Lemma: Exact decomposition

These faces partition the transverse graph without overlapping face interiors or omitted regions.

Proof. For $s\le 0$, apply Lemma lem:active. For $s>0$, in $\xi \ge \eta$ the equality $h=s+k$ is

joining $B_i$ to $C_i$. In $\eta \ge \xi$ it is

joining $C_i$ to $B_{i+1}$. The inner dominance region is exactly the union of the two triangles ${U_i}^{\pm }$. Since $\rho (C_i)=30s/11<3$, it is separated from the crest $\rho =3$. The remaining core region is the stated simple pentagon, and the cap gives the horizontal collar and exterior.