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Abstract
Let f 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), 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 f, 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.
Supplementary materials
Title
Technical companion to Long runs and rare patterns of a random completely multiplicative function
Description
This companion completes the proof package for the article, Version 3. Appendices A–E give the growing-degree Runge and bounded-height Pell calculations, prime cutoffs, scalar smoothing, marked and dictionary comparisons, conditional kernels, and the boundary and relative-bulk estimates. The common macroscopic square-product argument remains in the article.
Appendix F proves the full same-grid signed prefix comparison. It assembles the existing arithmetic inputs and an independently proved scalar tail, then gives the finite Palm–Stein inequality, the uniform reciprocal-pivot estimate and the exact one-word conditional coupling.
Appendix G is a separate methodological complement: regular target configurations, signed Palm-void identities and an arithmetic obstruction to global absolute cumulant smallness.
Its vanishing Palm deficit is a consequence of Appendix F, never an input to its proof.
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Title
Lean formalization
Description
This release provides the Lean 4 formalization of the paper and its companion.
The development covers critical Poisson fields and conditional transfers, word dictionaries and marked clusters, threshold staircases and Poisson–Gaussian limits, boundary and longest-run laws, microscopic–bulk crossover, microscopic and empirical fields, and Palm comparisons with cumulant obstructions.
The development uses Lean 4.34.0 and Mathlib v4.34.0. The seven selected interfaces were checked with Comparator, Lean and NanoDa.
Formal coverage is relative to seven explicit literature inputs: scalar Poisson Stein factors; finite Arratia–Goldstein–Gordon process comparison; the prime number theorem remainder; multivariate Poisson Stein solution bounds; the Laishram–Shorey prime-divisor estimate; Shorey's square-product exclusion; and the Nicolas–Robin divisor bound. In particular, the analytic Stein input used for companion F.2 is assumed in the formal theorem; the printed semigroup construction is outside that formal proof. Kernel verification does not discharge these premises or constitute independent human peer review.
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