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Dominant shapes naturally emerge in atomic nuclei from first principles, thereby establishing the shape-preserving symplectic Sp(3,\mathbb{R} $ℝ$) symmetry as remarkably ubiquitous and almost perfect symmetry in nuclei. We discuss the critical role of this emergent symmetry in enabling machine-learning descriptions of heavy nuclei, ab initio modeling of\alpha $\alpha$clustering and collectivity, as well as tests of beyond-the-standard-model physics. In addition, the Sp(3,\mathbb{R} $ℝ$) and SU(3) symmetries provide relevant degrees of freedom that underpin the ab initio symmetry-adapted no-core shell model with the remarkable capability of reaching nuclei and reaction fragments beyond the lightest and close-to-spherical species. 

We discuss emulators from the ab initio symmetry-adapted no-core shell-model framework for studying the formation of alpha clustering and collective properties without effective charges. We present a new type of an emulator, one that utilizes the eigenvector continuation technique but is based on the use of symplectic symmetry considerations. This is achieved by using physically relevant degrees of freedom, namely, the symmetry-adapted basis, which exploits the almost perfect symplectic symmetry in nuclei. Specifically, we study excitation energies, point-proton root-mean-square radii, along with electric quadrupole moments and transitions for 6 Li and 12 C. We show that the set of parameterizations of the chiral potential used to train the emulators has no significant effect on predictions of dominant nuclear features, such as shape and the associated symplectic symmetry, along with cluster formation, but slightly varies details that affect collective quadrupole moments, asymptotic normalization coefficients, and alpha partial widths up to a factor of two. This makes these types of emulators important for further constraining the nuclear force for high-precision nuclear structure and reaction observables.