\section{Boundary generation in genus three}\label{sec:05-genus-three-generation}
We prove that ordinary boundary Gysin classes generate genus-three
cohomology in degree fifteen. Reduction in the number of markings leaves
twelve symplectic local systems of weight thirteen. The corresponding
degree-two obstruction groups inject into cohomology on the elliptic
Satake stratum, where restriction to a proper collision curve detects
the relevant hypercohomology with its full derived coefficients.

\begin{proposition}[Bergstr\"om--Faber--Payne
{\cite[Proposition 2.1]{bergstrom2024}}]
\label{prop:cited-compact-support-bound}
For every stable $(g,n)$ with $g\geq1$,
\[
H_c^k(\Mopen_{g,n})=0\quad\text{if}\quad
\begin{cases}
k<2g,&n=0,1,\\
k<2g-2+n,&n\geq2.
\end{cases}
\]
\end{proposition}

\begin{definition}\label{def:compact-boundary-row}
For a stable $(g,F)$ and $r=0,1,2$, put
\[
C^r_{g,F}=
\bigoplus_{[\Gamma],\,|E(\Gamma)|=r}
\left(H^{15}(X_\Gamma)\otimes
\det(\mathbb Q^{E(\Gamma)})\right)^{\Aut_F(\Gamma)}.
\]
Let $\alpha:C^0_{g,F}\to C^1_{g,F}$ be boundary pullback.
On a labelled two-edge graph with ordered edges $(e_1,e_2)$, set
\[
(\beta x)_\Gamma=
\bigl(\xi_{e_1}^*x_{\Gamma/e_1}-
\xi_{e_2}^*x_{\Gamma/e_2}\bigr)\otimes(e_1\wedge e_2).
\]
Here $\xi_e:X_\Gamma\to X_{\Gamma/e}$ glues the flags of $e$.
Let $C^1_{\rm sep}$ be the direct sum of the separating one-edge summands.
For $g\geq2$, choose new flags $c,d$, set $t=(c\,d)$, and let
$\xi:\Mbar_{g-1,F\cup\{c,d\}}\to\Mbar_{g,F}$ glue them. Put
\[
N_{g,F}=H^{15}(\Mbar_{g-1,F\cup\{c,d\}})^{\langle t\rangle},
\quad
\mathscr Q_{g,F}=C^2_{g,F}/\beta(C^1_{\rm sep}),
\quad
\delta(x)=[\beta(0,x)].
\]
Thus $C^1_{g,F}=C^1_{\rm sep}\oplus N_{g,F}$ and
$\delta:N_{g,F}\to\mathscr Q_{g,F}$.
\end{definition}

\begin{proposition}[The compact restriction row]\label{prop:compact-restriction-row}
The maps of Definition~\ref{def:compact-boundary-row} satisfy
\[
\beta\alpha=0,\qquad
\ker\alpha\cong\operatorname{Gr}_{15}^W H_c^{15}(\Mopen_{g,F}),
\qquad
\ker\beta/\operatorname{im}\alpha
\cong\operatorname{Gr}_{15}^W H_c^{16}(\Mopen_{g,F}).
\]
For $g\geq1$, the map $\alpha$ is injective if $15<2g$ when
$|F|=0,1$, or if $15<2g-2+|F|$ when $|F|\geq2$.
\end{proposition}

\begin{proof}
Dualizing the spectral sequence in
Theorem~\ref{thm:cited-ordinary-weight-sequence} by Poincar\'e duality
gives
\[
 E_1^{r,q}=
 \bigoplus_{[\Gamma],\,|E(\Gamma)|=r}
 \left(H^q(X_\Gamma)\otimes\det(\mathbb Q^{E(\Gamma)})\right)^{
 \Aut_F(\Gamma)}
 \Longrightarrow H_c^{q+r}(\Mopen_{g,F}).
\]
Indeed $X_\Gamma$ has dimension $3g-3+|F|-r$; dualizing its ordinary
Gysin term cancels its codimension twist against the dualizing
twists of $X_\Gamma$ and $\Mbar_{g,F}$. Thus this restriction row
has no Tate twist. Labelled \'etale node charts give the same
normal-crossings resolution on the stack: the orientation of a
set of branches is precisely its edge determinant. Taking finite
graph invariants is exact, so the spectral sequences on these charts
descend to the displayed spectral sequence on the stack.

In weight fifteen the first two differentials are exactly $\alpha$
and $\beta$. On a two-edge chart the two composite restrictions
agree, whereas their exterior orientations have opposite signs.
This proves $\beta\alpha=0$. Row cohomology in degrees zero and
one gives the two displayed identifications. Finally
Proposition~\ref{prop:cited-compact-support-bound}, applied with
$k=15$, gives the claimed injectivity inequalities.
\end{proof}

\begin{lemma}[Low-degree symplectic coefficients]
\label{lem:low-degree-symplectic-vanishing}
Let $d\geq1$, $g\geq d+2$, and let $\lambda$ be a nonempty partition
of length at most $g$ with $\lambda_1>d$. Then
\[
H^d(\Mopen_g,\mathbb V_\lambda)=0.
\]
\end{lemma}

\begin{proof}
Let $\Gamma_g$ be the mapping class group of a closed oriented
genus-$g$ surface, let $I_g$ be its Torelli group, and denote the
rational fibre representation of $\mathbb V_\lambda$ by
$\mathbf V_\lambda$. The extension
$1\to I_g\to\Gamma_g\to\operatorname{Sp}_{2g}(\mathbb Z)\to1$ gives the
Lyndon--Hochschild--Serre spectral sequence
\[
 E_2^{a,q}=H^a(\operatorname{Sp}_{2g}(\mathbb Z);
 H^q(I_g;\mathbb Q)\otimes\mathbf V_\lambda)
 \Rightarrow H^{a+q}(\Gamma_g;\mathbf V_\lambda).
\]
For $a+q=d$ one has $q\le d\le g-2$, so by Gaifullin
\cite[Theorems A and B]{gaifullin2026} the homology
$H_q(I_g;\mathbb Q)$ is a finite-dimensional algebraic
$\operatorname{Sp}_{2g}(\mathbb Z)$-representation, the restriction
of a rational representation of $\operatorname{Sp}_{2g}$ defined
over $\mathbb Q$, isomorphic to a sum of irreducibles $\mathbf V_\mu$
with $\ell(\mu)\le g$ and $\mu_1\le q$, the empty partition allowed.
Universal coefficients and self-duality of symplectic
irreducibles give the same finite algebraic decomposition of
$H^q(I_g;\mathbb Q)$, with first parts at most $q$. None equals
$\lambda$, including the empty partition at $q=0$. Zariski density
therefore gives
\[
 (H^q(I_g;\mathbb Q)\otimes\mathbf V_\lambda)^{
 \operatorname{Sp}_{2g}(\mathbb Z)}
 =
 \operatorname{Hom}_{\operatorname{Sp}_{2g}}
 (\mathbf V_\lambda^\vee,H^q(I_g;\mathbb Q))=0.
\]
Complete reducibility and Borel's theorem \cite[Theorem 4.6]{miller2024},
that $H^k(\Gamma,V)=0$ for $0\leq k<g$ when $V$ is a nontrivial
irreducible real algebraic representation of $\operatorname{Sp}_{2g}$
and $\Gamma\subset\operatorname{Sp}_{2g}(\mathbb Q)$ an arithmetic
subgroup, imply the vanishing of $E_2^{a,q}$: rational-to-real scalar
extension is faithful, $\operatorname{Sp}_{2g}(\mathbb Z)$ is an
arithmetic subgroup, and the theorem applies since $0\le a\le d\le g-2<g$. Every term of total degree $d$ is
zero, so the finite filtration on the abutment gives
$H^d(\Gamma_g;\mathbf V_\lambda)=0$, without requiring degeneration.
Contractibility of Teichm\"uller space identifies this group with
$H^d(\Mopen_g,\mathbb V_\lambda)$.
\end{proof}


\begin{lemma}\label{lem:geometric-bases-supported-types}
Let $X$ be a smooth complex Deligne--Mumford stack of finite type,
and let $Z\subset X$ be a smooth closed substack of pure codimension
$c$.
\begin{enumerate}
\item Let $c\geq3$, and let $\mathscr V\subset H^{15}(X)$ be a pure
sub-Hodge structure with Hodge types contained in
$\{(13,2),(2,13)\}$. Then restriction is injective on $\mathscr V$:
\[
\mathscr V\longrightarrow H^{15}(X\setminus Z).
\]
\item If $W_{15-2c}H^{15-2c}(Z)=0$, then restriction
\[
W_{15}H^{15}(X)\longrightarrow W_{15}H^{15}(X\setminus Z)
\]
is injective.
\end{enumerate}
\end{lemma}

\begin{proof}
The localization sequence contains
\[
H^{15-2c}(Z)\otimes\Lmot^c\longrightarrow H^{15}(X)
\longrightarrow H^{15}(X\setminus Z).
\]
(1) The mixed Hodge structure on $H^{15-2c}(Z)$ is effective.
After tensoring with $\Lmot^c$, both Hodge indices are at least
$c$. Strictness of the Hodge and weight filtrations gives the same
bound for every subquotient of the first term. Its image has zero
intersection with $\mathscr V$, since each allowed type of
$\mathscr V$ has an index equal to two.

(2) The preceding term has weight-fifteen part
$W_{15-2c}H^{15-2c}(Z)\otimes\Lmot^c=0$. Taking weight fifteen in
localization, strictness of weights makes restriction injective in
weight fifteen.
\end{proof}

\begin{proposition}[The open locus with twelve markings]
\label{prop:genus-three-twelve-point-vanishing}
Let $U_{12}=\Mopen_{3,12}\setminus\operatorname{Hyp}_{3,12}$, and let
\[
Z_{12}=\{(C,p_1,\ldots,p_{12})\in U_{12}:
p_1+\cdots+p_{12}\sim3K_C\}.
\]
Then $Z_{12}$ is smooth of codimension three in $U_{12}$, and
$U_{12}\setminus Z_{12}$ has the Chow--K\"unneth generation property.
Moreover,
\[
\mathcal O_{3,n}=0\qquad(0\leq n\leq12).
\]
\end{proposition}

\begin{proof}
\emph{The exceptional locus.}
On the nonhyperelliptic locus the canonical model is a smooth plane
quartic and $\omega_C\simeq\mathcal O_C(1)$.  For
$D=\sum p_i$, restriction of quartic equations and $H^1(C,\omega_C^4)=0$
identify the evaluation cokernel with
\[
H^1(C,\omega_C^4(-D))\cong
H^0(C,\mathcal O_C(D)\otimes\omega_C^{-3})^\vee.
\]
The latter degree-zero line bundle has a section exactly when it is
trivial: a nonzero section has an effective divisor of degree zero and
is nowhere vanishing. Thus the locus where evaluation is not surjective
is precisely $Z_{12}$.
For the universal nonhyperelliptic curve $\pi_C$, Riemann--Roch gives
$h^0(C,3K_C)=10$ and $h^1(C,3K_C)=0$. Consequently
$\pi_{C*}\omega_{\pi_C}^3$ is a rank-ten vector bundle. The open locus of
simple divisors in its projectivization, followed by the finite \'etale
cover ordering the zeros, is $Z_{12}$. Scheme-theoretically this is
the fibre of the relative Abel map over the section $3K_C$: in degree
twelve that Abel map is a projective bundle. The separated relative
Picard space makes the condition closed. The base is smooth, so
$Z_{12}$ is smooth of codimension $12-9=3$.

\emph{Chow--K\"unneth generation on the complement.}
Let $\mathcal V$ be the rank-three bundle on $B\mathrm{SL}_3$ and set
$B_{\rm ev}=(\mathbb P\mathcal V)^{12}_{B\mathrm{SL}_3}$.
On the open locus where evaluation is surjective, the map
\[
 \operatorname{Sym}^4\mathcal V^\vee\longrightarrow
 \bigoplus_{r=1}^{12}\eta_r^*\mathcal O_{\mathbb P\mathcal V}(4)
\]
has a rank-three kernel. The open substack of its projectivization where the
quartic is smooth and the marked points distinct is a $\mu_3$-gerbe
over $U_{12}\setminus Z_{12}$. Projective bundles and open restriction
preserve the Chow--K\"unneth generation property, by
Proposition~\ref{prop:cited-chow-kunneth-operations}. The map from the
special-linear presentation to the projective-linear one is a finite
$\mu_3$-gerbe, so Canning--Larson \cite[Lemma 3.6]{canninglarson2022}
(finite gerbes preserve the property) implies
that $U_{12}\setminus Z_{12}$ has the Chow--K\"unneth generation property.

\emph{Lowest weight.}
The smooth hyperelliptic divisor has the same property by
Canning--Larson \cite[Theorem 6.2]{canninglarson2022}, by which
$\mathcal H^{\mathrm{irr}}_{g,n}\subset\overline{\mathcal M}_{g,n}$,
parametrizing smooth or irreducible nodal hyperelliptic curves, has
the Chow--K\"unneth generation property for $g\ge2$ and
$0\le n\le2g+6$, and the odd lowest-weight
vanishing in Proposition~\ref{prop:cited-chow-kunneth-operations},
whose compactification hypothesis holds because the open substacks
under consideration admit smooth proper compactifications obtained by
resolving their boundaries without modifying those substacks, gives
$W_{13}H^{13}(\operatorname{Hyp}_{3,12})=0$ and
$W_{15}H^{15}(U_{12}\setminus Z_{12})=0$.
Lemma~\ref{lem:geometric-bases-supported-types}(2), applied with
$c=1$, injects $\mathcal O_{3,12}$ into $W_{15}H^{15}(U_{12})$.
Proposition~\ref{prop:primitive-degree-fifteen-quotient} and
Theorem~\ref{thm:motivic-types} identify $\mathcal O_{3,12}$ as a pure quotient
with only types $(13,2)$ and $(2,13)$, so
Lemma~\ref{lem:geometric-bases-supported-types}(1), applied with
$c=3$, makes restriction to $U_{12}\setminus Z_{12}$ injective on the
injected quotient. Its target in weight fifteen is zero, so
$\mathcal O_{3,12}=0$.
For $n\le11$, Proposition~\ref{prop:cited-chow-kunneth-bases}
gives the assertion directly.
\end{proof}

\begin{definition}\label{def:genus-three-coefficients}
Set
\[
\begin{split}
\mathscr R_3={}&\{(13),(12,1),(11,2),(11,1,1),(10,3),(10,2,1),\\
&\hspace{12mm}(9,4),(9,3,1),(9,2,2),(8,5),(8,4,1),(8,3,2)\}.
\end{split}
\]
For $\lambda\in\mathscr R_3$, put
$\mathcal Q_\lambda=W_{15}H^2(\Mopen_3,\mathbb V_\lambda)$.
\end{definition}

\begin{proposition}[Reduction to twelve local systems]
\label{prop:genus-three-twelve-coefficient-reduction}
There is a canonical $\Sn_{13}$-equivariant isomorphism
\[
\mathcal O_{3,13}\cong
\bigoplus_{\lambda\in\mathscr R_3}
\mathcal Q_\lambda\otimes V_{\lambda^T}.
\]
For a finite set $F$ and $J\subseteq F$, let $\pi_J$ forget the
markings outside $J$. For every $F$ with $|F|\geq14$, the map
\[
\bigoplus_{\substack{J\subseteq F\\|J|=13}}\mathcal O_{3,J}
\longrightarrow\mathcal O_{3,F},\qquad
(x_J)_J\longmapsto\sum_J\pi_J^*x_J
\]
is surjective.
The sum of the ordinary one-edge Gysin maps is surjective in all
arities if and only if $\mathcal Q_\lambda=0$ for every
$\lambda\in\mathscr R_3$.
\end{proposition}

\begin{proof}
\emph{Nine rows in the ordinary source.}
We use the row bounds of Lemma~\ref{lem:row-bound-calculus}.
By Lemma~\ref{lem:row-bound-calculus}(3), the complete ordinary
source $B^{(1)}_{3,n}\otimes\Pmot$ of Proposition~\ref{prop:h13-input}
has row bound nine in every arity.

Theorems~\ref{thm:genus-one} and \ref{thm:genus-two} give that
bound for compact $H^{15}$ in genera one and two. In genus one
the remaining $\Sonesix$ hook has fifteen rows. For genus three
with at most twelve flags, Proposition~\ref{prop:genus-three-twelve-point-vanishing}
and Proposition~\ref{prop:primitive-degree-fifteen-quotient} give
surjectivity from the source with row bound nine, hence the same bound.
Genus-zero $H^{15}$ is zero by Theorem~\ref{thm:motivic-types}.

\emph{Eight rows at thirteen markings.}
Proposition~\ref{prop:compact-restriction-row} makes
$H^{15}(\Mbar_{3,13})\to C^1_{3,[13]}$ injective since $15<4+13$.
Its nonseparating term is
$H^{15}(\Mbar_{2,15})^{\Sn_2}$ and has row bound eight, by
Lemma~\ref{lem:row-bound-calculus}(1).
For a separating term, Theorem~\ref{thm:cited-low-odd-vanishing}
leaves exactly
\[
 (15,0),(0,15),(13,2),(2,13),(11,4),(4,11).
\]
The first two have bound eight after fixing the node whenever
the odd factor has genus one or two. If its genus is three,
the rational factor has $b\ge2$ external labels, so the odd
factor has $14-b\le12$ flags and the preceding bound applies.
An odd rational factor contributes zero. In the next two
sectors $H^{13}$ has row bound ten before fixing its node, hence nine
afterward. In the last two, $H^{11}$ is elliptic and has ten
rows after fixing its node, irrespective of the degree-four
factor. Induction and graph invariants preserve these bounds.
Thus the entire restriction target, and its compact subspace,
have at least eight rows.

\emph{The primitive quotient and higher markings.}
By Proposition~\ref{prop:primitive-degree-fifteen-quotient},
$\mathcal O_{3,13}$ is a quotient of this compact space. The span of the
images of pullbacks forgetting one marking is zero because
$\mathcal O_{3,12}=0$.
The same proposition gives
\[
 \mathcal O_{3,13}\cong
 \bigoplus_{\lambda\vdash13,\,\ell(\lambda)\le3}
 W_{15}H^2(\Mopen_3,\mathbb V_\lambda)\otimes V_{\lambda^T}.
\]
Since $\ell(\lambda^T)=\lambda_1$, the bound excludes
$\lambda_1\le7$. Writing the remaining parts in decreasing
order gives precisely the twelve partitions in $\mathscr R_3$.

At fourteen markings, every partition of length at most three
has first part at least five. Lemma~\ref{lem:low-degree-symplectic-vanishing}
with $(g,d)=(3,1)$ gives vanishing of its $H^1$ coefficient, so the primitive
quotient is zero. At arities at least fifteen it is zero by
Proposition~\ref{prop:primitive-degree-fifteen-quotient}.
Iterating the one-marking forgetful maps proves the stated
surjection from thirteen-element subsets. Vanishing of the twelve groups
therefore implies vanishing of every $\mathcal O_{3,n}$, including the lower
arities already treated. Conversely, surjectivity of the sum of the Gysin maps
implies vanishing of $\mathcal O_{3,13}$; its direct sum decomposition forces
each coefficient to vanish.
\end{proof}

\begin{definition}\label{def:genus-three-satake-coefficients}
Let $\mathcal A_g$ be the stack of principally polarized abelian
varieties, with coarse space $A_g$ and Satake compactification
$A_g^{\mathrm{Sat}}$. Put
\[
Y=A_3^{\mathrm{Sat}},\quad A=A_3,\quad B=A_2^{\mathrm{Sat}},
\quad Z=A_1^{\mathrm{Sat}},
\]
and denote the inclusions by $k:A\hookrightarrow Y$,
$i:B\hookrightarrow Y$, $z:Z\hookrightarrow B$, and
$v:A_2\hookrightarrow B$. Write $o=Z\setminus A_1$.
Let $\mathcal N=\Mopen_3\setminus\operatorname{Hyp}_3$, let
$U\subset A$ be its coarse Torelli image, and write
$p:\mathcal N\to U$ and $j:U\hookrightarrow A$.

Let $f:\mathcal M_3^{\mathrm{ct}}\to\mathcal A_3$ be compact-type
Torelli, $q:\mathcal A_3\to A$ the coarse map,
$a:\mathcal M_3^{\mathrm{ct}}\hookrightarrow\Mbar_3$ the open
immersion, and $t:\Mbar_3\to Y$ extended Torelli. Use
$\mathbb V_\lambda$ also for the weight-thirteen abelian variation
and its pullbacks. For $\lambda\in\mathscr R_3$, set
\[
\begin{aligned}
G_\lambda&=j_!p_*(\mathbb V_\lambda|_{\mathcal N}),&
C_\lambda&=i^*Rk_*G_\lambda,\\
F_\lambda&=q_*R^1f_*f^*\mathbb V_\lambda,&
E_\lambda&=i^*Rk_*F_\lambda,\\
M_\lambda&=i^*Rt_*Ra_*\mathbb V_\lambda,&
L_\lambda&=\mathcal H^1(C_\lambda)|_{A_1}.
\end{aligned}
\]
Coarse direct image
means exact rational direct image, including finite stabilizer
invariants. Fibre weights on a smooth stack stratum are the weights of
the corresponding geometric variation before this coarse direct image.
\end{definition}

\begin{proposition}[The Torelli correction]
\label{prop:genus-three-torelli-correction}
For every $\lambda\in\mathscr R_3$, one has
\[
H^0(A,F_\lambda)=H^1(A,F_\lambda)=0.
\]
There is a canonical distinguished triangle
\[
C_\lambda\longrightarrow M_\lambda\longrightarrow
E_\lambda[-1]\xrightarrow{+1}.
\]
The natural maps give isomorphisms of mixed Hodge structures
\[
H^2(A,G_\lambda)\cong
H^2(\mathcal M_3^{\mathrm{ct}},\mathbb V_\lambda)
\cong\mathbb H^2(B,C_\lambda),
\]
and restriction induces
\[
W_{15}H^2(\mathcal M_3^{\mathrm{ct}},\mathbb V_\lambda)
\cong\mathcal Q_\lambda.
\]
\end{proposition}

\begin{proof}
\emph{The fibres and their normalization.}
The Torelli morphism extends regularly from $\Mopen_3$ to the second
Voronoi compactification, and its composite with the Satake morphism
is the extended Satake Torelli map \cite[Theorem 3.4]{alexeev2011};
thus $t$ is proper, and its base change over $\mathcal A_3$ is $f$.
The Satake image of a stable
curve is the product of the Jacobians of its positive-genus
normalization components. Thus this base change is exactly the
compact-type locus. Every indecomposable principally polarized
abelian threefold is the Jacobian of a smooth projective curve of
genus three \cite[Corollary 7]{hain2002}; with Torelli this
identifies the coarse indecomposable locus with smooth
curve moduli: the proper quasi-finite coarse map is finite and
birational onto the normal indecomposable locus, hence an isomorphism.
In the framed fibre a nonhyperelliptic curve has two points exchanged
by the central abelian involution; a hyperelliptic curve has one.

Over $E_{\rm att}\times J(C)$ with $C$ smooth of genus two, the framed fibre
is $C$: attach $E_{\rm att}$ at a moving point of $C$, fixing its own attachment
by translation. The Jacobian framing retains that point, rather
than its quotient by the hyperelliptic involution. Over an ordered
product $E_1E_2E_3$, the fibre is
$E_1\cup E_2\cup E_3$ with their origins identified. On the
middle component $E_r$, the two attachments have a difference in
$E_r\setminus\{0\}$; at zero, the common stable limit is the rational
trivalent component with three elliptic tails. These exhaust the
graphs: a stable tree with three genus-one vertices has at most
one rational vertex, necessarily that trivalent vertex. The three
proper attachment maps are a proper bijection from this union to
the reduced fibre and hence a homeomorphism.

Let $\mathcal D_3\subset\mathcal A_3$ be the decomposable locus.
Its normalization is
$\nu_{12}:\mathcal A_1\times\mathcal A_2\to\mathcal D_3$.
Let $\iota_{111}:\mathcal T_{111}=\mathcal A_1^3/\Sn_3
\hookrightarrow\mathcal D_3$ be the triple locus, and let
$s_{111}:\mathcal T'_{111}=\mathcal A_1^3/\Sn_{\{2,3\}}
\to\mathcal T_{111}$ distinguish its first factor.
In framed Siegel coordinates there are exactly three normalization
branches at the triple locus, one per distinguished elliptic factor.
Repeated factors take the corresponding finite permutation quotient.
Write $H_r$ for the standard system of the $r$th elliptic factor,
and $H_{23}$ for that of $\mathcal A_2$. Define
\[
\begin{split}
 \mathscr K_{12}&=\mathbb V_\lambda(H_1\oplus H_{23})\otimes H_{23},\\
 \mathscr K_{111}&=\mathbb V_\lambda(H_1\oplus H_2\oplus H_3)
       \otimes(H_1\oplus H_2\oplus H_3)\otimes\operatorname{sgn}_{\Sn_3},
\end{split}
\]
where the second system is specified on the ordered cover, and
let $\mathscr K'_{111}$ be the restriction of $\mathscr K_{12}$
to $\mathcal T'_{111}$.

The sign comes from the attachment coordinates. Choose on middle
component $r$ the attachment to $r+2$ minus that to $r+1$, cyclically.
A transposition
reverses that difference, acting by $-1$ on first cohomology.
Attach the distinguished elliptic factor to the universal
genus-two compact-type curve on the normalization. At its node,
stable marking inserts the rational component described above.
This proper family gives the three degree-one pullbacks
\[
 P_1(v_1,v_2,v_3)=(v_2,-v_3),\quad
 P_2(v_1,v_2,v_3)=(-v_1,v_3),\quad
 P_3(v_1,v_2,v_3)=(v_1,-v_2).
\]
For example, on branch one the moving coordinate on $E_2$ agrees
with the cyclic difference and that on $E_3$ has the opposite
sign. Cyclic permutation gives the remaining formulas. A
transposition acts by the signed permutation on the source
and the usual branch exchange on the targets, so these are
equivariant maps. Proper base change consequently gives the
exact normalization sequence
\begin{equation}\label{eq:genus-three-generation-matching}
0\longrightarrow R^1f_*f^*\mathbb V_\lambda
\longrightarrow\nu_{12*}\mathscr K_{12}
 \oplus\iota_{111*}\mathscr K_{111}
\longrightarrow\iota_{111*}s_{111*}\mathscr K'_{111}
\longrightarrow0.
\end{equation}
The last map is normalization restriction minus the three $P_i$.
Off the triple locus the first arrow is an isomorphism onto the normalization
term. At a triple point the normalization part maps identically
onto the three branch spaces; its kernel is the graph of the
three $P_i$ on the actual fibre. This verifies exactness, including
finite stabilizers (Set-up~\ref{setup:standing}).

\emph{Degree zero and degree one of the correction.}
Write $\lambda=(a_1,a_2,a_3)$. There is no
$\operatorname{Sp}_2^3$ constituent of $\mathbb V_\lambda$ equal
to one standard factor with the others trivial. Indeed, by the
symplectic branching rule \cite[Corollary 3.3]{yacobi2010},
$\operatorname{Hom}_{\operatorname{Sp}_{2m-2}}
(\mathbf V^{(m-1)}_\mu,\mathbf V^{(m)}_\lambda)$ is zero unless
$\lambda_i\ge\mu_i\ge\lambda_{i+2}$ for $1\le i\le m-1$ (double
interlacing, with $\lambda_{m+1}=0$), and it is then the
representation
$\bigotimes_{i=1}^m\operatorname{Sym}^{x_i-y_i}\mathbb C^2$
of the commuting $\operatorname{Sp}_2$, where
$x_1\ge y_1\ge\cdots\ge x_m\ge y_m$ sorts the entries of
$(\lambda_1,\ldots,\lambda_m,\mu_1,\ldots,\mu_{m-1},0)$.
Thus the $\operatorname{Sp}_4$ weights admitting
$\operatorname{Sp}_2^2$ invariants are $(u,u)$.
Double interlacing requires $a_3\le u\le a_2$, and the remaining
rank-one multiplicity is
$\operatorname{Sym}^{a_1-a_2}\otimes\operatorname{Sym}^{a_3}$.
Every constituent has highest weight at least
$a_1-a_2-a_3=2a_1-13\ge3$, excluding the standard weight one.
Zariski density implies the vanishing of $H^0(\mathcal T'_{111},\mathscr K'_{111})$
and $H^0(\mathcal T_{111},\mathscr K_{111})$. A normalization
invariant would also be invariant after restriction to
$\operatorname{Sp}_2^3$, so it is zero. Sequence
\eqref{eq:genus-three-generation-matching} proves $H^0(A,F_\lambda)=0$.

For any finite algebraic coefficient $\mathscr E$ on $\mathcal A_2$,
Borel's theorem \cite[Theorem 4.6]{miller2024}, that for an arithmetic
subgroup $\Gamma\subset\operatorname{Sp}_{2g}(\mathbb Q)$ and
$0\leq k<g$ one has $H^k(\Gamma,V)=0$ for every nontrivial irreducible
real algebraic representation $V$ of $\operatorname{Sp}_{2g}$, while
for trivial coefficients in this range the cohomology is the
degree-$k$ part of the polynomial algebra on generators of degrees
$2,6,10,\ldots$, gives
$H^1(\mathcal A_2,\mathscr E)=0$: decompose semisimply, use its
nontrivial-coefficient assertion in degree $1<2$, and observe
that the trivial-coefficient algebra has no degree-one generator.
Rational-to-real scalar extension is faithful. On each
normalization branch, K\"unneth therefore places degree-one
cohomology only in the distinguished elliptic factor. Its
restriction $R_i$ to the triple cover is injective, since its
remaining degree-zero map is
$\mathscr E^{\operatorname{Sp}_4}\hookrightarrow
\mathscr E^{\operatorname{Sp}_2^2}$.

The degree-one sequence of \eqref{eq:genus-three-generation-matching}
injects $H^1(A,F_\lambda)$ into the kernel of the matching map.
On the ordered cover this map has equations
$P_i x-R_i b_i=0$. Split $x$ into its three K\"unneth cohomological
directions $x^{[r]}$. For $i\ne r$, the $[r]$ equation says
$P_i x^{[r]}=0$. The two kernels in question are distinct
direct coefficient summands, indexed by $H_i$, and intersect
trivially. Thus every $x^{[r]}=0$, and injectivity of the $R_i$
then gives every $b_i=0$. Exact permutation invariants prove
$H^1(A,F_\lambda)=0$ with the signed matching retained.

\emph{The truncation triangle.}
The central abelian involution acts as $(-1)^{13}=-1$ on
$\mathbb V_\lambda$. On each connected Torelli fibre it is
trivial on $H^0$ and $H^2$: on $C$ it is the hyperelliptic
involution, and on each elliptic parameter curve it is inversion.
Thus coarse invariants vanish in degree zero off $U$ and in degree
two everywhere. The fibres have dimension at most one. On $U$,
coarse degree zero is $p_*\mathbb V_\lambda$; adjunction from
extension by zero identifies the degree-zero sheaf with
$G_\lambda$. Consequently ordinary truncation gives
\begin{equation}\label{eq:genus-three-generation-torelli-truncation}
 G_\lambda\longrightarrow R(qf)_*\mathbb V_\lambda
 \longrightarrow F_\lambda[-1]\xrightarrow{+1}.
\end{equation}
Its degree-two sequence and the two established vanishings give
$H^2(A,G_\lambda)\cong
H^2(\mathcal M_3^{\mathrm{ct}},\mathbb V_\lambda)$.
Since $ta=kqf$, applying $i^*Rk_*$ gives the triangle of the
statement.

\emph{Comparison with the Satake boundary and with smooth curves.}
The coarse nonhyperelliptic locus $U$ is affine: it is the
discriminant-nonzero open in the projective GIT quotient of
plane quartics. A full symplectic level cover is a smooth
affine sixfold finite over $U$. On it the local system shifted
by six is perverse; Artin vanishing
\cite[Theorem 2.0.1]{decataldo2012},
\cite[Theorem 4.1.1 and Corollary 4.1.2]{bbd1982} gives
$H_c^p=0$ for $p<6$. Finite descent yields
$H_c^r(U,p_*\mathbb V_\lambda)=0$ for $r<6$.
Localization on proper $Y$ has adjacent terms $H_c^2(U,-)$
and $H_c^3(U,-)$, hence identifies
$H^2(A,G_\lambda)$ with $\mathbb H^2(B,C_\lambda)$.

Finally the complement of $\Mopen_3$ in $\mathcal M_3^{\mathrm{ct}}$
is a normal-crossings divisor whose normalization is
$\mathcal M_{1,1}\times\mathcal M_{2,1}^{\mathrm{ct}}$.
Its coefficient has no global sections: on the dense smooth
product, monodromy is Zariski dense in
$\operatorname{Sp}_2\times\operatorname{Sp}_4$, containing the
odd central involution. Thus its degree-two support cohomology
vanishes. In degree-three support cohomology the possible
one-branch term is $H^1(-,\mathbb V_\lambda)(-1)$, of weights
at least $13+1+2=16$ by
Theorem~\ref{thm:cited-geometric-weight-bound}; the two-branch
term is $H^0(-,\mathbb V_\lambda)(-2)$, of weights at least
seventeen. An $r$-branch intersection first contributes in support
degree $2r-(r-1)=r+1$, so no higher intersections intervene.
Localization and strictness give the asserted isomorphism on
$W_{15}$. Every comparison above is induced by algebraic
adjunction, proper direct image, or localization, and therefore
respects the geometric mixed Hodge structures.
\end{proof}

\begin{proposition}[Localization along the Satake strata]
\label{prop:genus-three-satake-strata}
For $\lambda\in\mathscr R_3$, one has
\[
\mathcal H^0(C_\lambda)=0,\qquad
\mathbb H_c^2(A_2,v^*C_\lambda)=0.
\]
Every ordinary cohomology sheaf of $C_\lambda$, $M_\lambda$, and
$E_\lambda$, restricted to any of
\[
\Mopen_2^{\mathrm{coarse}},\qquad
(\mathcal A_1^2/\Sn_2)^{\mathrm{coarse}},\qquad A_1,
\]
is the exact coarse direct image of a local system carrying a geometric
mixed Hodge variation on the indicated smooth stack. The same holds
for kernels and cokernels of their cohomology-sheaf maps.
All cohomology groups of the derived cusp stalk $(C_\lambda)_o$ are
mixed Tate. In particular, restriction and extension from the elliptic
stratum give
\[
\mathbb H^2(B,C_\lambda)\hookrightarrow
\mathbb H^2(Z,z^*C_\lambda),
\]
and, for every odd integer $b$ and every $r$,
\[
\operatorname{Gr}_b^W\mathbb H_c^r(A_1,C_\lambda|_{A_1})
\xrightarrow{\ \sim\ }
\operatorname{Gr}_b^W\mathbb H^r(Z,z^*C_\lambda).
\]
\end{proposition}

\begin{proof}
\emph{Degree-zero extension.}
The closure of the hyperelliptic Jacobian divisor contains $B$.
For a smooth genus-two curve, identify two distinct conjugate
non-Weierstrass points. The resulting hyperelliptic genus-three
curve is a limit obtained by colliding two branch points of a
double cover, and its Satake image is the genus-two Jacobian.
These images are dense in $B$.

Choose a connected normal analytic representative $V_b$ near
$b\in B$. Removing a nowhere-dense analytic subset leaves it
connected (Riemann extension on normal spaces
\cite[Theorem 0.2, p.~336]{saeki1992}).
Apply this to the complement of the automorphism-free locus
$U^\circ\subset U$, including the closure of the extra-inertia
locus and $Y\setminus U$. Thus $V_b\cap U^\circ$ is connected.
A section of $G_\lambda$ on $V_b\cap A$ vanishes near a
hyperelliptic point, since its stalk there is zero. That
neighborhood meets $U^\circ$, so local constancy forces the
section to vanish on all of $V_b\cap U^\circ$. Finite
uniformizing covers show that $p_*\mathbb V_\lambda$ has no
section supported on the extra-automorphism locus: its lifted
local-system section vanishes on a dense open. Hence the original
section is zero. Taking germs proves
$\mathcal H^0(C_\lambda)=0$. These complexes have no negative
ordinary degrees; the correction triangle also gives
$\mathcal H^0(M_\lambda)=0$.

\emph{The proper polar model.}
On a labelled \'etale deformation chart of $\Mbar_3$, write $t_e$
for the independent smoothing parameters of all nodes.
The compact-type open is obtained by requiring $t_e\ne0$
for every nonbridge edge. Contracting other edges preserves
a surviving nonbridge cycle, whereas a bridge remains separating.
Bridge parameters are therefore ordinary disks, including zero.
Blow up the noncompact-type divisor in real oriented directions:
$t_e=r_e\exp(i\theta_e)$ for the excluded parameters.
The resulting map $\rho:\Mbar_3^{\log}\to\Mbar_3$ is proper,
and node relabellings and units glue these charts. The variation
extends to a local system $\widetilde{\mathbb V}_\lambda$ on
this space, since small angular-interval charts have
contractible interiors. Consequently
\begin{equation}\label{eq:genus-three-generation-polar}
 Ra_*\mathbb V_\lambda=R\rho_*\widetilde{\mathbb V}_\lambda.
\end{equation}
Over a graph its complex is the full Koszul complex of the
commuting operators $T_e-1$. Each nonbridge contributes its own
exterior generator, even if two coedges coincide.

We verify the hypotheses of Thom's first isotopy lemma
\cite[Proposition 11.1]{mather1970} (properness on a Whitney-stratified
subset, submersivity on each stratum) for $t\rho$ over each
of the three listed smooth stack strata, ordering elliptic
factors when necessary. The retained normalization components
are respectively one genus-two curve, two elliptic curves, and
one elliptic curve; all others are rational. A graph stratum
is a product of distinct attachment configurations on these
curves and rational pointed moduli. Forgetting the attachments
is smooth, and freely varying the retained curves supplies all
base tangent directions. At collisions, rational insertions
give further graph strata with independent node parameters, so
the same surjectivity holds. Allowed contractions are exactly
those preserving the indicated normalization type. They impose
coordinate-vanishing conditions on node parameters and no
equations on the free moduli of retained curves.

The graph strata are thus normal-crossings coordinate
arrangements times smooth pointed-component deformations.
Their coordinate strata are Whitney; after polar blowup
they are products with radial faces and circles, and every
face still maps submersively. Embed the radial orthants in
real coordinate spaces to apply the isotopy lemma. Properness
holds because both $t$ and $\rho$ are proper. In finite
quotient charts average the lifting fields over stabilizers;
the linear lifting and tangency conditions are preserved.
The resulting local trivializations preserve all graph and
face strata. On a contractible base neighborhood the local
system on this product pulls back from the fibre.
Proper base change in \eqref{eq:genus-three-generation-polar}
therefore proves the assertion for every cohomology sheaf of
$M_\lambda$.

For $E_\lambda$, apply the same construction to the normalization
sequence \eqref{eq:genus-three-generation-matching}. The proper
models of its three terms are
\[
 \Mbar_{1,1}\times\Mbar_2\longrightarrow Y,\qquad
 \Mbar_{1,1}^{\,3}\longrightarrow Y,
\]
with the respective tensor coefficients
$\mathscr K_{12},\mathscr K_{111},\mathscr K'_{111}$
defined in the proof of Proposition~\ref{prop:genus-three-torelli-correction}.
Take their noncompact-type polar boundaries before permutation
invariants. Retained curves distribute among the factors in
finitely many ways, each with the same smooth attachment and
coordinate-face description. Their proper direct images are
locally constant on the stack strata. The matching long exact
sequence gives the assertion for $E_\lambda$, and the correction
triangle gives it for $C_\lambda$. Kernels and cokernels of
local-system maps are local systems. Since all original
operations are geometric mixed-Hodge operations, these
systems carry their geometric variations, with the same
assertion for weight subobjects and for the finite stabilizer
invariants taken by coarse direct image (Set-up~\ref{setup:standing}).

\emph{Vanishing on $A_2$.}
The indecomposable genus-two stratum has a smooth finite affine
level cover of dimension three: genus-two curves are double
covers branched at six points, and $\Mopen_{0,6}$ is affine.
The product-elliptic stratum has a smooth finite affine cover
of dimension two. On a smooth affine $d$-fold a local system
shifted by $d$ is perverse; Artin vanishing
\cite[Theorem 2.0.1]{decataldo2012} gives $H_c^p=0$ for $p<d$.
On either stratum $R$, $M_\lambda$
has no cohomology sheaves below degree one and $E_\lambda$ has none below
degree zero.
The compact-support spectral sequences therefore give
\[
 \mathbb H_c^2(R,M_\lambda)=0,\qquad
 \mathbb H_c^0(R,E_\lambda)=0.
\]
The correction triangle gives $\mathbb H_c^2(R,C_\lambda)=0$.
Localization between these two strata yields
$\mathbb H_c^2(A_2,v^*C_\lambda)=0$.

\emph{Mixed Tate cohomology at the cusp.}
For a product of the proper curve spaces just used, its
Satake cusp fibre has only rational normalization components.
A labelled graph stratum is a product
$\prod_v\Mopen_{0,F(v)}$. The normalization sequence for
the limiting standard coefficient on a graph $\Gamma$ gives
\[
 \operatorname{Gr}_0^W=H^1(\Gamma;\mathbb Q),\qquad
 \operatorname{Gr}_1^W=0,\qquad
 \operatorname{Gr}_2^W=H_1(\Gamma;\mathbb Q)(-1).
\]
These are constant on the labelled stratum. Tensor constructions,
duals, and direct summands have finite filtrations by constant
Tate systems. Their full link complex is
\[
 \Psi\mathbb V\longrightarrow
 \bigoplus_{e\ {\rm nonbridge}}\Psi\mathbb V(-1)
 \longrightarrow
 \bigoplus_{\{e,e'\}\ {\rm nonbridge}}\Psi\mathbb V(-2)
 \longrightarrow\cdots,
\]
with alternating logarithms $N_e$. The substitution
$T_e-1=N_e(1+N_e/2!+\cdots)$ is invertible, so this computes
the full torus cochains. Every node is retained independently.
Each $N_e$ has type $(-1,-1)$, and the complex has constant
Tate graded coefficients. Choosing three rational flags as
$0,1,\infty$ trivializes the marked tangent lines on each
labelled stratum. Changes of frames act through the nilpotent
operators on the extensions and trivially on these graded pieces.

Both ordinary and compactly supported cohomology of
$\Mopen_{0,m}$ are mixed Tate. To verify this directly, forget
one point after fixing $0,1,\infty$: the fibre is a punctured
projective line, with constant Tate $H^1$ generated by its
labelled punctures and their sum relation. Leray, induction,
and smooth duality prove the assertion. The coefficient
filtration, finite graph invariants, and localization over
the finite stratification by stable graphs now show that the proper
cusp-fibre hypercohomology is mixed Tate. Apply this to
$\Mbar_3$, $\Mbar_{1,1}\times\Mbar_2$, and
$\Mbar_{1,1}^3$. Proper base change and the full matching
sequence prove it for $(M_\lambda)_o$ and $(E_\lambda)_o$,
and the correction triangle proves it for $(C_\lambda)_o$.

Finally, the localization triangle for $A_2\subset B$ and
the degree-two vanishing proved above give the asserted injection
into $Z$. In the localization sequence for $A_1\subset Z$,
both adjacent cusp cohomology groups have only even weights.
Taking any odd $\operatorname{Gr}_b^W$ gives the stated isomorphism.
\end{proof}

\begin{proposition}[The elliptic localization injection]
\label{prop:genus-three-elliptic-localization}
For each $\lambda\in\mathscr R_3$, the localization maps induce a
canonical injection of pure weight-fifteen Hodge structures
\[
\mathcal Q_\lambda\hookrightarrow
\operatorname{Gr}_{15}^W H_c^1(A_1,L_\lambda).
\]
\end{proposition}

\begin{proof}
Theorem~\ref{thm:cited-geometric-weight-bound} gives weights at least
$13+2=15$ in $H^2(\Mopen_3,\mathbb V_\lambda)$, also after finite
level descent. Thus $\mathcal Q_\lambda$ is pure of weight fifteen.
The two preceding propositions inject it into
$\operatorname{Gr}_{15}^W\mathbb H_c^2(A_1,C_\lambda|_{A_1})$.
Every ordinary cohomology sheaf on this stratum is a coarse
local-system direct image. Such a sheaf has $H_c^0=0$:
on a finite smooth nonproper modular-curve cover a section of
a local system with compact support is zero. In
\[
 E_2^{p,q}=H_c^p(A_1,\mathcal H^q(C_\lambda)|_{A_1})
 \Longrightarrow\mathbb H_c^{p+q}(A_1,C_\lambda|_{A_1}),
\]
the $q=0$ row is zero and the $p=0$ column is zero.
Only $(p,q)=(1,1)$ remains in total degree two. No
differential enters it; the only possible outgoing term has
$q=0$ or negative $q$. This canonically identifies that
hypercohomology with $H_c^1(A_1,L_\lambda)$ and proves the
injection as a mixed-Hodge map.
\end{proof}

\begin{definition}\label{def:genus-three-elliptic-clutching}
Fix a smooth elliptic curve $E_0$, with Satake point $x_E\in A_1$.
Let $X_E=t^{-1}(x_E)$, and let $W_E$ be the fibre over $x_E$ of
the stabilized elliptic-Jacobian map
$\Mbar_{1,4}\to A_1^{\mathrm{Sat}}$. Put
\[
\mathfrak G=\langle(12),(34),(13)(24)\rangle\subset\Sn_4.
\]
Simultaneous clutching of the pairs $\{1,2\}$ and $\{3,4\}$ gives
$\nu:[W_E/\mathfrak G]\to X_E$ and $\widetilde\nu:W_E\to X_E$.
For $\lambda\in\mathscr R_3$, set
\[
K_{\lambda,E}=(Ra_*\mathbb V_\lambda)|_{X_E},\qquad
\delta_{\lambda,E}:\mathbb H^1(X_E,K_{\lambda,E})
\longrightarrow
\mathbb H^1(W_E,\widetilde\nu^*K_{\lambda,E})^{\mathfrak G}.
\]
\end{definition}

\begin{proposition}[Weights in the finite-clutching kernel]
\label{prop:genus-three-finite-clutching-kernel}
The morphism $\nu$ is finite, representable, and surjective. For every
$\lambda\in\mathscr R_3$,
\[
W_{13}\ker\delta_{\lambda,E}=\ker\delta_{\lambda,E}.
\]
\end{proposition}

\begin{proof}
\emph{The finite cover.}
A curve in $X_E$ has one elliptic normalization component
isomorphic to $E_0$, all other components rational, and graph
Betti number two. Choose a spanning tree and normalize at its
two complementary edges. Labelling the four preimages produces
a stable four-pointed genus-one curve in $W_E$: replacing a
node branch by a marking preserves stability. Gluing recovers
the curve, so $\nu$ is surjective. Its fibres are finite, since
one chooses two nodes from a finite set; the group
$\mathfrak G$ removes pair order and endpoint order.
An automorphism of such a partial normalization, allowing
these relabellings, that becomes the identity on the glued
curve fixes every normalized component and every branch.
It is the identity, proving representability. Since $W_E$
is proper and $X_E$ separated, $\nu$ is proper. A proper
representable quasi-finite morphism is finite.

\emph{Degree-zero link weights.}
At any point $x\in X_E$, the compact-type complement in a
labelled deformation chart punctures precisely its nonbridge
parameters. The commuting unipotent monodromies give
\[
 H^0((K_{\lambda,E})_x)=\bigcap_e\ker(T_e-1)
                         =\bigcap_e\ker N_e .
\]
Choose an algebraic smooth local curve through the chart
on which every node parameter vanishes to order one; its
punctured part smooths all nodes. Algebraic base change on
degree zero is the injective map
\[
 \bigcap_e\ker N_e\longrightarrow\ker N_{\rm tot},
 \qquad N_{\rm tot}=\sum_eN_e .
\]
It is restriction of local sections from the product of
punctured disks to the diagonal, and hence is a morphism of
geometric mixed Hodge structures. Bridge operators are zero.
By Schmid \cite[Lemma 6.4 and Theorem 6.16]{schmid1973}, the
one-variable limiting mixed Hodge structure of this polarized
weight-thirteen variation with unipotent monodromy has weight
filtration centered at thirteen, with
$N_{\rm tot}W_b\subset W_{b-2}$ and
$N_{\rm tot}^{\,r}:\operatorname{Gr}^W_{13+r}\xrightarrow{\sim}
\operatorname{Gr}^W_{13-r}(-r)$, and $N_{\rm tot}$ is of type
$(-1,-1)$; hence $\ker N_{\rm tot}\subset W_{13}$. The one-variable logarithmic complex
$\mathbb V_{\lambda,\infty}\xrightarrow{N_{\rm tot}}
\mathbb V_{\lambda,\infty}(-1)$ identifies degree zero with
these invariants with their induced filtration. Strictness
therefore bounds every $H^0((K_{\lambda,E})_x)$ by thirteen,
including its finite stabilizer invariants.

\emph{The adjunction cone.}
Let $\mathscr T_{\lambda,E}$ be the cone in
\[
 K_{\lambda,E}\longrightarrow R\nu_*\nu^*K_{\lambda,E}
 \longrightarrow\mathscr T_{\lambda,E}\xrightarrow{+1}.
\]
Finite proper base change makes the map on each degree-$q$
stalk the injective diagonal
$H^q(K_x)\to\bigoplus_{y\in\nu^{-1}(x)}H^q(K_x)$.
Thus $\mathscr T_{\lambda,E}$ has no negative cohomology sheaves and
\[
 H^0((\mathscr T_{\lambda,E})_x)=
 \operatorname{coker}\left(H^0(K_x)\longrightarrow
 \bigoplus_{y\in\nu^{-1}(x)}H^0(K_x)\right)
\]
has weights at most thirteen. Choose a finite constructible
stratification and a point on each connected stratum.
Evaluation gives an injection of mixed Hodge structures
\[
 \mathbb H^0(X_E,\mathscr T_{\lambda,E})\hookrightarrow
 \bigoplus_s H^0((\mathscr T_{\lambda,E})_{x_s}),
\]
because a section locally constant on a stratum is determined
at one point. The long exact sequence identifies
$\ker\delta_{\lambda,E}$ with an image of this bounded
degree-zero group. Strictness proves the assertion.
\end{proof}

\begin{definition}\label{def:genus-three-collision-curve}
Let $\overline W_E$ be the compactification of $E_0^4/E_0$ by stable
rational tails at collisions, with the elliptic normalization identified
with $E_0$. Thus $W_E=[\overline W_E/\operatorname{Aut}(E_0,0)]$.
For $S\subseteq\{1,2,3,4\}$ with $|S|\geq2$, let $D_S$ be its
collision divisor. Set
\[
\begin{split}
\mathcal B={}&\{13,14,23,24,123,124,134,234\},\\
O_E={}&\overline W_E\setminus\bigcup_{S\in\mathcal B}D_S,
\qquad D_E=D_{12}\cap D_{34}.
\end{split}
\]
Here a string denotes its set of labels. Let
$\widehat G_E=\operatorname{Aut}(E_0,0)\times\mathfrak G$, and let
$\widehat\nu:\overline W_E\to X_E$ be the clutching map.
Restriction to the invariant closed locus $D_E$ defines
\[
r_{D,E}:\mathbb H^1(W_E,\widetilde\nu^*K_{\lambda,E})^{\mathfrak G}
\longrightarrow
\mathbb H^1(D_E,\widehat\nu^*K_{\lambda,E}|_{D_E})^{\widehat G_E}.
\]
\end{definition}

\begin{proposition}[Detection on the proper collision curve]
\label{prop:genus-three-collision-detection}
For every smooth elliptic curve $E_0$ and $\lambda\in\mathscr R_3$,
the curve $D_E$ is proper and isomorphic to $E_0$, and $r_{D,E}$ is
injective. The Torelli correction triangle gives an exact sequence
\[
0\longrightarrow (L_\lambda)_{x_E}
\longrightarrow\mathbb H^1(X_E,K_{\lambda,E})
\xrightarrow{\ \gamma_{\lambda,E}\ }
\mathcal H^0(E_\lambda)_{x_E}.
\]
There is a morphism induced by restriction along the proper family of
middle-elliptic attachments,
\[
\eta_{\lambda,E}:\mathcal H^0(E_\lambda)_{x_E}
\longrightarrow
\mathbb H^1(D_E,\widehat\nu^*K_{\lambda,E}|_{D_E})^{\widehat G_E},
\]
such that
\[
r_{D,E}\delta_{\lambda,E}=
\eta_{\lambda,E}\gamma_{\lambda,E}.
\]
The displayed maps are the geometric maps on the full derived
coefficients and respect their mixed Hodge structures.
\end{proposition}

\begin{proof}
\emph{The collision compactification.}
Fix the stabilized fourth marking at zero. Then
$E_0^4/E_0=E_0^3$, and $\overline W_E$ is its blowup at the
full-collision point followed by the four proper triple-diagonal
transforms. The latter are disjoint after the first blowup:
their tangent directions are the three coordinate axes and
the diagonal direction. This is the wonderful compactification of
the diagonal building set \cite[Theorem 1.3]{li2009}, blown up in
order of decreasing cardinality; pair diagonals are Cartier, so
blowing them up changes nothing.

The identification with the compactification by stable marked curves
follows from its charts. A cluster has a rational component with attaching point
at infinity and coordinates given by rescaled differences,
modulo translation and dilation. Nested clusters repeat this
construction. Conversely, these relative coordinates and node
parameters recover the blowup charts. The full-collision
exceptional surface is $\mathbb P^2$ blown up at the four triple
directions; the triple exceptional fibres are $\mathbb P^1$.
Knudsen's stabilization of an extra section commutes with base
change \cite[Theorems 5.1--5.2]{knudsen2012}; this makes these
rational insertions compatible with families, also when an
extra section meets a node. Translation acts freely on the
stabilized fourth point and preserves the diagonal ideals.
This proves the description on their quotient and includes
every collision.

The eleven divisor classes $[D_S]$ are independent in
$H^2(\overline W_E)$. Project a relation to $H^2(E_0^3)$
using the blowup decomposition. For a symplectic basis
$u_E,v_E$ of $H^1(E_0)$, the six pair classes project to
\[
 u_{E,r}v_{E,r}\quad(1\le r\le3),\qquad
 (u_{E,r}-u_{E,s})(v_{E,r}-v_{E,s})\quad(1\le r<s\le3).
\]
Each off-diagonal K\"unneth component occurs in just its indicated
difference class, so all six coefficients vanish. The five
exceptional classes are independent because each connected
center of codimension at least two contributes a new
$H^0(\text{center})(-1)$ summand. Degree-one localization for
the eight removed divisors therefore gives
\[
 H^1(O_E;\mathbb Q)=H^1(\overline W_E;\mathbb Q)
 =\{(h_1,h_2,h_3,h_4)\in H^1(E_0)^4:\ \sum h_r=0\}.
\]
Indeed the residue map ends in the injective map
$\bigoplus_{S\in\mathcal B}\mathbb Q(-1)\to
H^2(\overline W_E)$, and the blowups preserve $H^1$.

The curve $D_E$ is parameterized by $z\in E_0$: place
the $12$-tail at $z$ and the $34$-tail at zero. When $z=0$
insert a three-pointed central rational component carrying
both tails and the elliptic attachment. This is the unique
additional stable point. Hence $D_E\simeq E_0$ is proper,
lies in $O_E$, and meets only the retained additional divisor
$D_{1234}$. Let
$\jmath=\iota_{E_0}(12)(34)$, where $\iota_{E_0}$ is inversion.
It is central in $\widehat G_E$. On constant $H^1(O_E)$ it
acts by minus the indicated permutation, and its minus
eigenspace is $\{(h,h,-h,-h)\}$. Restriction to
$z\mapsto(z,z,0,0)$ sends this vector to $2h$. Thus
\begin{equation}\label{eq:genus-three-generation-constant-detector}
 H^1(O_E;\mathbb Q)^{\jmath=-1}
 \xrightarrow{\sim}H^1(D_E;\mathbb Q).
\end{equation}

\emph{Stable normal lines and all corner complexes.}
Let $\mathcal T_r$ be the stable tangent line at marking $r$,
and put
\[
 \mathcal L_1=\mathcal T_1\otimes\mathcal T_2,\qquad
 \mathcal L_2=\mathcal T_3\otimes\mathcal T_4,\qquad
 \mathscr P_E=\mathcal L_1^\times|_{O_E}
                 \times_{O_E}\mathcal L_2^\times|_{O_E}.
\]
These are the smoothing lines of the two nodes obtained by pairwise
clutching: the equation
$xy=\tau$ transforms $\tau$ by the product of the derivatives
of the two branch coordinates. Orient their coedges as
$q_1,q_2$. A tail with label set $S$ has coedge
\[
 c_S=(1_S(1)-1_S(2))q_1+(1_S(3)-1_S(4))q_2 .
\]
This follows by closing each pair edge along its unique
path in the original rational tree. Exactly the sets in
$\mathcal B$ have nonzero coedge; the other three tails
are bridges after clutching.

Plumbing an annulus gives the Picard--Lefschetz
operator $N(c)v=\langle c,v\rangle c$ on the standard
coefficient: rotating its parameter makes a positive
Dehn twist, adding the intersection with its core.
The disjoint node operators commute. On $\mathbb V_\lambda$
use their differentiated tensor action; horizontal
symplectic projectors commute with this transport.

The stable and elliptic normal frames are related by
\begin{equation}\label{eq:genus-three-generation-stable-frame}
 \tau_r^{\rm ell}=u_r\tau_r^{\rm st}
     \prod_Ss_S^{m_r(S)},\qquad
 m_r(S)=|S\cap\{2r-1,2r\}|,\quad r=1,2.
\end{equation}
To verify it, at one tail attachment use $zy=s$ and
$y=1/w$ on the tail. Then $z=sw$, so $dz=s\,dw$
at a marked point. Along a tree the chain rule multiplies
these derivatives over its path to $E_0$, with an invertible
regular unit. Multiplying the two derivatives in a pair
gives precisely the displayed exponents, including exponent
two when both markings lie on the tail.
At fixed elliptic frame the tail logarithm is consequently
\[
 N(c_S)-\sum_{r=1}^2m_r(S)N(q_r).
\]

The full normal cochains have a separate exterior generator
for each punctured coordinate and differential
$\sum_eN_e e_e\wedge-$. The equality
$T_e-1=N_e(1+N_e/2!+\cdots)$ identifies them with the
unipotent circle resolution. Logarithmic differentiation of
\eqref{eq:genus-three-generation-stable-frame} makes the
triangular generator change
\[
 e_r=f_r-\sum_Sm_r(S)f_S,\qquad e_S=f_S.
\]
It is invertible in every exterior degree and gives exactly
the corrected tail operators. Restoring units adds
$-d\log u_r$ and the gauge transport
$\exp(-\log(u_r)N(q_r)/(2\pi i))$ in the same flat frame.
Changing a chosen logarithm changes both sides by the same
integral monodromy, so these local maps descend. At a bridge,
restriction of the unpunctured disk is inclusion into its
zero-operator circle complex before the change of frame.
Its correction is accounted for by
$d\iota_r+\iota_r d=N(q_r)$, with $\iota_r$ exterior
contraction. The contractions anticommute, giving the
compatible signs on iterated faces. Thus no bridge disk
is punctured in the compact-type open, and equal coedges
never identify their distinct smoothing coordinates.

Blow up $\overline W_E$ in oriented directions along the
eight removed divisors. On a chart its additional angles
are exactly the nonbridge-tail angles, with the two pair
angles supplied by $\mathcal L_1,\mathcal L_2$.
Equations \eqref{eq:genus-three-generation-polar} and
\eqref{eq:genus-three-generation-stable-frame} identify their
unit-circle bundle with the full pulled-back polar link,
including all nested and disjoint corners and horizontal
point transport. A collar gives a homotopy equivalence with
its interior relative to $D_E$: patch inward radial vector
fields on boundary neighborhoods disjoint from $D_E$,
average over the finite group, and lift by invariant
connections on the two lines. Small positive flow moves
all boundary faces inward and fixes $D_E$. Radial
contraction in each punctured line then gives
\begin{equation}\label{eq:genus-three-generation-link-comparison}
 \mathbb H^*(\overline W_E,\widehat\nu^*K_{\lambda,E})
 \cong H^*(\mathscr P_E,\mathscr V_E)
\end{equation}
as rational groups, with the full transported coefficient
$\mathscr V_E$ and its filtered subsystems, compatibly with
restriction to $D_E$.

\emph{Detection of the full coefficient.}
Both normal lines restrict trivially to $D_E$: their
three-pointed rational tails are fixed, including at the
coincident-attachment point. Thus
$\mathscr P_E|_{D_E}=E_0\times(\mathbb C^*)^2$.
The involution $\jmath$ preserves the complex orientation
of both normal circles. The torus-bundle Leray sequence
and \eqref{eq:genus-three-generation-constant-detector}
therefore give
\[
 H^1(\mathscr P_E;\mathbb Q)^{\jmath=-1}
 \xrightarrow{\sim}
 H^1(\mathscr P_E|_{D_E};\mathbb Q)^{\jmath=-1}.
\]
The vertical $H^1$ has $\jmath=1$ and contributes no
minus eigenspace. Both spaces are connected, so their
constant degree-zero minus eigenspaces vanish.

The normalization filtration on the standard limiting
coefficient has constant graded systems
$I_E,\underline{H^1(E_0)},I_E^\vee(-1)$, where
$I_E=\mathbb Qq_1\oplus\mathbb Qq_2$, of weights zero,
one, and two. Labels fix the graph basis and $E_0$ is fixed.
Point-pushing and meridian transports remain in the
extensions. Pair reversal negates the graph pieces and
elliptic inversion negates the elliptic piece, so $\jmath$
acts by $-1$ on all three. The degree-thirteen symplectic
tensor construction has a finite filtration by constant
quotients with coefficient action $\jmath=-1$; polarization
contractions have even degree and preserve this sign.

For each such constant quotient the preceding displayed map
is an isomorphism on invariant degree one, and invariant
degree zero vanishes on both spaces. In an extension
$0\to\mathscr A\to\mathscr V\to\mathscr B\to0$,
these degree-zero vanishings make the two maps from
$H^1(\mathscr A)^\jmath$ injective. A class in
$H^1(\mathscr P_E,\mathscr V)^\jmath$ restricting to zero
has zero image in the $\mathscr B$ group by induction,
so lifts to $\mathscr A$. Its restricted lift is zero
by that injectivity, and the inductive injectivity statement
then implies that the lift vanishes. This proves injectivity for the full
filtered system without splitting it. Further exact
$\widehat G_E$ invariants and
\eqref{eq:genus-three-generation-link-comparison} prove
injectivity of the actual map $r_{D,E}$.

\emph{The proper attachment and its boundary map.}
The correction triangle, $E_\lambda$ being nonnegative,
and proper base change for $t$ give the exact sequence of the statement.
On the ordered product $\mathcal Z_3=\mathcal A_1^3$
consider the proper universal elliptic curve
$r_0:\mathcal E_0\to\mathcal Z_3$ attaching the other two
factors at zero and at a moving point of its first factor.
The rational insertion at zero is included. Denote its
attachment by $c_0:\mathcal E_0\to\mathcal M_3^{\mathrm{ct}}$
and the product map followed by coarse moduli by
$h_0:\mathcal Z_3\to A$. Then $(qf)c_0=h_0r_0$.
Adjunction gives the restriction morphism
\[
 R(qf)_*\mathbb V_\lambda\longrightarrow
 Rh_{0*}Rr_{0*}c_0^*\mathbb V_\lambda.
\]
The simultaneous central involution acts as $(-1)^{13+j}$
on $\mathbb V_\lambda\otimes H^j(E_0)$, so exact coarse
direct image through the finite product map vanishes in degrees
zero and two. The target is thus a sheaf in degree one.
Negative Ext vanishing applied to
\eqref{eq:genus-three-generation-torelli-truncation}
factors this map uniquely through $F_\lambda[-1]$.
Its degree-one map is the middle-component
restriction in \eqref{eq:genus-three-generation-matching}.

This component restriction retains the matching term.
For the actual fibre $\mathcal T=E_0\cup E_1\cup E_2$
with common origin and normalization maps $\eta_i$,
every rational complex $\mathscr Q$ has the triangle
\[
 \mathscr Q\longrightarrow
 \bigoplus_{i=0}^2R\eta_{i*}\eta_i^*\mathscr Q
 \longrightarrow\mathscr Q|_0^{\oplus2}\xrightarrow{+1}.
\]
Its difference map is $(v_0,v_1,v_2)\mapsto
(v_1-v_0,v_2-v_0)$. The constant-sheaf sequence is
stalkwise $0\to\mathbb Q\to\mathbb Q^3\to\mathbb Q^2\to0$
at the origin, and tensoring proves the derived version.
Restriction to $E_0$ is its canonical first arrow followed
by that component, with the difference term preserved.
The signed cyclic descent above transports the entire triangle.

Compactify the attachment diagram by
\[
 \overline{\mathcal Z}_3=\Mbar_{1,1}^3,\qquad
 \overline{\mathcal P}_0=\Mbar_{1,2}\times\Mbar_{1,1}^2.
\]
The maps $\bar r_0,\bar c_0,\bar h_0$ are proper, and
$t\bar c_0=\bar h_0\bar r_0$. If
$b_0:\mathcal E_0\hookrightarrow\overline{\mathcal P}_0$,
then
$\bar r_0^{-1}(\mathcal Z_3)=
\bar c_0^{-1}(\mathcal M_3^{\mathrm{ct}})=\mathcal E_0$:
a loop on any elliptic factor remains nonseparating after
the separating attachments. There is the canonical algebraic
comparison, defined by adjunction from the identity on the open,
\begin{equation}\label{eq:genus-three-generation-open-comparison}
 \chi:\bar c_0^*Ra_*\mathbb V_\lambda
 \longrightarrow Rb_{0*}c_0^*\mathbb V_\lambda .
\end{equation}
We prove that it is an isomorphism near the component under consideration.

At a point with distinct attachments the target node chart
has coordinates $(t_1,t_2,\zeta_E,z',b_1,b_2)$, where $t_i$
smooth the two self-nodes, $b_i$ the separating attachments,
and $(\zeta_E,z')$ deform the middle elliptic curve and its
attachment difference. The source chart is
$(s_1,s_2,\zeta_E,z')$ and maps to
$(s_1,s_2,\zeta_E,z',0,0)$. At coincident attachments the target
has coordinates $(t_1,t_2,\zeta_E,b',b_1,b_2)$: $b'$ smooths
the new bridge to the central rational component. The
source's stable two-pointed first factor has coordinate
$z'$ smoothing this same bridge, so the map is again
$(s_1,s_2,\zeta_E,z')\mapsto(s_1,s_2,\zeta_E,z',0,0)$.
Normalizing the two ordered separating nodes recovers
the source factors and proves these coordinate
identifications without ramification in $s_1,s_2$.
These two chart types exhaust $D_E$.

The compact-type opens are $(\Delta^*)^2\times\Delta^2$
and $(\Delta^*)^2\times\Delta^4$. Contracting the last
two bridge disks gives a homotopy inverse of their
inclusion with the full local system. To check every
corner, let $I\subset\{1,2\}$ be the vanishing self-node
coordinates and let $V_{\rm loc}$ be the actual fibre.
Its cochains are
\[
 K_I^p=\bigoplus_{\substack{A'\subset I\\|A'|=p}}
 V_{\rm loc}e_{A'},\qquad
 d(v e_{A'})=\sum_{i\in I\setminus A'}(T_i-1)v\,
 e_i\wedge e_{A'}.
\]
In matched frames $\chi$ is identity on every coefficient
and every labelled generator. A generization to $I'\subset I$
keeps $e_{A'}$ if $A'\subset I'$ and kills it otherwise;
this commutes with the differential and with $\chi$.
Both orders of two generizations agree. Moving across
$z'=0$ removes no circle, since it is a bridge disk.
For arbitrary frames $t_i=u_i s_i$, polar angles change
by $\operatorname{Im}\log u_i$ with their actual lifted
transport. A change of logarithm multiplies both sides
by the same $T_i^{n_i}$. Permuting tails permutes the
operators and exterior generators together. Hence the
comparison holds on every overlap and corner for the
full coefficient. Since \eqref{eq:genus-three-generation-open-comparison}
is an algebraic mixed-Hodge morphism and its rational
realization is an isomorphism there, the same holds
in the geometric mixed-Hodge category.

There is a proper map $d_E:D_E\to(\bar h_0\bar r_0)^{-1}(x_E)$ obtained
by taking the stable two-pointed first factor and two
one-pointed nodal rational elliptic factors. At the
coincident point the first factor includes its rational
marked tail. Thus
$\bar c_0d_E=\widehat\nu|_{D_E}$. Restriction along
$d_E$ is the canonical adjunction map on the whole
proper fibre, with every other component and its
matching still in its source.

The composite of adjunction for $\bar c_0$ with
$R\bar c_{0*}\chi$ equals $Ra_*$ of adjunction for
$c_0$, since both are adjoint to the identity on
$c_0^*\mathbb V_\lambda$. Apply proper direct images
and proper base change for $t,\bar c_0,\bar h_0,\bar r_0$,
then restrict along $d_E$. The isomorphism $\chi|_{D_E}$
identifies this map with actual restriction from $X_E$
to $D_E$. Its factorization through $F_\lambda[-1]$
therefore gives $\eta_{\lambda,E}$ and exactly
$r_{D,E}\delta_{\lambda,E}=
\eta_{\lambda,E}\gamma_{\lambda,E}$.
The nodal branch reversals and interchange of the two
nodal factors are the pair symmetries in $\widehat G_E$,
so the maps descend with their stated invariants.
All Hodge comparisons in this last argument are the
algebraic maps just constructed; the nonproper open
comparison was proved on its corner complexes before
any proper base change was used.
\end{proof}

\begin{proposition}[The elliptic coefficient weight bound]
\label{prop:genus-three-elliptic-weight-bound}
For $\lambda\in\mathscr R_3$, one has $W_{13}L_\lambda=L_\lambda$.
Moreover, there is an algebraic mixed Hodge module $N_\lambda$ on
$A_1$ with
\[
\operatorname{rat}(N_\lambda)=L_\lambda[1],\qquad
W_{14}N_\lambda=N_\lambda.
\]
Its weights and rational realization agree with the geometric
coefficient defined above, including at the finite-quotient points.
\end{proposition}

\begin{proof}
\emph{The fibre bound.}
By Proposition~\ref{prop:genus-three-collision-detection}, the
image of $(L_\lambda)_{x_E}$ lies in $\ker\gamma_{\lambda,E}$
and hence restricts to zero on $D_E$. Injectivity of $r_{D,E}$
forces its finite-clutching pullback to vanish. It is therefore
a mixed-Hodge subobject of $\ker\delta_{\lambda,E}$, whose
weights are at most thirteen by
Proposition~\ref{prop:genus-three-finite-clutching-kernel}.
Strictness proves the fibre bound. All maps were constructed
equivariantly before finite elliptic stabilizer invariants;
the same calculation on each uniformizing elliptic chart
gives the geometric variation bound there. The local-system
assertion of Proposition~\ref{prop:genus-three-satake-strata}
then gives $W_{13}L_\lambda=L_\lambda$ with the stipulated
coarse interpretation.

\emph{An algebraic realization.}
After full finite level, the universal abelian scheme on
the smooth six-dimensional compact-type curve space gives
an algebraic first-cohomology Hodge module by proper direct
image. Tensor operations, the polarization, and the
symplectic idempotent construct $\mathbb V_\lambda^H$
inside the algebraic category. Finite descent defines
it on the stack. By \cite[Theorem 3.1]{peterssaito2008} a polarizable
variation of weight $w$ on a smooth $d$-dimensional base is a pure
smooth Hodge module of weight $w+d$ with rational perverse sheaf
$\mathbb V[d]$, and conversely; so its weight is $13+6=19$ and its
rational realization is $\mathbb V_\lambda[6]$. Put
$\mathscr V_\lambda^H=\mathbb V_\lambda^H[-6]$, with
ordinary rational realization $\mathbb V_\lambda$, and form
\[
 C_\lambda^H=i^*Rk_*j_!Rp_*
      (\mathscr V_\lambda^H|_{\mathcal N})
       \quad\text{in }D^b\operatorname{MHM}(B).
\]
These are the algebraic functors defining the
coefficient; rational coarse direct image, exact by
Set-up~\ref{setup:standing}, gives
$\operatorname{rat}(C_\lambda^H)=C_\lambda$.
For $\ell_1:A_1\hookrightarrow B$ define
\[
 N_\lambda={}^{p}\mathcal H^2(\ell_1^*C_\lambda^H).
\]
Every ordinary cohomology sheaf of $\ell_1^*C_\lambda$
is a coarse local-system direct image. On a curve such
a sheaf shifted by $[1]$ is perverse. The
ordinary-to-perverse cohomology spectral sequence has
only the row $ {}^{p}\mathcal H^1(\mathcal H^q)
=\mathcal H^q[1]$. Its index-two term is therefore
$L_\lambda[1]$, proving the required rational realization.

\emph{No punctual constituents.}
At a coarse elliptic orbifold point, pass to a finite
uniformizing disk. Each coefficient sheaf is the invariant
direct image of a local system on that disk. Its monodromy
on the punctured coarse disk has finite order and its
stalk is exactly that monodromy's invariant space. Shifted
by one, it is the middle extension, with no punctual
subquotient. The same description applies to weight
subobjects and graded quotients. At other points the
coefficient is a smooth variation.

A nonzero pure graded piece of $N_\lambda$ of weight $b$
thus restricts to a variation of weight $b-1$ on the
smooth one-dimensional elliptic stack, by the converse clause of
\cite[Theorem 3.1]{peterssaito2008} recalled above.
These are the fibre weights of the geometric complex
$C_\lambda^H$. If $b>14$, the fibre bound makes that
restriction zero. The graded piece would be punctual,
which the disk calculation excludes. Hence
$W_{14}N_\lambda=N_\lambda$.
\end{proof}

\begin{theorem}[Vanishing of the coefficient obstruction]
\label{thm:genus-three-coefficient-vanishing}
For every $\lambda\in\mathscr R_3$
(Definition~\ref{def:genus-three-coefficients}), one has
\[
W_{15}H^2(\Mopen_3,\mathbb V_\lambda)=0.
\]
\end{theorem}

\begin{proof}
The injection of Proposition~\ref{prop:genus-three-elliptic-localization}
places the obstruction inside
$\operatorname{Gr}_{15}^W H_c^1(A_1,L_\lambda)$.  By
Proposition~\ref{prop:genus-three-elliptic-weight-bound}, $N_\lambda$ has
upper derived weight at most fourteen.
For the structure map $s_1:A_1\to\operatorname{Spec}\mathbb C$,
compactly supported direct image preserves such upper weight bounds,
compatibly with the rational constructible functors
\cite[Definition 3.7 and Proposition 3.8]{tsaito2017}; this gives
weights at most fourteen on $\mathcal H^0(s_{1!}N_\lambda)$.
Its rational realization is $H_c^1(A_1,L_\lambda)$, because
$\operatorname{rat}(N_\lambda)=L_\lambda[1]$. Its weight-fifteen
graded piece is therefore zero, and the injection forces
$W_{15}H^2(\Mopen_3,\mathbb V_\lambda)=0$.
\end{proof}

\begin{definition}\label{def:genus-three-source}
Let $n\ge10$. Let $\mathcal G^{(2)}_{3,n}$ be the isomorphism classes of
stable genus-three graphs with two edges, and let
$\mathcal G^{(1;1,2)}_{3,n}$ be those with one separating edge and vertex
genera one and two. Define
\[
\begin{split}
 A_n={}&\bigoplus_{[\Gamma]\in\mathcal G^{(2)}_{3,n}}
 \left(\bigoplus_{v:g(v)=1}K_{F(v)}\right)^{\Aut(\Gamma)}\\
 &\quad\oplus\bigoplus_{[\Gamma]\in\mathcal G^{(1;1,2)}_{3,n}}
 K_{F(v_1)}\otimes\mathbb Q[F(v_2)].
\end{split}
\]
In the second sum, $g(v_i)=i$, and the vector indexed by $j\in F(v_2)$
corresponds to $\psi_j$, including the attaching flag. In the first sum,
automorphisms act on the whole direct sum and may move the decorated
vertex. Choose a rational basis in each graph summand by the hook basis
and the finite invariant equations. Let $a_1,\ldots,a_{N_n}$ be the
resulting basis and let $\rho_n:\Sn_n\to\operatorname{GL}_{N_n}(\mathbb Q)$
be the relabelling representation in this basis. Node flags are transported
by the induced graph isomorphisms.

For a decoration $a$ on $\Gamma$, let $\alpha_a(u)$ be its class on
$X_\Gamma$: it has the factor $\omega_{F(v)}(k,u)$ at its specified
genus-one vertex, a factor $\psi_j$ in the second family, and units on
the other vertices, extended linearly over the invariant decoration.
Define the Gysin map
\[
 \Phi_n:A_n\otimes\Pmot\longrightarrow H^{15}(\Mbar_{3,n}),\qquad
 \Phi_n(a,u)=\frac{\xi_{\Gamma*}\alpha_a(u)}{|\Aut(\Gamma)|}.
\]
The notation on the right suppresses the common Tate factor as above.
Both families have degree $11+2(|E(\Gamma)|+\deg f)=15$, with $f=1$
or $f=\psi_j$, respectively.
\end{definition}

\begin{proposition}[Genus-three boundary generation]
\label{prop:genus-three-generation}
For every $n\ge0$, the sum of the ordinary quotient-normalized one-edge
Gysin maps is surjective:
\[
 \bigoplus_{[\Gamma]\in\mathcal G^{(1)}_{3,n}}
 H^{13}(X_\Gamma)^{\Aut(\Gamma)}\otimes\Lmot
 \longrightarrow H^{15}(\Mbar_{3,n}).
\]
For every $n\ge10$, the map
$\Phi_n:A_n\otimes\Pmot\to H^{15}(\Mbar_{3,n})$ is surjective.
\end{proposition}

\begin{proof}
Proposition~\ref{prop:genus-three-twelve-coefficient-reduction}
and Theorem~\ref{thm:genus-three-coefficient-vanishing} give
surjectivity of the sum of the ordinary one-edge Gysin maps in every arity.

\emph{Degree-thirteen factors.}
For a separating edge its degree-thirteen K\"unneth sectors
are exactly $(13,0),(0,13),(11,2),(2,11)$, by
Theorem~\ref{thm:cited-low-odd-vanishing}.
The nonseparating term is $H^{13}(\Mbar_{2,n+2})$ with
the exchange invariants. In every $H^{13}$ factor,
Theorem~\ref{thm:cited-thirteen-cohomology} gives
generation by Gysin images of an elliptic $H^{11}$ class
tensored with the unit on its complement, including in genus
one. Composing this inner gluing with the outer edge
gives a stable two-edge genus-three graph, decorated
at its elliptic vertex by the hook of
Proposition~\ref{prop:eleven-hook-realizations}.
These graphs index the first family in $A_n$.

\emph{Picard factors.}
In the remaining sectors the $H^{11}$ factor has genus
one, so the other factor has genus two and carries a
Picard class. Proposition~\ref{prop:genus-three-picard-input}
generates $H^2$ by $\kappa_1$, the $\psi_j$, and the boundary
divisors, and its genus-two identity expresses $5\kappa_1$ through
$\sum_j\psi_j$, $\delta_{\rm irr}$, $\delta_0$, and $\delta_1$.
Thus every Picard class is a linear combination of flag $\psi$
classes and boundary divisors. The former give exactly
the second family, including the attaching flag. The
latter compose to graphs in the first family, also for
the irreducible boundary divisor.

\emph{The normalized span.}
Labelled and quotient-normalized pushforwards differ
only by their nonzero graph automorphism orders.
Given an invariant class in an intermediate source,
average any generating preimage over its finite group.
This proves the same spanning assertion with all
automorphisms, including those moving a decorated
vertex. The composed classes are exactly the images under
$\Phi_n$ of the generators in Definition~\ref{def:genus-three-source}.
It is therefore surjective for every $n\ge10$.
\end{proof}