S-COT State-Cycle Observation Theorem

We present a minimal, falsifiable rule for classifying a bounded scalar order parameter ξ(t) ∈ [0,1] into three regimes: Origin (ξ near a reference floor), Momentum (rapid change), and Position (ξ near a reference ceiling and locally stable), using two order-parameter thresholds, an activity threshold, a stability tolerance, and a smoothing window, all fixed before any dataset is examined. We prove the classification is well-posed under continuity and a monotone-crossing assumption (Theorem 1), extend it with hysteresis and a minimum-dwell-time filter for noisy and non-monotonic signals, and specify a domain-general reindexing operation R that turns a terminal Position state into a fresh Origin state for a following cycle. We benchmark the rule, using pre-registered default thresholds, against five standard alternatives: 2/3/4-state Gaussian hidden Markov models, PELT change-point detection, Bayesian online change-point detection, and continuous (global-mean and AR(1)) baselines, on three independently constructed simulated systems: a stochastically driven phase transition, a sequential Bayesian evidence-accumulation process, and an ecological succession model with disturbance. Scored by two-part minimum description length, the rule does not achieve the best compression in any of the three systems. A simple autoregressive model wins throughout, and we report this as a substantive negative result rather than concealing it. Scored by rolling-origin one-step-ahead prediction, the rule is competitive with or better than the change-point and hidden-Markov baselines in two of the three systems and is dominated by the autoregressive and change-point baselines in the smoothly drifting ecological system. We report a full sensitivity analysis over all five threshold parameters, a noise-robustness study, and a pre-registration-style protocol for testing the framework against real data. We conclude that the framework is an interpretable, zero-refit special case within the existing regime-switching and change-point literature, with a narrow, domain-conditional predictive advantage rather than a general-purpose forecasting advantage, and we state this scope explicitly rather than implying a broader claim.