{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:schur-number-recurrence",
  "title": "The Abbott-Hanson Recurrence for Schur Numbers",
  "canonical_statement": "The classical Abbott-Hanson recurrence gives $S(k+2) \\ge 9S(k)+4$ for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted $S$-templates, a more flexible form of Rowley's template construction, yield $S(k+2) \\ge 10S(k)+2$ and hence improved lower bounds.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:schur-number-recurrence:event",
      "problem_id": "vibemathed:schur-number-recurrence",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-16T00:00:00.000Z",
      "title": "arXiv:2607.15034 - Shifted S-templates and improved lower bounds for Schur numbers",
      "summary": "improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:schur-number-recurrence:attempt"
      ],
      "method_family_ids": [
        "vibemathed:schur-number-recurrence:method"
      ],
      "source_assertion_ids": [
        "vibemathed:schur-number-recurrence:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:schur-number-recurrence:attempt",
      "problem_id": "vibemathed:schur-number-recurrence",
      "model": "ChatGPT 5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-07-16T00:00:00.000Z",
      "human_collaborators": [
        "Nils Bengone",
        "Amine Brouk",
        "Max Grinsztajn",
        "Terence Helbert",
        "Bao Lugherini",
        "Arpad Rimmel",
        "Joanna Tomasik"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The abstract says the shifted $S$-template construction was discovered during a conversation with ChatGPT 5.5 Pro and then refined, verified and extended to multiple templates by the authors, and the conclusion describes the paper as building on an original idea of the model.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.15034 - Shifted S-templates and improved lower bounds for Schur numbers",
          "url": "https://arxiv.org/abs/2607.15034",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Abbott-Hanson Recurrence for Schur Numbers",
          "url": "https://vibemathed.com/problem/schur-number-recurrence",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "cost_usd": null,
      "independence": "unknown"
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      "id": "vibemathed:schur-number-recurrence:verification",
      "problem_id": "vibemathed:schur-number-recurrence",
      "solution_event_id": "vibemathed:schur-number-recurrence:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: The Abbott-Hanson Recurrence for Schur Numbers",
          "url": "https://vibemathed.com/problem/schur-number-recurrence",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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        {
          "label": "arXiv:2607.15034 - Shifted S-templates and improved lower bounds for Schur numbers",
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          "kind": "primary-mathematical-source",
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  "sources": [
    {
      "label": "arXiv:2607.15034 - Shifted S-templates and improved lower bounds for Schur numbers",
      "url": "https://arxiv.org/abs/2607.15034",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Abbott-Hanson Recurrence for Schur Numbers",
      "url": "https://vibemathed.com/problem/schur-number-recurrence",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:schur-number-recurrence:signal:recent_partial_result",
      "problem_id": "vibemathed:schur-number-recurrence",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}