{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R490",
  "content_hash": null,
  "slug": "maj8torus-claim-certified-numerator-interval",
  "type": "claim",
  "title": "The exact numerator remains open, with a certified interval of 128 through 18,446,744,073,709,551,106",
  "summary": "A symmetry orbit of one checked two-cycle witness supplies the lower bound, while 510 stripe configurations are fixed and supply the upper bound.",
  "relevance": "For Two-cycle probability for majority dynamics on the eight torus, record maj8torus-claim-certified-numerator-interval (“The exact numerator remains open, with a certified interval of 128 through 18,446,744,073,709,551,106”) records a bound, answer, status fact, or structural consequence. The record states: A symmetry orbit of one checked two-cycle witness supplies the lower bound, while 510 stripe configurations are fixed and supply the upper bound.",
  "relevance_source": "recorded",
  "body": "Write \\(N_2\\) for the number of initial configurations that reach a genuine two-cycle. The current certificate proves\n\\[\n128\\leq N_2\\leq 2^{64}-510\n=18{,}446{,}744{,}073{,}709{,}551{,}106.\n\\]\nThe lower bound comes from the 64 translations of the displayed witness and their color complements. The verifier checks that these 128 initial configurations are distinct. Majority update commutes with translations and complementation, so each reaches a genuine two-cycle.\n\nFor the upper bound, every configuration whose rows are individually monochromatic is fixed. East and west already give two neighbors of the cell's own color, so the vertical neighbors can never force a strict opposing majority. The same argument applies to monochromatic columns. These two families each contain \\(2^8=256\\) configurations and intersect in the two constant configurations. Their union therefore contains 510 distinct fixed points.\n\nColor complementation pairs the two-cycle basin without fixed configurations under the pairing, so \\(N_2\\) is even. Determining it exactly still requires an exhaustive symbolic basin count over all \\(2^{64}\\) starting states.",
  "status": "open",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all binary initial configurations on the labeled 8 by 8 torus under synchronous four-neighbor majority update with retention on a tie",
    "bounds": {
      "vertices": {
        "min": 64,
        "max": 64
      },
      "initial_configurations": {
        "min": 18446744073709552000,
        "max": 18446744073709552000
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0166-218X(81)90034-2",
      "locator": "Bounds and symmetry checks in maj8torus-artifact-witness-and-update-verifier"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0166-218X(81)90034-2",
    "locator": "Bounds and symmetry checks in maj8torus-artifact-witness-and-update-verifier"
  },
  "models": [],
  "relations": [
    {
      "slug": "R488",
      "title": "Deterministic majority-update and symmetry-orbit verifier",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R492",
      "title": "The candidate run reported 8,391 two-cycle outcomes among 50,000 starts",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R491",
      "title": "Every orbit eventually has period one or two",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R489",
      "title": "The period theorem is classical; the size-eight basin count was not located",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "majority-eight-torus-two-cycle-probability",
      "title": "majority eight torus two cycle probability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}