{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R215",
  "content_hash": null,
  "slug": "djs8-claim-certified-lower-bound",
  "type": "claim",
  "title": "The tent function certifies C_8 >= 1/2",
  "summary": "For u(x)=min(x,1-x), the midpoint defect is 1/2 and the best affine error is 1/4.",
  "relevance": "For Exact Jensen stability constant on the eighth dyadic grid, record djs8-claim-certified-lower-bound (“The tent function certifies C_8 >= 1/2”) records a bound, answer, status fact, or structural consequence. The record states: For u(x)=min(x,1-x), the midpoint defect is 1/2 and the best affine error is 1/4.",
  "relevance_source": "recorded",
  "body": "Take \\(u(x)=\\min(x,1-x)\\). Concavity makes every midpoint defect nonnegative, and its range lies in \\([0,1/2]\\). Thus \\(\\delta(u)\\leq1/2\\), with equality for the pair \\((0,1)\\).\n\nThe constant affine function \\(1/4\\) has uniform error \\(1/4\\). For any affine \\(\\ell\\), write \\(r=u-\\ell\\). Affineness gives\n\\[\nr(0)-2r(1/2)+r(1)=-1.\n\\]\nThe left side has absolute value at most \\(4\\|r\\|_\\infty\\), proving \\(d(u)\\geq1/4\\). Consequently\n\\[\n\\frac{d(u)}{\\delta(u)}=\\frac{1/4}{1/2}=\\frac12.\n\\]",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the sampled tent function on the denominator-256 grid",
    "bounds": {
      "dyadic_level": {
        "min": 8,
        "max": 8
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1515/dema-1989-0220",
      "locator": "Exact tent calculation reproduced in djs8-artifact-rational-certificates"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1515/dema-1989-0220",
    "locator": "Exact tent calculation reproduced in djs8-artifact-rational-certificates"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "djs8-problem-exact-constant",
      "title": "Determine the exact eighth-grid Jensen stability constant",
      "object_type": "problem",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R213",
      "title": "Exact rational lower and upper certificates",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "dyadic-jensen-stability-8",
      "title": "dyadic jensen stability 8",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}