Extrapolating dynamic aperture from a million to a billion turns

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 $D(N) = D_{\infty} + b/(\mathrm{log}\, N)^{\kappa}$. Determine whether such a law fitted for $N \le 10^{6}$ predicts $D(N)$ and the resulting intensity decay at $N \sim 10^{9}$ (about $10^{5}$ s of LHC operation) to within 10 percent, and derive $\kappa$ from the resonance structure of the map.

Direct tracking to $10^{9}$ turns of a realistic lattice compared with the extrapolation, plus a derivation of $\kappa$ from normal-form or resonance analysis.