot. The lists are placed in the booths, on the walls of the voting place (so you can find your canditate before going to booth, more important in bigger parliament elections with lot of canditates). Also, the numbers are assigned well beforehand and the canditates can use those in their campaigning.

No, I can't give you a proof that it works. Just show that it makes sense.