AI Mathematician as a Partner in Advancing Mathematical Discovery -- A Case Study in Homogenization Theory

Artificial intelligence (AI) has demonstrated impressive progress in mathematical reasoning, yet its integration into the practice of mathematical research remains limited. In this study, we investigate how the AI Mathematician (AIM) system can operate as a research partner rather than a mere problem solver. Focusing on a challenging problem in homogenization theory, we analyze the autonomous reasoning trajectories of AIM and incorporate targeted human interventions to structure the discovery process. Through iterative decomposition of the problem into tractable subgoals, selection of appropriate analytical methods, and validation of intermediate results, we reveal how human intuition and machine computation can complement one another. This collaborative paradigm enhances the reliability, transparency, and interpretability of the resulting proofs, while retaining human oversight for formal rigor and correctness. The approach leads to a complete and verifiable proof, and more broadly, demonstrates how systematic human-AI co-reasoning can advance the frontier of mathematical discovery.