Researchers reveal how large language models create discrete logical boundaries within continuous semantic spaces through geometric distortion mechanisms.

LLMs generalize smoothly across continuous semantic spaces but struggle with strict logical reasoning that requires discrete decision boundaries

Traditional linear isometric projection theories fail to explain how models transition between continuous and discrete logic

Study identifies a dual-modulation mechanism: class-agnostic topological preservation maintains global structure while algebraic divergence creates logical boundaries between concepts

Gram-Schmidt decomposition analysis of residual-stream activations reveals how task context operates as a non-isometric dynamical operator enforcing topological distortion

Findings validated across task gradient from simple mapping to complex primality testing, with causal evidence from targeted vector ablation