Square-lattice FK interfaces and nested loops for 1 <= q < 4

For every fixed $1\le q<4$, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal ${\mathrm{SLE}}_{\kappa (q)}$, where $\kappa (q)=4\pi /\arccos (-\sqrt{q}/2)$, confirming the Rohde–Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane ${\mathrm{CLE}}_{\kappa (q)}$ in a spherical matching topology retaining multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.