Spectral Invariants of Reasoning Trajectories: A Koopman‑DMD Dynamical View on LLM Internal Logic Geometry

Large language models produce multi‑step natural‑language reasoning chains, yet little is known about whether distinct logical inference types induce separable intrinsic geometric signatures within latent hidden‑state dynamics. Moving beyond static‑layer probing, we treat cumulative reasoning hidden‑state sequences as discrete phase‑space trajectories and adopt Koopman operator theory together with Exact Dynamic Mode Decomposition (DMD) to extract complex‑valued spectral fingerprints of internal reasoning dynamics. A major practical obstacle is that DMD eigenvalue sets from different reasoning traces vary in cardinality; naive zero‑padding introduces artificial mass at the origin and corrupts spectral comparisons. To address this, we develop a variable‑rank $W_2$ optimal‑transport metric for unaligned complex eigenvalue clouds, avoiding zero‑padding distortions. We state and empirically investigate the Spectral Invariant Hypothesis (SIH): spectral fingerprints from identical logic types exhibit smaller intra‑type $W_2$ distances than inter‑type distances. Evaluated on controlled logical reasoning benchmarks across multiple open‑weight models, our results confirm the hypothesis and reveal non‑monotonic scaling of dynamical manifold structure with model size. Our dynamical‑systems perspective provides a probe‑free diagnostic paradigm for analyzing LLM reasoning mechanisms.