UC Berkeley teaching professor Phillip Kerger asked GPT-5.6 Sol Pro to solve a specific problem in zeroth-order convex optimization: proving that d² function evaluations are necessary to minimize a convex function in d-dimensional space. The model worked for 2.5 hours and produced a complete argument. The proof was then machine-verified in Lean, a formal proof assistant.
The problem traces back to 1996, when mathematician Vladimir Protasov proved that d² evaluations were sufficient — an upper bound. But no one could prove that d² was also necessary. That left a gap between the best-known upper bound (d²) and the earlier best lower bound (d). For thirty years, the question sat open.
What makes this result notable is not just the mathematics but how it was produced. The model worked autonomously from a detailed prompt describing the problem and the expected proof structure. The output was not a plausible-sounding paragraph that collapsed under scrutiny — it was a rigorous argument that survived machine verification. Formal verification in Lean means every logical step is checked against the proof system's rules, leaving no room for hidden assumptions or reasoning leaps.
The proof concerns zeroth-order optimization, a class of problems where the solver can only evaluate a function at specific points without access to gradient information. This mirrors many real-world scenarios in engineering, finance, and scientific computing where gradients are unavailable or too expensive to compute. Proving that d² evaluations are necessary sets a fundamental limit on how efficient any such method can be.
GPT-5.6 Sol, released publicly on July 9, 2026, represents OpenAI's most advanced reasoning model. This result follows a pattern seen in earlier GPT iterations that independently produced novel mathematical proofs, but this marks the first time a machine-generated proof closed a long-standing open gap in a formal verification system.
Knowledge takeaway: GPT-5.6 Sol Pro proved that d² function evaluations are necessary for zeroth-order convex optimization — closing a 30-year gap between Protasov's 1996 upper bound and the prior d lower bound. The argument was produced in a single 2.5-hour session and machine-verified in Lean, demonstrating that LLMs can now contribute original, verifiable results in pure mathematics.