We show that computing a median under the Ulam distance is NP-hard even when the input consists of exactly four permutations. Previously, NP-hardness was known only for an unbounded number of input permutations (Fischer et al., ESA '25). Our result is tight, since an Ulam median of three permutations can be computed in polynomial time (Chakraborty--Das--Krauthgamer, SODA '21).

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