CMH: Axiomatic Foundations for a Quantitative Science of Large-Scale Social Systems

In December 1989, few analysts predicted the Soviet collapse within two years. The failure was not one of intelligence but of framework: available models did not distinguish structural forecasts (rising probability of instability) from event-level predictions (the exact date). This distinction, which Computational Macrohistory makes axiomatic, is the central contribution of this paper. We establish eight axioms (A1 through A8) as admissibility constraints for valid CMH models, and we extend the foundational treatment of the preceding versions with four formal theorems. Theorem 1 derives predictive decay from endogenous indeterminacy and isolates the linear divergence regime from the saturating accuracy baseline. Theorem 2 formalises the aggregation condition into an explicit sample-size threshold under intra-class correlation, yielding the CMH Herding Threshold: a correlation ceiling beyond which aggregation fails regardless of population size. Theorem 3 applies the Banach fixed-point theorem to reflexive feedback under the Wasserstein-1 metric, establishing conditions for self-consistent predictions and an empirical identifiability strategy via regression discontinuity. Theorem 4 anchors the falsifiability requirement to the Murphy decomposition of the Brier score. A formal bistable model with noise-induced switching is developed in Appendix C, with Feller boundary analysis confirming domain invariance. The axioms do not describe the world; they define the conditions under which historical systems become scientifically tractable.