\section{The ordinary Gysin image}\label{sec:07-ordinary-image} We describe the ordinary degree-fifteen boundary image in genus at least four as the image of four explicit Gysin maps. The induced map from the direct sum of their domains is injective: a rooted degree-four quotient separates the Picard and parallel-edge contributions, and further restrictions detect the elliptic degree-thirteen and path contributions. \begin{definition}\label{def:ordinary-image} For every stable genus and finite marking set put \[ \mathcal I_{g,F}=\operatorname{im}\Phi^{15}_{g,F,\mathrm B} \subset H^{15}(\Mbar_{g,F}). \] For $h\geq3$ and $q\in J$, let $\Pi^\circ_{h,J;q}\subset\Pi_{h,J}$ be the divisor multiplicity subspace spanned by $\kappa_1$, every $\psi_i$, $\delta_{\mathrm{irr}}$, and all stable separating divisors except those whose genus-zero side contains $q$. Complementary descriptions of a divisor are identified. \end{definition} \begin{definition}\label{def:ordinary-summands} Let $g\geq4$ and let $F$ be finite. Using distinct auxiliary flags, set \[ \begin{aligned} \mathsf D_{g,F}&=\bigoplus_{A\subseteq F}U_{1,A\cup\{p\}},\\ \mathsf W_{g,F}&=\bigoplus_{A\subseteq F} K_{A\cup\{p\}}\otimes\Pi^\circ_{g-1,(F\setminus A)\cup\{q\};q},\\ \mathsf R_{g,F}&=\bigoplus_{A\subseteq F} K_{A\cup\{u,v\}}^{\langle(u\,v)\rangle}. \end{aligned} \] Let $\mathsf T_{g,F}$ be the direct sum over stable three-vertex paths with legs $F$, a distinguished genus-one center carrying $A$, and two unordered positive-genus leaves of genera $r,s$ with $r+s=g-1$, of \[ K_{A\cup\{p_1,p_2\}}^{\Aut_F(\Gamma)}. \] The leaves carry units. All automorphism invariants are ordinary, without an edge determinant. The maps $\Gamma_D,\Theta,J_{\rm path}$ are the Gysin maps that respectively attach a genus-$(g-1)$ partner carrying the unit class to the genus-one degree-thirteen class, attach a Picard-decorated genus-$(g-1)$ partner to a hook class, and glue the two edges of the path. Let $\Pi_{\rm par}$ be the quotient-normalized gluing of the two parallel pairs on \[ [\Mbar_{1,A\cup\{u,v\}}\times \Mbar_{g-2,(F\setminus A)\cup\{u',v'\}}/\Sn_2], \] where $\Sn_2$ exchanges both pairs simultaneously. Put \[ \Omega_{g,F}=(\Gamma_D,\Theta,J_{\rm path},2\Pi_{\rm par}): (\mathsf D_{g,F}\oplus\mathsf W_{g,F}\oplus \mathsf T_{g,F}\oplus\mathsf R_{g,F})\otimes\Pmot \longrightarrow H^{15}(\Mbar_{g,F}). \] \end{definition} \begin{definition}\label{def:polito-essential-symbols} For $h\geq3$ and a finite set $J$, let $\mathcal E^{(4)}_{h,J}$ be the free rational vector space on Polito's Section~3 essential degree-four symbols, with realization \[ e_{h,J}:\mathcal E^{(4)}_{h,J}\longrightarrow H^4(\Mbar_{h,J};\mathbb Q). \] The symbols are Mumford monomials, one-edge classes decorated by a degree-two class on a normalization factor, and pure two-edge graph classes. Normalize a decoration by \[ p\mid\delta_\Gamma= \frac{\xi_{\Gamma,*}(p)}{|\operatorname{Aut}\Gamma|}. \] The presentation in Section~3 eliminates the $\psi$ and $\kappa_1$ decorations on genus-zero and genus-one factors and the $\kappa_1$ decorations on genus-two factors by their divisor relations. To reduce the pure graph symbols, the presentation uses Keel relations among trees with two rational vertices; choose the retained symbols as in that section. Every other pure two-edge graph is retained. In particular, for $h\geq3$ the Mumford symbols \[ \kappa_1^2,\quad\kappa_2,\quad\kappa_1\psi_i, \quad\psi_i^2,\quad\psi_i\psi_j\quad(i\ne j) \] are retained, as are the distinct nonseparating mixed symbols \[ \mathsf N^{\mathrm{irr}}_{h,J}=\frac12\xi_{\mathrm{irr},*}(\psi_u+\psi_v), \qquad \mathsf X^{\mathrm{irr}}_{i;h,J}=\frac12\xi_{\mathrm{irr},*}(\psi_i). \] Here $u,v$ are auxiliary node flags. The nonseparating $\kappa_1$ decoration is eliminated at $h=3$. Let $\operatorname{pr}_{M}$ project onto the free span of the Mumford symbols, and let $[m]r$ denote the coefficient of a symbol $m$ in $r\in\mathcal E^{(4)}_{h,J}$. \end{definition} \begin{theorem}[Polito {\cite[Section 5.3 and Theorem 10]{polito2003}}] \label{thm:cited-polito-projected-presentation} \begin{enumerate} \item Let $h\geq3$ be an integer and $J$ a nonempty finite marking set. \begin{enumerate} \item For $h\geq6$, the map $e_{h,J}$ is injective. \item For $h=5$, its kernel is spanned by a vector $R$ satisfying \[ [\kappa_2]R=-180. \] \item For $h=4$, its kernel is spanned by vectors $R_0$ and $R_i$ for $i\in J$, satisfying \[ \begin{array}{c|cc} & [\kappa_1^2] & [\kappa_2] \\\hline R_0 & 45/2 & -240 \\ R_i & 5 & -30 \end{array} \qquad [\kappa_1\psi_j]R_i= \begin{cases} -40,&j=i,\\ -10,&j\ne i. \end{cases} \] \item For $h=3$, its kernel is spanned by vectors $R_{00},R_{02}$, $R_i$ for $i\in J$, and $R_{ij}$ for unordered pairs $\{i,j\}\subset J$. There exist $a\in\mathbb Q$ and $c\in\mathbb Q\setminus\{0\}$ such that \[ \begin{array}{c|cc} & [\kappa_1^2] & [\kappa_2] \\\hline R_{00} & 1 & 0 \\ R_{02} & 0 & 1 \end{array} \] and, for all relevant $i,j$, \[ \begin{aligned} \operatorname{pr}_{M}(R_i)&=\kappa_1\psi_i+a\psi_i^2 -\sum_{j\in J\setminus\{i\}}\psi_i\psi_j, & [\mathsf N^{\mathrm{irr}}_{3,J}]R_i&=c,\\ \operatorname{pr}_{M}(R_{ij})&=\psi_i^2+\psi_j^2-\frac65\psi_i\psi_j. \end{aligned} \] \end{enumerate} \item On $\Mbar_{2,\{q\}}$ let $\gamma_{\mathrm{F}},\gamma_{\mathrm{E}},\gamma_{\mathrm{H},q}, \gamma_{\mathrm{H},\varnothing},\gamma_{\mathrm{G}}$ denote the ordinary pure two-edge classes with the following graphs: a rational vertex carrying $q$ and two loops; a rational vertex carrying $q$ joined by two parallel edges to an elliptic vertex; a looped rational vertex joined to an elliptic vertex with $q$ on the rational side; the same graph with $q$ on the elliptic side; and a rational center carrying $q$ joined to two elliptic leaves. The classes are normalized by the full finite graph-automorphism groups. Then these five classes are linearly independent in $H^4(\Mbar_{2,\{q\}};\mathbb Q)$, and \[ \psi_q^2=\frac1{120}\gamma_{\mathrm F} +\frac15\gamma_{\mathrm E} +\frac{13}{120}\gamma_{\mathrm H,q} -\frac1{120}\gamma_{\mathrm H,\varnothing} +\frac75\gamma_{\mathrm G}. \] \end{enumerate} \end{theorem} \begin{proposition}\label{prop:polito-forgetful-coordinates} For an integer $h\geq3$ and disjoint finite sets $P,K$ with stable target, let $f:\Mbar_{h,P\cup K}\to\Mbar_{h,P}$ forget $K$ and stabilize (a composite of forgetful maps as in Proposition~\ref{prop:external-stable-forgetting}). In the free-symbol pullback rules of Polito's Section~4.1 and Proposition~9, the Mumford parts of $f^*\kappa_1$ and $f^*\kappa_2$ are respectively $\kappa_1-\sum_{j\in K}\psi_j$ and $\kappa_2-\sum_{j\in K}\psi_j^2$. Products have the corresponding product Mumford part. Pure boundary symbols pull back to pure boundary symbols, and mixed symbols to mixed and pure boundary symbols. The essential reductions of boundary symbols are linear combinations of boundary symbols. The pullback of $\mathsf N^{\mathrm{irr}}_{h,P}$ has coefficient one at $\mathsf N^{\mathrm{irr}}_{h,P\cup K}$ and otherwise only pure boundary terms; the pullback of $\mathsf X^{\mathrm{irr}}_{i;h,P}$ has no $\mathsf N^{\mathrm{irr}}_{h,P\cup K}$ coefficient. \end{proposition} \begin{definition}\label{def:rooted-degree-four-quotient} Let $h\geq3$, $q\in J$, and $B=J\setminus\{q\}$. For each nonempty $A\subseteq B$, let \[ j_A:\Mbar_{0,\{q\}\cup A\cup\{x\}}\times \Mbar_{h,(B\setminus A)\cup\{y\}}\longrightarrow\Mbar_{h,J} \] glue $x,y$. Let $\mathcal N_{\rm tail}$ be the sum of $j_{A*}H^2(\operatorname{dom}j_A)$, and let $\mathcal N_{\rm rr}$ be the span of ordinary two-edge tree classes with exactly two genus-zero vertices. Put \[ Q^4_{h,J;q}=H^4(\Mbar_{h,J})/ (\mathcal N_{\rm tail}+\mathcal N_{\rm rr}). \] For every $A\subseteq B$, including the empty set, let $\zeta_A$ be the ordinary parallel-edge class with rational vertex carrying $\{q\}\cup A$ and genus-$(h-1)$ vertex carrying $B\setminus A$. \end{definition} \begin{lemma}[Rooted divisor and parallel detection] \label{lem:rooted-degree-four-detection} For every $h\geq3$ and rooted finite set $(J,q)$, the rational map \[ \Pi^\circ_{h,J;q}\oplus\bigoplus_{A\subseteq J\setminus\{q\}}\mathbb Q \longrightarrow Q^4_{h,J;q},\qquad (\eta,(c_A))\longmapsto \left[\psi_q\eta+\sum_Ac_A\zeta_A\right] \] is injective and equivariant for permutations fixing $q$. \end{lemma} \begin{proof} \smallskip\noindent\emph{Lifts of the subspaces being divided out.} In $\mathcal E^{(4)}_{h,J}$ let $\mathscr N$ be spanned by the following essential boundary symbols: mixed classes on a rational-tail divisor whose rational side contains $q$, decorated on its positive-genus side; pure graphs admitting a contraction to such a divisor; and retained two-edge trees with two rational vertices. Then \[ \mathcal N_{\rm tail}+\mathcal N_{\rm rr}\subseteq e_{h,J}(\mathscr N). \] Indeed, the degree-two K\"unneth decomposition on $\operatorname{dom}j_A$ has only the two factor summands, since the rational factor has no $H^1$ by Theorem~\ref{thm:cited-genus-zero-cycles}. Theorem~\ref{thm:cited-divisor-generation} expresses the class on the positive-genus factor by divisors. Its $\kappa_1$ and cotangent decorations give the indicated mixed symbols; its boundary decorations give pure graphs retaining the rational-tail edge. Divisor classes on the rational factor give trees with two rational vertices. Their essential reductions remain in that family. In particular, $\mathscr N$ has neither Mumford coordinates nor mixed coordinates on the nonseparating divisor. \smallskip\noindent\emph{Independent column coordinates.} Choose a lift of each displayed divisor generator multiplied by $\psi_q$. The products with $\kappa_1$ and $\psi_i$ are their Mumford symbols, and $\psi_q\delta_{\mathrm{irr}}$ is $\mathsf X^{\mathrm{irr}}_{q;h,J}$, with normalization $1/2$. For a separating divisor, specify its $q$-side as $(d,C)$, where $q\in C$. The retained cases have $d\geq1$. If $d\geq2$, the product is the essential external-$q$ mixed symbol on that side. These are distinct from one another and from every mixed symbol of $\mathscr N$. If $d=1$, let $x$ be the attaching flag on its elliptic factor. Proposition~\ref{prop:genus-three-picard-input} gives \[ \psi_q=\frac1{12}\delta_{\mathrm{irr}}+ \sum_{\substack{T\subseteq C\cup\{x\}\\q\in T,\ |T|\geq2}}\delta_{0,T}. \] Push this identity along the separating clutching. The term $T=\{q,x\}$ is the pure tree with rational center carrying exactly $q$, genus-one leaf carrying $C\setminus\{q\}$, and genus-$(h-1)$ leaf carrying $J\setminus C$. Denote its essential symbol by $\gamma_C$. Its coefficient is one: the leaf genera $1,h-1$ are distinct, and the graph has trivial automorphism group. The nonseparating term has coefficient $1/12$ in ordinary graph normalization, since its loop reversal contributes the same denominator $2$ as $\delta_{\mathrm{irr}}$ on the elliptic factor. The symbol $\gamma_C$ lies outside $\mathscr N$. Both its one-edge contractions put $q$ on a positive-genus component, and it has only one rational vertex. It occurs in no other elliptic column: to have rational center carrying exactly $q$, the above expansion must use $T=\{q,x\}$, which recovers $C$ from the genus-one leaf. The other terms have a rational endpoint, extra center markings, or a loop. Finally, each $\zeta_A$ is its retained parallel-graph symbol, normalized by the simultaneous edge exchange of order two. These distinct symbols have no separating edge and only one rational vertex. They occur in neither $\mathscr N$ nor any preceding column. Thus the chosen lifts are linearly independent modulo $\mathscr N$. Write $\mathscr V$ for their span. Every vector in $\mathscr V+\mathscr N$ has zero coordinate at \[ \kappa_1^2,\quad \kappa_2,\quad \kappa_1\psi_j,\ \psi_j^2\ (j\ne q),\quad \psi_i\psi_j\ (i,j\ne q,\ i\ne j),\quad \mathsf N^{\mathrm{irr}}_{h,J}. \] Its only possible Mumford coordinates are $\kappa_1\psi_q$ and $\psi_q\psi_i$. \smallskip\noindent\emph{Exclusion of the relation space.} We claim \[ \ker e_{h,J}\cap(\mathscr V+\mathscr N)=0. \] For $h\geq6$ this is Theorem~\ref{thm:cited-polito-projected-presentation}(1). For $h=5$, the coordinate $[\kappa_2]R=-180$ excludes every nonzero multiple of its relation. For $h=4$, suppose $\alpha R_0+\sum_{i\in J}\beta_iR_i\in\mathscr V+\mathscr N$ and put $b=\sum_i\beta_i$. The first two missing coordinates give \[ \frac{45}{2}\alpha+5b=0,\qquad -240\alpha-30b=0. \] The determinant is $525$, so $\alpha=b=0$. For $j\ne q$, the $\kappa_1\psi_j$ coordinate is \[ -40\beta_j-10\sum_{i\ne j}\beta_i=-30\beta_j, \] hence $\beta_j=0$. The equality $b=0$ then gives $\beta_q=0$. For $h=3$, write a relation as a combination of $R_{00},R_{02},R_i,R_{ij}$. The coordinates $\kappa_1^2,\kappa_2$ first remove $R_{00},R_{02}$. Next $\kappa_1\psi_j$, for $j\ne q$, removes $R_j$. For distinct $i,j\ne q$, the coordinate $\psi_i\psi_j$ is now $-6/5$ times the coefficient of $R_{ij}$, so all these coefficients vanish. For each $j\ne q$, the coordinate $\psi_j^2$ then removes $R_{qj}$. Only a multiple of $R_q$ remains. Its coefficient at $\mathsf N^{\mathrm{irr}}_{3,J}$ is the nonzero scalar $c$ in Theorem~\ref{thm:cited-polito-projected-presentation}(1), so it too vanishes. This argument includes $J=\{q\}$. These are free-symbol coordinate tests; by Proposition~\ref{prop:polito-forgetful-coordinates} forgetting adds no Mumford coordinate from boundary terms and preserves the node coordinate. \smallskip\noindent\emph{Passage to the cohomological quotient.} Suppose a linear combination of the proposed column lifts $v$ realizes to an element of $\mathcal N_{\rm tail}+\mathcal N_{\rm rr}$. Choose $w\in\mathscr N$ with $e_{h,J}(v)=e_{h,J}(w)$. Then $v-w\in\ker e_{h,J}\cap(\mathscr V+\mathscr N)$, so $v=w$. The distinct column coordinates force every coefficient to vanish. The constructions commute with all relabellings fixing $q$, proving the asserted equivariance. \end{proof} \begin{corollary}\label{cor:ordinary-image-rooted-picard-basis} Let $h\geq3$ and let $J$ be a finite set containing $q$. The divisor generators specified in Definition~\ref{def:ordinary-image} form a basis of $\Pi^\circ_{h,J;q}$. \end{corollary} \begin{proof} They span by definition, and they are a subset of the basis of $H^2(\Mbar_{h,J};\mathbb Q)$ given by the basis clause of Theorem~\ref{thm:cited-divisor-generation} for $h\geq3$ (all generators except the rational-tail divisors $\delta_{0,A}$ with $q\in A$). \end{proof} \begin{lemma}\label{lem:ordinary-image-cotangent-restrictions} Let $b\geq1$, let $I$ be a finite marking set containing $q$, and let $(b,|I|)$ be stable. For $b\geq2$, the restriction of $\psi_q$ to $\Mopen_{b,I}$ is nonzero. For $b=1$, let $\mathscr U_I$ be the open in $\Mbar_{1,I}$ obtained by removing all separating boundary divisors and all strata of codimension at least two. Then $\psi_q|_{\mathscr U_I}\ne0$. \end{lemma} \begin{proof} \smallskip\noindent\emph{Independence from the boundary classes.} Choose $k\geq3$ with $b+k\geq6$ and $k\ne b$, and a new marking $z$. Glue $q$ to a flag $q'$ on a genus-$k$ component carrying $z$: \[ \iota:\Mbar_{b,I}\times\Mbar_{k,\{q',z\}} \longrightarrow \Mbar_{b+k,(I\setminus\{q\})\cup\{z\}}. \] If $b\geq2$, the class $\iota_*(\psi_q\otimes1)$ is an essential node-cotangent mixed symbol. On the other hand, pushing any boundary divisor of the first factor gives pure two-edge symbols, whose essential reductions remain pure. Theorem~\ref{thm:cited-polito-projected-presentation}(1) is injective in this target genus, with its nonempty marking set. It therefore excludes a relation expressing $\psi_q$ as a combination of boundary divisors on $\Mbar_{b,I}$. If $b=1$, Proposition~\ref{prop:genus-three-picard-input} instead gives \[ \iota_*(\psi_q\otimes1) =\frac1{12}\iota_*(\delta_{\mathrm{irr}}\otimes1) +\sum_{\substack{T\subseteq I\\q\in T,\ |T|\geq2}} \iota_*(\delta_{0,T}\otimes1). \] The first term is $1/12$ times the essential bridge-and-loop graph with a rational looped vertex and a genus-$k$ vertex. Its automorphism group has order two, exactly the denominator of the elliptic $\delta_{\mathrm{irr}}$. All other terms are trees. The same injectivity therefore excludes a relation expressing $\psi_q$ using only separating divisors on $\Mbar_{1,I}$. \smallskip\noindent\emph{Restriction to the opens.} Remove first the strata with at least two nodes. This has codimension at least two and changes no $H^2$. The remaining boundary components are disjoint smooth divisors. The divisor localization sequence says that the kernel on $H^2$ when they are removed is their span of fundamental classes. Removing all components proves the claim for $b\geq2$; removing just the separating components proves it for $b=1$. \end{proof} \begin{lemma}\label{lem:ordinary-image-elliptic-restrictions} Let $I$ be a nonempty finite set, and let $\Mbar_{1,I}^{\mathrm{ct}}$ be the locus of curves with tree dual graph. The restriction maps \[ H^{11}(\Mbar_{1,I})\longrightarrow H^{11}(\Mopen_{1,I}), \qquad H^{13}(\Mbar_{1,I})\longrightarrow H^{13}(\Mbar_{1,I}^{\mathrm{ct}}) \] are injective. \end{lemma} \begin{proof} \smallskip\noindent\emph{The supports sequence.} For a union $\Delta$ of boundary components in a smooth proper stack $Y$, let $\Delta^{[p]}$ normalize its loci of $p$ distinct local branches, with branch-order sign local system $\epsilon_p$. The local Mayer--Vietoris complex of supports and purity in codimension $p$ give \[ E_1^{1-p,t}= H^{t-2p}(\Delta^{[p]},\epsilon_p)\otimes\Lmot^p \ \Longrightarrow\ H^{t+1-p}_\Delta(Y),\qquad p\geq1. \] Indeed, on a chart with ordered branches the term for a $p$-fold intersection is shifted by $p-1$ in the supports complex and by $2p$ under purity. The alternating complex descends under branch permutations, giving the displayed local system. The normalized intersections are finite quotients of products of smooth proper vertex moduli spaces. Thus each term has weight $t$ and Hodge indices at least $p$ in both coordinates. The first injectivity is Lemma~\ref{lem:elliptic-eleven-open-restriction}. \smallskip\noindent\emph{Degree thirteen on compact type.} Apply the construction with $Y=\Mbar_{1,I}$ and $\Delta_{\mathrm{ct}}=Y\setminus\Mbar_{1,I}^{\mathrm{ct}}$. On total supports degree thirteen the terms have $t=12+p$. For $p\geq2$ their weights are at least fourteen. For $p=1$, the possible weight-thirteen term is $H^{11}(\Delta_{\mathrm{ct}}^{[1]})\otimes\Lmot$. The normalization here is the quotient of $\Mbar_{0,I\cup\{u,v\}}$ by node exchange, so this term vanishes by Theorem~\ref{thm:cited-genus-zero-cycles}. Consequently $H^{13}_{\Delta_{\mathrm{ct}}}(Y)$ has no weight-thirteen part, and its map to the pure weight-thirteen group $H^{13}(Y)$ is zero. Localization gives the second injectivity. \end{proof} \begin{lemma}[Positive path detection]\label{lem:positive-path-detection} Let $\Gamma$ be a path of total genus $g$ with legs $F$, elliptic center, and positive leaves of genera $h_1,h_2\geq1$. Let $A,B_1,B_2$ be the external marking sets of its center and two leaves, with attaching flags $p_i$ on the center and $q_i$ on leaf $i$. If $h_1=h_2$, assume $B_1\ne B_2$, so the external labels distinguish the two leaves. Refine each genus-one leaf into a rational vertex with one loop, leave every other vertex unchanged, and call the result $\widetilde\Gamma$. Let $O_\Gamma$ be the open union of strata obtained by contracting edges of $\widetilde\Gamma$, and put \[ Z_\Gamma=Y_0\times Y_1\times Y_2,\qquad Y_0=\Mopen_{1,A\cup\{p_1,p_2\}}, \] where a leaf of genus at least two contributes its smooth moduli space $Y_i=\Mopen_{h_i,B_i\cup\{q_i\}}$, and a genus-one leaf contributes $Y_i=\mathscr U_{B_i\cup\{q_i\}}$ from Lemma~\ref{lem:ordinary-image-cotangent-restrictions}, the smooth or irreducible one-nodal curves. The restricted clutching map \[ i_\Gamma:Z_\Gamma\longrightarrow O_\Gamma \] is a closed regular embedding of codimension two. For every $z\in H^{11}(\Mbar_{1,A\cup\{p_1,p_2\}})$, \[ \operatorname{pr}_{11,2,2}i_\Gamma^*i_{\Gamma*} (z|_{Y_0}\otimes1\otimes1) =z|_{Y_0}\otimes\psi_{q_1}\otimes\psi_{q_2}, \] and this map is injective in $z$. The open $O_\Gamma$ meets no different two-edge tree stratum and no two-edge tree stratum with a rational vertex. If instead $h_1=h_2\geq2$ and $B_1=B_2=\emptyset$, so that $\Aut_F(\Gamma)$ has order two, the same conclusions hold for the induced map $[Z_\Gamma/\Aut_F(\Gamma)]\to O_\Gamma$, invariant cohomology being detected on the cover $Z_\Gamma$ by the same self-intersection formula, without sign from the edge exchange. \end{lemma} \begin{proof} \smallskip\noindent\emph{The labelled support.} At fixed genus and markings there are finitely many stable graph types. The types not obtained by contracting $\widetilde\Gamma$ are closed under specialization, so their union is closed. This proves that $O_\Gamma$ is open. Keeping the two original bridges in this open permits only the specified irreducible degenerations of the genus-one leaves. Consequently the inverse image of $O_\Gamma$ in the labelled path product is exactly $Z_\Gamma$. The center is distinguished by its two bridges. The leaf genera, or their different external label sets when both genera are equal, distinguish the leaves. The bridges are therefore uniquely recovered from the curve. The proper clutching map restricted to this open is a closed embedding, and its two independent node smoothing parameters show that it is regular of codimension two. Loop reversal on an irreducible leaf belongs to that leaf's moduli factor; it does not exchange the two bridges. Every loop must be contracted to obtain a tree from $\widetilde\Gamma$. To retain two edges one must then retain both bridges, giving exactly $\Gamma$. In fact $O_\Gamma$ is disjoint from the closure of every other two-edge tree stratum: a graph in such a closure would also contract to that tree. This proves the support assertions. \smallskip\noindent\emph{The self-intersection.} The normal classes are $-\psi_{p_i}-\psi_{q_i}$, so the self-intersection formula gives \[ i_\Gamma^*i_{\Gamma*}(z\otimes1\otimes1) =(z\otimes1\otimes1) (-\psi_{p_1}-\psi_{q_1})(-\psi_{p_2}-\psi_{q_2}), \] with $z$ restricted to the smooth center. Only the product of the two leaf classes has multidegree $(11,2,2)$. Lemma~\ref{lem:ordinary-image-elliptic-restrictions} makes restriction on the center injective, and Lemma~\ref{lem:ordinary-image-cotangent-restrictions} makes each leaf class nonzero. Tensoring with these two classes is injective. \smallskip\noindent\emph{Equal unmarked leaves.} Let $h_1=h_2\geq2$ and $B_1=B_2=\emptyset$. The pair of bridges is unique, up to the indicated leaf exchange; a second bridge with the same genus and marking partition would bound an unmarked stable rational tree with only two exterior flags. Summing $2g(v)-2+|F(v)|$ on such a tree gives zero, contradicting stability. The two node-smoothing parameters give regular local equations. The support assertions hold as before. The cover $Z_\Gamma\to[Z_\Gamma/\Aut_F(\Gamma)]$ has degree two; rational push-pull is multiplication by two, so invariant cohomology is detected on the cover by the same self-intersection formula. \end{proof} \begin{lemma}[Minimal elliptic tail detection] \label{lem:minimal-elliptic-tail-detection} Let $h\geq2$, let $J=A\sqcup B$ with $A\ne\varnothing$, and let $W_A\subseteq\Mbar_{h+1,J}$ be the open union of contractions of graphs obtained by gluing a compact-type genus-one curve on $A\sqcup\{p\}$ to a smooth curve of genus $h$ on $B\sqcup\{q\}$. Then the restricted clutching \[ j_A^{\mathrm{ct}}: \Mbar^{\mathrm{ct}}_{1,A\sqcup\{p\}} \times\Mopen_{h,B\sqcup\{q\}}\longrightarrow W_A \] is a regular closed embedding of codimension one, and for every $z\in H^{13}(\Mbar_{1,A\sqcup\{p\}})$, \[ \operatorname{pr}_{13,2}(j_A^{\mathrm{ct}})^* j_{A*}^{\mathrm{ct}}(z|_{\mathrm{ct}}\otimes1) =-z|_{\mathrm{ct}}\otimes\psi_q. \] This map is injective in $z$. A genus-one tail with external marking set $A'$ can meet $W_A$ only if $A'\subseteq A$. \end{lemma} \begin{proof} \smallskip\noindent\emph{The minimal tail support.} The contraction criterion used in the preceding proof makes $W_A$ open. An allowed uncontracted graph has one genus-one vertex, one smooth genus-$h$ vertex carrying every $B$ label, and only rational vertices on the genus-one side. A separating genus-one side therefore carries a subset of $A$: it must avoid the genus-$h$ vertex, and contractions preserve the partition of a retained edge. There cannot be two edges with the same genus-one side and marking set: the intervening unmarked rational tree would have only two external attachments, contradicting stability. This proves both the assertion about $A'$ and the uniqueness of the attachment of type $A$. It also identifies the inverse image of $W_A$ with the displayed product: a contraction retaining an edge of type $(1,A)$ retains that unique edge; every other edge is on the compact-type genus-one side, and the partner remains a single smooth vertex; conversely, every point of the product is in the defining family. The genera distinguish the two factors, so the restricted clutching map is a regular closed embedding of codimension one with normal class $-\psi_p-\psi_q$. \smallskip\noindent\emph{The self-intersection.} The self-intersection formula gives \[ (j_A^{\mathrm{ct}})^*j_{A*}^{\mathrm{ct}} (z|_{\mathrm{ct}}\otimes1) =-(z|_{\mathrm{ct}}\psi_p)\otimes1 -z|_{\mathrm{ct}}\otimes\psi_q. \] Only the second term has multidegree $(13,2)$. Its first factor depends injectively on $z$ by Lemma~\ref{lem:ordinary-image-elliptic-restrictions}; its second factor is nonzero by Lemma~\ref{lem:ordinary-image-cotangent-restrictions}, since $h\geq2$. \end{proof} \begin{lemma}\label{lem:ordinary-image-path-separation} Let $g\geq4$ and let $F$ be finite. The map $J_{\rm path}$ of Definition~\ref{def:ordinary-summands} is injective, and \[ \operatorname{im}J_{\rm path}\ \cap\ \operatorname{im}(\Gamma_D,\Theta,2\Pi_{\rm par})=0. \] \end{lemma} \begin{proof} \smallskip\noindent\emph{An open isolating one path.} Fix a path $\Gamma$ with center markings $A$ and leaves $(r,B_1),(s,B_2)$. Since $r+s=g-1\geq3$, at most one leaf has genus one, and leaves of equal genus have genus at least two; such leaves are distinguished by their labels or are both unmarked. Lemma~\ref{lem:positive-path-detection} therefore applies; we use its open $O_\Gamma$, product $Z_\Gamma=Y_0\times Y_1\times Y_2$ and embedding $i_\Gamma$. By its support assertions, every other path class with positive-genus leaves restricts to zero on $O_\Gamma$. Theorem~\ref{thm:cited-thirteen-cohomology} expresses each elliptic $H^{13}$ class by rational-tail Gysin classes. After the outer attachment, every $\Gamma_D$ class is supported on a two-edge tree with a rational vertex. Hence $\Gamma_D$ also restricts to zero. The same holds for $\Pi_{\rm par}$, since a contraction of a graph whose edges are bridges or self-loops cannot have two parallel edges. \smallskip\noindent\emph{Separation from the Picard Gysin images.} A genus-one tail divisor meets $O_\Gamma$ only by cutting off the genus-one leaf, when one exists. Suppose it is $Y_1$, so that $Y_2$ has genus $g-2\geq2$. Its restricted normalization is a regular divisor embedding $k$, through which $i_\Gamma$ factors. If $\beta$ is its degree-eleven leaf class and $\eta_{02}\in H^2(Y_0\times Y_2)$ is the restricted partner class, self-intersection gives, on $Z_\Gamma$, \[ i_\Gamma^*k_*(\beta\otimes\eta)= -(\psi_{p_1}+\psi_{q_1}) \operatorname{pr}_1^*\beta\,\operatorname{pr}_{02}^*\eta_{02}. \] For a K\"unneth component of $\eta_{02}$ of degrees $(d_0,d_2)$, where $d_0+d_2=2$, the two terms have respective degrees \[ (d_0+2,11,d_2),\qquad(d_0,13,d_2) \] in the fixed order $(Y_0,Y_1,Y_2)$. Their projection to degrees $(11,2,2)$ is zero. If both leaves have genus at least two, the supports of the Picard Gysin images are disjoint from the open. \smallskip\noindent\emph{Recovery of the path coefficient.} By Lemma~\ref{lem:positive-path-detection}, the $(11,2,2)$ component of the self-intersection of a center coefficient is injective in that coefficient; when $|\Aut_F(\Gamma)|=2$ the ordinary map is $\frac12\xi_{\Gamma*}$ on the invariant coefficient and the lemma detects it on the cover. Apply this detector to $J_{\rm path}(t)+\Gamma_D(d)+\Theta(w)+2\Pi_{\rm par}(\rho)=0$. The support conditions imply vanishing of the other path contributions, the elliptic degree-thirteen terms, and the parallel terms; the fixed K\"unneth projection vanishes on the Picard terms. This injectivity then forces the $\Gamma$ coefficient of $t$ to vanish. Repeating for all paths gives $t=0$, proving both assertions. \end{proof} \begin{lemma}\label{lem:ordinary-image-tail-parallel-injectivity} Let $g\geq4$ and let $F$ be finite. The map \[ (\Gamma_D,\Theta,2\Pi_{\rm par}): (\mathsf D_{g,F}\oplus\mathsf W_{g,F}\oplus\mathsf R_{g,F}) \otimes\Pmot\longrightarrow H^{15}(\Mbar_{g,F}) \] is injective. \end{lemma} \begin{proof} (We suppress the common Tate factors.) \smallskip\noindent\emph{The rooted detectors.} Put $h=g-1$. For $A\subseteq F$ with $|A|\geq10$, let \[ i_A:Y_A=\Mbar_{1,A\cup\{p\}}\times \Mbar_{h,(F\setminus A)\cup\{q\}}\longrightarrow\Mbar_{g,F} \] be the labelled separating clutching. It is a regular divisor embedding. Indeed, a node with this genus and marking partition is unique. Two such genus-one sides would contain the same nonempty marking set, hence be nested; their intervening region would be an unmarked rational tree with two exterior flags, excluded by stability. The normal class is $-\psi_p-\psi_q$. Define \[ d_A:H^{15}(\Mbar_{g,F})\longrightarrow H^{11}(\Mbar_{1,A\cup\{p\}})\otimes Q^4_{h,(F\setminus A)\cup\{q\};q} \] by $i_A^*$, the K\"unneth projection to $(11,4)$, and the quotient on the second factor. Let $C\subseteq F$ index another elliptic tail. Distinct nodes of types $(1,A)$ and $(1,C)$ meet transversely. After contracting all other edges, the possibilities are the three paths corresponding to $C\subsetneq A$, $A\subsetneq C$, or $A\cap C=\emptyset$. Partial overlap without containment is impossible: separating partitions are compatible, and two genus-one sides cannot cover a curve of genus at least four. Further degenerations are already included in these normalization products. For $\Theta_C(\alpha\otimes\eta)$, the diagonal self-intersection has $(11,4)$ part $-\alpha\otimes\psi_q\eta$. If $C\subsetneq A$, the first factor receives the inner Gysin and has degree at least thirteen. If $A\cap C=\emptyset$, the degree-eleven decoration stays on the second factor and the first receives degree at most two. Both cases disappear under the projection. If $A\subsetneq C$, the intersection on the partner is the $q$-containing rational-tail map $j_{C\setminus A}$. Its surviving class is the pushforward of a degree-two class on that normalization, hence belongs to $\mathcal N_{\rm tail}$. For $\Gamma_{D,C}(z)$, with $z$ the elliptic $H^{13}$ class, the diagonal terms have degrees $(15,0)$ and $(13,2)$. The cases $C\subsetneq A$ and $A\cap C=\emptyset$ disappear by the same degree bounds. If $A\subsetneq C$, only the $(11,2)$ part of the restriction of $z$ to the elliptic factor and rational middle can survive. Its degree-two rational class is pushed by $j_{C\setminus A}$ and again belongs to $\mathcal N_{\rm tail}$. Thus \[ d_A\Gamma_D=0,\qquad d_A\Theta_C=0\ (C\ne A),\qquad d_A\Theta_A(\alpha\otimes\eta)=-\alpha\otimes[\psi_q\eta]. \] \smallskip\noindent\emph{Parallel pullback and its factor.} On the source of $\Pi_{{\rm par},C}$, the two selected parallel edges remain nonseparating under every degeneration. A node of type $(1,A)$ must therefore be an additional node in a source factor. If it separates that factor's two selected flags, it becomes nonseparating after gluing. Otherwise the genus-one side must be the side away from the cycle. When this side lies in the genus-$(g-2)$ factor, the hook remains on the other side and the $(11,4)$ projection is zero. The remaining cases have $A\subseteq C$ and the fibre product component \[ \Mbar_{1,A\cup\{p\}}\times \left[ \Mbar_{0,(C\setminus A)\cup\{q,u,v\}}\times \Mbar_{g-2,(F\setminus C)\cup\{u',v'\}}/\Sn_2 \right]. \] The new node is distinct from the two selected nodes, so the intersection is transverse, of multiplicity one, with no excess line. Let \[ \rho_{C,A}:K_{C\cup\{u,v\}}^{\langle(u\,v)\rangle} \longrightarrow K_{A\cup\{p\}} \] be the rational-tail map of Definition~\ref{def:rational-tail-hook-map} that collapses $(C\setminus A)\cup\{u,v\}$ to $p$. By Proposition~\ref{prop:eleven-hook-realizations} this is the actual boundary restriction. Base change gives \[ d_A\Pi_{{\rm par},C}(\beta)= \begin{cases} \rho_{C,A}(\beta)\otimes[\zeta_{C\setminus A}],&A\subseteq C,\\ 0,&A\not\subseteq C. \end{cases} \] Here $\rho_{C,A}$ acts on the hook factor and the common $S$ is understood. Both the parallel source and the class $\zeta_{C\setminus A}$ divide the labelled pushforward by the same simultaneous flag exchange of order two. Thus this formula has coefficient one; the coefficient two in the map of the lemma remains an explicit factor two. The map $\rho_{C,C}$ is an isomorphism. To check the map itself, put $\mathscr A_I=\ker(\mathbb Q^I\xrightarrow{\sum}\mathbb Q)$. The tuple presentation identifies \[ K_I=\bigwedge^{10}\mathscr A_I. \] The alternating eleven-tuple relation verifies this identification, and the anchored tuples give a basis on both sides. Identifying $e_u,e_v$ with $e_p$ maps the invariant subspace of $\mathscr A_{C\cup\{u,v\}}$ isomorphically to $\mathscr A_{C\cup\{p\}}$. Its kernel is the anti-invariant line $\mathbb Q(e_u-e_v)$. Taking tenth exterior powers, the invariant part contains no copy of this line, since its square is zero. The induced map is therefore an isomorphism. On tuples it replaces a unique $u$ or $v$ by $p$ and kills a tuple containing both, exactly as $\rho_{C,C}$ does. This also shows that the invariant source vanishes for $|C|<10$. \smallskip\noindent\emph{Elimination of the Picard and parallel terms.} A linear relation among the images of the three Gysin maps gives, after applying $d_A$, \[ -(\operatorname{id}\otimes(\psi_q\cdot))w_A+ 2\sum_{C\supseteq A}\rho_{C,A}(\beta_C) \otimes[\zeta_{C\setminus A}]=0. \] Tensor Lemma~\ref{lem:rooted-degree-four-detection} with $H^{11}(\Mbar_{1,A\cup\{p\}})$. Different $C\supseteq A$ give different parallel columns, so $w_A=0$ and every $\rho_{C,A}(\beta_C)=0$. Taking $A=C$ and using the isomorphism just proved gives $\beta_C=0$. This covers every nonzero hook summand in the source. The relation is now $\Gamma_D(d)=0$. \smallskip\noindent\emph{Simultaneous detection of the elliptic classes.} Write this last relation as \[ \sum_{C\subseteq F}i_{C*}(z_C\otimes1)=0,\qquad z_C\in H^{13}(\Mbar_{1,C\cup\{p\}}). \] Suppose some $z_C$ is nonzero, and choose $A$ of smallest cardinality among its nonzero indices. Definition~\ref{def:low-genus-thirteen-modules} gives $|A|\geq11$. Let $W_A\subseteq\Mbar_{g,F}$ be the open of Lemma~\ref{lem:minimal-elliptic-tail-detection} for the partner genus $h=g-1\geq3$ and the decomposition $F=A\sqcup(F\setminus A)$. By that lemma, a genus-one tail of type $C$ meets $W_A$ only if $C\subseteq A$, so minimality of $A$ reduces the restricted relation to \[ j^{\mathrm{ct}}_{A*}(z_A|_{\mathrm{ct}}\otimes1)=0\quad\text{in }H^{15}(W_A), \] and the $(13,2)$ component of its self-intersection, $-z_A|_{\mathrm{ct}}\otimes\psi_q$, depends injectively on $z_A$, a contradiction. \end{proof} \begin{theorem}[Structure of the ordinary image]\label{thm:ordinary-image} For every $g\geq4$ and finite $F$, $\Omega_{g,F}$ is injective and its image is $\mathcal I_{g,F}$. Set $t_g=\lfloor(g-2)/2\rfloor$ and let $\epsilon_g$ be one for odd $g$ and zero for even $g$. The characteristic series of its four rational multiplicity summands, over $F=[n]$, are \[ \begin{aligned} \sum_n\ch_n(\mathsf D_{g,[n]})&=E\partial U,\\ \sum_n\ch_n(\mathsf W_{g,[n]})&=H(2E+(g-1)E^2),\\ \sum_n\ch_n(\mathsf T_{g,[n]})&= t_g(H+G)E^2+\epsilon_g(Hh_2[E]+Ge_2[E]),\\ \sum_n\ch_n(\mathsf R_{g,[n]})&=HE. \end{aligned} \] \end{theorem} \begin{proof} \smallskip\noindent\emph{Generation by the four Gysin images.} Proposition~\ref{prop:h13-input} gives the complete one-edge source. Its mixed K\"unneth terms are $H^{11}$ on a genus-one vertex and $H^2$ on its genus-$(g-1)$ partner. Divisor generation gives the full Picard source at that partner. A divisor omitted from $\Pi^\circ$ is $\delta_{0,\{q\}\cup B_0}$ for some nonempty $B_0\subseteq F\setminus A$. Its composite is the two-edge path with genera $1,0,g-1$ and the hook on its genus-one endpoint. Contracting the edge adjacent to that endpoint first gives an actual elliptic $H^{13}$ class, followed by the outer attachment. Thus its image lies in $\operatorname{im}\Gamma_D$. The retained Picard generators give $\operatorname{im}\Theta$. The separating $H^{13}$ sectors with a decorated genus-one vertex give $\Gamma_D$ directly. For every decorated vertex of genus at least two, Theorem~\ref{thm:cited-thirteen-cohomology} expresses its class as a sum of images under elliptic-tail Gysin maps. A separating outer edge then gives a two-edge tree with one hook-decorated genus-one vertex. If this vertex is an endpoint and the middle vertex has positive genus, contracting the other edge first expresses the class in the full Picard source, hence in $\operatorname{im}(\Gamma_D,\Theta)$. If the middle vertex is rational, contracting its edge to the decorated endpoint gives $\Gamma_D$. If the decorated vertex is the center, a rational leaf can likewise be contracted first to give $\Gamma_D$; two positive leaves give $J_{\rm path}$. These exhaust the tree cases. For a nonseparating outer edge, expand $H^{13}(\Mbar_{g-1,F\cup\{u,v\}})$ by the same theorem. If neither of $u,v$ is on the decorated elliptic vertex, the resulting loop is on its partner, giving the Picard decoration $\delta_{\mathrm{irr}}$. If both are on the elliptic vertex, first glue that loop to obtain an actual genus-two $H^{13}$ class. Expanding this class by the genus-two instance of the theorem reduces to the tree cases just listed. If exactly one of $u,v$ is on the elliptic vertex, the two edges are parallel, giving $\Pi_{\rm par}$. Here are the quotient factors in these reductions. All compositions on labelled products agree by functoriality of Gysin. A loop is divided by its reversal of order two. For a path with equal unmarked leaves, the labelled two-edge map $f_\Gamma$ satisfies $J_{\rm path}=\frac12f_{\Gamma*}$ on the invariant hook. A labelled one-edge factorization yields $f_{\Gamma*}$, and one half of that coefficient therefore represents $J_{\rm path}$. Otherwise a labelled path has trivial label-fixing group. For the parallel case, the two inner elliptic-tail coordinates containing exactly one outer node are exchanged by the outer loop flip. An invariant pair has the form $x+\tau x$, with $\tau$ the flag exchange. The outer factor $1/2$ makes its composite the labelled parallel pushforward of $x$. That pushforward vanishes on the anti-invariant part; on $x^+=(x+\tau x)/2$ it is \[ f_{\mathrm{par},*}(x^+)=2\Pi_{\rm par}(x^+). \] Thus the remaining parallel source is the ordinary invariant hook, with precisely the coefficient two used in $\Omega_{g,F}$. Every map defining $\Omega_{g,F}$ factors through an ordinary one-edge source by one of these contractions. Conversely, the preceding cases exhaust that source and place its image in $\operatorname{im}\Omega_{g,F}$. Consequently $\operatorname{im}\Omega_{g,F}=\mathcal I_{g,F}$. \smallskip\noindent\emph{Simultaneous injectivity.} Lemma~\ref{lem:ordinary-image-path-separation} shows that the path coefficient vanishes in every linear relation among the four Gysin images. Lemma~\ref{lem:ordinary-image-tail-parallel-injectivity} then shows that the remaining three coefficients vanish. Thus $\Omega_{g,F}$ is injective. The sources are the actual degree-thirteen, hook, and divisor multiplicity spaces; their maps are induced by ordinary Gysin pushforwards along algebraic clutching morphisms. They commute with relabelling, and the two codimension-one twists, or the single twist on a degree-thirteen source, give the common factor $\Pmot=\Lmot^2\otimes S$. \smallskip\noindent\emph{Character operations and the two hook restrictions.} Summing a tensor product over decompositions of a label set corresponds to induction from the product of the two symmetric groups, hence to multiplication of Frobenius series. The unit decoration on an arbitrary label set has series $E$. Fixing one additional flag corresponds to $\partial=\partial/\partial p_1$: in the power-sum formula, differentiation removes a fixed point and replaces its centralizer factor by that for the remaining cycle type. With $\mathscr A_a=\mathscr A_{[a]}$ as in the proof of Lemma~\ref{lem:ordinary-image-tail-parallel-injectivity}, the tuple identification used there gives the following restrictions of the hook: \[ \begin{aligned} K_{[a]\cup\{p\}}&\cong \bigwedge^{10}\mathscr A_a\oplus\bigwedge^9\mathscr A_a,\\ \operatorname{Res}^{\Sn_{a+2}}_{\Sn_a\times\Sn_2} \mathscr A_{a+2} &\cong \mathscr A_a\oplus\mathbf1\oplus\operatorname{sgn}_2. \end{aligned} \] The second decomposition follows by separating the two augmentation spaces of the ordinary-label block and the flag block; the remaining augmentation-zero direction is constant on each block. Taking its tenth exterior power shows that the plus and minus flag eigenspaces are \[ \begin{array}{c|c|c} &\text{representation on }[a]&\text{series over }a\\ \hline +&\bigwedge^{10}\mathscr A_a\oplus\bigwedge^9\mathscr A_a&H\\ -&\bigwedge^9\mathscr A_a\oplus\bigwedge^8\mathscr A_a&G . \end{array} \] Here $\bigwedge^j\mathscr A_a=V_{(a-j,1^j)}$ when nonzero. At $a=0$ all the hooks involved vanish; the formulas also vanish below their stated lower arities. \smallskip\noindent\emph{The elliptic degree-thirteen summand.} Definition~\ref{def:low-genus-thirteen-modules} gives the series \[ \sum_m\ch_m(U_{1,[m]}) =Es_{(2,1^{10})}+E_{\geq3}H=U. \] Fixing its attaching flag and allocating arbitrary complementary ordinary labels to the unit partner therefore gives \[ \sum_n\ch_n(\mathsf D_{g,[n]})=E\partial U. \] \smallskip\noindent\emph{The Picard summand.} Fix $h\geq3$ and write the partner markings as $B\cup\{q\}$. By Corollary~\ref{cor:ordinary-image-rooted-picard-basis}, we count the specified divisors as a basis. The three classes $\kappa_1,\psi_q,\delta_{\mathrm{irr}}$ give $3E$. The classes $\psi_i$, for $i\in B$, give $h_1E$. For a separating divisor, the side containing $q$ distinguishes the two sides even when the genera agree. There are $h-1$ choices of positive genera on both sides, each giving $E^2$. The remaining retained case has genus $h$ on the $q$-side and a rational opposite side with at least two ordinary labels, giving $EE_{\geq2}$. Hence \[ \sum_{b\geq0}\ch_b(\Pi^\circ_{h,[b]\cup\{q\};q}) =3E+h_1E+(h-1)E^2+E(E-1-h_1)=2E+hE^2. \] Taking $h=g-1$ and multiplying by the one-flag hook gives \[ \sum_n\ch_n(\mathsf W_{g,[n]})=H(2E+(g-1)E^2). \] \smallskip\noindent\emph{The positive paths.} There are $t_g$ unordered distinct positive-genus pairs $r