**11. Conclusions**

The present exploration of the interface between shell model and collective nuclear structure, which we term "emergence of nuclear collectivity", raises many questions. From a summary of systematic features in data, this paper has focused on the effective charge problem, which reveals itself already in the reduced transition strengths between the first-excited state and the ground state, *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ), in nuclei possessing two valence nucleons coupled to a doubly closed shell. A notable puzzle is the neutron effective charge needed for 18O compared to the well-known value of *en* ∼ +0.5*<sup>e</sup>* in 17O. It would appear that applying state-of-the-art shell model calculations beyond these simple structures needs great caution; and claims of successful descriptions in such nuclei deserve skepticism. Let us note the issue of spectroscopic factors as deduced from proton knockout by quaiselastic electron scattering [81] (see also [260,261]). The occupancies of particle configurations above the shell closures in doubly closed shell nuclei, shown in Figure 62, indicate that one is likely never dealing with simple shell model configurations when confronting data.

We have suggested directions in which shell model states should be explored as one moves away from closed shells, in the guise of seniority isomers (which involve pairing correlations). We have suggested criteria for exploring the validity of the language of deformation (proton–neutron correlations) in describing weakly deformed nuclei. Notably, nuclei that are termed "transitional" are severely neglected in the spectroscopic data base: we have outlined focal points for experimental study. We concluded with a sketch of details that leads shell model philosophy into the symplectic shell model: in the framework of this model, specific multi-shell configurations are emerging as a major clue to what is going on in low-energy nuclear excitations, and towards which state-of-the-art shell model activity needs to move.

We close with the view: "Data will have the last word in this Shakespearian drama" and "All the [nuclear] World's a [data] stage, and all the protons and neutrons merely players." (Adapted from *As You Like It* by W. Shakespeare). The message is that *one needs precision spectroscopy across the mass surface*, as well as pushing to exotic nuclei towards the limits of nuclear stability.

**Figure 62.** Quasiparticle strength for states just above the Fermi surface, observed in the reaction (*e*,*e p*) as a function of the target mass. All strengths are integrated to an excitation energy of about 20 MeV. Reprinted from [81], Copyright (1993), with permission from Elsevier. The language used in the original paper, from which this figure is taken, needs some clarification. 'Empty' orbits refers to shell model configurations above the shell closure, which are conventionally regarded as empty in doubly closed shell nuclei. However, in the (*e*,*e p*) studies, these configurations must have proton occupancy in the doubly closed shell target nuclei to explain the pattern of protons that are knocked out. Thus, one must conclude that the shells are not "closed".

An underlying theme that emerges in this look at nuclear structure is the role of algebraic structures in the quantum mechanics of the nuclear many-body problem. Two structures are widely manifested where many nucleon configurations are involved. In singly closed shell nuclei, the seniority coupling scheme dominates. This coupling scheme is explained by an su(2) algebra for correlated pairs in *j*-shell configurations. This stems from dominance of spin–orbit coupling imposed on a spherical mean-field independent-particle description. In open-shell nuclei, the Bohr unified model coupling scheme dominates. This can be traced to an sp(3,R) algebra with contraction on the very large quantum number values involved. Thus, we suggest that a way forward is to explore algebraic structures based on the shell model. This is being pursued, as noted, in the symmetry-adapted and symmetry-guided approaches [257–259], wherein effective charges are not needed.

**Author Contributions:** Conceptualization, J.L.W. and A.E.S.; methodology, J.L.W. and A.E.S.; validation, J.L.W. and A.E.S.; formal analysis, J.L.W. and A.E.S.; investigation, J.L.W. and A.E.S.; writing original draft preparation, J.L.W.; writing—review and editing, J.L.W. and A.E.S.; visualization, J.L.W. and A.E.S.; funding acquisition, A.E.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded in part by the Australian Research Council Grant No. DP170101673.

**Data Availability Statement:** Data available in the references given.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
