Geometric Control Manifolds for Hyperbolic Self-Organizing Intelligence

This work addresses a fundamental limitation in contemporary deep learning architectures: the Geometric Capacity Bottleneck—the intrinsic inability of Euclidean representational spaces to accommodate hierarchical information structures without catastrophic metric distortion. We propose a geometry-based architectural framework that resolves this bottleneck through precise alignment between the computational substrate and the structure of information. The system operates on hyperbolic Lorentzian manifolds, leveraging constant negative curvature to achieve exponential volume growth, thereby establishing a structural isomorphism with hierarchical semantic concepts. Main contributions: A Geometric Control Manifold (GCM) characterized by three quantifiable invariants Hyperbolic Neural Cellular Automata (H-NCA) operating on pentagrid tessellations Hyperbolic Kuramoto Oscillatory Neurons (H-AKOrN) for temporal binding Differentiable topological regularization via Persistence Landscapes A falsifiable validation protocol based on Gromov–Wasserstein divergence This work makes no claims regarding consciousness or phenomenology. All constructs are operationally defined and empirically testable.