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Sparse prime surgery in completely multiplicative sequences: Normality, rare-event realization and optimal repair

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.PR — Probabilitymath.DS — Dynamical Systems

Contributed by Brice Pouly ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

We study how changing prime signs affects fixed-word frequencies, rare-event laws and extremes of completely multiplicative sequences. The cost is dσ(f,g)=∑f(p)≠g(p)p−σd_\sigma(f,g)=\sum_{f(p)\ne g(p)}p^{-\sigma}. A known theorem of Frantzikinakis-Lemańczyk-de la Rue preserves normality at finite d1d_1; we give its elementary specialization. Using explicit arithmetic inputs from Long runs, we then establish three families of results. For each fixed 0<σ≤10<\sigma\le1 and almost every base, reversible surgery realizes every stationary finite-intensity marked point-process law along selected scales, with signs, exact excess lengths and converging actual intensity, at arbitrarily small dσd_\sigma cost and with a prescribed prefix unchanged. Almost surely over all integer scales, the minimum number of prime changes needed for a run of length ⌈clog⁡2N⌉\lceil c\log_2N\rceil is asymptotic to cαclog⁡2Nc\alpha_c\log_2N for 1<c<3/21<c<3/2, where αc∈(0,1/2)\alpha_c\in(0,1/2) satisfies H2(αc)=1−1/cH_2(\alpha_c)=1-1/c and H2(u)=−ulog⁡2u−(1−u)log⁡2(1−u)H_2(u)=-u\log_2u-(1-u)\log_2(1-u). At each scale, with probability tending to one, symbolic and private large-prime repair maxima coincide on the same random base for kN=o(log⁡N)k_N=o(\log N); unrestricted prime repair agrees when kN=o(log⁡log⁡N)k_N=o(\log\log N). This comparison transfers the classical lattice laws of runs with errors to adaptive prime repair. Elementary programming from arbitrary bases and an autonomous finite-range-dependent process construction complement these results.

Provenance statement

The following declaration describes the research and the published source edition: GPT-6 Astra provided substantial assistance in developing and consolidating arguments, mathematical exposition, literature comparison, and preparation of the manuscript and diagnostic programs. The author determines the claims and scope and retains responsibility for the arguments, references and any remaining errors. AI-assisted checks are not independent human peer review. 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.22959140. 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.

References

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026