Two exact frequency extrema for smooth sequences over {1,3}

Will Howe; Grant Huffman; Jensen Kohlmeyer; Liam Kruer

Two exact frequency extrema for smooth sequences over {1,3}

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

Using the frequency coding of Jamet, Marcovici, Poirier, and de la Rue, we determine the largest frequency of

1

in a smooth sequence over

\{1,3\}

and the largest such frequency below

1/2

. They are

1/2+1/(3+\sqrt{5})

and

1/2-1/(13-\sqrt{5})

, respectively. The unique attaining type directives are

1^\infty

and

0001^\infty

. The proof is an order comparison for two planar homographies. The first value corrects the numerical expression in the stated extremum question.

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 main one-sided frequency and extremizer statements; the separate bi-infinite identification in the exposition is outside that formalized scope. No independent referee acceptance is claimed.

Tools used

OpenAI: Codex
Lean: Lean

References

D. Jamet, I. Marcovici, L. Poirier, and T. de la Rue, Frequency of patterns in smooth sequences over the alphabet $\{1,3\}$ , arXiv:2604.11387v3, June 29, 2026, https://arxiv.org/abs/2604.11387v3.arXiv
D. Jamet, I. Marcovici, L. Poirier, and T. de la Rue, Frequency of Patterns in Smooth Sequences over the Alphabet $\{1,3\}$ , arXiv:2606.13846v1, June 11, 2026, https://arxiv.org/abs/2606.13846v1.arXiv

Version history

v1 · 6 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 10, 2026