Explain the critical eccentricity-ratio criterion

Determine the dynamical origin of the empirical relation $e_{\mathrm{s}}^{\mathrm{crit}}\approx5\sigma_{\mathrm{e}}^{\mathrm{crit}}$ governing the transition from stable to unstable apsidally aligned planetary systems.

Background

The simulations identify an empirical transition at which the instability timescale of initially apsidally aligned systems decreases sharply: the critical systematic eccentricity is approximately five times the critical dispersion eccentricity. The critical dispersion eccentricity is broadly consistent with mean-motion-resonance-overlap estimates, but the critical systematic eccentricity is not reproduced by those analytical predictions, suggesting that a distinct dynamical mechanism controls the breakdown of apsidal protection.

The paper therefore leaves unresolved why the ratio between the two critical eccentricities is approximately five and what dynamical process sets the systematic-eccentricity boundary.