\section{The special rays}\label{sec:rays}

Recall the points $\bone$ and $\btwo$ of \S\ref{ss:reduction}. At these two points the separation of \S\S\ref{sec:strata}--\ref{sec:curves} fails, and at $\bone$ it has to fail (\S\ref{sec:intro}). On the rays through these two mass vectors and their relabellings we therefore count the real central configurations directly, following Moczurad and Zgliczy\'nski \cite{MZ19,MZ26}: a branch and bound in a reduced system with interval arithmetic, exclusion tests derived from exact identities, and Krawczyk's test \cite{Kra69,Moo77,Neu90}. With the same method we also count the central configurations at one further mass point needed in \S\ref{sec:curves} (\S\ref{ss:twopoints}).

\subsection{Statements}\label{ss:raystatements}
In this section a configuration is $q=(q_1,\dots,q_5)\in(\R^2)^5$ outside the collision set $\Delta$, with $q_i=(\xi_i,\eta_i)$, $r_{ij}=|q_i-q_j|$, $M=\sum_im_i$ and
\[
F_i(q)=q_i-\sum_{j\ne i}m_j\,\frac{q_i-q_j}{r_{ij}^3}\qquad(i=1,\dots,5).
\]
A \emph{normalized central configuration} is a zero of $F=(F_1,\dots,F_5)$, that is, a solution of \eqref{eq:cc} with $\lambda=1$ and $c=0$. As in the proof of Lemma~\ref{lem:realpos}, every central configuration is similar, by an orientation-preserving similarity, to a normalized one, and two central configurations are orientation-preserving similar if and only if their normalizations lie in the same $SO(2)$-orbit. The group $SO(2)$ acts freely on normalized central configurations. Thus the central configurations up to orientation-preserving similarity correspond to the $SO(2)$-orbits of normalized central configurations. The reflection $(\xi,\eta)\mapsto(\xi,-\eta)$ acts on these orbits as an involution whose fixed orbits are the collinear ones. Hence, with $N$ orbits of which $N_c$ are collinear, there are $(N+N_c)/2$ central configurations up to similarity.

Rotations give $DF(q)\,Jq=0$ at a normalized central configuration $q$, where $J$ rotates each $q_i$ by $\pi/2$. Hence $\rank DF(q)\le9$. We call $q$ \emph{nondegenerate} if $\rank DF(q)=9$. Since $DF(q)$ is, up to the mass matrix, the Hessian of $V=\tfrac12\sum_im_i|q_i|^2+\sum_{i<j}m_im_j/r_{ij}$, this is the usual notion \cite{Moe14}: $q$ is a nondegenerate critical point of the potential on the normalized configurations, modulo rotations.

For a non-collinear configuration let $P$ be the convex hull of $q_1,\dots,q_5$. It is \emph{strictly convex} if all bodies are vertices of $P$, and of type $(4,1)$ or $(3,2)$ if exactly four, resp.\ three, bodies are vertices of $P$ and the others are interior points of $P$.

\begin{theorem}\label{thm:count1}
Let $m$ be a positive multiple of a permutation of $(1,4,4,4,4)$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item There are exactly $306$ central configurations up to orientation-preserving similarity, and exactly $183$ up to similarity.
\item Every central configuration is nondegenerate.
\item Of the $306$, exactly $60$ are collinear, $24$ strictly convex, $102$ of type $(4,1)$ and $120$ of type $(3,2)$. In every non-collinear central configuration each body is a vertex or an interior point of the convex hull, and each of the $24$ oriented cyclic orders of the five bodies is the counterclockwise order of exactly one strictly convex one.
\item There are exactly $612$ positive normalized central configurations in the sense of \cite[Def.~2]{AK12}.
\end{enumerate}
\end{theorem}

\begin{theorem}\label{thm:count2}
Let $m$ be a positive multiple of a permutation of $(1,1,4,4,4)$. Then statements \textup{(i)--(iv)} of Theorem~\ref{thm:count1} hold with the numbers $294$ and $177$ in \textup{(i)}; $60$, $24$, $90$ and $120$ in \textup{(iii)}; and $588$ in \textup{(iv)}.
\end{theorem}

The $60$ collinear configurations agree with Moulton's count \cite{Mou10}. Statement (iv) follows from (i) by Lemma~\ref{lem:realpos}. By (iv), the positive solutions of the system of \cite{AK12} at $(4,4,4,4,1)$ are finite in number, although the system has a continuum of real solutions there (\S\ref{sec:intro}). At this mass, finiteness therefore rests on the signs of the distances, as anticipated in \S\ref{sec:intro}.

\subsection{The reduced system and the a priori bounds}\label{ss:raysmethod}
Every normalized central configuration satisfies the following identities. Summing $m_iF_i$ gives $\sum_im_iq_i=0$. Taking the scalar product of $F_i=0$ with $m_iq_i$ and summing gives $I=U$, where $I=\sum_im_i|q_i|^2$ and $U=\sum_{i<j}m_im_j/r_{ij}$. With Lagrange's identity $MI=\sum_{i<j}m_im_jr_{ij}^2$, this is the virial identity $\sum_{i<j}m_im_j(r_{ij}^2/M-1/r_{ij})=0$. For a nonempty proper subset $C$ of the bodies, summing $m_iF_i$ over $C$ gives the cluster identity $\sum_{i\in C}m_iq_i=\sum_{i\in C,\,j\notin C}m_im_j(q_i-q_j)/r_{ij}^3$. Let $c_C$ be the centre of mass of $C$. The scalar products of $m_iF_i=0$ with $q_i-c_C$, summed over $C$, give the virial identity of the cluster $C$.

\begin{lemma}\label{lem:bounds}
Let $q$ be a normalized central configuration and $R=\max_i|q_i|$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item Some $r_{ij}$ is at least $M^{1/3}$. Hence $R\ge M^{1/3}/2$.
\item For every $\varepsilon>0$, $R<\max\{4\varepsilon,\ \max_{1\le s\le4}[(s-1)\varepsilon+W_s/\varepsilon^2]\}$, where $W_s$ is $M$ minus the sum of the $s$ smallest masses.
\item $r_{ij}\ge m_im_j/(MR^2)$ for all $i\ne j$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) If all $r_{ij}<M^{1/3}$, all terms of the virial identity are negative. (b) Let $G$ be the graph with an edge $ij$ when $r_{ij}\le\varepsilon$, and $C$ the component of a body $i_0$ with $|q_{i_0}|=R$, with $s$ bodies. If $s=5$, then $MR=|\sum_jm_j(q_{i_0}-q_j)|<4M\varepsilon$. Otherwise the scalar product of the cluster identity for $C$ with $q_{i_0}/R$ is at least $M_C(R-(s-1)\varepsilon)$ on the left and less than $M_C(M-M_C)/\varepsilon^2$ on the right. (c) $m_im_j/r_{ij}\le U=I\le MR^2$.
\end{proof}

Part (b) sharpens \cite[Thm.~11]{MZ19}: at $(1,4,4,4,4)$ that theorem gives about $10.11$, whereas ours gives $6.352$. Since the boxes $B_k$ below have sides set by this bound, the sharper bound shrinks the region searched. Table~\ref{tab:raycount} lists the values of $\varepsilon$ that we use and the resulting bounds.

Fix real constants $c_1,\dots,c_5$. A normalized central configuration lies in \emph{region} $k$ if $|q_k|^2+c_k\ge|q_i|^2+c_i$ for all $i$. Every one lies in some region. The constants only break ties: a symmetric configuration often has two bodies at the same distance from the centre of mass. With $c=0$ it would lie on the boundary of two regions. By Lemma~\ref{lem:bounds}(a), in region $k$ we have $|q_k|^2\ge\mu_k:=M^{2/3}/4+\min_ic_i-c_k$, which is positive for the constants we use. Hence $q_k\neq0$ and exactly one rotation puts $q_k$ on the positive $\xi$-axis. For a second body $e\ne k$ and the other three bodies $f_1,f_2,f_3$, the chart $\Phi_{k,e}\colon\R^7\to(\R^2)^5$, $v\mapsto q$ with
\[
q_k=(v_0,0),\qquad q_{f_a}=(v_{2a-1},v_{2a})\ (a=1,2,3),\qquad q_e=-\frac1{m_e}\sum_{i\ne e}m_iq_i,
\]
is an isomorphism onto $\{\sum_im_iq_i=0,\ \eta_k=0\}$, and the \emph{reduced system} $RS(v)$ consists of the seven components $F_{f_1},F_{f_2},F_{f_3}$ and the first component of $F_k$ at $\Phi_{k,e}(v)$ \cite[Thm.~2]{MZ26}, \cite[Thm.~3]{MZ22}.

\begin{lemma}\label{lem:RS}
Let $q=\Phi_{k,e}(v)\notin\Delta$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item If $RS(v)=0$ and $\xi_k\ne\xi_e$, then $q$ is a normalized central configuration. Conversely, every normalized central configuration with $\eta_k=0$ is $\Phi_{k,e}(v)$ for a zero $v$ of $RS$.
\item If $q$ is a normalized central configuration with $\xi_k\ne0$ and $DRS(v)$ is invertible, then $q$ is nondegenerate.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) For any configuration $\sum_im_iF_i=\sum_im_iq_i$ and $\sum_im_i\,q_i\times F_i=0$, with $a\times b=a_1b_2-a_2b_1$. If $RS(v)=0$, then $F_i=0$ for $i\ne k,e$. The first identity then gives $m_kF_k+m_eF_e=0$, and the second $m_k(q_k-q_e)\times F_k=0$. As $q_k-q_e=(\xi_k-\xi_e,-\eta_e)$ and the first component of $F_k$ vanishes, this is $m_k(\xi_k-\xi_e)F_{k,2}=0$. Hence $F_k=F_e=0$. The converse holds because the centre of mass of a normalized central configuration is $0$. (b) Differentiating the first identity gives $\sum_im_i(DF(q)u)_i=\sum_im_iu_i$ for all $u$. Since $\xi_k\neq0$, the vector $Jq$ does not lie in the image $L$ of $\Phi_{k,e}$. Hence $\R^{10}$ is the direct sum of $L$, the line spanned by $Jq$ and the translations. Let $u=\ell+\alpha Jq+\tau\in\ker DF(q)$ accordingly. Then $0=DF(q)\ell+\tau$, because $DF(q)Jq=0$ and $DF(q)\tau=\tau$. Taking the mass-weighted sum, we obtain $M\tau=0$. Thus $DF(q)\ell=0$, hence $DRS(v)\,\Phi_{k,e}^{-1}(\ell)=0$ and $\ell=0$. Therefore $\ker DF(q)$ is spanned by $Jq$.
\end{proof}

Consequently, in region $k$ the chart point of the rotated configuration lies in the box $B_k=[R_{\min,k},R_{\max}]\times[-R_{\max},R_{\max}]^6$, with $R_{\min,k}\le\sqrt{\mu_k}$ positive and $R_{\max}$ the bound of Lemma~\ref{lem:bounds}(b), and it is a zero of $RS$.

\subsection{The search and the count}\label{ss:raysearch}
We search the boxes $B_k$ by branch and bound in outward-rounded interval arithmetic: every operation of IEEE~754 binary64 arithmetic is computed in round-to-nearest, after which the lower endpoint is moved down and the upper endpoint up by one unit in the last place, and no multiplication is fused with an addition. Every computed interval therefore contains the exact value of the corresponding real expression on the box. We discard a box if one of the following tests shows that it contains no zero of $RS$ that is a normalized central configuration in region $k$:
\begin{enumerate}[label=\textup{(T\arabic*)},start=0]
\item an enclosure of $|q_i|^2+c_i-|q_k|^2-c_k$ is positive for some $i$;
\item the enclosure of the virial sum excludes $0$;
\item for some proper subset $C$ all of whose cross pairs are separated on the box, an enclosure of the difference between the two sides of the cluster identity excludes $0$;
\item for such a $C$ with $|C|\ge2$, the enclosures of the two sides of the virial identity for $C$ are disjoint;
\item[(MV)] for some component $F_r$ of $F\circ\Phi_{k,e}$, the mean-value enclosure $F_r(c)+\sum_t\partial_t(F_r\circ\Phi_{k,e})(X)\,(X_t-c_t)$ at the midpoint $c$ excludes $0$.
\end{enumerate}
Since the Newtonian kernel $(\xi,\eta)\mapsto\xi/(\xi^2+\eta^2)^{3/2}$ has no critical point, on a rectangle not containing the origin its extrema are attained at the corners, at the points with $\eta=0$ of its vertical sides, or at the critical points $(\pm|\beta|/\sqrt2,\beta)$ of its restrictions to the horizontal sides $\eta=\beta\ne0$, where it takes the values $\pm\frac{2}{3\sqrt3}\beta^{-2}$. The enclosure on a rectangle is the hull of its enclosed values at those of these points that lie in the rectangle. On a box on which all pairs are separated we apply Krawczyk's test, with the interval Jacobian of $RS$ and a floating-point approximate inverse $C$ of the Jacobian at the midpoint (\S\ref{ss:masscert}). If one component of the operator is disjoint from the box, the box contains no zero. If all components lie in the interior of the box, $RS$ has exactly one zero $v^*$ in it and $DRS(v^*)$ is invertible \cite{Kra69,Moo77,Neu90}. Boxes that are neither discarded nor certified are contracted by the operator or bisected, at an off-centre cut so that symmetric configurations do not lie on cut faces. We tighten a certified box by further Krawczyk steps. On the tightened box the enclosure of $\xi_k-\xi_e$ excludes $0$, and the enclosures of $|q_i|^2+c_i-|q_k|^2-c_k$ are negative for all $i\ne k$.

\begin{proposition}\label{prop:raycount}
Let $m$ be $(1,4,4,4,4)$ or $(1,1,4,4,4)$. Each certified box contains exactly one zero $v^*$ of $RS$, and $\Phi_{k,e}(v^*)$ is a nondegenerate normalized central configuration lying in region $k$ and in no other region. The map from certified boxes to $SO(2)$-orbits of normalized central configurations is a bijection. There are $306$ certified boxes for $(1,4,4,4,4)$ and $294$ for $(1,1,4,4,4)$, distributed over the regions as in Table~\ref{tab:raycount}.
\end{proposition}

\begin{proof}
For each mass vector we search the regions in subproblems whose initial boxes tile $B_k$. We verify the tiling, the inequalities $R_{\min,k}^2\le\mu_k$ and the bound $R_{\max}$ in exact rational arithmetic. In all subproblems each leaf box is discarded or certified, and no subproblem ends abnormally (\S\ref{ss:checks}). A box is discarded only if it contains no zero of $RS$ that is a normalized central configuration in region $k$, contractions keep every zero of $RS$, and bisections cover the box. Hence every chart point of a normalized central configuration in region $k$ lies in a certified box. By Krawczyk's test the certified box contains exactly one zero $v^*$, in its interior. It is a normalized central configuration by Lemma~\ref{lem:RS}(a), since $\xi_k\ne\xi_e$ on the tightened box, and nondegenerate by Lemma~\ref{lem:RS}(b), since $\xi_k\ge R_{\min,k}>0$. It lies strictly in region $k$, hence in no other region. Each orbit is thus the image of a certified box (surjectivity). Two certified boxes with the same orbit come from the same region and, since $q_k$ lies on the positive $\xi$-axis in both, give the same chart point, which lies in the interior of both boxes. The boxes of one subproblem are leaves in a tree of bisections and contractions. Boxes from different subproblems lie in different pieces of the tiling. The two boxes therefore coincide (injectivity). The counts are those of the search (Table~\ref{tab:raycount}).
\end{proof}

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}lllrrrrrrrr@{}}
\toprule
& & & \multicolumn{5}{c}{region} & & & boxes\\
\cmidrule(lr){4-8}
masses & $\varepsilon$ & bound & $1$ & $2$ & $3$ & $4$ & $5$ & total & collinear & examined\\
\midrule
$(1,4,4,4,4)$ & $\frac{397}{250}$ & $\frac{794}{125}$ & $168$ & $28$ & $38$ & $18$ & $54$ & $306$ & $60$ & $211{,}263{,}506$\\
$(1,1,4,4,4)$ & $\frac{1587}{1000}$ & $\frac{5332323001}{839523000}$ & $90$ & $150$ & $18$ & $24$ & $12$ & $294$ & $60$ & $123{,}212{,}100$\\
\bottomrule
\end{tabular}
\caption{The searches of Proposition~\ref{prop:raycount}: the value of $\varepsilon$ and the bound of Lemma~\ref{lem:bounds}(b) at it, which the binary64 number $R_{\max}$ is at least; the certified boxes per region; the collinear boxes; and the number of boxes examined.}\label{tab:raycount}
\end{table}

We certify the shapes in ball arithmetic (Arb, at $200$ bits, through python-flint 0.9.0) on the tightened boxes, after checking that all balls in these computations are finite, that all divisors exclude $0$ and that all square-root arguments are positive. We enclose the orientations of all triples of bodies on the box. If some orientation has a sign, the configuration is not collinear, and each body passes one of two tests: a body is a vertex of the convex hull if the orientations certify that the two hull edges proposed at it leave all other bodies strictly on their left. It is an interior point if they certify that it lies in the open triangle of three other bodies, or in the interior of a quadrilateral formed by the four others, as when the four bodies of mass $4$ form a square with the body of mass $1$ at its centre. If no orientation has a sign, we enlarge the box to one that is symmetric under the reflection $(v_2,v_4,v_6)\mapsto(-v_2,-v_4,-v_6)$ of the chart, which maps the zero set of $RS$ to itself, and Krawczyk's test succeeds on the enlarged box. Its unique zero then coincides with its mirror image. So all five bodies lie on the $\xi$-axis. In this way we show that every certified box is collinear, or non-collinear with a certified shape. Table~\ref{tab:raycount} gives the numbers of collinear boxes.

\begin{proof}[Proof of Theorems~\ref{thm:count1} and~\ref{thm:count2}]
By Proposition~\ref{prop:raycount} there are $306$, resp.\ $294$, orbits, all nondegenerate. With the $60$ collinear ones, we obtain $183$ and $177$ up to similarity. The shape counts are those of the classification. The reflection pairs the non-collinear orbits with equal shapes and reversed cyclic orders. As in the proof of Lemma~\ref{lem:slice}, positive scaling of the masses multiplies the normalized central configurations by $t^{1/3}$, and permutations of the masses relabel them. Since both preserve nondegeneracy, collinearity and shapes, the statements hold on the whole rays and for all relabellings.
\end{proof}

\begin{remark}
By nondegeneracy and the implicit function theorem, the numbers in Theorems~\ref{thm:count1} and~\ref{thm:count2} persist on open neighbourhoods of the two mass vectors that are invariant under positive scaling.
\end{remark}

\subsection{A further point}\label{ss:twopoints}
We treat the mass point $P_1=(9,36,36,36,4)$ of \S\ref{sec:curves} by the same count instead of the separation.

\begin{theorem}\label{thm:countP1}
Let $m$ be a positive multiple of a permutation of $P_1$. Then statements \textup{(i)--(iv)} of Theorem~\ref{thm:count1} hold with the same numbers.
\end{theorem}

The proof of Theorem~\ref{thm:countP1} is that of Theorem~\ref{thm:count1}. Here some coefficients of the chart, $\pm1/9$ and $\pm10/9$, are not binary64 numbers. So we enclose them by intervals. One certified box turns out to have its zero outside its region. This does not affect the count: we do not count that box, since the surjectivity in the proof of Proposition~\ref{prop:raycount} counts every orbit by a box of a region that contains it.

\begin{remark}\label{rem:P1hand}
The method of \cite{Tom26} also gives finiteness at $P_1$ without the count of Theorem~\ref{thm:countP1}. At every positive multiple $m$ of a permutation of $P_1$, the function $U$ vanishes identically on every irreducible curve in the slice $Y_m$ of \S\ref{ss:slice} \cite[Thm.~1.3]{Tom26}. So $m$ is settled, and it has finitely many central configurations up to similarity.
\end{remark}

\subsection{The complex solution sets on the rays}\label{ss:raysAB}
\begin{remark}\label{rem:AB}
At $m=(1,4,4,4,4)$ the slice $Y_m$ of \S\ref{ss:slice} is the union of $36$ pairwise disjoint irreducible curves and a finite set. Through the correspondence of Lemma~\ref{lem:slice}(a) between the points of $Y_m$ and the solutions of \eqref{eq:AK4}, the curves contain the modified Roberts continuum of \S\ref{sec:intro}, carried to these masses by the exchange of the bodies $1$ and $5$ (Lemma~\ref{lem:slice}(b)), and no point of the curves corresponds to a positive normalized central configuration. At $m=(1,1,4,4,4)$ the slice $Y_m$ is finite. Both statements are proved in \cite[Thms.~1.1 and~1.2]{Tom26}. They give a second proof, by a different method, that every point of the special rays has finitely many central configurations up to similarity, but they do not give the counts.
\end{remark}