# How Hard Does It Think? Analyzing Step-Aware Reasoning Energy in LLM Chain-of-Thought Trajectories

arXiv: https://arxiv.org/abs/2607.28674 · HTML: https://mlsanity.com/paper/2607.28674/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 19 of its 32 references that are in the map.
- First posted on arXiv 28 Jul 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, 19 of 19)

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- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
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- [Evaluation of OpenAI o1: Opportunities and Challenges of AGI](https://mlsanity.com/paper/2409.18486/) (2024)
- [Think Deep, Not Just Long: Measuring LLM Reasoning Effort via Deep-Thinking Tokens](https://mlsanity.com/paper/2602.13517/) (2026)
- [Bridging Internal Probability and Self-Consistency for Effective and Efficient LLM Reasoning](https://mlsanity.com/paper/2502.00511/) (2025)
- [Entropy trajectory shape predicts LLM reasoning reliability: A diagnostic study of uncertainty dynamics in chain-of-thought](https://mlsanity.com/paper/2603.18940/) (2026)
- [Cost-Awareness in Tree-Search LLM Planning: A Systematic Study](https://mlsanity.com/paper/2505.14656/) (2025)
- [Entropy Sentinel: Probing Entropy Traces for LLM Monitoring](https://mlsanity.com/paper/2601.09001/) (2026)
- [Oops, Wait: Discourse Tokens Matter in Reasoning Model](https://mlsanity.com/paper/2601.17421/) (2026)

## Abstract (arXiv)

Understanding how computational effort is allocated across individual chain-of-thought (CoT) reasoning steps remains an open challenge: existing interpretability methods rely on output-level signals or collapse processing depth into a single trajectory-level scalar, leaving step-wise effort opaque. We propose Step-Aware Reasoning Energy (SARE), a geometric framework that quantifies effort at the granularity of individual CoT steps via Centered Kernel Alignment (CKA) between Gram matrices of token hidden states across adjacent transformer layers, capturing inter-token relational structure without requiring eigenvector alignment or cluster correspondence. SARE further contextualizes this energy within reasoning's semantic progression by modeling CoT trajectories as transitions among latent semantic states. Across six reasoning benchmarks and three open-weight LLMs, we find that reasoning energy is highly non-uniform across step types, exhibiting phase-like transitions invisible to trajectory-level metrics; incorrect trajectories show systematically lower energy at critical reasoning junctions; and SARE-based features match or outperform output-based confidence baselines in most settings, indicating that internal geometric dynamics encode predictive information beyond surface-level signals.