{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R720",
  "content_hash": null,
  "slug": "sdw-claim-first-gap-50-and-prefix-maximum",
  "type": "claim",
  "title": "The first gap 50 occurs at P-position 42,762",
  "summary": "The first qualifying pair is (83,742, 83,792), and the maximum gap in the complete prefix is 67.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Counting \\((0,0)\\) as position 1, the first P-position with \\(b-a\\geq50\\) is\n\\[\nP_{42762}=(83742,83792),\\qquad 83792-83742=50.\n\\]\nThe preceding position has a smaller gap, and an exhaustive scan of positions 1 through 42,761 finds maximum gap 49. Thus this is the first threshold witness in the requested ordering.\n\nAcross all 100,000 positions, the maximum gap is 67. Its first occurrence is\n\\[\nP_{96868}=(190625,190692).\n\\]\nGap 67 occurs three times in the prefix. There are 8,421 positions with gap at least 50. The final pair is\n\\[\nP_{100000}=(196828,196850).\n\\]\nThe final ten pairs and every strict gap record are stored with the executable certificate. The 68th and last record is the gap-67 pair above, so no larger gap occurs in the finite scope.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the first 100,000 lexicographically ordered P-positions, counting (0,0) as position 1",
    "bounds": {
      "position_ordinal": {
        "min": 1,
        "max": 100000
      }
    },
    "exhaustive": true
  },
  "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": "Two independent exact generators in sdw-artifact-two-generator-certificate, run on 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://ajc.maths.uq.edu.au/pdf/44/ajc_v44_p043.pdf",
    "locator": "Two independent exact generators in sdw-artifact-two-generator-certificate, run on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R721",
      "title": "The mex construction generates exactly the P-positions",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R718",
      "title": "Two-generator certificate for 100,000 P-positions",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R719",
      "title": "The ruleset is an established Wyt_K specialization",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "square-diagonal-wythoff-gap-50",
      "title": "square diagonal wythoff gap 50",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}