The Alpöge counterexample to the Jacobian Conjecture: affine chart certificates and bounded rigidity computations

We study the counterexample to the Jacobian Conjecture in dimension 3 announced by L. Alpöge on July 20, 2026. In the affine chart Z ≠ 0 of the target we complement the projective global description of the anonymous Ulam preprint with explicit affine resultant–subresultant certificates: the abscissas of a fiber are exactly the roots of the cubic E = L·x³ + (4−3YZ)x − 2Z; for every root, its multiplicity in E equals the number of fiber points sharing that abscissa (so E = L·∏(x − x(p)) whenever L ≠ 0); the identities Δ_x(E) = −4·S²·L and Res_x(E, (3YZ−4)x + 3Z) = Z·S² locate the failure of the abscissa to separate fiber points exactly on {S = 0} ∩ F(ℂ³); and within the chart the non-properness set is {L = 0}. Our main contribution is a campaign of bounded rigidity computations: within explicitly parametrized families, the counterexample's data are unique up to precisely defined gauge actions, certified by explicit generating sets of the Rabinowitsch-saturated ideals (checked by mutual ideal inclusion against Gröbner bases); a graded analysis organizes viable kernels into an infinite tower whose rungs k = 2, 3, 4 are empty; and the candidate cone-degree sectors d = 4, 5, 6 are empty in the graded range, which — contrasted with the non-z-linear degree-d families of the Ulam preprint for d ≥ 4 — motivates the question whether every non-invertible z-linear Keller map must have generic degree 3. Every claim is labeled [proved], [verified] (exact symbolic computation, asserted by the accompanying scripts), or [bounded] (exhaustive within a stated finite family). Related work. This note complements the projective global description of the anonymous Ulam preprint (July 20, 2026) and the exposition by Gallagher with elementary affine certificates; the parallel preprint of Shaska (arXiv:2607.20210, July 22, 2026) approaches the counterexample through weighted G_m-equivariance and weighted projective geometry, with overlap confined to the elementary weight identities. The accompanying reproducibility package (PDF + LaTeX + reproduce.py + jc_hunt.py + requirements + README) asserts every identity, saturated-ideal certificate and computational sweep in the manuscript. Developed in collaboration with an AI assistant (Anthropic's Claude); revised across multiple rounds of independent LLM-assisted audit. The author takes full responsibility for the content.