Theorem: Admission of all three two-edge outputs

Every output in Proposition prop:existence belongs to ${\mathcal{P}}_{\Sigma }$, including all metric admission tests after arbitrary finite smooth directional insertions.

Proof. The scalar-only output has no metric derivatives.

For $G=G_{12,13}$, set

These expressions denote proper distributional pushforwards. Lemma lem:forest makes them compact smooth functions. Direct differentiation gives the actual linear metric kernel

It lies in ${\mathcal{M}}_1+{\mathcal{Q}}_{\Sigma }$.

A first metric directional insertion replaces $h$ and $w[h]$ in these formulae by the fixed direction and its smooth response. The same decomposition holds. Two directional insertions into the quadratic functional give a constant; a further metric derivative is zero. Scalar directions have no effect. This checks every finite direction list.

For $G_{13,23}$, integrate its tree to $y$ and to $x$ instead of $y$ and $x^{\prime }$, and repeat the same argument.