Proposition: Existence of the three canonical outputs

These are well-defined compact distributions on all of $X$. The labels in (eq:twolab) now define actual quadratic functionals $G_{12,23},G_{12,13},G_{13,23}$. Their compact field-variable coefficient kernels exist.

Proof. Apply Lemma lem:forest to leaves $x,x^{\prime }$, then $y,x^{\prime }$, then $x,y$.

For the scalar-only output, its coefficient kernel is

For $G_{12,13}$, first form

then transpose $L$ in the $x^{\prime }$-variable and symmetrize the metric slots. This transposition is defined: applying $L$ to a smooth joint test in one variable is continuous in the joint smooth topology, including parameter derivatives. Its transpose therefore acts on compact distributions. The third output uses the corresponding first-variable transpose. Finite propagation confines all field supports to the retained compact laboratory.