{
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  "source": "vibemath",
  "problem_id": "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding",
  "title": "A 32-leaf tree requiring six coordinates for an isometric $\\ell_\\infty$ embedding",
  "canonical_statement": "Every tree metric with $t$ leaves embeds isometrically into $\\ell_\\infty^{t-1}$, and the least dimension needed is at least $\\lceil \\log_2 t \\rceil$. Fitzpatrick and Nowakowski asked in 2000 whether the logarithmic bound is always attained, and Brigham, Chartrand, Dutton and Zhang conjectured in 2005 that every tree with $t$ leaves embeds isometrically into $\\ell_\\infty^{\\lceil \\log_2 t \\rceil}$, verifying it through $t = 21$. Aksoy, Kilic and Kocak posed a sharp leaf-threshold version for weighted metric trees in 2020.",
  "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:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding:event",
      "problem_id": "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-17T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Disproved. An explicit 32-leaf tree has least isometric $\\ell_\\infty$-dimension six, not the conjectured five, and keeps dimension six under every assignment of positive edge lengths, so the weighted sharp-threshold conjecture falls too. Every tree with at most 31 leaves does attain $\\lceil \\log_2 t \\rceil$, so 32 is the first failure; Brigham et al. had checked through 21. The proof is finite and exact: six explicit orientation masks cover all leaf pairs, and a recurrence on rooted branches rules out any five-orientation cover.",
      "ai_contribution": "ai_assisted",
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        "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding:attempt"
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        "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding:assertion"
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  "attempts": [
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      "id": "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding:attempt",
      "problem_id": "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding",
      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-17T00:00:00.000Z",
      "human_collaborators": [
        "Logan R. Chalmers"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper's disclosure: \"OpenAI's GPT-5.6 Sol assisted in implementing the computational search strategy and with drafting this manuscript. The author takes full responsibility for the content of the article.\" The search that found the tree and the proof that no five-orientation cover exists are the author's; the model wrote code and prose. That is the assisted tier on this site.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.16288",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: A 32-leaf tree requiring six coordinates for an isometric $\\ell_\\infty$ embedding",
          "url": "https://vibemathed.com/problem/a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
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    {
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      "problem_id": "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding",
      "solution_event_id": "vibemathed:a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Unreviewed. arXiv 2608.16288 (17 August 2026, six pages) read in full here; the abstract's claims match the theorems. The counterexample is an exact finite argument with the covering masks printed in the paper and ancillary verification code deposited on Zenodo, which this site has not run. No peer review, no independent expert statement.",
      "sources": [
        {
          "label": "VibeMathed: A 32-leaf tree requiring six coordinates for an isometric $\\ell_\\infty$ embedding",
          "url": "https://vibemathed.com/problem/a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding",
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          "url": "https://arxiv.org/abs/2608.16288",
          "kind": "primary-mathematical-source",
          "license": null
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  ],
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      "url": "https://arxiv.org/abs/2608.16288",
      "kind": "primary-mathematical-source",
      "license": null
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    {
      "label": "VibeMathed: A 32-leaf tree requiring six coordinates for an isometric $\\ell_\\infty$ embedding",
      "url": "https://vibemathed.com/problem/a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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      "generated_at": "2026-09-13T16:28:40.178Z"
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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."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}