{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R15",
  "content_hash": null,
  "slug": "apn256-claim-component-restrictions",
  "type": "claim",
  "title": "A hypothetical witness has tightly restricted Boolean components",
  "summary": "Every nonzero component is balanced and avoids partially bent or quadratic form; a cubic witness would place at least 85 components in a short classified list.",
  "relevance": "For An APN permutation of the 256-element field, record apn256-claim-component-restrictions (“A hypothetical witness has tightly restricted Boolean components”) records a bound, answer, status fact, or structural consequence. The record states: Every nonzero component is balanced and avoids partially bent or quadratic form; a cubic witness would place at least 85 components in a short classified list.",
  "relevance_source": "recorded",
  "body": "For \\(\\lambda\\ne0\\), write the Boolean component as\n\\[\nF_\\lambda(x)=\\lambda\\cdot F(x).\n\\]\nBijectivity makes all 255 such components balanced. Calderini, Sala, and Villa prove that an APN permutation in even dimension has no partially bent component. In particular, no component can be quadratic, so the vectorial algebraic degree is at least three.\n\nMusukwa, Sala, Villa, and Zaninelli prove a derivative restriction valid for every APN permutation: no nonzero component has an identically zero derivative in a nonzero direction, and each component has at most one direction whose derivative is identically one.\n\nTheir dimension-eight classification gives a sharper test for a cubic candidate. Every nonzero component must have degree three. For their invariant \\(\\mathcal M\\), each component satisfies either \\(\\mathcal M=192\\) or \\(\\mathcal M\\ge288\\). If\n\\[\n\\Lambda=\\{\\lambda\\ne0:\\mathcal M(F_\\lambda)\\le256\\},\n\\]\nthen\n\\[\n85\\le |\\Lambda|\\le252.\n\\]\nUp to EA-equivalence, every component indexed by \\(\\Lambda\\) belongs to the finite list in their Table 5. This restriction prunes cubic searches while leaving higher-degree candidates and the remaining cubic combinations unresolved.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "every hypothetical APN permutation on eight binary variables, with the sharper component restriction applied when its algebraic degree is three",
    "bounds": {
      "dimension": {
        "min": 8,
        "max": 8
      },
      "nonzero_components": {
        "min": 255,
        "max": 255
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s00009-024-02660-x",
      "locator": "Augustine Musukwa, Massimiliano Sala, Irene Villa, and Marco Zaninelli, On Second-Order Derivatives of Boolean Functions and Cubic APN Permutations in Even Dimension, Mediterranean Journal of Mathematics 21, 116 (2024), Theorem 51, Proposition 54, Theorem 56, and Remark 57; quadratic-component exclusion from Calderini, Sala, and Villa (2017), Theorem 3.3 and Corollary 3.4"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s00009-024-02660-x",
    "locator": "Augustine Musukwa, Massimiliano Sala, Irene Villa, and Marco Zaninelli, On Second-Order Derivatives of Boolean Functions and Cubic APN Permutations in Even Dimension, Mediterranean Journal of Mathematics 21, 116 (2024), Theorem 51, Proposition 54, Theorem 56, and Remark 57; quadratic-component exclusion from Calderini, Sala, and Villa (2017), Theorem 3.3 and Corollary 3.4"
  },
  "models": [],
  "relations": [
    {
      "slug": "R16",
      "title": "Existence of an APN permutation on F_256 remains open",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "outgoing"
    },
    {
      "slug": "apn-permutation-f256",
      "title": "apn permutation f256",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}