Hyperbolic-Native Large Language Model: Complete Mathematical Specification for Direct Implementation

This article contains a technical research document investigating Transformer-based neural architectures extended with hyperbolic geometric representations. The work explores how non-Euclidean geometry can be integrated into Transformer models to address limitations of standard Euclidean representations, particularly in hierarchical structure modeling and numerical stability. The document focuses on theoretical foundations, geometric constraints, and architectural considerations, rather than full empirical validation or complete implementation details. This release serves archival and reference purposes and does not disclose all components required for direct reproduction.