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  "problem_id": "vibemathed:dual-sequential-fat-shattering-bound",
  "title": "Dual Sequential Fat-Shattering and Tight Threshold Extraction",
  "canonical_statement": "Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in a result claimed by Jung, Kim and Tewari.",
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      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-23T00:00:00.000Z",
      "title": "arXiv:2607.21761 - Encoding orders and trees in real-valued functions",
      "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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      "provider": "OpenAI",
      "attempted_at": "2026-07-23T00:00:00.000Z",
      "human_collaborators": [
        "Gabriel Conant",
        "Caroline Terry"
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      "outcome": "The acknowledgements separate human and AI credit explicitly and attribute specific steps. Proposition 3.4 was provided by ChatGPT on direct request. The authors' original Theorem 1.11 gave only a uniform analogue with a weaker bound; asked whether the bound could be improved, the model proposed the two-parameter inductive scheme that gives the current one, and it identified that the uniformity hypotheses were an artifact of an imprecise formulation, letting the authors drop them. For Theorem 1.14 the authors handed over their draft with a description of the obstacles and asked for a proof; the model generated an argument following their induction scheme and two-case structure. The authors checked and thoroughly rewrote everything.",
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