Transport distance to joint mixability and quantitative characteristic-function gaps

Joint mixability asks whether prescribed marginals admit a coupling whose weighted sum is constant. We quantify the failure of this compatibility condition by the smallest possible \(L^p\) residual of the weighted sum. For fixed \(1\le p<\infty\) and each \(1\le q\le\infty\), we prove an exact identity between that residual \(J_{a,p}\) and the product-Wasserstein distance from the marginal tuple to the set of mean-preserving jointly mixable tuples, with the precise factor \(\|a\|_{\ell^{q^*}}\); both optima are attained. This is stronger than treating the residual merely as an upper or lower stability estimate. Under explicit frequency-isolation, Fourier log-concavity and quadratic phase-margin hypotheses, nonmixability yields a strictly positive quantitative gap below the scalar fixed-marginal characteristic-function profile. The hypotheses are verified for nondegenerate Gaussian marginals in every dimension \(d\ge3\), with an explicit positive lower bound expressed using a half-normal tail. An exact Bernoulli residual and smooth perturbative obstructions complement the theory. The paper does not determine the exact Gaussian coupling optimum; the projection identity and the quantitative Fourier transfer are its central contributions.