{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R491",
  "content_hash": null,
  "slug": "maj8torus-claim-eventual-period-at-most-two",
  "type": "claim",
  "title": "Every orbit eventually has period one or two",
  "summary": "The tie-retaining rule is a symmetric threshold map, so the finite threshold-network period-two theorem applies.",
  "relevance": "For Two-cycle probability for majority dynamics on the eight torus, record maj8torus-claim-eventual-period-at-most-two (“Every orbit eventually has period one or two”) records a bound, answer, status fact, or structural consequence. The record states: The tie-retaining rule is a symmetric threshold map, so the finite threshold-network period-two theorem applies.",
  "relevance_source": "recorded",
  "body": "Retention on a two-against-two neighbor tie can be written as ordinary majority among five inputs: the four neighbors and the vertex's current color. Thus the update is a binary threshold network with a symmetric interaction matrix, including a diagonal self-weight of one.\n\nGoles and Olivos proved that iteration of a finite symmetric binary threshold map reaches a fixed point or a cycle of length two. Their theorem applies to this torus and this tie convention. It justifies the fixed-point versus genuine-two-cycle dichotomy used by the candidate and the verifier. The theorem controls the possible eventual periods. It does not count either basin.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every initial configuration for the stated synchronous majority dynamics on the finite undirected 8 by 8 torus"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0166-218X(81)90034-2",
      "locator": "Eric Goles and Jorge Olivos, Comportement periodique des fonctions a seuil binaires et applications, Discrete Applied Mathematics 3 (1981), 93-105"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0166-218X(81)90034-2",
    "locator": "Eric Goles and Jorge Olivos, Comportement periodique des fonctions a seuil binaires et applications, Discrete Applied Mathematics 3 (1981), 93-105"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R490",
      "title": "The exact numerator remains open, with a certified interval of 128 through 18,446,744,073,709,551,106",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R488",
      "title": "Deterministic majority-update and symmetry-orbit verifier",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "majority-eight-torus-two-cycle-probability",
      "title": "majority eight torus two cycle probability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}