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A 30-year open problem in convex optimization is closed. UC Berkeley IEOR teaching professor Phillip Kerger fed GPT-5.6 Sol Pro a 10-page prompt on July 16, watched the model work for 2.5 hours without intervention, and received a proof — later machine-verified in Lean — that settles a question researchers had not been able to resolve since 1996.

This is a different result from OpenAI’s recently publicized cycle double cover conjecture proof. That was graph theory. This is computational complexity of optimization algorithms, with direct implications for how efficiently real-world systems can be solved.

The problem

The question is deceptively clean: when an algorithm can only observe the output value of a function — not its gradient, not its structure — how many such observations does it need to minimize a convex function of d variables?

This class of problems, called zeroth-order or derivative-free optimization, governs engineering design based on simulation outputs, hyperparameter tuning in machine learning, and optimization over physical measurements where only outcomes are observable.

Protasov established in 1996 that d² function evaluations are sufficient — his algorithm solves the problem in roughly d² queries. The question was whether you could do better. Could a smarter algorithm solve it with only d queries? Or is d² essentially the floor?

For three decades, the best lower bound proved was only d. Researchers knew you needed at least d queries; they could not prove you needed d². The gap between “d is necessary” and “d² is sufficient” meant the question was open: maybe there existed an algorithm that worked in O(d log d) or O(d^1.5) that nobody had found yet.

The proof

GPT-5.6 Sol Pro produced a proof that the lower bound is d² — up to logarithmic and constant factors. Protasov’s algorithm is therefore essentially optimal. No algorithm can solve zeroth-order convex optimization with substantially fewer than d² queries.

Kerger posted the result to Medium on July 16 and submitted it as an arXiv preprint. He had worked on the problem sporadically for years. Previous collaboration with other models, including extended attempts, yielded nothing. After the GPT-5.6 Sol release, he redesigned his prompt as a structured 10-page document — similar in architecture to the prompt OpenAI used in its recent graph theory work — and ran a single session.

Two and a half hours later, the model produced an argument. Kerger reviewed it for correctness, set it up in Lean, and the formal verifier passed. The proof is machine-confirmed.

What this means

Lean verification is the standard that distinguishes AI math results worth taking seriously from plausible-looking confabulations. The model produced an argument that survived formal logical checking, not just human inspection. Kerger’s own domain expertise — he is an IEOR faculty member, not an AI researcher using math as a demo — further validates the result.

For practitioners: the practical implication is that simulation-based design problems, hyperparameter tuning loops, and measurement-based optimization are computationally bounded at d² queries. There is no hidden algorithm waiting to be discovered that cuts this to something subquadratic in dimension. The field can now move on.

For the AI math capability story: the cycle double cover conjecture required a structured attack on a graph theory conjecture. The convex optimization result required crossing into complexity theory to establish a lower bound — a qualitatively different skill. Two unrelated research-grade results in the same model generation is not coincidence.

Key facts

• Model: GPT-5.6 Sol Pro
• Session length: 2.5 hours, single session, no human intervention during
• Prompt: 10-page structured document
• Result: d² lower bound for zeroth-order convex optimization (matching Protasov’s 1996 d² upper bound)
• Verification: Lean formal proof checker confirmed correct
• Researcher: Phillip Kerger, UC Berkeley IEOR teaching professor
• Published: Medium and arXiv preprint, July 16, 2026

The cycle double cover proof got more attention. This one is arguably more useful.