Pardon GW pushforward and J-independence

The domain stabilization and original marked evaluations define a continuous map $\tau$ to ${\bar{\mathcal{M}}}_{g,n}\times X^n$: forgetting/stabilizing is the actual domain morphism in [F2], while the marked values converge in its graph topology, including labels migrating to bubbles or transferred through a contracted ghost. By [F1] the constructed virtual functional exists. Define its pushforward by $G{W}^P(J)(a)=[\bar{\mathcal{M}}(X,J){]}^{\mathrm{v}\mathrm{i}\mathrm{r}}({\tau }^{\ast }a)$, exactly the cohomological pullback convention of [F5].

Choose the compatible reference $J_{\ast }$ from [F4]. Its eigenspace decomposition is $T_{\mathbb{C}}X=E_{\ast }^{+}\oplus E_{\ast }^{-}$. The Hermitian form $H(u,v)=-i\omega (u,\bar{v})$ has signature $(m,m)$, positive on $E_{\ast }^{+}$ and negative on $E_{\ast }^{-}$. For any tame $J$, its $i$-eigenspace $E_J^{+}$ is positive: writing $u=v-iJv$ gives $H(u,u)=2\omega (v,Jv)>0$. Projection to $E_{\ast }^{+}$ is therefore injective and, by equal dimension, an isomorphism. Thus $E_J^{+}$ is the graph of a smooth bundle map $A:E_{\ast }^{+}\to E_{\ast }^{-}$ with operator norm $\mathrm{\parallel }A\mathrm{\parallel }<1$. Conversely any such graph is positive, its conjugate is negative and complementary, and the two eigenspaces define a smooth real tame $J$.

Interpolate the graph maps of any two endpoints by $A_t=(1-t)A_0+t{A}_1$. The open unit operator-norm ball is convex, and compactness of $X$ gives a uniform bound below one. Hence the graph construction of step 2.1 gives a smooth tame joining path. Composing $t$ with a smooth cutoff constant near zero and one makes it constant in both endpoint collars, without altering the endpoints. This proves the required tame-path input rather than assuming compatible endpoints.

Apply [F3] to this path. Its base is compact by the uniform varying-domain result [F2]; its boundary orientations give endpoint $1$ minus endpoint $0$, and its endpoint atlas comparisons are the actual collar/common-refinement comparisons. Stabilization and evaluation give one continuous map $F$ on this entire bordism to the same target ${\bar{\mathcal{M}}}_{g,n}\times X^n$, restricting to the two $\tau$ maps.

For any closed degree-$d$ target cohomology class $a$, the boundary connector in [F1] gives $[\mathrm{\partial }W{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}(F{\mathrm{\mid }}_{\mathrm{\partial }W}^{\ast }a)=[W{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}(\delta F{\mathrm{\mid }}_{\mathrm{\partial }W}^{\ast }a)=0$, because restriction followed by the coefficient-sequence connector is zero. The signed boundary in step 4.1 is exactly $G{W}^P(J_1)(a)-G{W}^P(J_0)(a)$. This proves equality as rational Čech-cohomology-dual functionals for all inputs of the original Statement. The atlas and virtual cochains were constructed before this bordism argument; no additional descendant or degeneration formula is inferred.