Empirical variational principles for preimage entropies

What the math gives to ML

The paper provides a variational principle for the complexity of inverse trajectories in non-invertible dynamical systems: global preimage entropy equals the supremum of empirical preimage entropy over invariant empirical-measure classes. The transferable mechanism is to measure and control how many distinguishable hidden-state histories map to the same present state, while conditioning those histories on comparable long-run visitation statistics rather than counting all inverse branches indiscriminately. This suggests a preimage-entropy regularizer or diagnostic for recurrent state-space models and learned world models, with a falsifiable prediction that inverse-branch growth rates decompose by empirical state-distribution class and are maximized by one or more specific attractor-like visitation measures.