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The full list is known only up to five regions.","posed_since":"2015","precise":"For n boundary regions, the Ryu-Takayanagi entropies of all $2^n - 1$ unions form a polyhedral cone (Bao et al. 2015). Determine the complete list of facets (holographic entropy inequalities) for $n = 6$; the cone is known for $n \\le 5$.","problem_ref":null,"references":"","settled_by":"A complete list of $n = 6$ facets with a proof that each is valid and tight and that no others exist.","status_note":"$n = 5$ was completed in 2019; 2024-2026 work computed extreme rays of the 6-party strong-subadditivity-compatible cone and applied reinforcement-learning searches, without a complete 6-party facet list.","title":"Holographic entropy inequalities for six 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