Multi-agent coordination faces a fundamental divide between continuous Euclidean consensus, which fails under non-integrable constraints, and discrete symbolic logic, which collapses under open-world assumptions. This report presents a unified geometric and categorical framework bridging these paradigms. Agent states are modeled on homogeneous manifolds (Lie groups, Grassmannians) with consensus achieved via Riemannian center-of-mass flows. Clifford-algebraic representations (rotors, motors) enable singularity-free SE(3) pose synchronization. Network interactions are formalized as cellular sheaves, where heterogeneous stalks connected by linear restriction maps replace uniform weights; the sheaf Laplacian drives diffusion toward globally consistent sections. The Cartan connection encodes logical holonomy directly into restriction maps. Asynchronous nonlinear sheaf diffusion guarantees linear convergence to Dirichlet energy minimizers under bounded delays. Sheaf-Theoretic Planning (STP) models time as a Grothendieck topos, using intuitionistic logic and abductive repair for resilient temporal reasoning. Applications include discourse sheaves for opinion dynamics and knowledge sheaves for graph embedding. This synthesis establishes geometric consensus as a universal foundation for resilient multi-agent systems across physical, epistemic, and temporal domains.

翻译：多智能体协调面临连续欧几里得共识（在不可积约束下失效）与离散符号逻辑（在开放世界假设下崩溃）之间的根本分野。本报告提出一个统一几何与范畴框架以弥合这两种范式。智能体状态建模于齐次流形（李群、格拉斯曼流形）之上，通过黎曼质心流实现共识。克利福德代数表征（旋量、动量）实现无奇点的SE(3)位姿同步。网络交互形式化为细胞层，由线性限制映射连接的异质茎取代统一权重；层拉普拉斯算子驱动扩散过程朝向全局一致截面收敛。嘉当联络将逻辑完整直接编码进限制映射。异步非线性层扩散在有限时延下保证线性收敛至狄利克雷能量极小值。层论规划（STP）将时间建模为格罗滕迪克拓扑，利用直觉逻辑与溯因修复实现弹性时间推理。应用场景涵盖观点动力学话语层与图嵌入知识层。该综合框架确立了几何共识作为跨越物理、认知与时间域的弹性多智能体系统的统一基础。