An Explicit Counterexample to the Rank-Two Poisson Conjecture
resolvedconfidence 70%
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Precise 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.
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What AI did
ChatGPT 5.6 Sol; Claude Fable 5
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.
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.
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.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.