Stabilisation Dynamics VIII: Emergent Probability from Unstable Mode Projection in Stabilisation Dynamics

This paper identifies a concrete dynamical mechanism underlying probabilistic behaviour in stabilisation dynamics. It marks the transition from the geometric and operator framework developed in earlier work to a mechanism-based description of resolution and probability. Building on the continuum and operator formulation, we show that outcome selection in bistable systems is governed by projection of initial perturbations onto unstable modes of the linearised stabilisation operator. The sign of the dominant unstable-mode projection determines basin selection, while outcome probabilities arise from the distribution of these projections across an ensemble of initial conditions. For Gaussian perturbations, this yields a closed-form expression for outcome bias. Numerical simulations confirm deterministic basin selection at the level of individual realisations and emergent probability at the ensemble level. This work provides a dynamical realisation of the geometric stabilisation framework, linking basin structure, operator dynamics and statistical behaviour. It initiates a mechanism-based description of probabilistic behaviour in stabilisation dynamics, which is further developed through operator structure and statistical competition laws in subsequent papers. Part of Stabilisation Dynamics Framework