{
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  "source": "vibemath",
  "problem_id": "vibemathed:reiner-conjecture-corank-three",
  "title": "Reiner's Conjecture on Higher Bruhat Orders in Corank 3",
  "canonical_statement": "Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank $3$ it holds: the facial intervals of $B(n,n-3)$ are exactly the spherical intervals, and every other interval is contractible.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
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      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-23T00:00:00.000Z",
      "title": "arXiv:2607.21420 - Homotopy types of intervals in corank-three higher Bruhat orders",
      "summary": "corank 3; the general conjecture remains open, corank 2 being McConville's case",
      "ai_contribution": "ai_assisted",
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      "attempted_at": "2026-07-23T00:00:00.000Z",
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      "outcome": "The AI use declaration lists the model as search engine, proof assistant, vector graphics artist and editor, and states that all outputs of the proof assistant were thoroughly digested by a human and that the exposition is by and for humans.",
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      "sources": [
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          "label": "arXiv:2607.21420 - Homotopy types of intervals in corank-three higher Bruhat orders",
          "url": "https://arxiv.org/abs/2607.21420",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Reiner's Conjecture on Higher Bruhat Orders in Corank 3",
          "url": "https://vibemathed.com/problem/reiner-conjecture-corank-three",
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      "id": "vibemathed:reiner-conjecture-corank-three:verification",
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      "solution_event_id": "vibemathed:reiner-conjecture-corank-three:event",
      "level": "unreviewed",
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      "verifier": "VibeMathed (source-reported label)",
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      "note": "Single-author arXiv note; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Reiner's Conjecture on Higher Bruhat Orders in Corank 3",
          "url": "https://vibemathed.com/problem/reiner-conjecture-corank-three",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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        {
          "label": "arXiv:2607.21420 - Homotopy types of intervals in corank-three higher Bruhat orders",
          "url": "https://arxiv.org/abs/2607.21420",
          "kind": "primary-mathematical-source",
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  "sources": [
    {
      "label": "arXiv:2607.21420 - Homotopy types of intervals in corank-three higher Bruhat orders",
      "url": "https://arxiv.org/abs/2607.21420",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Reiner's Conjecture on Higher Bruhat Orders in Corank 3",
      "url": "https://vibemathed.com/problem/reiner-conjecture-corank-three",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:reiner-conjecture-corank-three:signal:recent_partial_result",
      "problem_id": "vibemathed:reiner-conjecture-corank-three",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}