Reciprocal LCM powers and Robin's criterion

For $\kappa >1$, we study the reciprocal LCM-power function $${\sigma }_{-1}^{[\kappa ]}(n):=\sum _{d\mid n}\mu (d){\sigma }_{-1}(n\mathrm{/}d{)}^{\kappa }\mathrm{.}$$ We determine its average and extremal orders. A natural Robin-scale normalization is strictly increasing in $\kappa$ and tends to ${\sigma }_{-1}$, yielding an RH-equivalent criterion. Unconditionally, the corresponding Robin inequality holds eventually on every bounded parameter interval. Directed interval arithmetic proves the inequality for each $n>5040$ and $1<\kappa \le \mathrm{100,000}$.