The Geometry of Reasoning: Flowing Logics in Representation Space
Authors: name: Yufa Zhou, name: Yixiao Wang, name: Xunjian Yin, name: Shuyan Zhou, name: Anru R. Zhang

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 work serves as both a conceptual
foundation and practical tools for studying reasoning phenomenon, offering a
new lens for interpretability and formal analysis of LLMs' behavior.