Constraining the Phase-Transition EoS using the Energy Dependence of Directed Flow

We propose a hybrid equation of state (VDF+MIT EoS) to describe the hadron-quark phase transition in dense nuclear matter. By coupling this EoS with the AMPT-HC transport model and comparing to recent experimental data on proton and $\Lambda$ directed flow $v_1$, we constrain the transition to likely occur near $5\rho_0$--$6\rho_0$, ruling out transitions below $3\rho_0$. Furthermore, we introduce the energy derivative of the mid-rapidity $v_1$ slope, $d(dv_1/dy)/d(\sqrt{s_{NN}})$, as a weakly model-dependent observable. Its zero crossing provides a direct signature of the phase transition critical point, offering a new tool for mapping the QCD phase diagram in future experiments.