{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies",
  "title": "Upper Bounds for High-Dimensional Sphere Packing",
  "canonical_statement": "How dense can a sphere packing in $\\mathbb{R}^n$ be as $n \\to \\infty$? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.",
  "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:sphere-packing-upper-bounds-cohn-elkies:event",
      "problem_id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-08-01T00:00:00.000Z",
      "title": "OpenAI: Ten advances in mathematics and theoretical computer science",
      "summary": "upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:sphere-packing-upper-bounds-cohn-elkies:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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      ]
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      "problem_id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies",
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      "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": [
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          "kind": "paper",
          "license": null
        },
        {
          "label": "Lean certificate (SpherePacking.lean)",
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          "kind": "lean-proof",
          "license": null
        },
        {
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          "kind": "transcript",
          "license": null
        }
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          "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: Upper Bounds for High-Dimensional Sphere Packing",
          "url": "https://vibemathed.com/problem/sphere-packing-upper-bounds-cohn-elkies",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
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      "independence": "unknown"
    }
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    {
      "id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies:verification",
      "problem_id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies",
      "solution_event_id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies: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: Upper Bounds for High-Dimensional Sphere Packing",
          "url": "https://vibemathed.com/problem/sphere-packing-upper-bounds-cohn-elkies",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "OpenAI: Ten advances in mathematics and theoretical computer science",
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          "kind": "primary-mathematical-source",
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  "sources": [
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      "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: Upper Bounds for High-Dimensional Sphere Packing",
      "url": "https://vibemathed.com/problem/sphere-packing-upper-bounds-cohn-elkies",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies:signal:recent_partial_result",
      "problem_id": "vibemathed:sphere-packing-upper-bounds-cohn-elkies",
      "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"
}