Phase transition and nuclear symmetry energy from neutron star observations: Constraints in light of PSR J0614--3329

The possible occurrence of a first-order hadron-quark phase transition (FOPT) in neutron-star interiors remains an open question. Whether such a transition can be directly tested with improved observations is a key challenge. Here, we incorporate the latest constraints, especially a new NICER radius measurement for PSR J0614–3329, into a nonparametric Gaussian Process (GP) EOS framework that explicitly includes a first-order transition. We find a Bayes factor of $B\approx 2.3$ when comparing models with and without an explicit phase transition, marginally favoring its presence. At $68\mathrm{\%}$ credibility, the transition onset density $n_{\mathrm{P}\mathrm{T}}$ is either below $2n_s$ (corresponding to masses $\lesssim 1M_{\odot }$, with density jump $\mathrm{\Delta }n\sim 0.5n_s$) or, more prominently, above $4n_s$ (near the central density of the heaviest NS, with $\mathrm{\Delta }n\sim 3n_s$), where $n_s$ represents the nuclear saturation density. In addition, by using symmetry-energy expansion at low densities ($<1.1n_s$), we infer a slope parameter $L=40.2_{-14.3}^{+19.3}$ MeV, in good agreement with nuclear-experiment values. Intriguingly, $L$ correlates positively with the radius difference between $1.4M_{\odot }$ and $2.0M_{\odot }$ stars.