Can artificial intelligence truly contribute to creative mathematical research, or does it merely automate routine calculations while introducing risks of error? We provide empirical evidence through a detailed case study: the discovery of novel error representations and bounds for Hermite quadrature rules via systematic human-AI collaboration. Working with multiple AI assistants, we extended results beyond what manual work achieved, formulating and proving several theorems with AI assistance. The collaboration revealed both remarkable capabilities and critical limitations. AI excelled at algebraic manipulation, systematic proof exploration, literature synthesis, and LaTeX preparation. However, every step required rigorous human verification, mathematical intuition for problem formulation, and strategic direction. We document the complete research workflow with unusual transparency, revealing patterns in successful human-AI mathematical collaboration and identifying failure modes researchers must anticipate. Our experience suggests that, when used with appropriate skepticism and verification protocols, AI tools can meaningfully accelerate mathematical discovery while demanding careful human oversight and deep domain expertise.

翻译：人工智能能否真正助力创造性数学研究，抑或仅能自动化执行常规计算，同时引入错误风险？我们通过详细的案例研究提供实证证据：在系统性人机协作中发现了埃尔米特求积公式的新型误差表达式与误差界。借助多个AI助手，我们拓展了纯人工研究未能触及的成果，在AI辅助下完成并证明了若干定理。协作过程既展现了卓越能力，也暴露了关键局限。AI在代数运算、系统化定理探索、文献综合及LaTeX排版方面表现优异，但每一步骤均需严格的人类验证、数学直觉支撑问题构建及战略方向把控。我们以异常透明的方式记录了完整研究流程，揭示了成功的人机数学协作模式，并识别出研究者必须警惕的失败情形。我们的经验表明，在适当保持审慎并建立验证机制的前提下，AI工具能显著加速数学发现进程，但始终需要人类严格监督与深厚的领域专长。