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The mechanism and the exact conditions under which this holds are not established.","posed_since":"1984","precise":"For a quantum system whose classical limit is fully chaotic (Anosov flow or ergodic billiard), the unfolded level statistics as $\\hbar \\to 0$ are expected to match GOE (time-reversal invariant) or GUE (no time-reversal symmetry). Determine the mechanism and the exact class of systems; known exceptions such as arithmetic surfaces show Poisson statistics. Answer: a characterization with proof; the sharper child problems are where work proceeds.","problem_ref":null,"references":"","settled_by":"A theorem identifying a class of chaotic systems with random-matrix spectral statistics and explaining the arithmetic exceptions.","status_note":"Periodic-orbit theory reproduces the random-matrix form factor to all orders in $\\tau$ non-rigorously (Muller, Heusler, Braun, Haake and Altland, 2004).","title":"Why chaotic quantum spectra follow random-matrix 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