Let $\mathcal L$ be a [minimal lamination](/page/Minimal%20Lamination) of $B(0,1)\setminus\{0\}\subset\mathbb R^3$, so its leaves are smooth embedded minimal surfaces. Suppose the leaves of $\mathcal L$ have locally bounded curvature near $0$ in the scale-invariant sense that
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\begin{align*}
\sup_{\mathcal L\cap B(0,r)\setminus B(0,r/2)} |A|^2 \le C r^{-2}
\end{align*}
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for all sufficiently small $r>0$. Then $\mathcal L$ extends across $0$ to a minimal lamination of $B(0,1)$.