For any $h^{\vee}>0$, small in the $\nu$-scale, the vector field $\mathfrak{X}_{h^{\vee}}$ induces a (straight line/ray) foliation $\mathcal{F}_{h^{\vee}}$ of $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda}$, smooth over ${\Sigma\backslash D}$, such that

1. (1)

   $\mathfrak{X}_{h^{\vee}}(q)=w$ for $q\in V_{w}^{\beta^{\vee}}$.

2. (2)

   The value of $\mathfrak{X}_{h^{\vee}}(q)$ is in $(\operatorname{carrier}\sigma)^{\vee}\subset\partial\Delta^{\vee}$ for $q\in F_{\sigma}\subset\partial\Delta^{\vee}_{\lambda}$. In particular, $\langle v,\mathfrak{X}_{h^{\vee}}\rangle=1$ in $U_{v}$.

3. (3)

   For any $q\in{\Sigma\backslash D}$, $\left|\nabla\mathfrak{X}_{h^{\vee}}(q)\right|\leq\frac{C|g_{ij}(q)|}{h^{\vee}}$, where $C$ is a constant independent of $\lambda$ and $\nu$, and the gradient and the metric $g_{ij}$ are taken in affine coordinates.