{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:chip-firing-middle-stair",
  "title": "The Middle Stair of Parallel Chip-Firing",
  "canonical_statement": "Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.",
  "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:chip-firing-middle-stair:event",
      "problem_id": "vibemathed:chip-firing-middle-stair",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-04T00:00:00.000Z",
      "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:chip-firing-middle-stair:attempt"
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      "id": "vibemathed:chip-firing-middle-stair:attempt",
      "problem_id": "vibemathed:chip-firing-middle-stair",
      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-04T00:00:00.000Z",
      "human_collaborators": [
        "Daniel Wang",
        "Nathan Lannan"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper presents the proof plainly as found by the model: \"The proof presented in Section 3 was found by the large language model GPT-5.6-Sol. The authors verified the resulting argument and take full responsibility.\"",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.04153",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Middle Stair of Parallel Chip-Firing",
          "url": "https://vibemathed.com/problem/chip-firing-middle-stair",
          "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:chip-firing-middle-stair:verification",
      "problem_id": "vibemathed:chip-firing-middle-stair",
      "solution_event_id": "vibemathed:chip-firing-middle-stair:event",
      "level": "independent_expert_verified",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Beyond the authors' own verification, the acknowledgments thank David Ji and Michael Li, two of the conjecture's posers, for assisting with reviewing the proof.",
      "sources": [
        {
          "label": "VibeMathed: The Middle Stair of Parallel Chip-Firing",
          "url": "https://vibemathed.com/problem/chip-firing-middle-stair",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.04153",
          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.04153",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Middle Stair of Parallel Chip-Firing",
      "url": "https://vibemathed.com/problem/chip-firing-middle-stair",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:chip-firing-middle-stair:signal:replay_ready",
      "problem_id": "vibemathed:chip-firing-middle-stair",
      "kind": "replay_ready",
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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"
}