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The task is to find such a system.","posed_since":"","precise":"Find a self-adjoint operator $H$ on a Hilbert space, defined independently of the zeros, with spectrum $\\{\\gamma_n\\}$ where $\\zeta(1/2 + i \\gamma_n) = 0$, and prove the spectral identity. The Berry-Keating proposal $H = xp$ with suitable boundary conditions reproduces the smooth counting function $N(E) \\sim (E/(2 \\pi)) \\operatorname{log}(E/(2 \\pi e)) + 7/8$ but not the individual zeros. Answer: an explicit operator with proof.","problem_ref":null,"references":"","settled_by":"A proof that the spectrum of an explicitly constructed self-adjoint operator equals the set of nontrivial zeros.","status_note":"Connes and collaborators proposed trace-formula and prolate-operator constructions (1998 to 2026) whose low spectra approach the zeros numerically; no proof of the identity.","title":"A self-adjoint operator whose eigenvalues are the Riemann 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