Units, polyhedra, and a conjecture of Satake

Let $F\mathrm{/}\text{QQ}$ be a totally real number field of degree $n$. We explicitly evaluate a certain sum of rational functions over a infinite fan of $F$-rational polyhedral cones in terms of the norm map $\text{Norm}\text{\hspace{0.17em}⁣}:F\to \text{QQ}$. This completes Sczech’s combinatorial proof of Satake’s conjecture connecting the special values of $L$-series associated to cusp singularities with intersection numbers of divisors in their toroidal resolutions.