{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R721",
  "content_hash": null,
  "slug": "sdw-claim-mex-construction",
  "type": "claim",
  "title": "The mex construction generates exactly the P-positions",
  "summary": "Unused coordinates and explicit square-diagonal exclusions give the losing pairs in lexicographic order.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Let \\(P_n=(a_n,b_n)\\), with \\(a_n\\leq b_n\\). Start with \\(P_0=(0,0)\\). After constructing the earlier pairs, let \\(a_n\\) be the least nonnegative integer absent from all their coordinates. Choose \\(b_n\\geq a_n\\) as the least unused integer such that\n\\[\n(a_n-s^2,b_n-s^2)\\notin\\{P_0,\\ldots,P_{n-1}\\}\n\\]\nfor every positive square \\(s^2\\leq a_n\\).\n\nNo two constructed pairs share a coordinate, so a single-heap move cannot connect them. The displayed exclusion prevents a legal square-diagonal move between them.\n\nIt remains to check that every other position can move to a constructed pair. If its smaller coordinate already occurs as the larger coordinate of an earlier pair, a single-heap reduction reaches that pair. Otherwise its smaller coordinate equals some \\(a_n\\). For a canonical position \\((a_n,y)\\), a value \\(y<b_n\\) was either used as the larger coordinate of an earlier pair or rejected because a square-diagonal move reaches an earlier pair. A value \\(y>b_n\\) can be reduced to \\(b_n\\) in one heap. These cases cover every position outside the constructed set. Hence the construction satisfies both defining P-position properties and lists the P-positions by increasing first coordinate.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every P-position of the two-heap game whose equal diagonal removals are positive squares"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://ajc.maths.uq.edu.au/pdf/44/ajc_v44_p043.pdf",
      "locator": "Duchene, Fraenkel, Gravier, and Nowakowski, Another bridge between Nim and Wythoff, Australasian Journal of Combinatorics 44 (2009), Definition 5 for Wyt_K; the recurrence proof specialized to square K is included here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://ajc.maths.uq.edu.au/pdf/44/ajc_v44_p043.pdf",
    "locator": "Duchene, Fraenkel, Gravier, and Nowakowski, Another bridge between Nim and Wythoff, Australasian Journal of Combinatorics 44 (2009), Definition 5 for Wyt_K; the recurrence proof specialized to square K is included here"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R720",
      "title": "The first gap 50 occurs at P-position 42,762",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R718",
      "title": "Two-generator certificate for 100,000 P-positions",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "square-diagonal-wythoff-gap-50",
      "title": "square diagonal wythoff gap 50",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}