The entropy of Gromov-Thurston manifolds and branched coverings

What the math gives to ML

The paper gives a variational decomposition of dynamical complexity into ordinary trajectory entropy plus an explicit branching-multiplicity potential. The transferable asset is that each base trajectory can represent many lifted trajectories, and this extra representational capacity contributes log(m) whenever the trajectory encounters a point with m valid branches. A neural analogue is a hierarchical router or latent rollout whose objective rewards both stochastic route entropy and the logarithm of the number of valid fine-grained realizations, rather than using an unstructured entropy bonus. This provides a falsifiable mechanism for encouraging expressive, well-utilized branching without simply increasing the number of experts or rollout particles.