Proposition: Reverse coefficient and its sign

The reverse ordered coefficient is

It belongs to ${\mathcal{P}}_{\Sigma }$, and

Proof. The real tensor coefficients in $H_{hh}$, together with $\overline{{\Delta }_0^{+}(z)}={\Delta }_0^{+}(-z)$, give reversal of the mixed covariance by kernel conjugation. The real zero-Cauchy operator preserves this relation. Therefore the reverse mixed component is $\overline{K(x^{\prime },y)}$.

The two slots of ${h_{00}}^2$ supply multiplicity $2$. There is one insertion $B_{f^{\prime }}$, so its defining minus sign gives $-2$. This proves (eq:QR). The formula immediately gives (eq:Qadjoint). Admission follows from IX-A's selected reverse-output result, or from its conjugation closure applied to (eq:Qadjoint) IXA (prop:knownoutputs, item 3; thm:closure).