Proposition: Parametrization dependence

Let $r=f(s)$ with $f^{\prime }>0$. Keeping the same unparametrized graph family,

Consequently, the unparametrized slices and the cone packet do not determine the branch-normal speed without the parameter calibration.

Proof. The spatial gradient is unchanged at corresponding slices, while ${\partial }_{r}F=({\partial }_{s}F)/f^{\prime }(s)$. Apply (eq:normal-speed). For example, $r=2s$ preserves the slices and their ordering but halves the speed. Lemma lem:factor gives the final assertion.