$β$-Decay Half-Lives Serve as Novel Evidence for the New Magic Number \(N=32\)
Conventional signatures of nuclear magic number, including low-lying quadrupole collectivity and mass systematics, face significant challenges when probing emergent shell closures near the drip line. However, $β$-decay half-lives are among the first experimental observables measurable following the discovery of neutron-rich isotopes. This letter demonstrates that $β$-decay half-lives provide evidence for the emergent magic number $N=32$. The observed half-life pattern around the $N=32$ can be attributed to the occupation probabilities of orbitals above this shell gap, which directly reflect the gap's magnitude. Our results reveal a pronounced $N=32$ shell gap in Ca isotopes and a weaker yet apparent gap in K isotopes, consistent with mass and electromagnetic transition data. Furthermore, the analysis indicates no prominent closed-shell signature at $N=32$ in Ar and Cl isotopes.