An Effective String Theory Toolbox for Quantum Hall Interfaces III: Open Worldsheets, Endpoint Conditions, and Branes

A freely moving quantum Hall (QH) interface may end on a physical edge or topological boundary, but fixed-edge theory cannot determine what endpoint data make such a termination consistent. Here we formulate an open-worldsheet junction framework in which the embedding, material charge, anomaly flow, and topological boundary condition are organized together. The endpoint is specified by a geometric support and variational boundary data, together with condensable topological sectors and any outgoing channels required to absorb or continue the worldsheet flux. This construction extends the charge--shape relation to an interval and shows why a lone chiral Majorana cannot terminate on a finite-dimensional endpoint degree of freedom. It gives an operational definition of a QH brane and a systematic basis for endpoint and network theories of dynamical QH interfaces.