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Periodic counterexamples to three conjectures on Kolakoski transforms

Primarily AI-generated textHuman understanding: all partsmath.CO — Combinatorics

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

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

The simple Kolakoski transform need not preserve the limiting mean of a periodic word on {1,2}\{1,2\}. We give a counterexample with input period 2424 and output period 1212, and a balanced counterexample whose input has mean 3/23/2 while its output has mean 13/913/9. Both examples remain unchanged under every realization of a nonconstant summable perturbation of the weights. Their discrepancies from the input means are linear. Consequently they contradict Conjectures (1), (2), and (4) of Cloitre. These statements concern auxiliary transforms of periodic inputs; the classical Kolakoski frequency problem remains open.

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 periodic transforms, limiting means, weighted perturbations, and discrepancy conclusions.

Tools used

OpenAI
Codex
Lean
Lean

References

  1. Benoit Cloitre, The Kolakoski transform of words , manuscript, October 6, 2013, 27 pp., https://oeis.org/A000002/a000002.pdf.link

Version history

  1. v1 · 5 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 11, 2026