🤖 AI Summary
This work addresses the fundamental disconnect between continuous geometric consensus and discrete symbolic reasoning in multi-agent systems under nonholonomic constraints and open-world conditions. To bridge this gap, the authors propose a unified geometric–categorical framework: agent states are modeled on homogeneous manifolds, with geometric consensus achieved via Riemannian centroid flows; interactions are formalized using cellular sheaves, and logical holonomies are embedded into restriction maps through Cartan connections; time is modeled as a Grothendieck topos, enabling intuitionistic logic and abductive repair. By integrally combining differential geometry, sheaf theory, and topos theory, the framework unifies physical, cognitive, and temporal dimensions. The approach yields Sheaf-Theoretic Planning and a singularity-free SE(3) synchronization method, with empirical validation on opinion dynamics and knowledge graph tasks demonstrating the efficacy of geometric consensus.
📝 Abstract
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.