Preliminary mathematical note · 20 July 2026
Explicit Weyl and Poisson lifts of Alpöge's Jacobian counterexample
Two classical lifts of the announced dimension-three Keller map: an injective, non-surjective endomorphism of the third Weyl algebra and an injective, non-surjective endomorphism of the canonical polynomial Poisson algebra.
Exact claim. For every field K of characteristic zero and every n ≥ 3:
Consequently, the generalized Dixmier and Poisson conjectures fail in every rank n ≥ 3. The rank-one and rank-two cases are untouched.
Status. This is a rapidly prepared preliminary note about a newly announced counterexample. The required polynomial identities are checked in exact arithmetic by two independent implementations, but the note has not undergone ordinary peer review.
Paper and verification artifacts
The SymPy implementation derives the inverse Jacobian, verifies the Weyl and Poisson relations, and generates the displayed operators. A second implementation checks the same polynomial identities using only Python's standard library.
- Paper Compiled preliminary note (PDF)
- LaTeX source Canonical standalone manuscript source
- Complete release archive Paper, source, verifiers, generated formulas, outputs, and manifest
- SymPy verifier Exact symbolic derivation and verification
- Standard-library verifier Independent exact-rational polynomial implementation
- One-command driver Runs both verifiers and reproducibility checks
- Operator generator Regenerates the explicit inverse-Jacobian formulas
- README Commands, software versions, provenance, and file guide
- SHA-256 manifest Checksums for the canonical release contents
Run locally
python -m pip install -r requirements.txt
python run_checks.pyThe successful final line is:
ALL EXACT VERIFICATION CHECKS PASSEDRelease archive checksum: weyl-poisson-lifts-release.zip.sha256
Public constructions and provenance
The note builds on Alpöge's announced three-dimensional map and records the classical induced Weyl and Poisson endomorphisms. Same-day independent Weyl-algebra write-ups are cited explicitly.
| Contributor | Public object | Scope |
|---|---|---|
| Alpöge / Fable 5 | Original Keller counterexample | Dimension 3 |
| T. Ranjos | Independent exact Weyl-algebra lift and verifier | A3 |
| Omniscience Project | Independent formal Weyl-algebra note | A3 |
| Present note | Combined explicit Weyl and Poisson treatment with two exact verifiers | All n ≥ 3 |
Contribution and AI disclosure
The counterexample map is due to Levent Alpöge, who credits Claude Fable 5 as a collaborator in its discovery. Harris Chan used Claude Fable 5, GPT-5.6 Sol, and Gemini 3.1 Pro as computational and editorial aids during the derivation, verification, and revision of this note. Their contributions included symbolic derivation and code generation, proof and presentation review, and preparation of the reproducibility package. Harris Chan reviewed the mathematical claims, executed the exact-arithmetic checks, selected and edited the final content, and takes responsibility for the work.
Citation
Please cite the PDF and include the access date while the note remains preliminary.
BibTeX
@misc{chan2026weylpoisson,
author = {Harris Chan},
title = {Explicit Weyl and Poisson Lifts of
Alpöge's Jacobian Counterexample},
year = {2026},
month = jul,
note = {Preliminary note},
url = {https://harrischan.com/docs/weyl-poisson-lifts/}
}