{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:planar-spin-systems-approximation",
  "title": "Approximate Counting for Spin Systems on Planar Graphs",
  "canonical_statement": "Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting $q$-colourings on planar graphs is NP-hard for every constant $q \\geq 4$, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.",
  "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:planar-spin-systems-approximation:event",
      "problem_id": "vibemathed:planar-spin-systems-approximation",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-06T00:00:00.000Z",
      "title": "arXiv",
      "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:planar-spin-systems-approximation:attempt"
      ],
      "method_family_ids": [
        "vibemathed:planar-spin-systems-approximation:method"
      ],
      "source_assertion_ids": [
        "vibemathed:planar-spin-systems-approximation:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:planar-spin-systems-approximation:attempt",
      "problem_id": "vibemathed:planar-spin-systems-approximation",
      "model": "GPT-5.6 Sol Ultra",
      "provider": "OpenAI",
      "attempted_at": "2026-08-06T00:00:00.000Z",
      "human_collaborators": [
        "Heng Guo",
        "Xinyuan Zhang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "\"The main ideas of all proofs in this paper were found by GPT-5.6 Sol Ultra. For consistency with standard mathematical exposition, the words we and our are used throughout the paper, including when presenting ideas that originate in the output of the model. The authors simplified, streamlined, and wrote all of the proofs.\" The paper singles out finding the right problem to reduce from as where the model was particularly helpful.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.06172",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Approximate Counting for Spin Systems on Planar Graphs",
          "url": "https://vibemathed.com/problem/planar-spin-systems-approximation",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:planar-spin-systems-approximation:verification",
      "problem_id": "vibemathed:planar-spin-systems-approximation",
      "solution_event_id": "vibemathed:planar-spin-systems-approximation:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint days old, with no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Approximate Counting for Spin Systems on Planar Graphs",
          "url": "https://vibemathed.com/problem/planar-spin-systems-approximation",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.06172",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.06172",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Approximate Counting for Spin Systems on Planar Graphs",
      "url": "https://vibemathed.com/problem/planar-spin-systems-approximation",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:planar-spin-systems-approximation:signal:watch_only",
      "problem_id": "vibemathed:planar-spin-systems-approximation",
      "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"
}