{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R217",
  "content_hash": null,
  "slug": "djs8-claim-exact-lp-formulation",
  "type": "claim",
  "title": "C_8 is an exact rational quotient-operator norm",
  "summary": "Chebyshev duality expresses C_8 as a maximum of finite rational l1 minimization problems.",
  "relevance": "For Exact Jensen stability constant on the eighth dyadic grid, record djs8-claim-exact-lp-formulation (“C_8 is an exact rational quotient-operator norm”) records a bound, answer, status fact, or structural consequence. The record states: Chebyshev duality expresses C_8 as a maximum of finite rational l1 minimization problems.",
  "relevance_source": "recorded",
  "body": "Index grid values by \\(u_0,\\ldots,u_{256}\\). Let \\(A\\) have one row for every \\(0\\leq p<q\\leq256\\) with \\(p\\equiv q\\pmod2\\):\n\\[\n(Au)_{p,q}=u_{(p+q)/2}-\\frac{u_p+u_q}{2}.\n\\]\nThere are \\(\\binom{129}{2}+\\binom{128}{2}=16384\\) rows. Its kernel is the two-dimensional affine subspace, since the rows with \\(q=p+2\\) force every second difference to vanish. Hence\n\\[\nC_8=\\|A^{-1}:A(\\mathbb R^{257}/\\mathrm{Aff})\\to\n\\mathbb R^{257}/\\mathrm{Aff}\\|.\n\\]\n\nFor \\(i<j<k\\), put\n\\[\nL_{ijk}(u)=u_j-\\frac{k-j}{k-i}u_i-\\frac{j-i}{k-i}u_k.\n\\]\nThe discrete Chebyshev alternation theorem gives\n\\[\nd(u)=\\frac12\\max_{i<j<k}|L_{ijk}(u)|.\n\\]\nLinear-programming duality now gives the finite exact formula\n\\[\nC_8=\\frac12\\max_{i<j<k}\n\\min\\{\\|y\\|_1:A^Ty=L_{ijk}\\}.\n\\]\nEvery coefficient is rational, so an optimizer and a matching dual solution would constitute a rational exact certificate.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "bounded",
    "statement": "the 257-point dyadic grid and all its 16384 nontrivial midpoint constraints",
    "bounds": {
      "dyadic_level": {
        "min": 8,
        "max": 8
      },
      "midpoint_constraints": {
        "min": 16384,
        "max": 16384
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1515/dema-1989-0220",
      "locator": "Finite-dimensional quotient-norm duality and Chebyshev alternation, specialized in this record"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1515/dema-1989-0220",
    "locator": "Finite-dimensional quotient-norm duality and Chebyshev alternation, specialized in this record"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "djs8-problem-exact-constant",
      "title": "Determine the exact eighth-grid Jensen stability constant",
      "object_type": "problem",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "dyadic-jensen-stability-8",
      "title": "dyadic jensen stability 8",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}