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