We introduce a graph-theoretic framework based on discrete sheaves to diagnose and localize inconsistencies in preference aggregation. Unlike traditional linearization methods (e.g., HodgeRank), this approach preserves the discrete structure of ordinal preferences, identifying which specific voter interactions cause aggregation failure -- information that global methods cannot provide -- via the Obstruction Locus. We formalize the Incompatibility Index to quantify these local conflicts and examine their behavior under stochastic variations using the Mallows model. Additionally, we develop a rigorous sheaf-theoretic pushforward operation to model voter merging, implemented via a polynomial-time constraint DAG algorithm. We demonstrate that graph quotients transform distributed edge conflicts into local impossibilities (empty stalks), providing a topological characterization of how aggregation paradoxes persist across scales.

翻译：我们提出了一种基于离散层（discrete sheaves）的图论框架，用于诊断和定位偏好聚合中的不一致性。与传统的线性化方法（如HodgeRank）不同，该方法保留了序数偏好的离散结构，通过障碍轨迹（Obstruction Locus）识别导致聚合失败的具体选民交互——这是全局方法无法提供的信息。我们形式化了不兼容指数（Incompatibility Index）以量化这些局部冲突，并利用Mallows模型研究了它们在随机变化下的行为。此外，我们开发了严格的层论推前（pushforward）操作来建模选民合并，并通过多项式时间约束有向无环图算法实现。我们证明了图商（graph quotients）将分布式的边冲突转化为局部不可能性（空茎，empty stalks），从而提供了聚合悖论如何在多尺度上持续存在的拓扑特征。