Maximal pattern complexity and structure of null systems

What the math gives to ML

The paper provides a quantitative characterization of null dynamical systems through maximal pattern complexity: vanishing sequence entropy is equivalent to polynomial growth of observable orbit patterns for every finite open cover, while equicontinuity corresponds to sublinear growth. Its transferable asset is a measurable complexity-growth certificate for recurrent or state-space neural dynamics, stronger than checking one-step Jacobian stability because it probes combinatorial diversity over arbitrary time subsets. A practical translation is to sample finite partitions of hidden-state trajectories, estimate maximal pattern counts over time subsets, and regularize or stop training when empirical growth exceeds a prescribed polynomial envelope. The main falsifiable prediction is a transition from polynomial to superpolynomial pattern growth near a dynamically chaotic or unstable regime.