\section{The genus-one and genus-two formulas}\label{sec:04-low-genus}

\begin{definition}\label{def:symplectic-coefficients-and-forgetful-spans}
For $g\geq2$, let $\mathbb H_g=R^1\pi_*\mathbb Q$ be the weight-one
variation associated with the universal smooth curve over $\Mopen_g$.
For a partition $\lambda$ of length at most $g$, let
$\mathbb V_\lambda$ be its irreducible symplectic variation, normalized
to weight $|\lambda|$; for $g=2$ we also write
$\mathbb V_{a,b}=\mathbb V_{(a,b)}$ for $a\geq b\geq0$. For $n>0$, let $\pi_i:\Mopen_{g,n}\to
\Mopen_{g,n-1}$ forget the $i$th marking, and put
\[
\mathcal F^k_{g,n}=\sum_{i=1}^n\pi_i^*W_kH^k(\Mopen_{g,n-1}),
\qquad
\Psi^k_{g,n}=\sum_{i=1}^n\psi_i\mathcal F^{k-2}_{g,n}.
\]
Set both spans to zero for $n=0$, and write
$\mathcal O_{g,n}=W_{15}H^{15}(\Mopen_{g,n})$.
\end{definition}

\begin{theorem}[Deligne {\cite[Section 3.2]{deligne1971}}]
\label{thm:cited-ordinary-weight-sequence}
Let $X$ be a smooth proper complex algebraic variety, let $D\subset X$
be a normal-crossings divisor, and put $U=X\setminus D$. Let
$D^{[p]}$ be the normalized locus of $p$ distinct local branches,
with the orientation local system $\epsilon_p$, and set $D^{[0]}=X$.
Then the ordinary weight spectral sequence is
\[
E_1^{-p,q}=H^{q-2p}(D^{[p]},\epsilon_p)\otimes\Lmot^p
\Longrightarrow H^{q-p}(U).
\]
Its first differential is alternating Gysin pushforward. It
degenerates at $E_2$, and $E_2^{-p,q}=\operatorname{Gr}_q^W H^{q-p}(U)$.
\end{theorem}

\begin{proposition}[The primitive degree-fifteen quotient]
\label{prop:primitive-degree-fifteen-quotient}
For $g\geq2$ and $n\geq0$, the restriction to the open moduli space gives
a right-exact sequence
\[
B^{(1)}_{g,n}\otimes\Pmot
\xrightarrow{\ \Phi^{15}_{g,n,\mathrm B}\ }
H^{15}(\Mbar_{g,n})\longrightarrow\mathcal O_{g,n}\longrightarrow0.
\]
Moreover, $W_{13}H^{13}(\Mopen_{g,n})=0$ and
\[
\mathcal O_{g,n}/\mathcal F^{15}_{g,n}
\cong\bigoplus_{\substack{\lambda\vdash n\\\ell(\lambda)\leq g}}
W_{15}H^{15-n}(\Mopen_g,\mathbb V_\lambda)\otimes V_{\lambda^T}.
\]
For $n\geq15$ one has $\mathcal O_{g,n}=\mathcal F^{15}_{g,n}$.
Consequently, the maps $\Phi^{15}_{g,n,\mathrm B}$ are surjective for
all $n$ if and only if
$W_{15}H^{15-n}(\Mopen_g,\mathbb V_\lambda)=0$ for all
$0\leq n\leq14$ and all $\lambda\vdash n$ of length at most $g$.
\end{proposition}
\begin{proof}
\smallskip\noindent\emph{The boundary quotient.}
Apply Theorem~\ref{thm:cited-ordinary-weight-sequence} to local
normal-crossings charts for the boundary of $\Mbar_{g,n}$.
The branch complex is invariant under changes of chart. Descending its
ordered branch covers and taking the alternating part gives the
orientation local system; finite stabilizer invariants are exact over
$\mathbb Q$ (Set-up~\ref{setup:standing}). Thus the same complex
computes the stack cohomology.
Its terms are pure of the indicated weight, so its higher differentials
vanish as in the variety theorem. At the right end of the weight-fifteen
row this gives
\[
H^{13}(\widetilde{\partial\Mopen_{g,n}})\otimes\Lmot
\longrightarrow H^{15}(\Mbar_{g,n})
\longrightarrow W_{15}H^{15}(\Mopen_{g,n})\longrightarrow0,
\]
where the normalization is taken with its quotient-stack stabilizers.
Proposition~\ref{prop:h13-input} identifies the first term and its map
with $B^{(1)}_{g,n}\otimes\Pmot$ and $\Phi^{15}_{g,n,\mathrm B}$.

\smallskip\noindent\emph{The marking quotient.}
By Theorem~\ref{thm:cited-thirteen-cohomology},
$H^{13}(\Mbar_{g,n})$ is spanned by boundary Gysin images. These restrict to zero
on the open stack, whereas restriction surjects onto its lowest-weight
cohomology. Hence $W_{13}H^{13}(\Mopen_{g,n})=0$ for every $n$.
It follows that $\mathcal F^{13}_{g,n}=0$ and
$\Psi^{15}_{g,n}=\sum_i\psi_i\mathcal F^{13}_{g,n}=0$.
By \cite[Lemma 3.1]{canning2024a}, for $g\geq2$ and $n,k\geq0$ the
quotient $W_kH^k(\Mopen_{g,n})/(\mathcal F^k_{g,n}+\Psi^k_{g,n})$ is
$\Sn_n$-equivariantly isomorphic to
$\bigoplus_{\lambda\vdash n,\ \ell(\lambda)\leq g}
W_kH^{k-n}(\Mopen_g,\mathbb V_\lambda)\otimes V_{\lambda^T}$, and
$W_kH^k(\Mopen_{g,n})=\mathcal F^k_{g,n}$ if $n>0$ and $n\geq k$.
With $k=15$ this gives
the asserted quotient and the equality
$\mathcal O_{g,n}=\mathcal F^{15}_{g,n}$ for $n\geq15$.

\smallskip\noindent\emph{The finite reduction.}
Surjectivity of the sum of the boundary Gysin maps in every arity is equivalent to
$\mathcal O_{g,n}=0$ for all $n$. This implies the listed coefficient
vanishings by the quotient formula. Conversely, assume those
vanishings. At $n=0$ the forgetful span is zero. If
$\mathcal O_{g,n-1}=0$, every summand defining
$\mathcal F^{15}_{g,n}$ is zero. The quotient formula therefore gives
$\mathcal O_{g,n}=0$ successively for $n\leq14$.
For $n\geq15$ the equality with the forgetful span continues the same induction.
\end{proof}

The function $\mu$ below is the number-theoretic M\"obius function.

\begin{definition}\label{def:low-genus-series}\label{def:low-genus-serre-series}
Work over $\widehat\Lambda\otimes\mathbb Q(\Lmot)$. Put
\[
\OpenSerre_0=\frac{\prod_{r\geq1}(1+p_r)^{\frac1r\sum_{d\mid r}
\mu(r/d)(1+\Lmot^d)}-1}{\Lmot^3-\Lmot}
-\frac{h_1}{\Lmot^2-\Lmot}-\frac{h_2}{\Lmot+1}.
\]
Let $\mathcal T=h_1+$ terms of arity at least two be the unique series with
$\mathcal T\circ(h_1-\partial\OpenSerre_0)=h_1$. For a Laurent variable
$w$, let
\[
\mathcal P(w)=\prod_{r\geq1}(1+p_r)^{\frac1r\sum_{d\mid r}\mu(r/d)
(1-w^d-\Lmot^dw^{-d}+\Lmot^d)}
\]
and define
\[
\mathcal C_{12}=\operatorname*{Res}_{w=0}
\left[\frac{\mathcal P(w)-1}{(1-w)(1-\Lmot/w)}
\frac{w^{10}}{\Lmot^{11}}\left(w-\frac{\Lmot}{w}\right)dw\right].
\]
Expand the products in the $p_r$, with $\deg p_r=r$. At each arity take
the ordinary Laurent residue at $w=0$ over $\mathbb Q(\Lmot)$; in
particular,
\[
\frac1{1-w}=\sum_{a\geq0}w^a,
\qquad
\frac1{1-\Lmot/w}=-\sum_{b\geq1}\frac{w^b}{\Lmot^b}.
\]

Let $\OpenSerre_1$ be the arity sum of the equivariant Serre
characteristics of $\Mopen_{1,n}$, and for $g=0,1$ let
$\CompactSerre_g$ be the corresponding sum for $\Mbar_{g,n}$, in each
case over stable arities. Serre
characteristics use compactly supported cohomology with sign $(-1)^i$.
Write $f'=\partial f$ and $\dot f=\partial f/\partial p_2$, with
$\partial$ and $\Psi_r$ as in Section~\ref{sec:02-notation}. For $k\geq2$ let
$\mathsf S_k$ denote the sum of the pure Hodge structures of weight
$k-1$ attached to cusp forms for $\mathrm{SL}_2(\mathbb Z)$; it is zero when the cusp space
is zero, in particular for odd $k$; thus $\mathsf S_{12}=\Soneone$
and $\mathsf S_{16}=\Sonesix$. The operator $[\Soneone]$ retains
all Tate powers. Let $\phi$ denote Euler's totient function.
\end{definition}

\begin{theorem}[Getzler; correction of Consani--Faber
{\cite[Section 3]{getzler1998}, \cite[Section 4]{consani2006}}]
\label{thm:cited-genus-one-serre}
With the arity expansions of Definition~\ref{def:low-genus-series},
$\OpenSerre_0$ is the arity sum of the equivariant Serre
characteristics of $\Mopen_{0,n}$, $\mathcal T=h_1+\CompactSerre_0'$, and
\[
\begin{split}
\OpenSerre_1=\operatorname*{Res}_{w=0}\Bigg[
&\frac{\mathcal P(w)-1}{(1-w)(1-\Lmot/w)}
\left(\sum_{k\geq1}\frac{\mathsf S_{2k+2}+1}{\Lmot^{2k+1}}w^{2k}-1\right)
\left(w-\frac{\Lmot}{w}\right)dw\Bigg],\\
\CompactSerre_1=\Bigg(&\OpenSerre_1-
\frac12\sum_{r\geq1}\frac{\phi(r)}r\log(1-\Psi_r(\OpenSerre_0''))
+\frac{\dot{\OpenSerre}_0^2+\dot{\OpenSerre}_0+
\tfrac14\Psi_2(\OpenSerre_0'')}{1-\Psi_2(\OpenSerre_0'')}
\Bigg)\circ\mathcal T.
\end{split}
\]
These are identities in the arity-completed Grothendieck ring of
equivariant Hodge structures.
\end{theorem}

\begin{proposition}[Genus-one coefficient extraction]
\label{prop:genus-one-coefficient}
For $n\geq1$, the genus-one Serre series satisfies
\[
[\Soneone]\CompactSerre_1=\mathcal C_{12}\circ\mathcal T,
\qquad
\ch(\En_{1,n})=
\bigl[-[\Lmot^2](\mathcal C_{12}\circ\mathcal T)\bigr]_n.
\]
Each homogeneous arity component of the first identity is a Laurent
polynomial in $\Lmot$, so the second formula is finite in each arity.
\end{proposition}
\begin{proof}
In the residue of Theorem~\ref{thm:cited-genus-one-serre}, the only term containing
$\Soneone$ is the $k=5$ term
\[
\Soneone\,\Lmot^{-11}w^{10}.
\]
The other factors contain no cusp structures. Taking this coefficient
inside the residue gives $[\Soneone]\OpenSerre_1=\mathcal C_{12}$.
The remaining terms in the compactification identity are genus-zero
terms. Genus-zero compact cohomology is Tate by
Theorem~\ref{thm:cited-genus-zero-cycles}, and the explicit
$\OpenSerre_0$ has only Tate coefficients. Differentiation, Adams
operations, products, and the displayed formal expansions preserve
this property. These terms have no $\Soneone$ coefficient. Since
$\mathcal T=h_1+\CompactSerre_0'$ also has Tate coefficients, the
compactification identity gives
\[
[\Soneone]\CompactSerre_1
=([\Soneone]\OpenSerre_1)\circ\mathcal T
=\mathcal C_{12}\circ\mathcal T.
\]

The coefficient of $\Lmot^2\Soneone$ in the equivariant Serre
characteristic of $\Mbar_{1,n}$ comes only from $H^{15}$, by purity.
It occurs with a negative sign in the Serre characteristic.
Theorem~\ref{thm:motivic-types} identifies
its multiplicity module with $\En_{1,n}$, giving the second formula.

The residue is finite with the stated convention.
In fixed arity the numerator is a Laurent polynomial in $w$ and
$\Lmot$, and its negative $w$ powers have a lower bound. Both denominator
expansions in Definition~\ref{def:low-genus-series} have nonnegative
$w$ powers, so only finitely many terms contribute to the coefficient of $w^{-1}dw$.
Their coefficients are Laurent polynomials in $\Lmot$.
The series $\mathcal T$ has no constant term and has polynomial Tate
coefficients. Composition therefore uses finitely many terms at each
arity and preserves Laurent-polynomial coefficients.
\end{proof}

\begin{theorem}[Genus one]\label{thm:genus-one}
With the series of Definition~\ref{def:low-genus-series}, one has
\[
F_1=-[\Lmot^2](\mathcal C_{12}\circ\mathcal T),\qquad
\ch_n(\En_{1,n})=[F_1]_n.
\]
This module vanishes for $n<13$ and
is nonzero for $n\geq13$. Together with the primitive term in
Theorem~\ref{thm:motivic-types}, this gives decompositions of Hodge
structures and $\ell$-adic Galois representations before semisimplification
in every stable genus-one arity.
The ordinary one-edge Gysin image is exactly
$\En_{1,n}\otimes\Pmot$.
\end{theorem}

\begin{proof}
The coefficient assertion is Proposition~\ref{prop:genus-one-coefficient}.

\smallskip\noindent\emph{The threshold.}
Equivariant Poincar\'e duality with $\dim_{\mathbb C}\Mbar_{1,n}=n$ gives
\[
H^{15}(\Mbar_{1,n})
\cong H^{2n-15}(\Mbar_{1,n})^\vee\otimes\Lmot^n.
\]
For $1\leq n\leq7$ the degree on the right is negative.
For $8\leq n\leq12$ it is one of $1,3,5,7,9$, so
Theorem~\ref{thm:cited-low-odd-vanishing} gives zero.
At $n=13$, Proposition~\ref{prop:eleven-hook-realizations} gives
\[
H^{11}(\Mbar_{1,13})\cong V_{(3,1^{10})}\otimes\Soneone.
\]
Specht self-duality and
$\Soneone^\vee\otimes\Lmot^{11}\cong\Soneone$ therefore give
\[
H^{15}(\Mbar_{1,13})\cong V_{(3,1^{10})}\otimes\Pmot.
\]

For $n\ge13$ the forgetful map $\pi_n:\Mbar_{1,n+1}\to\Mbar_{1,n}$ has
a section, so $\pi_n^*$ is injective.
The nonzero $\Pmot$ substructure at $n=13$ persists under these Hodge
morphisms. Theorem~\ref{thm:motivic-types} and the distinct Hodge types
in Notation~\ref{def:motivic-pieces} place it in the $\Pmot$ summand.

\smallskip\noindent\emph{The ordinary boundary image.}
The lowest-weight boundary sequence, obtained by the same branch
localization used in Proposition~\ref{prop:primitive-degree-fifteen-quotient},
is $\mathcal D^{13}_{1,n,\mathrm B}
\xrightarrow{\Phi^{15}_{1,n,\mathrm B}} H^{15}(\Mbar_{1,n})
\longrightarrow W_{15}H^{15}(\Mopen_{1,n})\longrightarrow0$.
By \cite[Proposition 2.2]{canning2024a},
$W_{15}H^{15}(\Mopen_{1,n})\cong\mathsf S_{16}\otimes V_{(n-14,1^{14})}$
for $n\geq15$ and $W_{15}H^{15}(\Mopen_{1,n})=0$ for $n<15$; this
identifies the last term
with $\Khook_n\otimes\Sonesix$. By Proposition~\ref{prop:h13-input}
the first term is entirely of type $\Pmot$.
The two simple Hodge structures are nonisomorphic. Hence the map to the open term
vanishes on the $\Pmot$ summand and is an isomorphism on the primitive
summand. Exactness proves
\[
\operatorname{im}\Phi^{15}_{1,n,\mathrm B}
=\En_{1,n}\otimes\Pmot.
\]

\smallskip\noindent\emph{The actual Galois realization.}
The boundary-image calculation satisfies the hypothesis of
Proposition~\ref{prop:h13-input}. The $\ell$-adic boundary image is
\[
(\En_{1,n}\otimes\mathbb Q_\ell)\otimes\Pmot_\ell.
\]
Lemma 8.3 of \cite{canning2024a} (recalled before
Theorem~\ref{thm:cited-primitive-hook}) and that theorem identify the
primitive summand $(\Khook_n\otimes\mathbb Q_\ell)\otimes\Sonesix_\ell$
via algebraic forgetful maps, with the stated hook relations.
The direct sum of the primitive inclusion and the boundary Gysin map
is surjective in the Betti realization by the preceding decomposition.
Comparison, compatible with these maps and the relabelling action,
makes its $\ell$-adic realization surjective. The dimensions of its
two images add to the target dimension, so their intersection is zero.
Consequently
\[
H^{15}_{\mathrm{et}}(\Mbar_{1,n,\overline{\mathbb Q}},\mathbb Q_\ell)
\cong
(\En_{1,n}\otimes\mathbb Q_\ell)\otimes\Pmot_\ell
\oplus(\Khook_n\otimes\mathbb Q_\ell)\otimes\Sonesix_\ell.
\]
The Hodge decomposition is the one in Theorem~\ref{thm:motivic-types}.
\end{proof}

\begin{theorem}[Saito {\cite[Section 4.5]{saito1990}}]
\label{thm:cited-geometric-weight-bound}
Let $X$ be a smooth complex algebraic variety and let $\mathbb V$ be
a geometric polarizable variation of pure Hodge structures of weight $w$.
Then $H^p(X,\mathbb V)$ has weights at least $p+w$ for every $p\geq0$.
\end{theorem}

\begin{definition}\label{def:low-genus-symplectic-coefficients}
Let $\mathcal A_2$ be the moduli stack of principally polarized abelian
surfaces, and put $D=\mathcal A_{1,1}=[\mathcal A_1^2/\Sn_2]$, whose
complement is $\Mopen_2$. For $a\geq b\geq0$, the irreducible
symplectic variation of highest weight $(a,b)$ on $\mathcal A_2$,
normalized to weight $a+b$, restricts on $\Mopen_2$ to the variation
$\mathbb V_{a,b}$ of
Definition~\ref{def:symplectic-coefficients-and-forgetful-spans}; we
use the same symbol on $\mathcal A_2$.
Let $s_k=\dim S_k(\mathrm{SL}_2(\mathbb Z))$, let $s_{j,k}$ be the
dimension of Siegel cusp forms of type $\operatorname{Sym}^j\otimes\det^k$
for $\operatorname{Sp}_4(\mathbb Z)$, and let $\overline{\mathsf S}_{j,k}$
be the weight-$(j+2k-3)$ part of their attached cohomological structure.
\end{definition}

\begin{theorem}[Petersen {\cite[Theorem 2.1 and Remark 2.2]{petersen2015}}]
\label{thm:cited-abelian-surface-coefficients}
Let $a\geq b\geq0$, assume $(a,b)\neq(0,0)$ and $a+b$ even.
Then $H_c^i(\mathcal A_2,\mathbb V_{a,b})=0$ for $i\notin\{2,3,4\}$,
and
\[
H_c^4(\mathcal A_2,\mathbb V_{a,b})=
\begin{cases}s_{a+b+4}\Lmot^{b+2},&a=b\text{ even},\\0,&\text{otherwise}.
\end{cases}
\]
The odd-weight part in degree two is
$\bigoplus_{r\ {\rm odd}}\operatorname{Gr}_r^W
H_c^2(\mathcal A_2,\mathbb V_{a,b})\cong\mathsf S_{b+2}$.
Moreover,
\[
\bigoplus_{r\ \mathrm{odd}}\operatorname{Gr}_r^W
H_c^3(\mathcal A_2,\mathbb V_{a,b})
\cong\overline{\mathsf S}_{a-b,b+3}
\oplus\mathsf S_{a-b+2}^{\oplus s_{a+b+4}}\otimes\Lmot^{b+1}
\oplus\mathsf S_{a+3}.
\]
The last formula is an isomorphism of weight-graded rational Hodge
structures and an isomorphism after semisimplification in the
$\ell$-adic category.
\end{theorem}

\begin{proposition}[Petersen {\cite[Proposition 3.4]{petersen2013}}]
\label{prop:cited-symplectic-branching}\label{def:low-genus-wreath-modules}
Write $V_r=\operatorname{Sym}^r\mathbb Q^2$ for the irreducible
$\operatorname{Sp}_2$-representation. On
$(\operatorname{Sp}_2\times\operatorname{Sp}_2)\rtimes\Sn_2$, let
$U_{r,s}$ restrict to $(V_r\boxtimes V_s)\oplus(V_s\boxtimes V_r)$
with exchanged summands, and let $U_r^\pm$ restrict to
$V_r\boxtimes V_r$ with positive or negative interchange of factors.
Let $a\geq b\geq0$ with $a+b$ even, and let the superscript
$(-1)^b$ mean $+$ for even $b$ and $-$ for odd $b$. For the irreducible
$\operatorname{Sp}_4$-representation $V_{a,b}$ one has
\[
\operatorname{Res}_{\operatorname{Sp}_2\wr\Sn_2}^{\operatorname{Sp}_4}V_{a,b}
=\bigoplus_{i=0}^b\left(
U_{(a-b)/2+i}^{(-1)^b}
\oplus\bigoplus_{0\leq j<(a-b)/2}U_{a-b+i-j,i+j}\right).
\]
\end{proposition}

\begin{proposition}\label{prop:genus-two-gysin-surjectivity}
For every $n\geq0$, the complete ordinary one-edge Gysin map
\[
\Phi^{15}_{2,n,\mathrm B}:\mathcal D^{13}_{2,n,\mathrm B}
\longrightarrow H^{15}(\Mbar_{2,n})
\]
is surjective, and $W_{15}H^{15}(\Mopen_{2,n})=0$.
\end{proposition}
\begin{proof}
The genus-two clauses of Proposition~\ref{prop:cited-chow-kunneth-bases} give
$\mathcal O_{2,n}=0$ for $n\leq10$.
At $n=12$, their compact boundary-generation statement makes
$\Phi^{15}_{2,12,\mathrm B}$ surjective. The right-exact sequence in
Proposition~\ref{prop:primitive-degree-fifteen-quotient} therefore gives
$\mathcal O_{2,12}=0$.

For an odd-weight symplectic variation on $\Mopen_2$, the central
hyperelliptic involution acts by $-1$. Rigidification by this order-two
automorphism computes its rational invariants
(Set-up~\ref{setup:standing}), which are zero.
All cohomology with coefficients in such a local system therefore vanishes.
The quotient in Proposition~\ref{prop:primitive-degree-fifteen-quotient},
first at $n=11$ and then at $n=13$, gives
$\mathcal O_{2,11}=\mathcal O_{2,13}=0$.

For $a+b=14$, localization along $D$ gives
\[
H^1(\mathcal A_2,\mathbb V_{a,b})
\longrightarrow H^1(\Mopen_2,\mathbb V_{a,b})
\longrightarrow H^0(D,\mathbb V_{a,b}|_D)\otimes\Lmot.
\]
The first term vanishes by
Theorem~\ref{thm:cited-abelian-surface-coefficients} and duality
from $H_c^5(\mathcal A_2,\mathbb V_{a,b})=0$.
In Proposition~\ref{prop:cited-symplectic-branching}, a trivial
representation of the wreath product can occur only when $a=b$,
in the $i=0$ equal-factor term. Interchange acts there by $(-1)^b$,
so it is invariant only for even $b$.
Here $a=b$ would force $a=b=7$. Thus the last term is also zero,
and $H^1(\Mopen_2,\mathbb V_{a,b})=0$ for every $a+b=14$.
Proposition~\ref{prop:primitive-degree-fifteen-quotient} now gives
$\mathcal O_{2,14}=0$.

Proposition~\ref{prop:primitive-degree-fifteen-quotient} (its induction
from $n=14$) now gives $\mathcal O_{2,n}=0$ and surjectivity in every arity.
\end{proof}

\begin{proposition}\label{prop:genus-two-eleven-one-gysin}
For $D=\mathcal A_{1,1}\subset\mathcal A_2$, the localization Gysin map
\[
\operatorname{Gr}_{15}^W
\bigl(H^1(D,\mathbb V_{11,1}|_D)\otimes\Lmot\bigr)
\longrightarrow
\operatorname{Gr}_{15}^W H^3(\mathcal A_2,\mathbb V_{11,1})
\]
is an isomorphism. Both its source and target are isomorphic to $\Pmot$.
\end{proposition}
\begin{proof}
\smallskip\noindent\emph{A sign-projected configuration space.}
Let $f:\Mopen_{2,12}\to\Mopen_{2,1}$ retain marking $12$, let
$\rho:\Mopen_{2,1}\to\Mopen_2$ be the universal proper curve, and put
$\mathbb V=\rho^*\mathbb H_2$.
The fiber of $f$ over $(C,p)$ is the ordered configuration space of
eleven points on $C\setminus\{p\}$.
Filter its complement in $(C\setminus\{p\})^{11}$ by collision
partitions. Each proper collision partition has a block of size at
least two, and a transposition in that block acts trivially on the
stratum. Its sign-isotypic compactly supported cohomology is therefore
zero. Localization for this finite stratification, followed by
relative duality, identifies the sign parts for the configuration
space and the product. This argument is functorial in $(C,p)$.

Since $H^\bullet(C\setminus\{p\})=\mathbb Q\oplus H^1(C)[-1]$,
the Koszul action gives
\[
\operatorname{Hom}_{\Sn_{11}}(\operatorname{sgn}_{11},R^qf_*\mathbb Q)
=
\begin{cases}
\operatorname{Sym}^{10}\mathbb V,&q=10,\\
\operatorname{Sym}^{11}\mathbb V,&q=11,\\
0,&\text{otherwise}.
\end{cases}
\]
Indeed, the alternating projection permits at most one degree-zero
factor and symmetrizes the odd factors.

In the sign-isotypic part of the Leray spectral sequence, the term
$E_2^{4,11}=H^4(\Mopen_{2,1},\operatorname{Sym}^{11}\mathbb V)$
has no incoming differential. Its only possible outgoing differential
has target $H^6(\Mopen_{2,1},\operatorname{Sym}^{10}\mathbb V)$.
Theorem~\ref{thm:cited-geometric-weight-bound} puts the target in
weights at least sixteen. To apply the variety statement, take a
finite level cover of the smooth moduli stack and then
finite-group invariants (Set-up~\ref{setup:standing}). Strictness
makes the differential zero on
weight fifteen. Thus the weight-fifteen part of $E_2^{4,11}$ survives.
It is a subquotient of $W_{15}H^{15}(\Mopen_{2,12})$, which is zero
by Proposition~\ref{prop:genus-two-gysin-surjectivity}. Hence
\[
\operatorname{Gr}_{15}^W
H^4(\Mopen_{2,1},\operatorname{Sym}^{11}\mathbb V)=0.
\]

\smallskip\noindent\emph{The $(11,1)$ coefficient.}
For $\rho$, the direct-image rows have coefficients
$\operatorname{Sym}^{11}\mathbb H_2$,
$\operatorname{Sym}^{11}\mathbb H_2\otimes\mathbb H_2$, and
$\Lmot\operatorname{Sym}^{11}\mathbb H_2$.
The hyperelliptic involution acts by minus the identity on the first
and last, so their base cohomology is zero. Leray therefore identifies
\[
H^4(\Mopen_{2,1},\operatorname{Sym}^{11}\mathbb V)
\cong H^3(\Mopen_2,\operatorname{Sym}^{11}\mathbb H_2\otimes\mathbb H_2).
\]
Symplectic Pieri gives the decomposition of variations
\[
\operatorname{Sym}^{11}\mathbb H_2\otimes\mathbb H_2
=\mathbb V_{12,0}\oplus\mathbb V_{11,1}
 \oplus\Lmot\mathbb V_{10,0}.
\]
All summands have weight twelve.
It follows that $\operatorname{Gr}_{15}^W
H^3(\Mopen_2,\mathbb V_{11,1})=0$.
Localization now makes the map in the statement surjective.

\smallskip\noindent\emph{The two rank-two objects.}
In the restriction of $\mathbb V_{11,1}$ to $D$, only the
unequal-factor term with one trivial factor contributes to $H^1$.
Here $H^1(\mathcal A_1,\mathbb Q)=0$, since the coarse space is the
$j$-line (Set-up~\ref{setup:standing}).
Proposition~\ref{prop:cited-symplectic-branching} gives that term as
$\Lmot$ times the two exchanged coefficients of degrees $(10,0)$.
The Tate factor restores the weight of the variation.
By Eichler--Shimura \cite[Section 5]{petersen2013}, for even $r>0$ and
the standard weight-one variation $\mathbb H_1$ on $\mathcal A_1$,
$H^i(\mathcal A_1,\operatorname{Sym}^r\mathbb H_1)=0$ for $i\neq1$, and
$H^1(\mathcal A_1,\operatorname{Sym}^r\mathbb H_1)$ has weights $r+1$
and $2r+2$, with
$\operatorname{Gr}_{r+1}^WH^1(\mathcal A_1,\operatorname{Sym}^r\mathbb H_1)
=\mathsf S_{r+2}$ and Eisenstein weight-$(2r+2)$ piece.
With $r=10$ this identifies
the weight-fifteen part after the localization twist with
$\Lmot^2\Soneone=\Pmot$.

Theorem~\ref{thm:cited-abelian-surface-coefficients} exhibits
$s_{16}\mathsf S_{12}\otimes\Lmot^2=\Pmot$ in
$\operatorname{Gr}_{15}^W H_c^3(\mathcal A_2,\mathbb V_{11,1})$;
here $s_{16}=1$. Duality on the three-dimensional stack has twist
$\Lmot^{15}$ for this weight-twelve coefficient, so it exhibits
$\Pmot^\vee\otimes\Lmot^{15}\cong\Pmot$ in the target of our map.
The target thus has dimension at least two, and surjectivity from
$\Pmot$ makes its dimension at most two. The map is an isomorphism.
\end{proof}

\begin{definition}\label{def:low-genus-ordinary-row}
For a stable $(g,n)$, let
\[
G_p^{15}(g,n)=\bigoplus_{[\Gamma],\,|E(\Gamma)|=p}
\left(H^{15-2p}(X_\Gamma)\otimes
\det(\mathbb Q^{E(\Gamma)})\right)^{A_\Gamma}\otimes\Lmot^p.
\]
The face map $d_p^{\mathrm{face}}$ is alternating Gysin contraction
of distinct ordered branches, descended by quotient-stack pushforward.
For an incidence stabilizer map $H\to K$, its transfer on a pushed
forward class $z$ is $|H|^{-1}\sum_{k\in K}k^*z$.
For a rational $\Sn_d$-module $W$ and $n\geq d$, write $I_n(W)$ for
the induced module
$\operatorname{Ind}_{\Sn_d\times\Sn_{n-d}}^{\Sn_n}(W\boxtimes\mathbf1)$
of Definition~\ref{def:low-genus-thirteen-modules}, and put $I_n(W)=0$
for $n<d$.
\end{definition}

\begin{proposition}[The genus-two ordinary row]
\label{prop:low-genus-boundary-rows}
Let $n\geq0$. Then $G_p^{15}(2,n)=0$ for $p\geq3$, and the row
\[
G_2^{15}(2,n)\xrightarrow{d_2^{\mathrm{face}}}
G_1^{15}(2,n)\xrightarrow{\Phi^{15}_{2,n,\mathrm B}}
H^{15}(\Mbar_{2,n})\longrightarrow0
\]
has canonical homology identifications
\[
\begin{split}
R_n=\ker d_2^{\mathrm{face}}
&\cong\operatorname{Gr}_{15}^W H^{13}(\Mopen_{2,n})
\cong\left(I_n(V_{(2,1^8)})\oplus I_n(V_{(3,1^8)})\right)\otimes\Pmot,\\
Q_n=\ker\Phi^{15}_{2,n,\mathrm B}/\operatorname{im}d_2^{\mathrm{face}}
&\cong\operatorname{Gr}_{15}^W H^{14}(\Mopen_{2,n})
\cong\bigwedge^{11}\mathbb Q^n\otimes\Pmot.
\end{split}
\]
The first decomposition into induced modules need not be canonical.
The map $d_2^{\mathrm{face}}$ is injective exactly for $n\leq9$;
the row is exact at $G_1^{15}(2,n)$ exactly for $n\leq10$.
\end{proposition}
\begin{proof}
\smallskip\noindent\emph{The ordinary boundary row.}
Apply Theorem~\ref{thm:cited-ordinary-weight-sequence} with the branch
descent described in Proposition~\ref{prop:primitive-degree-fifteen-quotient}.
Ordering the branches gives the determinant of the edge set.
On quotient charts, summing over the ordered branches gives the
finite transfer in Definition~\ref{def:low-genus-ordinary-row}.
Thus its weight-fifteen row is exactly the complex in the statement.
For $p\geq3$, a nonnegative degree $15-2p$ is one of $9,7,5,3,1$.
Every odd K\"unneth factor of such a degree vanishes by
Theorem~\ref{thm:cited-low-odd-vanishing}; negative degrees vanish as
well. Hence $G_p^{15}(2,n)=0$ for $p\geq3$.
Degeneration at $E_2$ identifies the kernel and middle homology with
\[
\operatorname{Gr}_{15}^W H^{13}(\Mopen_{2,n}),
\qquad
\operatorname{Gr}_{15}^W H^{14}(\Mopen_{2,n}),
\]
respectively. The last map is surjective by
Proposition~\ref{prop:genus-two-gysin-surjectivity}.

\smallskip\noindent\emph{Localization along the collision locus.}
The calculation uses cohomology with supports, since the collision
divisors need not have normal crossings.
Let $\mathcal C_2\to\Mopen_2$ be the universal proper curve, and put
$Y_n=\mathcal C_2^{\,n}$, with fiber products over $\Mopen_2$ and
$Y_0=\Mopen_2$. The open substack $\Mopen_{2,n}\subset Y_n$ is the
locus of pairwise distinct points. Write
$\Delta_{ij}\cong Y_{n-1}$ for a collision divisor and
$Z_n=\bigcup_{i<j}\Delta_{ij}$. A collision stratum is indexed by a
partition of $[n]$ into $k$ blocks; it is $\Mopen_{2,k}$ of codimension
$c=n-k$.

In genus two,
$W_mH^m(\Mopen_{2,k})=0$ for every odd $m\leq13$.
For $m\leq9$ this follows from
Theorem~\ref{thm:cited-low-odd-vanishing}, for $m=11$ from
Proposition~\ref{prop:eleven-hook-realizations}, and for $m=13$ from
Theorem~\ref{thm:cited-thirteen-cohomology} and restriction.
The same ordinary boundary sequence gives
\[
\begin{array}{ll}
\operatorname{Gr}_{m+2}^W H^m(\Mopen_{2,k})=0,
 &m\leq11\text{ odd},\\
\operatorname{Gr}_{m+1}^W H^m(\Mopen_{2,k})=0,
 &m\leq10\text{ even}.
\end{array}
\]
Indeed, the possible $E_1$ terms are respectively in column $-2$
with compact degree $m-2$, and column $-1$ with compact degree $m-1$.
Their odd degrees are at most nine.

Purity for a smooth collision stratum expresses its contribution to
degree-$r$ cohomology with supports as
$H^{r-2c}(\Mopen_{2,k})\otimes\Lmot^c$.
Filter a union of strata by codimension and use the localization exact
sequences. The preceding vanishings give
\[
\operatorname{Gr}_{15}^W H^{15}_{Z_n}(Y_n)=
\operatorname{Gr}_{15}^W H^{13}_{Z_n}(Y_n)=0.
\]
For a union of strata of codimension at least two they also give
vanishing of cohomology with supports in degree fourteen and weight fifteen.
The Mayer--Vietoris sequence for cohomology with supports, applied one
divisor at a time, consequently
gives
\[
\operatorname{Gr}_{15}^W H^{14}_{Z_n}(Y_n)
=\bigoplus_{i<j}\operatorname{Gr}_{13}^W H^{12}(\Delta_{ij})
 \otimes\Lmot.
\]
Localization is now the exact sequence
\begin{equation}\label{eq:low-genus-collision-sequence}
\begin{split}
0\longrightarrow \operatorname{Gr}_{15}^W H^{13}(Y_n)
&\longrightarrow R_n\\
&\longrightarrow
\bigoplus_{i<j}\operatorname{Gr}_{13}^W H^{12}(Y_{n-1})\otimes\Lmot\\
&\longrightarrow\operatorname{Gr}_{15}^W H^{14}(Y_n)
\longrightarrow Q_n\longrightarrow0.
\end{split}
\end{equation}

\smallskip\noindent\emph{Cohomology on the unmarked base.}
The rigidification of $\Mopen_2$ by the hyperelliptic involution is
$[\Mopen_{0,6}/\Sn_6]$.
Odd-weight coefficients have zero cohomology (proof of
Proposition~\ref{prop:genus-two-gysin-surjectivity}); for even
coefficients rational cohomology is the $\Sn_6$-invariant part.
The three forgetful fibrations
\[
\Mopen_{0,6}\longrightarrow\Mopen_{0,5}
\longrightarrow\Mopen_{0,4}\longrightarrow\Mopen_{0,3}
\]
have punctured projective lines as fibers. Each fiber has
cohomological dimension one for local systems, so the base has
cohomological dimension at most three.
After fixing three points at $0,1,\infty$,
$\Mopen_{0,6}$ is the complement of
$x_i=0$, $x_i=1$, and $x_i=x_j$ in affine three-space.
Deletion and restriction of these hyperplanes, whose intersections
are affine spaces, show that its cohomology is Hodge--Tate.
The same holds for the trivial coefficient on $\Mopen_2$.

For later use, we compute the relevant groups for
$\Lmot^t\mathbb V_{a,b}$, where $t\geq0$ and $w=a+b+2t$.
For $w=10$ or $12$ one has
\begin{equation}\label{eq:low-genus-base-degree-two}
\operatorname{Gr}_{w+3}^W
H^2(\Mopen_2,\Lmot^t\mathbb V_{a,b})
=
\begin{cases}
\Lmot\Soneone,&(a,b,t)=(10,0,0),\\
\Pmot,&(a,b,t)=(10,0,1),\\
0,&\text{otherwise}.
\end{cases}
\end{equation}
Here and below equalities between cohomology objects mean isomorphisms.
To prove this formula, first suppose $(a,b)\ne(0,0)$.
By Theorem~\ref{thm:cited-abelian-surface-coefficients} and duality,
$H^2(\mathcal A_2,\mathbb V_{a,b})$ is zero unless $a=b$ is even;
in that case it is $s_{2a+4}\Lmot^{a+1}$.
After the twist it has weight $w+2$, not $w+3$.
In $H^1(D,\mathbb V_{a,b}|_D)$,
Proposition~\ref{prop:cited-symplectic-branching} and
Eichler--Shimura (as recalled in the proof of
Proposition~\ref{prop:genus-two-eleven-one-gysin}), together
with K\"unneth, leave only the
unequal-factor term of degrees $(a-b,0)$, with twist $\Lmot^b$.
Terms with two positive symmetric degrees have zero $H^0$ on either
factor, and the equal trivial term has $H^1(\mathcal A_1)=0$.
After the localization twist, the cusp contribution is
\[
\Lmot^{t+b+1}\mathsf S_{a-b+2},
\]
of weight $w+3$. The Eisenstein contribution has even weight.
In the indicated range, a nonzero cusp contribution requires $a-b=10$:
the cusp spaces below weight twelve and in weight fourteen vanish.
Thus only $(a,b)=(10,0)$ and $(11,1)$ remain.
For $(10,0)$ the weight-thirteen part of
$H^3(\mathcal A_2,\mathbb V_{10,0})$ is zero:
the scalar coefficient is $s_{14}=0$, the odd elliptic coefficient is
$\mathsf S_{13}=0$, and $s_{10,3}=0$ since by
\cite[Section 2.1]{petersen2015} the level-one Siegel cusp dimensions
satisfy $\sum_{j\geq0}s_{j,3}x^j
=x^{36}/\bigl((1-x^6)(1-x^8)(1-x^{10})(1-x^{12})\bigr)$.
The same conclusion after a Tate twist shows that this boundary cusp
contribution lies in the kernel of the localization Gysin map.
For $(11,1)$ that kernel vanishes by the isomorphism in
Proposition~\ref{prop:genus-two-eleven-one-gysin}.
This proves \eqref{eq:low-genus-base-degree-two}; the trivial
coefficient was treated separately above.

Two further degrees enter the calculation of $R_n$. For $w=12$,
\[
\operatorname{Gr}_{15}^W H^1(\Mopen_2,\Lmot^t\mathbb V_{a,b})=0.
\]
For nontrivial coefficients, $H^1(\mathcal A_2,\mathbb V_{a,b})=0$
by duality from $H_c^5=0$; localization embeds $H^1$ of the complement
in $H^0(D,\Lmot^{t+1}\mathbb V_{a,b})$, which is pure of weight
fourteen. The trivial case is Hodge--Tate.
For $w=10$ one has
\begin{equation}\label{eq:low-genus-base-degree-three}
\operatorname{Gr}_{15}^W
H^3(\Mopen_2,\Lmot^t\mathbb V_{a,b})
=
\begin{cases}
\Pmot,&(a,b,t)=(9,1,0),\\
0,&\text{otherwise}.
\end{cases}
\end{equation}
In the localization sequence the $H^1(D)$ cusp contribution, including
the Gysin twist, has weight thirteen. Its Eisenstein contribution has
even weight. In $H^2(D)$, a cusp would require one symmetric degree at
least ten. Since the total symmetric degree is at most ten, the other
factor would be trivial and have zero $H^1$. All remaining
contributions have even weight. Thus the weight-fifteen group equals
that on $\mathcal A_2$.
Set $u=a+b=10-2t$. Duality identifies the desired part with weight
$u+1$ in $H_c^3(\mathcal A_2,\mathbb V_{a,b})$, followed by duality
and twist $\Lmot^{u+3+t}$.
The Siegel and endoscopic terms in
Theorem~\ref{thm:cited-abelian-surface-coefficients} have weight $u+3$.
The term $\mathsf S_{a+3}$ has weight $a+2=u+1$ exactly when $b=1$.
Then $a=9-2t$, and $\mathsf S_{12-2t}$ is nonzero only for $t=0$.
It gives $\Soneone^\vee\otimes\Lmot^{13}=\Pmot$, proving
\eqref{eq:low-genus-base-degree-three}.

\smallskip\noindent\emph{The universal-curve powers.}
The direct image $R^q(Y_n\to\Mopen_2)_*\mathbb Q$ is pure of weight
$q$. It is a direct sum of tensor products of $\mathbb H_2$ and
$\Lmot$, and its irreducible summands have the form just considered.
Odd $q$ gives zero base cohomology by the involution.
The cohomological dimension bound leaves only base degree two in
total degrees twelve and fourteen at weights thirteen and fifteen,
respectively. A base-degree-zero term has the weight of its fiber
degree and does not contribute. These degree-two terms survive Leray:
an incoming $d_2$ has an odd fiber row, higher incoming differentials
have negative base degree, and outgoing differentials exceed degree
three on the base.
For total degree thirteen and weight fifteen, the base-degree-one
term vanishes by the preceding calculation, leaving base degree
three and fiber degree ten. Its possible incoming $d_2$ has an odd
row; an incoming $d_3$ originates in base degree zero and pure weight
twelve, so is zero at weight fifteen.

Write $\mathbb Q^m$ with its permutation basis indexed by $[m]$.
Formula~\eqref{eq:low-genus-base-degree-two} gives
\[
\operatorname{Gr}_{13}^W H^{12}(Y_m)
=\bigwedge^{10}\mathbb Q^m\otimes\Lmot\Soneone.
\]
The ten degree-one factors are symmetrized as symplectic tensors.
Their Koszul action makes their marking action alternating.
Formula~\eqref{eq:low-genus-base-degree-three} similarly gives
\[
\operatorname{Gr}_{15}^W H^{13}(Y_n)
=I_n\bigl(V_{(2,1^8)}\bigr)\otimes\Pmot.
\]
Indeed, $\mathbb V_{9,1}$ first occurs in tensor degree ten, with
ordinary Schur--Weyl multiplicity $V_{(9,1)}$; tensoring by the
Koszul sign conjugates this partition.

Let $M_n$ be the multiplicity space in
$\operatorname{Gr}_{15}^W H^{14}(Y_n)=M_n\otimes\Pmot$.
There are two types of generators. For an oriented ten-subset $T$ and
$i\notin T$, let $B_{T;i}$ denote ten degree-one factors and the
degree-two class in slot $i$. For an unordered pair $\{i,j\}$ disjoint
from $T$, let $A_{T;ij}$ denote twelve degree-one factors, with the
symplectic tensor inserted in slots $i,j$.
These generators are independent and exhaustive.
To check the latter assertion for twelve degree-one slots, the
multiplicity of $\Lmot\operatorname{Sym}^{10}\mathbb H_2$ is
$\binom{12}{2}$. In the symplectic tensor rule, a path from the empty
partition to $(10)$ in twelve moves has eleven additions and one
removal. There are eleven paths staying in one row. The other paths
add a second-row box at a time $i$ and remove it at a time $j$, with
$2\leq i<j\leq12$, giving $\binom{11}{2}$ paths. A third row cannot
be removed in the single removal. The total is $11+55=66$.
The corresponding $66$ insertion maps are independent: in a
symplectic basis $x,y,z,w$, the coefficient with $z,w$ in a prescribed
pair and $x$ in every other slot detects just that pair.
Before the Koszul sign their permutation module is induced from
$\mathbf1_{10}\boxtimes\operatorname{sgn}_2$; after it, the module is
induced from $\operatorname{sgn}_{10}\boxtimes\mathbf1_2$.
This is precisely the orientation and symmetry specified for $A$.

\smallskip\noindent\emph{The collision Gysin map.}
Normalize the fiber point class $\nu_i$ by $\int_C\nu_i=1$.
Put $\Omega_{ij}=[\Delta_{ij}]-\nu_i-\nu_j$. The two diagonal identities are
\[
\Delta_*1=\nu_i+\nu_j+\Omega_{ij},
\qquad
\Delta_*h=h_i\nu_j+\nu_i h_j\quad(h\in H^1(C)).
\]
Choose the insertion generators $A$ using $\Omega_{ij}$.
These identities determine the middle map in
\eqref{eq:low-genus-collision-sequence}, since the Leray pieces
identified above are its only weight-fifteen pieces.

A source generator is an oriented ten-subset of the labels after
merging $i,j$ to a label $*$. If it excludes $*$, its image is
\[
A_{T;ij}+B_{T;i}+B_{T;j}.
\]
If it contains $*$, put $*$ last and write $T$ for the preceding
oriented nine-subset. Its image is
\[
B_{(T,i);j}+B_{(T,j);i}.
\]
The two signs agree because the point class has even degree.
Each $A$ occurs once in the first family, with coefficient one.
Eliminating all these generators leaves the second family of
relations among the $B$'s.
The map
\[
B_{T;i}\longmapsto
\left(\bigwedge_{t\in T}e_t\right)\wedge e_i
\]
annihilates these relations. For a fixed eleven-element support the
relations identify the eleven distinguished-slot choices with their
exterior signs, leaving one alternating line. Hence
\[
Q_n\cong\bigwedge^{11}\mathbb Q^n\otimes\Pmot.
\]

The kernel can be read from the same presentation. The projection of
the first source family onto the $A$ generators is an isomorphism,
so a kernel vector uses only the second family. On each
eleven-element support its source and target characters are
\[
e_9h_2=s_{(3,1^8)}+s_{(2,1^9)},\qquad
e_{10}h_1=s_{(2,1^9)}+e_{11}.
\]
The cokernel is the alternating line just found. Thus the image is
$V_{(2,1^9)}$ and the kernel is $V_{(3,1^8)}$.
Inducing over the support gives
$I_n(V_{(3,1^8)})\otimes\Pmot$.
The left term of \eqref{eq:low-genus-collision-sequence} contributes
$I_n(V_{(2,1^8)})\otimes\Pmot$.
Pure polarizable Hodge structures are semisimple, and averaging over
$\Sn_n$ makes a splitting equivariant (Set-up~\ref{setup:standing}).
Consequently $R_n$ is their
direct sum, with no preferred splitting.

Its left homology first occurs on ten markings, and its
middle homology first occurs on eleven. Both induced modules remain
nonzero in every larger arity. This proves the two exactness
thresholds.
\end{proof}

\begin{lemma}\label{lem:low-genus-pointed-elliptic-divisors}
Let $B$ be a finite set and let $q\notin B$. Put
$\lambda_1=\delta_{\mathrm{irr}}/12$ on $\Mbar_{1,B\cup\{q\}}$.
Then $\lambda_1$ and the classes $\delta_{0,T}$ for
$T\subseteq B\cup\{q\}$, $|T|\geq2$, form a basis of
$H^2(\Mbar_{1,B\cup\{q\}};\mathbb Q)$.
As $B$ varies, the Frobenius characteristic of this permutation basis is
\[
E+E E_{\geq2}+E E_{\geq1}=E(2E-1-h_1).
\]
\end{lemma}

\begin{proof}
Proposition~\ref{prop:genus-three-picard-input} expresses all generators
in Theorem~\ref{thm:cited-divisor-generation} in the displayed classes.
We prove independence by showing that
$H^1(\Mopen_{1,B\cup\{q\}};\mathbb Q)=0$.

Let $\mathcal C_1\to\Mopen_{1,1}$ be the universal elliptic curve with
origin $q$, and let $m=|B|$. Inside $\mathcal C_1^m$, the open moduli
stack removes the zero sections in each slot and all pair diagonals.
The base has zero rational $H^1$ (as at
Proposition~\ref{prop:genus-two-eleven-one-gysin}). The invariant part
of $R^1(\mathcal C_1^m\to\Mopen_{1,1})_*\mathbb Q$ is zero: inversion
acts by $-1$. Leray gives $H^1(\mathcal C_1^m)=0$.
Intersections of distinct removed divisors have codimension at least
two. The localization exact sequence therefore injects $H^1$ of the
complement into the span of the divisor classes, with image equal to
the kernel of their map to $H^2(\mathcal C_1^m)$.
These classes are independent already after restriction to one smooth
fiber $C^m$. The zero sections give the independent point classes
$\nu_i$. Each diagonal has, in addition to $\nu_i+\nu_j$, a nonzero
component in the distinct K\"unneth summand
$H^1(C)_i\otimes H^1(C)_j$. These components first force every diagonal
coefficient to vanish, and the point classes then force the remaining
coefficients to vanish. Thus the open $H^1$ is zero.

Localization for $\Mopen_{1,B\cup\{q\}}\subset
\Mbar_{1,B\cup\{q\}}$ now injects the span of its irreducible boundary
divisors into compact $H^2$. They are $\delta_{\mathrm{irr}}$ and the
listed $\delta_{0,T}$, proving independence.
The basis element $\lambda_1$ contributes $E$.
A tail not containing $q$ contributes $E E_{\geq2}$; a tail containing
$q$ contributes $E E_{\geq1}$. Their sum is the stated characteristic.
\end{proof}

\begin{proposition}\label{prop:genus-two-character-row}
Let $D_+=\tfrac12\partial^2+\partial/\partial p_2$.
Let $B_2$ and $C_2$ be the arity sums of the rational multiplicity
characters of $G_1^{15}(2,n)$ and $G_2^{15}(2,n)$ after removing
$\Pmot$. Then
\[
\begin{split}
B_2={}&D_+U+E_{\geq2}\partial(EH)+E\partial U+HE(2E-1-h_1),\\
C_2={}&(2H+G)EE_{\geq2}+HEE_{\geq1}+HE
 +(H+G)E_{\geq2}+GE_{\geq1},\\
\sum_{n\geq0}\ch(\En_{2,n})={}&B_2-C_2
 +E\bigl(s_{(2,1^8)}+s_{(3,1^8)}\bigr)-Ee_{11}.
\end{split}
\]
\end{proposition}
\begin{proof}
\smallskip\noindent\emph{One edge.}
Proposition~\ref{prop:h13-input} requires both K\"unneth families in the
degree-thirteen source. A nonseparating genus-one normalization
contributes $D_+U$: the two gluing flags are unordered and invariant,
so their operator is
$h_2^\perp=\frac12\partial^2+\partial/\partial p_2=D_+$.
A separating divisor of genera zero and two contributes
$E_{\geq2}\partial(EH)$, since the rational vertex needs at least two
ordinary markings. For a divisor of genera one and one, the
$H^{13}\otimes H^0$ family contributes $E\partial U$.
The $H^{11}\otimes H^2$ family contributes
$H E(2E-1-h_1)$ by
Lemma~\ref{lem:low-genus-pointed-elliptic-divisors}.
Here $H$ is the hook module with one distinguished attaching flag,
by Proposition~\ref{prop:eleven-hook-realizations}.
The odd decoration distinguishes the two elliptic vertices, so no
further symmetric-square operation occurs. This gives
$B_2$.

\smallskip\noindent\emph{Two edges.}
The compact degree is eleven. By
Proposition~\ref{prop:eleven-hook-realizations}, precisely one vertex
has genus one and carries $H^{11}$; all other factors are units.
Let
\[
\mathcal K^{11}=\sum_{m\geq11}s_{(m-10,1^{10})}.
\]
Restriction of the exterior hook gives
\[
\partial\mathcal K^{11}=H,\qquad
\partial H=H+G,\qquad
h_2^\perp\mathcal K^{11}=H,\qquad
e_2^\perp\mathcal K^{11}=G.
\]

First consider a path with three vertices. If the decorated vertex
is central, the remaining genera are zero and one, giving
$(H+G)E E_{\geq2}$. If it is an endpoint, a genus-one central vertex
and a rational endpoint give $H E E_{\geq2}$; a rational central
vertex and a genus-one endpoint give $H E_{\geq1}E$.
Together these are
\[
(2H+G)E E_{\geq2}+H E E_{\geq1}.
\]
For a loop joined by a bridge, a loop on the rational unit vertex
gives $HE$. That vertex has three flags and needs no ordinary marking.
A loop on the decorated elliptic vertex gives
$(H+G)E_{\geq2}$.
The flip of a loop fixes its edge and hence acts trivially on the
edge determinant; its two half-edges take invariants.
Finally, two parallel edges between the decorated elliptic vertex
and a rational vertex contribute $G E_{\geq1}$.
The rational vertex needs one ordinary marking, and interchanging
the two edges acts by $-1$ on their determinant, so the hook takes
anti-invariants. A single vertex with two loops has genus zero and
has no odd cohomology. This exhausts the connected two-edge graphs
of total genus two, and gives $C_2$.

\smallskip\noindent\emph{The Euler characteristic.}
For a surjection at the right end,
\[
[H^{15}(\Mbar_{2,n})]
=[G_1^{15}(2,n)]-[G_2^{15}(2,n)]+[R_n]-[Q_n].
\]
All four source and homology terms have common Hodge factor $\Pmot$.
Proposition~\ref{prop:low-genus-boundary-rows} gives their left and
middle homology multiplicity series, respectively, as
\[
E\bigl(s_{(2,1^8)}+s_{(3,1^8)}\bigr),\qquad E e_{11}.
\]
Taking Frobenius characteristics proves the asserted formula.
\end{proof}

\begin{theorem}[Genus two]\label{thm:genus-two}
With the series of Definition~\ref{def:common-series},
\[
F_2=E(1+E)s_{(2,1^{10})}+E^2(e_{11}+s_{(2,1^9)})
 +H(E_{\geq1}+(E+1)E_{\geq3}+2EE_{\geq2})+EGE_{\geq3}.
\]
The homogeneous degree-$n$ part is $\ch(\En_{2,n})$, it is nonzero exactly
for $n\geq11$, and it gives the Hodge and $\ell$-adic
realizations before Galois semisimplification.
\end{theorem}
\begin{proof}
\smallskip\noindent\emph{A positive expression.}
Put $Z=s_{(2,1^{10})}$, $V=s_{(2,1^9)}$,
$X=s_{(2,1^8)}$, and $Y=s_{(3,1^8)}$ for this calculation.
Definition~\ref{def:common-series} and Pieri's rule give
$U=EZ+E_{\geq3}H$, $H=Ee_{10}$, and $G=Ee_9$.
First use $\partial(EH)=E(2H+G)$ in
Proposition~\ref{prop:genus-two-character-row}. Cancellation gives
\[
F_2=D_+U+E\partial U+H(E-1)E_{\geq2}
 -G(E_{\geq2}+E_{\geq1})+E(X+Y-e_{11}).
\]
The necessary differentiation identities are
\[
\begin{gathered}
\partial Z=e_{11}+V,\qquad D_+Z=e_{10},\\
D_+E=E,\qquad \partial E=E,\qquad D_+H=H+G,\\
\partial E_{\geq3}=E_{\geq2},\qquad D_+E_{\geq3}=E_{\geq1}.
\end{gathered}
\]
They follow by deleting one or two boxes, with horizontal two-strips
for $D_+=h_2^\perp$.
Its product rule is
$D_+(fg)=(D_+f)g+(\partial f)(\partial g)+fD_+g$.
Consequently
\[
\begin{split}
D_+U={}&E(Z+e_{11}+V+e_{10})
 +H(E_{\geq1}+E_{\geq2}+E_{\geq3})
 +G(E_{\geq2}+E_{\geq3}),\\
E\partial U={}&E^2(Z+e_{11}+V)
 +E E_{\geq2}H+E E_{\geq3}(H+G).
\end{split}
\]
Substitution makes the $H$ coefficient
\[
E_{\geq1}+(E+1)E_{\geq3}+2E E_{\geq2}
\]
and the remaining $G$ coefficient
$(E+1)E_{\geq3}-E_{\geq1}$.
The terms not yet included in these two groups are
\[
E(Z+V+e_{10}+X+Y)+E^2(Z+e_{11}+V).
\]
Pieri gives
$e_9h_1=e_{10}+X$ and $e_9h_2=V+Y$. Thus
\[
G E_{\geq1}
=E(e_{10}+X+V+Y)+G E_{\geq3}.
\]

\smallskip\noindent\emph{The threshold.}
Every factor in that expression is Schur-positive.
The series $H$ starts in degree ten, $G$ in degree nine, and $Z$ in
degree twelve. It follows that every term starts in degree at least
eleven. The degree-eleven part is
$2(e_{11}+V)$, so it is nonzero.
For every $n\geq11$, the summand $E^2e_{11}$ contains
$e_{11}h_{n-11}$, a nonzero Schur-positive character.
Hence no later arity vanishes.

\smallskip\noindent\emph{The actual realizations.}
The Hodge decomposition follows from Theorem~\ref{thm:motivic-types}
and the multiplicity character just computed.
Proposition~\ref{prop:genus-two-gysin-surjectivity} proves that the
ordinary boundary image is the whole Betti target.
Proposition~\ref{prop:h13-input} therefore identifies its actual
$\ell$-adic image with
$(\En_{2,n}\otimes\mathbb Q_\ell)\otimes\Pmot_\ell$ and makes this
image the whole target.
\end{proof}