A direct inductive proof of the sharp Merino--Welsh threshold for matroids

Motivated by the Merino–Welsh conjecture, we consider the smallest $c\ge 0$, denoted by $c_{\ast }$, for which the inequality $T_M(c,0)T_M(0,c)\ge T_M(1,1{)}^2$ holds for every loopless and coloopless finite matroid $M$. The counterexamples constructed by Beke, Csáji, Csikvári, and Pituk [\emph{Adv. Math.} $\mathbf{\text{446}}$ (2024), 109674] give the lower bound $x_0$, where $x_0\approx 2.22668$ is the largest real root of the polynomial $x^3-9(x-1)$. Later, Csikvári [\emph{European J. Combin.} $\mathbf{\text{137}}$ (2026), 104402] improved the known upper bound for this constant to $2.35$ and then conjectured that the above inequality holds at $c=x_0$. This conjecture was recently proved by Liu (2026). We give an alternative direct inductive proof that $c_{\ast }=x_0$, without computer-assisted finite verification.