Let $n \geq 3$ and fix a standard cutoff-cap construction in dimension $n$: a smooth rotationally symmetric cap profile and a smooth cutoff function, both independent of the surgery scale. For every integer $m \geq 0$ there exist constants $\delta_m > 0$ and $C_m < \infty$, depending only on $n$, $m$, and the fixed cutoff-cap construction, with the following property.

Let $0 < \delta \leq \delta_m$, let $h > 0$, and suppose a $\delta$-cutoff surgery at scale $h$ is performed on a strong $\delta$-neck. Let $g_+$ denote the post-surgery metric, and let $\mathcal{C}_h$ denote the union of the inserted cap and the transition region. Then, on $\mathcal{C}_h$,

\begin{align*} |\nabla_{g_+}^m \operatorname{Rm}_{g_+}|_{g_+} \leq C_m h^{-2-m}. \end{align*}

In particular, for $m = 0$,

\begin{align*} |\operatorname{Rm}_{g_+}|_{g_+} \leq C_0 h^{-2}. \end{align*}