Stabilisation Dynamics XII: Spatial Structure and Stability of Early-Time Prediction in Stabilisation Dynamics

This paper extends the projection-based predictive framework for stabilisation dynamics developed in Stabilisation Dynamics XI to spatially coupled systems and to the analysis of prediction stability under perturbations. Building on the unstable-mode projection mechanism and its operator formulation, we show that early-time observations determine not only outcome identity, probability structure, temporal evolution, and predictive reliability, but also spatial organisation and robustness. In spatially extended systems, mode amplitudes evolve under both exponential growth and coupling, transporting early-time projection structure across space. We demonstrate that evolving early-time observations under the stabilisation operator accurately recovers final spatial outcome patterns, substantially outperforming local baselines that neglect coupling. Both sitewise accuracy and global domain structure are shown to be determined by early-time projection geometry. We further examine prediction stability under perturbations of the observed state. Numerical results show that sensitivity decreases with observation time, revealing a transition from a high-sensitivity regime to a stable prediction regime as separation between competing modes increases. Ensemble-level probability structure remains robust even at early times, while sample-level outcome and timing predictions stabilise progressively. Together, these results extend the predictive hierarchy established in Stabilisation Dynamics XI, demonstrating that predictive structure is generated early, transported across space, and stabilised over time. Outcome, probability, timing, spatial organisation, and robustness are shown to arise from a common mechanism of unstable-mode competition governed by operator dynamics and spectral separation. Part of the Stabilisation Dynamics Framework.