Lorentzian Hyperbolic Compression: A Family of Geometric Models for Euclidean-to-Hyperbolic Representation Transfer

This paper contains the formal methodological specification of a family of geometric representation models based on the Lorentz (hyperboloid) formulation of hyperbolic space. The work provides a self-contained mathematical framework for projecting high-dimensional Euclidean embeddings into low-dimensional hyperbolic manifolds while preserving structural similarity relations. The contribution is methodological rather than experimental. The document rigorously defines the geometric substrate, manifold operations, loss formulations, and optimization procedures required to support independent reimplementation, without relying on any specific software framework or implementation details. The model family is organized as a structured progression of variants, each introducing an additional level of geometric governance. These variants are designed to isolate and study the effects of capacity, structural alignment, topological regularization, and control mechanisms within hyperbolic representation learning. The framework enables principled ablation by construction, supporting analytical comparison across configurations. This work does not propose a new application, decoder, or end-to-end system. Instead, it focuses on formalizing the representational layer itself, serving as a theoretical and methodological foundation for future empirical studies in areas such as semantic representation learning, hierarchical embedding, and geometry-aware compression. All definitions are explicitly stated, and the presentation is intended for researchers familiar with differential geometry, representation learning, and metric learning. No