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

arXiv: https://arxiv.org/abs/2603.10384 · HTML: https://mlsanity.com/paper/2603.10384/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 41 of its 67 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 41)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [Can LLMs Express Their Uncertainty? An Empirical Evaluation of Confidence Elicitation in LLMs](https://mlsanity.com/paper/2306.13063/) (2023)
- [Reasoning Models Don't Always Say What They Think](https://mlsanity.com/paper/2505.05410/) (2025)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [Layer by Layer: Uncovering Hidden Representations in Language Models](https://mlsanity.com/paper/2502.02013/) (2025)
- [The Illusion of Thinking: Understanding the Strengths and Limitations of Reasoning Models via the Lens of Problem Complexity](https://mlsanity.com/paper/2506.06941/) (2025)
- [Prompting GPT-3 To Be Reliable](https://mlsanity.com/paper/2210.09150/) (2022)
- [Making Large Language Models Better Reasoners with Step-Aware Verifier](https://mlsanity.com/paper/2206.02336/) (2022)
- [TheoremQA: A Theorem-driven Question Answering dataset](https://mlsanity.com/paper/2305.12524/) (2023)
- [Phi-4-reasoning Technical Report](https://mlsanity.com/paper/2504.21318/) (2025)
- [DeepSeek-R1 Thoughtology: Let's think about LLM Reasoning](https://mlsanity.com/paper/2504.07128/) (2025)
- [Small Language Models Need Strong Verifiers to Self-Correct Reasoning](https://mlsanity.com/paper/2404.17140/) (2024)
- [SEAL: Steerable Reasoning Calibration of Large Language Models for Free](https://mlsanity.com/paper/2504.07986/) (2025)
- [Attention Satisfies: A Constraint-Satisfaction Lens on Factual Errors of Language Models](https://mlsanity.com/paper/2309.15098/) (2023)
- [Latent Space Chain-of-Embedding Enables Output-free LLM Self-Evaluation](https://mlsanity.com/paper/2410.13640/) (2024)
- [CoT-Kinetics: A Theoretical Modeling Assessing LRM Reasoning Process](https://mlsanity.com/paper/2505.13408/) (2025)
- [Understanding Aha Moments: from External Observations to Internal Mechanisms](https://mlsanity.com/paper/2504.02956/) (2025)
- [Look Before You Leap: An Exploratory Study of Uncertainty Measurement for Large Language Models](https://mlsanity.com/paper/2307.10236/) (2023)
- [Detection and Mitigation of Hallucination in Large Reasoning Models: A Mechanistic Perspective](https://mlsanity.com/paper/2505.12886/) (2025)
- [Uncertainty Estimation in Autoregressive Structured Prediction](https://mlsanity.com/paper/2002.07650/) (2020)
- [The Geometry of Reasoning: Flowing Logics in Representation Space](https://mlsanity.com/paper/2510.09782/) (2025)
- [Verifying Chain-of-Thought Reasoning via Its Computational Graph](https://mlsanity.com/paper/2510.09312/) (2025)
- [Adaptive Multi-Subspace Representation Steering for Attribute Alignment in Large Language Models](https://mlsanity.com/paper/2508.10599/) (2025)
- [A Hopfieldian View-based Interpretation for Chain-of-Thought Reasoning](https://mlsanity.com/paper/2406.12255/) (2024)
- [Understanding How Value Neurons Shape the Generation of Specified Values in LLMs](https://mlsanity.com/paper/2505.17712/) (2025)

## Built on by (most cited first, 5 of 5)

- [Semantic Step Prediction: Multi-Step Latent Forecasting in LLM Reasoning Trajectories via Step Sampling](https://mlsanity.com/paper/2604.18464/) (2026)
- [Density Ridge Selective Prediction for LLM and VLM Hallucination Detection under Calibration Label Scarcity](https://mlsanity.com/paper/2606.10198/) (2026)
- [GeoFaith: A Spatio-Temporal Dual View of Faithful Chain-of-Thought](https://mlsanity.com/paper/2605.26893/) (2026)
- [GLIDE: Generalized Layer-wise Intrinsic Distributional Evaluation for Heterogeneous LLM Agents](https://mlsanity.com/paper/2609.32295/) (2026)
- [Hallucination Across the Reasoning Lifecycle: Interface Visibility, Causal Evidence, and Release Control in Large Reasoning Models](https://mlsanity.com/paper/2610.05472/) (2026)

## Abstract (arXiv)

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.