Rare events in multiplicative sequences: Poisson laws, clusters, and observation
Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.PR — Probability
Contributed by Brice Pouly ↗
Version 1 / Oct 18, 2026 / CC BY 4.0
Abstract
A multiplicative identity can persist at every scale while disappearing from a rare-event count. Another identity can survive as a cluster whose points lie far apart. Which distinction matters depends on what is observed. We explain this through one completely calculated model: a uniform germ can trigger two occurrences in disjoint spatial windows. Its two marginal counts become Poisson, but their sum need not do so. A coupling identifies the correct same-grid target and separates the mean number of points from the mean number of active germs. We then isolate the additional work required by multiplicative sources: private prime coordinates, weighted relations between windows, corrections to prime-site intensities, and relations visible at the target scale. A final passage from marked configurations to records explains the roles of ties, recognition and observation clocks. Selected results are stated with their hypotheses; a short concordance leads to their proofs. The elementary calculations are proved here, while the arithmetic and functional limit theorems are taken from the four cited research articles.
Provenance statement
The following declaration describes the research and the published source edition:
The author organized and directed this work with substantial assistance from GPT-6 Astra (OpenAI), the only language-model system used for this article, for exposition, elementary calculations, source checks, diagnostics and file preparation. The author determines the claims and accepts responsibility for the text, references and remaining errors. AI-assisted checks and finite tests are not independent human peer review and do not replace the proofs.
For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review.
The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.22880994.
At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.
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Version history
- v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026