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

arXiv: https://arxiv.org/abs/2609.32924 · HTML: https://mlsanity.com/paper/2609.32924/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 16 of its 20 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 16 of 16)

- [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)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Learning to Reason Across Parallel Samples for LLM Reasoning](https://mlsanity.com/paper/2506.09014/) (2025)
- [FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving](https://mlsanity.com/paper/2310.18021/) (2023)
- [Autoformalizing Euclidean Geometry](https://mlsanity.com/paper/2405.17216/) (2024)
- [Heimdall: test-time scaling on the generative verification](https://mlsanity.com/paper/2504.10337/) (2025)
- [Shrinking the Generation-Verification Gap with Weak Verifiers](https://mlsanity.com/paper/2506.18203/) (2025)
- [ROC-n-reroll: How verifier imperfection affects test-time scaling](https://mlsanity.com/paper/2507.12399/) (2025)
- [FGeo-HyperGNet: Geometric Problem Solving Integrating FormalGeo Symbolic System and Hypergraph Neural Network](https://mlsanity.com/paper/2402.11461/) (2024)
- [Reliable Proof Generation with LLMs via Analogical Retrieval and Symbolic Verification: A Case Study in Euclidean Geometry](https://mlsanity.com/paper/2505.14479/) (2025)
- [Budget-aware Test-time Scaling via Discriminative Verification](https://mlsanity.com/paper/2510.14913/) (2025)

## Abstract (arXiv)

Repeated sampling can reveal a correct numerical answer without yielding either a reliable system output or a supported derivation. We present Coverage, Realization, and Validity Evidence (CRV), an evaluation protocol for sampled large language model (LLM) reasoning over formal geometry states. Coverage is answer availability, realization is readout accuracy on the frozen candidate pool, and validity evidence is a label-blinded critic judgment of derivational support rather than a proof certificate. CRV freezes each candidate pool before comparing readouts and analyzes covered failures by correct-answer multiplicity and within-problem discrimination. On HardShift441, a 441-problem set for which a reference solver leaves 406 problems unsolved, a LoRA-adapted Qwen2.5-7B generator obtains 24.2% average single-sample accuracy and 68.9% pass@16, whereas verifier-weighted self-consistency (WSC) reaches 38.0%. Readout accuracy is particularly low when the correct answer occurs only once or twice in the pool. In a separate constructed audit of 195 covered problems, the critic labels 12 correct-answer representatives as supported, 181 as refuted, and two as uncertain. These results show that coverage, realization, and validity evidence from the critic are distinct quantities and should be reported separately.