{
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  "source": "vibemath",
  "problem_id": "vibemathed:bruhat-hypercube-intervals",
  "title": "Large Hypercube Intervals in Bruhat Order",
  "canonical_statement": "How large can a Bruhat interval in $S_n$ that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension $O(n \\log n)$ for $n$ a power of 2, matching the largest possible dimension up to a constant - in the problem circle around the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.",
  "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",
  "solution_events": [
    {
      "id": "vibemathed:bruhat-hypercube-intervals:event",
      "problem_id": "vibemathed:bruhat-hypercube-intervals",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-01-03T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:bruhat-hypercube-intervals:attempt"
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      "model": "AlphaEvolve",
      "provider": "Google DeepMind",
      "attempted_at": "2026-01-03T00:00:00.000Z",
      "human_collaborators": [
        "Jordan Ellenberg",
        "Nicolas Libedinsky",
        "David Plaza",
        "José Simental",
        "Geordie Williamson"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "AlphaEvolve, searching evolutionarily rather than exhaustively, \"produced a pattern which performed well for the n tested, and which we show works well for general n\" - the agent found the construction, the humans proved it works in general.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2601.01235",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Large Hypercube Intervals in Bruhat Order",
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        }
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      "independence": "unknown"
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      "level": "unreviewed",
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": null,
      "sources": [
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          "label": "VibeMathed: Large Hypercube Intervals in Bruhat Order",
          "url": "https://vibemathed.com/problem/bruhat-hypercube-intervals",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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          "label": "arXiv",
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          "kind": "primary-mathematical-source",
          "license": null
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    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2601.01235",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Large Hypercube Intervals in Bruhat Order",
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      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:bruhat-hypercube-intervals:signal:recent_partial_result",
      "problem_id": "vibemathed:bruhat-hypercube-intervals",
      "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"
}