{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R146",
  "content_hash": null,
  "slug": "ch31-claim-certified-bracket",
  "type": "claim",
  "title": "The sharp fourth-power norm lies between 1.5693 and 1.6453",
  "summary": "A rational zero-mean witness gives the lower endpoint, while the exact Fourier multiplier and the convolution row norm give the upper endpoint.",
  "relevance": "For Sharp fourth-power norm of the cyclic Hilbert transform at order 31, record ch31-claim-certified-bracket (“The sharp fourth-power norm lies between 1.5693 and 1.6453”) records a bound, answer, status fact, or structural consequence. The record states: A rational zero-mean witness gives the lower endpoint, while the exact Fourier multiplier and the convolution row norm give the upper endpoint.",
  "relevance_source": "recorded",
  "body": "Let\n\\[\nC_{31}=\\sup_{f\\in\\mathbb R^{C_{31}},\\,\\sum_x f(x)=0,\\,f\\ne0}\\frac{\\|Hf\\|_4}{\\|f\\|_4}.\n\\]\nThe certified interval in this fixture is\n\\[\n1.5693<C_{31}<1.6453.\n\\]\nThe lower inequality comes from the explicit rational vector in `ch31-artifact-interval-certificate`. Its interval-evaluated ratio is approximately 1.5693754353.\n\nFor the upper inequality, the convolution kernel is\n\\[\nh(j)=\\frac{2}{31}\\sum_{k=1}^{15}\\sin\\frac{2\\pi kj}{31},\\qquad h(0)=0.\n\\]\nIts row \\(\\ell^1\\)-norm is enclosed near 2.70676032010662 and is strictly below 2.707. The multiplier has modulus one away from the zero mode, so \\(\\|H\\|_{2\\to2}=1\\). The convolution estimate gives \\(\\|H\\|_{\\infty\\to\\infty}\\leq\\|h\\|_1<2.707\\). Riesz-Thorin interpolation therefore yields\n\\[\nC_{31}\\leq\\sqrt{\\|h\\|_1}<\\sqrt{2.707}<1.6453.\n\\]\nThe endpoints remain separated. This entry supplies a certified bracket rather than the requested ten-place sharp value.",
  "status": "partial",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the cyclic Hilbert transform on all real zero-mean functions on C_31, with the stated Fourier multiplier convention",
    "bounds": {
      "group_order": {
        "min": 31,
        "max": 31
      },
      "exponent": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Explicit witness and interval checks in ch31-artifact-interval-certificate; interpolation proof in ch31-claim-interpolation-bound"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Explicit witness and interval checks in ch31-artifact-interval-certificate; interpolation proof in ch31-claim-interpolation-bound"
  },
  "relations": [
    {
      "slug": "R144",
      "title": "Sage interval certificate for both numerical endpoints",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R147",
      "title": "The fourth-power norm is at most the square root of the kernel row norm",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R145",
      "title": "Local optimization supplies an incumbent without a global certificate",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "cyclic-hilbert-l4-norm-31",
      "title": "cyclic hilbert l4 norm 31",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}