1. Three Problems, One Invariant
Buffon’s Needle
Random orientation sampling
Linear transport intersecting parallel constraints
π appears as orientation phase measure
Parseval’s Theorem
Representation change
Time ↔ mode transport conservation
π² emerges as phase-space area
The Basel Problem
Mode density summation
Discrete phase packing
π²/6 fixed by transport geometry
↓
2. The Shared Structure
3. Why This Matters
These are not separate discoveries.
They are the same transport experiment conducted in different representations:
- Physical space (Buffon)
- Function space (Parseval)
- Mode space (Basel)
π appears because phase does not close under transport.
Nothing else is required.