Let $\Gamma$ be the graph with vertices the maximal cones of $\Sigma$ (hence the maximal cones of $\widehat{\Sigma}$) and with an edge between $\sigma$ and $\sigma^{\prime}$ if and only if $\sigma\cap\sigma^{\prime}$ is a common face of codimension one. Note that since $Z$ is proper, if $\mathbb{S}\subset N_{\mathbb{R}}$ is a sphere with center the origin, then $\Sigma\cap\mathbb{S}$ is a triangulation of $\mathbb{S}$. In particular, $\Gamma$ is the 1-skeleton of the dual complex of a triangulation of the sphere, and is thus connected.

Let $\sigma_{0}\in\Sigma$ be a maximal cone, and $p_{0}\in\Gamma$ the corresponding vertex, that we will use as a reference point. We fix a tuple of sections $s^{\sigma_{0}}$ of $W^{\sigma_{0}}$ as in Section 2.3.
Let $\sigma\in\Sigma$ be a maximal cone, and $p\in\Gamma$ the corresponding vertex. By connectedness of $\Gamma$, there exists a path $\gamma$ from $p_{0}$ to $p$, hence a sequence of maximal cones $\sigma_{0},\ldots,\sigma_{q}=\sigma$ such that $\sigma_{h}\cap\sigma_{h+1}$ is a codimension one face of both $\sigma_{h}$ and $\sigma_{h+1}$, for $h=0,\ldots,q-1$. The construction of Section 2.4 allows us to construct inductively along $\gamma$ a tuple of sections $s^{\sigma_{h}}$ of $W^{\sigma_{h}}$.

This provides us with a tuple of sections $s^{\sigma}$ of $W^{\sigma}$ for each maximal cone $\sigma\in\Sigma$, and the function

$$w_{\sigma}=t\cdot\prod_{j\in J}(s_{j}^{\sigma})^{-1}\cdot\prod_{i\in L_{\sigma}}(s_{i}^{\sigma})^{-1}\,\in\mathcal{O}(\mathfrak{X}_{\sigma})^{\times}.$$

By Eq. 2.4.4 the $w_{\sigma}$ glue to an invertible function $w$ on $\mathfrak{X}$; $w$ admits a $(n+1)$-th root on $Z$, since it is constant, and by Hensel’s lemma we obtain an invertible function $w^{\prime}$ on $\mathfrak{X}$ such that $(w^{\prime})^{n+1}=w$. We use the sections $s^{\sigma}$ and the function $w^{\prime}$ to define a morphism

$$f_{\sigma}:\mathfrak{X}_{\sigma}\longrightarrow\mathfrak{N}_{\sigma}$$

as follows. Denoting by $((\varepsilon_{i})_{i\in L_{\sigma}},(\varepsilon_{j})_{j\in J})$ the dual basis to $((v_{i})_{i\in L_{\sigma}},(v_{j})_{j\in J})$, the toric chart $\mathfrak{N}_{\sigma}$ has the following explicit description:

$$\mathfrak{N}_{\sigma}=\Spf R[\chi^{\varepsilon_{i}},i\in L_{\sigma}][[\chi^{\varepsilon_{j}},j\in J]]/\{t-\chi^{\sum_{i\in L_{\sigma}}\varepsilon_{i}+\sum_{j\in J}\varepsilon_{j}}\}.$$

Indeed, $\mathfrak{N}_{\sigma}$ is the formal completion along $Z$ of $\mathcal{N}_{\hat{\sigma}}\times_{\mathbb{A}^{1}}R$, where $\mathcal{N}_{\hat{\sigma}}=\Spec k[(\hat{\sigma})^{\vee}\cap\hat{N}]$; since $\ord(t)=\sum_{i\in L_{\sigma}}\varepsilon_{i}+\sum_{j\in J}\varepsilon_{j}$ on $\widehat{\sigma}$, the relation $t=\chi^{\sum_{i\in L_{\sigma}}\varepsilon_{i}+\sum_{j\in J}\varepsilon_{j}}$ holds.
The map $f_{\sigma}$ is now defined at the level of function rings by

	$\displaystyle f^{\#}_{\sigma}:\mathcal{O}(\mathfrak{N}_{\sigma})$	$\displaystyle\longrightarrow\mathcal{O}(\mathfrak{X}_{\sigma})$
	$\displaystyle\chi^{\varepsilon_{i}}$	$\displaystyle\mapsto w^{\prime}s^{\sigma}_{i}\;\textrm{ \quad for }i\in L_{\sigma}$
	$\displaystyle\chi^{\varepsilon_{j}}$	$\displaystyle\mapsto w^{\prime}s_{j}^{\sigma}\;\textrm{ \quad for }j\in J$

where the sections $s^{\sigma}$ are viewed as functions on $\mathfrak{X}_{\sigma}$ thanks to the proof of Lemma 2.3.1.

This concludes the proof of Theorem B: $\mathscr{X}$ is toric along $Z$.