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For deterministically minimizing a convex 1-Lipschitz function on the $d$-dimensional ball using only exact function values, the query complexity sat between $\Omega(d)$ and $O(d^2 \log^2 d)$ since 1996. The paper proves a near-quadratic lower bound $\Omega(d^2 / \log(d+1))$, closing the gap: $Q(d, \sim d^{-1/2}) = \Theta(d^2)$, a polynomial separation from full first-order information.
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Kerger reports that GPT-5.6 Sol Pro solved the problem rather than the author, following a workflow like OpenAI's Cycle Double Cover effort. It first proved a $\tilde{\Omega}(d^2)$ lower bound at accuracy of order $d^{-3}$ (after ~148 minutes), which was then refined to the order-$d^{-1/2}$ result via a further ~230-minute run. The author verified the arguments by hand and takes full responsibility.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
arXiv preprint 2607.13335 (14 Jul 2026) by Phillip Kerger (UC Berkeley), not yet peer-reviewed. The weaker-accuracy $\tilde{\Omega}(d^2)$-at-$d^{-3}$ lower bound was formally verified in Lean (github.com/PhillipKerger/zero-order-bounds-lean-verification); the headline improvement to accuracy $d^{-1/2}$ is not yet Lean-formalized (it needs convex-geometry results like Urysohn's inequality absent from current Lean libraries) and rests on the author's hand verification.
Correctness: unknown · statement fidelity: unaudited · peer review: none
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.