Geometric Learning Dynamics

We present a unified geometric framework for modeling learning dynamics in physical, biological, and machine learning systems. The theory reveals three fundamental regimes, each emerging from the power-law relationship $g \propto \kappa^\alpha$ between the metric tensor $g$ in the space of trainable variables and the noise covariance matrix $\kappa$. The quantum regime corresponds to $\alpha = 1$ and describes Schr\"odinger-like dynamics that emerges from a discrete shift symmetry. The efficient learning regime corresponds to $\alpha = \tfrac{1}{2}$ and describes very fast machine learning algorithms. The equilibration regime corresponds to $\alpha = 0$ and describes classical models of biological evolution. We argue that the emergence of the intermediate regime $\alpha = \tfrac{1}{2}$ is a key mechanism underlying the emergence of biological complexity.