Discuss Findings

overall mean_peak_p_stay +0.65856 +/- 0.00011 and mean_interior_peak_excess +0.33260 +/- 0.00011 with interior_peak_rate +0.65900 +/- 0.00035; peak-group breakdowns show a=0.14 with k={0,1,2,4,12} gives peak_p={0.710,0.695,0.700,0.704,0.708} and excess={0.262,0.395,0.332,0.298,0.274} (all 95% CI widths <= 0.001), while a=0.28 gives peak_p={0.617,0.611,0.612,0.613,0.615} and excess={0.316,0.410,0.369,0.344,0.326} (all 95% CI widths <= 0.001); representative condition means are a=0.14,k=1,p=0.65 cv_tau 2.051 +/- 0.001, mean_tau 15.853 +/- 0.006, extinct_frac 0.970 +/- 0.000; a=0.28,k=1,p=0.65 cv_tau 1.893 +/- 0.001, mean_tau 13.900 +/- 0.005, extinct_frac 0.990 +/- 0.000; a=0.28,k=12,p=0.97 cv_tau 1.281 +/- 0.000, mean_tau 12.392 +/- 0.003, extinct_frac 0.767 +/- 0.000

CONFIRMED. The interior resonance is not a fragile single-parameter artifact: every amplitude-dispersion block retains a finite-memory peak in extinction-time heterogeneity, with the ensemble peak locked near p_stay approximately 0.66. Raising environmental amplitude shifts the preferred persistence lower, from about 0.70 at amplitude 0.14 to about 0.61 at amplitude 0.28, which is consistent with stronger environmental swings requiring less temporal memory to maximize lineage-to-lineage timing spread. Moderate offspring overdispersion is the strongest amplifier, with k=1 producing the largest interior-peak excess at both amplitudes, while heavier overdispersion gradually damps but does not erase the effect. The mechanism is a burstiness-memory coupling: negative-binomial offspring variability broadens extinction-time distributions most when environmental runs are long enough to create coherent good and bad stretches, but not so long that trajectories collapse into near-deterministic fast-die or long-survive regimes.