Entropies of compact subsets and supported measures

What the math gives to ML

The paper provides a precise entropy-amplification mechanism under the probability-measure lift of a dynamical system: a compact state set with positive upper-capacity entropy induces infinite upper-capacity entropy on the space of all probability measures supported on that set. The transferable asset is a construction showing that distribution-valued states can amplify trajectory distinctions into arbitrarily many distinguishable mixture states. A neural implementation is a measure-valued latent or world model represented by particles or weighted prototypes, with the learned state map applied independently to particles and an auxiliary objective preserving ensemble diversity. The theorem predicts a sharp zero-versus-unbounded transition that can be tested by measuring separated trajectory counts as particle count and horizon increase.