Zariski cancellation and affine fibrations over the complex numbers

What OpenAI claims

Constructs an integral complex affine fourfold X≇𝔸⁴ with X×𝔸¹ ≅ 𝔸⁵, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections X → 𝔸¹ and 𝔸⁵ → 𝔸² have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial. (Lean)

Paper

1. An explicit failure of complex affine-space cancellation

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We construct an explicit integral complex affine fourfold X with X×𝔸¹ ≅ 𝔸⁵ but X≇𝔸⁴. This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸3-fibrations over 𝔸1 and 𝔸2 that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.