GPT-5.6 used a prompt to close a 30-year gap in convex optimization

A post on the r/math forum reports that GPT-5.6 Sol Pro produced the missing half of a complexity result in convex optimization that had been open since 1996, in a single session of 148 minutes. The author of the accompanying preprint says the argument has been formally verified in Lean, and that the result has not yet been peer reviewed.

What was actually open

The problem is deterministic zeroth-order convex optimization. An algorithm may query any point in the unit ball in R^d and receives only the exact value of a convex, 1-Lipschitz function — no gradients — while being otherwise unrestricted in computation and memory. Such function-value-only settings arise when an objective is evaluated by a physical experiment or a simulator, and the natural question is how many evaluations are fundamentally required. In 1996, Protasov gave an algorithm showing that on the order of d² evaluations suffice. The matching lower bound was missing: the strongest previously applicable result, Ω(d), was inherited from the stronger first-order model in which gradients are available, which left a linear gap in d and no certainty that gradients help at all. The proof the model supplied closes that gap — no algorithm can do better than order d² evaluations, so Protasov's method is optimal.

A ten-page prompt, 148 minutes

The author, a teaching professor in industrial engineering and operations research at UC Berkeley, had worked on the problem sporadically for about a year and had tried GPT-5.4 and GPT-5.5 without success. After OpenAI announced its Cycle Double Cover proof, he wrote a prompt of roughly ten pages in the same style — it is attached to the end of the preprint — and asked for the quadratic lower bound at accuracy of order d⁻⁴. After 148 minutes of uninterrupted work the model returned a proof resolving the quadratic dimension dependence at accuracy of order d⁻³. The author checked the argument himself and formally verified it in Lean. The construction, a maximum of affine functions, is closely related to the one behind Nemirovsky and Yudin's tight bound for first-order convex optimization.

What it means for research

The preprint, the Lean repository, the full prompt and the original chat logs are linked from the post. The author is careful about the size of the claim: the proof does not introduce fundamentally new techniques in convex geometry, and he argues that if a result is reachable with existing methods, current AI systems will reach it too. He does not expect mathematicians to become obsolete, but says it will stop making sense to work on low-hanging or even moderately hard problems, leaving researchers the ones that need genuinely new ideas. Commenters report comparable results from the same model, including a proof of Sabidussi's compatibility conjecture and an open problem in error-correcting codes, both posted to arXiv with Lean formalizations. Asked about cost, the author puts the project between twenty and two hundred dollars of subscription time, with no more than about fifteen hours of model use.