Diagnosing Sampled LLM Reasoning in Formal Geometry: Coverage, Realization, and Validity Evidence

Large language models (LLMs) are increasingly evaluated on mathematical reasoning, yet their robustness to equivalent problem representations remains poorly understood. In geometry, identical problems can be expressed in Euclidean, coordinate, or vector forms, but existing benchmarks report accuracy on fixed formats, implicitly assuming representation invariance and masking failures caused by representational changes alone. We propose GeoRepEval, a representation-aware evaluation framework that measures correctness, invariance, and consistency at the problem level across parallel formulations, combining strict answer matching, bootstrap confidence intervals, paired McNemar tests, representation-flip analyses, and regression controls for surface complexity. We prove that our Invariance@3 metric decomposes accuracy into robust and fragile components and is bounded by the weakest representation. Evaluating eleven LLMs on 158 curated high-school geometry problems (474 instances), we find accuracy gaps of up to 14 percentage points induced solely by representation choice. Vector formulations emerge as a consistent failure point, with Invariance@3 as low as 0.044 even after controlling for length and symbolic complexity. A convert-then-solve prompting intervention improves vector accuracy by up to 52 percentage points for high-capacity models, suggesting that failures reflect representation sensitivity rather than inability; however, low-capacity models show no gains, indicating deeper limitations. These results suggest that current models rely on representation-specific heuristics rather than abstract geometric reasoning. All datasets, prompts, and scripts are released at https://github.com/vedjaw/GeoRepEval.

Evaluating the symbolic reasoning of large language models (LLMs) calls for geometry benchmarks that require multi-step proofs grounded in both text and diagrams. However, existing benchmarks are often limited in scale and rarely provide visually grounded multiple-choice questions, limiting reliable evaluation of complex reasoning. We introduce GeoChallenge, a dataset of 90K automatically generated multiple-choice geometry proof problems, each requiring multi-step reasoning over aligned textual descriptions and diagrams. GeoChallenge provides fine-grained complexity ratings and formal language annotations to enable controlled evaluation. Experiments on multiple advanced LLMs show a clear performance gap between models and humans (the best-performing model, GPT-5-nano, achieves 75.89 exact match vs. 94.74 for humans). Further analysis also reveals three common failure patterns of LLMs: (1) exact match failures under the multiple-choice setting; (2) weak visual reliance; and (3) overextended reasoning without convergence.

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 structure, 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.