Distinct-angle perturbation conjecture for the layered geodesic-net construction

For the finite sequence of layer angles $\varphi_i(\varphi)$ in the construction $G_n(\varphi)$, take a sufficiently small nonzero parameter $\varphi\in(-\epsilon,\epsilon)$. Distinct-angle perturbation conjecture. The sequence $\varphi_i(\varphi)$ never repeats itself, and consequently $G_n(\varphi)$ is a geodesic net whose edges all have weight one. This is presented as a consequence of the preceding derivative nonrepetition conjecture, rather than as an independent claim; it remains conditional on that conjecture.