Corollary: Parameter regularity of labelled data

On each incidence epoch and its one-sided germ, the finite coefficient, ray, vertex, gradient, chart, length, and area tables are Lipschitz in $\mu$, uniformly over the allowed $s$. The graphs are jointly continuous on the common topological template.

Proof. The finite table formulas are smooth in $\mu$ on a neighbourhood of $I$. Denominators stay away from zero by Lemma lem:bounds. Each active edge radical has a positive continuous coefficient on the compact interval, so its square root is smooth there. Born lengths are smooth coefficients times $s$ and born areas are smooth coefficients times $s^2$. Bounded derivatives prove the uniform Lipschitz assertions. Joint graph continuity follows directly from continuous minimum and maximum on the glued coordinate template.