\section{Proof of Theorem~\ref{thm:main}}\label{sec:proof}

Let $m\in(\R_{>0})^5$. If $m$ lies on a special ray, Theorems~\ref{thm:count1} and~\ref{thm:count2} give exactly $306$ or $294$ central configurations up to orientation-preserving similarity. Otherwise Corollary~\ref{cor:S} gives finitely many up to similarity. In its proof we apply Theorem~\ref{thm:S} to the sorted scaled image $\hat m$ of $m$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item if $\Lambda(\hat m)$ is certified in the sense of \S\ref{ss:nearlabels}, Proposition~\ref{prop:cert} with $V=\Vtwo^{\mathrm w}$ settles $\hat m$, the pieces $\pc{d2}$, $\pc{d9}$, $\pc{d14b}$ and $\pc{d15}$ being removed by Proposition~\ref{prop:nod2}, Theorem~\ref{thm:D9}, Theorem~\ref{thm:D14} and Corollary~\ref{cor:nod15};
\item if $\Lambda(\hat m)$ contains a curve set, Proposition~\ref{prop:farT9} or Theorem~\ref{thm:curves} applies, and at the counted points the latter rests on Theorem~\ref{thm:countP1};
\item if $\hat m$ is $\mathbf q_*$ or $\mathbf p_*$, Proposition~\ref{prop:cert} with $V=\VN$ settles $\hat m$, by Proposition~\ref{prop:Spoints}. \qed
\end{enumerate}

\medskip
Section~\ref{sec:verification} lists the computations on which the proof rests and how they are checked.