Shell and within-shell bounds

certificate. Localizing by any fixed smooth partition from Section sec:audit preserves the order $64$. On each of its four observation regions, the shell and within-shell bounds are therefore $C_{K,\rho ,m}(n+1)2^{-n}$. The constants may depend on the localization but not on $n$. For intersection neighborhoods they require no condition $|t-t^{\prime }|\ge \delta >0$.