\begin{table}[t]
\centering\small
\caption{Real dimensions; $r_i$ is the number of entries of $d_i(\xi)$ with cube one, and
every $r_i=0$ at a nonresonant parameter.}\label{tab:dimensions}
\begin{tabular}{@{}lp{1.72in}ll@{}}
dimension&space&count&proved in\\\hline
$27$, $26$&$J$, $J_0$&$27=9+18$&Section~\ref{sec:families}\\
$1053$&configurations $J^{39}$&$39\cdot27$&\eqref{eq:frame-equations}\\
$2184$&residual coordinates of $F$&$1053+39+351+741$&\eqref{eq:frame-equations}\\
$52$&$F_4$; orbit tangent space $\mathcal O$&&Lemma~\ref{lem:orbit}\\
$8$&pointwise stabilizer of a quartet&$SU(3)$&Section~\ref{sec:families}\\
$18$&parameters of \eqref{eq:anchored-family}&$8+8+2$&Section~\ref{sec:families}\\
$18+2r$&first-order matrix solutions&$6+12+2r$&Lemma~\ref{lem:fourier}\\
$8$&first-order deformations of a quartet inside $H_3(\mathbb C)$&$\mathfrak{su}(3)$&Lemma~\ref{lem:quartet}\\
$26+2r_i$&$\mathcal T^{K_i}$&$8+(18+2r_i)$&Lemma~\ref{lem:TH}\\
$8$&$\mathcal T^K$&$2+6$&Lemma~\ref{lem:TK}\\
$16$, $4$&$\mathcal O^{K_i}$, $\mathcal O^K$&$8+8$, $2+2$&Lemma~\ref{lem:orbit}\\
$80+2\sum r_i$&$\mathcal T=\ker DF_{P(\xi)}$&$\sum_i(26+2r_i)-24$&\eqref{eq:fullkernel}\\
$1001$&slice $S$&$1053-52$&Lemma~\ref{lem:slice}\\
$28$&$\overline{\mathcal T}=\mathcal T/\mathcal O$, nonresonant&$80-52=4+4\cdot6$&\eqref{eq:Qsplit}\\
$4$, $10$, $6$&$\overline{\mathcal T}^K$, $\overline{\mathcal T}^{K_i}$, $W_i$, nonresonant&$8-4$, $26-16$, $10-4$&\eqref{eq:Qsplit}\\
$62$&$F_4$-orbits of one family&$52+18-8$&Lemma~\ref{lem:families}\\
$10$, $4$&$M_i$, $M_*$ in the slice&$10=4+6$&Lemma~\ref{lem:families}\\
$973$&complement of $\overline{\mathcal T}$ in the slice&$1001-28$&Prop.~\ref{prop:simultaneous}\\
$92$, $40$&$\mathcal T$, $\overline{\mathcal T}$ at the original code, $(r_i)=(3,1,1,1)$&$80+2\cdot6$, $92-52$&\eqref{eq:fullkernel}
\end{tabular}
\end{table}