HEXAGONbeta

An exact moment criterion for survival under renewed spatial catastrophes

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

Contributed by Man Huang ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

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.

Author comments: Based on Zenodo v1 (3 November 2026), https://doi.org/10.5281/zenodo.23911961. Mathematical text and references unchanged; portable font and provenance updates only. Original source and finite diagnostics: same DOI.

Provenance statement

The mathematical text, abstract and references reproduce Zenodo v1, first made public on 3 November 2026 (https://doi.org/10.5281/zenodo.23911961). Repository-only changes replace the macOS-specific PT Serif and Menlo fonts with TeX-distributed Latin Modern fonts and add a repository notice and provenance disclosure; no mathematical claims or proofs are changed. The contributor confirms that the text was primarily generated by AI and then adjusted by the contributor. The contributor independently understands some arguments but has not mastered the full argument. The public manuscript records OpenAI GPT-6.1 Sol and xAI Grok 4.7 assistance for literature synthesis, development of theoretical arguments, expansion of propositions into drafts, language revision, clarification of definitions and examination of proof steps. Additional AI assistance was used for this repository package and metadata. The original companion archive documents exact finite diagnostics on the listed finite configurations and coefficient identities. These finite checks do not prove the general or infinite-volume results. Completed author-performed line-by-line verification and independent human expert verification of the full proof are not asserted. The author takes responsibility for the manuscript; no AI system is an author or contributor. This packaging performs compilation, version-integrity and layout checks, not a new mathematical proof review. The original source archive and finite diagnostics remain available through the Zenodo DOI. Private author records, correspondence, local build products and third-party papers are excluded from the source upload.

Tools used

OpenAI
GPT modelsVersion GPT-6.1 Sol
xAI
GrokVersion Grok 4.7

References

No bibliography entries.

Version history

  1. v1 · 17 pagesSubmitted by Man HuangBased on Zenodo v1 (3 November 2026), https://doi.org/10.5281/zenodo.23911961. Mathematical text and references unchanged; portable font and provenance updates only. Original source and finite diagnostics: same DOI.CurrentNov 10, 2026