You shouldn't have to prove anything - ask your neighbor why is it that the chlorine from bleach doesn't move around but the chlorine from pucks does. Does the chlorine from pucks have legs? Is it "smarter" chlorine? If (when) they cannot answer the question then it should be on them to find the answer from their pool guy. Your pool is proof enough to show that you know what you're doing. If they don't want to follow your advice, then it's their loss.

Peter