Proposition: Bounded-aperture generator remainder

In a model with bounded self-adjoint position operator $\Vert \hat{x}\Vert \le L$,

Proof. Evenness writes the generator as $\int [\cos (k_e{\mathrm{ad}}_x)-\mathrm{id}]Q_{r_{\mathrm{c}}}(\mathrm{d}k)$. The cosine family is the average of two unitary-conjugation trace-norm isometries. Taylor's integral remainder after the quadratic term is bounded by $|k_e{|}^4\Vert {\mathrm{ad}}_x{\Vert }^4/24$. Since $\Vert {\mathrm{ad}}_x{\Vert }_{1\to 1}\le 2L$, integration proves the result.