MathBode: Understanding LLM Reasoning with Dynamical Systems

• URL: http://arxiv.org/abs/2509.23143v3
• Date: Tue, 28 Oct 2025 05:44:55 GMT
• Title: MathBode: Understanding LLM Reasoning with Dynamical Systems
• Authors: Charles L. Wang,
• Abstract summary: We present MathBode, a dynamic diagnostic for mathematical reasoning in large language models (LLMs)<n>MathBode treats each parametric problem as a system: we drive a single parameter sinusoidally and fit first-harmonic responses of model outputs and exact solutions.<n>Across five closed-form families, the diagnostic surfaces systematic low-pass behavior and growing phase lag that accuracy alone obscures.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper presents MathBode, a dynamic diagnostic for mathematical reasoning in large language models (LLMs). Instead of one-shot accuracy, MathBode treats each parametric problem as a system: we drive a single parameter sinusoidally and fit first-harmonic responses of model outputs and exact solutions. This yields interpretable, frequency-resolved metrics -- gain (amplitude tracking) and phase (lag) -- that form Bode-style fingerprints. Across five closed-form families (linear solve, ratio/saturation, compound interest, 2x2 linear systems, similar triangles), the diagnostic surfaces systematic low-pass behavior and growing phase lag that accuracy alone obscures. We compare several models against a symbolic baseline that calibrates the instrument ($G \approx 1$, $\phi \approx 0$). Results separate frontier from mid-tier models on dynamics, providing a compact, reproducible protocol that complements standard benchmarks with actionable measurements of reasoning fidelity and consistency. We open-source the dataset and code to enable further research and adoption.

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