Revisiting the Possibility of a Sharp Phase Transition in Cold Neutron Stars

First-order phase transitions (FOPTs) in cold neutron stars (NSs) have been extensively studied and have provided valuable insights into the behavior of the densest matter visible in our Universe, although a strong consensus has yet to emerge. Revisiting the possibility of a hadron-quark FOPT from a new perspective, we examine the interplay between the coupled nature of gravity and microscopic interactions in Tolman--Oppenheimer--Volkoff (TOV) equations and the fundamental requirements of thermodynamic consistency in NSs. We demonstrate that a sharp FOPT manifested as a plateau in the equation of state (EOS) $P(\varepsilon)$, i.e., pressure $P$ versus energy density $\varepsilon$, is intrinsically incompatible with the regularity conditions of the TOV solutions. Although numerical integrations of the TOV equations with EOSs incorporating FOPTs may yield seemingly reasonable mass-radius relations consistent with current observations, such results can mask underlying inconsistencies. Our analysis thus establishes a structural consistency criterion for constraining dense-matter EOSs using NS observables, complementing existing studies of possible phase transitions in NS interiors.