Stabilisation Dynamics XIII: Predictability and Operator Structure in Stabilisation Dynamics

This paper develops a unified operator-based interpretation of predictability in stabilisation dynamics. Building on the projection-based predictive framework established in Stabilisation Dynamics XI–XII and on the operator and projection structure developed in Stabilisation Dynamics VII and IX, we show that outcome selection, probability structure, temporal prediction, spatial organisation and predictive robustness arise from a common mechanism of unstable-mode competition governed by operator dynamics and spectral separation. Early-time projection onto unstable modes selects competing directions of evolution, while operator dynamics transport and amplify this structure across time and space. Spectral gaps suppress subdominant modes, stabilising prediction and reducing sensitivity to perturbations over time. Within this framework, predictive behaviour is interpreted as a structural consequence of projection-based spectral competition rather than as an independent statistical feature. This paper provides a unified synthesis of the prediction framework developed in earlier work and establishes a common operator-based interpretation linking outcome, probability, timing, spatial structure and robustness in stabilisation dynamics. Part of Stabilisation Dynamics Framework.