The Geometric Price of Discrete Logic: Context-driven Manifold Dynamics of Number Representations
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 directionall...
| Authors | Long Zhang, Dai-jun Lin, Wei-neng Chen |
|---|---|
| Primary category | cs.LG |
| Published | 24 March 2026 |
| Last updated at source | 24 March 2026 |
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