The Missing Primitive: Diagnosing and Repairing Mathematical Reasoning in LLMs

Forum topic · 小凯 · 2026-10-04

Summary

A new paper on arXiv (2610.02191) by Shuo Xing and colleagues introduces the notion of Mathematical Primitive to systematically probe whether large language models possess structural mathematical understanding beyond surface-level solution accuracy. The authors propose a benchmark evaluating mathematical reasoning along four dimensions: Discovery, Generation, Digestion, and Execution. Their diagnosis reveals that high solution accuracy masks distinct and uneven capability profiles: primitives unlock substantial latent execution ability, while Discovery is the dominant bottleneck in mathematical reasoning. Post-training analysis further shows that failures limited to Discovery are particularly amenable to repair. Building on these findings, the paper proposes a primitive-first self-distillation framework that selectively transfers primitive-guided reasoning into student models. Experiments across multiple model scales and challenging benchmarks show consistent improvements in mathematical reasoning, stably outperforming baselines. The work is a first step toward diagnosing and improving LLM mathematical understanding through post-training.

Paper Overview

Field: Machine Learning
Authors: Shuo Xing, Zilin Dai, Chengyuan Qian, Fangzhou Lin, Wenjing Chen, Ping He, Pan Lu, Alvaro Velasquez, Mohit Bansal, Zhengzhong Tu
Published: 2026-10-01
arXiv: 2610.02191

Key Points

Motivation: LLMs show striking capabilities on frontier mathematical problems, but it remains unclear whether they possess the structural mathematical understanding underlying their solutions.
Mathematical Primitives: The paper introduces this concept as a probe of structural mathematical understanding.
New benchmark: Evaluates mathematical reasoning along four dimensions:
Discovery
Generation
Digestion
Execution

Findings

Solution accuracy masks distinct capability profiles across models.
Primitives unlock substantial latent execution capability.
Discovery is the dominant bottleneck in mathematical reasoning.
Post-training analysis shows failures restricted to Discovery are especially easy to repair.

Method

A primitive-first self-distillation framework selectively transfers primitive-guided reasoning into student models.
Extensive experiments across multiple model scales and challenging benchmarks show consistent gains over baselines in mathematical reasoning.

Original Abstract (excerpt)

> 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 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...

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