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The answer shows whether the arrow of time is tied to cosmic expansion or only to the initial state.","posed_since":"1962","precise":"For a closed Friedmann universe with matter and radiation that recollapses, with low-entropy initial conditions at the big bang and no imposed final condition, determine whether coarse-grained entropy (including a gravitational contribution) increases monotonically to the big crunch, and whether two-time boundary conditions with low entropy at both ends are consistent with the statistical mechanics of matter plus gravity. Answer: yes or no for monotonic increase, with conditions.","problem_ref":null,"references":"","settled_by":"A statistical-mechanical calculation of entropy evolution through recollapse for a specified matter content, including structure formation and black holes.","status_note":"Hawking argued for reversal in 1985 and later withdrew the claim; no complete statistical treatment exists.","title":"Does the thermodynamic arrow reverse in a recollapsing 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