1. Real scaling
2. Shear
3. Spiral drift
4. Rotation / complex phase
Counter-spiral cancellation
The paper first pairs expanding and contracting real spirals: r₊(θ)=Re^{κθ}, r₋(θ)=Re^{-κθ}. Their product remains R² and their geometric mean remains R. This cancels radial drift, but it is not yet complex phase. Rotation supplies the bounded returning branch.
Generator test
| Motion | Generator G | Key property | Expected result |
|---|---|---|---|
| Scaling | κI | e^{κt} changes norm | Unbounded or decaying; no nontrivial return |
| Shear | [[0,s],[0,0]] | G²=0 | Linear drift; no return |
| Spiral | κI+ωJ | Rotation plus radial drift | Winds inward/outward; no closure when κ≠0 |
| Rotation | ωJ | J²=−I | Bounded, norm-preserving, periodic return |
Interpretation
A successful numerical result does not prove that all physical phase must arise this way. It verifies the limited mathematical statement implemented here: among these standard two-dimensional one-parameter branches, pure rotation preserves radius and returns periodically, while real scaling, shear, and nonzero radial spiral drift do not.