Geometric Analysis of Reasoning Trajectories: A Phase Space Approach to Understanding Valid and Invalid Multi-Hop Reasoning in LLMs

TL;DR

This work introduces a physics-inspired framework that treats multi-hop reasoning in embedding spaces as Hamiltonian dynamics in a phase space, with $H_R(q,p) = T(p) - V(q)$ guiding the balance between exploration and targeted reasoning. By embedding states with $q$ and momentum $p = q_{t+1}-q_t$, the approach enables energy-based diagnostics and geometric analyses using differential geometry and Frenet-Serret framing to distinguish valid from invalid chains. Empirical results on OpenBookQA reveal that valid reasoning tends to exhibit lower energy, smoother trajectories and stronger conservation of the Hamiltonian, though energy alone is not a definitive predictor. The framework offers a novel diagnostic lens and visualization tools for LLM reasoning, while acknowledging the mapping remains largely metaphorical and requiring further validation across tasks and domains. Overall, the paper provides a principled pathway to quantify and interpret reasoning dynamics, with potential implications for explainability and algorithmic improvements in large language models.

Abstract

This paper proposes a novel approach to analyzing multi-hop reasoning in language models through Hamiltonian mechanics. We map reasoning chains in embedding spaces to Hamiltonian systems, defining a function that balances reasoning progression (kinetic energy) against question relevance (potential energy). Analyzing reasoning chains from a question-answering dataset reveals that valid reasoning shows lower Hamiltonian energy values, representing an optimal trade-off between information gathering and targeted answering. While our framework offers complex visualization and quantification methods, the claimed ability to "steer" or "improve" reasoning algorithms requires more rigorous empirical validation, as the connection between physical systems and reasoning remains largely metaphorical. Nevertheless, our analysis reveals consistent geometric patterns distinguishing valid reasoning, suggesting this physics-inspired approach offers promising diagnostic tools and new perspectives on reasoning processes in large language models.

Figures (22)

• Figure 1: Canonical Transformations in Reasoning Space
• Figure 2: Phase plots for focused and multi-concept reasoning in a two-dimensional Hamiltonian system
• Figure 3: Representation of curvature with Frenet frame field.
• Figure 4: Representation of curvature for a large e-commerce company chatbot example.
• Figure 5: Velocity, acceleration and trajectory angle in a curve using Frenet frame.