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The conjecture is that $\\lambda$ can never be smaller than a fixed number set only by the number of spacetime dimensions.","posed_since":"2006","precise":"Distance conjecture (Ooguri and Vafa 2006): along a geodesic of length D in moduli space, measured in Planck units, a tower of states has masses $m \\sim \\exp(-\\lambda D)$. Sharpened version (Etheredge, Heidenreich, Kaya, Qiu, Rudelius 2022): the lightest tower obeys $\\lambda \\ge 1/\\sqrt{d-2}$ in d spacetime dimensions. Prove the bound for all infinite-distance limits of string compactifications, or find a limit with $\\lambda < 1/\\sqrt{d-2}$.","problem_ref":null,"references":"","settled_by":"A proof over all infinite-distance limits of string compactifications in each $d$, or an explicit limit with $\\lambda < 1/\\sqrt{d-2}$.","status_note":"Proposed in 2022 with the bound preserved under dimensional reduction and saturated in many string and M-theory compactifications.","title":"Sharp lower bound on the exponent in the distance 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