Proposition: The admitted partial coefficient

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

It is an actual compact-distribution bilinear functional and belongs to ${\mathcal{P}}_{\Sigma }$. Thus

is a well-defined specified first-stage output. This does not define a quantum product on the full carrier.

Proof. Existence and full carrier admission of the selected contraction output are the input in Assumption ass:carrier, inherited from IX-A-ST. Multiplying by fixed smooth compact tests preserves its compact coefficient support.

For its normalization, differentiation of $W_g$ supplies the factor $2$. The tensor identity in Assumption ass:tensor selects $K(x,y){\partial }_t\varphi (x)$. There is one defining minus sign from $B_f$, giving $-2$. Classical closure supplies $B_f{W}_g$, so the stated formal first-stage polynomial is well defined.