{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R415",
  "content_hash": null,
  "slug": "iqfd-artifact-exact-factorization-through-12",
  "type": "artifact",
  "title": "Exact factorization sweep through the degree-4096 iterate",
  "summary": "Full factorization over F_2 through n=12 gives only the predicted squarefree factor degrees and counts.",
  "relevance": "This supplies a computation, dataset, or executable check used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Python 3.9.6 with SymPy 1.14.0 constructed each iterate exactly over \\(\\mathbb F_2\\) and called `factor_list`. For \\(n=1,\\ldots,12\\), the pairs (common factor degree, number of factors) were\n\\[\n(2,1),(4,1),(4,2),(8,2),(8,4),(8,8),(8,16),(16,16),(16,32),(16,64),(16,128),(16,256).\n\\]\nEvery returned exponent was 1. The final polynomial has degree 4096. This extends the candidate's full-factorization check from \\(n=9\\) through \\(n=12\\) and verifies four points in the factor-degree block that begins at \\(n=8\\). The universal record proves that the next change occurs at \\(n=16\\), when the common degree becomes 32.",
  "status": "available",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every compositional iterate f^[n] for 1 <= n <= 12",
    "bounds": {
      "n": {
        "min": 1,
        "max": 12
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "runtime": "Python 3.9.6 with SymPy 1.14.0",
    "citation": {
      "url": "https://arxiv.org/abs/1810.03142",
      "locator": "Exact one-off Python and SymPy computation run by TheoremDB entry research on 2026-07-24"
    },
    "inline_source": "from sympy import Poly, symbols\nx = symbols('x')\nF = Poly(x, x, modulus=2)\nfor n in range(1, 13):\n    F = Poly(F.as_expr()**2 + F.as_expr() + 1, x, modulus=2)\n    factors = F.factor_list()[1]\n    d = 1 << n.bit_length()\n    assert all(e == 1 and q.degree() == d for q, e in factors)\n    assert sum(e*q.degree() for q, e in factors) == F.degree()\n    print(n, F.degree(), d, len(factors))",
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1810.03142",
    "locator": "Exact one-off Python and SymPy computation run by TheoremDB entry research on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R416",
      "title": "Every factor has the predicted degree",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "R417",
      "title": "The third iterate factorization is published",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "iterated-quadratic-factor-degree",
      "title": "iterated quadratic factor degree",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}