# Explicit cubic-homogeneous Jacobian counterexample in dimension 19

Version 0.5, 21 July 2026. Preliminary draft.

This release gives an explicit map

\[
\widehat G=X+H:\mathbb C^{19}\to\mathbb C^{19}
\]

for which every nonzero component of \(H\) is cubic homogeneous,
\((JH)^{18}=0\) but \((JH)^{17}\neq0\), and three distinct rational points
have the same image. Hence \(\det J\widehat G=1\) identically while
\(\widehat G\) is non-injective.

Canonical landing page:

<https://harrischan.com/docs/yagzhev-19/>

## Files

- `yagzhev19-note.pdf` - compiled preliminary manuscript.
- `yagzhev19-note.tex` - manuscript source.
- `yagzhev19_appendices.tex` - generated appendix formulas; checked byte-for-byte by the SymPy verifier.
- `yagzhev19_map.json` - canonical machine-readable map, collision points, and common image.
- `verify_yagzhev19_sympy.py` - derives the 19-variable object from the posted 11-variable input and checks the global identities.
- `verify_yagzhev19_stdlib.py` - independent exact verifier using only the Python standard library.
- `SHA256SUMS` - hashes for the distributed release files.
- `LICENSE.md` - licensing terms.
- `CITATION.bib` and `CITATION.cff` - citation metadata.
- `VERSION` - release version.

## Run the verifiers

Tested with Python 3.13.5 and SymPy 1.14.0.

```bash
python3 -m pip install 'sympy==1.14.0'
python3 verify_yagzhev19_sympy.py
python3 verify_yagzhev19_stdlib.py yagzhev19_map.json
```

The expected successful checks include:

```text
(JH)^18 = 0
((JH)^17)_[4,19] = -18*t^16*x^10*y^6*z^2
```

The paper uses one-based matrix indices; the Python key is `(3, 18)`.

To regenerate the appendix formulas from the checked symbolic expressions:

```bash
python3 verify_yagzhev19_sympy.py --write-generated
```

A normal run checks that the distributed `yagzhev19_appendices.tex` matches the regenerated text byte-for-byte.

## Core hashes

```text
f88a0883ce9509339c53a6a721390053b424039a8dd6c41ebc7770aeb955cced  yagzhev19_map.json
82a169f01211110d8c55bbc03fe18f45c57a9b4bdebaeb07584851d8584baf15  verify_yagzhev19_sympy.py
9ab62f3ccc3b149e9b72676399173eb5a8b1f61ee36bd0107ecabf2c1e8c5814  verify_yagzhev19_stdlib.py
```

See `SHA256SUMS` for the core release manifest; the archive itself is excluded to avoid a circular checksum.

## Attribution

The original three-variable map was announced by Levent Alpoge, whose post credits Claude Fable 5. Ali Demirci posted a 79-dimensional Yagzhev realization, `u/Distinct-Soft-3991` posted a 17-variable degree-three reduction, and Joe Atkins-Turkish posted the factor-aware 11-variable reduction used here. The Atkins-Turkish Gist states that its explicit construction, simplification, and verification code were generated by ChatGPT.

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.

## License

See `LICENSE.md`. In brief: the Python code is under the MIT License; the manuscript and canonical map are under CC BY 4.0.