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Also, just because someone has a parking lot, are they then able to micromanage the contents of my vehicle?
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If the parking lot in question is on someone's private property, yes, they can 'micromanage' what they tolerate on their property, be it in a vehicle or not. More correctly, they can dictate which vehicles are allowed on their property and which aren't.
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Also, isn't my car MY property? My own little moving sphere of influence.
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If the local nuclear power plant puts their radioactive waste on wheels, does that mean they can put it in my backyard? Isn't it their own "little moving sphere of influence"?
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That's fine. Like I said, I can see both sides. There are some states that treat the car as an extension of your home, so my example is not that far fetched.
http://www.packing.org/state/new_mexico/#statecar_law
Your example is a bit silly, but if you allowed parking in your backyard, and the beta particles (or the gun) never left the vehicle, and no harm was done, then maybe that's ok.