\section{Generic exclusion of the cells}\label{sec:exclusion}

We now exclude the listed cells at generic masses, except $43$ of them. Let $D$ be the representative of a class and $\Pi$ the polyhedron of a listed cell of $D$. Recall from Definition~\ref{def:cell} that a cell consists of a pair of trees and of an argmin set for each base identity. Throughout, $m\in(\R_{>0})^6$ and $C\subseteq Y_m$ is a nontrivial curve.

For an identity of \S\ref{ss:identities} we say that its \emph{argmin set is constant on the cell} if there is a set $S$ of its terms such that at every point of $\Pi$ the terms in $S$ have equal exponents and all other terms have larger exponents. Besides the base identities, whose argmin sets are part of the cell, we use the subset sums for $2\le|J|\le4$, the torques $T_J$ for the sets $J\ni1$ with $|J|\le5$, the partial virials for all nonempty $J\ne\{1,\dots,6\}$, the differences, the identity $U=\sum_km_kz_kw_k$ and the identity $\sum_km_kz_kw_k=E$. Whether an argmin set is constant on the cell is decided by linear programs, and each decision on which the conclusion depends is certified by Lemma~\ref{lem:farkas}.

\begin{lemma}\label{lem:single}
If some identity has on the whole of $\Pi$ a single term of least exponent, no singular place of $C$ lies in the cell.
\end{lemma}

\begin{proof}
At such a place, the least exponent of the identity is attained once, contradicting Lemma~\ref{lem:initial}. The coefficients $c_i(m)$ are rational multiples of products of masses, or $E$, or $ME$, and do not vanish.
\end{proof}

Let the cell have no identity as in Lemma~\ref{lem:single}. Its \emph{initial system} is the following set of polynomials in the node coordinates $X_{v,c}$, $Y_{v,c}$ ($c\ne c_0$), in variables $P_{kl}$ for the leading coefficients of the $p_{kl}$, and in $E$, with coefficients in $\Q[m]$: for each identity whose argmin set $S$ is constant on the cell, the initial form $\sum_{i\in S}c_i\,\lc{g_i}{}$, where $\lc{g_i}{}$ is the product of the leading coefficients of the factors of $g_i$, written through Lemma~\ref{lem:tree} ($\lc{z_p-z_q}{}=X_{v,c(p)}-X_{v,c(q)}$ and $\lc{p_{kl}}{}=P_{kl}$); and for each pair $kl$ the polynomial $P_{kl}^2\lc{z_{kl}}{}\lc{w_{kl}}{}-1$. At a place that lies in the cell, the leading coefficients are a common zero of the initial system (Lemma~\ref{lem:initial} and $p_{kl}^2z_{kl}w_{kl}=1$) at which the product $\Theta$ of the following quantities does not vanish: the node coordinates $X_{v,c}$, $Y_{v,c}$ with $c\ne c_0$ and their pairwise differences at each node, the $P_{kl}$, and $E$.

\begin{lemma}\label{lem:core}
Let $G$ be a subset of the initial system of a cell.
\begin{enumerate}[label=\textup{(\alph*)}]
\item If the saturation of the ideal generated by $G$ in $\Q(m_1,\dots,m_6)[X,Y,P,E]$ by $\Theta$ is the unit ideal, there is a nonzero $h\in\Z[m_1,\dots,m_6]$ such that no singular place of $C$ lies in the cell when $h(m)\ne0$.
\item The same conclusion holds if the saturation is the unit ideal over $\Q(m_2,\dots,m_6)$ after the substitution $m_1=1$, or after the same substitution for another mass.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) The hypothesis gives $\Theta^N=\sum_ig_if_i$ with $f_i\in G$ and $g_i$ polynomials over $\Q(m)$. Clearing denominators gives $h\,\Theta^N=\sum_ig_i'f_i$ with $0\ne h\in\Z[m]$ and $g_i'$ polynomials over $\Q[m]$. At the leading coefficients of a place in the cell the right side vanishes and $\Theta\ne0$, so $h(m)=0$. (b) The isomorphism $Y_m\to Y_{m/m_1}$ of Lemma~\ref{lem:slice}(b) maps $C$ to a nontrivial curve and preserves exponents, so a place in the cell at $m$ gives one at $m/m_1$. Part (a), applied over $\Q(m_2,\dots,m_6)$, gives $h_1\ne0$ with $h_1(m_2/m_1,\dots,m_6/m_1)=0$, and $h=m_1^{\deg h_1}h_1(m_2/m_1,\dots,m_6/m_1)$ is a nonzero polynomial.
\end{proof}

A subset $G$ as in Lemma~\ref{lem:core} is a \emph{core} of the cell. The program searches for cores with the divide-and-conquer procedure QuickXplain of \cite{Jun04}, testing with Singular, modulo the prime $32003$ and at random masses, whether a subset generates the unit ideal. This search only proposes candidates. Each proposed core is then certified by a standard basis in Singular, over $\Q(m)$ or, in the second pass below, over the field of rational functions of the other masses after one mass that occurs in the core is set to $1$. Both steps ran without saturation (Section~\ref{sec:verification}), so each certified core generates the unit ideal, which is more than Lemma~\ref{lem:core} needs. Each cell with a core also has an explicit identity $\sum_ic_if_i=h$, in which the $f_i$ are the elements of a core of the cell, the $c_i$ are polynomials over $\Z$ and $h\in\Z[m]$ is not zero. These identities are checked in Lean (\S\ref{ss:lean}).

\begin{proposition}\cert\label{prop:exclusion}
Of the $91{,}273$ cells of the list, $52{,}432$ satisfy the hypothesis of Lemma~\ref{lem:single}. Of the others, $37{,}808$ have a core certified as in Lemma~\ref{lem:core}\textup{(a)}, and $990$ further cells have a core certified as in Lemma~\ref{lem:core}\textup{(b)}, found by repeating the search with random orders of the initial forms. The remaining $43$ cells belong to the diagrams $64$ \textup{(}$11$ cells\textup{)}, $66$ \textup{(}$11$\textup{)}, $88$ \textup{(}$9$\textup{)}, $89$ \textup{(}$1$\textup{)}, $93$ \textup{(}$1$\textup{)}, $94$ \textup{(}$9$\textup{)} and $98$ \textup{(}$1$\textup{)}.
\end{proposition}

The first pass left $1{,}033$ cells without a certified core, and the second pass certified $990$ of them (Section~\ref{sec:verification}). Neither pass produced a certified core for the remaining $43$ cells. For these we need Sections~\ref{sec:ak50} and~\ref{sec:k4}, which treat their seven diagrams at the level of places and so cover all cells of these diagrams. For diagram~$89$ the torque of Theorem~\ref{thm:d89} adds nothing to the initial system where $b_4-2<1+\frac12a_1$, since there its initial form is a multiple of that of the equation of body~$4$ (Remark~\ref{rem:d89lead}).

Of the $24$ diagrams left unresolved in \cite{CC23}, the eleven numbered $5$, $6$, $65$, $67$, $69$, $71$, $73$, $74$, $75$, $79$ and $80$ have no nonpositive cell, so that a place with one of these diagrams has $\ell(x)>0$ and is never the place of Corollary~\ref{cor:sum}. For the six numbered $8$, $52$, $63$, $81$, $82$ and $90$, Proposition~\ref{prop:exclusion} excludes all listed cells. The remaining seven are the diagrams of Proposition~\ref{prop:exclusion} that contain the $43$ cells, treated in Sections~\ref{sec:ak50} and~\ref{sec:k4}.