Reasoning-step curvature tracks logical structure, not surface form

Zhou et al. model each reasoning step as a mean-pooled final-layer hidden state and compute the Menger curvature of the step sequence across 2,430 sequences that vary logic, topic, and language independently [zhou-etal-2026-geometry-of-reasoning] Curvature similarity is consistently higher for logic-matched pairs than topic- or language-matched pairs (e.g. Qwen3-0.6B: logic 0.53 vs topic 0.11 vs language 0.13; Llama3-8B: 0.58 vs 0.13 vs 0.17) [zhou-etal-2026-geometry-of-reasoning] Raw position similarity is instead dominated by language (Qwen3-0.6B: language 0.85 vs logic 0.26), so second-order curvature, not position, is organized by logical structure [zhou-etal-2026-geometry-of-reasoning] A step-shuffle control collapses velocity and curvature similarity toward zero while leaving position similarity high, and no causal intervention is performed [zhou-etal-2026-geometry-of-reasoning]