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 < p <= p0. Thus the primary object is the complete signed tail family lambda: it is realized at every admissible finite scale and can be carried along families of witness copulas with p0 decreasing to 0. Partial, noisy, or inconsistent specifications are treated through linear-feasibility and weighted-l1 recovery problems in the same parametrization. The representation separates the p0-free incidence/Moebius layer from finite-threshold realization and provides tools for realization, simulation, calibration, completion, repair, and scenario design.

翻译：我们研究了完全和部分带符号尾部族的多元尾部相依性相容性，将下尾、上尾及混合配置统一纳入一个由活跃坐标集和符号模式索引的几何证据表示中。对于完全带符号尾部族，形如w = (w_{I,sigma})的证据生成器权重提供了线性关联参数化，并通过显式三角反演得到恢复。排除几何尺度p0后，完全情形需要使用3^d - 1个生成器权重，这与完全带符号尾部系数的数量一致；对于部分规范，仅需指定选定的目标系数。在固定阈值p0 ∈ (0, 1/2)处，该反演识别出任意实现联结函数的归一化非中心三元胞质量。因此，有限阈值相容性由非负恢复生成器权重、单点归一化以及残余中心质量约束共同刻画。这就在证据框架内产生了一种完整的莫比乌斯型综合。若恢复的增量非负且单点归一化成立，则S(w)= sum(w)决定了容许的有限尺度范围，且每个容许的p0都能给出精确的证据实现。在正则射线几何中，这样的实现在0 < p ≤ p0范围内保持相同的完全带符号尾部族。因此，核心对象是完全带符号尾部族λ：它在每个容许的有限尺度上均可实现，并可沿p0递减至0的证据联结函数族进行传递。部分、含噪或不一致的规范通过同一参数化下的线性可行性及加权l1恢复问题进行处理。该表示将p0无关的关联/莫比乌斯层与有限阈值实现分离开来，并为实现、模拟、校准、补全、修复及情景设计提供了工具。