[2510.09782] The Geometry of Reasoning: Flowing Logics in Representation Space

Computer Science > Artificial Intelligence arXiv:2510.09782 (cs) [Submitted on 10 Oct 2025 (v1), last revised 3 Mar 2026 (this version, v2)] Title:The Geometry of Reasoning: Flowing Logics in Representation Space Authors:Yufa Zhou, Yixiao Wang, Xunjian Yin, Shuyan Zhou, Anru R. Zhang View a PDF of the paper titled The Geometry of Reasoning: Flowing Logics in Representation Space, by Yufa Zhou and 4 other authors View PDF Abstract:We study how large language models (LLMs) ``think'' through their representation space. We propose a novel geometric framework that models an LLM's reasoning as flows -- embedding trajectories evolving where logic goes. We disentangle logical structure from semantics by employing the same natural deduction propositions with varied semantic carriers, allowing us to test whether LLMs internalize logic beyond surface form. This perspective connects reasoning with geometric quantities such as position, velocity, and curvature, enabling formal analysis in representation and concept spaces. Our theory establishes: (1) LLM reasoning corresponds to smooth flows in representation space, and (2) logical statements act as local controllers of these flows' velocities. Using learned representation proxies, we design controlled experiments to visualize and quantify reasoning flows, providing empirical validation of our theoretical framework. Our findings indicate that training solely via next-token prediction can lead LLMs to internalize logical invariants as h...