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 metamodel EOSs with nine microscopic parameters, we analyze mock radius measurements R 1.4 = 11.9 ± σ R km with σ R = 0.9 and 0.1 km for canonical NSs. Rather than relying only on marginalized posterior probability distributions, 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. Resolving the mappings into four mass–radius topologies, Connected, Disconnected, Both, and No-Quark-Matter, reveals a clear hierarchy of information. The symmetry-energy parameters L (slope) and K sym (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 ρ 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 EOSs.