TWO-POINT EXTREMALS FOR CHARACTERISTIC-FUNCTION DEPENDENCE WITH FIXED MARGINALS

We study the maximal characteristic-function dependence defect when the one-dimensional marginal laws are fixed. The central mechanism is a two-point extremal principle: convex-order relaxation, quantile-split fusions, and two-tangent certificates reduce a high-dimensional coupling problem to explicit one-dimensional extremal problems. For broad classes of symmetric unimodal marginals we obtain exact constants, including the uniform and triangular laws for all \(d\ge3\), and additional compactly supported families in the stated ranges. Under strict hypotheses, attainment is characterized by complete mixability, while Gaussian marginals provide a natural non-attainment example. For general bounded marginals we derive a large-\(d\) expansion in which the leading correction is governed by \[ \frac{\operatorname{Var}X}{(E|X-\operatorname{med}X|)^2}, \] interpreted as the price of fixing the marginal. We also analyze what changes when symmetry is removed, including a sharp \(9/7\) inequality for unimodal laws and the emergence of asymmetric extremal marginals. The paper develops the fixed-marginal branch of the theory introduced in the companion paper on the universal free-law profile.