{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R439",
  "content_hash": null,
  "slug": "l32peak-claim-certified-interval",
  "type": "claim",
  "title": "The minimum peak lies between 1064^(1/4) and 7.7174713",
  "summary": "Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.",
  "relevance": "For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-claim-certified-interval (“The minimum peak lies between 1064^(1/4) and 7.7174713”) records a bound, answer, status fact, or structural consequence. The record states: Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.",
  "relevance_source": "recorded",
  "body": "Write\n\\[\nC_k=\\sum_{j=0}^{31-k}\\varepsilon_j\\varepsilon_{j+k}.\n\\]\nThe normalized fourth moment is\n\\[\n\\|P\\|_4^4=32^2+2\\sum_{k=1}^{31}C_k^2.\n\\]\nFor odd \\(k\\), the integer \\(C_k\\) is odd. For even \\(k\\), it is even. The parity floor is therefore \\(\\sum C_k^2\\geq16\\). Equality would make every odd-shift correlation equal to \\(\\pm1\\) and every even-shift correlation zero, which is a Barker sequence of length 32. Turyn proved that an even Barker length greater than four must have the form \\(4u^2\\). Since 32 has no such form, equality is impossible. The next possible increase in the integer energy is four, so\n\\[\n\\sum_{k=1}^{31}C_k^2\\geq20.\n\\]\nSince \\(\\|P\\|_\\infty\\geq\\|P\\|_4\\), every polynomial in the candidate family satisfies\n\\[\n\\|P\\|_\\infty\\geq1064^{1/4}=5.7113057054\\ldots.\n\\]\n\nThe interval artifact proves that the displayed 32-sign polynomial has circle maximum below \\(7.7174713\\). Hence the present certified result is\n\\[\n\\boxed{1064^{1/4}\\leq\\min_P\\|P\\|_\\infty<7.7174713}.\n\\]\nThe exact minimum remains open in this record.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all 32-term Littlewood polynomials on the complex unit circle",
    "bounds": {
      "terms": {
        "min": 32,
        "max": 32
      },
      "coefficient_choices": {
        "min": 4294967296,
        "max": 4294967296
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1109/PROC.1963.2526",
      "locator": "R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256; exact fourth-moment calculation and interval certificate l32peak-artifact-fixed-point-circle-bound"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/PROC.1963.2526",
    "locator": "R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256; exact fourth-moment calculation and interval certificate l32peak-artifact-fixed-point-circle-bound"
  },
  "models": [],
  "relations": [
    {
      "slug": "R437",
      "title": "Exact fixed-point interval certificate for the incumbent",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R438",
      "title": "Symmetry leaves 536,887,296 reversal orbits",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "littlewood-32-minimum-peak",
      "title": "littlewood 32 minimum peak",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}