# Failure Modes of Large Language Models on Research-Level Mathematics: A Taxonomy and an Empirical Characterisation

arXiv: https://arxiv.org/abs/2606.24902 · HTML: https://mlsanity.com/paper/2606.24902/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 9 references that are in the map.
- First posted on arXiv 12 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Survey of Hallucination in Natural Language Generation](https://mlsanity.com/paper/2202.03629/) (2022)
- [Improving Factuality and Reasoning in Language Models through Multiagent Debate](https://mlsanity.com/paper/2305.14325/) (2023)
- [First Proof](https://mlsanity.com/paper/2602.05192/) (2026)
- [RealMath: A Continuous Benchmark for Evaluating Language Models on Research-Level Mathematics](https://mlsanity.com/paper/2505.12575/) (2025)
- [IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation](https://mlsanity.com/paper/2509.26076/) (2025)

## Abstract (arXiv)

The "First Proof" benchmark [1] posed ten research-level mathematics questions to the strongest publicly available LLMs and found them consistently wrong-not silent, but confidently, fluently wrong. This paper asks why. Working from the per-question post-mortems in First Proof's Appendix A, I identify four failure modes: citation fabrication (F1), premise smuggling (F2), silent problem reformulation (F3), and local-to-global compatibility gaps (F4). I then audit eight one-shot proofs generated by Gemini 2.5 Flash on Questions 1, 2, and 5 of the benchmark, using two instruments built specifically to surface F1 and F2. The central finding is uncomfortable for anyone who sees retrieval-augmented generation (RAG) as the obvious fix: not one of the eight proofs contained a confirmed fabricated citation, yet every single one contained at least one load-bearing claim asserted as a "fundamental result" or "standard argument" with no justification attached. That failure mode-F2, premise smuggling-is invisible to citation verification by design. A premise-audit instrument I introduce flags it at 100% precision (5/5 judge-confirmed flags are true positives) and 50% proof-level recall in this corpus. The taxonomy and the audit together suggest that the right long-term objective is building inference-time pipelines that prevent these failure modes from occurring, not just detecting them after the fact. Index Terms--Large language models, mathematical reasoning, hallucination, premise smuggling, failure-mode taxonomy.