{
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  "source": "vibemath",
  "problem_id": "vibemathed:integer-domination-root-conjecture",
  "title": "The Integer Domination Root Conjecture",
  "canonical_statement": "Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial $D(G, x)$, proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at $x = -4$, built from an S-unit branch cancellation mechanism.",
  "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:integer-domination-root-conjecture:event",
      "problem_id": "vibemathed:integer-domination-root-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-31T00:00:00.000Z",
      "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:integer-domination-root-conjecture:attempt"
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      "method_family_ids": [
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      "model": "Claude Fable 5",
      "provider": "Anthropic",
      "attempted_at": "2026-07-31T00:00:00.000Z",
      "human_collaborators": [
        "Saeid Alikhani",
        "Max Griswold"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "\"The concept and theoretical formulation of the S-Unit Branch Cancellation mechanism were generated by Claude Fable 5.\" The authors verified all formal proofs and carried out independent computational validations. Alikhani is one of the conjecture's original posers.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.00109",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Integer Domination Root Conjecture",
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      "independence": "unknown"
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      "id": "vibemathed:integer-domination-root-conjecture:verification",
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      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Author-verified by exact enumeration, and one author co-posed the conjecture being refuted; no external review yet.",
      "sources": [
        {
          "label": "VibeMathed: The Integer Domination Root Conjecture",
          "url": "https://vibemathed.com/problem/integer-domination-root-conjecture",
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          "label": "arXiv",
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          "kind": "primary-mathematical-source",
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      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.00109",
      "kind": "primary-mathematical-source",
      "license": null
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    {
      "label": "VibeMathed: The Integer Domination Root Conjecture",
      "url": "https://vibemathed.com/problem/integer-domination-root-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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    {
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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"
}