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# Release audit: Alexander–Gröbner 0.4.1

## Summary

The previously announced version 0.3.0 distribution was not available in complete form in the audit environment and could not be re-audited file by file. Version 0.4.0 was therefore rebuilt from the complete version 0.2.0 source tree that remained available for inspection.

During preparation of the English distribution, the 0.4.0 source archive was run against its own non-regression tests. Two provenance defects were found:

1. user-supplied metadata could overwrite computed PD provenance fields;
2. `input_record_sha256`, which was intended to describe the normalized input record, still depended on `source_file_sha256` and hence on whitespace or formatting in the JSON file.

Both defects were corrected in version 0.4.1. The fixes do not alter the mathematical presentation, Fox matrix, Fitting ideal, Laurent contraction, or reduced Gröbner basis.

## Mathematical pipeline checked

The implementation uses

```text
relative Alexander module = Coker(A.T)
E_k = I_{n-k}(A),
```

where `A` is the abelianized Fox matrix and `n` is its number of columns. Laurent exponents are cleared by multiplying entire rows by Laurent monomial units. The contraction is computed by

```text
(J Lambda) intersect P = J:(t1*...*tell)^infinity
                        = (J, 1-z*t1*...*tell) intersect P,
```

followed by a reduced Gröbner basis in lex order `t1 > ... > tell`.

## Tests

After the two fixes, all 28 non-regression tests passed. All 16 frozen Section 3 cases passed when run from the source tree, and all 16 passed again from the installed 0.4.1 wheel. The wheel and the source distribution both passed the built-in `doctor` checks; in the offline audit environment the source distribution was installed with `--no-build-isolation` because build dependencies could not be downloaded.

The audit includes checks of the Fox fundamental identity, input validation, row-wise Laurent denominator clearing, saturation, reduced-basis stability, and independent reducedness/Buchberger tests. The machine-readable source and wheel reproduction records are retained with the release audit materials.

## Scope and limitations

This release supports reproduction from frozen inputs and application of the same algorithm to new inputs. It does not independently certify external link-table labels, decide link equivalence, reproduce an unarchived exhaustive table scan, or replace the paper’s proofs.

The reference audit environment is Linux with Python 3.13.5 and SymPy 1.14.0. The Windows batch files still require a final execution test on an actual Windows machine before public release. A completely independent second-CAS implementation is also outside the present audit.