{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:purdy-inequality-hyperplane-arrangements",
  "title": "Purdy's Inequality for Hyperplane Arrangements",
  "canonical_statement": "For an arrangement of $n$ hyperplanes in $\\mathbb{P}^3_{\\mathbb{C}}$ with $\\ell$ intersection lines and $p$ intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects $p - \\ell + n + 2 \\ge 0$. An explicit arrangement built from roots of unity violates it.",
  "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:purdy-inequality-hyperplane-arrangements:event",
      "problem_id": "vibemathed:purdy-inequality-hyperplane-arrangements",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-09T00:00:00.000Z",
      "title": "arXiv:2607.08463 - A counterexample to Purdy's inequality for hyperplane arrangements",
      "summary": "the refined form for essential arrangements in projective three-space",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:purdy-inequality-hyperplane-arrangements:attempt"
      ],
      "method_family_ids": [
        "vibemathed:purdy-inequality-hyperplane-arrangements:method"
      ],
      "source_assertion_ids": [
        "vibemathed:purdy-inequality-hyperplane-arrangements:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:purdy-inequality-hyperplane-arrangements:attempt",
      "problem_id": "vibemathed:purdy-inequality-hyperplane-arrangements",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-07-09T00:00:00.000Z",
      "human_collaborators": [
        "Mateusz Michalek",
        "Piotr Pokora"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The acknowledgement credits the model with generating the computer programs used to carry out the symbolic computations over configurations of points and planes, which is how the violating arrangement was checked.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.08463 - A counterexample to Purdy's inequality for hyperplane arrangements",
          "url": "https://arxiv.org/abs/2607.08463",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Purdy's Inequality for Hyperplane Arrangements",
          "url": "https://vibemathed.com/problem/purdy-inequality-hyperplane-arrangements",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:purdy-inequality-hyperplane-arrangements:verification",
      "problem_id": "vibemathed:purdy-inequality-hyperplane-arrangements",
      "solution_event_id": "vibemathed:purdy-inequality-hyperplane-arrangements:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The counterexample is an explicit root-of-unity arrangement whose point, line and plane counts the paper works out in closed form, so the violation is a finite check. arXiv preprint, not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Purdy's Inequality for Hyperplane Arrangements",
          "url": "https://vibemathed.com/problem/purdy-inequality-hyperplane-arrangements",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.08463 - A counterexample to Purdy's inequality for hyperplane arrangements",
          "url": "https://arxiv.org/abs/2607.08463",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.08463 - A counterexample to Purdy's inequality for hyperplane arrangements",
      "url": "https://arxiv.org/abs/2607.08463",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Purdy's Inequality for Hyperplane Arrangements",
      "url": "https://vibemathed.com/problem/purdy-inequality-hyperplane-arrangements",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:purdy-inequality-hyperplane-arrangements:signal:watch_only",
      "problem_id": "vibemathed:purdy-inequality-hyperplane-arrangements",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "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"
}