Non-Lipschitz Wasserstein stability of Brenier potentials on a smooth planar dumbbell

We give an explicit three-atom obstruction to Lipschitz stability in the target's 2-Wasserstein distance for source-mean-zero Brenier potentials. The source is uniform on one fixed bounded connected smooth planar dumbbell, and all targets lie in the closed unit ball. The three sites remain uniformly separated. A small atom has a source cell consisting of two disk caps, one in each room. Moving this atom while preserving all three masses gives potential distance asymptotic to a positive constant times h^(3/2), whereas the target distance is asymptotic to a positive constant times h^(7/4). Consequently no target-uniform estimate with exponent greater than 6/7 holds on this fixed source. The value 6/7 is the exact exponent along the constructed family, not a claimed global stability endpoint. We also show directly why a convex-source cell-moment inequality cannot extend to this source. The sharp width dependence of the global W_2^(1/2) potential constant remains open; a separate axial target calculation attains the conjectured width scale within that subclass.