OpenAI · 047-01 · 2026-09-23 · Algebraic and complex geometry
An explicit failure of complex affine-space cancellation
preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026
Family 047. Complex counterexamples to cancellation and affine fibrations.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Constructs an integral complex affine fourfold with , disproving affine-space cancellation over in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections and have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.
We construct an explicit integral complex affine fourfold X with but . 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 047: Zariski cancellation and affine fibrations over the complex numbers. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.