HEXAGONbeta

Finite-preparation response under heavy-tailed catastrophes: exact amplitudes and separation from Poisson coverage

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

Contributed by Man Huang ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

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 νr∼r−pL(r)\nu_r\sim r^{-p}L(r), p>3p>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.

Author comments: Mathematical content and references follow Zenodo v1 (3 November 2026), https://doi.org/10.5281/zenodo.23912425. AI-use and verification-status declaration clarified; no mathematical results changed. Supplement and reproducibility archive: same DOI.

Provenance statement

The text was primarily generated by AI and then adjusted by the author. The author reports independently understanding some arguments, but not having mastered the full argument. This repository package reproduces the mathematical content and bibliography of Zenodo v1, published on 3 November 2026 (https://doi.org/10.5281/zenodo.23912425). The repository version note and AI-use/verification declaration are the only prose changes; no new mathematical review was performed for this packaging. AI assisted theoretical development, literature synthesis, proof drafting and exposition. The manuscript records OpenAI GPT-6.1 Sol and xAI Grok 4.7; AI assistance also included deterministic computations. The Hexagon package and metadata were prepared with additional AI assistance. Available internal records contain AI-assisted scoped proof checks, finite diagnostics and certificate replays. These do not verify the full proof. Completed author-performed line-by-line verification or independent human expert verification of the full manuscript is not asserted. General-gamma quantitative constants have not all been independently reconstructed. The original-source access gap for Harris (1960), reference [10], remains. The inherited Zenodo version used a generic sentence that all AI-assisted output had been reviewed, verified and edited by the author. For this Hexagon package, that sentence is replaced by the scoped description above; it should not be interpreted as completed line-by-line mathematical verification. The author takes responsibility for the work. No AI system is an author or contributor. The technical supplement and reproducibility archive remain publicly available at the Zenodo DOI. Internal author records and third-party literature full texts are excluded from this source package.

Tools used

OpenAI
GPT modelsVersion GPT-6.1 Sol
xAI
GrokVersion Grok 4.7

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Version history

  1. v1 · 36 pagesSubmitted by Man HuangMathematical content and references follow Zenodo v1 (3 November 2026), https://doi.org/10.5281/zenodo.23912425. AI-use and verification-status declaration clarified; no mathematical results changed. Supplement and reproducibility archive: same DOI.CurrentNov 11, 2026