{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:permanent-formula-lower-bounds",
  "title": "Lower Bounds for the Permanent in Arithmetic Circuits",
  "canonical_statement": "How large must arithmetic circuits and formulas computing the $n \\times n$ permanent be? New lower bounds include an arithmetic-formula bound of order $n^4/\\log n$, far beyond the quadratic barrier that stood for decades.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:permanent-formula-lower-bounds:event",
      "problem_id": "vibemathed:permanent-formula-lower-bounds",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-08-01T00:00:00.000Z",
      "title": "OpenAI: Ten advances in mathematics and theoretical computer science",
      "summary": "an n^4/log n formula lower bound; VP vs VNP remains wide open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:permanent-formula-lower-bounds:attempt"
      ],
      "method_family_ids": [
        "vibemathed:permanent-formula-lower-bounds:method"
      ],
      "source_assertion_ids": [
        "vibemathed:permanent-formula-lower-bounds:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:permanent-formula-lower-bounds:attempt",
      "problem_id": "vibemathed:permanent-formula-lower-bounds",
      "model": "Astra (internal preview)",
      "provider": "OpenAI",
      "attempted_at": "2026-08-01T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Manuscripts (ten-proofs paper, PDF)",
          "url": "https://cdn.openai.com/pdf/ten-proofs-oai.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Lean certificate (Permanent.lean)",
          "url": "https://github.com/openai/ten-proofs/blob/main/Permanent.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Reasoning walkthroughs (PDF)",
          "url": "https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf",
          "kind": "transcript",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "OpenAI: Ten advances in mathematics and theoretical computer science",
          "url": "https://openai.com/index/ten-advances-in-mathematics/",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Lower Bounds for the Permanent in Arithmetic Circuits",
          "url": "https://vibemathed.com/problem/permanent-formula-lower-bounds",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": 182,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:permanent-formula-lower-bounds:verification",
      "problem_id": "vibemathed:permanent-formula-lower-bounds",
      "solution_event_id": "vibemathed:permanent-formula-lower-bounds:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.",
      "sources": [
        {
          "label": "VibeMathed: Lower Bounds for the Permanent in Arithmetic Circuits",
          "url": "https://vibemathed.com/problem/permanent-formula-lower-bounds",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "OpenAI: Ten advances in mathematics and theoretical computer science",
          "url": "https://openai.com/index/ten-advances-in-mathematics/",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "OpenAI: Ten advances in mathematics and theoretical computer science",
      "url": "https://openai.com/index/ten-advances-in-mathematics/",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Lower Bounds for the Permanent in Arithmetic Circuits",
      "url": "https://vibemathed.com/problem/permanent-formula-lower-bounds",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:permanent-formula-lower-bounds:signal:recent_partial_result",
      "problem_id": "vibemathed:permanent-formula-lower-bounds",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}