[2603.23577] The Geometric Price of Discrete Logic: Context-driven Manifold Dynamics of Number Representations

Computer Science > Machine Learning arXiv:2603.23577 (cs) [Submitted on 24 Mar 2026] Title:The Geometric Price of Discrete Logic: Context-driven Manifold Dynamics of Number Representations Authors:Long Zhang, Dai-jun Lin, Wei-neng Chen View a PDF of the paper titled The Geometric Price of Discrete Logic: Context-driven Manifold Dynamics of Number Representations, by Long Zhang and 1 other authors View PDF HTML (experimental) Abstract:Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses par...