This paper derives the operator origin of projection-based probability in stabilisation dynamics. Building on the continuum and operator framework developed in earlier work, and on the projection-based mechanism identified in Stabilisation Dynamics VIII, we show that the projection coordinates governing outcome selection arise directly from 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 not an additional assumption but a direct consequence of the operator formulation. It provides the second core mechanism underlying prediction in stabilisation dynamics and forms the foundation for subsequent work on statistical competition and finite-temperature structure. Part of Stabilisation Dynamics Framework
Luke Found (Fri,) studied this question.