One-dimensional quadratic transport uniqueness criterion ($\pi \{x\le a,y\le b\}=\min \{F(a),G(b)\}$)

For atomless source and target probabilities with continuous strictly increasing cumulative distributions, the squared-distance optimal coupling is unique. Two-point optimality forbids crossed support pairs. This makes every pair of source and target lower-tail events nested up to null sets and forces the displayed joint distribution, which is the common-quantile coupling and is induced by $G^{-1}F$.