# Explicit Weyl and Poisson lifts of Alpöge's Jacobian counterexample

This archive contains the paper, a standalone LaTeX source, a generated copy of the displayed operator formulas, and two independent exact-arithmetic verification implementations.

## Canonical files

- `main.tex`: compilation entry point.
- `weyl-poisson-lifts.tex`: canonical standalone manuscript source. It has no `operators.tex` dependency.
- `weyl-poisson-lifts.pdf`: compiled paper.
- `operators.tex`: generated copy of the explicit formulas retained for reproducibility.
- `generate_operators.py`: regenerates `operators.tex` and checks the inline manuscript block.
- `sympy_verify_lifts.py`: SymPy verification of the identities shared by the Weyl and Poisson lifts.
- `independent_verify_lifts.py`: no-CAS exact-rational polynomial verification of the same identities.
- `run_checks.py`: one-command verification driver.
- `SHA256SUMS.txt`: SHA-256 manifest for the release contents, excluding the manifest itself.

## Verify the mathematics

Tested with Python 3.13.5 and SymPy 1.14.0.

```bash
python -m pip install -r requirements.txt
python run_checks.py
```

Expected final line:

```text
ALL EXACT VERIFICATION CHECKS PASSED
```

The driver verifies:

- the Jacobian determinant;
- the three-point collision;
- both inverse-Jacobian products;
- all nine commuting-vector-field coefficients;
- the Weyl relations;
- the Poisson relations obtained from the same identities;
- the correspondence between `generate_operators.py` and `operators.tex`;
- the correspondence between the generator and the inline formulas in the standalone manuscript source.

The two verifier outputs are recorded as `sympy_verify_lifts_output.txt` and `independent_verify_lifts_output.txt`. The complete driver output is recorded as `verification_output.txt`.

## Compile the paper

Tested with pdfTeX 3.141592653-2.6-1.40.26 and LaTeX2e 2024-11-01.

```bash
pdflatex -interaction=nonstopmode -halt-on-error main.tex
pdflatex -interaction=nonstopmode -halt-on-error main.tex
```

The generated file is `main.pdf`. The archive also includes the same compiled document as `weyl-poisson-lifts.pdf`.

The canonical source also compiles directly:

```bash
pdflatex -interaction=nonstopmode -halt-on-error weyl-poisson-lifts.tex
pdflatex -interaction=nonstopmode -halt-on-error weyl-poisson-lifts.tex
```

## Provenance of the explicit formulas

`generate_operators.py` reconstructs Alpöge's map, computes `A = (JF)^{-1}` exactly in SymPy, and formats the displayed generators. The only presentational operation is line breaking of the longest coefficient.

Regenerate the derivative file:

```bash
python generate_operators.py --output operators.tex
```

Check both the derivative file and the inline manuscript block:

```bash
python generate_operators.py --check \
  --check-source weyl-poisson-lifts.tex
```

## Checksums

`SHA256SUMS.txt` identifies all files inside the archive. The SHA-256 checksum for the complete ZIP is distributed alongside it as `weyl-poisson-lifts-release.zip.sha256` and may also be listed on the project page.

## Credit and responsibility

Formal author and responsible reviewer: Harris Chan.

Claude Fable 5, GPT 5.6 Sol, and Gemini 3.1 Pro are disclosed in the paper as computational and editorial aids. The note builds on Levent Alpöge's explicit Jacobian counterexample and cites the same-day community Weyl-algebra write-ups. The author independently reviewed the mathematical claims and takes full responsibility for the work.

## License

Copyright 2026 Harris Chan. No reuse license is granted in this release. All rights reserved. This conservative status can be replaced by an explicit paper and code license after the author confirms the desired terms and any applicable employer policy.