Sharp Wasserstein stability of a multivariate characteristic-function defect

For a law on the closed unit polydisk, consider the difference between its product moment and the product of its coordinate means. The sharp universal bound is attained by a common fair sign on two phases, up to coordinate rotations. We prove that, for every fixed \(d≥3\) and finite \(p≥1\), a deficit \(Δ\) from this bound controls the Wasserstein distance to the entire equality orbit by \(Kd,pΔ1/(2p)\) whenever \(0≤Δ≤δ_d\), for an explicit positive threshold \(δ_d\). The argument separates location from mass: an exact telescoping inequality propagates the full deficit to partial products; three coordinates synchronize their centered fluctuations to one almost fair sign; restoring fairness transports the excess mass across a nonzero two-point gap. The exponent is optimal for every \(p\), even after optimizing all coordinate phases. We compute its leading coefficient on biased extremizers and an exact formula for their \(W_2\) distance to the full orbit. A structured discrepancy that keeps location and mass errors separate has the sharper, and optimal, square-root rate.