{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-planar-unit-distance",
  "title": "Erdős's Planar Unit Distance Conjecture",
  "canonical_statement": "Erdős conjectured an $n^{1+o(1)}$ upper bound on the number of unit-distance pairs among $n$ points in the plane. A construction with more than $n^{1.014}$ pairs disproves that conjectured asymptotic upper bound; the exact extremal unit-distance problem remains open.",
  "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:erdos-planar-unit-distance:event",
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      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-05-27T00:00:00.000Z",
      "title": "Remarks on the disproof of the unit distance conjecture",
      "summary": "The asymptotic conjecture was disproved; this does not determine the exact unit-distance extremal function.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-planar-unit-distance:attempt"
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      "problem_id": "vibemathed:erdos-planar-unit-distance",
      "model": "OpenAI frontier model (specific version not disclosed)",
      "provider": "OpenAI",
      "attempted_at": "2026-05-27T00:00:00.000Z",
      "human_collaborators": [
        "Noga Alon",
        "Thomas Bloom",
        "Timothy Gowers",
        "Daniel Litt",
        "Will Sawin",
        "Jacob Tsimerman",
        "Melanie Matchett Wood"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Model-assisted construction of a point configuration reported to have more than $n^{1.014}$ unit-distance pairs.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "Remarks on the disproof of the unit distance conjecture",
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          "license": null
        },
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          "label": "VibeMathed: Erdős's Planar Unit Distance Conjecture",
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          "kind": "source-assertion",
          "license": "CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
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      "level": "independent_expert_verified",
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "VibeMathed reports human expert verification and an arXiv write-up; VibeMath preserves that label as a source assertion.",
      "sources": [
        {
          "label": "VibeMathed: Erdős's Planar Unit Distance Conjecture",
          "url": "https://vibemathed.com/problem/erdos-planar-unit-distance",
          "kind": "source-assertion",
          "license": "CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)"
        },
        {
          "label": "Remarks on the disproof of the unit distance conjecture",
          "url": "https://arxiv.org/abs/2605.20695",
          "kind": "primary-mathematical-source",
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        }
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  "sources": [
    {
      "label": "Remarks on the disproof of the unit distance conjecture",
      "url": "https://arxiv.org/abs/2605.20695",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős's Planar Unit Distance Conjecture",
      "url": "https://vibemathed.com/problem/erdos-planar-unit-distance",
      "kind": "source-assertion",
      "license": "CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)"
    }
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  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
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      "id": "vibemathed:erdos-planar-unit-distance:signal:replay_ready",
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      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "generated_at": "2026-08-17T15:54:39.560Z"
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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."
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  "generated_at": "2026-08-17T15:54:39.560Z"
}