I think it's sort of a mistake to try to come up with a "real" definition of private property. It's not a physical truth about the universe we can discover if we only try hard enough; it's an agreement we can make amongst ourselves. I mean, there are better and worse ways of defining it, but the goal should be "useful" (like, for social/legal purposes, such that it's fairly clear to everyone what IS considered theirs) rather than "philosophically airtight".

Yes. Even if you do come up with a definitive proof of something you still have the problem of some geezer with a shotgun ignoring all of your logic.