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

Let $F_0\in\Z[m_1,\dots,m_6]$ be the product of the following nonzero polynomials:
\begin{itemize}
\item for each listed cell that has a core in Proposition~\ref{prop:exclusion}, the polynomial $h$ of Lemma~\ref{lem:core};
\item $P_{98}$ of Theorem~\ref{thm:d98}, and $P_{93}(m_1,m_2,m_3)$ with $P_{93}$ as in Theorem~\ref{thm:d93};
\item $P_{88}(m_1,\dots,m_4;m_5,m_6)$ and $P_{94}(m_1,\dots,m_4;m_5,m_6)$ of Theorem~\ref{thm:d88};
\item a nonzero $h_E\in\Z[\mu]$ vanishing on $B_E$, evaluated at $(m_3,m_4,m_5,m_6)$, and a nonzero $h_C\in\Z[\mu]$ vanishing on $B_C$, evaluated at $(m_3,m_4,m_5,m_6)$ (Propositions~\ref{prop:E4} and~\ref{prop:C4}).
\end{itemize}
Such $h_E$ and $h_C$ exist because $B_E$ and $B_C$ are proper Zariski-closed subsets defined over $\Q$. Let $F(m)=\prod_{g\in S_6}F_0(g\cdot m)$, where $g$ permutes the masses. Then $F\ne0$.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
We argue by contradiction. Let $m\in(\R_{>0})^6$ with $F(m)\ne0$, and suppose that the planar six-body problem with masses $m$ has infinitely many central configurations up to similarity. By Corollary~\ref{cor:curve}, $Y_m$ contains a nontrivial curve $C$, and by Corollary~\ref{cor:sum}, $C$ has a singular place $P$ with $\ell(x(P))\le0$. By Lemma~\ref{lem:rules} and Proposition~\ref{prop:enum}, the diagram of $P$ is the image of the representative $D$ of its class under a relabelling, possibly followed by the exchange of the colours. By Lemma~\ref{lem:symm}, the inverse relabelling and, if needed, the involution $\iota$ give a nontrivial curve $C'\subseteq Y_{m'}$, where $m'$ is a permutation of $m$, and a singular place $P'$ of $C'$ with diagram $D$ whose type is the type of $P$ with the pairs permuted. In particular $\ell(x(P'))\le0$, so the trees and argmin sets of $P'$ form a nonpositive cell of $D$, which is in the list of Proposition~\ref{prop:cells}.

If the cell is one of the $43$ cells of Proposition~\ref{prop:exclusion}, then $D$ is one of the diagrams $64$, $66$, $88$, $89$, $93$, $94$, $98$, up to relabelling and exchange of the colours. Theorem~\ref{thm:d89} excludes $P'$ for diagram $89$. For the other six diagrams, Theorems~\ref{thm:d98}, \ref{thm:d93}, \ref{thm:d88}, \ref{thm:d64} and~\ref{thm:d66} show that a polynomial of the list above vanishes at a permutation of $m$, contradicting $F(m)\ne0$. Otherwise, by Proposition~\ref{prop:exclusion} and Lemmas~\ref{lem:single} and~\ref{lem:core}, either no place of $C'$ lies in the cell, contradicting the fact that $P'$ lies in it, or $h(m')=0$ for the polynomial $h$ of the cell, again contradicting $F(m)\ne0$. This contradiction proves the theorem.
\end{proof}

The proof shows that $Y_m$ contains no nontrivial curve when $m\in(\R_{>0})^6$ and $F(m)\ne0$. But it does not show that $Y_m$ is finite, since curves on which $U$ vanishes identically are not excluded; by Lemma~\ref{lem:U}(c) they contain no point given by a real central configuration.

The positivity of the masses enters the proof in the following places.
\begin{itemize}
\item Lemma~\ref{lem:U}(c) and Corollary~\ref{cor:curve}.
\item The nonvanishing of partial sums of the masses: of $M$ in Lemma~\ref{lem:U}(a), (b) and in the identity $\sum_{k<l}m_km_lz_{kl}w_{kl}=ME$, of the sums $M_A$, $M_B$ of the two triangles when Lemma~\ref{lem:three} is applied in Theorems~\ref{thm:d98} and~\ref{thm:d93}, and of other partial sums in Lemmas~\ref{lem:rules}, \ref{lem:constraints}, \ref{lem:common}, \ref{lem:d64type} and~\ref{lem:d64lead}.
\item $m_i^3+m_j^3\ne0$ in Lemma~\ref{lem:binary}.
\item $t=1/m_1>0$, when Lemma~\ref{lem:core}(b) uses Lemma~\ref{lem:slice}(b).
\item The moduli in Lemma~\ref{lem:Ttest}.
\item The real points of the open sets in the proof of Theorem~\ref{thm:d88}.
\end{itemize}
Elsewhere only $m_k\ne0$ is used.