{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:lukic-conjecture",
  "title": "The Lukic Conjecture",
  "canonical_statement": "Let $\\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\\alpha$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\\alpha$ into components localized at those points. A counterexample with two critical points of multiplicity three refutes it: the sequence satisfies the decomposition conditions while the corresponding weighted entropy is $-\\infty$.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:lukic-conjecture:event",
      "problem_id": "vibemathed:lukic-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-29T00:00:00.000Z",
      "title": "arXiv:2607.26419 - A Mixed-Resonance Counterexample to the Lukic Conjecture",
      "summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:lukic-conjecture:attempt"
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      "method_family_ids": [
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      "id": "vibemathed:lukic-conjecture:attempt",
      "problem_id": "vibemathed:lukic-conjecture",
      "model": "GPT-5.6",
      "provider": "OpenAI",
      "attempted_at": "2026-07-29T00:00:00.000Z",
      "human_collaborators": [
        "Jun Yan"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The counterexample - two phase modes with common power-decay exponent $3/20$ - was found by GPT-5.6, as the abstract states directly; the author developed and verified the construction.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
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          "label": "arXiv:2607.26419 - A Mixed-Resonance Counterexample to the Lukic Conjecture",
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
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      "note": "Single-author arXiv preprint with the counterexample stated explicitly; not yet peer-reviewed.",
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          "label": "arXiv:2607.26419 - A Mixed-Resonance Counterexample to the Lukic Conjecture",
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      "label": "arXiv:2607.26419 - A Mixed-Resonance Counterexample to the Lukic Conjecture",
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      "kind": "primary-mathematical-source",
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      "label": "VibeMathed: The Lukic Conjecture",
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      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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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.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}