The topological classification of the one-loop Weinberg operator at dimension-5 enables a systematic categorization of radiative neutrino mass models. Among these, the category consisting loop-extended seesaw frameworks is theoretically appealing but conventional discrete or continuous symmetries (\emph{ e.g.}, $U(1)$ or $\mathbb{ Z}_M$) cannot genuinely forbid the corresponding tree-level contributions, making loop dominance difficult to realize. We show that \textit{ non-invertible selection rules} (NISRs) naturally enforce the absence of tree-level terms while ensuring a dominant one-loop contribution. Intriguingly, the same non-invertible structure also stabilizes the dark matter candidate, providing a unified radiative origin of neutrino mass and dark sector stability. In particular, we focus on the T4-2-$i$ topology which embodies a type-II one-loop seesaw and demonstrate its natural realization from $Z_{ 7}$ Tambara--Yamagami (TY) fusion algebra.
Categories: hep-ph