Corollary: Positive finite-time perturbation budget

Fix $K\ge 1$ and $0<\epsilon <1$. Let

Use $K{N}_{\ast }$ serial events of duration $\tau >0$. If

then ${\epsilon }_{\mathrm{implemented}}\le \epsilon$.

The allowed set contains nonzero bare-energy-conserving perturbations, for example constant event perturbations

Proof. $K{r}^{N_{\ast }}\le \epsilon /2$, and the accumulated implementation allowance is at most $\epsilon /2$. Apply (eq:robustrequested). Since ${\Vert S_e\Vert }_{\infty }=1$ and $[S_e,H_0]=0$, the displayed perturbations have the claimed norm and conservation.