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  "problem_id": "vibemathed:quantum-monge-kantorovich-distance-disproof",
  "title": "The Quantum Wasserstein Semidistance Is Not a Distance",
  "canonical_statement": "Friedland and coauthors proposed a quantum analogue of the $p$-Wasserstein distance and conjectured that, though only a semidistance in general, it is a true distance for a particular quantum cost matrix and for cost matrices near it. The paper disproves both conjectures with an explicit family of triples of states violating the triangle inequality.",
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      "occurred_at": "2026-07-08T00:00:00.000Z",
      "title": "arXiv",
      "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.",
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      "human_collaborators": [
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      "outcome": "The acknowledgements state the counterexamples were found with the assistance of ChatGPT 5.",
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.07764",
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