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

2026-03-11

AI Summary

This work proposes TRACED, a novel framework that addresses the limitations of scalar-probability-based evaluation methods in capturing the structural dynamics and reliability of large language model (LLM) reasoning. By introducing geometric dynamics into LLM reasoning analysis for the first time, TRACED models reasoning trajectories along two geometric dimensions—“progress” (displacement) and “stability” (curvature)—revealing fundamental topological distinctions between correct reasoning and hallucination. The framework establishes cognitive mappings between “hesitation loops” and high curvature, as well as between “certainty accumulation” and displacement, offering a new physical perspective on machine cognition. Experimental results demonstrate that TRACED achieves strong performance and robustness across multiple benchmarks, effectively discriminating between correct reasoning and hallucinatory behavior.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Large Multimodal Models (LMMs)
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

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.

Problem

• LLM reasoning
• reasoning evaluation
• structural dynamics
• hallucination detection
• reasoning reliability

Innovation

• geometric reasoning
• reasoning stability
• progress trajectory
• hallucination detection
• cognitive geometry