Beyond Scalars: Evaluating and Understanding LLM Reasoning via Geometric Progress and Stability

Evaluating LLM reliability via scalar probabilities often fails to capture the structural dynamics of reasoning. We introduce TRACED, a framework that assesses reasoning quality through theoretically grounded geometric kinematics. By decomposing reasoning traces into Progress (displacement) and Stability (curvature), we reveal a distinct topological divergence: correct reasoning manifests as high-progress, stable trajectories, whereas hallucinations are characterized by low-progress, unstable patterns (stalled displacement with high curvature fluctuations). Leveraging these signatures, our probabilistic framework achieves competitive performance and superior robustness across diverse benchmarks. Crucially, TRACED bridges geometry and cognition by mapping high curvature to "Hesitation Loops'' and displacement to ''Certainty Accumulation'', offering a physical lens to decode the internal dynamics of machine thought.

We introduce TRACED, a method that evaluates AI reasoning by tracking the "physics" of its internal thoughts. We found that correct AI reasoning looks like a runner moving smoothly and confidently forward. But when an AI hallucinates or gets confused, its thoughts resemble a driver trapped in a "hesitation loop"—spinning the steering wheel wildly without actually going anywhere. By measuring these physical patterns—specifically how far the thoughts travel (progress) and how smoothly they move (stability)—we can instantly catch AI mistakes as they happen. This gives us a new, visual way to understand the hidden mechanics of machine thought.