Dense Matter Beyond Nuclear Saturation: Hybrid Stars, the Hadron–Quark Phase Transition, the Hyperon Puzzle, and the Speed-of-Sound Signature of Quark Matter Cores

This paper constitutes a direct extension of our earlier work on nuclear symmetry energy and neutron star constraints, advancing the discussion from the nuclear saturation region into the supra-nuclear density regime where the internal composition of neutron stars remains deeply contested. We systematically examine three interconnected problems: (i) the hadron–quark phase transition and its imprint on the equation of state (EoS) and neutron star observables; (ii) the hyperon puzzle and its modern proposed resolutions, including momentum-dependent potentials, repulsive hyperon–vector-meson couplings, and the quarkyonic matter framework; and (iii) the speed-of-sound profile cs²(ρ) as a diagnostic observable for the onset of quark degrees of freedom. The non-monotonic behavior of cs² — rising above the conformal limit of 1/3 before declining at higher densities — has emerged as a robust prediction across diverse theoretical frameworks and constitutes indirect evidence for a crossover transition toward quark matter in the cores of massive neutron stars. Combining the Relativistic Mean Field (RMF) framework for hadronic matter, the MIT bag model and Nambu–Jona-Lasinio (NJL) model for the quark phase, and current multimessenger constraints, we present a unified analysis of hybrid star sequences, twin star configurations, and the gravitational-wave signatures of first-order phase transitions. The paper further situates these discussions within the framework of the 2025 lattice QCD constraints on the EoS at isospin-dense conditions and the emerging color-superconducting quark matter picture. Our conclusions consistently favor a smooth hadron–quark crossover near (2–3)ρ₀ as the most observationally compatible scenario, while a strong first-order transition is disfavored by the combined NICER and GW170817 dataset at 90% confidence.