{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R416",
  "content_hash": null,
  "slug": "iqfd-claim-exact-factor-degrees",
  "type": "claim",
  "title": "Every factor has the predicted degree",
  "summary": "For d=2^ceil(log_2(n+1)), the polynomial f^[n] is the product of 2^n/d distinct irreducible polynomials of degree d.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Let \\(\\sigma\\) be Frobenius on \\(\\overline{\\mathbb F}_2\\), so \\(\\sigma(a)=a^2\\), and put \\(D=\\sigma+1\\). Since \\(f(a)=D(a)+1\\) and \\(D(1)=0\\), induction on composition gives\n\\[\nf^{[n]}(a)=D^n(a)+1.\n\\]\nSet \\(d=2^{\\lceil\\log_2(n+1)\\rceil}\\). Equivalently, \\(d\\) is the least power of two greater than \\(n\\). If \\(\\alpha\\) is a root of \\(f^{[n]}\\), then \\(D^n(\\alpha)=1\\). Apply \\(D^{d-n}\\). The exponent is positive, so \\(D^{d-n}(1)=0\\), and therefore \\(D^d(\\alpha)=0\\). Because \\(d\\) is a power of two, the binomial identity in characteristic two gives\n\\[\nD^d=(\\sigma+1)^d=\\sigma^d+1.\n\\]\nHence \\(\\alpha^{2^d}=\\alpha\\), placing every root in \\(\\mathbb F_{2^d}\\).\n\nLet \\(m=[\\mathbb F_2(\\alpha):\\mathbb F_2]\\). Then \\(m\\mid d\\). If \\(m<d\\), the power-of-two condition forces \\(m\\mid d/2\\), so \\(\\alpha\\in\\mathbb F_{2^{d/2}}\\). This would give \\(D^{d/2}(\\alpha)=\\alpha^{2^{d/2}}+\\alpha=0\\). The minimality of \\(d\\) gives \\(d/2\\leq n\\), and thus \\(D^n(\\alpha)=0\\), contradicting \\(D^n(\\alpha)=1\\). Every root consequently has degree \\(d\\) over \\(\\mathbb F_2\\).\n\nThe coefficient of \\(x\\) in the linearized polynomial \\(D^n(x)\\) is 1, so the derivative of \\(f^{[n]}\\) is 1. Its \\(2^n\\) roots are distinct. It follows that there are exactly \\(2^n/d\\) irreducible factors, each with multiplicity one and degree \\(d\\). Any root generates \\(\\mathbb F_{2^d}\\), which is the splitting field.\n\nOre developed the symbolic-composition framework for linearized polynomials. The argument above specializes it and proves the candidate formula. A focused search located the third-iterate factorization in Ahmadi and Monsef-Shokri, while the exact formula for every iterate was not located. Its novelty therefore remains unverified.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every compositional iterate f^[n] of f(x)=x^2+x+1 over F_2, for every integer n at least 1"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1211.5475",
      "locator": "Wu and Liu, Linearized polynomials over finite fields revisited, equations (1)-(2), section 2.1, and Theorems 2.1-2.2 for the Frobenius and composition algebra; the specialized all-n proof is complete in this record"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1211.5475",
    "locator": "Wu and Liu, Linearized polynomials over finite fields revisited, equations (1)-(2), section 2.1, and Theorems 2.1-2.2 for the Frobenius and composition algebra; the specialized all-n proof is complete in this record"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R417",
      "title": "The third iterate factorization is published",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R415",
      "title": "Exact factorization sweep through the degree-4096 iterate",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "iterated-quadratic-factor-degree",
      "title": "iterated quadratic factor degree",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}