{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:wow-conjecture-103",
  "title": "Written on the Wall II, Graph Conjecture 103",
  "canonical_statement": "For every connected graph $G$, is $\\alpha(G) \\le \\lfloor b(G) - \\log(\\operatorname{ecc}_{avg}(G)) \\rfloor$, where $b(G)$ is the largest induced-bipartite-subgraph order? An $11$-vertex counterexample - a triangle with four leaves on each of two vertices - has $\\alpha = 9$ against bound $8$.",
  "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-103:event",
      "problem_id": "vibemathed:wow-conjecture-103",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-22T00:00:00.000Z",
      "title": "formal-conjectures PR #4482 - Disprove WOWII Conjecture 103",
      "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-103:attempt"
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  "attempts": [
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      "id": "vibemathed:wow-conjecture-103:attempt",
      "problem_id": "vibemathed:wow-conjecture-103",
      "model": "ChatGPT + Codex",
      "provider": "OpenAI",
      "attempted_at": "2026-07-22T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The counterexample was found with ChatGPT and Codex and verified in Lean, alongside exhaustive subset enumeration.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "formal-conjectures PR #4482 - Disprove WOWII Conjecture 103",
          "url": "https://github.com/google-deepmind/formal-conjectures/pull/4482",
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
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      "note": "Lean-checked counterexample merged into the google-deepmind/formal-conjectures repository.",
      "sources": [
        {
          "label": "VibeMathed: Written on the Wall II, Graph Conjecture 103",
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      "url": "https://github.com/google-deepmind/formal-conjectures/pull/4482",
      "kind": "primary-mathematical-source",
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    },
    {
      "label": "VibeMathed: Written on the Wall II, Graph Conjecture 103",
      "url": "https://vibemathed.com/problem/wow-conjecture-103",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:wow-conjecture-103:signal:replay_ready",
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      "kind": "replay_ready",
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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."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}