[proofplan] We prove the estimate by reducing the surgery region to unit scale. After rescaling by $h^{-2}$, the strong $\delta$-neck is a small $C^{m+2}$ perturbation of the round cylinder, while the inserted cap and cutoff profile come from a fixed smooth model. Curvature and its covariant derivatives are smooth universal expressions in finitely many derivatives of the metric, so they are uniformly bounded at unit scale once $\delta$ is small. Finally, rescaling back by $h^2$ multiplies the $m$-th covariant derivative of curvature by $h^{-2-m}$. [/proofplan]

[step:Rescale the surgery region to unit neck radius]Let $g$ denote the pre-surgery metric on the strong $\delta$-neck, and define the rescaled metric \begin{align*} \tilde g := h^{-2} g. \end{align*} The strong $\delta$-neck hypothesis means that, after choosing cylindrical coordinates through a neck parametrisation \begin{align*} \Phi: S^{n-1} \times I \to N, \end{align*} where $I \subset \mathbb{R}$ is an interval containing the surgery transition interval, the pulled-back metric $\Phi^* \tilde g$ is $\delta$-close in $C^{m+2}$ to the product round cylinder metric \begin{align*} g_{\mathrm{cyl}} := g_{S^{n-1}} + dz^2 \end{align*} on the relevant unit-scale cylindrical subregion. Here $g_{S^{n-1}}$ is the round metric of sectional curvature $1$ on $S^{n-1}$, and $z$ is the coordinate on $I$. The post-surgery rescaled metric is \begin{align*} \tilde g_+ := h^{-2} g_+. \end{align*} On the inserted cap and transition region at unit scale, $\tilde g_+$ is obtained by combining the fixed standard cap metric with the pulled-back neck metric $\Phi^*\tilde g$ using the fixed cutoff function. Thus its coordinate coefficients and their partial derivatives up to order $m+2$ are bounded by a constant depending only on $n$, $m$, the fixed cap profile, and the cutoff construction, provided $\delta \leq \delta_m$.[/step]

[guided]The reason for introducing $\tilde g = h^{-2}g$ is that the surgery scale $h$ should disappear from the local geometry. A neck of radius $h$ becomes a neck of radius $1$ under this rescaling. In cylindrical coordinates, the model metric is therefore not $h^2(g_{S^{n-1}} + dz^2)$ but exactly \begin{align*} g_{\mathrm{cyl}} = g_{S^{n-1}} + dz^2. \end{align*} The strong $\delta$-neck assumption supplies a parametrisation \begin{align*} \Phi: S^{n-1} \times I \to N \end{align*} such that $\Phi^*\tilde g$ is close to $g_{\mathrm{cyl}}$ in $C^{m+2}$ on the part of the cylinder where the cutoff and cap are attached. The order $m+2$ is the relevant one because $\nabla^m \operatorname{Rm}$ contains two derivatives from curvature and then $m$ further covariant derivatives. The surgery construction itself uses no new scale after this rescaling: the cap profile is fixed, and the cutoff function is fixed. Hence the coordinate coefficients of $\tilde g_+$ are formed from fixed smooth functions and from the coefficients of $\Phi^*\tilde g$. Since $\Phi^*\tilde g$ is $C^{m+2}$-close to the round cylinder, every partial derivative of the coefficients of $\tilde g_+$ up to order $m+2$ is bounded by a constant depending only on $n$, $m$, and the chosen standard cap construction, once $\delta$ is chosen below a threshold $\delta_m$.[/guided]

[step:Bound unit-scale curvature derivatives by smooth dependence on the metric] Work in the unit-scale cap and transition coordinate charts used in the construction. In any such chart, the components of $\operatorname{Rm}_{\tilde g_+}$ are universal polynomial expressions in the components of $\tilde g_+^{-1}$, the first partial derivatives of $\tilde g_+$, and the second partial derivatives of $\tilde g_+$. Likewise, the components of $\nabla_{\tilde g_+}^m \operatorname{Rm}_{\tilde g_+}$ are universal polynomial expressions in the components of $\tilde g_+^{-1}$ and the partial derivatives of $\tilde g_+$ up to order $m+2$. Because $\tilde g_+$ remains uniformly positive definite for $\delta \leq \delta_m$, the components of $\tilde g_+^{-1}$ are uniformly bounded. The preceding step gives uniform bounds for the required metric derivatives. Therefore there is a constant $A_m < \infty$, depending only on $n$, $m$, and the fixed cutoff-cap construction, such that \begin{align*} |\nabla_{\tilde g_+}^m \operatorname{Rm}_{\tilde g_+}|_{\tilde g_+} \leq A_m \end{align*} throughout the unit-scale inserted cap and transition region. [/step]

[step:Scale the unit estimate back to radius $h$] Since $g_+ = h^2 \tilde g_+$, the Levi-Civita connections of $g_+$ and $\tilde g_+$ agree, while tensor norms scale according to the number of covariant tensor slots. The curvature tensor $\operatorname{Rm}$ has scaling order $h^{-2}$ as a curvature quantity, and each additional covariant derivative contributes one further factor of $h^{-1}$. Hence \begin{align*} |\nabla_{g_+}^m \operatorname{Rm}_{g_+}|_{g_+} = h^{-2-m} |\nabla_{\tilde g_+}^m \operatorname{Rm}_{\tilde g_+}|_{\tilde g_+}. \end{align*} Using the unit-scale bound from the previous step gives \begin{align*} |\nabla_{g_+}^m \operatorname{Rm}_{g_+}|_{g_+} \leq A_m h^{-2-m}. \end{align*} Set $C_m := A_m$. This proves the desired estimate for every fixed integer $m \geq 0$, and the case $m=0$ gives the asserted curvature bound \begin{align*} |\operatorname{Rm}_{g_+}|_{g_+} \leq C_0 h^{-2}. \end{align*} [/step]