Upper bounds for the Chvátal–Sankoff constants $\gamma_k$, $k\ge3$

Lueker (J. ACM 56(3), 2009) constructed a family of finite-state machines $\mathcal{M}_h$ whose associated transfer matrices yield upper bounds on the Chvátal–Sankoff constants $\gamma_k$, and proved (his Theorem 3.26) that these bounds converge to $\gamma_k$ for every alphabet size $k$. He published values only for $k=2$. For $k\ge3$ the best upper bounds we were able to find in the literature in September 2026 were still those of Dančík's 1994 thesis: subsequent work generalising Lueker has generalised his Section 2, the lower bound. We implement $\mathcal{M}_h$ for arbitrary $k$. One ingredient has no binary analogue — the generating function of a truncation class when both sides of a minimal match pair are truncated — and we derive it. We then give a reduction by relative forms under which the transition multiplicities of the transfer matrix are falling factorials in $k$; one alphabet-free construction consequently determines the exact matrix for every $k$. We obtain improved upper bounds for $k=3,\dots,15$, each supported by a Collatz–Wielandt certificate verified in exact rational arithmetic, among them $\gamma_5\le0.645033$. A note added on 31 October 2026 compares these values with those of a repository made public by L. Jack on 25 October 2026.