Contiguity of unions of perfect matchings and random regular uniform hypergraphs

For fixed integers s≥3 and r satisfying log(r!)>r(s-1)slog r, the union of r independent uniform perfect matchings of the complete s-uniform hypergraph is mutually contiguous to the uniform simple r-regular s-uniform hypergraph, as n→∞ through multiples of s. In particular, r= eˢ suffices. The main input is the sharp entropy inequality for random permutations of Bristiel and Caputo. It proves the global maximization needed for the second moment of the number of complete factorizations. The remaining calculation, including its lattice factor, matches the short-cycle contribution required by small-subgraph conditioning.\\ This is an expanded version of a proof provided by ChatGPT-Astra.