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  "problem_id": "vibemathed:derivative-free-convex-oracle-gap",
  "title": "Oracle-Complexity Gap in Derivative-Free Convex Optimization",
  "canonical_statement": "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.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
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      "type": "proved",
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      "occurred_at": "2026-07-14T00:00:00.000Z",
      "title": "arXiv:2607.13335",
      "summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
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      "problem_id": "vibemathed:derivative-free-convex-oracle-gap",
      "model": "GPT-5.6 Sol Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-07-14T00:00:00.000Z",
      "human_collaborators": [
        "Phillip Kerger"
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      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "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.",
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      "note": "arXiv preprint 2607.13335 (14 Jul 2026) by Phillip Kerger (UC Berkeley), not yet peer-reviewed.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.",
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