Time evolution of e_p (top left), a_p (bottom left), m_p (top right), and R_p (bottom right) for an example system that forms a hot Jupiter through a single quadrupole cycle. Curves show evolution with varying fractions of angular momentum return, including f_ret = 0 (orange curves), f_ret = 0.5 (blue curves), and f_ret = 1 (gray curves). The a_p panel (bottom left) shows a zoomed-in view of the final circularization locations of the hot Jupiter (inset panel corresponds to the gray rectangle), with dashed black lines showing analytical predictions from Equation (14). The system parameters are m_* = 1.427M_⊙, m_p = 0.857M_J, m_c = 0.838M_⊙, a_p = 1.621 au, a_c = 82.846 au, e_p = 0.077, e_c = 0.084, ω₁ = 46﹩\mathop{.}\limits^{\unicode{x000b0}}﹩075, ω₂ = 212﹩\mathop{.}\limits^{\unicode{x000b0}}﹩893, and i = 84﹩\mathop{.}\limits^{\unicode{x000b0}}﹩210.