真·博士水平！GPT-5首次给出第四矩定理显式收敛率，数学教授只点拨了一下

Core Insights - GPT-5 has successfully extended the qualitative fourth moment theorem to a quantitative form with explicit convergence rates, marking a significant advancement in mathematical research [1][2][10]. Group 1: Research Achievements - The original theorem indicated that convergence would occur but did not specify the speed of convergence; GPT-5's contribution clarifies this aspect [2]. - OpenAI co-founder Greg Brockman expressed satisfaction with the progress made using GPT-5 in mathematical research [4]. - GPT-5 Pro improved known boundary values in convex optimization from 1/L to 1.5/L within minutes, showcasing its capabilities [8]. Group 2: Research Methodology - A controlled experiment was conducted by three mathematics professors using the Malliavin–Stein framework to test GPT-5's ability to generalize the fourth moment theorem [9][10]. - Initial prompts were based on a paper that established a qualitative fourth moment theorem applicable to two Wiener–Itô integrals with differing parity [11]. - GPT-5 provided a generally correct conclusion but made errors in reasoning that could jeopardize the proof's validity [13][14]. Group 3: Iterative Improvement - Upon identifying errors, researchers prompted GPT-5 to check its formulas and provide detailed derivations, leading to further corrections [15]. - GPT-5 was able to format the results into a research paper structure, including an introduction, main theorem statements, and a complete proof process [17]. - The AI suggested that the method could be extended to non-Gaussian frameworks, indicating its potential for broader applications [20]. Group 4: Further Exploration - Researchers aimed to extend the findings to Poisson cases, recognizing structural differences between Gaussian and Poisson scenarios [21][24]. - GPT-5 initially overlooked a critical fact regarding non-negativity in Poisson cases but was able to correct itself after specific guidance from researchers [26][28]. Group 5: Publication Challenges - The authors initially intended to list GPT-5 as a co-author but were informed by arXiv that AI cannot be credited as an author [29]. - Ultimately, the paper was submitted without GPT-5 listed as an author, reflecting ongoing discussions about AI's role in academic contributions [30].