{
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  "source": "vibemath",
  "problem_id": "vibemathed:courtade-kumar-coordinatewise",
  "title": "Courtade and Kumar's Coordinate-wise Mutual Information Question",
  "canonical_statement": "The Courtade-Kumar conjecture (2014) posits that dictatorship functions maximize mutual information between a Boolean function's output and a noisy input. The paper resolves an open question posed by Courtade and Kumar themselves - a sharp bound of $1-H(\\alpha)$ on the sum of coordinate-wise mutual informations for arbitrary bias - and extends the proven high-noise range of the main conjecture via optimal entropy bounds.",
  "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:courtade-kumar-coordinatewise:event",
      "problem_id": "vibemathed:courtade-kumar-coordinatewise",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-01-14T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Fully resolves the posed coordinate-wise question; the main Courtade-Kumar conjecture itself remains open outside the extended high-noise range.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:courtade-kumar-coordinatewise:attempt"
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      "method_family_ids": [
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    {
      "id": "vibemathed:courtade-kumar-coordinatewise:attempt",
      "problem_id": "vibemathed:courtade-kumar-coordinatewise",
      "model": "Gemini Deep Think (larger internal version)",
      "provider": "Google DeepMind",
      "attempted_at": "2026-01-14T00:00:00.000Z",
      "human_collaborators": [
        "Adel Javanmard",
        "David P. Woodruff"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "\"The results in this paper were obtained with significant interaction with a larger version of Google's Deep Think Gemini-based model. The authors verified the entire paper and take full responsibility.\" The acknowledgments thank the Deep Think team by name.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2601.09679",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Courtade and Kumar's Coordinate-wise Mutual Information Question",
          "url": "https://vibemathed.com/problem/courtade-kumar-coordinatewise",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:courtade-kumar-coordinatewise:verification",
      "problem_id": "vibemathed:courtade-kumar-coordinatewise",
      "solution_event_id": "vibemathed:courtade-kumar-coordinatewise:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": null,
      "sources": [
        {
          "label": "VibeMathed: Courtade and Kumar's Coordinate-wise Mutual Information Question",
          "url": "https://vibemathed.com/problem/courtade-kumar-coordinatewise",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2601.09679",
          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2601.09679",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Courtade and Kumar's Coordinate-wise Mutual Information Question",
      "url": "https://vibemathed.com/problem/courtade-kumar-coordinatewise",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:courtade-kumar-coordinatewise:signal:recent_partial_result",
      "problem_id": "vibemathed:courtade-kumar-coordinatewise",
      "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"
}