Working papers and introductory guides

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.

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.

The Computational Macrohistory (CMH) framework, as developed in the preceding papers of this series, operates within a temporal horizon of roughly five to fifteen years for quantitative probabilistic forecasts. This constraint is not a technological artefact to be overcome with faster computation or larger datasets: it is a structural consequence of chaotic dynamics formalised in Axioms A5 and A7 of the CMH axiomatic system, and known in the broader literature on dynamical systems as the Lyapunov Wall. The present paper asks whether, and to what extent, mathematical strategies exist that can push this horizon further without abandoning the epistemological rigour on which CMH is founded. A central claim of this paper is that the question of horizon extension cannot be answered without first specifying what kind of prediction is being considered. We distinguish three levels of the CMH prediction space: event prediction (the trajectory-level forecast of system state at a future time, subject to the full force of the Lyapunov Wall); regime probability (the probability that the system will be operating in a given dynamical regime over a future interval, subject to a substantially slower decay); and spectral structure persistence (the identification of invariant or near-invariant cyclic structures in the system’s dynamics, whose predictive content is not subject to exponential decay but to linear phase uncertainty and structural stability constraints). The five methodological strategies surveyed in this paper operate at different levels of this hierarchy, and the comparison is meaningful only when the level is held constant. We survey five candidate approaches — Ensemble Methods, Slow Manifold theory combined with Critical Slowing Down indicators, Analog Forecasting, Reservoir Computing, and Koopman Operator theory implemented through Dynamic Mode Decomposition (DMD) — organising them into three methodological classes: uncertainty propagation strategies, regime detection strategies, and spectral decomposition strategies. We identify the Koopman Operator / DMD framework as the most promising candidate for integration with CMH at the level of spectral structure persistence, owing to its natural connection with the ergodicity assumption formalised in CMH Axiom A2, its compatibility with the panel structure of historical datasets, and its capacity to produce empirically falsifiable predictions about persistent cyclic structures. The paper concludes with a description of the historical datasets most suited to a first empirical test of Koopman-CMH, including explicit criteria for dataset selection, and an announcement of the companion paper that will develop the full mathematical treatment.

The paper distinguishes three levels of the CMH prediction space: event prediction, regime probability, and spectral structure persistence. Five candidate approaches are surveyed (Ensemble Methods, Slow Manifold theory with Critical Slowing Down indicators, Analog Forecasting, Reservoir Computing, and Koopman Operator theory implemented through Dynamic Mode Decomposition), organised into three methodological classes. The Koopman/DMD framework emerges as the most promising candidate for integration with CMH at the level of spectral structure persistence, and the paper concludes with explicit criteria for the datasets best suited to a first empirical test of Koopman-CMH.

This paper extends the Computational Macrohistory (CMH) framework to eleven countries of the Middle East and North Africa over 2000 to 2012, addressing the structural preconditions of the Arab Spring. Building on the three-country proof of concept that preceded it, we construct a five-component Systemic Stress Index (SSI) over demographic pressure, inequality, youth unemployment, regime type, and connectivity. The SSI is a reduced proxy for testing whether the framework’s structural signals are detectable in historical data; it does not implement the full dynamic system. At the 2010 benchmark the index ranks the instability countries above the stability countries in twenty of twenty-eight pairs, an area under the curve of 0.714 with an exact permutation probability of 0.158, and separates the two groups by 0.290 index units. Threshold classification assigns eight of eleven correctly; with the threshold fixed in advance that accuracy carries an exact probability of 0.197, but against its own threshold-optimised null it sits at the median and carries no evidential weight. The ranking is stable across weighting schemes except under heavy connectivity weighting. Leave-one-component-out analysis locates the discrimination in youth unemployment and the anocracy transformation of the Polity score, which carry half the index weight; the other three contribute nothing or degrade it, and the inequality series is model-imputed across the panel. Three misclassified cases, Jordan, Egypt, and Syria, are examined, each a specifiable limit of the specification. The paper states the conditions under which the expanded programme, at thirty or more validated country-cases, would constitute statistical validation.

We apply the CMH operational framework to three Arab Spring cases: Tunisia (revolution), Egypt (revolution) and Saudi Arabia (stability). Using data from 2000-2012, we construct a Systemic Stress Index comprising five structural variables: youth bulge, income inequality, youth unemployment, regime type and internet penetration. The SSI correctly ranks all three countries by outcome severity: Tunisia (0.52) > Egypt (0.10) > Saudi Arabia (-0.09), and a threshold of SSI > 0 achieves perfect discrimination in this sample. Component decomposition reveals regime type as the primary discriminating variable, and counterfactual analysis indicates that an anocratic Saudi Arabia would have entered the revolutionary zone. Economic stress factors prove necessary but not sufficient for revolution.

Document II of the CMH series translates the axiomatic foundations of Document I into a concrete mathematical apparatus for empirical analysis. It defines a 25-dimensional state space spanning demographic, economic, political, social and psychological-collective variables, each with operationalization protocols and data source specifications. The document presents coupled dynamic equations governing system evolution, derives composite indices including the Systemic Stress Index and the Elite Pressure Index, and specifies probabilistic event functions for instability episodes and regime transitions. Complete derivations, reference parameter values and variable specifications are provided in appendices.

This document establishes the minimal, non-redundant set of axioms necessary for the foundation of Computational Macrohistory. Eight axioms (A1-A8) are formulated as admissibility constraints that any valid CMH model must satisfy, covering statistical aggregation of collective behavior, conditioned historical ergodicity, structural causality, continuous temporal dynamics, endogenous indeterminacy, bounded reflexivity, predictive decay and computational falsifiability. The axioms do not describe the world as it is: they define the conditions under which historical systems become scientifically tractable. A minimal toy model demonstrates that the axiom set is jointly satisfiable.