{
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  "problem_id": "vibemathed:an-explicit-counterexample-to-the-rank-two-poisson-conjecture",
  "title": "An Explicit Counterexample to the Rank-Two Poisson Conjecture",
  "canonical_statement": "Let $\\mathcal{P}_{2}=\\mathbb{C}[x,q,p,z]$ carry the canonical Poisson bracket determined by $\\{p,x\\}=\\{z,q\\}=1$ and by the vanishing of the other brackets between distinct generators. Here and throughout, “rank two” means two canonical pairs in the standard indexing of the canonical Poisson algebras; thus there are four polynomial generators and the Poisson tensor has geometric rank four. We give explicit polynomials $R,T,D,S\\in\\mathbb{Q}[x,q,p,z]$ satisfying $\\{D,R\\}=1,\\qquad\\{S,T\\}=1,\\qquad\\{R,S\\}=\\{R,T\\}=\\{D,S\\}=\\{D,T\\}=0,$ while $R=x(2-3xq)$. Consequently, the assignment $(x,q,p,z)\\mapsto(R,T,D,S)$ defines a Poisson endomorphism of $\\mathcal{P}_{2}$ that is not an automorphism. This disproves the Poisson Conjecture for two canonical pairs, and hence for every number of canonical pairs at least two.",
  "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",
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    {
      "id": "vibemathed:an-explicit-counterexample-to-the-rank-two-poisson-conjecture:event",
      "problem_id": "vibemathed:an-explicit-counterexample-to-the-rank-two-poisson-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-22T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The paper constructs explicit $R,T,D,S\\in\\mathbb Q[x,q,p,z]$ defining a Poisson endomorphism of the canonical rank-two Poisson algebra $\\mathcal P_2$ that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant $1$, and has an explicit fiber of exactly three points. Thus $\\mathrm{PC}(2)$ is false, and stabilization gives failure of $\\mathrm{PC}(n)$ for every $n\\ge2$. An appendix further constructs an explicit nonautomorphic endomorphism of the fourth Weyl algebra, proving $\\mathrm{DC}(4)$ false.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:an-explicit-counterexample-to-the-rank-two-poisson-conjecture:attempt"
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      "problem_id": "vibemathed:an-explicit-counterexample-to-the-rank-two-poisson-conjecture",
      "model": "ChatGPT 5.6 Sol; Claude Fable 5",
      "provider": "OpenAI; Anthropic",
      "attempted_at": "2026-07-22T00:00:00.000Z",
      "human_collaborators": [
        "Christopher D. Long"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The four-variable rank-two Poisson construction, including its Hamiltonian correction, was produced during an interactive research session with ChatGPT 5.6 Sol. The model also assisted with organizing the differential-form proof, exact symbolic verification, literature checking, and manuscript drafting. Claude Fable 5 subsequently performed independent algebraic audits and supplied editorial comments. Christopher Long checked the mathematics and assumes responsibility for the final paper.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Dixmier conjecture",
          "url": "https://en.wikipedia.org/wiki/Dixmier_conjecture",
          "kind": "wikipedia",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.23777",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: An Explicit Counterexample to the Rank-Two Poisson Conjecture",
          "url": "https://vibemathed.com/problem/an-explicit-counterexample-to-the-rank-two-poisson-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
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      "problem_id": "vibemathed:an-explicit-counterexample-to-the-rank-two-poisson-conjecture",
      "solution_event_id": "vibemathed:an-explicit-counterexample-to-the-rank-two-poisson-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Unreviewed. A preprint with no peer review and no proof-assistant verification. What it does have is unusual for the tier: every Poisson identity is verified directly in the paper, there is an exact symbolic audit of the polynomial identities, and the noninjectivity is exhibited as a fiber of exactly three explicit points. All of that is four polynomials in four variables, so any reader with a computer algebra system can check the whole claim in minutes. Claude Fable 5 supplied an independent algebraic audit, which is not independent human expert review. Nobody has done that on the record.",
      "sources": [
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          "label": "VibeMathed: An Explicit Counterexample to the Rank-Two Poisson Conjecture",
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          "kind": "source-assertion",
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          "url": "https://arxiv.org/abs/2608.23777",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
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      "url": "https://arxiv.org/abs/2608.23777",
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    {
      "label": "VibeMathed: An Explicit Counterexample to the Rank-Two Poisson Conjecture",
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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"
}