Working papers and introductory guides

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 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, known in the literature on dynamical systems as the Lyapunov Wall. This paper asks whether mathematical strategies exist that can push the horizon further without abandoning the epistemological rigour on which CMH is founded.

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.