Proposition: Actual two-edge distributions

Each product in (eq:E1223)–(eq:E1323) exists canonically as a distribution on all of $X$. Each has assignment multiplicity $2$, positive insertion sign, and residual field degree two.

Proof. For (eq:E1223), the leaves $x,x^{\prime }$ give the product criterion. For (eq:E1213), the leaves are $y,x^{\prime }$. For (eq:E1323), they are $x,y$. Lemma lem:forest applies in each case.

Both sharp insertions contribute a minus sign, whose product is positive. There are two assignments of the middle metric slots, hence the displayed factor $2$, not $4$. Two edges remove four slots from the initial six.