Discuss Findings

across 1124 numeric sanitized payloads, the rescue-correlation signal is concentrated at rescue_amplitude=1.6 and p_stay=0.85: good-phase-correlated rescue has peak survival excess 0.03217 [0.03210, 0.03224], anti-correlated rescue only 0.00081 [0.00080, 0.00082], and the peak-gap is 0.03136 [0.03128, 0.03143]; at p_stay=0.85 the final survival fraction is 0.95306 [0.95301, 0.95310] for good-correlated rescue versus 0.99903 [0.99902, 0.99903] for bad-correlated rescue, but the phase-survival gap flips sign, +0.05994 [0.05977, 0.06012] versus -0.00359 [-0.00361, -0.00356]; the same ordering is already visible at p_stay=0.50 and becomes much larger by p_stay=0.95, where the good-correlated phase gap reaches +0.17045 [0.17009, 0.17080].

CONFIRMED Rescue immigration coupled to good phases preserves a strong interior-persistence structure, while bad-phase rescue almost completely flattens it. The key mechanism is not just higher raw survival: anti-correlated rescue keeps populations alive in bad epochs and therefore erases the persistence-sensitive contrast between good and bad stretches. Good-correlated rescue does the opposite, reinforcing long favorable runs and producing a pronounced phase-survival asymmetry that peaks at intermediate persistence. The result shows that rescue timing relative to environmental state is a first-order control parameter for near-critical branching, not a small perturbation.