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\begin{align*}
D(e)= &(d-1-8r) \dim S^r_{2r}(\sigma')+\sum_{\delta=0}^{2r}(\dim S^r_{2r}(\sigma')- \dim S^r_{\delta} (\sigma'))\\
  =&(d-6r) \dim S^r_{2r}(\sigma') -
 \sum_{\delta=0}^{2r}\dim S^r_{\delta} (\sigma')
\\
=&(d-6r) \dim S^r_{2r}(\sigma') -\sum_{\delta=0}^{r}\binom{\delta+2}{2}
-\sum_{\delta=r+1}^{2r}\dim S^r_{\delta} (\sigma')\\
=&(d-6r) \dim S^r_{2r}(\sigma') -\binom{r+3}{3}
-\sum_{\delta=r+1}^{2r}\dim S^r_{\delta} (\sigma')
\end{align*}
 where $\sigma'$ is a 2D cell in the cross-section of $O(e)$ by a plane perpendicular to $e$.
 If $e$ is a boundary edge, then
 
\begin{align*}
D(e)=& (d-6r)\left(\binom{2r+2}{2}+(n_e-2)\binom{r+1}{2}\right)-\binom{r+3}{3}\\
-&\sum_{\delta=r+1}^{2r}\left(\binom{\delta+2}{2}+(n_e-2)\binom{\delta-r+1}{2}\right)\\
=&(d-6r)\left(\binom{2r+2}{2}+(n_e-2)\binom{r+1}{2}\right)-\binom{2r+3}{3}\\
-&(n_e-2)\sum_{\delta=r+1}^{2r}\binom{\delta-r+1}{2}
\\
=(d-6r)&\left(\binom{2r+2}{2}+(n_e-2)\binom{r+1}{2}\right)-\binom{2r+3}{3}
-(n_e-2)\binom{r+2}{3}
\end{align*}




 If $e$ is an interior edge, then
\begin{align*}
D(e)=& (d-6r)\left(\binom{r+2}{2}+n_e \binom{r+1}{2}+\sigma_{2r,e}\right)-\binom{r+3}{3}
\\
-&\sum_{\delta=r+1}^{2r}\left(\binom{r+2}{2}+n_e\binom{\delta-r+1}{2}+\sigma_{\delta,e}\right)\\
 =&(d-6r)\left(\binom{r+2}{2}+n_e \binom{r+1}{2}+\sigma_{2r}\right)-\binom{r+3}{3}\\
-&r\binom{r+2}{2}-n_e\binom{r+2}{3}-\sum_{\delta=r+1}^{2r}\sigma_{\delta,e}\\
\end{align*}
 
 where $\sigma_{d,e}:=\sum_{j=1}^{d-r}(r+j+1-jn_e)_+$.
 From Lemma 7.1 
 \begin{align*}
 \sigma_{d,e}=&\binom{d+2}{2}-\binom{r+2}{2}-n_e\binom{d+1-r}{2}\\
 +&b_e\binom{d+2-q_e}{2}+a_e\binom{d+1-q_e}{2}
 \end{align*}
Then
\begin{align*}
\sum_{\delta=r+1}^{2r}\sigma_{\delta,e}
=&-r\binom{r+2}{2}+\sum_{\delta=r+1}^{2r}\binom{\delta+2}{2}-n_e\sum_{\delta=r+1}^{2r}\binom{\delta+1-r}{2}\\
&+b_e\sum_{\delta=r+1}^{2r}\binom{\delta+2-q_e}{2}+a_e\sum_{\delta=r+1}^{2r}\binom{\delta+1-q_e}{2}\\
=&-r\binom{r+2}{2}+\binom{2r+3}{3}-\binom{r+3}{3}-n_e\binom{r+2}{3}\\
+&b_e\left(\binom{2r+3-q_e}{3}-\binom{r+3-q_e}{3} \right)\\
+&a_e\left(\binom{2r+2-q_e}{3}-\binom{r+2-q_e}{3} \right)
\end{align*}
Additionally, from the same lemma
\begin{align*}
 \sigma_{2r,e}=&\binom{2r+2}{2}-\binom{r+2}{2}-n_e\binom{r+1}{2}\\
 +&b_e\binom{2r+2-q_e}{2}+a_e\binom{2r+1-q_e}{2}
 \end{align*}
 
 We now substitute it into $D(e)$ for the interior edge $e$
  \begin{align*}
D(e)
 =&(d-6r)\left(\binom{2r+2}{2}+b_e\binom{2r+2-q_e}{2}+a_e\binom{2r+1-q_e}{2}\right)\\
-&\binom{2r+3}{3}-b_e\left(\binom{2r+3-q_e}{3}-\binom{r+3-q_e}{3} \right)\\
-&a_e\left(\binom{2r+2-q_e}{3}-\binom{r+2-q_e}{3} \right)
\end{align*}
 