Discrete Prime-Aligned Hex-Hive Topologies, Continuous Hopf Phase Transport, and Thermodynamic Stability in Recurrent Dynamical Systems

The historical scaling of deep learning models and decentralized multi-agent routing networks has relied almost exclusively on linear operations mapped within flat, Euclidean geometric spaces. However, as the dimensionality of representations and the topological complexity of agentic networks scale, these Cartesian coordinate systems suffer from severe spatial crowding, topological distortion, and unsustainable computational bottlenecks. In high-dimensional vector spaces, Euclidean distances fail to capture the hierarchical, scale-free branching structures native to semantic datasets, leading to representations that crowd near the boundaries of flat manifolds.