Title: X-RAY: Charting the Reasoning Competence of LLMs Using Formalized and Calibrated Probes

Abstract

Despite the impressive performance metrics of Large Language Models (LLMs), their underlying reasoning mechanisms remain largely opaque. Current assessment methods predominantly focus on task-level accuracy, a practice that frequently confuses simple pattern recognition with genuine reasoning ability. To address this gap, we introduce X-RAY, an interpretable analysis framework that visualizes LLM reasoning capabilities through formally verified and calibrated probes.

In our approach, reasoning capability is defined as a function of structure, specifically operationalized via formal properties including constraint interaction, reasoning depth, and the geometry of the solution space. X-RAY employs formal tools to generate probes with controlled structural variations. This methodology allows for the precise isolation of incremental structural information, facilitated by rigorous formal calibration and verification processes.

We benchmark state-of-the-art LLMs against problems spanning from junior-level to advanced complexity in the fields of mathematics, physics, and chemistry. Our findings highlight a systematic asymmetry in how LLMs reason: while models demonstrate relative robustness when constraints are refined—thereby shrinking an existing solution space—they experience significant performance degradation when the solution space is restructured, meaning modifications that alter the fundamental structural form of the solution manifold.

Furthermore, these calibrated formal probes can distinguish between models that appear statistically identical on standard benchmarks. They also uncover failure modes that are structurally interpretable, moving beyond opaque explanations. Beyond mere evaluation, our framework is designed to be free from data contamination and is suitable for both the training and testing of reasoning models.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC