{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:ramachandra-natarajan-correlation-gap",
  "title": "The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture",
  "canonical_statement": "Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.",
  "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:ramachandra-natarajan-correlation-gap:event",
      "problem_id": "vibemathed:ramachandra-natarajan-correlation-gap",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-06-18T00:00:00.000Z",
      "title": "arXiv:2606.19663 - Counterexample to a conjecture on the pairwise independent correlation gap using AI",
      "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_discovered",
      "attempt_ids": [
        "vibemathed:ramachandra-natarajan-correlation-gap:attempt"
      ],
      "method_family_ids": [
        "vibemathed:ramachandra-natarajan-correlation-gap:method"
      ],
      "source_assertion_ids": [
        "vibemathed:ramachandra-natarajan-correlation-gap:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:ramachandra-natarajan-correlation-gap:attempt",
      "problem_id": "vibemathed:ramachandra-natarajan-correlation-gap",
      "model": "GPT-5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-06-18T00:00:00.000Z",
      "human_collaborators": [
        "Arjun Ramachandra",
        "Karthik Natarajan"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The abstract credits the counterexample to GPT-5.5 Pro in its first sentence. The authors add that earlier attempts with free tiers of ChatGPT and Claude made no progress, which is a useful data point on where the capability threshold sat. They are refuting their own conjecture.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Lean 4 / Mathlib formalization of the counterexample (Codex, author-contributed)",
          "url": "https://github.com/Olangu/pairwise-correlation-gap-lean",
          "kind": "lean-proof",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv:2606.19663 - Counterexample to a conjecture on the pairwise independent correlation gap using AI",
          "url": "https://arxiv.org/abs/2606.19663",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture",
          "url": "https://vibemathed.com/problem/ramachandra-natarajan-correlation-gap",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:ramachandra-natarajan-correlation-gap:verification",
      "problem_id": "vibemathed:ramachandra-natarajan-correlation-gap",
      "solution_event_id": "vibemathed:ramachandra-natarajan-correlation-gap:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The refutation is an explicit counterexample, so it is a finite check. Short arXiv note, not peer-reviewed.\n\nA Lean 4 / Mathlib formalization of the counterexample was contributed in August 2026 by its author, produced with Codex. Curator source audit: all 579 lines read, with no sorry, admit, native_decide, unsafe declaration, user-declared axiom, implemented_by or partial def anywhere; finite checks go through kernel decide and rational identities through norm_num, and Mathlib is pinned to an exact revision on toolchain v4.33.0-rc2. The curator has not compiled it, and it is the work of the same person who reported the result, so it is not third-party corroboration.\n\nWhat the formalization does and does not settle is worth stating exactly. Its final theorem is a seven-part conjunction certifying the witness and its bounds: the three-atom distribution attains the target marginals with expected coverage 4, no distribution exceeds 4, the product distribution is pairwise feasible, every pairwise-feasible distribution is bounded by $479/160$, and $4 \\div (479/160) = 640/479 > 4/3$. Both bounds are universally quantified rather than spot-checked. What the file never states is the Ramachandra-Natarajan conjecture itself, so the step from this instance to the conjecture being refuted stays informal and rests on the conjectured bound really being $4/3$. That is the difference between a kernel-checked artifact and an audited claim, and why this sits on the unaudited Lean rung.",
      "sources": [
        {
          "label": "VibeMathed: The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture",
          "url": "https://vibemathed.com/problem/ramachandra-natarajan-correlation-gap",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2606.19663 - Counterexample to a conjecture on the pairwise independent correlation gap using AI",
          "url": "https://arxiv.org/abs/2606.19663",
          "kind": "primary-mathematical-source",
          "license": null
        }
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  ],
  "sources": [
    {
      "label": "arXiv:2606.19663 - Counterexample to a conjecture on the pairwise independent correlation gap using AI",
      "url": "https://arxiv.org/abs/2606.19663",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture",
      "url": "https://vibemathed.com/problem/ramachandra-natarajan-correlation-gap",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:ramachandra-natarajan-correlation-gap:signal:replay_ready",
      "problem_id": "vibemathed:ramachandra-natarajan-correlation-gap",
      "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"
}