Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions

Abstract

We study when geometry-induced soft state abstractions admit accurate finite-dimensional linear dynamics. Each state is represented by simplex-valued coordinates obtained from class-specific Kernel Affine Hull Machine (KAHM) reconstruction scores, and a matrix is used to predict the next-state coordinates. Our main result is a computable lower confidence bound on the minimum root-mean-square prediction error over all matrices satisfying a prescribed spectral-norm limit. The bound combines within-class variation of successor coordinates with the deviation of soft coordinates from their one-hot reference labels, and can be evaluated from independent state-successor pairs without fitting a prediction matrix. For fixed coordinates and evaluation distribution, the certificate converges almost surely to a population lower bound as the sample size grows; any tolerance below this limit is eventually certified unattainable. Reconstruction-score margins further control the soft-to-hard assignment error. Under deterministic dynamics and exact coordinate closure, eigenvectors of the closure matrix and its reduced transpose induce Koopman and adjoint Koopman eigenfunctions, respectively. A four-state KAHM construction shows that identical soft coordinates can permit exact closure under one dynamics map yet force positive prediction error under another. Experiments on Duffing, Van der Pol, CartPole, MountainCar, and Acrobot compare direct soft-coordinate prediction with state-space DMD/EDMD baselines and report prediction, representation-variation, and spectral diagnostics. The benchmarks assess fitted models but do not numerically evaluate the exclusion certificate.