Theorem: Polynomial smoothing growth of the complete contraction

For every compact $L⋐U$, there are $C_L>0$ and a continuous coefficient seminorm $p_{L,4}$ such that

Here $q$ is any fixed order in (eq:cross-order).

The estimate applies to arbitrary joint tests meeting $N$, to the complete finite vertex operation, and in particular to the named retained sum $K_{\mathrm{ret}}$. It proves a finite order-five, polynomial-growth bound. It does not prove a bound independent of $\delta$.

Proof. Apply the order-four internal estimate (eq:internal-order) to the smooth test appearing in (eq:vertex-test).

Four derivatives of $gZ(C_{\delta }){\partial }_y\Phi$ require at most five derivatives of $C_{\delta }$ and at most five derivatives of $\Phi$. The mass contribution needs at most four derivatives of $C_{\delta }$, and is bounded by the same right-hand side. The fixed linear map $\mathfrak{f}$, metric coefficients, smooth $g$, and finite component sums contribute fixed constants.

Lemma lem:smooth-bounds, with $r=5$, gives ${\delta }^{-8-q-5}={\delta }^{-q-13}$. All internal blocks are included in the reconstruction estimate, so this is an estimate of the complete regularized expression. No cancellation or branch nonvanishing assumption is used.