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Palindromic sections in two Kolakoski phase actions

Primarily AI-generated textHuman understanding: all partsmath.DS — Dynamical Systems

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

We prove the two specified-phase orbit statements in Shen's Conjectures 3.4 and 3.7. The all-ones phase for the alphabet {1,n}\{1,n\}, with nn positive and even, and both alternating phases for {−1,n}\{-1,n\}, with nn any even integer, have orbit length 2⌈d/2⌉2^{\lceil d/2\rceil} at depth dd. The proof uses unreduced signed words and a palindrome class preserved by taking the section of the square at two phase coordinates. It also gives the all-ones statement for nonpositive even nn under Shen's displayed signed recursion. These are finite phase-orbit results; they do not imply the density conjecture for the classical Kolakoski sequence.

Provenance statement

The mathematical draft and exposition were produced primarily with OpenAI Codex under the direction of the listed contributors. Lean was used to verify the exact specified-phase periods, all-return-times formulation, and identification with the original recursion. No independent referee acceptance is claimed.

Tools used

OpenAI
Codex
Lean
Lean

References

  1. Noah MacAulay, Group theory of the Kolakoski sequence , preprint, 2026, arXiv:2605.14234v1, https://arxiv.org/abs/2605.14234v1.arXiv
  2. Bobby Shen, The Kolakoski sequence and related conjectures about orbits , Experimental Mathematics 29 (2020), no. 1, 54--65, https://doi.org/10.1080/10586458.2018.1434703. The numbering used here is that of arXiv:1702.08156v2 (2017), https://arxiv.org/abs/1702.08156v2.arXivDOI

Version history

  1. v1 · 5 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 10, 2026