\section{Ordinary lifting}\label{sec:15-ordinary-lifting}

We recover ordinary boundary classes from their nonseparating restrictions.
The two ingredients are descent for rooted divisor classes and detection
of parallel classes modulo the complete separating image.

\begin{definition}\label{def:rooted-picard-quotient}
For $h\geq2$ and $q\in J$, set
\[
\mathcal B_{h,J;q}
=\operatorname{span}_{\mathbb Q}
\{\delta_{0,A}:q\in A\subseteq J,\ |A|\geq2\},
\qquad
\overline{\mathcal P}_{h,J;q}
=H^2(\Mbar_{h,J})/\mathcal B_{h,J;q}.
\]
Only stable divisors are included. For $h\ge3$ the quotient map
restricts to an isomorphism
$\Pi^\circ_{h,J;q}\otimes\Lmot\xrightarrow{\sim}\overline{\mathcal P}_{h,J;q}$,
by the basis clause of Theorem~\ref{thm:cited-divisor-generation} and
Corollary~\ref{cor:ordinary-image-rooted-picard-basis}.
\end{definition}

\begin{lemma}[Rooted Picard descent]\label{lem:rooted-picard-descent}
For $h\geq4$, nonseparating pullback gives an injection
\[
\xi_{cd}^*:\overline{\mathcal P}_{h,J;q}\longrightarrow
\overline{\mathcal P}_{h-1,J\cup\{c,d\};q}^{\langle(c\,d)\rangle}.
\]
For $h\geq5$, choose further distinct flags $a,b$, let $\nu_{ab}$
glue $a,b$, and put $s=(a\,c)(b\,d)$. The sequence
\[
0\longrightarrow\overline{\mathcal P}_{h,J;q}
\xrightarrow{\ \xi_{cd}^*\ }
\overline{\mathcal P}_{h-1,J\cup\{c,d\};q}^{\langle(c\,d)\rangle}
\xrightarrow{\ (1-s^*)\nu_{ab}^*\ }
\overline{\mathcal P}_{h-2,J\cup\{a,b,c,d\};q}
\]
is exact at its first two nonzero terms.
\end{lemma}

\begin{proof}
All genera in the first comparison are at least three; the same holds
for all three genera in the second.

\smallskip\noindent\emph{One-loop injectivity.}
By Theorem~\ref{thm:cited-nonseparating-divisor-pullbacks},
$\xi_{cd}^*\delta_{0,A}=\delta_{0,A}$ for every omitted rational-root
tail, so the quotient map is well-defined. The coefficients of
$\kappa_1$, of $\psi_i$ for $i\in J$, and of $\delta_{\mathrm{irr}}$
are preserved. If a class pulls back to zero, these coefficients vanish.

Distinguish the side carrying $q$ in each remaining source divisor. This
side has positive genus. If its other side has positive genus, normalize
there; the $q$-side remains positive. If the other side is rational,
normalize on the $q$-side, whose genus decreases from $h$ to $h-1$.
In either case the result is a stable retained target divisor. Gluing
$c,d$ recovers the unique source divisor, so distinct source coefficients
cannot cancel. The image is fixed by $(c\,d)$.

\smallskip\noindent\emph{Reduction to separating divisors.}
Assume $h\ge5$. Gluing the two pairs commutes, so the displayed maps
compose to zero, also after passing to the rooted quotients. Let $\eta$
be in the kernel of the second map, and write
\[
 \eta=\lambda\kappa_1+\sum_{i\in J}\mu_i\psi_i
       +\varepsilon(\psi_c+\psi_d)
       +\gamma\delta_{\mathrm{irr}}+\eta_{\mathrm{sep}},
\]
where $\eta_{\mathrm{sep}}$ is a linear combination of the retained separating divisor classes.
The coefficient of $\psi_c$ in $(1-s^*)\nu_{ab}^*\eta$ is
$\varepsilon+\gamma$: the new cotangent coefficients in the pullback
of $\delta_{\mathrm{irr}}$ are both $-1$. Thus
$\varepsilon=-\gamma$. Subtract the pullback of
$\lambda\kappa_1+\sum_{i\in J}\mu_i\psi_i+
\gamma\delta_{\mathrm{irr}}$ from genus $h$. The remainder is a pure
separating combination and still satisfies the exchange equation.

A retained divisor separating $c$ from $d$ has a retained pullback
obtained by the choice of vertex used above. This term places $a,b$
together and keeps $c,d$ apart. Every exchanged pullback places $c,d$
together. The target term uniquely recovers its source by gluing $a,b$,
and its coefficient is one. Hence every coefficient separating $c,d$
vanishes.

\smallskip\noindent\emph{Reconstruction of the divisor orbits.}
Put $m=h-1$. Each remaining source divisor is uniquely specified by its
side containing $c,d$ and has the form
\[
 D_{r,A}=\delta_{r,A\cup\{c,d\}},\qquad A\subseteq J.
\]
Its stable indices are $0\le r\le m-1$, together with $r=m$ exactly
when $|J\setminus A|\ge2$. The omitted indices are precisely
\[
 r=0,\ q\in A,\qquad\text{or}\qquad r=m,\ q\notin A.
\]
For $0\le r\le m-1$, normalizing the side without $c,d$ gives the
opposite-pair target with sides
$(r,A\cup\{c,d\})$ and $(m-1-r,(J\setminus A)\cup\{a,b\})$.
This target is omitted exactly when
$r=0,q\in A$ or $r=m-1,q\notin A$. These are exactly the cases in
which one member of the involution
\[
 (r,A)\longleftrightarrow(m-1-r,J\setminus A)
\]
is omitted. When both members are retained, the exchange equation equates
their coefficients. When just one is retained, it imposes no condition
on that coefficient. The other pullback terms put all four labels on
one side and are fixed individually by $s$; they give no further equations.

For each orbit with retained coefficient $f$, take
$f\delta_{r+1,A}$ in genus $h$, using either representative when both
are retained. Its two sides have positive genus, and its pullback is
\[
 f\bigl(D_{r,A}+D_{m-1-r,J\setminus A}\bigr).
\]
If one summand is omitted, this realizes the required single coefficient
in the quotient. The choice of representative does not matter, by
complementarity in genus $h$. Since $q\in J$, these involution orbits
have no fixed point. Finally, a retained $D_{m,A}$ lifts through
$\delta_{0,J\setminus A}$; its rational side excludes $q$.
All formulas count complementary divisors once; the unmarked equal-genus
exception is absent because the root is retained.
\end{proof}

\begin{definition}\label{def:relative-parallel-source}
For $h\ge3$ and finite $L$, let $\mathcal R^{\mathrm{par}}_h(L)$ and
$p_h:\mathcal R^{\mathrm{par}}_h(L)\to H^{15}(\Mbar_{h,L})$ be as in
Definition~\ref{def:g5-parallel-source}; thus
$\mathcal R^{\mathrm{par}}_h(L)=\mathsf R_{h,L}\otimes\Pmot$ and
$p_h=2\Pi_{\rm par}$ in the notation of
Definition~\ref{def:ordinary-summands}. For $h\geq5$ and distinct
$x,y\notin L$, let
\[
H_{xy}:\mathcal R_h^{\mathrm{par}}(L)
\longrightarrow\mathcal R_{h-1}^{\mathrm{par}}(L\sqcup\{x,y\})
\]
place both new markings on the unit partner, acting identically on
the displayed elliptic coefficient.
For $g\geq5$, choose $a,b,c,d\notin F$, set $I=F\sqcup\{c,d\}$,
$I^{(4)}=I\sqcup\{a,b\}$, $t=(c\,d)$, and $s=(a\,c)(b\,d)$, write
$\xi=\xi_{cd}$, let $\nu_{ab}:\Mbar_{g-2,I^{(4)}}\to\Mbar_{g-1,I}$
glue $a,b$, put $\mathfrak q=(1-s^*)\nu_{ab}^*$, and put
\[
\mathcal R^{\mathrm{rel}}_{g,F}
=\mathcal R_{g-1}^{\mathrm{par}}(I)^{\langle t\rangle},
\qquad
\mathcal S_{g,F}=\operatorname{Sep}_{g-1,I}^{\langle t\rangle}.
\]
Let $\mathcal R^{\mathrm{rel},0}_{g,F}$ be the allocation subspace
with neither $c$ nor $d$ on the decorated elliptic vertex.
Define
\[
e^{\mathrm{rel}}_{g,F}:
\mathcal R^{\mathrm{rel}}_{g,F}
\longrightarrow\mathscr Q_{g,F}/\delta(\mathcal S_{g,F}),
\qquad
r\longmapsto[\delta(p_{g-1}(r))].
\]
This is Definition~\ref{def:g5-parallel-source} with $4$ replaced by
$g-1$: for $g=5$ these are $\mathcal R^{\mathrm{par}}_F$,
$B_{\mathrm{sep},F}$, $\mathcal R^{\mathrm{par}}_{0,F}$ of that
definition and the map $e_F$ of
Proposition~\ref{prop:g5-parallel-left-inverse}.
\end{definition}

\begin{lemma}\label{lem:ordinary-lifting-separating-coordinates}
Let $h\ge4$ and let $L$ be a finite marking set.
\begin{enumerate}
\item The complete separating image is
\[
 \operatorname{Sep}_{h,L}
 =\Omega_{h,L}\left(
  (\mathsf D_{h,L}\oplus\mathsf W_{h,L}\oplus\mathsf T_{h,L})
  \otimes\Pmot\right),
\]
\item The map
\[
 \mathcal R_h^{\mathrm{par}}(L)\longrightarrow
 H^{15}(\Mbar_{h,L})/\operatorname{Sep}_{h,L},
 \qquad r\longmapsto[p_h(r)]
\]
is injective.
\end{enumerate}
\end{lemma}
\begin{proof}
The K\"unneth decomposition in Proposition~\ref{prop:h13-input}
consists of $H^{13}\otimes H^0$ and $H^{11}\otimes H^2$, with both
placements. For the latter, the odd factor has genus one. The full
degree-two partner splits as the retained subspace
$\Pi^\circ\otimes\Lmot$ and the span of the rational divisors containing
its attaching flag. This is a direct splitting by the basis clause of
Theorem~\ref{thm:cited-divisor-generation}. The first summand
is exactly the coefficient used in $\mathsf W$. A divisor in the second
summand composes with the elliptic factor to give a
degree-thirteen elliptic Gysin class, hence a $\mathsf D$ coordinate.

An $H^{13}$ coefficient of genus one itself gives $\mathsf D$.
For higher genus, Theorem~\ref{thm:cited-thirteen-cohomology} expresses
the entire coefficient by elliptic $H^{11}$ tails with a positive-genus
unit partner. If the outer attaching flag lies on that partner, the
composition is an endpoint-decorated path, hence a boundary decoration
in the full Picard partner just considered. If the flag lies on the
elliptic factor, two positive leaves give $\mathsf T$, while a rational
outer leaf composes into $\mathsf D$. These exhaust the placements of
the outer flag.

Conversely, the maps realizing $\mathsf D$ and $\mathsf W$ are separating Gysin maps, and a
path class is obtained by composing two separating Gysin maps. The same statement
holds for interchangeable leaves after rational transfer from the ordered
cover. The map $p_h$ is the $\mathsf R\otimes\Pmot$ part of
$\Omega_{h,L}$. Its injectivity modulo the first three summands now
follows from Theorem~\ref{thm:ordinary-image}.
\end{proof}

\begin{proposition}[Relative parallel detection]
\label{prop:relative-parallel-detection}
For $h\geq5$, finite $L$, and distinct new markings $x,y$,
nonseparating pullback satisfies
\[
[\xi_{xy}^*p_h(r)]
=[p_{h-1}(H_{xy}r)]
\quad\hbox{in}\quad
H^{15}(\Mbar_{h-1,L\sqcup\{x,y\}})
  /\operatorname{Sep}_{h-1,L\sqcup\{x,y\}}.
\]
For every $g\geq5$ and finite $F$,
\[
\ker e^{\mathrm{rel}}_{g,F}
=\mathcal R^{\mathrm{rel},0}_{g,F}.
\]
\end{proposition}

\begin{proof}
\smallskip\noindent\emph{Cohomological excess and the four branches.}
Let $A,B$ be stable graphs of the same type and let
$\eta\in H^*(X_B;\mathbb Q)$ be invariant under $\Aut(B)$.
For a common refinement $(\Gamma,f_A,f_B)$, also write $f_A$ and $f_B$
for the induced maps from $X_\Gamma$ to the two vertex products.
The geometric fibre product and excess bundle in
Theorem~\ref{thm:genus-three-excess-cited} give the identity
\begin{equation}\label{eq:ordinary-lifting-odd-pullback}
 \xi_A^*[B,\eta]=\frac1{|\Aut(B)|}
 \sum_{[\Gamma,f_A,f_B]\in\mathcal R(A,B)}
 (f_A)_*\left(f_B^*\eta
   \prod_{\{u_0,v_0\}\in H(f_A,f_B)}
       (-\psi_{u_0}-\psi_{v_0})\right).
\end{equation}
To justify this identity for arbitrary cohomology classes $\eta$, choose an
\'etale atlas with the selected nodes ordered. On this atlas, Gysin pushforward is multiplication by
their Thom classes, followed by extension from a tubular neighborhood.
On each component of the fibre product, transverse branches retain their
Thom classes and common branches contribute the Euler class of the excess
bundle. Its summands are the smoothing lines
$L_{u_0}^{\vee}\otimes L_{v_0}^{\vee}$, whose first Chern classes are
$-\psi_{u_0}-\psi_{v_0}$. This construction gives base change for the
Gysin maps themselves and is compatible with descent on the atlas.
Thom and Euler classes have even degree, so the argument applies to odd
$\eta$ with its coefficient order unchanged. Rational transfer gives
$1/|\Aut(B)|$. The target is the labelled $X_A$, and the common
refinements have fixed target identifications, so there is no further
$1/|\Aut(A)|$ or $1/|\Aut(\Gamma)|$ factor.

Apply \eqref{eq:ordinary-lifting-odd-pullback} with $A$ the one-loop
graph and $B$ the parallel graph, first on the cover ordering its two
edges. The normalized node is one of the following.
\begin{enumerate}
\item A selected edge. The remaining edge becomes separating.
The excess class is $-(\psi_x+\psi_y)$, multiplying the pulled-back
degree-eleven coefficient. The result is an ordinary separating Gysin
with a degree-thirteen coefficient. Both selected edges and both
assignments of $x,y$ are included.
\item A nonseparating node inside the elliptic factor. The coefficient
pulls back to $H^{11}$ in genus zero and vanishes by
Theorem~\ref{thm:cited-genus-zero-cycles}.
\item A nonseparating node inside the genus-$(h-2)$ unit factor. That
factor becomes genus $h-3$ and receives $x,y$. Its smoothing parameter
is independent of the selected-edge parameters, so the labelled
coefficient is one. The simultaneous edge-exchange group is the same
order-two group in both genera. Since $p_h$ and $p_{h-1}$ are both
$2\Pi_{\rm par}$, this branch is exactly $p_{h-1}(H_{xy}r)$.
\item A separating node inside a vertex. It is nonseparating in the
glued curve precisely when the two old attaching flags lie on opposite
sides. After normalization, the selected edges form a three-vertex path
and are both separating. Its class is in the complete separating image.
This includes every stable genus split, every marking allocation, and
the full compact restriction of the odd coefficient.
\end{enumerate}
A node is either selected or internal to one of the two compact vertex
factors, and an internal node is separating or nonseparating. Thus the
list is exhaustive.

The same base-change argument on a separating one-edge source gives
\begin{equation}\label{eq:ordinary-lifting-separating-pullback}
 \xi_{xy}^*\operatorname{Sep}_{h,L}
 \subseteq\operatorname{Sep}_{h-1,L\sqcup\{x,y\}}.
\end{equation}
Indeed, its selected edge is a bridge and stays separating in every
degeneration of its connected vertex factors. It cannot be the normalized
nonseparating node. Base change along it is transverse, and the remaining
bridge carries the full pulled-back degree-thirteen coefficient, which
proves \eqref{eq:ordinary-lifting-separating-pullback}.

For $g=5$ the second assertion is the last assertion of
Proposition~\ref{prop:g5-parallel-left-inverse}, by the identification
in Definition~\ref{def:relative-parallel-source}.

\smallskip\noindent\emph{The quotient detector.}
Fix $g\ge6$, with $I^{(4)}$, $s$, $\nu_{ab}$ of
Definition~\ref{def:relative-parallel-source}.
Project $C^2_{g,F}$ onto the graph with one genus-$(g-2)$
vertex and two loops. On its labelled cover, trivialize the determinant
by $e_{ab}\wedge e_{cd}$, and call this coefficient projection
$\operatorname{pr}_{\mathrm{dl}}$. Each individual pair reversal is
positive, while the exchange $s$ is negative. The definition of $\beta$
in Definition~\ref{def:compact-boundary-row} gives
\[
 \operatorname{pr}_{\mathrm{dl}}\beta(0,u)
   =(1-s^*)\nu_{ab}^*u\qquad(u\in N_{g,F}).
\]
Both contractions of this graph are nonseparating. Consequently
$\operatorname{pr}_{\mathrm{dl}}\beta(C^1_{\rm sep})=0$, with the
entire compact separating source included.
By \eqref{eq:ordinary-lifting-separating-pullback}, the image of
$\operatorname{pr}_{\mathrm{dl}}\beta(0,\mathcal S_{g,F})$ is contained
in $\operatorname{Sep}_{g-2,I^{(4)}}$. Hence the map
\[
 \begin{aligned}
 \mathcal D_{g,F}:\mathscr Q_{g,F}/\delta(\mathcal S_{g,F})
 &\longrightarrow
 H^{15}(\Mbar_{g-2,I^{(4)}})/\operatorname{Sep}_{g-2,I^{(4)}},\\
 [v]&\longmapsto[\operatorname{pr}_{\mathrm{dl}}v]
 \end{aligned}
\]
is well-defined, where $v$ is a representative in $C^2_{g,F}$.

The first assertion, applied with $h=g-1$, and relabelling equivariance give
\[
 \mathcal D_{g,F}e^{\mathrm{rel}}_{g,F}(r)
 =\left[p_{g-2}\bigl((1-s^*)H_{ab}r\bigr)\right].
\]
Lemma~\ref{lem:ordinary-lifting-separating-coordinates} makes $p_{g-2}$
injective in this quotient because $g-2\ge4$.
Split $\mathcal R^{\mathrm{rel}}_{g,F}$ into
$\mathcal R^{\mathrm{rel},0}_{g,F}$ and the allocation subspace
$\mathcal R^{\mathrm{rel},+}_{g,F}$ whose elliptic marking set meets
$\{c,d\}$. On the first subspace all four labels lie on the unit partner
after $H_{ab}$, so $s^*H_{ab}=H_{ab}$. On the second, the image of
$H_{ab}$ has elliptic marking set meeting $\{c,d\}$ and avoiding
$\{a,b\}$, while the image of $s^*H_{ab}$ meets $\{a,b\}$ and avoids
$\{c,d\}$. These are disjoint supports in the labelled direct sum.
Projection onto the former support recovers $H_{ab}r$, and $H_{ab}$
is the identity on each displayed coefficient. Thus
$(1-s^*)H_{ab}$ is injective on $\mathcal R^{\mathrm{rel},+}_{g,F}$.
Taking $t$-invariants pairs the one-$c,d$ allocations or takes invariants
within the two-$c,d$ allocations; it never identifies the two supports
just distinguished. It follows that
$\ker e^{\mathrm{rel}}_{g,F}\subseteq\mathcal R^{\mathrm{rel},0}_{g,F}$.

\smallskip\noindent\emph{Lifts of the remaining allocations.}
For $r_0\in\mathcal R^{\mathrm{rel},0}_{g,F}$, let
$\widetilde r_0\in\mathcal R_g^{\mathrm{par}}(F)$ be its identical
elliptic coefficients, so $H_{cd}\widetilde r_0=r_0$, and put
$z=p_g(\widetilde r_0)$. The first assertion in genus $g$ gives
\[
 \xi_{cd}^*z-p_{g-1}(r_0)\in\mathcal S_{g,F}.
\]
The difference is $t$-invariant since both terms are. Write
$\alpha(z)=(v_{\rm sep},\xi_{cd}^*z)$ in the full compact row.
Proposition~\ref{prop:compact-restriction-row} yields
$\beta\alpha(z)=0$, whence
$\delta(\xi_{cd}^*z)=[-\beta(v_{\rm sep},0)]=0$.
Hence $e^{\mathrm{rel}}_{g,F}(r_0)=0$.
\end{proof}

\begin{theorem}[Ordinary lifting]\label{thm:ordinary-lifting}
For every $g\geq6$ and finite $F$, nonseparating pullback preserves
the ordinary image, and the sequence
\[
0\longrightarrow\mathcal I_{g,F}\xrightarrow{\xi^*}
\mathcal I_{g-1,F\cup\{c,d\}}^{\langle t\rangle}
\xrightarrow{\delta}\mathscr Q_{g,F}
\]
is exact at the first two nonzero terms.
\end{theorem}

\begin{proof}
\smallskip\noindent\emph{The four image subspaces.}
For every $h\ge4$ and finite $L$, define the subspaces of
$H^{15}(\Mbar_{h,L})$
\[
\begin{aligned}
 D_h(L)&=\Gamma_D(\mathsf D_{h,L}\otimes\Pmot),&
 W_h(L)&=\Theta(\mathsf W_{h,L}\otimes\Pmot),\\
 T_h(L)&=J_{\rm path}(\mathsf T_{h,L}\otimes\Pmot),&
 R_h(L)&=p_h(\mathcal R_h^{\mathrm{par}}(L)).
\end{aligned}
\]
The domains and maps here are those of
Definition~\ref{def:ordinary-summands}. In particular, for an elliptic
allocation $A\subseteq L$, the degree-two partner coefficient in
$W_h(L)$ lies in
\[
 \Pi^\circ_{h-1,(L\setminus A)\cup\{q\};q}\otimes\Lmot
 \subset H^2(\Mbar_{h-1,(L\setminus A)\cup\{q\}})
\]
(Definition~\ref{def:rooted-picard-quotient}).
An omitted rational
divisor containing $q$ is excluded from the domain of $\Theta$. Gluing
its rational component to the elliptic factor first gives an
$H^{13}$ class on the resulting genus-one component; its remaining
attachment is the $\Gamma_D$ map.

Theorem~\ref{thm:ordinary-image} and
Lemma~\ref{lem:ordinary-lifting-separating-coordinates} now give
\[
 \begin{aligned}
 \mathcal I_{h,L}&=D_h(L)\oplus W_h(L)\oplus T_h(L)\oplus R_h(L),\\
 \operatorname{Sep}_{h,L}&=D_h(L)\oplus W_h(L)\oplus T_h(L).
 \end{aligned}
\]
All subsequent coordinate projections are projections in this cohomological
direct sum, with the cohomology coefficients and their Tate factors
retained.

\smallskip\noindent\emph{Pullback preserves the ordinary image.}
Let $h\ge5$ and let $v,w$ be new flags. For the graphs defining
$D_h,W_h,T_h$, every selected edge is a bridge. A selected bridge,
or a separating node inside a vertex of this tree, cannot be the
normalized nonseparating node. Formula
\eqref{eq:ordinary-lifting-odd-pullback} therefore gives transverse base
change along the selected edges. The possible internal nonseparating
normalizations give the following complete list.

On $D_h$, the pair $v,w$ lies on the unit partner and the elliptic
$H^{13}$ coefficient is unchanged. The other possible branch pulls that
coefficient to genus zero and vanishes. On $W_h$, the pair lies on the
Picard partner, and its coefficient is the nonseparating $H^2$
pullback. Its retained projection is the map on
$\overline{\mathcal P}$; every omitted rational-root term is a
$D_{h-1}$ correction by the preceding composition. Normalization of the
elliptic $H^{11}$ factor again vanishes. Both vanishings follow from
Theorem~\ref{thm:cited-genus-zero-cycles}.

On a path in $T_h$, normalization of a unit leaf of genus at least two
lowers its genus by one and puts $v,w$ there, leaving the elliptic
coefficient unchanged. A genus-one unit leaf becomes rational; composing
it into the center gives an $H^{13}$ elliptic coefficient in $D_{h-1}$.
Normalization of the decorated genus-one center vanishes. Every surviving
branch has coefficient one on the ordered cover. If the two leaves
upstairs are identical, the two branches are exchanged by the graph
involution; division by its order gives one copy of the invariant
coefficient. If they are distinct, each branch occurs once.

Finally, Proposition~\ref{prop:relative-parallel-detection} gives the
$R_h$ pullback modulo the full separating image. Consequently the
pullbacks have the triangular form
\[
\begin{aligned}
 D_h&\longrightarrow D_{h-1},&
 W_h&\longrightarrow W_{h-1}+D_{h-1},\\
 T_h&\longrightarrow T_{h-1}+D_{h-1},&
 R_h&\longrightarrow R_{h-1}+\operatorname{Sep}_{h-1},
\end{aligned}
\]
with marking sets as above.

\smallskip\noindent\emph{Parallel lifts and the two-loop equation.}
$\delta\circ\xi_{cd}^*=0$ on $\mathcal I_{g,F}$
(Proposition~\ref{prop:compact-restriction-row}, as at the end of the
proof of Proposition~\ref{prop:relative-parallel-detection}).

Let $x^{(0)}\in\mathcal I_{g-1,I}^{\langle t\rangle}$ satisfy
$\delta(x^{(0)})=0$. Its parallel coefficient
$\varrho\in\mathcal R^{\mathrm{rel}}_{g,F}$ satisfies
$e^{\mathrm{rel}}_{g,F}(\varrho)=0$ because all other coordinates lie in
$\mathcal S_{g,F}$. Proposition~\ref{prop:relative-parallel-detection}
therefore places $\varrho$ in $\mathcal R^{\mathrm{rel},0}_{g,F}$.
Let $\widetilde\varrho\in\mathcal R_g^{\mathrm{par}}(F)$ have those same
coefficients and set $z_R=p_g(\widetilde\varrho)$.
Its pullback has parallel coefficient $\varrho$ and separating
corrections. Thus
\[
 \begin{gathered}
 x^{(1)}=x^{(0)}-\xi_{cd}^*z_R,\qquad \delta(x^{(1)})=0,\\
 x^{(1)}\in
 (D_{g-1}(I)\oplus W_{g-1}(I)\oplus T_{g-1}(I))^{\langle t\rangle}.
 \end{gathered}
\]
The two-loop projection annihilates $\beta(C^1_{\rm sep})$, as in
the proof of Proposition~\ref{prop:relative-parallel-detection}.
Its value on $\beta(0,x^{(1)})$ is $\mathfrak q(x^{(1)})$.
Hence $\mathfrak q(x^{(1)})=0$. Each further subtraction below is a
pullback and preserves both kernel conditions.

\smallskip\noindent\emph{Picard lifts.}
Project $\mathfrak q(x^{(1)})$ to $W_{g-2}(I^{(4)})$; only
the $W$ coordinate of $x^{(1)}$ contributes. If its elliptic allocation
$A\subseteq I$ meets $\{c,d\}$, the $\nu_{ab}^*$ image meets
$\{c,d\}$ and avoids $\{a,b\}$ on the elliptic factor, while its
exchanged image has the opposite support. On this allocation the
coefficient map is the identity on $H^{11}$ tensored with rooted
Picard pullback from partner genus $g-2\ge4$. It is injective by
Lemma~\ref{lem:rooted-picard-descent}. The disjoint supports force all
such $W$ coefficients to vanish. Rational tensor products and the
finite $t$-invariants preserve this injection.

For $A\subseteq F$, put $M_A=(F\setminus A)\cup\{q\}$. The elliptic
coefficient is unchanged by all four flags, and the $W$ equation is
exactly the equalizer of Lemma~\ref{lem:rooted-picard-descent} with
higher partner genus $g-1\ge5$ and root $q$. Choose the unique retained
representative of each lifted divisor coefficient and apply $\Theta$ in
genus $g$. Their sum is a class $z_W\in W_g(F)$. The pullback
$\xi_{cd}^*z_W$ has all the required $W$ coefficients, no $T$
coefficient, and its full rational-root $D$ corrections. Thus
$x^{(2)}=x^{(1)}-\xi_{cd}^*z_W$ lies in $D+T$ and still satisfies
$\mathfrak q(x^{(2)})=0$.

\smallskip\noindent\emph{Path lifts.}
Put $k=g-2\ge4$. A path in $T_{g-1}(I)$ has positive leaf genera
summing to $k$. If $c,d$ are not together on one unit leaf, normalize
a leaf of genus at least two. This gives a nonzero $T_{g-2}$ term
with $a,b$ on that leaf and with the original placement of $c,d$.
No exchanged term has this support, since every exchanged $T$ term
has $c,d$ on a single unit leaf. Gluing $a,b$ uniquely recovers the
source path, so no other source coefficient cancels it. These allocations
therefore vanish in the $T$ coordinate of $x^{(2)}$.

Distinguish the leaf carrying $c,d$ in each remaining path. Write its
center markings as $A$, its distinguished leaf as
$(r,B\cup\{c,d\})$, and its other leaf as $(k-r,C)$, where
$F=A\sqcup B\sqcup C$ and $1\le r\le k-1$. Write
\[
 z_{r,A,B,C}\in H^{11}(\Mbar_{1,A\cup\{p_1,p_2\}})(-2)
\]
for its coefficient, with $p_1$ attaching to the distinguished leaf.
That leaf is distinguished by its labels, so this source has no
leaf-exchange automorphism. For $1\le r\le k-2$, normalize the other
leaf. Comparing the target with its exchanged target gives exactly
\[
 z_{r,A,B,C}=(p_1\,p_2)^*z_{k-1-r,A,C,B}.
\]
The other $T$ targets put all four flags on one leaf and are individually
fixed by $s$. At $r=k-1$, normalization of the other, genus-one leaf
is a $D$ term and gives no additional $T$ equation.

For each two-element orbit of the displayed equation, take the genus-$g$
path with center $A$ and ordered leaves $(r+1,B)$ and $(k-r,C)$,
with coefficient $z_{r,A,B,C}$. Both leaf genera are at least two;
its two normalization branches give exactly the prescribed orbit.
Changing the orbit representative exchanges the attaching flags and
leaves the ordinary class unchanged. A fixed orbit occurs precisely for
$B=C=\varnothing$ and $2r=k-1$. Its coefficient is invariant under
$(p_1\,p_2)$, exactly the ordinary automorphism condition on the
two equal unmarked leaves upstairs. The quotient pullback has coefficient
one by the transfer calculation above.

For each remaining $r=k-1$ coordinate, use the genus-$g$ path with
leaves $(k,B)$ and $(1,C)$. Normalizing the first leaf gives the required
$T$ coefficient. Normalizing the genus-one leaf gives a rational leaf
with markings $C\cup\{c,d\}$; composing its Gysin with the center
produces a $D$ correction, which is retained. Summing these path
classes gives $z_T\in T_g(F)$. Thus
$x^{(3)}=x^{(2)}-\xi_{cd}^*z_T$ has only $D$ coordinates and
$\mathfrak q(x^{(3)})=0$.

\smallskip\noindent\emph{Elliptic-tail lifts and exactness.}
On $D_{g-1}(I)$, the map $\nu_{ab}^*$ preserves each elliptic
$H^{13}$ coefficient and places $a,b$ on its unit partner. The
allocations meeting $\{c,d\}$ and their exchanged allocations have
disjoint supports in the $D_{g-2}(I^{(4)})$ direct sum.
Thus $\mathfrak q(x^{(3)})=0$ forces all those coefficients to vanish.
The remaining allocations are contained in $F$ and are exactly the
pullbacks of the corresponding genus-$g$ elliptic tails. Let $z_D$
be their $\Gamma_D$ image. Adding the four pullbacks gives
\[
 x^{(0)}=\xi_{cd}^*(z_R+z_W+z_T+z_D).
\]
Every separating correction was included in the pullback being
subtracted. This proves the required kernel inclusion; the converse was
established by $\beta\alpha=0$ above.

\smallskip\noindent\emph{Injectivity.}
Suppose $z\in\mathcal I_{g,F}$ has $\xi_{cd}^*z=0$. The parallel
projection of this pullback is the unchanged coefficient placed by
$H_{cd}$, so its injectivity and the direct ordinary decomposition force
the $R$ coefficient of $z$ to vanish. The $W$ projection is then the
rooted Picard pullback at each elliptic allocation, which is injective
by Lemma~\ref{lem:rooted-picard-descent}; hence the $W$ coefficient
vanishes. A path in the higher genus has leaf genera summing to $g-1\ge5$, so
at least one leaf has genus at least two. Normalizing it gives a
$T$ target that uniquely determines the original path by gluing $c,d$,
with unchanged coefficient. Equal unmarked leaves are detected with
coefficient one on their invariant subspace. Therefore the $T$ coefficient
also vanishes. Finally the $D$ pullback is the identity on its coefficients
in distinct allocations, so $z=0$.
\end{proof}