Beyond the Lyapunov Wall: Extending the Predictive Horizon of Computational Macrohistory. A Methodological Roadmap

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. Version 2 (October 2026). This version carries a Revision note that lists every change from version 1 (doi:10.5281/zenodo.20111272, 10 May 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, dates and version number, references to the series by concept DOI, typesetting in the series template, two entries of the reference list. Content notes, which leave the passages as published and mark them: (C1) Conjecture 1 is now Theorem 1 of WP-2026-006. (C2) In stochastic systems, persistent modes decay slowly; they do not lie on the unit circle. (C3) The test proposed in §5 did not have sufficient power. (C4) The Lyapunov Wall and Theorem 1 of WP-2026-001. (C5) Two citations in §3.2 misstate their sources. (C6) Two figures in §3.3 and §5 do not match. (C7) A result on reservoir computing in §3.3 is misstated. (C8) The reference for kernel DMD in §4.4.