Theorem: Quantitative compact-test construction

There is a finite constant $C$, depending only on the fixed support geometry, $m,T,R$, and a fixed spatial cutoff used in Appendix app:PDE, such that

The derivative orders are sufficient and are not claimed optimal. Consequently the map to $v$ is continuous in the stated input controls, uniformly on bounded families.

Proof. The detailed Cauchy and advanced estimates are proved in Appendix app:PDE. They bound $u_f$ through four derivatives by $C{p}_6(f)$, and $w[h]$ on $K$ through three derivatives by $C{H}_6(h)$. Applying the product rule to (eq:q) gives the first estimate. Lemma lem:mass-shell, with common compact support $K\cup L$, gives the second.