Hadamard contractions preserve microcausal polynomial functionals

Fix a contraction order $r\ge 1$, a field $\varphi$, and put $A=F^{(r)}(\varphi )\otimes G^{(r)}(\varphi )$ and $B=W^{\otimes r}$. By [F3], each wavefront component of $B$ is either zero or a Hadamard pair $(k_i,-{\mathrm{\ell }}_i)$ with both $k_i,{\mathrm{\ell }}_i$ future null. If a nonzero covector of $\mathrm{WF}(A)$ cancelled one of $B$, then in the $F$ block every contracted entry would be past null or zero and in the $G$ block every contracted entry would be future null or zero. Both full tuples would lie in the forbidden cones of [F1], contrary to microcausality. Hence $\mathrm{WF}(A)$ has no covector opposite to $\mathrm{WF}(B)$, and [F3] defines $A\cdot B$. Its support is compact because it is contained in $\mathrm{supp}A$.

Push the product distribution forward over the $r$ contracted pairs, integrating against ${\mu }_g^{\otimes 2r}$. The projection is proper on its compact support, so [F3] makes the result a compactly supported distribution in the remaining variables. For each homogeneous term, the two original compact coefficient kernels satisfy the microcausal condition, since $F^{(m)}(0)=m!f_m$ and likewise for $G$. The contracted pairs are disjoint singular cross edges between those two blocks; all remaining field factors are smooth. Thus [F4] proves that the full product and each intermediate integration give this same canonical kernel. Differentiating this finite expression in $\varphi$ uses the finite external-slot Leibniz rule of [F4]: for any order $s$, each term assigns a subset of the $s$ labeled directions to $F$ and the rest to $G$, producing contractions of $F^{(r+a)}(\varphi )$ and $G^{(r+s-a)}(\varphi )$ with the same $W^{\otimes r}$. This defines every output derivative as a finite sum of the same proper pushforwards. Since the polynomial degrees are finite, all contractions with $r>\min (\mathrm{deg}F,\mathrm{deg}G)$ vanish and the displayed sum is finite.

Fix an output derivative order $s\ge 1$ and one term from step 2.1, with $a$ external directions assigned to $F$ and $s-a$ to $G$. Put $A=F^{(r+a)}(\varphi )\otimes G^{(r+s-a)}(\varphi )$ and $B=W^{\otimes r}$. Suppose a nonzero wavefront covector of the pushed-forward term were entirely future causal. By [F3] it lifts to a covector of $A\cdot B$ with zero covectors in all contracted directions. The product wavefront bound in [F3] has three cases. A contribution from $\mathrm{WF}(A)$ alone would give an all-future nonzero tuple in one of the two functional derivatives, forbidden by [F1]; a contribution from $\mathrm{WF}(B)$ alone has zero external covectors and cannot give the nonzero output. In a sum contribution from $\mathrm{WF}(A)+\mathrm{WF}(B)$, the vertical zero condition makes each contracted $F$ entry past null or zero and each contracted $G$ entry future null or zero. The external entries of both blocks are the assumed future output entries. The full $G$ tuple is therefore nonzero and all future, forbidden by [F1]. If the output covector were all past, the same vertical equations make the full $F$ tuple nonzero and all past. Thus every output derivative kernel avoids both forbidden cones. The case $r=0$ is covered by the $\mathrm{WF}(A)$-alone case.

If $K$ is smooth, multiplying $F^{(2)}(\varphi )$ by $K$ is allowed by the smooth-factor case of [F3] and adds no wavefront directions. Pairing the result against the cutoff metric density $\chi {\mu }_g^{\otimes 2}$ is defined because the distribution has compact support. For each output derivative, the corresponding wavefront covectors of $F$ have zero entries in the two contracted slots; if the output had an all-future or all-past nonzero covector, adjoining those zeros would put $F^{(s+2)}(\varphi )$ in a forbidden cone from [F1]. Hence ${\mathrm{\Gamma }}_{K}F$ is microcausal. Its degree is at most $\mathrm{deg}F-2$ because two functional derivative slots were contracted; constants and degree-one functionals give zero.