{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:binary-code-upper-bounds",
  "title": "Upper Bounds for Binary and Spherical Codes",
  "canonical_statement": "What is the maximum size of a binary code of given minimum distance? The linear-programming bounds of McEliece, Rodemich, Rumsey and Welch (1977) resisted improvement for half a century. The new upper bounds are exponentially stronger at every prescribed distance, with analogous results for high-dimensional spherical codes.",
  "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:binary-code-upper-bounds:event",
      "problem_id": "vibemathed:binary-code-upper-bounds",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-08-01T00:00:00.000Z",
      "title": "OpenAI: Ten advances in mathematics and theoretical computer science",
      "summary": "exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:binary-code-upper-bounds:attempt"
      ],
      "method_family_ids": [
        "vibemathed:binary-code-upper-bounds:method"
      ],
      "source_assertion_ids": [
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      ]
    }
  ],
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    {
      "id": "vibemathed:binary-code-upper-bounds:attempt",
      "problem_id": "vibemathed:binary-code-upper-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 (MetricCodes.lean)",
          "url": "https://github.com/openai/ten-proofs/blob/main/MetricCodes.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Reasoning walkthroughs (PDF)",
          "url": "https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf",
          "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 Binary and Spherical Codes",
          "url": "https://vibemathed.com/problem/binary-code-upper-bounds",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": 182,
      "independence": "unknown"
    }
  ],
  "verifications": [
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      "id": "vibemathed:binary-code-upper-bounds:verification",
      "problem_id": "vibemathed:binary-code-upper-bounds",
      "solution_event_id": "vibemathed:binary-code-upper-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: Upper Bounds for Binary and Spherical Codes",
          "url": "https://vibemathed.com/problem/binary-code-upper-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: Upper Bounds for Binary and Spherical Codes",
      "url": "https://vibemathed.com/problem/binary-code-upper-bounds",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:binary-code-upper-bounds:signal:recent_partial_result",
      "problem_id": "vibemathed:binary-code-upper-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"
}