I have a knock-off bottle, and I don't think the top is the same size as a Nalgene (mine didn't fit like you suggested). It has a diameter of 6.5mm with an area of 30 sq. mm once you subtract the 2mm center. In comparison, the 2"x3" Ultimate Survival mirror is 7x4.5mm with an area of 28.5 sq. mm once you subtract the targeting circle and the corner lanyard notch.
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Experience is a hard teacher because she gives the test first, the lesson afterwards.