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An explicit failure of complex affine-space cancellation
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Abstract
We construct an explicit integral complex affine fourfold X with $X\times\mathbb A^1\cong\mathbb A^5$ but $X\not\cong\mathbb A^4$. 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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- queelius (fork of piiiico/openai-math-counterexamples), openai-math-counterexamples (GitHub repository, accessed 2026-10-21). Its row 047 reports an independent exact-arithmetic recomputation of the identities in the stabilization step (the route to $A[w]\cong\mathbb C^{[5]}$) as "holds", and states that the non-polynomiality of $A$ (Sections 3–6) was not checked. [AI-generated per source]
Formalization
- OpenAI, lean/docs/047.md: Zariski cancellation and affine fibrations over the complex numbers (Lean scope note) (OpenAI Math Release, 2026-10-06). The note says the Lean formalization covers the explicit four-dimensional cancellation counterexample (Theorem 1.1) but not the stable-coordinate consequence or the general line-bundle lifting statement. [family-level]
- OpenAI, lean/ComparatorChallenges/ComplexCancellation.lean (openai/math, commit adc7f12). This is the comparator statement
MainStatementfor the explicit ring $A=\mathbb C[p,s,u,F,J]/(H)$: a finitely generated domain of Krull dimension 4 with $A[t]$ polynomial in 5 variables and $A$ not polynomial in 4 variables; as usual for comparator challenge statements, its proof is left assorryin this file. - google-deepmind/formal-conjectures, FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean (formal-conjectures, 2026). The Zariski cancellation statement from Gupta's ICM survey is restated here as false in characteristic zero, with a variant
false_complex_fourfor $\mathbb C^{[4]}$; both cite this paper, are tagged "research solved", and link to OpenAI's Lean development asformal_proof. - mo271, feat: mark problems solved in the OpenAI math release and link formal proofs (PR #6898) (google-deepmind/formal-conjectures, opened 2026-10-07, merged 2026-10-11). This pull request marked Zariski cancellation as refuted ("false over ℂ, n = 4") and added the
false_complex_fourvariant linked to the release's Lean proof; none of the visible review comments discuss this result. [family-level]
Related prior literature mentioned by others
- Neena Gupta, The Zariski Cancellation Problem and related problems in Affine Algebraic Geometry (arXiv:2208.14736, 2022-08-31). The AI Proved What? exposition links this survey for the characteristic-zero background to the claimed fourfold counterexample; it is also in the manuscript's bibliography.
- Neena Gupta, On Zariski's Cancellation Problem in positive characteristic (arXiv:1309.1368, 2013-09-05). The AI Proved What? exposition cites this earlier failure of cancellation in every dimension at least three in positive characteristic when explaining the paper's complex case; it is also in the manuscript's bibliography.
Possibly relevant literature (found by the agent, not checked)
- A. Dubouloz and P. Ghosh, Algebraic families of higher dimensional $\mathbb A^1$-contractible affine varieties non-isomorphic to affine spaces (arXiv:2501.09613, January 2025). This paper builds characteristic-zero smooth $\mathbb A^1$-contractible varieties in every dimension $\ge 4$ that are not affine spaces, presented as candidate counterexamples to Zariski cancellation; this bears on the introduction's account of the characteristic-zero problem as open in dimensions $\ge 3$.
- P. Ghosh and A. Pal, A new family of counterexamples to the Zariski Cancellation Problem in positive characteristic (arXiv:2404.13803, April 2024). It gives a further positive-characteristic family of counterexamples in dimension $\ge 3$, distinct from Gupta's; the paper's history of positive-characteristic counterexamples cites only Asanuma and Gupta.
- P. Das and A. K. Dutta, A Note on Residual Variables of an Affine Fibration (arXiv:2002.02128; J. Pure Appl. Algebra 218 (2014), 1792–1799). It proves stable polynomiality over $R[W]$ for residual variables of affine fibrations over Noetherian domains, which relates to the residual-coordinate stabilization the paper takes from El Kahoui–Ouali and Dutta–Lahiri.
- C. F. Venegas R. and H. J. Venegas R., Cancellation problem via locally nilpotent derivations (arXiv:2512.15590, December 2025, revised February 2026). This survey covers locally nilpotent derivation and Makar-Limanov invariant methods for cancellation, the general technique the paper uses to show $A\not\cong\mathbb C^{[4]}$.
Digests and expositions
- AI Proved What?, Cancel a line? Zariski cancellation over the complex numbers (episode 047, October 2026). This video and transcript outline the explicit equation, stabilization, degeneration and additive-action obstruction, and the claimed stable-coordinate and affine-fibration consequences. [AI-generated per source] [family-level]
News and blogs
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (audit report, findings dated 2026-10-10). The report lists family 047's Dolgachev–Weisfeiler affine-fibration claims among conclusions broader than the formal statement checked in the release. [family-level]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 047
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026,
author = {{OpenAI}},
title = {{An explicit failure of complex affine-space cancellation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf}{OAI:An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/047.md. The Lean statements have not been checked by people against the paper's claims.
Family 047: Zariski cancellation and affine fibrations over the complex numbers
Constructs an integral complex affine fourfold $X\not\cong\mathbb A^4$ with $X\times\mathbb A^1\cong\mathbb A^5$, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections $X\to\mathbb A^1$ and $\mathbb A^5\to\mathbb A^2$ have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.
This is the only paper in this family.