{
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  "source": "vibemath",
  "problem_id": "vibemathed:classical-smith-ward-problem",
  "title": "The Classical Smith-Ward Problem",
  "canonical_statement": "The Smith-Ward theorem realizes the first $k$ essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of $k$, equivalently whether the identity map on a three-dimensional operator system $\\mathrm{span}\\{1,q(D),q(K)\\}$ in the Calkin algebra always lifts. It need not: an explicit three-dimensional hyperrigid operator system has no unital completely positive lift, and its dual is the first three-dimensional operator system that fails to be exact.",
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      "problem_id": "vibemathed:classical-smith-ward-problem",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-13T00:00:00.000Z",
      "title": "arXiv:2607.04274 - A Three-Dimensional Operator System without the Smith-Ward Property",
      "summary": "Harris had settled the generalized problem in dimension four; this reaches dimension three",
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      "label": "arXiv:2607.04274 - A Three-Dimensional Operator System without the Smith-Ward Property",
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}