Transport distance to joint mixability and quantitative characteristic-function gaps

A constant-sum coupling is an exact compatibility condition on marginal laws. We measure its failure by the least possible \(L^p\) residual of a weighted sum with the individual marginal means fixed. This residual admits an exact projection identity with the set of jointly mixable marginal tuples in a product Wasserstein metric, extending a variance-stability inequality already known in the \(L^2\) setting. For identical nonnegative marginals, the mean-preserving positive-support projection is also exact in \(W_1\). A classical dependent-rounding example yields an exact discrete residual, while smooth perturbations give explicit lower bounds even when a support–mean test is inconclusive. We then transfer this distance to a quantitative gap below the fixed-marginal characteristic-function profile. The transfer requires specified frequency isolation, strong Fourier log-concavity, and a quadratic phase margin; the coefficient is given by scalar data that do not involve the unknown coupling optimum. These conditions hold for Gaussian marginals in every dimension at least three. A half-normal tail integral supplies an explicit positive lower bound for the Gaussian gap. The result is a quantitative obstruction, not a computation of the exact Gaussian extremum or an optimal stability constant.