Exact boundary of the reachable set for an open qubit

We establish the exact boundary structure of reachable sets for open two-level quantum systems governed by Gorini-Kossakowski-Lindblad-Sudarshan (GKSL) master equations with coherent and incoherent controls. We present an exact analytical characterization of the reachable set's geometric structure. The boundary is proven to be a smooth surface of revolution about the $Or_x$ axis with two conical singularities. We derive explicit control protocols achieving boundary states with arbitrary precision and reveal a nested structure featuring a torus-like domain of local controllability.