{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R216",
  "content_hash": null,
  "slug": "djs8-claim-certified-upper-bound",
  "type": "claim",
  "title": "Recursive midpoint interpolation proves C_8 <= 711/128",
  "summary": "The tent function and recursive midpoint interpolation certify \\(1/2\\le C_8\\le711/128\\); the exact rational value of \\(C_8\\) remains open.",
  "relevance": "For Exact Jensen stability constant on the eighth dyadic grid, record djs8-claim-certified-upper-bound (“Recursive midpoint interpolation proves C_8 <= 711/128”) records a bound, answer, status fact, or structural consequence. The record states: The tent function and recursive midpoint interpolation certify \\(1/2\\le C_8\\le711/128\\); the exact rational value of \\(C_8\\) remains open.",
  "relevance_source": "recorded",
  "body": "Let \\(\\ell_i=((256-i)u_0+iu_{256})/256\\). Recursively bisect an interval \\([a,b]\\) containing \\(i\\), and write\n\\[\ne_{a,b}=u_{(a+b)/2}-\\frac{u_a+u_b}{2}.\n\\]\nIf \\(i<(a+b)/2\\), the coefficient added for \\(e_{a,b}\\) is \\(2(i-a)/(b-a)\\); on the right it is \\(2(b-i)/(b-a)\\). Stop when \\(i\\) is an endpoint or midpoint. Substitution proves an exact identity\n\\[\nu_i-\\ell_i=\\sum_{a,b}c_{a,b}(i)e_{a,b},\\qquad c_{a,b}(i)\\geq0.\n\\]\nThe executable rational certificate checks all 257 identities and finds\n\\[\n\\max_i\\sum_{a,b}c_{a,b}(i)=\\frac{711}{128},\n\\]\nattained at \\(i=85,171\\). The endpoint chord is an admissible affine approximant, so\n\\[\nd(u)\\leq\\max_i|u_i-\\ell_i|\\leq\\frac{711}{128}\\delta(u).\n\\]",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all real-valued functions on the denominator-256 dyadic grid",
    "bounds": {
      "dyadic_level": {
        "min": 8,
        "max": 8
      },
      "grid_points": {
        "min": 257,
        "max": 257
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1515/dema-1989-0220",
      "locator": "Exact coefficient identities 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 coefficient identities 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"
    }
  ]
}