Theorem: Filtered compact packet bridge

The compact test coefficient $l_L=k(b_L)$ satisfies

For $\omega \in {\mathfrak{S}}_C$,

Proof. For any free field solution, integration by parts gives

The conserved Cauchy symplectic pairing and $\int {\eta }_T^{\prime }=1$ make this identity exact. Applying $-{{\partial }_t}^2$ to the test multiplies its coefficient by ${\omega }^2$.

The coefficient difference is therefore

Use $|z-\mathrm{i}\omega w{|}^2\le 2|z{|}^2+2{\omega }^2|w{|}^2$ and $1+2n_{\mathrm{H}}\le 1+2/(\beta \omega )$. Spectral unitarity gives (eq:packeterror).

The cutoffs converge in $H^2$ for $p$ and $H^3$ for $q$. This follows from derivative tails and ${{\partial }_x}^j\chi (x/L)=O(L^{-j})$. The bounded smooth potential gives convergence of all displayed graph norms. For an ordinary-derivative bound one can use

and ${{\delta }_L}^2\le (1+2/\beta )(D_p+D_q)$. The fractional-power inequalities follow from ${\lambda }^{3/2},\lambda \le 1+{\lambda }^2$ and ${\lambda }^{5/2}\le 1+{\lambda }^3$. Finally apply Lemma lem:weylcontinuity.