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.

In legal practice, judges apply the trichotomous dogmatics of criminal law, sequentially assessingthe elements of the offense, unlawfulness, and culpability to determine whether an individual’s conduct constitutes a crime. Although current legal large language models (LLMs) show promising accuracy in judgment prediction, they lack trichotomous reasoning capabilities due to the absence of an appropriate benchmark dataset, preventing them from predicting innocent outcomes. As a result, every input is automatically assigned a charge, limiting their practical utility in legal contexts. To bridge this gap, we introduce LJPIV, the first benchmark dataset for Legal Judgment Prediction with Innocent Verdicts. Adhering to the trichotomous dogmatics, we extend three widely-used legal datasets through LLM-based augmentation and manual verification. Our experiments with state-of-the-art legal LLMs and novel strategies that integrate trichotomous reasoning into zero-shot prompting and fine-tuning reveal: (1) current legal LLMs have significant room for improvement, with even the best models achieving an F1 score of less than 0.3 on LJPIV; and (2) our strategies notably enhance both in-domain and cross-domain judgment prediction accuracy, especially for cases resulting in an innocent verdict.