Invertibility at the boundary minimum-degree condition

Prove that the signless Laplacian matrix of a k-uniform (1,γ)-Dirac hypergraph remains invertible when the minimum 1-degree satisfies the boundary equality δ_1(G)=\frac{n}{k}\binom{n-2}{k-2}, thereby extending the entropy lower bound of Theorem 1-degree to the non-strict boundary case.

Background

The proof of the paper’s 1-degree entropy bound uses strict positive mean-dominance of the signless Laplacian matrix, which follows from the strict inequality δ_1(G)>\frac{n}{k}\binom{n-2}{k-2}. At equality, strict positive mean-dominance and the automatic invertibility argument no longer apply. The authors reduce possible singularity to partially bipartite hypergraphs and explain that the boundary case can be handled for some small uniformities, but leave the general invertibility issue unresolved.