{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:jungic-rainbow-ap-order",
  "title": "The Order of Long Rainbow Arithmetic Progressions",
  "canonical_statement": "Let $T_k$ be the least $t$ such that every equinumerous $t$-coloring of $[tn]$ contains a rainbow $k$-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured $T_k = \\Theta(k^2)$; Conlon, Fox and Sudakov proved $T_k = O(k^2 \\log k)$. The matching lower bound $T_k = \\Omega(k^2 \\log k)$ holds, so $T_k = \\Theta(k^2 \\log k)$ and the conjectured order is wrong.",
  "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:jungic-rainbow-ap-order:event",
      "problem_id": "vibemathed:jungic-rainbow-ap-order",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-16T00:00:00.000Z",
      "title": "arXiv:2607.15116 - The order of long rainbow arithmetic progressions",
      "summary": "the true order is determined, and it is not the conjectured one",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:jungic-rainbow-ap-order:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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    }
  ],
  "attempts": [
    {
      "id": "vibemathed:jungic-rainbow-ap-order:attempt",
      "problem_id": "vibemathed:jungic-rainbow-ap-order",
      "model": "Codex (GPT-5.6), Claude Code (Fable 5)",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-07-16T00:00:00.000Z",
      "human_collaborators": [
        "Jesse Geneson"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The acknowledgement says the two systems were used for proof exploration, proof criticism, exposition and revision. Proof exploration and criticism are mathematical work rather than prose work, but no specific step is attributed, so the lowest tier applies.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.15116 - The order of long rainbow arithmetic progressions",
          "url": "https://arxiv.org/abs/2607.15116",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Order of Long Rainbow Arithmetic Progressions",
          "url": "https://vibemathed.com/problem/jungic-rainbow-ap-order",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:jungic-rainbow-ap-order:verification",
      "problem_id": "vibemathed:jungic-rainbow-ap-order",
      "solution_event_id": "vibemathed:jungic-rainbow-ap-order:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Single-author arXiv preprint; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: The Order of Long Rainbow Arithmetic Progressions",
          "url": "https://vibemathed.com/problem/jungic-rainbow-ap-order",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.15116 - The order of long rainbow arithmetic progressions",
          "url": "https://arxiv.org/abs/2607.15116",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.15116 - The order of long rainbow arithmetic progressions",
      "url": "https://arxiv.org/abs/2607.15116",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Order of Long Rainbow Arithmetic Progressions",
      "url": "https://vibemathed.com/problem/jungic-rainbow-ap-order",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:jungic-rainbow-ap-order:signal:watch_only",
      "problem_id": "vibemathed:jungic-rainbow-ap-order",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}