The complete grouped species sum

For a block-diagonal even line $L$, define ${\mathrm{Contr}}_L^{\gamma \varphi }$ and ${\mathrm{Contr}}_L^{\varphi \varphi }$ to mean the two-slot contractions using, respectively, $L_{\gamma \gamma }\otimes L_{\varphi \varphi }$ and $L_{\varphi \varphi }\otimes L_{\varphi \varphi }$, with the indicated derivative blocks.

Where the contractions are defined, species expansion of (eq:grouped) gives

On the unresolved domain this is a formal specification only.

The factor $i$ in the mixed channel includes its two equivalent slot orderings. There is no two-metric channel because ${{\mathcal{V}}_g^{\prime \prime }}_{\gamma \gamma }=0$. No metric–scalar line has been added.

Equation (eq:species), together with (eq:VHessian)–(eq:OHessian), is the complete tensor-valued algebraic sum. It does not replace the requested microlocal composition theorem.