Sharp asymptotics for permutation uncertainty in the Gaussian location model

We derive the sharp two-term asymptotic for the total permutation uncertainty n(1−κ(μ)), the sum of Bernoulli variances of all pairwise rank indicators, in the equally-spaced Gaussian location model as the signal-to-noise ratio δ→∞. The leading constant is Θ(n), not the O(n3) suggested by a uniform-pair bound; the overestimate by a factor Θ(n2) stems from ignoring a concentration-of-pairs phenomenon in which only the n−1 nearest-neighbour pairs contribute at leading order, while all other pairs are exponentially suppressed. The exact second-order correction −2/δ2, arising from the Mills ratio expansion, is non-negligible for moderate signal-to-noise ratios and has direct implications for temperature initialisation in differentiable sorting.