[proofplan] The annular curvature hypothesis is first converted into the pointwise scale-invariant estimate $|A|(x)|x|\le M$ near the puncture. This is exactly the hypothesis needed in the local removable singularity theorem for minimal laminations of a punctured three-ball, a global result whose content is stronger than a pointwise graphical theorem: it rules out spiraling, proves compatible local product structure, and supplies the extension across the puncture. Applying that theorem gives a minimal lamination in a smaller ball, and then the extension is glued to the original lamination away from the puncture. [/proofplan]

[step:Derive the pointwise hypothesis of the local removable singularity theorem]Let $r_0 \in (0,1)$ be such that the stated estimate holds for every $r \in (0,r_0)$. For $x\in \bigcup_{L\in\mathcal L}L$, let $|A_{\mathcal L}|(x)$ denote the norm of the second fundamental form of the leaf of $\mathcal L$ through $x$. Let $x\in (\bigcup_{L\in\mathcal L} L)\cap B(0,r_0/2)$ with $x\ne 0$, and set $s:=|x|$. Then $x\in B(0,2s)\setminus B(0,s)$, so the curvature hypothesis applied with $r=2s$ gives \begin{align*} |A_{\mathcal L}|^2(x) \le C(2s)^{-2}. \end{align*} Define the curvature constant $M>0$ by \begin{align*} M:=\frac{\sqrt{C}}{2}. \end{align*} Taking square roots yields \begin{align*} |A_{\mathcal L}|(x)|x|\le M. \end{align*} Thus the second fundamental form of every leaf satisfies the scale-invariant pointwise estimate required by the local removable singularity theorem for minimal laminations in a punctured three-ball.[/step]

[guided]The global graphical structure near the puncture cannot be obtained from the local bounded-curvature graphical theorem alone. That theorem gives graphs only in balls whose radii are controlled by the local curvature scale; it does not by itself rule out spiraling, prove that different local graphs use a common limiting tangent plane, or show that local sheets patch into punctured disks. We therefore reduce the annular estimate to the standard global removable-singularity criterion for minimal laminations. For $x\in \bigcup_{L\in\mathcal L}L$, the notation $|A_{\mathcal L}|(x)$ denotes the norm of the second fundamental form of the leaf of $\mathcal L$ through $x$. Let $x\in (\bigcup_{L\in\mathcal L} L)\cap B(0,r_0/2)$ with $x\ne 0$, and define $s:=|x|$. Since $s<r_0/2$, the radius $2s$ lies in $(0,r_0)$, so the curvature hypothesis is available at radius $2s$. Also $|x|=s$ implies \begin{align*} x\in B(0,2s)\setminus B(0,s). \end{align*} Applying the hypothesis with $r=2s$ gives \begin{align*} |A_{\mathcal L}|^2(x) \le \sup_{y\in \mathcal L\cap B(0,2s)\setminus B(0,s)} |A_{\mathcal L}|^2(y) \le C(2s)^{-2}. \end{align*} Define \begin{align*} M:=\frac{\sqrt{C}}{2}. \end{align*} Then the preceding inequality is equivalent to \begin{align*} |A_{\mathcal L}|(x)|x|\le M. \end{align*} This is the correct scale-invariant pointwise form of the curvature hypothesis. It is stronger than a merely local graphical statement because it is the input to the removable-singularity theorem, whose conclusion includes the missing global sheet compatibility and lamination product structure.[/guided]

[step:Apply the local removable singularity theorem for minimal laminations] Choose $\varepsilon\in(0,r_0/2)$. The restriction $\mathcal L\big|_{B(0,\varepsilon)\setminus\{0\}}$ is a minimal lamination of the punctured ball $B(0,\varepsilon)\setminus\{0\}\subset\mathbb R^3$. By the previous step, its leaves satisfy $|A_{\mathcal L}|(x)|x|\le M$ at every point of the punctured ball. We use the local removable singularity theorem of Meeks, Pérez, and Ros for minimal laminations in the following form: if $\mathcal M$ is a minimal lamination of a punctured Euclidean three-ball $B(p,\rho)\setminus\{p\}\subset\mathbb R^3$ and the norm $|A_{\mathcal M}|$ of the second fundamental form satisfies $|A_{\mathcal M}|(y)|y-p|\le M_0$ for all $y\in \bigcup_{L\in\mathcal M}L$, for some constant $M_0>0$, then $\mathcal M$ extends uniquely near $p$ to a minimal lamination of $B(p,\rho)$ with compatible local product structure. Its hypotheses hold here with $p=0$, $\rho=\varepsilon$, $\mathcal M=\mathcal L\big|_{B(0,\varepsilon)\setminus\{0\}}$, and $M_0=M$: minimality is inherited by restriction, the ambient set is a punctured Euclidean three-ball, and the required pointwise curvature bound was proved above. Therefore the theorem gives a minimal lamination $\mathcal L^{\mathrm{ext}}$ of $B(0,\varepsilon)$ such that \begin{align*} \mathcal L^{\mathrm{ext}}\big|_{B(0,\varepsilon)\setminus\{0\}}=\mathcal L\big|_{B(0,\varepsilon)\setminus\{0\}}. \end{align*} The conclusion of that theorem includes the local product structure at $0$, the closedness of the extended leaf collection in a neighbourhood of $0$, and the compatibility of sheets that approach the puncture; these are not consequences of the local graphical theorem alone. [/step]

[step:Glue the local extension to the original lamination] Define $\widetilde{\mathcal L}$ on $B(0,1)$ by \begin{align*} \widetilde{\mathcal L}\big|_{B(0,\varepsilon)}:=\mathcal L^{\mathrm{ext}}, \end{align*} and by \begin{align*} \widetilde{\mathcal L}\big|_{B(0,1)\setminus\{0\}}:=\mathcal L. \end{align*} This definition is consistent on the overlap $B(0,\varepsilon)\setminus\{0\}$ because $\mathcal L^{\mathrm{ext}}$ restricts there to the original lamination. The lamination charts away from $0$ are the original charts of $\mathcal L$, and the lamination charts in $B(0,\varepsilon)$ are those supplied by $\mathcal L^{\mathrm{ext}}$. Since both chart systems describe the same leaves on their punctured overlap, they are compatible. Hence $\widetilde{\mathcal L}$ is a minimal lamination of $B(0,1)$ and satisfies \begin{align*} \widetilde{\mathcal L}\big|_{B(0,1)\setminus\{0\}} = \mathcal L. \end{align*} This is the required extension across $0$. [/step]