GPT-5 Pro独立做数学研究！读论文后给出更精确边界，OpenAI总裁：这是生命迹象

Core Viewpoint - The article discusses the capabilities of OpenAI's GPT-5 Pro in independently exploring and proving mathematical concepts, specifically in the field of convex optimization, highlighting its potential as a significant breakthrough in AI research [1][9][42]. Group 1: GPT-5 Pro's Achievements - GPT-5 Pro provided a more precise threshold and corresponding proof for a boundary problem in convex optimization compared to the original paper [2][26]. - The model was able to refine the boundary from 1/L to 1.5/L using advanced inequality techniques in just 17.5 minutes, while human verification took 25 minutes [27][28]. - OpenAI's president referred to this achievement as a "sign of life," indicating the model's advanced capabilities [9]. Group 2: Convex Optimization Insights - The original paper titled "Are Optimization Curves Convex?" investigates whether the optimization curve generated by gradient descent on smooth convex functions is convex [10][11]. - The paper concludes that the convexity of the optimization curve depends on the choice of step size, with specific ranges ensuring convexity [14][17]. - Key findings include that for step sizes in the range (0, 1/L], the optimization curve is guaranteed to be convex, while in the range (1.75/L, 2/L), it may not be convex even if gradient descent converges [17][26]. Group 3: Comparison of Approaches - GPT-5 Pro's proof approach differed from the updated version of the original paper, demonstrating its ability to independently discover and prove mathematical rules [41][42]. - The original authors later updated their paper to establish 1.75/L as an exact boundary, closing previously unexplored intervals [41][42].