{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R436",
  "content_hash": null,
  "slug": "ksd6-claim-min-plus-certificate",
  "type": "claim",
  "title": "A 4,096-state transfer proves the infinite tail",
  "summary": "The reachable min-plus cost vector obeys f_25=f_19+4, and transfer homogeneity propagates that identity forever.",
  "relevance": "For A period-six recurrence for domination on the three-row knight graph, record ksd6-claim-min-plus-certificate (“A 4,096-state transfer proves the infinite tail”) records a bound, answer, status fact, or structural consequence. The record states: The reachable min-plus cost vector obeys f_25=f_19+4, and transfer homogeneity propagates that identity forever.",
  "relevance_source": "recorded",
  "body": "Represent a set of rows in one column by a three-bit mask. For a mask \\(m\\), let \\(L_1(m)\\) be the rows reached one column away by a knight move, so rows 0 and 2 are interchanged and row 1 reaches nothing. Let \\(L_2(m)\\) be the rows reached two columns away, so each selected row reaches its adjacent rows.\n\nAfter processing a prefix, use the state\n\\[\n(a,b,u,v)\\in\\{0,\\ldots,7\\}^4.\n\\]\nHere \\(a,b\\) are the selected masks in the last two columns and \\(u,v\\) are their currently dominated masks. Choosing mask \\(x\\) in the next column is allowed exactly when\n\\[\nu\\mathbin{\\vert}L_2(x)=7,\n\\]\nbecause this is the final chance to dominate the older column. The new state is\n\\[\n(b,x,\\ v\\mathbin{\\vert}L_1(x),\\ x\\mathbin{\\vert}L_1(b)\\mathbin{\\vert}L_2(a)),\n\\]\nwith cost \\(|x|\\). Start at \\((0,0,7,7)\\), using two already-satisfied padding columns. After \\(n\\) actual transitions, two forced zero-mask transitions enforce domination in the final two actual columns. Every set of board squares gives one path, and every accepted path gives a dominating set.\n\nLet \\(f_n(s)\\) be the minimum path cost to state \\(s\\), with infinity for unreachable states. The state space has \\(8^4=4096\\) elements. Exactly 351 states are reachable at width 6, and this reachable set remains fixed. Its transfer graph has 2,046 labeled edges. Exact evaluation gives\n\\[\nf_{25}(s)=f_{19}(s)+4\n\\]\nfor all 351 finite entries, with the same states infinite on both sides. If \\(T\\) is the min-plus transfer, then \\(f_{n+1}=Tf_n\\), and \\(T(z+4)=Tz+4\\). Applying \\(T^k\\) proves \\(f_{25+k}=f_{19+k}+4\\) for every \\(k\\geq0\\). The two-zero finalization also commutes with an additive constant, so \\(\\gamma(n+6)=\\gamma(n)+4\\) for every \\(n\\geq19\\). Direct exact evaluation covers \\(9\\leq n\\leq18\\).",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "universal",
    "statement": "all widths of the three-row knight graph under ordinary closed-neighborhood domination"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Transition derivation and componentwise identity checked by ksd6-artifact-transfer-certificate"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Transition derivation and componentwise identity checked by ksd6-artifact-transfer-certificate"
  },
  "relations": [
    {
      "slug": "R435",
      "title": "The period-six recurrence holds from n=9",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R433",
      "title": "Executable min-plus periodicity certificate",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "knight-strip-domination-period-six",
      "title": "knight strip domination period six",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}