Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions

For the empirical measure of the first $n$ points of one infinite sequence of independent uniform samples in the unit square, we identify the leading constant in the bounded-Lipschitz distance to area measure, equivalently in $W_1$, both in expectation and almost surely along all sample sizes. The constant is $c_{\ast }={\kappa }_1\mathrm{/}(2\sqrt{\pi })\in (0,1\mathrm{/}4]$, where ${\kappa }_1$ is the linear-growth Gaussian variational constant of Armegioiu–Goldman–Grotto–Trevisan. This answers the two-dimensional bounded-Lipschitz convergence question posed by Yukich in 1989. We give the complete endpoint argument, square–torus comparison, empirical reduction, and full-sequence probability argument.