{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:purely-prime-not-purely-maximal-counterexample",
  "title": "Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals",
  "canonical_statement": "Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every purely-prime ideal is purely-maximal. False: there is a commutative ring with a purely-prime ideal that is not purely-maximal.",
  "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:purely-prime-not-purely-maximal-counterexample:event",
      "problem_id": "vibemathed:purely-prime-not-purely-maximal-counterexample",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-14T00:00:00.000Z",
      "title": "A counterexample to a question on the maximality of purely-primes",
      "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:purely-prime-not-purely-maximal-counterexample:attempt"
      ],
      "method_family_ids": [
        "vibemathed:purely-prime-not-purely-maximal-counterexample:method"
      ],
      "source_assertion_ids": [
        "vibemathed:purely-prime-not-purely-maximal-counterexample:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:purely-prime-not-purely-maximal-counterexample:attempt",
      "problem_id": "vibemathed:purely-prime-not-purely-maximal-counterexample",
      "model": "ChatGPT Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-08-14T00:00:00.000Z",
      "human_collaborators": [
        "Abolfazl Tarizadeh"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "In the author's words: after years without progress on his own conjecture, \"by using an advanced model of AI (ChatGPT Pro), a counterexample is found to this conjecture\". The counterexample is the paper's content; the conjecture's own poser credits the model with finding it.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "A counterexample to a question on the maximality of purely-primes",
          "url": "https://arxiv.org/abs/2608.14251",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals",
          "url": "https://vibemathed.com/problem/purely-prime-not-purely-maximal-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:purely-prime-not-purely-maximal-counterexample:verification",
      "problem_id": "vibemathed:purely-prime-not-purely-maximal-counterexample",
      "solution_event_id": "vibemathed:purely-prime-not-purely-maximal-counterexample: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:2608.14251): the abstract and introduction say what the entry says, including that the conjecture is the author's own published Conjecture 5.8 which resisted for years. The construction was not checked here. Days-old preprint by the conjecture's poser, no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals",
          "url": "https://vibemathed.com/problem/purely-prime-not-purely-maximal-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "A counterexample to a question on the maximality of purely-primes",
          "url": "https://arxiv.org/abs/2608.14251",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "A counterexample to a question on the maximality of purely-primes",
      "url": "https://arxiv.org/abs/2608.14251",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals",
      "url": "https://vibemathed.com/problem/purely-prime-not-purely-maximal-counterexample",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:purely-prime-not-purely-maximal-counterexample:signal:watch_only",
      "problem_id": "vibemathed:purely-prime-not-purely-maximal-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"
}