Stabilisation Dynamics IX: Operator Origin of Projection-Based Probability in Stabilisation Dynamics

This paper derives the operator origin of projection-based probability in stabilisation dynamics. Building on the projection-based mechanism identified in Stabilisation Dynamics VIII, it further develops the mechanism-based description of resolution by linking projection coordinates directly to the spectral structure of the linearised stabilisation operator. Near an unstable equilibrium, dynamics reduce to exponential growth along unstable eigenmodes. Projection of the initial perturbation onto these modes determines the direction of evolution, while nonlinear saturation preserves this dominance, leading to outcome selection. Probability arises as a geometric measure over regions of unstable-mode space induced by the preparation ensemble. This establishes a direct link between operator spectrum, projection geometry and statistical behaviour, showing that projection-based probability is a consequence of the operator formulation rather than an additional assumption. It provides the second core mechanism underlying prediction in stabilisation dynamics and introduces the connection to statistical competition structures developed in subsequent work. Part of Stabilisation Dynamics Framework