Lemma: Explicit test-order bound

For $\Phi \in {\mathcal{D}}_A(X)$,

with a combinatorial constant depending only on dimension and $d$. Differential operators and multiplication do not enlarge the test support.

Proof. Apply every derivative of order at most $64$ to the finite sum defining $D^{\mathrm{t}}(a\Phi )$, then use the product rule on $b_{\beta }a\Phi$. The total number of derivatives distributed among the three factors is at most $64+|\beta |\le 64+d$. Taking suprema and summing the finitely many multinomial coefficients gives (eq:leibniz).