# The Missing Primitive: Diagnosing and Repairing Mathematical Reasoning in Large Language Models

arXiv: https://arxiv.org/abs/2610.02191 · HTML: https://mlsanity.com/paper/2610.02191/

## 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 27 of its 44 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Placed in

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

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

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [gpt-oss-120b & gpt-oss-20b Model Card](https://mlsanity.com/paper/2508.10925/) (2025)
- [Humanity's Last Exam](https://mlsanity.com/paper/2501.14249/) (2025)
- [Self-Distilled Reasoner: On-Policy Self-Distillation for Large Language Models](https://mlsanity.com/paper/2601.18734/) (2026)
- [FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI](https://mlsanity.com/paper/2411.04872/) (2024)
- [Beyond Benchmarks: MathArena as an Evaluation Platform for Mathematics with LLMs](https://mlsanity.com/paper/2605.00674/) (2026)
- [Self-Distilled RLVR](https://mlsanity.com/paper/2604.03128/) (2026)
- [Privileged Information Distillation for Language Models](https://mlsanity.com/paper/2602.04942/) (2026)
- [Self-Distillation Zero: Self-Revision Turns Binary Rewards into Dense Supervision](https://mlsanity.com/paper/2604.12002/) (2026)
- [Early science acceleration experiments with GPT-5](https://mlsanity.com/paper/2511.16072/) (2025)
- [Rethinking On-Policy Self-Distillation for Thinking Models](https://mlsanity.com/paper/2607.05184/) (2026)
- [Purified OPSD: On-Policy Self-Distillation Without Losing How to Think](https://mlsanity.com/paper/2607.02234/) (2026)
- [Large Language Models and Mathematical Reasoning Failures](https://mlsanity.com/paper/2502.11574/) (2025)
- [HLE-Verified: A Systematic Verification and Structured Revision of Humanity's Last Exam](https://mlsanity.com/paper/2602.13964/) (2026)
- [Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning](https://mlsanity.com/paper/2412.15184/) (2024)
- [Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof](https://mlsanity.com/paper/2510.23513/) (2025)
- [Decomposing Elements of Problem Solving: What "Math" Does RL Teach?](https://mlsanity.com/paper/2505.22756/) (2025)
- [Mismatch Matters: On-Policy Distillation Beyond Token Agreement](https://mlsanity.com/paper/2608.09836/) (2026)
- [AVSD: Adaptive-View Self-Distillation by Balancing Consensus and Teacher-Specific Privileged Signals](https://mlsanity.com/paper/2605.20643/) (2026)
- [DemoPSD: Disagreement-Modulated Policy Self-Distillation](https://mlsanity.com/paper/2607.02502/) (2026)
- [Diagnosing and Mitigating Thinking Collapse in On-Policy Self-Distillation](https://mlsanity.com/paper/2607.10805/) (2026)
- [Executable Functional Abstractions: Inferring Generative Programs for Advanced Math Problems](https://mlsanity.com/paper/2504.09763/) (2025)
- [Trust Region Policy Distillation](https://mlsanity.com/paper/2607.04751/) (2026)
- [Learning to Reason with Insight for Informal Theorem Proving](https://mlsanity.com/paper/2604.16278/) (2026)
- [Unmasking Reasoning Processes: A Process-aware Benchmark for Evaluating Structural Mathematical Reasoning in LLMs](https://mlsanity.com/paper/2602.00564/) (2026)

## Abstract (arXiv)

While Large Language Models (LLMs) have demonstrated striking capabilities on frontier mathematical problems, it remains unclear whether they possess the structural mathematical understanding underlying their solutions. In this paper, we take a first step toward systematically studying mathematical understanding in LLMs, from diagnosing its distinct capabilities to leveraging these findings to improve post-training. First, we introduce the notion of Mathematical Primitive to probe structural mathematical understanding and propose hlei, a novel benchmark that evaluates mathematical reasoning along four distinct dimensions: Discovery, Generation, Digestion, and Execution. Second, our systematic diagnosis shows that solution accuracy masks distinct capability profiles, primitives unlock substantial latent execution capacity, and Discovery is the dominant bottleneck in mathematical reasoning. Our post-training analysis further shows that discovery-limited failures are particularly amenable to repair. Finally, building on these findings, we introduce abs, a primitive-privileged self-distillation framework that selectively transfers primitive-guided reasoning into the student model. Extensive experiments demonstrate that abs consistently improves mathematical reasoning over baselines across model scales and challenging benchmarks.