Multimodal Large Language Models (MLLMs) struggle with complex geometric reasoning, largely because "black box" outcome-based supervision fails to distinguish between lucky guesses and rigorous deduction. To address this, we introduce a paradigm shift towards subgoal-level evaluation and learning. We first construct GeoGoal, a benchmark synthesized via a rigorous formal verification data engine, which converts abstract proofs into verifiable numeric subgoals. This structure reveals a critical divergence between reasoning quality and outcome accuracy. Leveraging this, we propose the Sub-Goal Verifiable Reward (SGVR) framework, which replaces sparse signals with dense rewards based on the Skeleton Rate. Experiments demonstrate that SGVR not only enhances geometric performance (+9.7%) but also exhibits strong generalization, transferring gains to general math (+8.0%) and other general reasoning tasks (+2.8%), demonstrating broad applicability across diverse domains.
@article{arxiv.2601.05073,
title = {Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward},
author = {Jianlong Chen and Daocheng Fu and Shengze Xu and Jiawei Chen and Yuan Feng and Yue Yang and Junchi Yan and Hongyuan Zha and Renqiu Xia},
journal= {arXiv preprint arXiv:2601.05073},
year = {2026}
}