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Determine whether such a law fitted for $N \\le 1e6$ predicts $D(N)$ and the resulting intensity decay at $N \\sim 1e9$ (about 1e5 s of LHC operation) to within 10 percent, and derive $\\kappa$ from the resonance structure of the map.","problem_ref":null,"references":"","settled_by":"Direct tracking to 1e9 turns of a realistic lattice compared with the extrapolation, plus a derivation of $\\kappa$ from normal-form or resonance analysis.","status_note":"Fitted scaling laws described LHC intensity and luminosity evolution in several fills (2018-2019 papers); a derivation of the exponent and a test at 1e9 turns are missing.","title":"Extrapolating dynamic aperture from a million to a billion 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