Excursion Lengths in a Uniform Eulerian Circuit of the Complete Graph: Aldous's Conjecture Holds if and only if i = o(n^(3/2))

Let N_i be the number of excursions of length i from a fixed vertex in a uniformly random Eulerian circuit of the complete graph K_n in which every edge is replaced by two opposite arcs. Aldous and Yu (Open Problems in Mathematics, 2014) stated as the natural conjecture that E N_i is asymptotic to exp(-i/n). We prove that for every fixed K, uniformly in 2 <= i <= K n^(3/2), E N_i = exp(-i/n - i^2/(2n^3))(1 + O(n^(-1/2) log n)), and that E N_i exp(i/n) tends to zero when i/n^(3/2) tends to infinity. Consequently E N_i is asymptotic to exp(-i/n) if and only if i = o(n^(3/2)). This range contains fixed i, the scale i proportional to n and essentially all excursions, and the length of a uniformly chosen excursion, divided by n, converges in distribution to the standard exponential law. At the scale i proportional to y n^(3/2) the ratio E N_i/exp(-i/n) tends to exp(-y^2/2), so the conjecture read literally for all i is false. The proof combines the uniform-spanning-tree construction of a uniform Eulerian circuit (the BEST theorem, in the form used by Kandel, Matias, Unger and Winkler), a hazard representation of the first excursion, and a convexity bound. Exact computations for n <= 6 and Monte Carlo simulations up to n = 6400 agree with the results. Scope: This result answers example (a), the complete bidirected graph, of Aldous and Yu's random Eulerian circuits note, corresponding to corpus record AMR-096-0015. It proves the conjectured excursion-count asymptotics in the sharp range i = o(n^(3/2)) and disproves the literal all-i formulation. The torus conjecture, the Hamming-cube example, and the broader random-Eulerian-circuit programme are not solved here. Monte Carlo computations are evidence only and are not used as proof. No absolute priority claim is made. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-096-0015. Public paper page: https://eulersolve.org/papers/amr-096-0015/