Self-dual random-cluster interfaces below one

For every fixed $0<q<1$, we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLE${}_{\kappa }$, where $\kappa =4\pi /\arccos (-\sqrt{q}/2)\in (6,8)$. The approximating domains are simple closed nearest-neighbor lattice polygons with distinct marked vertices, whose marked boundary parametrizations converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence in the uniform metric on oriented curves modulo increasing reparametrization. The proof combines finite connection comparisons below one, localization in irregular tiled disks, and a boundary observable that determines the limiting Loewner driver.