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Direct transfer to $^{46,48}$K as a survey of the $π(s_{1/2})$$-ν(sdpf)$ interaction

The collapse of the canonical $N=28$ magic number in nuclei with $Z<20$ has drawn significant interest as it relates to the emergence of an island of inversion centered on $^{42}$Si and $^{44}$S. In particular, interactions between the $πs_{1/2}$ orbital -- empty in $^{42}$Si and full in $^{44}$S -- and the neutron orbitals just above and below the $N=28$ gap are expected to be critical in this region, but remain relatively unexplored. In this paper, we expand upon the results of our previous study of the direct transfer reaction $^{47}$K(d,p$γ$)$^{48}$K [C.\,J.~Paxman \textit{et al.}, Phys. Rev. Lett. 134, 162504 (2025)] with the results of the complementary $^{47}$K(d,t$γ$)$^{46}$K reaction. Through this study, we present a comprehensive scan of the interaction between the critical $πs_{1/2}$ orbital and a broad range of neutron orbitals spanning nearly two full shells. We identify several discrepancies between the experimental results and state-of-the-art shell model calculations, which suggest a deficiency of the shell model to fully capture the complex proton configuration mixing in this region, highlighting a significant challenge for single-particle descriptions of the island of inversion.