A Geometric Witness Framework for Signed Multivariate Tail-Dependence Compatibility: Asymptotic Structure and Finite-Threshold Synthesis

We study multivariate tail-dependence compatibility for complete and partial signed tail families, treating lower-tail, upper-tail, and mixed configurations in one geometric witness representation indexed by active coordinate sets and sign patterns. For a complete signed tail family, witness generator weights w = (w_{I,sigma}) give a linear incidence parametrization and are recovered by explicit triangular inversion. Excluding the geometric scale p0, the complete case uses 3^d - 1 generator weights, matching the number of complete signed tail coefficients; for partial specifications, only selected target coefficients need be prescribed. At a fixed threshold p0 in (0, 1/2), the inversion identifies the normalized noncentral ternary cell masses of any realizing copula. Hence finite-threshold compatibility is characterized by nonnegative recovered generator weights, singleton normalization, and the residual central-mass constraint. This yields a complete Moebius-type synthesis within the witness framework. If the recovered increments are nonnegative and singleton normalization holds, then S(w) = sum(w) determines the admissible finite-scale range, and every admissible p0 gives an exact witness realization. In the canonical ray geometry, such a realization preserves the same complete signed tail family throughout 0