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

Soft state representations assign states nonnegative class weights that sum to one, but agreement with reference labels does not guarantee accurate prediction of the next weight vector. We derive lower confidence bounds on the minimum population root-mean-square prediction error over linear predictors with a prescribed spectral-norm limit. The bounds compare within-class successor variation with retained within-class input variation, with an alternative based on reference-label deviation. They are estimated from independent transition pairs without fitting a prediction matrix to the evaluation data. A bound above a chosen error tolerance excludes that accuracy for every admissible predictor at the stated confidence level; a bound of zero does not establish whether an accurate predictor exists. For coordinates constructed from Kernel Affine Hull Machine reconstruction scores, we relate the reference class's affinity advantage over competitors to label agreement. The class-centered population bound equals the optimum in a four-state example. Using 117,000 previously generated evaluation datasets from fixed four-state designs, we compare the finite-sample certificates with an empirical-risk-minimization baseline. For one Van der Pol representation selected before evaluation, the label-based bound is positive in all 64 independent evaluation datasets at the fitted matrix's spectral norm. Separately, exact matrix propagation under deterministic dynamics yields a Koopman interpretation of the coordinate-function span. The certificates help identify forecasting targets that cannot be reached by refitting a fixed representation within a specified predictor class and evaluation distribution.