{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R438",
  "content_hash": null,
  "slug": "l32peak-attempt-symmetry-and-literature-audit",
  "type": "attempt",
  "title": "Symmetry leaves 536,887,296 reversal orbits",
  "summary": "Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.",
  "relevance": "For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-attempt-symmetry-and-literature-audit (“Symmetry leaves 536,887,296 reversal orbits”) documents a concrete method, search boundary, or failed route. The record states: Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.",
  "relevance_source": "recorded",
  "body": "Global sign change, substitution \\(z\\mapsto-z\\), and coefficient reversal preserve the circle peak. Global sign and alternation select a unique representative with \\(\\varepsilon_0=\\varepsilon_{31}=1\\), leaving \\(2^{30}\\) sequences. Reversal has \\(2^{15}\\) fixed representatives in this normalization. Burnside's lemma therefore leaves\n\\[\n\\frac{2^{30}+2^{15}}2=536{,}887{,}296\n\\]\nreversal orbits.\n\nThis count was checked by the artifact, though those orbits were not enumerated. The 500 seeded one-flip descents recorded in the candidate found the displayed incumbent. They carry no exclusion force.\n\nBalister, Bollobás, Morris, Sahasrabudhe, and Tiba prove the existence of uniformly flat Littlewood polynomials at every length, with absolute constants. Erdélyi surveys the sup-norm flatness problem and the Rudin-Shapiro construction. Turyn's even-Barker restriction supplies the finite lower bound used here. Focused searches for fixed-degree tables, length 32, degree 31, minimum circle maximum, and Littlewood sup norm found no primary-source table giving this exact optimum. A complete proof can continue with the 536,887,296 normalized reversal orbits, using interval lower bounds to discard every orbit below the incumbent.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "symmetries and primary literature relevant to the exact 32-term Littlewood circle-peak problem",
    "bounds": {
      "terms": {
        "min": 32,
        "max": 32
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.4007/annals.2020.192.3.6",
      "locator": "Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4007/annals.2020.192.3.6",
    "locator": "Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256"
  },
  "models": [],
  "relations": [
    {
      "slug": "R439",
      "title": "The minimum peak lies between 1064^(1/4) and 7.7174713",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "littlewood-32-minimum-peak",
      "title": "littlewood 32 minimum peak",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}