Topological Bridges and Precision-Aware Geometry: Eradicating Deterministic Numerical Instability in Lorentz Manifold Deep Learning

This work presents a rigorous investigation into deterministic numerical instabilities arising in the computational implementation of Lorentz (hyperboloid) manifolds under finite-precision arithmetic (IEEE 754). While hyperbolic geometry has become a cornerstone for modeling hierarchical data structures in modern machine learning—particularly in Hyperbolic Neural Networks (HNNs), Hyperbolic Graph Convolutional Networks (HGCNs), and related architectures—its practical deployment remains severely constrained by persistent and poorly understood numerical failures. Contrary to common assumptions that such instabilities are merely stochastic artifacts of floating-point noise, this study demonstrates that they are in fact deterministic consequences of the interaction between hyperbolic geometry and finite-precision computation. Through a detailed analytical and empirical examination, the work identifies and formalizes three critical failure modes: (i) violation of self-distance identity (d(x,x) ≠ 0), (ii) divergence in exponential-logarithmic map round-trip consistency, and (iii) breakdown of tangency constraints in the logarithmic map. The analysis reveals that these failures originate from a combination of catastrophic cancellation in the Minkowski inner product, irreversible precision loss during float32 ↔ float64 transitions (termed Precision Boundary Rupture), and the intrinsic singular behavior of the inverse hyperbolic cosine function near unity. A key theoretical result shows that small perturbations introduced by floating-point quantization are systematically amplified according to the relation acosh(1 + ε) ≈ √(2ε), leading to macroscopic geometric errors. To conceptualize these phenomena, the paper introduces three novel constructs: Topological Drift, describing the deviation of points from the manifold due to quantization; Precision Boundary Rupture, characterizing irreversible geometric corruption during type casting; and the Geometric Amplification Loop, which explains the propagation and escalation of numerical errors through manifold operations. As a solution, the work proposes a robust computational framework termed Topological Bridging + Precision-Aware Geometry, which enforces manifold consistency through strategic reprojection, precision-isolated computation zones (float64), and stable mathematical reformulations (e.g., Taylor expansions and log1p-based implementations). The proposed approach reduces error magnitudes from ~1e-4 to machine-level precision (~1e-8), restoring numerical stability and enabling reliable training of hyperbolic machine learning models. This contribution bridges a critical gap between theoretical differential geometry and practical deep learning systems, establishing a new foundation for precision-aware geometric computation. The findings have direct implications for the scalability and reliability of hyperbolic representation learning across domains such as knowledge graphs, phylogenetics, and large-scale hierarchical modeling.