{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample",
  "title": "Bondal-Polishchuk Conjecture for a Smooth Projective Variety",
  "canonical_statement": "Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form $D^b(X)$ for a smooth projective variety was known. A particular weak Fano threefold provides one.",
  "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:bondal-polishchuk-smooth-projective-counterexample:event",
      "problem_id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-28T00:00:00.000Z",
      "title": "arXiv:2607.25391 - A smooth projective counterexample to Bondal-Polishchuk's conjecture",
      "summary": "first counterexample of the form D^b(X) for X smooth projective",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:bondal-polishchuk-smooth-projective-counterexample:attempt"
      ],
      "method_family_ids": [
        "vibemathed:bondal-polishchuk-smooth-projective-counterexample:method"
      ],
      "source_assertion_ids": [
        "vibemathed:bondal-polishchuk-smooth-projective-counterexample:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample:attempt",
      "problem_id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample",
      "model": "ChatGPT 5.6",
      "provider": "OpenAI",
      "attempted_at": "2026-07-28T00:00:00.000Z",
      "human_collaborators": [
        "Anya Nordskova"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The AI disclosure is carefully bounded and worth quoting in full: the model most significantly found numerous mistakes in earlier attempted constructions, leading to the successful modifications, and it located the reference containing the threefold used. The author states that it did not produce the actual counterexample and was not used to generate any of the text.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.25391 - A smooth projective counterexample to Bondal-Polishchuk's conjecture",
          "url": "https://arxiv.org/abs/2607.25391",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Bondal-Polishchuk Conjecture for a Smooth Projective Variety",
          "url": "https://vibemathed.com/problem/bondal-polishchuk-smooth-projective-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample:verification",
      "problem_id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample",
      "solution_event_id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Single-author arXiv preprint; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Bondal-Polishchuk Conjecture for a Smooth Projective Variety",
          "url": "https://vibemathed.com/problem/bondal-polishchuk-smooth-projective-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.25391 - A smooth projective counterexample to Bondal-Polishchuk's conjecture",
          "url": "https://arxiv.org/abs/2607.25391",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.25391 - A smooth projective counterexample to Bondal-Polishchuk's conjecture",
      "url": "https://arxiv.org/abs/2607.25391",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Bondal-Polishchuk Conjecture for a Smooth Projective Variety",
      "url": "https://vibemathed.com/problem/bondal-polishchuk-smooth-projective-counterexample",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample:signal:watch_only",
      "problem_id": "vibemathed:bondal-polishchuk-smooth-projective-counterexample",
      "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"
}