{
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  "source": "vibemath",
  "problem_id": "vibemathed:complementary-bell-fibers",
  "title": "Finitude of the Fibers of Complementary Bell Numbers",
  "canonical_statement": "Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers $f(n) = B_n(-1)$ take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result whose techniques connect to Wilf's conjecture on the vanishing of $f(n)$.",
  "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:complementary-bell-fibers:event",
      "problem_id": "vibemathed:complementary-bell-fibers",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-01T00:00:00.000Z",
      "title": "arXiv",
      "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:complementary-bell-fibers:attempt"
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      "problem_id": "vibemathed:complementary-bell-fibers",
      "model": "GPT-5.6 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-08-01T00:00:00.000Z",
      "human_collaborators": [
        "John M. Campbell"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Solved \"through our extensive interactions with GPT-5.6 Pro\" during the exploratory and proof-development stages; all AI-generated suggestions were substantially revised, corrected and independently verified by the author.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.00575",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Finitude of the Fibers of Complementary Bell Numbers",
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      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
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      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": null,
      "sources": [
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          "label": "VibeMathed: Finitude of the Fibers of Complementary Bell Numbers",
          "url": "https://vibemathed.com/problem/complementary-bell-fibers",
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          "label": "arXiv",
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          "kind": "primary-mathematical-source",
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      "url": "https://arxiv.org/abs/2608.00575",
      "kind": "primary-mathematical-source",
      "license": null
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    {
      "label": "VibeMathed: Finitude of the Fibers of Complementary Bell Numbers",
      "url": "https://vibemathed.com/problem/complementary-bell-fibers",
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      "license": "https://vibemathed.com/data-license"
    }
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  "opportunity_signals": [
    {
      "id": "vibemathed:complementary-bell-fibers:signal:watch_only",
      "problem_id": "vibemathed:complementary-bell-fibers",
      "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."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}