# X-RAY: Mapping LLM Reasoning Capability via Formalized and Calibrated Probes

arXiv: https://arxiv.org/abs/2603.05290 · HTML: https://mlsanity.com/paper/2603.05290/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 27 of its 44 references that are in the map.
- First posted on arXiv 5 Mar 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 27 of 27)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [DeepSeek-V3 Technical Report](https://mlsanity.com/paper/2412.19437/) (2024)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Faith and Fate: Limits of Transformers on Compositionality](https://mlsanity.com/paper/2305.18654/) (2023)
- [LiveBench: A Challenging, Contamination-Limited LLM Benchmark](https://mlsanity.com/paper/2406.19314/) (2024)
- [SciBench: Evaluating College-Level Scientific Problem-Solving Abilities of Large Language Models](https://mlsanity.com/paper/2307.10635/) (2023)
- [Impact of Pretraining Term Frequencies on Few-Shot Reasoning](https://mlsanity.com/paper/2202.07206/) (2022)
- [Holistic Evaluation of Language Models](https://mlsanity.com/paper/2211.09110/) (2022)
- [DyVal: Dynamic Evaluation of Large Language Models for Reasoning Tasks](https://mlsanity.com/paper/2309.17167/) (2023)
- [NPHardEval: Dynamic Benchmark on Reasoning Ability of Large Language Models via Complexity Classes](https://mlsanity.com/paper/2312.14890/) (2023)
- [ARB: Advanced Reasoning Benchmark for Large Language Models](https://mlsanity.com/paper/2307.13692/) (2023)
- [Dynamic Evaluation of Large Language Models by Meta Probing Agents](https://mlsanity.com/paper/2402.14865/) (2024)
- [MACM: Utilizing a Multi-Agent System for Condition Mining in Solving Complex Mathematical Problems](https://mlsanity.com/paper/2404.04735/) (2024)
- [From Entropy to Epiplexity: Rethinking Information for Computationally Bounded Intelligence](https://mlsanity.com/paper/2601.03220/) (2026)
- [Knowledge Boundary of Large Language Models: A Survey](https://mlsanity.com/paper/2412.12472/) (2024)
- [Achieving >97% on GSM8K: Deeply Understanding the Problems Makes LLMs Better Solvers for Math Word Problems](https://mlsanity.com/paper/2404.14963/) (2024)
- [LLLMs: A Data-Driven Survey of Evolving Research on Limitations of Large Language Models](https://mlsanity.com/paper/2505.19240/) (2025)
- [How and Why LLMs Generalize: A Fine-Grained Analysis of LLM Reasoning from Cognitive Behaviors to Low-Level Patterns](https://mlsanity.com/paper/2512.24063/) (2025)
- [Evaluating the Logical Reasoning Abilities of Large Reasoning Models](https://mlsanity.com/paper/2505.11854/) (2025)

## Abstract (arXiv)

Large language models (LLMs) achieve promising performance, yet their ability to reason remains poorly understood. Existing evaluations largely emphasize task-level accuracy, often conflating pattern matching with reasoning capability. We present X-RAY, an explainable reasoning analysis system that maps the LLM reasoning capability using calibrated, formally verified probes. We model reasoning capability as a function of extractable textitstructure, operationalized through formal properties such as constraint interaction, reasoning depth, and solution-space geometry. X-Ray generates probes via formal tools with controlled structural variations, enabling precise isolation of incremental structural information through formal calibration and verification. We evaluate state-of-the-art LLMs on problems ranging from junior-level to advanced in mathematics, physics, and chemistry. Our analysis reveals a systematic asymmetry in LLM reasoning: models are relatively robust to constraint refinement, where additional conditions shrink an existing solution space, but degrade sharply under solution-space restructuring, where modifications alter the underlying structural form of the solution manifold. Moreover, calibrated formal probes differentiate models that appear indistinguishable on standard benchmarks and reveal failure modes that are structurally interpretable rather than opaque. Beyond evaluation, our framework is contamination-free and supports the training and testing of reasoning models.