Stabilisation Dynamics XIV: Deterministic Outcome Selection via Operator and Lyapunov Structure

This paper develops a rigorous operator-based framework for deterministic outcome selection within a class of gradient-like stabilisation dynamics. Building on the operator and projection structure introduced in earlier stabilisation dynamics papers, we establish a local-to-global mechanism linking unstable-mode competition near a hyperbolic equilibrium to final attractor selection. Projection onto unstable modes partitions the local state space into dominance regions associated with competing directions of evolution. We show that this dominance is preserved under nonlinear dynamics within a local neighbourhood, that trajectories exit through mode-dependent boundary sets, and that, under Lyapunov-induced basin separation, these exit sets lie within forward-invariant regions containing unique attractors. Together, these results establish a deterministic outcome-selection mechanism in which initial projection onto unstable modes determines the final attractor for sufficiently small perturbations. Outcome selection is therefore interpreted as a structural consequence of the interaction between local operator dynamics and global energy geometry. This paper provides the first formal mathematical foundation of the stabilisation dynamics framework and establishes the deterministic selection mechanism that underlies subsequent developments of probability, predictability, spectral hierarchy and robustness in later papers of the series. Part of Stabilisation Dynamics Framework.