Prescribed-marginal dependence synthesis with moment and Fourier constraints

We construct joint laws with prescribed marginals while imposing finite moment constraints, spectral zeros and several Fourier targets simultaneously. One centered-tensor lemma gives both a rational lattice construction and a local complex interpolation theorem for fixed marginal densities bounded below on intervals. The lattice parameters have an exact positivity body; a weighted absolute-value bound describes only a guaranteed inner region. A convolution family attains every pure Fourier defect between zero and one while concealing dependence from any preassigned finite moment depth and preserving two algebraic contacts. Its sharp radius extends to arbitrary real signed measures with the same finite support and prescribed zero multiplicities. A three-sign model provides a complementary global calibration: we determine the exact dominant regime for unequal pair correlations and the full profile for equal pair correlations. At the transition, strictly positive laws can have quartic contact along a distinguished direction, while the contact is quadratic in transverse directions. The constructions use the classical centered-product mechanism; the point is the simultaneous realization and the precise scope of its constraints. For unequal three-sign laws, the largest pair coefficient is the global Fourier defect exactly when \(2q^2≥ R_2+τ^2\), leaving a sharply delimited interior regime.