\section{Curves of normalized central configurations and their places}\label{sec:setup}

In this section we set up the six-body versions of the definitions and lemmas of \cite[\S\S2--3]{Tom26}. Since their proofs there do not use the number of bodies, except where noted, we cite them. We use no result of \cite{AK12} as an input: the rules of \cite[\S3]{AK12} that we need are proved at places in Lemma~\ref{lem:rules}, and the arguments of \cite[\S\S5.0, 8.2, 8.6]{AK12} that we adapt will be proved in Sections~\ref{sec:ak50} and~\ref{sec:k4}. Throughout, $m=(m_1,\dots,m_6)$ and $M=m_1+\dots+m_6$. Pairs of bodies are written $kl$.

\subsection{The system and the slice}\label{ss:system}
Following \cite[p.~540]{AK12}, a \emph{normalized central configuration} is a solution $(x,y,\delta)\in\C^6\times\C^6\times\C^{15}$ of
\begin{equation}\label{eq:AK}
x_k=\sum_{l\ne k}m_l\,\delta_{kl}^{3}(x_k-x_l),\quad y_k=\sum_{l\ne k}m_l\,\delta_{kl}^{3}(y_k-y_l),\quad \delta_{kl}^{2}\big((x_k-x_l)^2+(y_k-y_l)^2\big)=1,\quad y_1=y_2,
\end{equation}
for $k\ne l$ in $\{1,\dots,6\}$, with $\delta_{kl}=\delta_{lk}$. For $m\in(\R_{>0})^6$, every central configuration with masses $m$ is similar, by an orientation-preserving similarity, to exactly two real normalized central configurations with positive $\delta_{kl}$ \cite[Lemma~2.1]{Tom26}. Hence finiteness of the solution set of \eqref{eq:AK} implies finiteness of the central configurations up to similarity.

We put $z=x+iy$, $w=x-iy$, $z_{kl}=z_k-z_l$, $w_{kl}=w_k-w_l$ and $r_{kl}^2=z_{kl}w_{kl}$. As in \cite[\S2.2]{Tom26}, the unknowns are $u=(u_1,\dots,u_5)$, $v=(v_1,\dots,v_5)$ and $p=(p_{kl})_{k<l}$, with $u_6=v_6=0$, $u_{kl}=u_k-u_l$, $v_{kl}=v_k-v_l$ and $p_{lk}=p_{kl}$. With $F_k(h)=h_k-\sum_{l\ne k}m_l\,p_{kl}^3(h_k-h_l)$, the $25$ \emph{defining polynomials} are
\[
D_{kl}=p_{kl}^2u_{kl}v_{kl}-1,\qquad Z_k=F_k(u)-F_6(u),\qquad W_k=F_k(v)-F_6(v)\qquad(k\le5).
\]
Let $X_m\subset\C^{25}$ be their zero set and $Y_m=X_m\cap\{u_1=1\}$, the \emph{slice}. On $X_m$ the functions $u_k$, $v_k$, $u_{kl}$, $v_{kl}$ and $p_{kl}$ vanish nowhere.

\begin{lemma}[{\cite[Lemma~2.2]{Tom26}}]\label{lem:slice}
\begin{enumerate}[label=\textup{(\alph*)}]
\item The group $\C^*$ acts freely on $X_m$ by $s\cdot(u,v,p)=(su,s^{-1}v,p)$, the map $\C^*\times Y_m\to X_m$, $(s,y)\mapsto s\cdot y$, is an isomorphism, and \eqref{eq:AK} has exactly $2|Y_m|$ solutions.
\item For $t>0$ there is an isomorphism $Y_m\to Y_{tm}$ that multiplies the $p_{kl}$ by $t^{-1/3}$. Relabelling the bodies together with their masses, followed by a translation and a rotation, is an isomorphism $Y_m\to Y_{m'}$ for the permuted masses $m'$.
\end{enumerate}
\end{lemma}

On $X_m$ we put $z_k=u_k-F_6(u)$ and $w_k=v_k-F_6(v)$ $(k\le6)$. Then $z_k-z_l=u_{kl}$, $w_k-w_l=v_{kl}$, and the equations $D_{kl}=Z_k=W_k=0$ say that
\begin{equation}\label{eq:bodies}
z_k=\sum_{l\ne k}m_l\,Z_{kl},\qquad w_k=\sum_{l\ne k}m_l\,W_{kl},\qquad Z_{kl}=p_{kl}^3z_{kl},\quad W_{kl}=p_{kl}^3w_{kl}
\end{equation}
for $k=1,\dots,6$. These are the \emph{equations of the bodies}. Summing them with the weights $m_k$ gives $\sum_km_kz_k=\sum_km_kw_k=0$.

\begin{lemma}\label{lem:exchange}
The map $\iota(u,v,p)=(v/v_1,\,v_1u,\,p)$ is an involution of $Y_m$.
\end{lemma}

\begin{proof}
The defining polynomials are permuted by $(u,v,p)\mapsto(v,u,p)$, which fixes the $D_{kl}$ and exchanges $Z_k$ and $W_k$. Composing with the rotation by $s=v_1^{-1}$ of Lemma~\ref{lem:slice}(a) gives a point with $u_1=1$. Applying $\iota$ twice gives $(u,v,p)$ back, since $u_1=1$.
\end{proof}

Let
\[
U=\sum_{k<l}m_km_l\,p_{kl},\qquad I=\frac1M\sum_{k<l}m_km_l\,u_{kl}v_{kl}.
\]

\begin{lemma}[{\cite[Lemmas~2.3 and~2.4]{Tom26}}]\label{lem:U}
\begin{enumerate}[label=\textup{(\alph*)}]
\item On $X_m$, $U=\sum_km_kz_kw_k$, and $U=I$ if $M\ne0$.
\item If $M\ne0$, $U$ is constant on every irreducible component of $Y_m$.
\item For $m\in(\R_{>0})^6$, $U>0$ at the point of $Y_m$ given by a real central configuration.
\end{enumerate}
\end{lemma}

\begin{corollary}[{\cite[Corollary~2.5]{Tom26}}]\label{cor:curve}
Let $m\in(\R_{>0})^6$. If the planar six-body problem with masses $m$ has infinitely many central configurations up to similarity, then $Y_m$ contains an irreducible curve $C$ on which $U$ is a nonzero constant and some $r_{kl}^2$ is not constant.
\end{corollary}

The proof in \cite{Tom26} bounds the fibres of $(u,v,p)\mapsto(r_{kl}^2)_{k<l}$ on $Y_m$. For six bodies the bound is $2^{19}$ instead of $2^{13}$: on a fibre, $u_1=1$ determines $v_1$, each of $u_2,\dots,u_5$ is a root of a quadratic equation, and the fifteen $p_{kl}$ are determined up to sign. We call a curve as in Corollary~\ref{cor:curve} \emph{nontrivial}, and we write $E$ for the value of $U$ on it, so that $E\ne0$.

\subsection{Places, exponents and balancing}\label{ss:places}
Let $C\subseteq Y_m$ be an irreducible curve, $\bar C$ its smooth projective model, and $P$ a point of $\bar C$ not over $C$, a \emph{place at infinity}. We put
\[
\cZ=(z_1,\dots,z_6,Z_{12},\dots,Z_{56}),\qquad \cW=(w_1,\dots,w_6,W_{12},\dots,W_{56}).
\]
We call $z$ and $w$ the two \emph{colours}. A function on $C$ is of \emph{$z$-type} if the rotation $s\cdot$ multiplies it by $s$ (as $u_{kl}$, $z_k$, $z_{kl}$, $Z_{kl}$), of \emph{$w$-type} if it multiplies it by $s^{-1}$, and \emph{neutral} if it is invariant (as $p_{kl}$, $r_{kl}^2$). With $\nu=\val_P$ and $\nu$ of a vector the minimum over its components, the place is \emph{singular} if $\nu(\cZ)+\nu(\cW)<0$. Then we put $d=-\frac12(\nu(\cZ)+\nu(\cW))>0$, $\gamma=\frac12(\nu(\cW)-\nu(\cZ))$, and define the \emph{exponent}
\[
e(f)=\tfrac2d\big(\nu(f)+\gamma\big),\qquad e(f)=\tfrac2d\big(\nu(f)-\gamma\big),\qquad e(f)=\tfrac2d\,\nu(f)
\]
of a function $f$ of $z$-type, $w$-type or neutral type, with $e(f)=+\infty$ if $f$ vanishes identically on $C$.

\begin{lemma}[{\cite[\S3.1 and Lemma~3.1]{Tom26}}]\label{lem:place}
Let $P$ be a singular place. Every component of $\cZ$ and of $\cW$ has exponent at least $-2$, and each of the two vectors has a component of exponent $-2$. Let $t$ be a local parameter at $P$ and $\vartheta^2=t$. After the point of $C$ near $P$ is rotated by $s=\vartheta^{2\gamma}$, every function $f$ of the three kinds that does not vanish identically on $C$ becomes $f=\vartheta^{d\,e(f)}y_f(\vartheta)$ with $y_f\in\C\{\vartheta\}$, $y_f(0)\ne0$ and $d\,e(f)\in\Z$.
\end{lemma}

For a function $f$ of one of the three kinds and a number $e\le e(f)$ we write $\lc fe$ for the coefficient of $\vartheta^{de}$ in this expansion. Thus $\lc f{e(f)}=y_f(0)\ne0$, $\lc fe=0$ for $e<e(f)$, $\lc{fg}{e+e'}=\lc fe\lc g{e'}$ and $\lc{f+g}e=\lc fe+\lc ge$. We write
\[
a_k=e(z_k),\quad b_k=e(w_k),\quad \zeta_{kl}=e(z_{kl}),\quad \omega_{kl}=e(w_{kl}),\quad x_{kl}=\tfrac12(\zeta_{kl}+\omega_{kl}).
\]
Then $e(p_{kl})=-x_{kl}$, $e(Z_{kl})=-\frac12(\zeta_{kl}+3\omega_{kl})$, $e(W_{kl})=-\frac12(3\zeta_{kl}+\omega_{kl})$ and $\zeta_{kl}\ge\min(a_k,a_l)$, with equality if $a_k\ne a_l$. The body $k$ is \emph{$z$-circled} if $a_k=-2$, and $kl$ is a \emph{$z$-stroke} if $e(Z_{kl})=-2$, and likewise for $w$. The vector $x(P)=(x_{kl})_{k<l}\in\Q^{15}$ is the \emph{type} of $P$.

\begin{lemma}[{\cite[\S2.4 and (3.1)]{Tom26}}]\label{lem:balance}
Let $C$ be a nontrivial curve. Then $C$ has at least one singular place, and
\begin{equation}\label{eq:balx}
\sum_{P\ \mathrm{singular}}d_P\,x(P)=0 .
\end{equation}
\end{lemma}

The proof uses that the $p_{kl}$ are units on $C$, so that their divisors have degree zero, and that $\nu(p_{kl})=0$ at a place that is not singular. We write $\ell(x)=\sum_{k<l}x_{kl}$.

\begin{corollary}\label{cor:sum}
Let $C$ be a nontrivial curve. Some singular place $P$ of $C$ has $\ell(x(P))\le0$.
\end{corollary}

\begin{proof}
Apply $\ell$ to \eqref{eq:balx}. Since all $d_P$ are positive and there is a singular place, the values $\ell(x(P))$ cannot all be positive.
\end{proof}

Since $x_{kl}=\nu(r_{kl}^2)/d$, we have $\ell(x(P))=\nu(\Delta)/d$ for the unit $\Delta=\prod_{k<l}r_{kl}^2$ of $C$, so Corollary~\ref{cor:sum} says that $\Delta$ does not tend to zero at some singular place.

\begin{lemma}[{\cite[Lemma~3.2]{Tom26}}]\label{lem:orders}
At a singular place, for every pair $kl$:
\begin{enumerate}[label=\textup{(O\arabic*)}]
\item $\zeta_{kl},\omega_{kl}\ge-2$, $\zeta_{kl}+3\omega_{kl}\le4$ and $3\zeta_{kl}+\omega_{kl}\le4$. Hence $\zeta_{kl},\omega_{kl}\le2$ and $-2\le x_{kl}\le1$.
\item $kl$ is a $z$-stroke if and only if $\zeta_{kl}+3\omega_{kl}=4$, and a $w$-stroke if and only if $3\zeta_{kl}+\omega_{kl}=4$.
\item On a $z$-stroke, $\zeta_{kl}=3x_{kl}-2$, $\omega_{kl}=2-x_{kl}$ and $0\le x_{kl}\le1$. On a $w$-stroke, $\omega_{kl}=3x_{kl}-2$ and $\zeta_{kl}=2-x_{kl}$.
\item $x_{kl}=1$ if and only if $kl$ is both a $z$-stroke and a $w$-stroke \textup{(}a \emph{$zw$-edge}\textup{)}.
\item In every triangle $klj$ the minimum of $\zeta_{kl},\zeta_{lj},\zeta_{jk}$ is attained at least twice, and the same holds for $\omega$.
\item If $k$ is $z$-circled and $l$ is not, then $\zeta_{kl}=-2$. If $\zeta_{kl}=2$, then $\omega_{kl}=-2$ and $kl$ is a $w$-stroke.
\end{enumerate}
The same holds with $z$ and $w$ exchanged.
\end{lemma}

In \cite[\S3]{Tom26} the masses are positive and the curve is nontrivial, but the proofs of Lemmas~\ref{lem:place} and~\ref{lem:orders} use neither assumption. We will use the two lemmas for every $m$ and every irreducible curve $C\subseteq Y_m$.

\begin{lemma}\label{lem:symm}
Let $P$ be a singular place of a curve $C\subseteq Y_m$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item A relabelling of the bodies as in Lemma~\ref{lem:slice}\textup{(b)} maps $C$ to a curve in $Y_{m'}$ and $P$ to a singular place whose exponents, circles and strokes are those of $P$, relabelled.
\item The involution $\iota$ of Lemma~\ref{lem:exchange} maps $C$ to a curve in $Y_m$ and $P$ to a singular place at which $a_k,b_k,\zeta_{kl},\omega_{kl}$ are the values $b_k,a_k,\omega_{kl},\zeta_{kl}$ at $P$. The type is unchanged.
\end{enumerate}
Both maps preserve $U$ and permute the functions $r_{kl}^2$, so they map nontrivial curves to nontrivial curves.
\end{lemma}

\begin{proof}
(a) The relabelling multiplies $\cZ$ and $\cW$ by $s$ and $s^{-1}$ for a function $s$ and permutes their components, and it preserves the $p_{kl}$ up to the permutation. (b) Since $F_6$ is linear, $z_k\circ\iota=w_k/v_1$ and $w_k\circ\iota=v_1z_k$, so $\cZ\circ\iota=v_1^{-1}\cW$ and $\cW\circ\iota=v_1\cZ$, while the $p_{kl}$ are unchanged. At the image place the numbers $d$ and $\gamma$ become $d$ and $\nu(v_1)-\gamma$. So the exponent of $z_k$ there is $\frac2d(\nu(w_k)-\nu(v_1)+\nu(v_1)-\gamma)=e(w_k)$, and likewise for the other functions.
\end{proof}

\subsection{Identities}\label{ss:identities}
The following identities hold on $X_m$. On a nontrivial curve $C$ we also have $U=E$. For a set $J$ of bodies, $J^c$ is its complement.
\begin{itemize}
\item \emph{Subset sums}: $\sum_{k\in J}m_kz_k=\sum_{k\in J,\,l\in J^c}m_km_l\,Z_{kl}$, and the same in $w$. For $J=\{k\}$ this is the equation of the body $k$.
\item \emph{Torques}: $T_J:=\sum_{k\in J,\,l\in J^c}m_km_l\,p_{kl}^3(z_kw_l-z_lw_k)=0$.
\item \emph{Partial virials}: $\sum_{k\in J}m_kz_kw_k-\sum_{k<l\in J}m_km_lp_{kl}-\frac12\sum_{k\in J,\,l\in J^c}m_km_lp_{kl}^3(w_kz_{kl}+z_kw_{kl})=0$.
\item \emph{Differences}: $z_{kl}=\sum_{j\ne k}m_jZ_{kj}-\sum_{j\ne l}m_jZ_{lj}$, and the same in $w$.
\item $U=\sum_km_kz_kw_k$, \ $\sum_{k<l}m_km_l\,z_{kl}w_{kl}=MU$, \ $p_{kl}^2z_{kl}w_{kl}=1$.
\end{itemize}

The subset sums and the differences follow from \eqref{eq:bodies}, since the terms of pairs inside $J$ cancel. For the torques, $w_k$ times the $z$-equation of $k$ minus $z_k$ times its $w$-equation gives $\sum_{l\ne k}m_lp_{kl}^3(z_kw_l-z_lw_k)=0$. Multiplying by $m_k$ and summing over $k\in J$, the pairs inside $J$ cancel because $p_{kl}^3(z_kw_l-z_lw_k)$ is antisymmetric. For the partial virials, the half sum of $w_k$ times the $z$-equation of $k$ and $z_k$ times its $w$-equation gives $m_kz_kw_k=\frac12\sum_{l\ne k}m_km_lp_{kl}^3(w_kz_{kl}+z_kw_{kl})$, and the pairs inside $J$ contribute $m_km_lp_{kl}^3z_{kl}w_{kl}=m_km_lp_{kl}$. The last line is Lemma~\ref{lem:U}(a) and the equations $D_{kl}=0$. For $J=\{1,\dots,6\}$ the partial virial is $\sum_km_kz_kw_k=U$. The torque identity is the constraint of \cite[\S5.0]{AK12} on the external forces of a set of bodies.

Each identity is a sum $\sum_ic_ig_i=0$ in which the $c_i$ are nonzero polynomials in the masses (or the constant $E$) and each $g_i$ is a product of the functions $z_k,w_k,z_{kl},w_{kl},p_{kl}$, and all $g_i$ are multiplied by the same power of $s$ under the rotation. At a singular place the exponent of $g_i$ is the sum of the exponents of its factors. We omit the terms with a factor that vanishes identically on $C$. Only $z_k$ and $w_k$ can do so.

\begin{lemma}\label{lem:initial}
Let $C\subseteq Y_m$ be an irreducible curve and $\sum_ic_ig_i=0$ an identity on $C$ in which the $c_i$ are nonzero complex numbers, each $g_i$ is a product of functions $z_k,w_k,z_{kl},w_{kl},p_{kl}$ that do not vanish identically on $C$ \textup{(}possibly the empty product $1$\textup{)}, and all $g_i$ are multiplied by the same power of $s$ under the rotation. Then at every singular place the least exponent $e_0$ of the terms is attained by at least two terms, and $\sum_{e(g_i)=e_0}c_i\lc{g_i}{e_0}=0$, where $\lc{g_i}{e_0}$ is the product of the leading coefficients of the factors of $g_i$.
\end{lemma}

Examples are the identities above at masses with $c_i(m)\ne0$ for all $i$, and, if $M\ne0$, the identity $U-c=0$, where $c$ is the constant value of $U$ on $C$ (Lemma~\ref{lem:U}(b)); if $c=0$, this identity has no constant term.

\begin{proof}
By Lemma~\ref{lem:place} each term is $\vartheta^{d\,e(g_i)}$ times a unit, after the common rotation. The coefficient of $\vartheta^{de_0}$ of the sum is the displayed sum, which vanishes. A single term of least exponent would make it nonzero.
\end{proof}

We call $\sum_{e(g_i)=e_0}c_i\lc{g_i}{e_0}$ the \emph{initial form} of the identity at the place.

\subsection{Diagrams and the rules of Albouy and Kaloshin}\label{ss:rules}
The \emph{diagram} of a singular place consists of the $z$-strokes, the $w$-strokes and the $z$- and $w$-circled bodies. A pair is an \emph{edge} if it is a stroke of some colour, and its \emph{kind} is then $z$, $w$ or $zw$. The \emph{$z$-component} of a body is its connected component in the graph of the $z$-strokes. The following rules are the exponent forms of rules of \cite[\S3]{AK12}, which are stated there for any number of bodies whose masses have no vanishing partial sum.

\begin{lemma}\label{lem:rules}
Let $m\in(\R_{>0})^6$ and let $P$ be a singular place of an irreducible curve $C\subseteq Y_m$. The diagram of $P$ satisfies the following rules, and the same rules with $z$ and $w$ exchanged.
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item There is a $z$-stroke.
\item If $ab$ and $bc$ are $zw$-edges, then $ac$ is a $zw$-edge.
\item The three edges of a triangle of edges have the same kind, equal $\zeta$ and equal $\omega$.
\item From each $zw$-edge $ab$ another $z$-stroke leaves $a$ or $b$.
\item The number of $zw$-edges is not $1$.
\item If $ab$, $bc$, $cd$ and $da$ are edges, then $ab$ and $cd$ have equal $\zeta$, equal $\omega$ and the same kind, and so have $bc$ and $da$.
\end{enumerate}
\begin{enumerate}[label=\textup{(C\arabic*)}]
\item A body without a $z$-stroke is not $z$-circled, and a body with exactly one $z$-stroke is $z$-circled.
\item No $w$-stroke joins a $z$-circled body to a body that is not $z$-circled.
\item If $k$ is $z$-circled, some $z$-circled body of the $z$-component of $k$ lies outside the $w$-component of $k$.
\item If $k$ and $j$ are $z$-circled, $l$ is not, and $kl$ and $lj$ are $z$-strokes, then $kj$ is a $z$-stroke.
\end{enumerate}
\end{lemma}

The rules correspond to Rule~1a (E1, C1), the circling method based on Rule~1b and Estimate~2 (C2), Rule~1d (C3), the corollary of Rule~2c together with Rule~2e (C4), Rules~2b, 2e, 2a (E2, E3, E4), the corollary of Rule~2h (E5) and a strengthening of Rule~2g (E6) of \cite{AK12}. Rule~(C3) implies that a $z$-component never contains exactly one $z$-circled body. Besides $m_k\ne0$, only (E4), (C3) and (E5) use the positivity of the masses. They use it through the nonvanishing of partial sums, and (E5) through $M\ne0$ in Lemma~\ref{lem:U}(b). Only (E5) uses $U$, through its constancy on $C$.

\begin{proof}
On a $z$-stroke that is not a $zw$-edge, $\zeta=3x-2<1<\omega$; on a $w$-stroke that is not a $zw$-edge, $\omega<1<\zeta$; on a $zw$-edge, $\zeta=\omega=1$ (Lemma~\ref{lem:orders}). So the kind of an edge is determined by $\zeta$, and on edges $\omega=f(\zeta)$ for the strictly decreasing function $f(\zeta)=(4-\zeta)/3$ $(\zeta\le1)$, $f(\zeta)=4-3\zeta$ $(\zeta\ge1)$.

(C1) The terms of exponent $-2$ in the equation $z_k=\sum_lm_lZ_{kl}$ are $z_k$ if $k$ is circled and $Z_{kl}$ for the $z$-strokes $kl$. By Lemma~\ref{lem:initial} there cannot be exactly one such term. (E1) Some component of $\cZ$ has exponent $-2$ (Lemma~\ref{lem:place}). If it is $z_k$, (C1) gives a $z$-stroke at $k$.

(C2) On a $w$-stroke $kl$, $\zeta_{kl}=2-x_{kl}\ge1>-2$, so by (O6) the bodies $k$ and $l$ are both $z$-circled or both not.

(E2) On $zw$-edges $\zeta=\omega=1$, so (O5) gives $\zeta_{ac},\omega_{ac}\ge1$. Hence $x_{ac}=1$, and $ac$ is a $zw$-edge by (O4).

(E3) Let $\zeta_1=\zeta_2\le\zeta_3$ be the values of $\zeta$ on the three edges, ordered by (O5). If $\zeta_1>1$, all three edges are $w$-edges. If $\zeta_1=1$, the first two are $zw$-edges, and so is the third by (E2). If $\zeta_1<1$, the first two are $z$-edges with $\omega_1=\omega_2=f(\zeta_1)>1$; if the third were not a $z$-edge, $\omega_3\le1$ would be the unique minimum of $\omega$, contradicting (O5). In each case the three edges have the same kind. Then $\omega=f(\zeta)$ and (O5) for $\omega$ give $\zeta_3=\zeta_1$, hence equal $\omega$.

(E4) Suppose no other $z$-stroke leaves $a$ or $b$. Since $\zeta_{ab}=1>-2$, the bodies $a$ and $b$ are both $z$-circled or both not (O6), and $\lc{z_a}{-2}=\lc{z_b}{-2}$. If they are circled, the subset sum for $J=\{a,b\}$ has the terms $m_az_a,m_bz_b$ of exponent $-2$ and all other terms of larger exponent, so $(m_a+m_b)\lc{z_a}{-2}=0$ by Lemma~\ref{lem:initial}, a contradiction. If they are not circled, $m_bZ_{ab}$ is the only term of exponent $-2$ in the equation of $a$, contradicting (C1).

(E5) Let $c$ be the constant value of $U$ on $C$. The terms $m_km_lp_{kl}$ of $U-c=0$ have exponents $-x_{kl}\ge-1$, with equality exactly at the $zw$-edges (O4). A single $zw$-edge would give a single term of least exponent, contradicting Lemma~\ref{lem:initial}.

(E6) The identity $z_{ab}+z_{bc}+z_{cd}+z_{da}=0$ shows that the minimum of $\zeta$ over the four edges is attained at least twice, and the same identity in $w$, with $\omega=f(\zeta)$, shows that the maximum of $\zeta$ is attained at least twice. If the four values are not all equal, two edges carry the minimum $\alpha$ and the other two the maximum $\beta>\alpha$. Suppose $ab$ and $bc$ carry $\alpha$. Then (O5) in the triangles $acd$ and $abc$ gives $\zeta_{ac}\ge\beta$ and $\omega_{ac}\ge f(\alpha)$. If $\alpha<1$, then $\zeta_{ac}+3\omega_{ac}\ge\beta+4-\alpha>4$; if $\alpha\ge1$, then $3\zeta_{ac}+\omega_{ac}\ge3\beta+4-3\alpha>4$. Both contradict (O1). So opposite edges carry equal values of $\zeta$, hence of $\omega$, and have the same kind.

(C3) Let $K$ be the $z$-component of $k$ and $L$ its $w$-component. The subset sum for $K$ has the terms $m_jz_j$, $j\in K$, of exponent $-2$ exactly when $j$ is circled, and its other terms are $m_jm_iZ_{ji}$ with $j\in K$, $i\notin K$, which are not $z$-strokes and have exponent greater than $-2$. So $\sum_jm_j\lc{z_j}{-2}=0$ over the circled $j\in K$. On a $w$-stroke $\zeta\ge1>-2$, so all bodies of $L$ have the same coefficient $\lc{z_j}{-2}$, which is $\lc{z_k}{-2}\ne0$. If all circled bodies of $K$ were in $L$, the sum would be $\lc{z_k}{-2}$ times a nonempty partial sum of masses, which is not zero.

(C4) By (O6), $\zeta_{kl}=\zeta_{lj}=-2$, so $x=0$ and $\omega_{kl}=\omega_{lj}=2$ (O3). By (O5), $\omega_{kj}\ge2$, hence $\omega_{kj}=2$, and (O1) gives $\zeta_{kj}\le-2$. So $\zeta_{kj}+3\omega_{kj}=4$, and $kj$ is a $z$-stroke by (O2).
\end{proof}

If all six pairs of a set of four bodies are edges, (E3) applied to its four triangles shows that the six pairs have the same kind, equal $\zeta$ and equal $\omega$. This is the conclusion of Rule~2f of \cite{AK12} for $p=4$, under the stronger hypothesis that all six pairs are edges.