{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R88",
  "content_hash": null,
  "slug": "c31m-claim-diagonal-upper-certificate",
  "type": "claim",
  "title": "A weighted Cauchy-Schwarz certificate gives N_31 squared at most 1916477/720720",
  "summary": "A single radial weight sequence majorizes every centered averaging row and remains diagonal after summing over the cycle.",
  "relevance": "For Sharp L2 norm of the centered maximal operator on C_31, record c31m-claim-diagonal-upper-certificate (“A weighted Cauchy-Schwarz certificate gives N_31 squared at most 1916477/720720”) records a bound, answer, status fact, or structural consequence. The record states: A single radial weight sequence majorizes every centered averaging row and remains diagonal after summing over the cycle.",
  "relevance_source": "recorded",
  "body": "Replace \\(f\\) by \\(x=|f|\\), which preserves the input norm and the maximal function. Set\n\\[\nd_0=1,\\qquad d_k=d_{-k}=\\frac1{4k}\\quad(1\\leq k\\leq15).\n\\]\nFor a radius \\(r\\), the exact identity\n\\[\n\\sum_{k=-r}^{r}\\frac{1}{(2r+1)^2d_k}\n=\\frac{1+8\\sum_{k=1}^{r}k}{(2r+1)^2}=1\n\\]\nturns weighted Cauchy-Schwarz into\n\\[\n\\left(\\frac1{2r+1}\\sum_{k=-r}^{r}x_{j+k}\\right)^2\n\\leq \\sum_{k=-r}^{r}d_kx_{j+k}^2\n\\leq \\sum_{k=-15}^{15}d_kx_{j+k}^2.\n\\]\nThe right side is independent of the selected radius, so it also bounds \\((Mx(j))^2\\). Summing over all cyclic centers makes every input coordinate receive the coefficient\n\\[\nd_0+2\\sum_{k=1}^{15}d_k\n=1+\\frac12H_{15}\n=\\frac{1916477}{720720}.\n\\]\nTherefore \\(\\|Mx\\|_2^2\\leq(1916477/720720)\\|x\\|_2^2\\) for every input. The artifact checks all sixteen coefficient identities over the rationals.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "universal",
    "statement": "every real-valued function on C_31"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2005.03146",
      "locator": "Symbolic proof in this record and exact coefficient replay in c31m-artifact-exact-replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2005.03146",
    "locator": "Symbolic proof in this record and exact coefficient replay in c31m-artifact-exact-replay"
  },
  "relations": [
    {
      "slug": "R87",
      "title": "The L2 norm lies between 1.33030427059916347 and 1.63067915195310467",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R85",
      "title": "Exact lower-witness and upper-certificate replay",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "c31-centered-maximal-l2-norm",
      "title": "c31 centered maximal l2 norm",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}