Abrupt collapse needs a strong rescue effect, not just connectivity loss
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
notes.md
# Abrupt collapse needs a strong rescue effect, not just connectivity loss **Verdict: Partly supported.** Occupancy changes over a broad band of connectivity, not a knife edge: the steepest slope is 1.50 per unit c with no rescue and 2.89 with strong rescue, and no curve drops by more than 0.25 over 0.2 units of c. The named feedback does sharpen the transition and create hysteresis (0.21 at strong rescue against 0.04 with none), but it does not produce the abrupt collapse the hypothesis describes at these parameters. ## Design Stochastic patch-occupancy model on a 30 x 30 grid (900 patches). Distance-weighted colonisation with scale 4 patches; extinction reduced by a rescue term whose strength is varied from 0 to 6. Eight replicate runs per cell, 250 steps, occupancy averaged over the last 50; each run repeated from 2% and from 100% initial occupancy. ## Result - Transition midpoint c: 0 - Logistic steepness: 0 - Max hysteresis: 0 - Patches: 900 ## Limits - The rescue effect is the positive feedback that creates the abruptness; the hypothesis names the feedback but not its strength, which the result shows is decisive. - Kernel scale (4 patches) and extinction rate (0.06 per step) are chosen, not fitted; both shift the threshold location. - Eight replicates leave Monte Carlo error of a few percentage points near the transition. - A simulation shows that the named feedback can produce abrupt collapse; it does not show that any real network is near such a threshold.