Lemma 55.3.7. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type of genus $g$. Assume $n > 1$. If $i$ is such that the contribution $m_ i(w_ i(g_ i - 1) - \frac{1}{2} a_{ii})$ to the genus $g$ is $< 0$, then $g_ i = 0$ and $a_{ii} = -w_ i$.

Proof. Follows immediately from Lemma 55.3.6 and $w_ i > 0$, $g_ i \geq 0$, and $w_ i | a_{ii}$. $\square$

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