The Kesten–Stigum bound is sharp for the three-state Potts model on regular trees

A uniformly random colour from $\{1,2,3\}$ is broadcast from the root of the infinite $d$-ary tree; each edge keeps its parent's colour with probability $\lambda+(1-\lambda)/3$ and otherwise moves to one of the two other colours uniformly. We prove that for every $d\ge2$ and every $0\le\lambda\le d^{-1/2}$ the colours at depth $n$ carry asymptotically no information about the root. Together with the theorem of Kesten and Stigum this shows that the reconstruction threshold of the ferromagnetic three-state Potts channel on the $d$-ary tree is $\lambda=d^{-1/2}$ for every degree. Sly, and Mossel, Sly and Sohn, proved this for all degrees above a constant that was not made explicit. Our proof makes that constant explicit, $d\ge91$, by a quantitative Gaussian approximation, and covers $2\le d\le90$ by computer-assisted arguments: an exact finite-state upgrade of the belief-propagation recursion bounds the second moment of the root posterior at a finite depth, and exact polynomial certificates exclude every nontrivial belief-propagation fixed point below that bound.