The KIDS framework for the nuclear equation of state (EoS) and energy density functional (EDF) offers the possibility to explore symmetry-energy (SE) parameters such as J (value at saturation density), L (slope), Ksym (curvature) and so on independently of each other and of assumptions about the effective mass. Here we examine the performance of EoSs with different SE parameters in reproducing nuclear properties and astronomical observations in an effort to constrain especially L and Ksym or the droplet-model counterpart Ktau. Assuming a standard EoS for symmetric matter, we explore several points on the hyperplane of (J,L,Ksym or Ktau) values. For each point, the corresponding EDF parameters and a pairing parameter are obtained for applications in spherical even-even nuclei. This is the first application of KIDS EDFs with pairing correlations. The EoSs are tested successively on properties of closed-shell nuclei, along the Sn isotopic chain, and on astronomical observations, in a step-by-step process of elimination and correction. A small regime of best-performing parameters is determined. The results strongly suggest that Ksym is negative and no lower than -200MeV, that Ktau lies between roughly -400 and -300MeV and that L lies between 40 and 65MeV with L<55MeV more likely. Correlations between symmetry-energy parameters are critically discussed. Predictions for the position of the neutron drip line and the neutron skin thickness of selected nuclei are reported. They are only weakly affected by the choice of effective mass values. Parts of the drip line can be sensitive to the SE parameters. The results underscore the role of Ktau and of precise astronomical input. Better constraints are possible with precise fits to nuclear energies and, in the future, more-precise input from astronomy.
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