{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:t-edge-balanced-existence",
  "title": "Existence of $t$-Edge-Balanced Graphs for $t \\ge 3$",
  "canonical_statement": "A graph $G$ on $n$ vertices with $k$ edges is $t$-edge-balanced if every graph on $n$ vertices with $t$ edges is contained in exactly the same number of subgraphs of $K_n$ isomorphic to $G$. Infinite families were known for $t = 2$, but no example was known for any $t \\ge 3$. Resolved in both directions: $3$-edge-balanced graphs exist, and no nontrivial $t$-edge-balanced graphs exist for $t \\ge 4$.",
  "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:t-edge-balanced-existence:event",
      "problem_id": "vibemathed:t-edge-balanced-existence",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-05-16T00:00:00.000Z",
      "title": "arXiv:2605.16840 - On $t$-edge-balanced graphs",
      "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_assisted",
      "attempt_ids": [
        "vibemathed:t-edge-balanced-existence:attempt"
      ],
      "method_family_ids": [
        "vibemathed:t-edge-balanced-existence:method"
      ],
      "source_assertion_ids": [
        "vibemathed:t-edge-balanced-existence:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:t-edge-balanced-existence:attempt",
      "problem_id": "vibemathed:t-edge-balanced-existence",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-05-16T00:00:00.000Z",
      "human_collaborators": [
        "Yeow Meng Chee"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Tooling rather than mathematics: the simulated annealing search was implemented in C++ with the code developed with the assistance of ChatGPT. The search then found zero-score graphs for 11 parameter sets including the ten smallest. The arithmetic conditions on the parameters and the nonexistence proof for $t \\ge 4$ are the author's.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.16840 - On $t$-edge-balanced graphs",
          "url": "https://arxiv.org/abs/2605.16840",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Existence of $t$-Edge-Balanced Graphs for $t \\ge 3$",
          "url": "https://vibemathed.com/problem/t-edge-balanced-existence",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:t-edge-balanced-existence:verification",
      "problem_id": "vibemathed:t-edge-balanced-existence",
      "solution_event_id": "vibemathed:t-edge-balanced-existence:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The examples are explicit graphs, published with source code, and each candidate had its subgraph count profile recomputed from scratch as an independent check. That makes the existence half verifiable by anyone who reruns the count. arXiv preprint, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Existence of $t$-Edge-Balanced Graphs for $t \\ge 3$",
          "url": "https://vibemathed.com/problem/t-edge-balanced-existence",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.16840 - On $t$-edge-balanced graphs",
          "url": "https://arxiv.org/abs/2605.16840",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2605.16840 - On $t$-edge-balanced graphs",
      "url": "https://arxiv.org/abs/2605.16840",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Existence of $t$-Edge-Balanced Graphs for $t \\ge 3$",
      "url": "https://vibemathed.com/problem/t-edge-balanced-existence",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:t-edge-balanced-existence:signal:watch_only",
      "problem_id": "vibemathed:t-edge-balanced-existence",
      "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"
}