Determine the topology produced by collapse of frustrated interface patches

Determine whether collapse of an interface patch identified by the minimum-residual construction produces a 4-line or another specific subsequent topology in a fully connected interfacial network.

Background

The paper introduces a minimum-residual construction for non-constructible four-node junctions by minimizing the Herring residuals over triple-line directions while enforcing a non-degeneracy constraint. When one or more pairwise separation measures approach zero, the corresponding interface patches tend toward local collapse, potentially indicating a topological rearrangement. However, the construction is local and contains neither kinetic information nor the surrounding network connectivity, so it does not resolve which topology follows the collapse. Establishing this outcome would require coupling the local geometric analysis to network-scale geometry and dynamics.

References

The approach of one or more $\rho_{ij}$ to zero therefore identifies interface patches that tend toward local collapse in the minimum-residual construction and may indicate precursors to topological rearrangement. However, because the construction contains neither kinetics nor information about network connectivity away from the node, it does not determine whether such a collapse produces a 4-line or any other specific subsequent topology.

— Frustrated junctions in interfacial networks  (2609.29030 - Hallberg et al., 24 Sep 2026) in Supplementary Text, subsection “Supplementary minimum-residual construction of non-constructible 4-nodes”