We will exhibit a vector field on $\mathbb{R}^{d}\backslash Q^{\lambda}_{\{0\}}(\epsilon)$ with values in $\partial\Delta^{\vee}_{\lambda}\subset\mathbb{R}^{d}$. Its integral curves yield the desired foliation $\mathcal{F}$. Recall from Lemma 2.3 that there is a subdivision of $\partial\Delta^{\vee}_{\lambda}$ which is combinatorially isomorphic to $\operatorname{bsd}(S)\times T$, restricted to $|\Sigma|$. In this context, the projection $p_{2}$ can be thought of as defined $\partial\Delta^{\vee}_{\lambda}\rightarrow{\partial\Delta^{\vee}}$, and can be interpreted as a vector field on $\partial\Delta^{\vee}_{\lambda}$. We will deform $p_{2}$, and extend the deformed vector field to $\mathbb{R}^{d}\backslash Q^{\lambda}_{\{0\}}(\epsilon)$.

For the deformation part, we use that the faces of $\operatorname{bsd}(T,T)$ are parameterized by pairs $(\tau,$ $\mathfrak{T}$ $=\tau_{0}\prec\ldots\prec\tau_{r})\in T\times\operatorname{bsd}(T)$ with $\tau\preceq\tau_{0}$ (cf. § 2.2).

We will apply Lemma 3.5 for $K=p_{2}(\partial\mathcal{V})$, and $N=p_{2}(N_{2}(\partial\mathcal{V}))$, where $N_{2}(\partial\mathcal{V})$ is a neighborhood of $\partial\mathcal{V}$ in $\Sigma$.

Now we are ready to define the vector field $\mathfrak{X}^{\delta}$ on $\partial\Delta^{\vee}_{\lambda}$ as the composition $\mathfrak{d}^{\delta}p_{2}\colon\partial\Delta^{\vee}_{\lambda}\rightarrow{\partial\Delta^{\vee}}$. In order to extend $\mathfrak{X}^{\delta}$ to both sides of $\partial\Delta^{\vee}_{\lambda}$, we present a polyhedral subdivision of a neighborhood of $\partial\Delta^{\vee}_{\lambda}$ whose trace on $\partial\Delta^{\vee}_{\lambda}$ realizes the restriction of $\operatorname{bsd}(S)\times T$ to $|\Sigma|$.

Denote by $\lambda^{\epsilon}\in\mathbb{R}^{\Delta\cap(\mathbb{Z}^{d})^{*}}$ the vector given by $\lambda^{\epsilon}(0)=\lambda(0)$, and $\lambda^{\epsilon}(v)=\lambda(v)+\epsilon$ for $v\in\operatorname{vert}(S)$. Suppose that $\epsilon>0$ is small enough to ensure that $\lambda$ and $\lambda^{\epsilon}$ induce the same triangulation. Then $\Delta^{\vee}_{\lambda^{\epsilon}}\subset\Delta^{\vee}_{\lambda}$ are combinatorially equivalent. For a simplex $\mathfrak{S}$ $\in\operatorname{bsd}(S)$, denote $\mathfrak{S}$_ϵ the corresponding simplex of $\operatorname{bsd}(\Delta^{\vee}_{\lambda^{\epsilon}})$. (I.e., $\mathfrak{S}$ $\subset(\mathbb{R}^{d})^{*}$, while $\mathfrak{S}$${}_{\epsilon}\subset\mathbb{R}^{d}$.) Given a simplex $\tau\in T$ with $\langle$$\mathfrak{S}$$,\tau\rangle=1$, we can form the Minkowski sum $\mathfrak{S}$${}_{\epsilon}+\mathbb{R}_{\geq\epsilon/2}\tau$. These fit together to form a complex of (unbounded) polyhedra which subdivides $\mathbb{R}^{d}$ outside $\Delta^{\vee}_{\lambda^{\epsilon/2}}$. (It actually refines the subdivision into $Q^{\lambda^{\epsilon/2}}_{\sigma}$’s provided by the non-Archimedean amoeba for $\lambda^{\epsilon/2}$.)

We need to argue that this determines a continuous vector field. The problem may arise only when we try to assign a vector to a point $n\in$ $\mathfrak{S}$${}_{\epsilon}+\mathbb{R}_{\geq\epsilon/2}\tau$ such that there are two points $n_{1},n_{2}$ which already have a vector assigned to them, so that both $n-n_{1}$ and $n-n_{2}$ are a multiple of $\widehat{\tau}$. Then $n_{1}\in$ $\mathfrak{S}$${}_{\epsilon}+[\epsilon/2,\epsilon]\tau_{1}$ for some face $\tau_{1}\prec\tau$, and $\mathfrak{X}^{\delta}(n_{1})$ is defined as $\mathfrak{X}^{\delta}(n_{1}^{\prime})$, where $n_{1}^{\prime}\in\partial\Delta^{\vee}_{\lambda}$, and $n_{1}-n_{1}^{\prime}$ is a multiple of $\widehat{\tau}_{1}$. Furthermore, $n_{2}\in\partial\Delta^{\vee}_{\lambda}$, and $n_{2}-n_{1}^{\prime}$ is a multiple of $\widehat{\tau}-\widehat{\tau}_{1}$. Here we use the assumption that $\delta<1/2$ to conclude that $\mathfrak{X}^{\delta}(n_{2})=\mathfrak{d}^{\delta}p_{2}(n_{2})=\mathfrak{d}^{\delta}p_{2}(n_{1}^{\prime})=\mathfrak{X}^{\delta}(n_{1}^{\prime})$.

The integral curves of $\mathfrak{X}^{\delta}$ foliate $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda^{\epsilon/2}}$. The following lemma summarizes the main properties of $\mathfrak{X}^{\delta}$ and the foliation $\mathcal{F}$.