GPT-5.6 Reportedly Proves 30-Year-Old Convex Optimization Lower Bound

On July 17, 2026, a researcher shared findings on Reddit's r/math community claiming that GPT-5.6 Sol, guided by a carefully constructed prompt, produced a proof establishing an Omega(d^2) lower bound for convex optimization over a standard function class. This lower bound matches the upper bound of an algorithm published roughly 30 years ago, effectively closing a long-standing complexity gap and confirming that no faster algorithm can exist for this problem class. The computation reportedly took approximately 148 minutes of sustained reasoning, and the researcher noted the result followed over a year of failed attempts using earlier AI models. The finding carries broader significance because convex optimization underpins the gradient descent methods used to train modern neural networks, including the very models that produced the proof. The announcement gained traction on Hacker News with over 500 points, though mainstream coverage of GPT-5.6 that same week remained focused largely on consumer features and pricing.