Existence of epichristoffel words of arbitrary lengths for fixed k-letter alphabets

Determine whether, for every fixed integer k ≥ 3 and every positive integer n, there exists an epichristoffel word of length n over a k-letter alphabet.

Background

Paquin provided an algorithm (via the operator T) to decide existence of epichristoffel k-tuples (letter counts), but a general length-existence theorem across all alphabets of size k ≥ 3 was left open. This paper introduces epichristoffel trees and corresponding Stern–Brocot constructions to generate infinite families of lengths and proves specific existence results for k=3 (e.g., Theorem 5.3). Nonetheless, the fully general claim—existence for every length n and each fixed k ≥ 3—remains unresolved.