# Legal Mathematical Reasoning with LLMs: Procedural Alignment through Two-Stage Reinforcement Learning

arXiv: https://arxiv.org/abs/2504.02590 · HTML: https://mlsanity.com/paper/2504.02590/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning for agents, planning and code, one of 17 directions in LLM reasoning.
- Builds on 21 of its 26 references that are in the map.
- First posted on arXiv 3 Apr 2025.

## Placed in

- LLM reasoning / [Reasoning for agents, planning and code](https://mlsanity.com/lr/reasoning-for-agents-planning-and-code/)

## Builds on (most cited first, 21 of 21)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [ZeRO: Memory Optimizations Toward Training Trillion Parameter Models](https://mlsanity.com/paper/1910.02054/) (2019)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [O1-Pruner: Length-Harmonizing Fine-Tuning for O1-Like Reasoning Pruning](https://mlsanity.com/paper/2501.12570/) (2025)
- [CAIL2018: A Large-Scale Legal Dataset for Judgment Prediction](https://mlsanity.com/paper/1807.02478/) (2018)
- [Rethinking the Bounds of LLM Reasoning: Are Multi-Agent Discussions the Key?](https://mlsanity.com/paper/2402.18272/) (2024)
- [DISC-LawLLM: Fine-tuning Large Language Models for Intelligent Legal Services](https://mlsanity.com/paper/2309.11325/) (2023)
- [JEC-QA: A Legal-Domain Question Answering Dataset](https://mlsanity.com/paper/1911.12011/) (2019)
- [LEVEN: A Large-Scale Chinese Legal Event Detection Dataset](https://mlsanity.com/paper/2203.08556/) (2022)
- [Explainable Legal Case Matching via Inverse Optimal Transport-based Rationale Extraction](https://mlsanity.com/paper/2207.04182/) (2022)
- [Mixed Distillation Helps Smaller Language Model Better Reasoning](https://mlsanity.com/paper/2312.10730/) (2023)
- [Enabling Discriminative Reasoning in LLMs for Legal Judgment Prediction](https://mlsanity.com/paper/2407.01964/) (2024)
- [CitaLaw: Enhancing LLM with Citations in Legal Domain](https://mlsanity.com/paper/2412.14556/) (2024)
- [Explicitly Integrating Judgment Prediction with Legal Document Retrieval: A Law-Guided Generative Approach](https://mlsanity.com/paper/2312.09591/) (2023)
- [Evaluating Test-Time Scaling LLMs for Legal Reasoning: OpenAI o1, DeepSeek-R1, and Beyond](https://mlsanity.com/paper/2503.16040/) (2025)
- [Beyond Guilt: Legal Judgment Prediction with Trichotomous Reasoning](https://mlsanity.com/paper/2412.14588/) (2024)
- [Logic Rules as Explanations for Legal Case Retrieval](https://mlsanity.com/paper/2403.01457/) (2024)

## Built on by (most cited first, 7 of 7)

- [LegalDelta: Enhancing Legal Reasoning in LLMs via Reinforcement Learning with Chain-of-Thought Guided Information Gain](https://mlsanity.com/paper/2508.12281/) (2025)
- [PrLM: Learning Explicit Reasoning for Personalized RAG via Contrastive Reward Optimization](https://mlsanity.com/paper/2508.07342/) (2025)
- [LRAS: Advanced Legal Reasoning with Agentic Search](https://mlsanity.com/paper/2601.07296/) (2026)
- [PoliLegalLM: A Technical Report on a Large Language Model for Political and Legal Affairs](https://mlsanity.com/paper/2604.17543/) (2026)
- [Relation Reasoning with LLMs in Expensive Optimization](https://mlsanity.com/paper/2605.02933/) (2026)
- [Time as Structure: Temporal Dependency Graphs for Verifiable Deadline Computation over Legal Documents](https://mlsanity.com/paper/2608.15270/) (2026)
- [LexReward: A Taxonomy-Driven Reward Framework for Legal Language Models](https://mlsanity.com/paper/2609.39071/) (2026)

## Abstract (arXiv)

Legal mathematical reasoning is essential for applying large language models (LLMs) in high-stakes legal contexts, where outputs must be both mathematically accurate and procedurally compliant. However, existing legal LLMs lack structured numerical reasoning, and open-domain models, though capable of calculations, often overlook mandatory legal steps. To address this, we present LexNum, the first Chinese legal mathematical reasoning benchmark, covering three representative scenarios where each instance reflects legally grounded procedural flows. We further propose LexPam, a two-stage reinforcement learning framework for efficient legal reasoning training. Leveraging curriculum learning, we use a stronger teacher model to partition data into basic and challenging subsets. A lightweight 1.5B student model is then fine-tuned with Group Relative Policy Optimization, which avoids costly value networks and enables stable training from sparse, end-of-sequence rewards. The first stage improves accuracy and format; the second introduces a novel reward to guide procedural alignment via task-specific legal elements. Experiments show that existing models perform poorly on LexNum, while LexPam enhances both mathematical accuracy and legal coherence, and generalizes effectively across tasks and domains.