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 Insights - OpenAI's GPT-5 Pro demonstrated the ability to independently derive new mathematical conclusions, specifically in the field of convex optimization, by providing a more precise threshold and proof for a boundary problem compared to the original paper [1][4]. Group 1: GPT-5 Pro's Achievements - GPT-5 Pro was able to refine the boundary condition from 1/L to 1.5/L using advanced inequality techniques within 17.5 minutes, while human researchers took 25 minutes to verify the proof [21][22]. - The model's approach involved transforming the optimization curve's convexity problem into proving the monotonic decrease of function value differences, utilizing Bregman divergence and co-coercivity inequalities [24][34]. Group 2: Research Context - The original paper discussed whether the optimization curve generated by gradient descent on smooth convex functions is convex, concluding that the convexity depends on the choice of step size [6][9]. - Key findings included specific ranges for step sizes that guarantee convexity and conditions under which non-convexity may occur [9][10]. Group 3: Competitive Dynamics - Following GPT-5 Pro's findings, the original authors updated their paper to establish a precise boundary of 1.75/L, closing previously unexplored intervals [34]. - This update indicates a competitive landscape in mathematical research where AI can contribute significantly, but human researchers can still respond and refine findings rapidly [4][34].