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 \(d≥3\), 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 \[ VarX/(E|X-medX|)^2, \] 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.