Computational Macrohistory: Operational Framework — State Space, Variables, and Dynamic Equations

This paper translates the axiomatic foundations of Computational Macrohistory (CMH), established in Document I, into an operational mathematical structure for empirical application. It defines a 25-dimensional state space organised into five categories of variables: demographic (D), economic (E), political (P), social (S), and psychological-collective (Ψ), with canonical definitions, measurement protocols, and documented data limitations for each. System evolution is specified through a coupled system of stochastic differential equations, interpreted throughout in the Itô sense, whose drift terms are theoretically motivated by structural-demographic and collective-action theory rather than fitted post hoc. Critical events are modelled through threshold functions that yield probability distributions, never point forecasts. Relative to the first version of this framework, three developments are integrated. The four theorems of Document I now supply quantitative admissibility diagnostics, including the aggregation threshold with the Herding Threshold correlation ceiling and the reflexivity contraction condition. The three-level prediction hierarchy of the Koopman programme (event prediction, regime probability, spectral structure persistence) locates each framework output at its proper epistemic level. The empirical results of the Arab Spring case studies inform a corrected canonical youth-bulge definition, a graded ladder of operationalisation levels connecting the full dynamic system to the reduced Systemic Stress Index, and the specification of the planned fifteen-variable SSI extension.