Proposition: The cochain kernel and admission

The family (eq:C-definition) is even, has ghost number zero, is linear in $\xi$, and has field degree $(1,1)$ in $(\gamma ,\varphi )$. Its unsymmetrized kernel is

where $y$ is the auxiliary slot replaced by the metric slot. After normalized species symmetrization it is compact and satisfies $\mathrm{WF}(c_{\xi })\subset Z_2$. Hence ${\mathcal{C}}_{\xi }\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$.

On the complete free shell,

Proof. The source functional is bilinear and parameter-linear. Replacing its first smooth argument by $D\gamma$ gives the transpose formula (eq:C-kernel). A differential operator does not enlarge support or wavefront set. The finite species symmetrization preserves the sum-zero bound.

A nonzero tuple in either all-future or all-past closed causal cone cannot have zero covector sum. The bound $Z_2$ therefore implies the inherited microcausal condition Hormander,XIIA. The field species are even and have ghost number zero.

On ${\mathcal{C}}_{\mathrm{sol}}$, the equation $b=D\gamma$ holds. Substitution gives (eq:C-cancel).