Ptolemy's Equant Equates to a Universal Dynamical Clock via Machine Learning

What the math gives to ML

The paper offers a data-driven dynamical-clock construction for attractive limit-cycle systems: learn a scalar phase function whose directional derivative along the unknown vector field is constant, \(\nabla\phi\cdot F=\omega\). Its transferable asset is a coordinate in which complicated high-dimensional oscillations become approximately uniform rotation, separating phase progression from amplitude and embedding distortions. A practical neural-network transfer is to regularize recurrent or latent-state dynamics with a learned clock coordinate and use phase-velocity variance as a quantitative monitor for long-horizon instability and regime transitions.