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An Explicit Counterexample to the Rank-Two Poisson Conjecture

Let P2=C[x,q,p,z]\mathcal{P}_{2}=\mathbb{C}[x,q,p,z] carry the canonical Poisson bracket determined by {p,x}={z,q}=1\{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∈Q[x,q,p,z]R,T,D,S\in\mathbb{Q}[x,q,p,z] satisfying {D,R}=1,{S,T}=1,{R,S}={R,T}={D,S}={D,T}=0,\{D,R\}=1,\qquad\{S,T\}=1,\qquad\{R,S\}=\{R,T\}=\{D,S\}=\{D,T\}=0, while R=x(2−3xq)R=x(2-3xq). Consequently, the assignment (x,q,p,z)↦(R,T,D,S)(x,q,p,z)\mapsto(R,T,D,S) defines a Poisson endomorphism of P2\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.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI-discovered
Method
Construction
Field
Poisson algebra
Posed by
—
Year posed
—
Years open
—
Solved
2026-07-22
Model
ChatGPT 5.6 Sol; Claude Fable 5
Vendor
OpenAI; Anthropic
Collaborators
Christopher D. Long
Verification
Unreviewed
Publication
Preprint
Significance
35 / 100
Disclosed cost
—
Wikipedia
4 languages

What was actually shown

The paper constructs explicit R,T,D,S∈Q[x,q,p,z]R,T,D,S\in\mathbb Q[x,q,p,z] defining a Poisson endomorphism of the canonical rank-two Poisson algebra P2\mathcal P_2 that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant 11, and has an explicit fiber of exactly three points. Thus PC(2)\mathrm{PC}(2) is false, and stabilization gives failure of PC(n)\mathrm{PC}(n) for every n≥2n\ge2. An appendix further constructs an explicit nonautomorphic endomorphism of the fourth Weyl algebra, proving DC(4)\mathrm{DC}(4) false.

What the AI did

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.

Verification

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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