A common finite-density charge-symmetry-breaking response in mirror displacement energies and charge radii

Mirror charge radii can constrain neutron skins only if nuclear isospin-symmetry breaking beyond the Coulomb interaction is controlled. We present a differential Skyrme energy-density-functional analysis in which mirror displacement energies (MDEs) and mirror charge-radius differences are generated by the same class-III charge-symmetry-breaking (CSB) functional. Starting from self-consistent Coulomb-only SLy4 and SkM* baselines, we map Coulomb-subtracted MDE residuals onto volume and surface-gradient CSB terms. The MDEs determine the nearly degenerate volume-surface direction, while the measured anchor radii select the point on that direction used for target predictions, without introducing radius-specific parameters. Measured pairs ${}^{34}$Ar-${}^{34}$S, ${}^{36}$Ca-${}^{36}$S, ${}^{38}$Ca-${}^{38}$Ar, and ${}^{54}$Ni-${}^{54}$Fe form compact surface-gradient-like response classes, with ${\lambda }_B=1.32\pm 0.04{\mathrm{f}\mathrm{m}}^{-2}$ for SLy4 and $1.20\pm 0.05{\mathrm{f}\mathrm{m}}^{-2}$ for SkM*. MDEs therefore constrain mainly the effective combination $t_0^{\mathrm{I}\mathrm{I}\mathrm{I}}-{\lambda }_{\mathrm{c}\mathrm{l}}{C}_{\mathrm{\Delta }}^{\mathrm{I}\mathrm{I}\mathrm{I}}$, rather than the two couplings separately. A joint volume-plus-surface-gradient fit gives MDE and $\mathrm{\Delta }{R}_{\mathrm{c}\mathrm{h}}^{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}}$ RMS residuals of $0.0529$ MeV and $0.0031$ fm in SLy4; SkM* recovers the response class but gives a larger $\mathrm{\Delta }{R}_{\mathrm{c}\mathrm{h}}^{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}}$ residual. Calibrated response changes proton-rich mirror skins at the ${10}^{-2}$ fm level and yields charge radius predictions for ${}^{40}$Ti, ${}^{42}$Ti, ${}^{46}$Cr, and ${}^{50}$Fe, with an SLy4 - SkM* spread of up to $0.021$ fm. Surface-gradient-sensitive CSB corrections must therefore be quantified before mirror charge-radius differences are used as clean neutron-skin or symmetry-energy probes. The agreement of SLy4 and SkM* in the response-class assignment is more robust than their absolute radius corrections, which remain EDF dependent.