Successor right-special strings with few Burrows--Wheeler Transform runs

We study successor right-special strings over an alphabet $\mathrm{\Sigma }$ of size $\sigma$, a minimal-branching analogue of de Bruijn strings, and ask how few Burrows–Wheeler transform (BWT) runs are possible. In a de Bruijn string of order $k$, every $(k-1)$-context has all $\sigma$ right-extensions; here, every context is still right-special but has exactly two right-extensions, chosen by a successor rule. For order $3$, we construct an explicit family $B_{\sigma }^{(3)}$, for every $\sigma \ge 2$, whose cyclic BWT has $r_c={\sigma }^2+2$ runs. A suitable terminated linearization has the same run count, $r=r_c={\sigma }^2+2$, while the smallest suffixient set has size $\chi =2{\sigma }^2+1$. The ratio $\chi \mathrm{/}r=2-3\mathrm{/}({\sigma }^2+2)$ then quantifies how nearly this forced branching saturates the known bound $\chi \mathrm{/}r\le 2$, which we have previously shown to be asymptotically tight. Compared with our earlier alphabet-growing construction, this improves the gap from $O(1\mathrm{/}\sigma )$ to $O(1\mathrm{/}{\sigma }^2)$. We also show that the order-$3$ pattern appears as a blockwise two-row projection of normalized linear-feedback shift register (LFSR) de Bruijn sequences over ${\mathbb{F}}_q$, when such primitive trinomials $x^3-x+c$ exist. For higher orders, we analyze the natural boundary-merged candidate $L_{\sigma ,k}$ using the last-to-first (LF) permutation: it fails for $k=4$ and all $\sigma \ge 3$, while verified $k=5$ instances for $\sigma \in 3,4$ yield $\chi \mathrm{/}r$ ratios exceeding $1.96$.