math.PR — Probability
Finite-preparation response under heavy-tailed catastrophes: exact amplitudes and separation from Poisson coverage
We determine the terminal response of a one-dimensional binary recovery process to a finite preparation history under independently renewed Poisson interval catastrophes. For regularly varying interval weights , , and fixed small positive intensities, we prove an exact leading response when preparation and observation durations grow proportionally. The main result expresses the response as an explicit functional of the actual recovery-front speeds accumulated along each fixed finite-stage preparation history and of the selected target-phase open density. Its weak-intensity second variation is a strictly positive quadratic form in every nonzero fixed signed temporal modulation, including equal-dose redistributions. A separate four-update theorem identifies local cooperation through an explicit positive equal-dose contrast for every nonzero interval law with finite first moment. These positive contrasts separate the process from conditional Poisson Boolean coverage with fixed grain kernels, a fixed environment distribution and affine intensity control, whose contrast is nonpositive at every finite time. As an effective extension, relative power-law tail bounds yield a conservative computable sufficient threshold valid at every later integer time. The core proof combines nonstationary front transfer, uniform background forgetting and exact resetting at intensity switches.