It's why I said "at least theoretically" -- the vaccine cards have a date and vaccine lot number on them. I'm making the assumption that this has been recorded somewhere, such that you could at least check that a certain batch was administered on a certain day to a certain person.

Of course, this could all fail under the "medical records in the USA are garbage" issue.

Still, if nothing else, it's better for proving it than "I took a COVID test at home, trust me".