Lemma 55.3.3. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type. Then the expression
\[ g = 1 + \sum m_ i(w_ i(g_ i - 1) - \frac{1}{2} a_{ii}) \]
is an integer.
Proof. To prove $g$ is an integer we have to show that $\sum a_{ii}m_ i$ is even. This we can see by computing modulo $2$ as follows
\begin{align*} \sum \nolimits _ i a_{ii} m_ i & \equiv \sum \nolimits _{i,\ m_ i\text{ odd}} a_{ii}m_ i \\ & \equiv \sum \nolimits _{i,\ m_ i\text{ odd}} \sum \nolimits _{j \not= i} a_{ij}m_ j \\ & \equiv \sum \nolimits _{i,\ m_ i\text{ odd}} \sum \nolimits _{j \not= i,\ m_ j\text{ odd}} a_{ij}m_ j \\ & \equiv \sum \nolimits _{i < j,\ m_ i\text{ and }m_ j\text{ odd}} a_{ij}(m_ i + m_ j) \\ & \equiv 0 \end{align*}
where we have used that $a_{ij} = a_{ji}$ and that $\sum _ j a_{ij}m_ j = 0$ for all $i$. $\square$
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