In the July 2026 discussion of the counterexample to the Jacobian conjecture, the source variety of the geometric construction, the complement of the ramification divisor and the hyperplane preimage in P¹ × P², was widely reported to be affine three-space by an explicit change of coordinates relayed by Will Sawin from ChatGPT: on the affine slice where av + bu = 1 and av² − buv + cu² = 1, the map (v, u, a, b, c) ↦ (u, 2vb − uc, 12vb² − 6ubc + 4b² + 2a − 3ac) was said to "give an isomorphism from the complement of R and H to A³." That statement is the recorded instance on claim a21e1db1-f393-4eb0-9543-7411544c2015.

Direct symbolic computation, recorded in the claim's current assessment, shows the formula as transcribed is not an isomorphism. Writing D = 2vb − uc, the two defining equations force v = 1 + uD, and the coordinates (u, D, E) with E = −2D²a − 4Db + c (proposed by the commenter "bruno" later in the same thread) do give a polynomial isomorphism with polynomial inverse, the relevant 3×3 matrix having determinant exactly 2. Expressed in these coordinates, Sawin's third component is quadratic in E (for u = 1 and D = 0 it equals (5E² − 8E + 4)/2), so the map (u, D, F) is neither injective nor of constant Jacobian.

The conclusion that the complement is affine three-space is nonetheless correct and verified, resting on the corrected coordinates, on the bundle-tower arguments of Sawin, Litt, and van Dobben de Bruyn, on Speyer's finite-field point count, and on Theorem A.7 of arXiv:2608.27341. The point is narrow: anyone taking the first circulated formula as the explicit polynomial change of variables should use the corrected one instead.