The cop said, "If you can talk to me and SHOW me correct proof , you're good to go. "
evidence, testimony . . .: somewhere in between 証明（しょうめい: proof） and 証拠 （しょうこ: phsyical evidence）. You'd typically use 証 when you're talking about evidence of god such as miracles. Or you'd challenge someone "You say you're honest and give to charity, but do you have any evidence of this?" Unlike 証拠、 証 does not have to be physical, it can be verbal.
physical evidence, such as a footprint, or "This green tap-water is evidence of pollution." Also this is the word one uses about evidence at trials, not 証.
a legal document certifying something: that you're married, that you have a job, that you paid tax, that you ...