Minimal Wigner-$SU(4)$ Interaction in Microscopic Cluster Models for $\alpha$-Conjugate Nuclei

We present a minimalist, symmetry-guided interaction for microscopic cluster models based on Wigner-$SU(4)$ symmetry. Retaining only an $SU(4)$-invariant two-body attraction and a local three-body repulsion, this framework is implemented via the generator coordinate method (GCM) to describe $\alpha$--$\alpha$ scattering phase shifts, the low-lying spectrum and transition properties of $^{12}\mathrm{C}$, and the cluster spectrum of $^{16}\mathrm{O}$. We show that the long-standing structural tension between the $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$ ground states can be mitigated within this restricted $SU(4)$ operator space without introducing additional phenomenological complexity. These results indicate that Wigner-$SU(4)$ symmetry provides an effective organizing principle for $N\alpha$ clustering, offering a more fundamental baseline for understanding complex cluster structures.