2025ApJ...985....1V

The equation of state is fundamental in describing matter under the extreme conditions characteristic of neutron stars and is central to advancing our understanding of dense matter physics. A critical challenge, however, lies in accurately modeling first-order phase transitions while ensuring thermodynamic consistency and aligning with astrophysical observations. This study explores two frameworks for constructing equations of states with first-order phase transitions: the polytropic interpolation method and the randomized speed-of-sound interpolation approach. It is found that the mass-radius relation and pressure versus energy density relation are blind toward the thermodynamic consistency check. The polytropic interpolation method can exhibit discontinuities in the chemical potential for first-order phase transition, raising concerns regarding potential causality violations and thermodynamic inconsistencies. In contrast, the speed of sound interpolation approach ensures continuity in the chemical potential, offering a more thermodynamically consistent and reliable framework. Moreover, the sound speed method effectively captures the softer segment of the mass-radius spectrum, having an early phase transition, a capability not achieved by the consistent piecewise-polytropic approach due to its monotonic stiffness constraints. The speed of sound definition involving number density and chemical potential reveals the thermodynamic inconsistency, making it a more consistent and robust definition. The sound speed interpolation method also generated a sufficient number of twins, which the piecewise method failed. These findings underscore the importance of thermodynamic consistency in the equation of state construction and highlight the advantages of the randomized speed-of-sound method for modeling phase transitions in dense matter.