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 $\mathrm{\Lambda }$ 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, $d(d{v}_1\mathrm{/}dy)\mathrm{/}d(\sqrt{s_{NN}})$, 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.