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This is the half of the mobility-edge problem that has no proof at all.","posed_since":"1958","precise":"For $H = \\Delta + \\lambda V$ on $\\ell^2(Z^3)$, with $\\Delta$ the nearest-neighbour hopping operator (spectrum $[-6, 6]$ at $\\lambda = 0$), i.i.d. bounded $V$ and $\\lambda$ small, prove that the almost-sure spectrum has a nonzero absolutely continuous component in some energy interval inside $(-6, 6)$, or at least that eigenfunctions in that interval are not exponentially localized. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of absolutely continuous spectrum, or of delocalized eigenfunctions in finite volume uniformly in the volume, for $d = 3$.","status_note":"Absolutely continuous spectrum at weak disorder is proven on tree graphs (Klein, 1998, and later work of Aizenman and Warzel), not on $Z^d$.","title":"Absolutely continuous spectrum of the 3D Anderson model at weak 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