True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics

The paper defines Dstsp as a KL divergence over state space and DH as a Hellinger distance over power spectra, and argues empirically that (i) treating dimensions independently degrades long-run geometric fidelity and (ii) context parroting yields nearly periodic spectra that disagree with chaotic ground truth. However, it does not provide a formal impossibility statement. The candidate solution supplies a clean measure-theoretic argument: for any factorized generator ν, D(μ||ν) ≥ Iμ > 0 when the true invariant law μ is non-product; and if the true spectrum is absolutely continuous but the generator is periodic (discrete spectrum), the Hellinger distance on that coordinate is 1, forcing DH > 0. These conclusions align with the paper’s definitions of Dstsp (eq. 12–13) and DH (eq. 14) and its qualitative failure modes (dimension misalignment and context parroting), but extend them into a rigorous proof of the stated claim, which the paper itself does not include .