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It said he was shooting 80 meters from his neighbors house. Which is probably close enough to annoy someone, which is the illegal part

Not in any legislation I've seen. There is the guideline of 'not within 500meters of a track on DoC land' which isn't in the legislation but the Conservation Act allows Hunting Permits to be issued...

I highly doubt that it would stand up in court if you refused to let an officer who showed up on your doorstep do a spot inspection then t hey took that as grounds to revoke your license and search...

There's this but it's not related to a routine inspection

I've never found anything in the Arms Act about random inspections, does anyone know what legislation this is covered under if at all?

If not specifically mentioned surely a random inspection...

Specifically,