An exact moment criterion for survival under renewed spatial catastrophes

We prove an exact moment criterion for survival in a binary growth process subject to independently renewed spatial catastrophes. Occupied sites spread through a finite set of allowed increments on the integer lattice, and Poisson catastrophes with unbounded cube sizes empty all occupied sites in each affected cube. Under finite one-site hitting intensity, a positive finite-seed survival phase exists if and only if the cube-size measure has a finite moment of order one more than the spatial dimension. The divergent endpoint yields extinction at every positive intensity. The proof centres on a one-dimensional positive-phase construction for connected interval errors under a finite second moment. Extremal-path resampling gives a conditional correlation inequality, which controls exact interval insertion counts and a finite-history potential. Coordinate restriction and envelope covering then establish the criterion in every dimension. We also prove the bulk-to-seed reversal identity and extend the criterion to uniformly fat shapes. For the one-sided one-dimensional rule, the stationary hole density has a quadratic dilute expansion under the same second-moment condition. Its excess over independently eroded Poisson events is strictly positive at every positive intensity, with an explicit quadratic coefficient in the dilute limit.