Mass-transfer rate (upper panel) and BH spin χ_BH (lower panel) as functions of the model number. The dashed line indicates the standard Eddington accretion rate (1.0 ̇M_Edd). In the binary sequence shown, the initial masses of the BH and He-rich star are 3.6 M_⊙ and 6.0 M_⊙, respectively, with an initial orbital period of 0.08 days. Different MT phases are highlighted in various colors: Case BA (green), case BB (blue), and case BC (gray).