Preliminary mathematical note - version 0.5 - 21 July 2026
An explicit cubic-homogeneous counterexample in dimension 19
A sparse Segre-Yagzhev lift of the factor-aware 11-variable degree-three map posted by Joe Atkins-Turkish, together with two exact, independently implemented verification scripts.
Exact claim. There is an explicit polynomial map
such that every nonzero component of H is homogeneous of degree three,
and three distinct rational points have the same image. Consequently det JG = 1 identically, while G is not injective.
Status. This is a rapidly prepared preliminary draft about a newly announced counterexample. The displayed 19-dimensional object is independently machine-checkable by finite exact-arithmetic certificates, but the note has not undergone ordinary peer review.
Paper and verification artifacts
The final object is stored in a canonical JSON representation. The SymPy script derives the map from the 11-variable input and checks the global identities; the second script independently checks the same JSON using only Python's standard library.
- Paper Compiled preliminary draft (PDF)
- LaTeX source Complete source for the note
- Canonical map Machine-readable H, collision points, and image
- SymPy verifier Derivation and exact symbolic certificate
- Standard-library verifier Independent dictionary-polynomial implementation
- Generated appendices Typeset formulas generated and byte-for-byte checked by the SymPy verifier
- README Commands, versions, hashes, and attribution notes
- Versioned release archive Exact version 0.5 bundle
- SHA-256 manifest Hashes for the core release files
- Licenses MIT for code; CC BY 4.0 for manuscript and map
Run locally
python3 -m pip install 'sympy==1.14.0'
python3 verify_yagzhev19_sympy.py
python3 verify_yagzhev19_stdlib.py yagzhev19_map.jsonTested with Python 3.13.5 and SymPy 1.14.0.
Canonical JSON SHA-256: f88a0883ce9509339c53a6a721390053b424039a8dd6c41ebc7770aeb955cced
Public constructions and provenance
The note does not claim authorship of the 11-variable input map. It records the public chain of constructions and isolates the new 19-dimensional endpoint and its exact certificate.
| Contributor | Public object | Dimension |
|---|---|---|
| Alpöge / Fable 5 | Original Keller counterexample | 3 |
| Ali Demirci | Generic explicit Yagzhev realization | 79 |
| u/Distinct-Soft-3991 | Degree-three Keller map | 17 |
| Joe Atkins-Turkish | Posted factor-aware degree-three Keller map; Gist credits ChatGPT | 11 |
| Present note | Sparse Yagzhev lift with global symbolic certificate | 19 |
Contribution and AI disclosure
The dimension-19 lift and its verification were developed through iterative prompting and symbolic computation using Claude Fable 5, GPT-5.6 Sol, and Gemini 3.1 Pro. Harris Chan curated the resulting construction, checked the generated artifacts, and prepared the exposition. The systems are acknowledged as computational tools rather than authors; Harris Chan takes responsibility for the mathematical claims, citations, attribution decisions, and conclusions.
Citation
Please cite the versioned PDF and include the access date while the note remains preliminary.
BibTeX
@misc{chan2026yagzhev19,
author = {Harris Chan},
title = {An Explicit Cubic-Homogeneous Counterexample
to the Jacobian Conjecture in Dimension 19},
year = {2026},
month = jul,
note = {Preliminary draft, version 0.5},
url = {https://harrischan.com/docs/yagzhev-19/}
}