This paper develops a projection-based predictive framework for stabilisation dynamics, extending the mechanism-based description of outcome selection and probability established in Stabilisation Dynamics VIII–X. Building on the unstable-mode projection mechanism and its operator formulation, we show that early-time observations determine not only the final outcome of the system, but also the full probability distribution over outcomes, the time to dominance, and the reliability of predictions. Near an unstable equilibrium, projection onto unstable eigenmodes defines competing growth directions. Outcome selection is determined by the dominant projection, while probabilistic structure arises from the distribution of projection vectors across ensembles, consistent with the dominance-region formulation introduced in Stabilisation Dynamics IX. Extending this framework, we show that temporal prediction follows from exponential growth and spectral separation, and that predictive reliability is encoded in the separation between leading projection amplitudes. Numerical results demonstrate that a projection-based predictor matches the performance of fitted statistical models while outperforming standard baselines, with softmax-like behaviour emerging as an effective approximation to the underlying projection geometry. These results establish a hierarchical predictive structure in which outcome, distribution, timing and reliability all arise from a common mechanism of unstable-mode competition. This paper marks the transition from mechanism to prediction within the Stabilisation Dynamics framework, showing that predictive behaviour emerges directly from the geometry and spectral structure of the underlying operator. Part of Stabilisation Dynamics Framework.
Luke Found (Mon,) studied this question.