{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:psd-permanent-optimal-approximation",
  "title": "The Optimal Approximation Ratio for Permanents of PSD Matrices",
  "canonical_statement": "What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation $\\widehat P(A)$ satisfies $e^{-\\gamma n}\\widehat P(A) \\le \\mathrm{per}(A) \\le \\widehat P(A)$, giving a deterministic $e^{(\\gamma+\\varepsilon)n}$-approximation for every $\\varepsilon > 0$ and matching the known $e^{(\\gamma-\\varepsilon)n}$ hardness, where $\\gamma$ is the Euler-Mascheroni constant.",
  "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:psd-permanent-optimal-approximation:event",
      "problem_id": "vibemathed:psd-permanent-optimal-approximation",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-05-21T00:00:00.000Z",
      "title": "arXiv:2605.21946 - Optimal Approximation of the Permanent of Positive Semidefinite Matrices",
      "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_co_developed",
      "attempt_ids": [
        "vibemathed:psd-permanent-optimal-approximation:attempt"
      ],
      "method_family_ids": [
        "vibemathed:psd-permanent-optimal-approximation:method"
      ],
      "source_assertion_ids": [
        "vibemathed:psd-permanent-optimal-approximation:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:psd-permanent-optimal-approximation:attempt",
      "problem_id": "vibemathed:psd-permanent-optimal-approximation",
      "model": "GPT 5.5 Pro Extended",
      "provider": "OpenAI",
      "attempted_at": "2026-05-21T00:00:00.000Z",
      "human_collaborators": [
        "Nima Anari",
        "Farzam Ebrahimnejad"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The authors describe two different interaction styles converging on the same result: the first author's interaction was one-shot, the second author's involved high-level guidance. Both state they verified the theorem and proof themselves. Codex was used separately to assemble and typeset the manuscript, and the disclosure keeps that clerical use distinct from the mathematics.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.21946 - Optimal Approximation of the Permanent of Positive Semidefinite Matrices",
          "url": "https://arxiv.org/abs/2605.21946",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Optimal Approximation Ratio for Permanents of PSD Matrices",
          "url": "https://vibemathed.com/problem/psd-permanent-optimal-approximation",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:psd-permanent-optimal-approximation:verification",
      "problem_id": "vibemathed:psd-permanent-optimal-approximation",
      "solution_event_id": "vibemathed:psd-permanent-optimal-approximation:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Both authors state they verified the theorem and proof. The result is a sandwich inequality around an explicit concave maximisation, so it is checkable by following the argument. arXiv preprint, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: The Optimal Approximation Ratio for Permanents of PSD Matrices",
          "url": "https://vibemathed.com/problem/psd-permanent-optimal-approximation",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.21946 - Optimal Approximation of the Permanent of Positive Semidefinite Matrices",
          "url": "https://arxiv.org/abs/2605.21946",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2605.21946 - Optimal Approximation of the Permanent of Positive Semidefinite Matrices",
      "url": "https://arxiv.org/abs/2605.21946",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Optimal Approximation Ratio for Permanents of PSD Matrices",
      "url": "https://vibemathed.com/problem/psd-permanent-optimal-approximation",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:psd-permanent-optimal-approximation:signal:watch_only",
      "problem_id": "vibemathed:psd-permanent-optimal-approximation",
      "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"
}