{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:line-graph-signature-unbounded",
  "title": "Signature of Connected Line Graphs",
  "canonical_statement": "Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A $14$-vertex witness has signature $2$, and chaining copies gives connected line graphs of signature $k + 1$ for every $k \\ge 1$ - the signature is unbounded.",
  "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:line-graph-signature-unbounded:event",
      "problem_id": "vibemathed:line-graph-signature-unbounded",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-30T00:00:00.000Z",
      "title": "arXiv:2607.22874 - The signature of connected line graphs is unbounded",
      "summary": "no constant-bound repair of the conjecture is possible",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:line-graph-signature-unbounded:attempt"
      ],
      "method_family_ids": [
        "vibemathed:line-graph-signature-unbounded:method"
      ],
      "source_assertion_ids": [
        "vibemathed:line-graph-signature-unbounded:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:line-graph-signature-unbounded:attempt",
      "problem_id": "vibemathed:line-graph-signature-unbounded",
      "model": "ChatGPT-5.6 Pro, Claude Fable 5",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-07-30T00:00:00.000Z",
      "human_collaborators": [
        "Luke Francis",
        "Trevor Uptain"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The 14-vertex witness came from a ChatGPT-assisted search; Claude assisted an independent 48-vertex search and the development of the unbounded family. The authors reproduced everything with separately coded exact-arithmetic audits.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.22874 - The signature of connected line graphs is unbounded",
          "url": "https://arxiv.org/abs/2607.22874",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Signature of Connected Line Graphs",
          "url": "https://vibemathed.com/problem/line-graph-signature-unbounded",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:line-graph-signature-unbounded:verification",
      "problem_id": "vibemathed:line-graph-signature-unbounded",
      "solution_event_id": "vibemathed:line-graph-signature-unbounded:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Exact finite certificates with independent audit implementations; revised arXiv preprint, not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Signature of Connected Line Graphs",
          "url": "https://vibemathed.com/problem/line-graph-signature-unbounded",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.22874 - The signature of connected line graphs is unbounded",
          "url": "https://arxiv.org/abs/2607.22874",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.22874 - The signature of connected line graphs is unbounded",
      "url": "https://arxiv.org/abs/2607.22874",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Signature of Connected Line Graphs",
      "url": "https://vibemathed.com/problem/line-graph-signature-unbounded",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:line-graph-signature-unbounded:signal:watch_only",
      "problem_id": "vibemathed:line-graph-signature-unbounded",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}