A $2$-branching construction for the $χ\leq 2r$ bound

Abstract: The string repetitiveness measures $χ$ (the size of a smallest suffixient set of a string) and $r$ (the number of runs in the Burrows--Wheeler Transform) are related. Recently, we have shown that the bound $χ\leq 2r$, proved by Navarro et al., is asymptotically tight as the size $σ$ of the alphabet increases, but achieving near-tight ratios for fixed $σ> 2$ remained open. We introduce a \emph{2-branching property}: a cyclic string is 2-branching at order~$k$ if every $(k{-}1)$-length substring admits exactly two $k$-length extensions. We show that 2-branching strings of order~$k$ yield closed-form ratios $χ/r = (2σ^{k-1}+1)/(σ^{k-1}+4)$. For order~$3$, we give an explicit construction for every $σ\geq 2$, narrowing the gap to~$2$ from $O(1/σ)$ to $O(1/σ^2)$. For $σ\in {3,4}$, we additionally present order-$5$ instances with ratios exceeding~$1.91$.