{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:gaussian-moments-conjecture",
  "title": "Gaussian Moments Conjecture",
  "canonical_statement": "The Gaussian Moments Conjecture asks whether, for complex polynomials $P,Q$ in $n$ independent standard real Gaussian variables, $\\mathbb{E}(P^m)=0$ for all $m\\geq 1$ forces $\\mathbb{E}(QP^m)=0$ for all large $m$. Explicit counterexamples with $\\mathbb{E}(P^m)=0$ and $\\mathbb{E}(QP^m)=m!\\neq 0$ exist in three variables (a five-term quartic $P$) and four variables, so the conjecture is false in every dimension $n\\geq 3$.",
  "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:gaussian-moments-conjecture:event",
      "problem_id": "vibemathed:gaussian-moments-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-20T00:00:00.000Z",
      "title": "arXiv:2607.18186",
      "summary": "Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:gaussian-moments-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:gaussian-moments-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:gaussian-moments-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:gaussian-moments-conjecture:attempt",
      "problem_id": "vibemathed:gaussian-moments-conjecture",
      "model": "GPT-5.6 Sol Pro, Claude Fable 5",
      "provider": "OpenAI, Anthropic",
      "attempted_at": "2026-07-20T00:00:00.000Z",
      "human_collaborators": [
        "Christopher D. Long"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Per the paper's AI-provenance section, the four-variable construction was produced by ChatGPT 5.6 Sol Pro without human intervention after the initial prompt, which told it the Jacobian conjecture had been disproved and asked whether a small Gaussian-moments counterexample might follow; shown that example, Claude Fable 5 found the three-variable construction and supplied independent algebraic checks. The author bears responsibility for the mathematics and exposition.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Wilson, proof of the two-variable case (no AI involved)",
          "url": "https://arxiv.org/abs/2607.23887",
          "kind": "paper",
          "license": null
        }
      ],
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        {
          "label": "arXiv:2607.18186",
          "url": "https://arxiv.org/abs/2607.18186",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Gaussian Moments Conjecture",
          "url": "https://vibemathed.com/problem/gaussian-moments-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:gaussian-moments-conjecture:verification",
      "problem_id": "vibemathed:gaussian-moments-conjecture",
      "solution_event_id": "vibemathed:gaussian-moments-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Re-derived by the site on 2026-08-02: both counterexamples were rebuilt from the paper's stated polynomials and evaluated in exact rational arithmetic against the standard Gaussian moment rules (E(W^a Z^b) = a! when a = b, else 0; E(T^c) the double factorial), independently of the paper's own algebra. For m = 1 through 10 both give E(P^m) = 0 and E(QP^m) = m! exactly, and the term counts and degrees match the paper (five terms of degree four in three variables, six of degree three in four). The surrounding exposition has had no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Gaussian Moments Conjecture",
          "url": "https://vibemathed.com/problem/gaussian-moments-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.18186",
          "url": "https://arxiv.org/abs/2607.18186",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.18186",
      "url": "https://arxiv.org/abs/2607.18186",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Gaussian Moments Conjecture",
      "url": "https://vibemathed.com/problem/gaussian-moments-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
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      "kind": "watch_only",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "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"
}