{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R357",
  "content_hash": null,
  "slug": "h65537-artifact-complete-p1009-calibration",
  "type": "artifact",
  "title": "Complete p=1009 cycle decomposition",
  "summary": "The calibration space has 1,011 cycles and maximum period 6,724; its period histogram has a fixed SHA-256 digest.",
  "relevance": "For Largest cycle of a Hénon permutation over the 65537 field, record h65537-artifact-complete-p1009-calibration (“Complete p=1009 cycle decomposition”) supplies evidence or a replay used to check the packet. The record states: The calibration space has 1,011 cycles and maximum period 6,724; its period histogram has a fixed SHA-256 digest.",
  "relevance_source": "recorded",
  "body": "A visited bitmap decomposes all \\(1009^2=1{,}018{,}081\\) states into disjoint cycles. It finds 1,011 cycles. The maximum period is 6,724, and the first representative in lexicographic state order is \\((0,132)\\).\n\nThe program serializes the period histogram in increasing order as one `period:multiplicity` line per entry. Its SHA-256 digest is `404eaecab6ff5302910b9986568d2362856415376fcb3b49c4b30efe972f50f3`. The digest also follows from the affine conjugacy with the kicked-map calibration, since conjugacy and inversion preserve every cycle length.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the complete state space F_1009 squared for H(x,y)=(y,y^2+1-x)",
    "bounds": {
      "prime": {
        "min": 1009,
        "max": 1009
      },
      "states": {
        "min": 1018081,
        "max": 1018081
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_exact_computation",
    "entrypoint": "Execute code with python3",
    "runtime": "Python 3.9 or newer, standard library only",
    "citation": {
      "url": "https://doi.org/10.1088/0951-7715/18/5/015",
      "locator": "Complete enumeration reported in candidate record 155; histogram checksum cross-checked through the conjugate kicked-map fixture"
    },
    "inline_source": "from collections import Counter\nfrom hashlib import sha256\np = 1009\nseen = bytearray(p*p)\nhist = Counter()\nbest = 0\nwitness = None\nfor index in range(p*p):\n    if seen[index]:\n        continue\n    x, y = divmod(index, p)\n    x0, y0 = x, y\n    period = 0\n    while True:\n        state_index = x*p + y\n        assert not seen[state_index]\n        seen[state_index] = 1\n        x, y = y, (y*y + 1 - x) % p\n        period += 1\n        if (x, y) == (x0, y0):\n            break\n    hist[period] += 1\n    if period > best:\n        best, witness = period, (x0, y0)\npayload = ''.join(f'{length}:{hist[length]}\\n' for length in sorted(hist)).encode()\ndigest = sha256(payload).hexdigest()\nassert (sum(hist.values()), best, witness) == (1011, 6724, (0, 132))\nassert sum(length*count for length, count in hist.items()) == p*p\nassert digest == '404eaecab6ff5302910b9986568d2362856415376fcb3b49c4b30efe972f50f3'\nprint(f'states={p*p} cycles={sum(hist.values())} maximum={best} witness={witness} histogram_sha256={digest}')\n",
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1088/0951-7715/18/5/015",
    "locator": "Complete enumeration reported in candidate record 155; histogram checksum cross-checked through the conjugate kicked-map fixture"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R358",
      "title": "Exact cycle replay with an orbit digest",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "henon-65537-max-cycle",
      "title": "henon 65537 max cycle",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}