Proposition: Sign and multiplicity of the admitted partial output

The selected one-contraction coefficient for the ordered pair $(B_f,W_g)$ is

It is an actual continuous bilinear functional with a compact distributional kernel and belongs to ${\mathcal{P}}_{\Sigma }$. Its admission is supplied by IX-A; (eq:Qpartial) fixes the previously unspecified combinatorial coefficient and sign.

Proof. IX-A's selected contraction-output proposition, item 3, establishes existence and admission of this two-factor coefficient IXA (prop:knownoutputs, item 3). It remains to identify the coefficient in that admitted expression.

The metric derivative of $W_g$ supplies the factor $2$ from its two metric slots. The inherited identity (eq:tensor) selects the $0$-component of $Lh$ when contracted with $h_{00}$, giving $K(x,y){\partial }_t\varphi (x)$. There is exactly one insertion $B_f$ in this partial coefficient, so its single defining minus sign gives the sign $-2$. This proves (eq:Qpartial) without re-proving admission.