Author name disambiguation is a critical challenge in academic search systems, often addressed through from-scratch and real-time disambiguation approaches. However, current algorithms remain vulnerable to cumulative errors of paper-author assignments and overlook inconsistent assignments across different sources. Resorting to expert annotation is resource-intensive. To this end, this paper explores a new perspective for author name disambiguation: cross-source correction by leveraging inconsistent assignments across sources. We propose CrossND, a full-stack framework that integrates data refinement, cross-source reasoning, and test-time scaling. First, a chain-of-refinement pipeline denoises author profiles and produces more accurate paper-author matching probabilities. Second, a supervised fine-tuning process incorporates these refined signals and a probabilistic soft logic-based cross-correction module to infer the assignments of which sources are incorrect. Third, test-time scaling further enhances the accuracy and robustness of the predictions. Experiments on real-world datasets indicate that CrossND consistently outperforms 17 baselines by leveraging cross-source reasoning without human intervention.