On the Global Image, Fibre Structure, and Weyl-Algebra Lift of the Alpöge–Fable Map

### OverviewThis record contains the complete formal manuscript and independent exact verification script analyzing the polynomial Keller map introduced by Alpöge and Fable, its induced endomorphism on the three-dimensional Weyl algebra A_3, the global image geometry, fiber stratifications, and its application to the Dixmier Conjecture D_3. ### AbstractWe present a self-contained mathematical analysis and computational certification of the exact Alpöge–Fable polynomial Keller map F: C^3 -> C^3. We prove that the induced endomorphism Phi_F: A_3 -> A_3 on the complex Weyl algebra preserves differential order via a field extension argument on the associated graded algebra gr(A_3). Evaluating on a non-trivial triple fiber certified symbolically establishes x_1 not in Phi_F(A_3), directly refuting the Dixmier Conjecture D_3. Furthermore, we determine the exact global image Im(F) = C^3 \ C using a rigorous two-chart analysis (c = 0 and c != 0), providing full fiber stratifications and proving the non-finiteness of C[x,y,z] as a C[F]-module. An independent Python certification script (SymPy) is provided in the Appendix and as a standalone file. ### Files Included in this Record- `paper_completo.tex`: Full formal manuscript containing definitions, lemmata, theorems, proofs, and the embedded verification script.- `verify.py`: Independent Python script providing automated exact symbolic and numerical verification of the Jacobian determinant, triple fiber, eliminant discriminant, and image properties.- `README.md`: Repository documentation and usage instructions.