Fourth-order diffusion approximation for three-player gambler's ruin

For three-player fair gambler's ruin, we prove that the probability of each elimination order differs from its Brownian scaling limit by $O((nN)^{-4})$ when fixed positive capitals with total $N$ are multiplied by $n$. This is Conjecture 4.2(b) of Diaconis and Ethier. The error is antisymmetric when the two losing players are exchanged. On the resulting half triangle, the Brownian harmonic function has a sixth-order stencil defect with the angular factor $\sin(6\theta)$. The explicit barrier $r^{-4}\sin(6\theta)$ and a finite discrete comparison argument give the claimed rate.