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 nu_r ~ r^(-p)L(r), p > 3, 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. The record includes the main manuscript (36 pages), technical supplement (ESM_1.pdf, 19 pages), reproducibility code and complete outputs (ESM_2.zip), matching source archives, README, licence notice and SHA-256 checksums. The Python programs use the standard library. No empirical dataset is used. AI-use, funding, competing-interest and availability declarations are included in the manuscript. Keywords: interacting particle systems; spatial catastrophes; regularly varying tails; recovery fronts; finite preparation; Boolean coverage. MSC 2020: 60K35; 60G55; 60K37. Related preprint: An exact moment criterion for survival under renewed spatial catastrophes, https://doi.org/10.5281/zenodo.23911961. Shared foundations and distinct results are identified in the introduction; the necessary proofs are included here.