{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:wowii-graph-conjecture-322",
  "title": "Written on the Wall II, Graph Conjecture 322",
  "canonical_statement": "Let $G$ be a simple connected graph on $n\\geq 5$ vertices. If the maximum over all vertices $v$ of $\\ell(v)$ - the independence number of the subgraph induced by the open neighborhood $N(v)$ - is at most $1$, must $G$ be well totally dominated? Answered affirmatively; the Lean proof in fact needs only $n\\geq 2$, and retains the conjecture's $n\\geq 5$ to state the source faithfully.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:wowii-graph-conjecture-322:event",
      "problem_id": "vibemathed:wowii-graph-conjecture-322",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-08-02T00:00:00.000Z",
      "title": "formal-conjectures PR #4686 - prove Conjecture 322",
      "summary": "The formalization proves the statement under the weaker hypothesis n >= 2; the pull request marking the conjecture solved is open, not merged",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:wowii-graph-conjecture-322:attempt"
      ],
      "method_family_ids": [
        "vibemathed:wowii-graph-conjecture-322:method"
      ],
      "source_assertion_ids": [
        "vibemathed:wowii-graph-conjecture-322:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:wowii-graph-conjecture-322:attempt",
      "problem_id": "vibemathed:wowii-graph-conjecture-322",
      "model": "Aristotle",
      "provider": "Harmonic",
      "attempted_at": "2026-08-02T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The pull request marking the conjecture solved credits the proof to Aristotle, Harmonic's prover; a human contributor prepared and filed the formalization.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Lean proof",
          "url": "https://github.com/MrBrain295/formal-conjectures/blob/322/FormalConjectures/WrittenOnTheWallII/GraphConjecture322.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Written on the Wall II conjecture list",
          "url": "http://cms.dt.uh.edu/faculty/delavinae/research/wowII/",
          "kind": "problem-record",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "formal-conjectures PR #4686 - prove Conjecture 322",
          "url": "https://github.com/google-deepmind/formal-conjectures/pull/4686",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Written on the Wall II, Graph Conjecture 322",
          "url": "https://vibemathed.com/problem/wowii-graph-conjecture-322",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:wowii-graph-conjecture-322:verification",
      "problem_id": "vibemathed:wowii-graph-conjecture-322",
      "solution_event_id": "vibemathed:wowii-graph-conjecture-322:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Sorry-free Lean 4 proof filed against google-deepmind/formal-conjectures, which flips the conjecture's attribute from `research open` to `research solved` and links the proof. Unlike the site's WOWII 217 entry it needs no native_decide: the argument is conceptual, showing every neighborhood is a clique and deducing well-total-domination. Not independently reviewed, and the pull request is still open.",
      "sources": [
        {
          "label": "VibeMathed: Written on the Wall II, Graph Conjecture 322",
          "url": "https://vibemathed.com/problem/wowii-graph-conjecture-322",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "formal-conjectures PR #4686 - prove Conjecture 322",
          "url": "https://github.com/google-deepmind/formal-conjectures/pull/4686",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "formal-conjectures PR #4686 - prove Conjecture 322",
      "url": "https://github.com/google-deepmind/formal-conjectures/pull/4686",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Written on the Wall II, Graph Conjecture 322",
      "url": "https://vibemathed.com/problem/wowii-graph-conjecture-322",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:wowii-graph-conjecture-322:signal:verify_now",
      "problem_id": "vibemathed:wowii-graph-conjecture-322",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "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"
}