Distributional results for the shortest distance between trajectories of different dynamics

What the math gives to ML

The paper introduces a co-dimension for several invariant measures that quantifies how rapidly their local mass neighborhoods overlap as the radius shrinks. This is transferable as a geometric compatibility statistic for latent distributions produced by different augmentations, environments, model branches, or learned dynamics. A practical adaptation is a multiscale overlap regularizer computed on embeddings, paired with variance and covariance constraints to prevent the trivial collapsed solution. The extracted material does not include the paper's extremal-index formula, so the proposed transfer uses the constructive co-dimension definition.