Geometric Flows of Logic in LLM Representation Space

The October 10, 2025 Duke University academic paper introduces a **novel geometric framework** that views Large Language Model (LLM) reasoning as continuous, evolving trajectories—or **flows**—within the model's representation space. The core hypothesis posits that while surface semantics determine the position of these representations, the **underlying logical structure** acts as a **local differential controller** that governs the flow's velocity and curvature. To validate this, the researchers created a dataset that systematically disentangles formal logic skeletons (from natural deduction) from their semantic carriers (such as topics and languages). Empirical results using LLMs like Qwen3 and LLaMA3 demonstrate that **velocity and Menger curvature similarities** remain high for reasoning flows sharing the same logical structure, even when surface topics or languages vary significantly, supporting the conclusion that LLMs internalize abstract logic beyond mere linguistic form.Source:https://arxiv.org/pdf/2510.09782

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