{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R435",
  "content_hash": null,
  "slug": "ksd6-claim-exact-recurrence",
  "type": "claim",
  "title": "The period-six recurrence holds from n=9",
  "summary": "The exact values satisfy gamma(n+6)=gamma(n)+4 for every n at least 9.",
  "relevance": "For A period-six recurrence for domination on the three-row knight graph, record ksd6-claim-exact-recurrence (“The period-six recurrence holds from n=9”) records a bound, answer, status fact, or structural consequence. The record states: The exact values satisfy gamma(n+6)=gamma(n)+4 for every n at least 9.",
  "relevance_source": "recorded",
  "body": "The answer is yes. The initial values are\n\\[\n(\\gamma(1),\\ldots,\\gamma(14))\n=(3,4,4,4,4,4,6,8,8,8,8,8,10,11).\n\\]\nFor every \\(n\\geq9\\),\n\\[\n\\gamma(n+6)=\\gamma(n)+4.\n\\]\nEquivalently, write \\(n=6q+r\\), where \\(0\\leq r<6\\). Then for \\(n\\geq9\\),\n\\[\n\\gamma(n)=4q+c_r,\n\\qquad (c_0,c_1,c_2,c_3,c_4,c_5)=(0,2,3,4,4,4).\n\\]\n\nThe finite-state proof is recorded in `ksd6-claim-min-plus-certificate`. Its dynamic program computes the optimum over every column-mask sequence, so each displayed value includes both a construction and a lower bound. The scalar recurrence is checked directly for \\(9\\leq n\\leq18\\). A componentwise min-plus vector identity at widths 19 and 25 propagates the result through every larger width.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "universal",
    "statement": "every three-row knight graph K_n, with the recurrence asserted for every integer n at least 9"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Exact finite-state proof and executable artifact ksd6-artifact-transfer-certificate, 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Exact finite-state proof and executable artifact ksd6-artifact-transfer-certificate, 2026-07-24"
  },
  "relations": [
    {
      "slug": "R436",
      "title": "A 4,096-state transfer proves the infinite tail",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R433",
      "title": "Executable min-plus periodicity certificate",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R434",
      "title": "Targeted knight-domination literature search",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "knight-strip-domination-period-six",
      "title": "knight strip domination period six",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}