{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:feiges-conjecture",
  "title": "Feige's Conjecture",
  "canonical_statement": "Let $X_1,\\ldots,X_n$ be independent nonnegative random variables with $\\mathbb{E}X_i \\le 1$, and let $S$ be their sum. Is $\\mathbb{P}(S < \\mathbb{E}S + 1) \\ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\\delta \\ge 1$.",
  "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:feiges-conjecture:event",
      "problem_id": "vibemathed:feiges-conjecture",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-27T00:00:00.000Z",
      "title": "arXiv:2607.23980 - Sharp small-deviation inequalities for sums of independent nonnegative random variables",
      "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:feiges-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:feiges-conjecture:method"
      ],
      "source_assertion_ids": [
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  ],
  "attempts": [
    {
      "id": "vibemathed:feiges-conjecture:attempt",
      "problem_id": "vibemathed:feiges-conjecture",
      "model": "ChatGPT 5.6 Pro, GPT-5.6 Sol, Codex",
      "provider": "OpenAI",
      "attempted_at": "2026-07-27T00:00:00.000Z",
      "human_collaborators": [
        "Weibo Fu",
        "Yanjun Han",
        "Guanyang Wang",
        "Jun Yan",
        "Peng Zhang",
        "Zhengqing Zhou",
        "Zipei Nie",
        "Jiaye Wei",
        "Mark Stander"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The primary paper states plainly that the proof was found by ChatGPT 5.6 Pro, combining the Vlassis-Thomas Dirichlet calibration theorem with Grünbaum-type convex geometry; the authors checked, revised and rewrote the argument, and the accompanying Lean formalization was developed with Codex. The independent second proof by Nie and Wei was obtained with the assistance of GPT-5.6 Sol. A further independent proof was found by Stander.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Independent second proof (Nie and Wei), arXiv:2607.24528",
          "url": "https://arxiv.org/abs/2607.24528",
          "kind": "independent",
          "license": null
        },
        {
          "label": "Lean formalization of the e^-1 conjecture",
          "url": "https://github.com/pengzhang91/Feige",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Stander's proof",
          "url": "https://doi.org/10.5281/zenodo.21622950",
          "kind": "paper",
          "license": null
        }
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      "sources": [
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          "label": "arXiv:2607.23980 - Sharp small-deviation inequalities for sums of independent nonnegative random variables",
          "url": "https://arxiv.org/abs/2607.23980",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Feige's Conjecture",
          "url": "https://vibemathed.com/problem/feiges-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:feiges-conjecture:verification",
      "problem_id": "vibemathed:feiges-conjecture",
      "solution_event_id": "vibemathed:feiges-conjecture:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "An end-to-end Lean formalization of the $e^{-1}$ conjecture accompanies the primary paper, formalizing the Vlassis-Thomas theorem, Grünbaum's centroid theorem and the combining argument. Three independent AI-assisted proofs appeared within days; neither preprint is peer-reviewed yet.",
      "sources": [
        {
          "label": "VibeMathed: Feige's Conjecture",
          "url": "https://vibemathed.com/problem/feiges-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.23980 - Sharp small-deviation inequalities for sums of independent nonnegative random variables",
          "url": "https://arxiv.org/abs/2607.23980",
          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "arXiv:2607.23980 - Sharp small-deviation inequalities for sums of independent nonnegative random variables",
      "url": "https://arxiv.org/abs/2607.23980",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Feige's Conjecture",
      "url": "https://vibemathed.com/problem/feiges-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:feiges-conjecture:signal:replay_ready",
      "problem_id": "vibemathed:feiges-conjecture",
      "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"
}