{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R358",
  "content_hash": null,
  "slug": "h65537-artifact-exact-witness-replay",
  "type": "artifact",
  "title": "Exact cycle replay with an orbit digest",
  "summary": "Standard-library Python checks the first return, selected states, the predecessor, and the period again with Floyd's algorithm.",
  "relevance": "For Largest cycle of a Hénon permutation over the 65537 field, record h65537-artifact-exact-witness-replay (“Exact cycle replay with an orbit digest”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python checks the first return, selected states, the predecessor, and the period again with Floyd's algorithm.",
  "relevance_source": "recorded",
  "body": "The program performs exactly 294,672 direct modular updates. It rejects any early return to the start and checks the first four states, the predecessor, and the final return. A second pass uses Floyd's constant-memory algorithm and requires preperiod zero and period 294,672.\n\nDuring the direct pass, each cycle state is encoded as x followed by y, each a four-byte little-endian unsigned integer. SHA-256 of the resulting 2,357,376-byte logical stream is printed as `trace_sha256`. This supplies a portable orbit checksum for independent runs.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "every state in the 294672-cycle through (0,2) over F_65537, checked by direct replay and Floyd cycle finding",
    "bounds": {
      "prime": {
        "min": 65537,
        "max": 65537
      },
      "cycle_updates": {
        "min": 294672,
        "max": 294672
      },
      "independent_cycle_methods": {
        "min": 2,
        "max": 2
      }
    },
    "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": "Self-contained Python 3 standard-library replay derived from candidate record 155"
    },
    "inline_source": "from hashlib import sha256\np = 65537\nstart = (0, 2)\ntarget = 294672\n\ndef step(state):\n    x, y = state\n    return y, (y*y + 1 - x) % p\n\nexpected = {\n    0: (0, 2),\n    1: (2, 5),\n    2: (5, 24),\n    3: (24, 572),\n    target - 1: (65536, 0),\n}\nh = sha256()\nstate = start\nfor time in range(target):\n    if time in expected:\n        assert state == expected[time]\n    h.update(state[0].to_bytes(4, 'little'))\n    h.update(state[1].to_bytes(4, 'little'))\n    state = step(state)\n    assert time + 1 == target or state != start\nassert state == start\n\ntortoise = step(start)\nhare = step(step(start))\nwhile tortoise != hare:\n    tortoise = step(tortoise)\n    hare = step(step(hare))\nmu = 0\ntortoise = start\nwhile tortoise != hare:\n    tortoise = step(tortoise)\n    hare = step(hare)\n    mu += 1\nperiod = 1\nhare = step(tortoise)\nwhile tortoise != hare:\n    hare = step(hare)\n    period += 1\nassert (mu, period, tortoise) == (0, target, start)\nprint(f'period={period} predecessor={expected[target-1]} trace_sha256={h.hexdigest()}')\n",
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1088/0951-7715/18/5/015",
    "locator": "Self-contained Python 3 standard-library replay derived from candidate record 155"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R360",
      "title": "A certified cycle has length 294,672",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "R357",
      "title": "Complete p=1009 cycle decomposition",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "henon-65537-max-cycle",
      "title": "henon 65537 max cycle",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}