Let $(M^3,g)$ be a closed Riemannian $3$-manifold. Fix constants $A_0>0$ and $g_0\in\mathbb N$. Let $\{\Sigma_k\}$ be a sequence of closed embedded minimal surfaces in $M$ such that

\begin{align*} \mathcal H^2(\Sigma_k) \le A_0, \qquad \operatorname{genus}(\Sigma_k)\le g_0. \end{align*}

Assume the following two local compactness inputs. First, the Choi-Schoen curvature-concentration estimate for closed embedded minimal surfaces with area bound $A_0$ and genus bound $g_0$ in $(M,g)$ gives a finite curvature-concentration set and locally uniform second-fundamental-form bounds away from it. Second, at any point of that finite set where the limiting punctured minimal lamination satisfies a stated removable-singularity criterion, such as the local curvature, stability, properness, or one-sided hypotheses required by that criterion, the lamination extends smoothly across the point. Then there is a subsequence, a finite set $\mathcal S\subset M$, and a smooth embedded minimal surface or minimal lamination $\mathcal L_\infty$ in $M\setminus\mathcal S$ such that $\Sigma_k$ converges smoothly locally, possibly with multiplicity, to $\mathcal L_\infty$ on compact subsets of $M\setminus\mathcal S$. At every point of $\mathcal S$ where the removable-singularity criterion holds for $\mathcal L_\infty$, the lamination limit extends smoothly across that point.