Proposition: Reverse coefficient and its sign

The reverse first-stage coefficient is

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

Proof. The locked reverse covariance is the conjugate mixed kernel. The middle observable has two metric slots, giving factor $2$; the single sharp insertion supplies one minus sign. Conjugation of (eq:Q) gives the displayed reversal. Admission follows from the carrier's conjugation closure.