Geometric Dynamics in Neural Architectures: The Integration of Hyperbolic Adaptive Kuramoto Oscillators into Large Language Models

This work presents H-AKORN (Hyperbolic Adaptive Kuramoto Oscillatory Representational Neurons), a fully specified architecture for hyperbolic-native Large Language Models (H-LLMs).The model addresses two fundamental limitations of conventional Euclidean neural architectures: the Geometric Capacity Bottleneck, which prevents faithful representation of hierarchical structures, and the Temporal Binding Problem, which limits dynamic feature grouping in static attention mechanisms. By embedding token representations in a Lorentz hyperbolic manifold, the architecture leverages exponential volume growth to naturally encode deep semantic hierarchies without over-squashing. Static dot-product attention is replaced by a dynamic phase-synchronization process, inspired by the Kuramoto model, where attention emerges from temporal coherence between oscillatory token states. The specification provides a complete mathematical formulation of the geometric substrate, H-AKORN dynamics, tangent-space feed-forward networks, Riemannian normalization, hybrid optimization strategies, and validation criteria.This work is intended as a reproducible, falsifiable foundation for future hyperbolic-native language models and geometric deep learning research.