Detecting Slow Cycles in Historical Panels: An In Vitro Study

Secular cycles, oscillations of political instability with periods of one to three centuries, are a central claim of structural-demographic theory and the target of the Koopman programme of this series (WP-2026-005, WP-2026-006). Before a historical test can be read, one must know whether it could have seen a cycle if there were one. We answer this question in vitro. We plant a slow cycle of known period and coherence in simulated societies, and we measure the power of a size-calibrated pooled Hankel-DMD test against a null twin without the cycle. We also measure the forecast value of the detected cycle for crisis onsets 10 to 200 years ahead. The design, the hypotheses and the decision rules were preregistered, and the verification was run once on fresh seeds. For a 200-year cycle with coherence time three periods and moderate observation noise, observed at decadal cadence with a 70-year delay window, power 0.8 needs about 350–390 pooled decadal columns, however they are split between societies and years: 32 × 200 years, 8 × 500, 4 × 1,000 or 2 × 2,000. The strict historical panel of independent states since 1789 sits at the edge, with power 0.79 [0.73, 0.84] (a post hoc interval; the preregistered rule counts detections, §6). A noisy proxy of the slow driver lifts it to 1.00. The persistence criterion used in WP-2026-006 has power at most 0.07 on its own premise. Once detected, the cycle's phase beats a renewal model by 0.16 of Brier skill at 100 years, as preregistered; in an exploratory comparison it falls short of a hazard model on the observed state, which keeps positive skill up to 200 years. In vitro, long-horizon skill comes from the observed state and the model class, not from spectral detection as such. Deposited with the paper (see README.md): the simulation and analysis code of the study at its verification tag, with the fixed seeds; the verified result tables; the scripts of Tables T2 and A1 and of Figures F1–F5; the preregistered protocol, frozen version 1.0 (24 September 2026), with the register of the changes made after the freeze and before the first verification run; in a separate folder marked as exploratory, a robustness check run outside the protocol and the output of the linear benchmark cited by WP-2026-006 version 2; and, in a folder marked as post hoc, the script and output of the paired differences of forecast skill quoted in §7. The paper is in vitro throughout; its historical counterpart is WP-2026-008 (in preparation).