I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear about your dilemma.
If your lease is just for you or limits the number of nights a visitor can stay over (which is common), then your landlord has a right to ask you to remove any unauthorized guests. If they reasonably believe that a third party is living with you, they can give you a notice to remove that person or vacate, and they can sue for eviction. In court, you'd have the burden of proving that your guest lives elsewhere - and the way to do that would be to show proof that the visitor lives somewhere else.
So, while technically you have a right to refuse to produce that information, it could actually save you a lot of time and trouble with your landlord if you give it to them. The other option is to have your visitor leave or come over less often. I realize that the request feels like an invasion of privacy, but if they're worried about someone unauthorized in the property, showing that the person lives somewhere else really is probably the best way for you to resolve it.
The other option would be to have them sign a request to be added to your lease, if they don't have a separate home.
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