Disentangling Topology from Measure: Certified Koopman Spectral Learning

Data-driven Koopman models learn finite-dimensional representations of nonlinear dynamics from trajectories sampled under a particular state-visitation distribution, yet deployment may change which regions of state space are visited while leaving the underlying dynamics fixed. Standard smooth dictionaries entangle this distribution-dependent fitting geometry with the transition structure of the dynamics inside the same Galerkin matrices. We ask which representation structures can separate these two sources of variation and remain reusable and certifiable under sampling shift. We show that measure-universal Gram diagonality is equivalent to pairwise disjoint support; on a compact connected state space, a dense continuous dictionary cannot retain this property, while indicators of refining generating partitions provide a canonical realization (Theorem 1). This representation-level separation yields the Topological Koopman Certificate (TKC), with non-asymptotic residual and pseudospectral guarantees (Theorems 2,3), a TV-linear operator-perturbation bound with sparse support-preserving reweighting under measure shift (Theorem 4), and an $\mathcal{O}(KN)$ Krylov pseudospectral solver in place of a dense $\mathcal{O}(N^3)$ generalized eigenvalue computation. We turn the structural principle into a data-driven construction through Generative Koopman Decoupling (GKD, Algorithm 1), which uses either an analytic or learned preimage oracle to form forward–backward itinerary partitions. Proposition 1 links oracle error to partition resolution in the uniformly expanding regime. Controlled low-dimensional systems evaluate the certificates under matched assumptions, while Lorenz-63 and Hénon test the learned representation and oracle mechanism empirically beyond that regime. On Lorenz-63, the learned-itinerary representation attains the lowest in-distribution prediction MSE in our comparison using only about $25$ occupied itineraries; on Hénon, learned- and analytic-oracle constructions give statistically indistinguishable predictions.