Keane’s Conjecture on the 1-density in the Kolakoski sequence K(1,2)

Prove Keane’s conjecture that the asymptotic frequency of the symbol 1 in the Kolakoski sequence K(1,2) exists and equals 1/2. Equivalently, determine whether the limiting density of 1s in K(1,2) exists and is 1/2.

Background

The paper studies structural properties of the Kolakoski sequence K(1,3) and contrasts them with the long-standing difficulties of the classical Kolakoski sequence K(1,2). In this context, it recalls a central unresolved problem about K(1,2) attributed to Keane.

Keane’s conjecture concerns the asymptotic frequency (density) of the symbol 1 in K(1,2), asserting that this limit exists and equals 1/2. Despite extensive study, the existence of this limit and its value remain unproven, making it a benchmark open problem in the area.