By Lemma \ref{lemma-bound-neighbours} it suffices to show $m_i|a_{ii}| \leq 768g$ for all $i$. Let $J \subset \{1, \ldots, n\}$ be the set of non-$(-2)$-indices as in Lemma \ref{lemma-bound-heart}. Observe that $J$ is nonempty as $g \geq 2$. Also $m_j|a_{jj}| \leq 6g$ for $j \in J$ by the lemma. \medskip\noindent Suppose we have $j \in J$ and a sequence $i_1, \ldots, i_7$ of $(-2)$-indices such that $a_{ji_1}$ and $a_{i_1i_2}$, $a_{i_2i_3}$, $a_{i_3i_4}$, $a_{i_4i_5}$, $a_{i_5i_6}$, and $a_{i_6i_7}$ are nonzero. Then we see from Lemma \ref{lemma-bound-neighbours} that $m_{i_1}w_{i_1} \leq 6g$ and $m_{i_1}a_{ji_1} \leq 6g$. Because $i_1$ is a $(-2)$-index, we have $a_{i_1i_1} = -2w_{i_1}$ and we conclude that $m_{i_1}|a_{i_1i_1}| \leq 12g$. Repeating the argument we conclude that $m_{i_2}w_{i_2} \leq 12g$ and $m_{i_2}a_{i_1i_2} \leq 12g$. Then $m_{i_2}|a_{i_2i_2}| \leq 24g$ and so on. Eventually we conclude that $m_{i_k}|a_{i_ki_k}| \leq 2^k(6g) \leq 768g$ for $k = 1, \ldots, 7$. \medskip\noindent Let $I \subset \{1, \ldots, n\} \setminus J$ be a maximal connected subset. In other words, there does not exist a nonempty proper subset $I' \subset I$ such that $a_{i'i} = 0$ for $i' \in I'$ and $i \in I \setminus I'$ and $I$ is maximal with this property. In particular, since a numerical type is connected by definition, we see that there exists a $j \in J$ and $i \in I$ with $a_{ij} > 0$. Looking at the classification of such $I$ in Proposition \ref{proposition-classify-subgraphs} and using the result of the previous paragraph, we see that $w_i|a_{ii}| \leq 768g$ for all $i \in I$ unless $I$ is as described in Lemma \ref{lemma-long} or Lemma \ref{lemma-Dn}. Thus we may assume the nonvanishing of $a_{ii'}$, $i, i' \in I$ has either the shape $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{..}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet } $$ (which has 3 subcases as detailed in Lemma \ref{lemma-long}) or the shape $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{..}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \\ & & & & \bullet } $$ We will prove the bound holds for the first subcase of Lemma \ref{lemma-long} and leave the other cases to reader (the argument is almost exactly the same in those cases). \medskip\noindent After renumbering we may assume $I = \{1, \ldots, t\} \subset \{1, \ldots, n\}$ and there is an integer $w$ such that $$ w = w_1 = \ldots = w_t = a_{12} = \ldots = a_{(t - 1) t} = -\frac{1}{2} a_{i_1i_2} = \ldots = -\frac{1}{2} a_{(t - 1) t} $$ The equalities $a_{ii}m_i + \sum_{j \not = i} a_{ij}m_j = 0$ imply that we have $$ 2m_2 \geq m_1 + m_3, \ldots, 2m_{t - 1} \geq m_{t - 2} + m_t $$ Equality holds in $2m_i \geq m_{i - 1} + m_{i + 1}$ if and only if $i$ does not ``meet'' any indices besides $i - 1$ and $i + 1$. And if $i$ does meet another index, then this index is in $J$ (by maximality of $I$). In particular, the map $\{1, \ldots, t\} \to \mathbf{Z}$, $i \mapsto m_i$ is concave. \medskip\noindent Let $m = \max(m_i, i \in \{1, \ldots, t\})$. Then $m_i|a_{ii}| \leq 2mw$ for $i \leq t$ and our goal is to show that $2mw \leq 768g$. Let $s$, resp.\ $s'$ in $\{1, \ldots, t\}$ be the smallest, resp.\ biggest index with $m_s = m = m_{s'}$. By concavity we see that $m_i = m$ for $s \leq i \leq s'$. If $s > 1$, then we do not have equality in $2m_s \geq m_{s - 1} + m_{s + 1}$ and we see that $s$ meets an index from $J$. In this case $2mw \leq 12g$ by the result of the second paragraph of the proof. Similarly, if $s' < t$, then $s'$ meets an index from $J$ and we get $2mw \leq 12g$ as well. But if $s = 1$ and $s' = t$, then we conclude that $a_{ij} = 0$ for all $j \in J$ and $i \in \{2, \ldots, t - 1\}$. But as we've seen that there must be a pair $(i, j) \in I \times J$ with $a_{ij} > 0$, we conclude that this happens either with $i = 1$ or with $i = t$ and we conclude $2mw \leq 12g$ in the same manner as before (as $m_1 = m = m_t$ in this case).