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 Rényi-2 genuine multi-entropy 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 ξ. For partitions with a single junction, GM^{\(mathtt{q}\)}₂ exhibits a universal scaling crossover in L/ξ, growing for Lllξ and saturating to a ξ-dependent constant for Lggξ, up to mathcal{O}\(e^-L/ξ\) corrections. In sharp contrast, for partitions without a junction, GM^{\(mathtt{q}\)}₂ is exponentially suppressed in L/ξ and drops below numerical resolution once Lggξ. 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.