Lemma: Crossed derivative placement

On either $47$ or $5+$, every term in the exact differentiated seed expression has crossed derivative order at most three. The $t$-derivatives in the seed operators never differentiate the crossed factor.

Proof. For $47$,

On the crossed factor,

Each printed operator term has one ${\partial }_s$ and one $\mathcal{B}$. The kinetic vertex adds one crossed derivative.

For $5+$,

Thus ${\partial }_t=0$ and ${\partial }_s=-{\partial }_2$. There are two possible ${\partial }_s$-actions on the crossed factor, and the kinetic vertex adds one. All other derivatives act on coefficients or joint tests.