We perform a Bayesian inference of the dense-matter equation of state (EOS) within a unified framework that incorporates hadronic matter, quark matter, and a smooth hadron-quark crossover. The EOS is constrained using physical consistency conditions, gravitational wave data from GW170817, NICER mass-radius measurements, and hypothetical future high-precision radius observations. In contrast to most previous studies that assume a sharp first-order phase transition or fix part of the EOS, we simultaneously infer hadronic, quark, and crossover parameters within a single statistical framework. We find that current observations strongly constrain the density dependence of the nuclear symmetry energy, particularly its slope and curvature, while the highest-density hadronic parameters and quark matter properties remain only weakly constrained. The posterior distributions favor a crossover centered at an energy density $ϵ\sim (4–6){ϵ}_0$ with a width $\mathrm{\Gamma }\sim (0.5–1.0){ϵ}_0$. A pronounced peak in the speed of sound emerges naturally near the crossover region, typically around $4{ϵ}_0$, and often coincides with the central densities of $\sim 2M_{\odot }$ neutron stars. We further show that the trace anomaly exhibits a remarkably universal behavior across the accepted EOS ensemble and remains largely insensitive to current observational constraints. This indicates that present data primarily probe the low- to intermediate-density EOS, while robust inference of quark matter and genuinely high-density physics will require next-generation precision radius measurements or complementary observables.

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• Nuclear matter in neutron stars