A uniform Kolmogorov bound for Jack measures

For the normalized transposition statistic under Jack measure on partitions of $n$, we prove a Kolmogorov bound $10\max\{n^{-1/2},\sqrt\alpha/n\}$ for every $n\ge3$ and $\alpha\ge1$. This removes the logarithmic loss in the known bound, including the transition $\alpha\asymp n$, and proves the uniform estimate conjectured by Fulman. The proof retains cancellation in the conditional third moment of a Jack growth exchangeable pair. Its expansion in three orthogonal Jack characters gives the required parameter dependence.