The Entry of π: The Conditions That Force Compact Closure

The Entry of π: The Conditions That Force Compact Closure asks not where π appears, but what conditions make π unavoidable. The paper argues that π is usually defined too late, after the circle has already appeared, and instead proposes that π enters when distinction becomes boundary-bearing, when boundary creates an inside/outside relation, and when persistence requires completed return. The central claim is that π is the compact-closure invariant of maintained distinction. A bare difference do