Proposition: Existence and admission of the one-edge outputs

All three functionals in (eq:oneedges) exist and belong to ${\mathcal{P}}_{\Sigma }$. They are homogeneous of field degree four.

Proof. Each factor in (eq:oneedges) is already admitted. Classical pointwise closure in IX-A gives the products and their full admission criterion, including directional descendants IXA (thm:closure). The degree is the sum of the two residual degrees. No new singular-variable multiplication is used in this argument.