**4. Conclusions**

In this paper, an almost complete survey of proton and neutron shell evolution for atomic mass number 25 - *A* - 55 neutron-rich nuclei is performed on the basis of shellmodel calculations, in order to understand how well the observed evolution is explained with a simple monopole-based universal interaction, *V*MU.

On the proton side, the observed one-proton removal spectroscopic distributions in 40,48Ca were very well reproduced with shell-model calculations, pointing to a ∼2 MeV change of *πd*5/2-*πd*3/2 spin–orbit splitting. Since this change is caused almost solely by the tensor force, this agreement quantitatively confirms the validity of a *π* + *ρ* meson exchange tensor force in the *V*MU interaction. The 1/2<sup>+</sup> <sup>1</sup> -3/2<sup>+</sup> <sup>1</sup> level difference in K isotopes changes the sign twice, with *ν f*7/2 filled and with *νp*3/2 filled.

As discussed, this change is caused by the "reinversion" of single-particle level ordering between *πd*3/2 and *πs*1/2 as a result of the non-monotonic evolution of these level spacings. Such a manner of evolution cannot be produced by one-body potential models, and therefore it is strong evidence for the dominant role of two-body forces in shell evolution. In this particular case, the non-monotonic evolution observed in K isotopes is driven by the central force.

On the neutron side, the neutron-number *N* = 28 shell gap is reduced with protons removed from the *d*3/2 and *d*5/2 orbitals, dominated by the central force. The relevant single-particle-like levels are well reproduced by the shell-model calculation. In addition, the central force causes the enhancement of the *N* = 34 shell gap for the atomic-number *Z* < 20 isotopes. This effect well accounts for the recently observed 2<sup>+</sup> <sup>1</sup> level in 52Ar.

In this way, the present scheme, based on *V*MU, provides a successful description of the shell evolution. Neutron shell evolution in exotic nuclei is often argued in the context of weak binding. In the present study, we were successful in obtaining not only neutron shells but also proton shells that were free from weak binding. Thus, such a unified description strongly indicates the dominance of the effective interaction in shell evolution, as far as the region of the present study is concerned, including the narrowing *N* = 28 shell gap toward a neutron-rich nuclei.

**Funding:** This work was supported in part by JSPS KAKENHI Grant Numbers 20K03981 and 15K05094.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The author thanks Takaharu Otsuka for valuable discussions on shell evolution and thank Noritaka Shimizu for his help with large-scale shell-model calculations.

**Conflicts of Interest:** The author declares no conflict of interest.
