{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R555",
  "content_hash": null,
  "slug": "otpc2q-claim-sharp-optimum",
  "type": "claim",
  "title": "The unique optimal node is (8 sqrt(14)-29)/6",
  "summary": "Peano-kernel integration gives the exact minimizer, sharp error, and every extremizing function.",
  "relevance": "For Optimal symmetric three-point quadrature for a bounded second derivative, record otpc2q-claim-sharp-optimum (“The unique optimal node is (8 sqrt(14)-29)/6”) records a bound, answer, status fact, or structural consequence. The record states: Peano-kernel integration gives the exact minimizer, sharp error, and every extremizing function.",
  "relevance_source": "recorded",
  "body": "Write\n\\[\nE_a(f)=\\int_0^1f(x)\\,dx-Q_a(f).\n\\]\nThis functional kills affine functions. Twice integrating \\(f''\\), followed by Fubini, gives\n\\[\nE_a(f)=\\int_0^1K_a(t)f''(t)\\,dt,\n\\quad\nK_a(t)=\\frac{(1-t)^2}{2}-\\frac{(a-t)_++(1/2-t)_++(1-a-t)_+}{3}.\n\\]\nHence\n\\[\nC(a)=\\int_0^1|K_a(t)|\\,dt. \\tag{1}\n\\]\nThe kernel is symmetric about \\(1/2\\). On the left half,\n\\[\nK_a(t)=\\begin{cases}\nt^2/2,&0\\leq t\\leq a,\\\\\n(3t^2-2t+2a)/6,&a\\leq t\\leq1/2.\n\\end{cases} \\tag{2}\n\\]\nFor \\(a\\leq1/6\\), the second quadratic has roots\n\\[\nr_\\pm(a)=\\frac{1\\pm\\sqrt{1-6a}}3.\n\\]\nThe upper root crosses \\(1/2\\) at \\(a=1/8\\), and the roots meet at \\(a=1/6\\). Put \\(u=\\sqrt{1-6a}\\). Splitting (1) at these roots gives\n\\[\nC(a)=\\begin{cases}\n(1+32u^3-6u^4)/648,&0\\leq a\\leq1/8,\\\\\n(3-24u^2+64u^3-6u^4)/648,&1/8\\leq a\\leq1/6,\\\\\n(-8a^2+8a-1)/24,&1/6\\leq a\\leq1/2.\n\\end{cases} \\tag{3}\n\\]\nThe derivatives of the three branches are\n\\[\n\\frac{u(u-4)}9,\n\\qquad\n\\frac{u^2-8u+2}{9},\n\\qquad\n\\frac{1-2a}{3}. \\tag{4}\n\\]\nThe first branch decreases. In the middle branch the derivative changes from negative to positive at \\(u_*=4-\\sqrt{14}\\). The last branch increases before its right endpoint. The branches agree at \\(1/8\\) and \\(1/6\\), so the unique global minimizer and sharp error are\n\\[\na_*=\\frac{8\\sqrt{14}-29}{6}\n=0.155543182365255180778331643089\\ldots,\n\\]\n\\[\nC(a_*)=\\frac{3355-896\\sqrt{14}}{648}\n=0.003819415818747713760742493587\\ldots.\n\\]\nAt \\(a_*\\), the left-half kernel zeros are\n\\[\n\\rho_-=\\frac{\\sqrt{14}}3-1,\n\\qquad\n\\rho_+=\\frac{5-\\sqrt{14}}3.\n\\]\nEquality in the \\(L^\\infty\\)-\\(L^1\\) bound occurs exactly when\n\\[\nf''(t)=\\varepsilon\\operatorname{sgn}K_{a_*}(t)\n\\quad\\text{almost everywhere},\n\\qquad \\varepsilon\\in\\{-1,1\\}.\n\\]\nThus every extremizer has the form\n\\[\nf(x)=\\varepsilon\\int_0^x(x-t)\\operatorname{sgn}K_{a_*}(t)\\,dt+\\beta x+\\gamma,\n\\qquad \\beta,\\gamma\\in\\mathbb R.\n\\]\nThese piecewise-quadratic functions attain the displayed sharp error.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "family",
    "statement": "the equal-weight symmetric rules Q_a(f)=(f(a)+f(1/2)+f(1-a))/3 for every a in [0,1/2], acting on real W^{2,infinity}([0,1]) functions",
    "family": "one-parameter symmetric three-point quadrature rules"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Complete independent Peano-kernel proof in this record; exact replay otpc2q-artifact-symbolic-check"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Complete independent Peano-kernel proof in this record; exact replay otpc2q-artifact-symbolic-check"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R554",
      "title": "Exact symbolic Peano-kernel check",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "optimal-three-point-c2-quadrature",
      "title": "optimal three point c2 quadrature",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}