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：人工智能究竟能否真正促进创造性数学研究，抑或仅仅是自动化常规计算并引入错误风险？我们通过一项详细案例研究提供实证证据：通过系统性人机协作，为埃尔米特求积规则发现了新颖的误差表示与边界。借助多个AI助手，我们拓展了人工研究取得的成果，在AI协助下构建并证明了若干定理。该协作既展现出卓越能力，也暴露出关键局限。AI在代数运算、系统性证明探索、文献综述与LaTeX文档准备方面表现优异，但每个步骤都需要严格的人工验证、用于问题构建的数学直觉以及战略方向指引。我们以罕见的透明度记录了完整研究流程，揭示了成功的人机数学协作模式，并指出了研究者必须预见的失效情形。我们的经验表明，在保持适当怀疑态度并采用验证协议的前提下，AI工具能够实质性加速数学发现进程，但始终需要严谨的人工监督与深厚的领域专业知识。