{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:wow-conjecture-109",
  "title": "Written on the Wall II, Graph Conjecture 109",
  "canonical_statement": "Must every connected graph satisfy the proposed upper bound on its independence number in terms of residue and largest induced-bipartite-subgraph order? The family $\\overline{K}_{2r+1} \\vee (K_r \\sqcup K_r)$ violates it for every $r \\ge 3$.",
  "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:wow-conjecture-109:event",
      "problem_id": "vibemathed:wow-conjecture-109",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-28T00:00:00.000Z",
      "title": "google-deepmind/formal-conjectures (WrittenOnTheWallII)",
      "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_discovered",
      "attempt_ids": [
        "vibemathed:wow-conjecture-109:attempt"
      ],
      "method_family_ids": [
        "vibemathed:wow-conjecture-109:method"
      ],
      "source_assertion_ids": [
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      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:wow-conjecture-109:attempt",
      "problem_id": "vibemathed:wow-conjecture-109",
      "model": "GPT-5.6 Sol Max (Codex)",
      "provider": "OpenAI",
      "attempted_at": "2026-07-28T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The infinite counterexample family was found with GPT-5.6 Sol Max running in Codex; verified in Lean plus independent Python and C++ enumeration.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "google-deepmind/formal-conjectures (WrittenOnTheWallII)",
          "url": "https://github.com/google-deepmind/formal-conjectures",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Written on the Wall II, Graph Conjecture 109",
          "url": "https://vibemathed.com/problem/wow-conjecture-109",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
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  "verifications": [
    {
      "id": "vibemathed:wow-conjecture-109:verification",
      "problem_id": "vibemathed:wow-conjecture-109",
      "solution_event_id": "vibemathed:wow-conjecture-109:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean-checked disproof in the google-deepmind/formal-conjectures repository, with independent computational enumeration.",
      "sources": [
        {
          "label": "VibeMathed: Written on the Wall II, Graph Conjecture 109",
          "url": "https://vibemathed.com/problem/wow-conjecture-109",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
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          "label": "google-deepmind/formal-conjectures (WrittenOnTheWallII)",
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          "kind": "primary-mathematical-source",
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  ],
  "sources": [
    {
      "label": "google-deepmind/formal-conjectures (WrittenOnTheWallII)",
      "url": "https://github.com/google-deepmind/formal-conjectures",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Written on the Wall II, Graph Conjecture 109",
      "url": "https://vibemathed.com/problem/wow-conjecture-109",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:wow-conjecture-109:signal:replay_ready",
      "problem_id": "vibemathed:wow-conjecture-109",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "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"
}