\section{The nine-row coefficients}\label{sec:12-nine-row-coefficients}

We prove the vanishing of the two coefficients with first part nine
and of $\mathcal C_{(8,3)}$, using the residual interface of
Proposition~\ref{prop:residual-interface}. The Picard test detects
$\mathcal C_{(9,2)}$ and $\mathcal C_{(8,3)}$ simultaneously; the Picard
and parallel tests together detect $\mathcal C_{(9,1,1)}$.

\begin{proposition}\label{prop:g5-coefficient-nine-two}
Let $F=[12]$ and $\tau=(2,2,2,1^6)$. Then
$\mathcal R_\tau=M_\tau(\mathcal R_F^{\mathrm{Pic}})\simeq\Pmot$ is
the single Picard tail copy described in
Lemma~\ref{lem:genus-four-source-multiplicities}; and the residual map
$\varepsilon_\tau:\mathcal R_\tau\to\mathscr Q_\tau/\delta_\tau(\mathcal U_\tau)$
of Proposition~\ref{prop:residual-interface} is injective. Its kernel
being canonically
$(\mathcal C_{(9,2)}\otimes\mathcal B_{\tau,(9,2)})
 \oplus(\mathcal C_{(8,3)}\otimes\mathcal B_{\tau,(8,3)})$
with one-dimensional branching spaces,
$\mathcal C_{(9,2)}=\mathcal C_{(8,3)}=0$.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:genus-four-source-multiplicities}, $\mathcal R_\tau$
is the single tail copy, with $|I\setminus A|=3$ and divisor in $Z_B$.
Exactness of $M_\tau$ (Set-up~\ref{setup:standing}) identifies the
residual target with $M_\tau(\mathscr Q_F^{\mathrm{Pic}})$ and
$\varepsilon_\tau$ with $M_\tau$ of the map
$f_F:\mathcal R_F^{\mathrm{Pic}}\to\mathscr Q_F^{\mathrm{Pic}}$ of
Proposition~\ref{prop:g5-picard-injection}. That proposition makes
$f_F$ injective; its hypotheses cover the contributing allocation, whose
endpoint has an old flag and whose partner has three markings besides
its attaching flag. Hence the kernel in the statement is zero.
\end{proof}

\begin{proposition}\label{prop:g5-coefficient-nine-one-one}
Let $F=[11]$ and $\mu=(3,1^8)$, and put $U=\mathcal U_\mu$,
$R_W=M_\mu(\mathcal R_F^{\mathrm{Pic}})$,
$R_{\mathrm{par}}=M_\mu(\mathcal R^{\mathrm{par}}_{+,F})$, so that
$\mathcal R_\mu=R_W\oplus R_{\mathrm{par}}$;
let $\Phi_W=M_\mu(\Phi_F^{\mathrm{Pic}})$ and let
$\Phi_{\mathrm{par}}$ be the restriction of $M_\mu(p_4)$ to
$R_{\mathrm{par}}$. Then the residual map $\varepsilon_\mu$ of
Proposition~\ref{prop:residual-interface} is injective. Its kernel
being canonically $\mathcal C_{(9,1,1)}$, one has
$\mathcal C_{(9,1,1)}=0$. Moreover,
$\operatorname{rank}_{\mathbb Q}\delta_\mu=394$.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:genus-four-source-multiplicities},
\[
 N_\mu=U\oplus\Phi_W(R_W)\oplus\Phi_{\mathrm{par}}(R_{\mathrm{par}}),
 \quad U\simeq\Pmot^{182},\quad R_W\simeq\Pmot^{17},\quad
 R_{\mathrm{par}}\simeq\Pmot^6,
\]
all with an old flag on the decorated vertex.
Proposition~\ref{prop:residual-interface}
gives $J_\mu=\ker(\delta_\mu|_U)\simeq\Pmot^8$ and
$\delta_\mu(U)\simeq\Pmot^{174}$, and identifies
the residual kernel with $\mathcal C_{(9,1,1)}$.

\emph{The Picard residual.}
Exactness of $M_\mu$ (Set-up~\ref{setup:standing}) identifies the
residual target with
$M_\mu(\mathscr Q_F^{\mathrm{Pic}})$. Proposition~\ref{prop:g5-picard-injection}
therefore makes the restriction to $R_W$ injective. Its hypotheses
cover every contributing allocation: the endpoint has an old flag
and the partner has at most three markings besides its attaching
flag, as established in the source count.

\emph{Separation of the parallel residual.}
Put $B_\mu=M_\mu(B_{\mathrm{sep},F})$.
Every map defining $U$ is a separating Gysin or a composition of
separating Gysins, and every map defining $\Phi_W$ is a separating
Gysin. Hence
\[
 U+\Phi_W(R_W)\subseteq B_\mu.
\]
The complete degree-thirteen coefficient on a separating parent
contains each of these factorizations. There is consequently a further
quotient map
\[
 \mathscr Q_\mu/\delta_\mu(U)
 \longrightarrow\mathscr Q_\mu/\delta_\mu(B_\mu)
 \cong M_\mu\bigl(
 \mathscr Q_{5,F}/\delta_F(B_{\mathrm{sep},F})\bigr).
\]
The subspace by which we take this last quotient is the image under $\delta_F$ of the
full ordinary separating image in genus four. It is distinct from the
image of the separating term of the compact row already used to define
$\mathscr Q_{5,F}$.

Proposition~\ref{prop:g5-parallel-left-inverse} supplies the left
inverse $\mathcal L\operatorname{pr}_{\rm dl}$ after precisely this
further quotient. Its kernel $\mathcal R^{\rm par}_{0,F}$ has zero
$\mu$-multiplicity (Lemma~\ref{lem:genus-four-source-multiplicities}).
Thus the induced left inverse detects all of $R_{\mathrm{par}}$.

Suppose the residual image of $(w,r)$ is zero. Passing to the further
quotient kills the Picard contribution, and applying the parallel
left inverse gives $r=0$. The Picard injection then gives $w=0$.

Finally $\delta_\mu(U)\cong\Pmot^{174}$ and the residual source is
$\Pmot^{17+6}$. The exact sequence of images therefore gives
\[
 \rank_{\mathbb Q}\delta_\mu
   =2\cdot174+2(17+6)=394.
\]
\end{proof}