{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:facial-distance-patterns-planar",
  "title": "Facial Distance Patterns in Planar Graphs",
  "canonical_statement": "For a designated face of an undirected unweighted planar graph, how many distinct distance patterns can vertices have? Li and Parter (STOC 2019) proved an upper bound; Mozes, Wallheimer and Weimann conjectured the true answer matches their lower bound. Proved, closing the gap.",
  "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:facial-distance-patterns-planar:event",
      "problem_id": "vibemathed:facial-distance-patterns-planar",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-07T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:facial-distance-patterns-planar:attempt"
      ],
      "method_family_ids": [
        "vibemathed:facial-distance-patterns-planar:method"
      ],
      "source_assertion_ids": [
        "vibemathed:facial-distance-patterns-planar:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:facial-distance-patterns-planar:attempt",
      "problem_id": "vibemathed:facial-distance-patterns-planar",
      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-07T00:00:00.000Z",
      "human_collaborators": [
        "Viktor Fredslund-Hansen",
        "Shay Mozes",
        "Oren Weimann"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "\"The simple proof was found by OpenAI's GPT 5.6-Sol model\", and the paper says it came from a single prompt describing the state of the art and asking for any improvement on the upper bound. The authors are candid about what that means: \"It is surprising (not to say embarrasing) that this open problem has such a simple proof, which has eluded the community despite the human efforts invested in it.\" Section 2 of the paper is titled The ChatGPT Proof.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.07187",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Facial Distance Patterns in Planar Graphs",
          "url": "https://vibemathed.com/problem/facial-distance-patterns-planar",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "cost_usd": null,
      "independence": "unknown"
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  ],
  "verifications": [
    {
      "id": "vibemathed:facial-distance-patterns-planar:verification",
      "problem_id": "vibemathed:facial-distance-patterns-planar",
      "solution_event_id": "vibemathed:facial-distance-patterns-planar: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. The argument is short and self-contained, turning on the observation that a pattern's entries must sum to plus or minus one because the face is a cycle.",
      "sources": [
        {
          "label": "VibeMathed: Facial Distance Patterns in Planar Graphs",
          "url": "https://vibemathed.com/problem/facial-distance-patterns-planar",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.07187",
          "kind": "primary-mathematical-source",
          "license": null
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      ]
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      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.07187",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Facial Distance Patterns in Planar Graphs",
      "url": "https://vibemathed.com/problem/facial-distance-patterns-planar",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:facial-distance-patterns-planar:signal:watch_only",
      "problem_id": "vibemathed:facial-distance-patterns-planar",
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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."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}