Sharp Wasserstein stability of a multivariate characteristic-function defect

For laws \(P\) on the closed unit polydisk, let \(D(P)\) denote the product-moment factorization defect and \(\Delta(P)=C_d-D(P)\) its deficit from the sharp universal constant. We prove a quantitative inverse theorem: for each fixed \(d\ge3\) and every finite \(p\ge1\), there are positive constants such that \(\operatorname{dist}_{W_p}(P,\mathcal E_d)\le K_{d,p}\Delta(P)^{1/(2p)}\) whenever the deficit is sufficiently small. Here \(\mathcal E_d\) is the entire equality orbit of common fair-sign two-point laws, including coordinate rotations. The exponent \(1/(2p)\) is optimal even after minimizing over all phase rotations. A family of biased two-point extremizers gives matching asymptotics and an exact formula for its \(W_2\)-distance to the equality orbit. The proof identifies distinct position and mass errors; when these are measured separately, a natural structured discrepancy admits the sharper and again optimal square-root rate. We also treat nonidentical prescribed mean moduli under explicit hypotheses. The statement is a stability theorem for near-extremizers, not an unqualified global Wasserstein estimate.