WITHDRAWN: A One-Parameter Family of Jacobian Counterexamples Deforming the July 2026 Counterexample

Abstract

WITHDRAWN (2026-08-21) This note is withdrawn. A reader pointed out, and we have independentlyconfirmed by exact symbolic computation, that the family G(tau) isisotrivial: with s = 1 - 3*tau and J = (J1,J2,J3) denoting the July 2026counterexample, the componentwise identity G(tau)(x,y,z) = ( s*J1 , J2 , J3 ) at ( x, y + tau*x*z, s*z ) holds for all tau. Hence every member of the family is equivalent to theJuly map under an explicit polynomial coordinate change (shear + diagonalscaling; the shear absorbs 1 + xy + tau*x^2*z = 1 + x*(y + tau*x*z), andthe varying determinant -2*(1-3*tau)^2 is exactly the product of theJacobian factors of the two coordinate changes). The degeneration attau = 1/3 is the pole of this coordinate change, not a new phenomenon. Consequently the family contains no new counterexample, and the z-degreeevidence recorded in Question 1 of the note does not survive (z-degree isnot preserved by the shear). Theorems A-D remain formally correct asstatements about this parametrization, and the verification script inAppendix A still runs, but the framing of the note as a new family ofcounterexamples is retracted. We thank the reader who identified the trivialization. Priority for thefirst counterexample was, and remains, entirely with Alpoge and Fable.The classification question for the weight-(1,-1,-2) Wronskian engine(Question 2 of the note) is unaffected by this withdrawal and remainsunder study.