Koopman–CMH: A Formal Treatment of Spectral Structure Persistence in Historical Dynamics

This paper proves the Koopman–A2 Correspondence (Theorem 1) — the formal result announced as Conjecture 1 of WP-2026-005 — and conducts the first pre-registered empirical test of its substantive antecedent. Theorem 1 shows that shared ergodic-class membership implies shared leading Koopman spectrum, conditional on Axioms A2, A3, A4, and a forcing-homogeneity assumption that refines the antecedent stated in WP-2026-005. The empirical test applies a canonical Hankel-DMD estimator to a panel of eight Western European polities across two structurally stationary sub-windows (1820–1913, 1945–2020). The pre-registered binary criterion rejects Question 1 in both sub-windows: no eigenvalue beyond the invariant-measure cluster meets the persistence threshold. A post-hoc power analysis finds effectively zero power at the available effective sample size — the post-war effective sample size (N_eff = 210) falls below the operator's parameter count — so the verdict constrains detectability, not existence. A credible test requires N_eff ≈ 1,000. Two methodological results give the paper positive content. The invariant-measure mode is recovered with the directional attenuation Proposition 4.2 predicts, on the one mode whose true value the axiomatic structure fixes a priori. A Juglar-band correspondence recurs across the inter-war break but is not statistically distinguishable from noise (37% match rate in surrogate null panels). The paper recommends canonical Tu et al. (2014) DMD with Gavish-Donoho truncation as the default for macrohistorical panel data. Version 2 (October 2026). This version carries a Revision note that lists every change from version 1 (doi:10.5281/zenodo.20679000, 13 June 2026), which was published under the author's pen name, Galen Fontaise; author and institute are the same. Changes of form (F1–F5): author and institute, date and version number, references to the series, keywords and JEL classification (as in version 1), the ORCID, now given on the title page, and typesetting in the series template (mathematical notation that version 1 printed as raw LaTeX code is now typeset). Corrections, which leave the passages as published and mark them: (N1) The requirement "N_eff ≈ 1,000" is withdrawn. (N2) Question 1, as preregistered, is not a valid test for slow cycles. (N3) The statement of Proposition 4.2 is corrected. The evidence for the corrections comes from a preregistered simulation study with planted slow cycles, reported in WP-2026-007. Theorem 1 and its proof, the data, and the preregistered verdict on Question 1 as a record of what was tested are not changed.