{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:green-open-problem-90",
  "title": "Ben Green's Open Problem 90",
  "canonical_statement": "For $A \\subset \\mathbb{F}_p$ of density $1/2$, call $A$ almost affine invariant under $\\varphi(x) = ax+b$ if $|A \\triangle \\varphi(A)| = o(p)$. Problem 90 asks for the threshold $K$ below which $A$ can be almost affine invariant simultaneously under all such $\\varphi$ with $|a|, |b| \\le K$ and $a \\ne 0$. The threshold is $K = o(\\log p)$.",
  "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.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:green-open-problem-90:event",
      "problem_id": "vibemathed:green-open-problem-90",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-05-13T00:00:00.000Z",
      "title": "arXiv:2605.13454 - Almost Affine Invariance Over Prime Fields: Green Problem 90",
      "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.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:green-open-problem-90:attempt"
      ],
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    {
      "id": "vibemathed:green-open-problem-90:attempt",
      "problem_id": "vibemathed:green-open-problem-90",
      "model": "ChatGPT 5.4",
      "provider": "OpenAI",
      "attempted_at": "2026-05-13T00:00:00.000Z",
      "human_collaborators": [
        "Jie Ma",
        "Quanyu Tang",
        "Max Wenqiang Xu"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "One of the more honest disclosures in the catalog, because it itemises what the model got right and says plainly what it got wrong. The authors credit two specific ideas as mostly due to AI: considering the $q$-adic valuation formulation, which is what yields the sharp $o(\\log p)$ upper bound in the final step, and using the amenability of the affine group in an earlier version of one lemma. They also record that the original AI-produced arguments contained many logical mistakes and gaps across the iterative process. Both halves belong in the record: real mathematical ideas, arriving inside output that needed human repair.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.13454 - Almost Affine Invariance Over Prime Fields: Green Problem 90",
          "url": "https://arxiv.org/abs/2605.13454",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Ben Green's Open Problem 90",
          "url": "https://vibemathed.com/problem/green-open-problem-90",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "independence": "unknown"
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      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint, not peer-reviewed. The authors state the model's original arguments contained many logical mistakes and gaps, so the published proof is their reconstruction rather than model output.",
      "sources": [
        {
          "label": "VibeMathed: Ben Green's Open Problem 90",
          "url": "https://vibemathed.com/problem/green-open-problem-90",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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        {
          "label": "arXiv:2605.13454 - Almost Affine Invariance Over Prime Fields: Green Problem 90",
          "url": "https://arxiv.org/abs/2605.13454",
          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "arXiv:2605.13454 - Almost Affine Invariance Over Prime Fields: Green Problem 90",
      "url": "https://arxiv.org/abs/2605.13454",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Ben Green's Open Problem 90",
      "url": "https://vibemathed.com/problem/green-open-problem-90",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_exposure": "result_only",
  "opportunity_signals": [
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      "id": "vibemathed:green-open-problem-90:signal:watch_only",
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      "generated_at": "2026-09-13T16:28:40.178Z"
    }
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}