Fix the choice of $N,A$. Suppose we are given a sequence of Lagrangians $L^{(j)}$ with potential satisfying $(N,A)$-potential clustering, and assume $L_{i}^{(j)}\to L_{i}$ as currents for $1\leq i\leq N$, all $f_{L_{i}}^{(j)}$ have uniform $L^{\infty}$ bounds, and $f_{L_{i}^{(j)}}L^{(j)}\to f_{L_{i}}L_{i}$ as currents. Then the limit $L=\sum_{1}^{N}L_{i}$ with its potential $f_{L}$ also satisfies $(N,A)$-potential clustering.