{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:son-pham",
  "title": "Huneke-Wiegand Conjecture",
  "canonical_statement": "The Huneke–Wiegand Conjecture: Let $R$ be a one-dimensional Gorenstein local domain, and let $M$ be a finitely generated, non-zero, torsion-free $R$-module. If the tensor product $M \\otimes_R M^*$ is torsion-free, then $M$ is a projective (hence free) $R$-module.",
  "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:son-pham:event",
      "problem_id": "vibemathed:son-pham",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-29T00:00:00.000Z",
      "title": "GitHub",
      "summary": "Verified by author of the conjecture",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:son-pham:attempt"
      ],
      "method_family_ids": [
        "vibemathed:son-pham:method"
      ],
      "source_assertion_ids": [
        "vibemathed:son-pham:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:son-pham:attempt",
      "problem_id": "vibemathed:son-pham",
      "model": "GPT-5.6-Pro",
      "provider": null,
      "attempted_at": "2026-07-29T00:00:00.000Z",
      "human_collaborators": [
        "Son Pham & Craig Huneke"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Came up with the counterexample on shot. GPT-share chat for proof:\n\nhttps://chatgpt.com/c/6a6529a6-fb04-83ea-a397-a64ffed0b3d6",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "GitHub",
          "url": "https://github.com/sonpham-org/huneke-wiegand-candidate-verification",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Huneke-Wiegand Conjecture",
          "url": "https://vibemathed.com/problem/son-pham",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:son-pham:verification",
      "problem_id": "vibemathed:son-pham",
      "solution_event_id": "vibemathed:son-pham:event",
      "level": "independent_expert_verified",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The proposed data are:\n\nΓ = ⟨56,57,58,63,64,70,71,72,73,74,75,76,77,78,79,80,81,82,83, 87,89,90,93,95,96,97⟩,\n\nR = ℚ[t^Γ]_𝔪\n\nI = (t^56,t^70)R\n\nFull link: https://github.com/sonpham-org/huneke-wiegand-candidate-verification\nContains my own counter example proof and an independent verification by the conjecture author",
      "sources": [
        {
          "label": "VibeMathed: Huneke-Wiegand Conjecture",
          "url": "https://vibemathed.com/problem/son-pham",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "GitHub",
          "url": "https://github.com/sonpham-org/huneke-wiegand-candidate-verification",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "GitHub",
      "url": "https://github.com/sonpham-org/huneke-wiegand-candidate-verification",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Huneke-Wiegand Conjecture",
      "url": "https://vibemathed.com/problem/son-pham",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:son-pham:signal:replay_ready",
      "problem_id": "vibemathed:son-pham",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "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"
}