For $0<\delta<1/2$, the vector field $\mathfrak{X}^{\delta}\colon\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda^{\epsilon/2}}\rightarrow{\partial\Delta^{\vee}}\subset\mathbb{R}^{d}$ is the unique vector field which agrees with $\mathfrak{d}^{\delta}p_{2}$ on $\partial\Delta^{\vee}_{\lambda}$, and is invariant with respect to $\widehat{\tau}$ on the polyhedron $\mathfrak{S}$${}_{\epsilon}+\mathbb{R}_{\geq\epsilon/2}\tau$.