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 (Nₑff = 210) falls below the operator's parameter count — so the verdict constrains detectability, not existence. A credible test requires Nₑff ≈ 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.