Extrapolating dynamic aperture from a million to a billion turns
In plain words
Computers can track particles for about a million turns, but colliders store beams for billions; the rule used to extrapolate the stable region is a conjecture.
Precise statement
The dynamic aperture $D(N)$, the largest initial amplitude stable for $N$ turns of the 4D or 6D symplectic one-turn map of a hadron collider, is fitted by Nekhoroshev-type laws D(N) = D_inf + b/(log N)^kappa. 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.
What would settle it
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 in the literature
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.