A UC Berkeley mathematician used GPT-5.6 Sol Pro to solve a complexity problem in convex optimization that has remained open since 1996. In a 148-minute session, the model produced a proof establishing a quadratic lower bound for deterministic zeroth-order convex optimization—effectively closing a three-decade gap in the field.
The problem concerns the oracle complexity of optimizing convex functions when algorithms can only query function values, not gradients. While a 1996 algorithm by Protasov demonstrated that d² function evaluations are sufficient, no matching lower bound existed. The new result proves that no algorithm can do better than order d² evaluations—establishing that Protasov’s approach is essentially optimal.
From CDC Proof to Optimization Breakthrough
The achievement follows OpenAI’s recent demonstration where GPT-5.6 Sol Pro proved the Cycle Double Cover Conjecture, a 50-year-old topology problem. The Berkeley researcher adapted the same prompting methodology, constructing a ten-page prompt following OpenAI’s approach.
“Modern AI methods will solve problems attainable with existing techniques,” the researcher noted. “What this means is that it no longer makes sense to work on low-hanging fruit. We’ll be needed for problems where novel approaches are truly required.”
Verified in Lean
The proof was formally verified in Lean, a proof assistant that mechanically checks mathematical reasoning. This verification adds confidence in the result’s correctness—a crucial step for AI-assisted mathematical proofs.
The work suggests frontier language models are becoming capable of genuine mathematical research, not just pattern matching on training data. As formal verification tools improve, AI-assisted proofs may become increasingly common in pure mathematics.