The singleton hypergraph is extremal for the Isolation Lemma

Let $H$ be an inclusion-free hypergraph on $n$ vertices. A weight assignment $w:[n]\to[d]$ is isolating if there is a unique edge $e$ whose weight $w(e) = \sum_{i \in e} w(i)$ is minimum. We show that the number of isolating weight assignments is at least $$n\sum_{j=0}^{d-1} j^{n-1},$$ a bound which is attained with equality by the hypergraph consisting of the $n$ singleton edges. This proves the conjecture stated in Faber & Harris (2018). We also prove the bound for a more general class of edge-weight objectives, including arbitrary edge offsets.