We construct $Q^{n}$ inductively by dimension $n$. If $n=0$ then $H^{\circ}$ is a point and $Q^{0}=H^{\circ}$. Assume that $Q^{k}$, $k<n$ is already constructed. Consider the simplex

Each its $k$-dimensional face is dual to a $(n+1-k)$-cell of $\Sigma_{n}$. Fix a sufficiently large number $R_{n}>0$.

First we define $Q^{n}\cap\operatorname{Log}^{-1}(\partial\Delta(R_{n}))$. Each $k$-face of $\Delta(R_{n})$ is contained in a unique affine $k$-space $A$ in $\mathbb{R}^{n+1}$. Furthermore, the adjoint faces cut the polyhedron $\Delta_{k-1}(R_{n})\subset A$. Thus we may identify $A$ with $\mathbb{R}^{k}$ and, therefore, $\operatorname{Log}^{-1}(A)$ with $(\mathbb{C}^{*})^{k}$. By the induction assumption we already have $Q^{k-1}\subset(\mathbb{C}^{*})^{k}\to\mathbb{R}^{k}$. We define $Q^{n}\cap\operatorname{Log}^{-1}(\partial\Delta(R_{n}))$ to be equal to the union of these $Q^{k}$ for all faces of $\partial\Delta(R_{n})$. By the induction hypothesis (and since $R_{n}$ was large enough) the choices over different faces agree.

Our next step is to extend $Q^{n}$ to the complement of $\operatorname{Log}^{-1}(\Delta(R_{n}))$. For each face $\Delta^{\prime}$ of $\partial\Delta(R_{n})$ consider its outer normal cone $C_{\Delta^{\prime}}\subset\mathbb{R}^{n+1}$ (e.g. if $\Delta^{\prime}$ is a facet then $C_{\Delta^{\prime}}$ is a ray). We define

In other words, we span the region above the normal cone of a $k$-face $\Delta^{\prime}$ by the translates of the manifold $Q^{k}$.

We set $Q^{n}\cap\operatorname{Log}^{-1}(\Delta(R_{n}-1))=H^{\circ}\cap\operatorname{Log}^{-1}(\Delta(R_{n}-1))$. By now we have defined $Q^{n}$ everywhere, but $\operatorname{Log}^{-1}(\Delta(R_{n})\smallsetminus\Delta(R_{n}-1))$.

Consider a facet $\Delta^{\prime}$ of $\partial\Delta(R_{n}-1)$, e.g. the one sitting in the hyperplane $A=\{x_{n+1}=R_{n}-1\}$. Since $R_{n}$ is large enough, $z_{n+1}^{R_{n}-1}$ is small enough and the intersection $H^{\circ}\cap\operatorname{Log}^{-1}(A)$ is close enough to the zero set of $z_{1}+\dots+z_{n}+1=0$. By the induction hypothesis this zero set can be deformed to $Q^{n-1}$. We define $Q^{n}\cap\{\operatorname{Log}|z_{n+1}|=t\}$, $-R_{n}\leq t\leq-R_{n}+1$ using this deformation. We repeat the same procedure for all other facets of $\Delta(R_{n}-1)$. ∎