Where Multipartite Entanglement Localizes: The Junction Law for Genuine Multi-Entropy

We uncover a "junction law" for genuine multipartite entanglement, suggesting that in gapped local systems multipartite entanglement is controlled and effectively localized near junctions where subsystem boundaries meet. Using the R\'enyi-2 genuine multi-entropy $\mathrm{GM}^{(\mathtt{q})}_2$ as a diagnostic of genuine $\mathtt{q}$-partite entanglement, we establish this behavior in $(2+1)$-dimensional gapped free-fermion lattices with correlation length $\xi$. For partitions with a single junction, $\mathrm{GM}^{(\mathtt{q})}_2$ exhibits a universal scaling crossover in $L/\xi$, growing for $L\ll\xi$ and saturating to a $\xi$-dependent constant for $L\gg\xi$, up to $\mathcal{O}(e^{-L/\xi})$ corrections. In sharp contrast, for partitions without a junction, $\mathrm{GM}^{(\mathtt{q})}_2$ is exponentially suppressed in $L/\xi$ and drops below numerical resolution once $L\gg\xi$. We observe the same pattern for $\mathtt{q}=3$ (tripartite) and $\mathtt{q}=4$ (quadripartite) cases, and further corroborate this localization by translating the junction at fixed system size. We also provide a geometric explanation of the junction law in holography. Altogether, these results show that in this gapped free-fermion setting genuine multipartite entanglement is localized within a correlation-length neighborhood of junctions.