Lemma: Crossed order bound

On compact sets, $W_m,D_{T,m}$ have order at most four. The compact arrays $\chi C_{\pm }$ have order at most eight, with an overall factor ${\hslash }^2$.

Proof. For a compact smooth test $\psi (t,\mathbf{x})$, spatial integration by parts gives

uniformly on its bounded time support. Both oscillatory time factors have absolute value one. Hence their pairings are bounded by

The integral converges, including at massless zero momentum.

For the tensor products, perform the integration by parts in each spatial variable separately. This uses at most eight total derivatives and gives an integrable double-momentum majorant. Fixed cutoff derivatives and finite tensor factors preserve the order bound.