Lemma: Active region

For all transverse points,

Thus:

1. for $s\le 0$, $F_s=s+\min \{2,h\}$;

2. for $s>0$, the branch $s+k$ can dominate $h$ only where $\rho \le 30s/11<3$;

3. for every parameter and $\rho \ge 3$, $F_s=2+s$.

Proof. Use $\min \{\xi ,\eta \}\le \rho /2$. For $s\le 0$, $s+k\le h$. For $s>0$, $s+k\ge h$ implies $s\ge 11\rho /30$. Finally $h\ge 2$ when $\rho \ge 3$.