{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R718",
  "content_hash": null,
  "slug": "sdw-artifact-two-generator-certificate",
  "type": "artifact",
  "title": "Two-generator certificate for 100,000 P-positions",
  "summary": "A forbidden-gap mex generator and a direct pair-attack generator agree on the complete prefix.",
  "relevance": "This supplies a computation, dataset, or executable check used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The optimized generator observes that a diagonal move preserves \\(b-a\\). For each new mex coordinate \\(a\\), it checks every square \\(s^2\\leq a\\), reads the gap of the earlier pair whose first coordinate is \\(a-s^2\\), and forbids that gap. The independent generator stores complete prior pairs and tests each candidate \\((a,b)\\) directly against every shifted pair \\((a-s^2,b-s^2)\\). Both also exclude previously used coordinates.\n\nThe generators return identical lists of 100,000 pairs. Serializing that list as compact JSON with no spaces or trailing newline gives 1,486,349 bytes and SHA-256 digest\n```\nf5cacfeedfe15d229189db324a2a8457c4b765007c9f575184549578be15d55b\n```\nThe compact 68-row record-gap table has SHA-256 digest `6a8a5adb0eb6794be663251ab7babce82198d8f95eb81466c45ecb59d094101a`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all of the first 100,000 P-positions and their gaps",
    "bounds": {
      "position_ordinal": {
        "min": 1,
        "max": 100000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "entrypoint": "join source_lines with newline and run with python3",
    "runtime": "CPython 3, standard library only; about 35 seconds on the recorded run",
    "citation": {
      "url": "https://ajc.maths.uq.edu.au/pdf/44/ajc_v44_p043.pdf",
      "locator": "Self-contained CPython computation independently executed on 2026-07-24"
    },
    "inline_source": [
      "from hashlib import sha256",
      "from json import dumps",
      "LIMIT=100000",
      "CAP=250000",
      "PATH_HASH='f5cacfeedfe15d229189db324a2a8457c4b765007c9f575184549578be15d55b'",
      "RECORD_HASH='6a8a5adb0eb6794be663251ab7babce82198d8f95eb81466c45ecb59d094101a'",
      "def optimized():",
      "    used=bytearray(CAP)",
      "    first_gap=[-1]*CAP",
      "    path=[]; mex=0; squares=[]; root=1",
      "    for _ in range(LIMIT):",
      "        while used[mex]: mex+=1",
      "        a=mex",
      "        while root*root<=a:",
      "            squares.append(root*root); root+=1",
      "        forbidden=set()",
      "        for square in squares:",
      "            gap=first_gap[a-square]",
      "            if gap>=0: forbidden.add(gap)",
      "        b=a",
      "        while used[b] or b-a in forbidden: b+=1",
      "        path.append((a,b))",
      "        first_gap[a]=b-a",
      "        used[a]=used[b]=1",
      "    return path",
      "def direct():",
      "    used=bytearray(CAP)",
      "    prior=set()",
      "    path=[]; mex=0; squares=[]; root=1",
      "    for _ in range(LIMIT):",
      "        while used[mex]: mex+=1",
      "        a=mex",
      "        while root*root<=a:",
      "            squares.append(root*root); root+=1",
      "        b=a",
      "        while True:",
      "            if not used[b] and all((a-square,b-square) not in prior for square in squares): break",
      "            b+=1",
      "        pair=(a,b)",
      "        path.append(pair); prior.add(pair)",
      "        used[a]=used[b]=1",
      "    return path",
      "path=optimized()",
      "assert path==direct()",
      "records=[]; best=-1",
      "for ordinal,(a,b) in enumerate(path,1):",
      "    gap=b-a",
      "    if gap>best:",
      "        best=gap; records.append([ordinal,a,b,gap])",
      "qualified=[(i+1,a,b,b-a) for i,(a,b) in enumerate(path) if b-a>=50]",
      "maxima=[(i+1,a,b) for i,(a,b) in enumerate(path) if b-a==67]",
      "raw=dumps(path,separators=(',',':')).encode()",
      "record_raw=dumps(records,separators=(',',':')).encode()",
      "assert len(raw)==1486349 and sha256(raw).hexdigest()==PATH_HASH",
      "assert len(record_raw)==1438 and sha256(record_raw).hexdigest()==RECORD_HASH",
      "assert qualified[0]==(42762,83742,83792,50) and len(qualified)==8421",
      "assert records[-1]==[96868,190625,190692,67] and len(maxima)==3",
      "assert path[-1]==(196828,196850) and max(map(max,path))==196880",
      "print(len(path),qualified[0],records[-1],path[-1],PATH_HASH)"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://ajc.maths.uq.edu.au/pdf/44/ajc_v44_p043.pdf",
    "locator": "Self-contained CPython computation independently executed on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R720",
      "title": "The first gap 50 occurs at P-position 42,762",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "R721",
      "title": "The mex construction generates exactly the P-positions",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "square-diagonal-wythoff-gap-50",
      "title": "square diagonal wythoff gap 50",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}