{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R671",
  "content_hash": null,
  "slug": "rr8gc-attempt-literature-audit",
  "type": "attempt",
  "title": "General cover-time theorems settle the finite upper bound",
  "summary": "The sources fix the update convention and prove bounds for every Eulerian graph; the exact 8 by 8 clockwise extremum was absent from the located papers.",
  "relevance": "For Longest rotor-router cover time on the eight by eight grid, record rr8gc-attempt-literature-audit (“General cover-time theorems settle the finite upper bound”) documents a concrete method, search boundary, or failed route. The record states: The sources fix the update convention and prove bounds for every Eulerian graph; the exact 8 by 8 clockwise extremum was absent from the located papers.",
  "relevance_source": "recorded",
  "body": "Holroyd and Propp describe the retrospective rotor convention on pages 2 and 3: the rotor advances to the next arc and the particle follows that arc. This matches the problem after each boundary vertex receives its shortened clockwise neighbor list.\n\nFlorescu, Levine, and Peres prove \\(t_{\\mathrm{vertex}}\\leq D|E|\\) for every finite Eulerian directed graph, arbitrary rotor mechanism, and arbitrary initial configuration. Their proof decomposes the walk into excursions. Applied to the bidirected grid, it gives 3,136. They credit Yanovski, Wagner, and Bruckstein with the earlier edge-cover bound \\(2D|E|\\). Friedrich and Sauerwald give other general vertex and edge cover-time techniques and work out several graph families.\n\nFocused searches used `rotor-router cover time grid`, `clockwise rotor walk finite square`, and the exact dimensions 8 by 8. The sources found asymptotic grid results and general finite-graph bounds. None supplied the exact extremum under this boundary-order convention. Novelty of the finite question remains unverified.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.4169/amer.math.monthly.123.7.627",
      "locator": "Florescu, Levine, and Peres, The range of a rotor walk, Section 6, Theorem 6.1; Holroyd and Propp, Rotor walks and Markov chains, arXiv:0904.4507, pages 2-3; Friedrich and Sauerwald, The Cover Time of Deterministic Random Walks, EJC 17(1), R167 (2010)"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4169/amer.math.monthly.123.7.627",
    "locator": "Florescu, Levine, and Peres, The range of a rotor walk, Section 6, Theorem 6.1; Holroyd and Propp, Rotor walks and Markov chains, arXiv:0904.4507, pages 2-3; Friedrich and Sauerwald, The Cover Time of Deterministic Random Walks, EJC 17(1), R167 (2010)"
  },
  "relations": [
    {
      "slug": "R673",
      "title": "The worst cover time lies between 1,282 and 3,136 moves",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "rotor-router-eight-grid-cover",
      "title": "rotor router eight grid cover",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}