Our biggest claim to the arctic land and waters surrounding that land is the fact that for a few decades now we've maintained an Arctic Ranger patrol group. Basically, the military gives some of the locals free weapons and ammo that they use to 'protect our sovereignty' (and hunt for free) and they keep records of what they see, etc, during their hunting expeditions. The hope is that when the day comes and we find ourselves in an international court over this issue these records will help assert our claim to the arctic territory (as no other country has anything like them I believe). These coupled with some of the other stuff we do like fly a jet up there every few years, etc. However, unless Harper actually does something about it we don't have any sort of military might to back up our claim as of now.
I guess one time during one of the arctic patrols one of the rangers saw what he thought to be a submarine surfaced through the ice but when he brought a larger party out to confirm this it was gone

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