How Neutron Star Radii Encode the Dense-Matter Equation of State and Hadron-Quark Transition

We investigate how future high-precision neutron star (NS) radius measurements encode microscopic information about the dense-matter equation of state (EOS), focusing on a possible first-order hadron–quark phase transition and the resulting mass–radius topology. Within a Bayesian framework using meta-model EOSs with nine microscopic parameters, we analyze mock radius measurements $R_{1.4}=11.9\pm {\sigma }_R$ km with ${\sigma }_R=0.9$ and $0.1$ km for canonical NSs. We introduce inverse EOS–radius mappings by studying the conditional mean of each EOS parameter as a function of the model-predicted canonical radius $R_{1.4}$. Their slope measures the sensitivity of an EOS parameter to the radius, while their curvature determines the leading precision dependence of its posterior mean through the Jensen expansion. The symmetry-energy parameters $L$ (slope) and $K_{\mathrm{s}\mathrm{y}\mathrm{m}}$ (curvature) are strongly encoded in $R_{1.4}$ and their posterior means shift appreciably with improved radius precision, whereas the higher-order hadronic parameters show stronger topology dependence. Among the transition parameters, the transition density ${\rho }_t$ is the most strongly encoded in $R_{1.4}$, while the energy-density jump and quark-matter sound speed are more strongly associated with the topology of the full mass–radius sequence. Since the different topologies have strongly overlapping $R_{1.4}$ distributions, even precise radius measurements cannot by themselves identify the topology or uniquely determine the high-density transition properties. These results provide a parameter-dependent hierarchy for assessing the scientific return of future high-precision radius measurements and complementary probes of high-density NS EOS.