Logarithmic Geometry as a Foundational Representation for Multi-Scale Neural Computation

This work investigates neural network design under a logarithmic coordinate system rather than the conventional linear–affine formulation. Motivated by the scale-invariant structure observed in biological perception, physical measurement systems, and information geometry, the paper formalizes logarithmic geometry as an alternative representational substrate for neural computation. Instead of treating logarithmic operations as isolated nonlinearities, the proposed framework treats log-space as the primary computational domain in which addition, composition, and depth correspond to multiplicative and ratio-preserving interactions in the original signal space. The document develops the geometric and informational foundations of logarithmic representations, analyzes their implications for gradient flow, activation concentration, and dynamic range compression, and contrasts these properties with those of standard linear networks. A fully logarithmic neural architecture is defined, including forward mappings, inverse mappings, and training dynamics, without relying on normalization heuristics or architectural shortcuts. Rather than claiming universal superiority, the work aims to establish feasibility, identify regimes where logarithmic geometry provides structural advantages, and clarify open problems related to optimization, numerical stability, and computational cost. The results suggest that logarithmic coordinate systems constitute a viable and conceptually distinct design axis for neural models, particularly in settings involving multi-scale data, multiplicative structure, or extreme dynamic ranges.