This paper presents MathBode, a dynamic diagnostic tool for mathematical inference of large-scale language models (LLMs). Instead of relying on one-time accuracy, MathBode treats each parameter as a system, driving a single parameter sinusoidally and fitting the first harmonic response of the model output to the exact solution. This produces interpretable frequency-decomposed metrics, such as gain (amplitude tracking) and phase (delay), which form a Bode-style fingerprint. For five closed-form families—linear solver, rate/saturation, compounding, 2x2 linear systems, and pseudo-triangles—the diagnostic reveals systematic lowpass behavior and increasing phase delay that are difficult to detect based on accuracy alone. The model is compared to a symbol-based baseline ($G \approx 1$, $\phi \approx 0$) that calibrates the device. The results dynamically distinguish state-of-the-art from intermediate models and provide a concise and reproducible protocol that complements standard benchmarks with actionable measures of inference fidelity and consistency. The dataset and code are open-sourced to support further research and adoption.
