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

Abstract

Companion material for the preprint: PDF (7 pages), HTML source, and a self-contained SymPy verification script (Appendix A of the paper). The preprint exhibits an explicit family G(τ): ℂ³ → ℂ³ of polynomial maps with det Jac G(τ) = −2(1−3τ)², non-injective for every τ ≠ 1/3 (three closed-form preimages over an entire target plane), specializing at τ = 0 component by component to the three-variable counterexample to the Jacobian conjecture announced by L. Alpöge in July 2026. Reproduce every claim: pip install sympy && python verify_tau_family.py → all checks passed Author: Zhuofei Wei (independent researcher). AI collaboration disclosure: Kimi K3 (Moonshot AI) carried out the symbolic computations and supported the drafting; the author takes full responsibility for all statements.