{
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  "source": "vibemath",
  "problem_id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements",
  "title": "Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements",
  "canonical_statement": "Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for oriented-poset lattices. False: the order dimension of the poset of regions can exceed the rank - the Coxeter arrangements $H_4$ and $E_6$ satisfy $\\dim W(H_4) \\ge 5$ and $\\dim W(E_6) \\ge 7$.",
  "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:order-dimension-beyond-rank-simplicial-arrangements:event",
      "problem_id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-14T00:00:00.000Z",
      "title": "Order dimension beyond rank for simplicial hyperplane arrangements",
      "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:order-dimension-beyond-rank-simplicial-arrangements:attempt"
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  "attempts": [
    {
      "id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements:attempt",
      "problem_id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements",
      "model": "ChatGPT 5.6 Sol Ultra",
      "provider": "OpenAI",
      "attempted_at": "2026-08-14T00:00:00.000Z",
      "human_collaborators": [
        "Daria Poliakova"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The declaration, in full: \"The small obstruction subgraphs were found by ChatGPT 5.6 Sol Ultra. The human input was the belief that the rank guess is incorrect, and one should look for counterexamples.\" The obstruction subgraphs are the entire content of the disproof, so the model produced the central objects under human direction.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "Order dimension beyond rank for simplicial hyperplane arrangements",
          "url": "https://arxiv.org/abs/2608.14092",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements",
          "url": "https://vibemathed.com/problem/order-dimension-beyond-rank-simplicial-arrangements",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
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  ],
  "verifications": [
    {
      "id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements:verification",
      "problem_id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements",
      "solution_event_id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14092): the declaration is verbatim, and the problem attribution is precise - Reading's Problem 9.3 (2016) with the H_4/E_6 background from his 2003 computations, plus Segovia's analogous question. The obstruction subgraphs were not re-verified here. Days-old preprint, no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements",
          "url": "https://vibemathed.com/problem/order-dimension-beyond-rank-simplicial-arrangements",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Order dimension beyond rank for simplicial hyperplane arrangements",
          "url": "https://arxiv.org/abs/2608.14092",
          "kind": "primary-mathematical-source",
          "license": null
        }
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  "sources": [
    {
      "label": "Order dimension beyond rank for simplicial hyperplane arrangements",
      "url": "https://arxiv.org/abs/2608.14092",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements",
      "url": "https://vibemathed.com/problem/order-dimension-beyond-rank-simplicial-arrangements",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements:signal:watch_only",
      "problem_id": "vibemathed:order-dimension-beyond-rank-simplicial-arrangements",
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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"
}