Confront your landlord and tell him/her that you’re taking them to court; remind him/her that, in a court of law, the accuser (i.e., your landlord) is required to produce evidence of their claims against the accused (i.e., you).

Time to start a VLOG

Correct, OP; there is no way to prove a negative, duh! Have the cops stop by to fill out a search report, perhaps? Good luck!

Take video and/or photographic evidence of every square millimeter of your place. Provide it to them as proof.

You’re being ghosted in the other way, aka, haunted!

Ask for proof that someone else is living there.

Well, at least if you move you will finally get away from that ghost (and his rent increasing ways)

The onus is on the landlord to prove that someone is living with you, not the other way around. In any event, you can't prove the absence of something. Good luck OP. Don't let them push you around!

Get a nanny cam. each new apartment my dad would tell us pay for a change of locks. maybe the last renter comes by?