{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:lamplighter-return-probability",
  "title": "Return Probability for the Lamplighter Walk on a Tree",
  "canonical_statement": "For the switch-walk-switch lamplighter walk on $\\mathbb{Z}_2 \\wr T_d$, prove the sharp asymptotic $p_{2n}(e,e) = \\rho_d^{2n} \\exp[-(\\pi^2 (\\log(d-1))^2 + o(1)) \\frac{n}{\\log^2 n}]$ with $\\rho_d = \\frac{2\\sqrt{d-1}}{d}$.",
  "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:lamplighter-return-probability:event",
      "problem_id": "vibemathed:lamplighter-return-probability",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-05-15T00:00:00.000Z",
      "title": "arXiv:2605.21744 - Return probability for the switch-walk-switch lamplighter walk",
      "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:lamplighter-return-probability:attempt"
      ],
      "method_family_ids": [
        "vibemathed:lamplighter-return-probability:method"
      ],
      "source_assertion_ids": [
        "vibemathed:lamplighter-return-probability:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:lamplighter-return-probability:attempt",
      "problem_id": "vibemathed:lamplighter-return-probability",
      "model": "QED (GPT-5.5 Pro)",
      "provider": "OpenAI",
      "attempted_at": "2026-05-15T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The QED multi-agent system produced the proof from the problem statement alone, through multiple rounds of decomposition and refinement.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.21744 - Return probability for the switch-walk-switch lamplighter walk",
          "url": "https://arxiv.org/abs/2605.21744",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Return Probability for the Lamplighter Walk on a Tree",
          "url": "https://vibemathed.com/problem/lamplighter-return-probability",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:lamplighter-return-probability:verification",
      "problem_id": "vibemathed:lamplighter-return-probability",
      "solution_event_id": "vibemathed:lamplighter-return-probability:event",
      "level": "independent_expert_verified",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Verified by the contributing domain expert who posed the problem; public preprint.",
      "sources": [
        {
          "label": "VibeMathed: Return Probability for the Lamplighter Walk on a Tree",
          "url": "https://vibemathed.com/problem/lamplighter-return-probability",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.21744 - Return probability for the switch-walk-switch lamplighter walk",
          "url": "https://arxiv.org/abs/2605.21744",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2605.21744 - Return probability for the switch-walk-switch lamplighter walk",
      "url": "https://arxiv.org/abs/2605.21744",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Return Probability for the Lamplighter Walk on a Tree",
      "url": "https://vibemathed.com/problem/lamplighter-return-probability",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:lamplighter-return-probability:signal:replay_ready",
      "problem_id": "vibemathed:lamplighter-return-probability",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "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"
}