Edges of the uniform random forest of K_n are pairwise negatively correlated for every n

Let Fₙ be uniform on all forests of the simple complete graph Kₙ, with isolated vertices allowed. A conjecture of Kahn and of Winkler, studied by Grimmett and Winkler, asserts that any two distinct edges of any finite graph are negatively correlated under the uniform forest measure. Stark proved this for Kₙ once n is sufficiently large, but did not furnish an explicit threshold. We prove it for every n ≥ 2, strictly whenever two distinct edges exist. The difficulty is concentrated in the disjoint-edge orbit, whose correlation ratio tends to one. We remove this cancellation before estimating anything: the desired inequality becomes an exact comparison among the first two moments of the component count and the expected sum of squared degrees. When the component count fluctuates, this comparison contains a variance term absent from the fixed-component identities of Tang and Zhang. Component marking, tail elimination, and effective Stirling bounds control the three moments for n ≥ 651; exact integer recurrences cover the remaining values. This is version v1 of the preprint and its reproducibility companion. It has not been peer reviewed. The source archive contains the LaTeX manuscript, bibliography, standalone exact verifier, build instructions, citation metadata, and licence information. The maintained companion repository is https://github.com/agupta/uniform-forests-complete-graphs.