\section{Cells of exponents}\label{sec:cells}

In this section we describe the exponents at a singular place with a given diagram by finitely many cells, and we enumerate the cells on which $\ell(x)\le0$ is possible. Throughout, $m\in(\R_{>0})^6$, $C\subseteq Y_m$ is a nontrivial curve and $P$ is a singular place of $C$.

\subsection{Trees}\label{ss:trees}
Let $O$ be a further point with $z_O=w_O=0$, the centre of mass, and let $B=\{1,\dots,6,O\}$. For $p\ne q$ in $B$ the exponent $e(z_p-z_q)$ is finite unless $\{p,q\}=\{k,O\}$ for a body $k$ with $z_k\equiv0$ on $C$. At most one body has this property, since $z_k-z_l$ vanishes nowhere on $X_m$.

\begin{definition}\label{def:tree}
A \emph{tree} is a rooted tree whose leaves carry the points of $B$, one point on each leaf, except that the leaf of $O$ may carry one body as well. Every internal node has at least two children and a rational \emph{height}, and the height of an internal node is smaller than the heights of its internal children. For points $p,q$ on different leaves, $h(p,q)$ is the height of the lowest common ancestor of their leaves. For points on the same leaf, $h(p,q)=+\infty$.
\end{definition}

\begin{lemma}\label{lem:tree}
There is a unique tree $T_z$ with $e(z_p-z_q)=h(p,q)$ for all $p\ne q$ in $B$, and likewise a tree $T_w$ for $w$. For each internal node $v$ of $T_z$ with children $c_0,\dots,c_r$ there are pairwise distinct numbers $X_{v,c_0}=0,X_{v,c_1},\dots,X_{v,c_r}\in\C$ such that
\[
\lc{z_p-z_q}{h_v}=X_{v,c(p)}-X_{v,c(q)}
\]
for all points $p$, $q$ below $v$ in different children $c(p)$, $c(q)$ of $v$, where $h_v$ is the height of $v$.
\end{lemma}

\begin{proof}
On functions of $z$-type, $e$ is an increasing affine function of the valuation. Hence $e(z_p-z_r)\ge\min(e(z_p-z_q),e(z_q-z_r))$ for all $p,q,r\in B$, so $-e(z_p-z_q)$ is an ultrametric on the set of leaves, with the convention above for a body at $O$. The closed balls of a finite ultrametric space form a hierarchy, and the tree is the tree of this hierarchy. Its internal nodes are the balls with at least two leaves, and the height of such a ball is the least exponent $e(z_p-z_q)$ of two of its points. This tree is the only one with the stated property. For the second statement, fix a point $o$ below $c_0$ and put $X_{v,c(p)}=\lc{z_p-z_o}{h_v}$ for $p$ below $v$. This depends only on the child $c(p)$, since points below one child have differences of exponent larger than $h_v$, and it is $0$ for $c(p)=c_0$. For $p$, $q$ in different children, $z_p-z_q$ has exponent exactly $h_v$, so the difference $X_{v,c(p)}-X_{v,c(q)}$ is nonzero.
\end{proof}

We call the $X_{v,c}$ and the analogous $Y_{v,c}$ of $T_w$ the \emph{node coordinates}. The exponents at $P$ are read off the trees: $a_k=h_z(k,O)$, $\zeta_{kl}=h_z(k,l)$, and likewise $b_k$ and $\omega_{kl}$ from $T_w$.

\begin{lemma}\label{lem:constraints}
The trees of $P$ satisfy the following conditions.
\begin{enumerate}[label=\textup{(\alph*)}]
\item The root heights are at least $-2$. For each pair $kl$, $\zeta_{kl}+3\omega_{kl}\le4$, with equality if and only if $kl$ is a $z$-stroke, and $3\zeta_{kl}+\omega_{kl}\le4$, with equality if and only if $kl$ is a $w$-stroke. A body $k$ has $a_k=-2$ if and only if it is $z$-circled, and $b_k=-2$ if and only if it is $w$-circled.
\item The root of each tree has at least three children.
\item In each of the twelve equations of the bodies and in each of the identities $U=E$ and $\sum_{k<l}m_km_lz_{kl}w_{kl}=ME$, the least exponent of the terms is attained by at least two terms.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) is Lemma~\ref{lem:place}, (O1), (O2) and the definitions, since the root height is an exponent of a position or of a separation. (b) Let $r$ be the root of $T_z$ and $c_O$ its child containing $O$. A body below another child $c$ has $a_k=h_r$ and $\lc{z_k}{h_r}=X_{r,c}-X_{r,c_O}$ (Lemma~\ref{lem:tree}), and a body below $c_O$ has $a_k>h_r$. The coefficient of exponent $h_r$ in $\sum_km_kz_k=0$ is therefore $\sum_{c\ne c_O}(X_{r,c}-X_{r,c_O})M_c$, where $M_c$ is the sum of the masses below $c$. If $r$ had only the two children $c_O$ and $c$, this would be $(X_{r,c}-X_{r,c_O})M_c\ne0$. The same holds for $T_w$. (c) is Lemma~\ref{lem:initial}, since the $m_k$, $E$ and $M$ are nonzero.
\end{proof}

Note that condition (c) for the second identity contains \cite[Lemma~3.3]{Tom26}: if all $x_{kl}$ were positive, the constant term $ME$ would be the only term of least exponent.

\subsection{Cells}\label{ss:celldef}
The equations of the bodies, $U=E$ and $\sum_{k<l}m_km_lz_{kl}w_{kl}=ME$ are the fourteen \emph{base identities}. Their terms are $z_k$ (omitted if $k$ shares the leaf of $O$) and $m_lZ_{kl}$ in the $z$-equation of $k$, likewise in $w$, the constant and the $m_km_lp_{kl}$ in $U=E$, and the constant and the $m_km_lz_{kl}w_{kl}$ in the last identity. Their exponents are linear forms in the heights.

\begin{definition}\label{def:cell}
Let $D$ be a labelled diagram. A \emph{cell} of $D$ consists of a pair of trees without heights satisfying Lemma~\ref{lem:constraints}(b), and of a set of at least two terms for each base identity, its \emph{argmin set}, such that the following set $\Pi$ of height vectors is not empty: the heights increase strictly along the trees, they satisfy Lemma~\ref{lem:constraints}(a) for $D$, and in each base identity the terms of the argmin set have equal exponents and all other terms have larger exponents. The cell is \emph{nonpositive} if $\ell(x)\le0$ at some point of $\Pi$.
\end{definition}

The set $\Pi$ is a relatively open polyhedron. By Lemma~\ref{lem:constraints}, the trees of $P$ and the sets of terms of least exponent at $P$ form a cell of the diagram of $P$, and the heights of $P$ lie in its set $\Pi$. We then say that $P$ \emph{lies in} this cell. When the diagram forces terms of exponent $-2$ in a base identity (a circle or a stroke in an equation of a body), the argmin set consists of these terms, by (O1). If the diagram has $zw$-edges, the argmin set of $U=E$ consists of their terms, of exponent $-1$, by (O1) and (O4). The relabellings and $\iota$ act on the cells of all diagrams compatibly with Lemma~\ref{lem:symm}, so it suffices to enumerate the cells of one representative of each class.

\begin{proposition}\cert\label{prop:cells}
The enumeration below lists $91{,}273$ cells of the representatives of the $191$ classes of Proposition~\ref{prop:enum}, in $69$ classes. Every nonpositive cell of a representative is in the list.
\end{proposition}

\begin{proof}
The program starts from the trees with the single leaf $O$ and inserts the bodies one at a time into both trees, in all ways: below an internal node, by subdividing an edge, as a new root, or on the leaf of $O$. It discards a pair of partial trees if the linear conditions among the points inserted so far are infeasible together with the relaxation of $\ell(x)\le0$ in which each pair not yet inserted contributes its least possible value of $x$: $-2$ in general, $0$ on a stroke and $1$ on a $zw$-edge (Lemma~\ref{lem:orders}). Restricting two trees to fewer points gives partial trees visited by the search, so no pair of trees is lost. Each tree arises exactly once: without pruning, the insertions produce $55{,}720$ distinct trees, the $39{,}208$ trees with seven leaves and the $16{,}512$ trees with a body on the leaf of $O$. For complete pairs satisfying Lemma~\ref{lem:constraints}(b) it chooses the argmin sets one base identity at a time and discards infeasible choices. Apart from linear programs, the search discards a pair or a choice for only four reasons: the root of a complete tree has two children (Lemma~\ref{lem:constraints}(b)), a body circled in the colour of a tree shares the leaf of $O$ in that tree, a base identity has a single forced term of least exponent, or fewer than two terms remain for an argmin set. Every infeasibility is certified by Lemma~\ref{lem:farkas} below. Feasibility is decided by the floating-point solver without a certificate (see Section~\ref{sec:verification}). The search for each class ran in $384$ parts. Each part repeats the insertions of the first two bodies, and a counter over the insertions of the third body assigns each of them to one part. This division is consistent because all parts reach the same decisions on the first two insertions: the discarded pairs are certified in every part, and each kept pair was confirmed by an exact rational solution of its conditions. So every node of the search lies in some part.
\end{proof}

A linear system as in the proof has rows $\mathrm{lo}_i\le a_i\cdot h+c_is\le\mathrm{hi}_i$, where $h$ is the vector of heights, either bound may be absent, and a strict inequality $f>0$ is written $f-s\ge0$ with a common slack $s$. Each row has $c_i\in\{-1,0,1\}$, and the rows with $c_i\ne0$ remain valid when $s$ decreases. The system has a solution with $s>0$ exactly when the strict system is feasible.

\begin{lemma}\label{lem:farkas}
Let $y_i$ be rational numbers with $y_i>0$ only if $\mathrm{hi}_i$ exists and $y_i<0$ only if $\mathrm{lo}_i$ exists. Put $b_i=\mathrm{hi}_i$ for $y_i>0$ and $b_i=\mathrm{lo}_i$ for $y_i<0$, $g=\sum_iy_ic_i$, $\beta=\sum_iy_ib_i$ and $g^-=\max(0,-g)$. Suppose that $\sum_iy_ia_i=0$ and $\beta+g^-\le0$, and that $g+g^->0$ or $\beta+g^-<0$. Then the system has no solution with $s>0$.
\end{lemma}

\begin{proof}
Let $(h,s)$ be a solution with $s>0$. Decreasing $s$ keeps the rows valid, so we may assume $s\le1$. Adding the rows with the weights $y_i$ gives $gs\le\beta$, and adding $g^-s\le g^-$ gives $(g+g^-)s\le\beta+g^-\le0$. Since $g+g^-\ge0$ and $s>0$, both $g+g^-$ and $\beta+g^-$ vanish, contrary to the hypothesis.
\end{proof}

By the transposition theorem of Motzkin, multipliers as in Lemma~\ref{lem:farkas} exist for any infeasible system of this form. The program obtains them by rounding the dual solution or the dual ray of the floating-point solver, and checks them in rational arithmetic. When rounding fails, it computes them with an exact simplex method. The counts are in Table~\ref{tab:verif}.