Alpöge's counterexample in dimension three: a machine-checked certificate for the induced endomorphism of A_3(C), and the arithmetic of its fibre cubic

Alpöge's polynomial map F from C^3 to C^3 with det DF = -2 refutes the Jacobian Conjecture in dimension three and, through the Bass-Connell-Wright transfer, the Dixmier Conjecture for A_3(C). This deposit contains a short note (12 pages) and a single self-contained Python script that together form an executable certificate for that transfer and for the arithmetic that surrounds it. Nothing in this deposit is claimed to be new, and the note says so in five explicit disclaimers. The counterexample is entirely Alpöge's. The passage from F to the induced endomorphism of the third Weyl algebra is classical and appears in the Secret Blogging Seminar account of 20 July 2026, including the non-surjectivity witness reproduced here. The reduction of the generic fibre to a one-variable cubic is the reduction of Gallagher (jacobianfun.org, doi:10.5281/zenodo.21479195, 22 July 2026) in a different normalisation; the note exhibits the change of variable w = R t / 2 identifying the two, and checks the resulting identity symbolically. The local maximality criteria for the attached cubic rings are those of Davenport and Heilbronn (1971, Lemma 6) in the algorithmic form of Belabas (Math. Comp. 66 (1997), Lemma 1.2), here only specialised to one family. What is offered is an artifact. Fourteen checks, run in a single command, certify every algebraic identity involved: the constant Jacobian determinant, the polynomiality of the inverse Jacobian, the mixed relations, the commutation of the derivations, the Piola identity, the symmetry, the collision points, a finite witness for the non-surjectivity of the endomorphism that replaces the usual appeal to a general theorem, the fibre cubic and its dictionary with Gallagher's, the Levi-Delone-Faddeev multiplication table and the index [R : Z[theta]] = |r|, the covariant identities behind the specialisation, and the maximality conditions cross-validated against Round 2 of Zassenhaus and against sympy's round_two. Files: paper_alpoge.pdf (the note), paper_alpoge.tex (source), verify_alpoge.py (the certificate). The script requires only Python 3.8+ and SymPy, and runs in under a minute. The text is released under CC BY 4.0, the script under the MIT License. Author: Marcos José Valenzuela Mena, ORCID 0009-0002-2316-7981, Zaragoza, Spain, holamarcos137@protonmail.com, 10-08-2026.