Result: π measures residual phase from non-closure under transport.
1. The Classical Story
π is defined as the ratio of a circle’s circumference to its diameter:
This definition is descriptive, not explanatory.
2. Why π Appears Everywhere
π occurs in:
- waves
- orbits
- probability
- quantization
- rotation
This ubiquity is impossible if π is merely geometric.
3. Transport Loops Do Not Close
Consider any motion transported around a loop.
If the frame itself evolves during transport, closure fails.
The residual phase is what we call π.
4. π as Holonomy
In differential geometry, holonomy measures how much a vector fails to return to itself after parallel transport.
π is the scalar holonomy of planar transport.
It measures rotational memory.
5. Buffon’s Needle Revisited
The probability result:
arises from averaging phase over orientations.
No circles are involved.
6. Orbits and π
Orbital motion repeatedly transports velocity through turning frames.
Each loop accumulates phase.
π encodes that accumulation.
7. Waves and π
A wave is linear motion observed through periodic phase slicing.
The factor of $2\pi$ converts transport into phase.
8. Why Circles Are a Special Case
A circle is the only path where curvature is constant.
It produces uniform phase accumulation.
Geometry inherits π — it does not define it.
9. Two Independent Derivations
Method A: Transport Holonomy
π measures phase residue after loop transport.
Method B: Statistical Phase Averaging
π arises from uniform angular measure.
Both agree.
10. Transport-First Summary
π is not geometry.
π is memory.
It measures how motion fails to close.