Current distributed data fabrics lack a rigorous mathematical foundation, often relying on ad-hoc architectures that struggle with consistency, lineage, and scale. We propose a mathematical framework for data fabrics, unifying heterogeneous data management in distributed systems through a hypergraph-based structure \( \mathcal{F} = (D, M, G, T, P, A) \). Datasets, metadata, transformations, policies, and analytics are modeled over a distributed system \( Σ= (N, C) \), with multi-way relationships encoded in a hypergraph \( G = (V, E) \). A categorical approach, with datasets as objects and transformations as morphisms, supports operations like data integration and federated learning. The hypergraph is embedded into a modular tensor category, capturing relational symmetries via braided monoidal structures, with geometric analogies to Hurwitz spaces enriching the algebraic modeling. We prove the NP-hardness of critical tasks, such as schema matching and dynamic partitioning, and propose spectral methods and symmetry-based alignments for scalable solutions. The framework ensures consistency, completeness, and causality under CAP and CAL theorems, leveraging sparse incidence matrices and braiding actions for fault-tolerant operations.

翻译：当前分布式数据织物缺乏严格的数学基础，通常依赖临时架构，难以应对一致性、数据沿袭和可扩展性等挑战。本文提出一种数据织物的数学框架，通过基于超图的结构 \( \mathcal{F} = (D, M, G, T, P, A) \) 统一分布式系统中的异构数据管理。数据集、元数据、转换、策略和分析均建模于分布式系统 \( Σ= (N, C) \) 之上，其中多路关系编码于超图 \( G = (V, E) \) 中。采用范畴化方法——以数据集为对象、转换操作为态射——支持数据集成与联邦学习等运算。该超图被嵌入模张量范畴，通过辫子幺半结构捕捉关系对称性，其与Hurwitz空间的几何类比进一步丰富了代数建模。我们证明了模式匹配与动态分区等关键任务的NP难解性，并提出基于谱方法及对称性对齐的可扩展解决方案。该框架利用稀疏关联矩阵与辫子作用实现容错操作，在CAP定理与CAL定理约束下确保一致性、完备性与因果性。