We prove that Assumption A1 of Abbott (2026b)---the existence of a positive eigenvalue gap $\lambda_1(W_{T_0}) - \lambda_2(W_{T_0}) \geq \delta_0 > 0$ at the ergodic-to-annealing transition---holds as a theorem rather than an assumption, eliminating the sole remaining external hypothesis from the CGHL framework. The proof proceeds in three steps. We show that the $N$-step transition kernel of the weight process is absolutely continuous on the cone of positive definite matrices $S^N_+$ (Lemma~1), that the stationary measure $\pi$ therefore assigns zero mass to the degenerate set $\{\lambda_1 = \lambda_2\}$ (Lemma~2), and that Harris recurrence then guarantees almost sure finite-time entry into any set of positive gap bounded away from zero (Theorem). The P\'eclet mechanism---the coherence gate's preferential amplification of the dominant eigendirection---is identified as the qualitative driver of gap magnitude, consistent with empirical ratios $\lambda_1/\lambda_2 \in [4.4, 13.2]$ across 20 independent trajectories. Together with Papers 1 and 2, this completes a fully assumption-free mathematical theory of coherence emergence and directional memory consolidation under CGHL.