This paper develops the finite–temperature structure of emergent probability in stabilisation dynamics. Building on the continuum and operator framework developed in earlier work, and on the projection-based mechanism established in Stabilisation Dynamics IX, we show that statistical outcome laws arise from a unified geometric competition mechanism acting on unstable mode projections. Near an unstable equilibrium, outcome selection is determined by projection of the initial perturbation onto unstable eigenmodes. This induces a geometric partition of projection space into dominance regions. We show that both hard selection (dominant projection) and quadratic weighting of amplitudes arise as limiting cases of a one-parameter family of exponential competition laws acting on squared projection amplitudes. This family introduces an effective temperature parameter controlling the sharpness of competition between modes. In the zero-temperature limit, hard selection is recovered, while finite-temperature regimes produce distributed outcomes. Quadratic weighting is shown to emerge as an effective, expectation-level description of this finite-temperature competition under Gaussian ensembles. Numerical results demonstrate that exponential competition provides a highly accurate approximation to quadratic weighting across a range of dimensions, anisotropies and correlated ensembles, while also revealing systematic deviations associated with finite-temperature structure. This establishes a unified account of probabilistic behaviour as arising from geometric competition between unstable modes, linking operator-derived projection coordinates to statistical laws through a finite-temperature framework. It provides the third core mechanism underlying prediction in stabilisation dynamics and completes the mechanism layer preceding the prediction series. Part of Stabilisation Dynamics Framework
Luke Found (Tue,) studied this question.