{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R632",
  "content_hash": null,
  "slug": "qorp-artifact-independent-witness-verification",
  "type": "artifact",
  "title": "Independent cycle and primality verification",
  "summary": "A standard-library Python check reproduces the collision with both a first-visit map and Floyd's algorithm, then verifies primality.",
  "relevance": "This supplies a computation, dataset, or executable check used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The verifier constructs the orbit with a Python dictionary and obtains total 5145, preperiod 2520, period 2625, and repeated residue 817978. Floyd's cycle algorithm uses constant auxiliary space and independently returns the same preperiod, period, and cycle entry. The last new residue is 1039286, which maps to 817978.\n\nA direct divisibility test through 1108 certifies primality independently of the sieve. The verifier also checks the displayed Pocklington congruences and the orbit digest. Its first ten residues are 0, 1, 2, 5, 26, 677, 458330, 848734, 540676, and 769059.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the orbit and primality certificate for p=1229237",
    "bounds": {
      "p": {
        "min": 1229237,
        "max": 1229237
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "runtime": "Python 3.9.6 standard library",
    "citation": {
      "url": "https://arxiv.org/abs/1701.02707",
      "locator": "Self-contained Python 3 standard-library verification run by TheoremDB entry research on 2026-07-24"
    },
    "inline_source": "from hashlib import sha256\nfrom math import gcd, isqrt\nimport json\np = 1229237\nassert all(p % d for d in range(2, isqrt(p)+1))\nf = lambda x: (x*x+1) % p\nseen, orbit, x = {}, [], 0\nwhile x not in seen:\n    seen[x] = len(orbit); orbit.append(x); x = f(x)\nmu, lam = seen[x], len(orbit)-seen[x]\nassert (len(orbit), mu, lam, x) == (5145, 2520, 2625, 817978)\nt, h = f(0), f(f(0))\nwhile t != h: t, h = f(t), f(f(h))\nmu2, t = 0, 0\nwhile t != h: t, h, mu2 = f(t), f(h), mu2+1\nlam2, h = 1, f(t)\nwhile t != h: h, lam2 = f(h), lam2+1\nassert (mu2, lam2, t) == (2520, 2625, 817978)\nassert sha256(json.dumps(orbit,separators=(',',':')).encode()).hexdigest() == '9cd1f30ec1c06adb76c1c437bccaabb81a6f023f65ca1b81f459704b85f51db3'\nassert p-1 == 2**2 * 17 * 18077 and pow(2,p-1,p) == 1\nfor q, residue in ((2,1229236),(17,1054755),(18077,163446)):\n    assert all(q % d for d in range(2, isqrt(q)+1))\n    assert pow(2,(p-1)//q,p) == residue and gcd(residue-1,p) == 1\nprint(len(orbit), mu, lam, x)",
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1701.02707",
    "locator": "Self-contained Python 3 standard-library verification run by TheoremDB entry research on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R633",
      "title": "The first qualifying prime is 1229237",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "quadratic-orbit-record-prime-10m",
      "title": "quadratic orbit record prime 10m",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}