Zariski cancellation and affine fibrations over the complex numbers
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
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. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
The algebra A in (1.2) is an integral complex algebra of dimension four. For an independent variable w,
A[w]\simeq\kk\poly5,
\qquad
A\not\simeq\kk\poly4.
Consequently the Zariski cancellation problem over the complex numbers has a negative answer in dimension four.
Corollary 1.2(Failure of the Stable Coordinate Conjecture)
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
The polynomial H in (1.2) is not a coordinate of P=\kk[p,s,u,F,J], but it is a coordinate of P[w] for one independent variable w. Thus the Stable Coordinate Conjecture over \kk fails in ambient dimension five after one stabilization.
No Lean statement in the release is paired with this theorem.
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Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
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.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
ComplexCancellation.lean, the release lists it as a main result.
defMainStatement : Prop := Algebra.FiniteType ℂ A ∧ IsDomain A ∧ ringKrullDim A = 4 ∧ Nonempty (Polynomial A ≃ₐ[ℂ] MvPolynomial (Fin 5) ℂ) ∧ ¬ Nonempty (A ≃ₐ[ℂ] MvPolynomial (Fin 4) ℂ)⋮theoremmain : MainStatement := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:Cancellation counterexample (dimension four) fully stated; the Dolgachev-Weisfeiler affine-fibration counterexamples (smooth surjections with every fibre A^3, not Zariski-locally trivial) are not stated.
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name:The Dolgachev–Weisfeiler clause (smooth surjections X → 𝔸¹ and 𝔸⁵ → 𝔸² with all residue-field fibers 𝔸³ that are not Zariski-locally trivial) is not stated.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
Affine-space cancellation asks whether $X\times\mathbb A^1\cong\mathbb A^{n+1}$ forces $X\cong\mathbb A^n$. The formalized result gives an explicit finite-type complex domain $A$ of Krull dimension four with $A[w]\cong\mathbb C[x_1,\ldots,x_5]$ but $A\not\cong\mathbb C[x_1,\ldots,x_4]$. Thus adjoining one variable erases a genuine algebraic distinction.
The separate stable-coordinate and general line-bundle lifting consequences are not included.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an explicit complex domain A of dimension four with A[w] isomorphic to a polynomial ring in five variables and A not isomorphic to one in four. The stable-coordinate corollary is not included.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Proposition 2.2 of An explicit failure of complex affine-space cancellation, 23 September 2026
Reported by
cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the cylinder isomorphism A[w] = C^[5] (Proposition prop:stabilization), by the paper’s printed maps with inverses; NOT A not = C^[4], which carries the counterexample”
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the proof for Zariski cancellation over the complex numbers.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 047The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 047Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In An explicit failure of complex affine-space cancellation, 23 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 386 s; peak memory 6.5 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Proposition 2.2 of An explicit failure of complex affine-space cancellation, 23 September 2026
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the cylinder isomorphism A[w] = C^[5] (Proposition prop:stabilization), by the paper’s printed maps with inverses; NOT A not = C^[4], which carries the counterexample”
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.