HEXAGONbeta

Long runs and rare patterns of a random completely multiplicative function

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.PR — Probability

Contributed by Brice Pouly ↗

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

Let ff be a random completely multiplicative function with independent symmetric signs at the primes. We study rare constant runs and prescribed words at logarithmic lengths, despite long-range multiplicative identities. A uniform weighted two-window square-relation estimate, with critical bound Oε(N5/3+ε)O_\varepsilon(N^{5/3+\varepsilon}), separates exact rational relations from residual components. At critical intensity, this estimate gives Poisson comparisons retaining positions, exact excess lengths and signs. For dictionaries, uniform bounds under size and overlap conditions are complemented by typical bounds obtained by averaging the distance after fixing the dictionary. In a quantitatively controlled regime of diverging intensity, an exact one-word coupling by largest-odd-prime pivots compares the full signed run field under prescribed small-prime conditioning. For fixed admissible window parameters and almost every fixed realization of ff, run-start counts in prescribed near-macroscopic windows satisfy an empirical Poisson law along dyadic scales. In the exceptionally rare regime, a relative marked comparison separates the boundary event from bulk occurrences. Conditional on existence, the first location, sign and overshoot have a two-source lattice law with scale-dependent weights. The two overshoots use different clocks: prime rank at the boundary and integer distance in the bulk.

Provenance statement

The research was organized and directed by the human contributor with substantial assistance from language model systems, including Claude Fable 5 and Claude Opus 4.8 (Anthropic), and GPT-5.6 Sol and GPT-6 Astra (OpenAI). These systems were used to explore candidate lemmas and proof routes, draft and reorganize manuscripts, assist with the separate Lean development, write diagnostic programs, check references and build artifacts, and conduct adversarial reviews through separate instances.

Formalizations

Tools used

Anthropic
Claude OpusVersion 4.8
Anthropic
Claude FableVersion 5
OpenAI
ChatGPTVersion 5.6 Sol
OpenAI
ChatGPTVersion 6 Astra

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 13, 2026