Noncoordinate polynomial in every dimension at least four
Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.AbhyankarSathaye.exists_noncoordinate_polynomial
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
No confident match with a paper statement
No paper theorem is paired with this Lean statement.
Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements in An explicit noncoordinate polynomial with affine three-space zero fibre come first because the release’s name for this Lean statement shares words with that paper’s title.
Other possible paper matches, closest wording first
Theorem 1.1 of An explicit noncoordinate polynomial with affine three-space zero fibre. Words of the Lean statement, its names split into words, that also appear in the theorem: polynomial and coordinate. Found in the paper’s title, which its main theorem states: noncoordinate, affine and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: cong.
Corollary 1.2 of An explicit noncoordinate polynomial with affine three-space zero fibre. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, polynomial and coordinate. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: cong.
Proposition 2.3 of A stable coordinate that is not a coordinate in four variables. Words of the Lean statement, its names split into words, that also appear in the theorem: ring and polynomial. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: quotient.
Proposition 2.2 of A stable coordinate that is not a coordinate in four variables. Words of the Lean statement, its names split into words, that also appear in the theorem: ring, polynomial and coordinates.
Theorem 1.1 of A stable coordinate that is not a coordinate in four variables. Found in the paper’s title, which its main theorem states: four and coordinate.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 30 s; peak memory 2.1 GiB).
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 049’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: StableCoordinateFour.principal_package unfolds (PrincipalPackage = PolynomialPackage ∧ FiberPackage) to: the explicit f (same Q and f as the paper's Theorem thm:main) has totalDegree 5, is a coordinate of ℂ[x0..x4] after rename Fin.castSucc, and is not a coordinate of ℂ[x0..x3]; moreover every fibre f = t has coordinate ring ≅ ℂ[y1,y2,y3] and no automorphism carries (f − t) onto (x0). The only steps are definition unfolding (and f − C 0 = f for the Abhyankar–Sathaye reading at t = 0), so both the stable-coordinate claim and the fibre/Abhyankar–Sathaye claim are stated; AbhyankarSathaye (all n ≥ 4) and CommutingDerivations go beyond the headline.
Definitions the group asks readers to compare with the paper: IsCoordinate: ∃ ℂ-algebra automorphism φ of R n with φ g = X 0; equivalent to g being the image of a variable (or a member of a polynomial generating set), standard ; IsOneStableCoordinate: φ (stabilize g) = X 0 for an automorphism φ of R 5, with stabilize = rename Fin.castSucc; standard (adjoin one variable) ; IsAffineThreeSpaceFiber / FiberRing: the scheme-theoretic fibre ring R 4 ⧸ span {f − C t} is ℂ-algebra isomorphic to R 3; standard ; IsRectifiableFiber: some ℂ-automorphism φ of R 4 maps the ideal (f − C t) onto (X 0); standard rectifiability, equivalent to f − t being a coordinate up to a nonzero scalar (the paper's cor:fibers proof) ; StableCoordinateFour.PolynomialPackage includes f.totalDegree = 5 as an extra conjunct, matching the paper's 'explicit degree-five polynomial' ; CommutingDerivations.fourInclusion / extendedF / blocksToAmbient / blocksToOutput: index bookkeeping; unfolding Equiv.sumComm, finSumFinEquiv (castAdd / natAdd) and finCongr (value-preserving Fin.cast) shows that for i < n−4 derivation number 3+i must equal ∂/∂x_(4+i) (the ordinary extra partials); derivations 0, 1, 2 are constrained only by the other conjuncts (they are not required to ignore the added variables) ; CommutingDerivations.LocallyNilpotent: ∀ a, ∃ n, D^[n] a = 0; standard ; CommutingDerivations: LinearIndependent (MvPolynomial (Fin n) ℂ) d uses Mathlib's R-module structure on Derivation ℂ R R (Derivation.instModule), i.e. independence over the polynomial ring; standard ; CommutingDerivations S, P, h, u, v, w, x, y, s, p, F: the explicit F = h − P(x,y,s) − 1 of the September paper (matches eq:xyz and eq:F symbol by symbol); it is a different polynomial from StableCoordinateFour's degree-five f ; AbhyankarSathaye: no custom definitions; F is existentially quantified, not pinned to the explicit polynomial
OpenAI’s own note on what this Lean statement covers
“For every $n\ge4$, the formalized counterexample gives $F\in\mathbb C[x_1,\ldots,x_n]$ with quotient $\mathbb C[x_1,\ldots,x_n]/(F)\cong\mathbb C[y_1,\ldots,y_{n-1}]$, although $F$ is not a coordinate.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
21 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 0 to its own declarations, 14 to Mathlib’s documentation. 1 declared name also links to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
import Mathlib.Algebra.MvPolynomial.CommRingimport Mathlib.Basic.Complex.Basicimport Mathlib.RingTheory.Ideal.Quotient.Operationsnamespace OAInamespace AbhyankarSathaye/-- In every dimension at least four, a polynomial can define affine spacewithout being a coordinate of the ambient polynomial ring. -/theoremexists_noncoordinate_polynomial (n : ℕ) (hn : 4 ≤ n) : ∃ F : MvPolynomial (Fin n) ℂ,Nonempty ((MvPolynomial (Fin n) ℂ ⧸ Ideal.span {F}) ≃ₐ[ℂ]MvPolynomial (Fin (n - 1)) ℂ) ∧ ¬ ∃ (equiv : MvPolynomial (Fin n) ℂ ≃ₐ[ℂ] MvPolynomial (Fin n) ℂ) (index : Fin n), equiv (MvPolynomial.X index) = F := bysorryend AbhyankarSathayeend OAI
Every extracted statement of the accompanying papers
The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.
The polynomial $F$ defined in \eqref{eq:xyz}--\eqref{eq:F} satisfies
\[
R/(F)\cong\CC^{[3]},
\qquad
\grad F(2,0,-1/2,1/2)=0.
\]
Consequently $F$ is not a coordinate of $R$.
Corollary
For every integer n\ge4, there is a polynomial f\in\CC[z_1,\ldots,z_n] such that \CC[z_1,\ldots,z_n]/(f)\cong\CC^{[n-1]} and f is not a coordinate.
For every integer $n\ge4$, there is a polynomial
$f\in\CC[z_1,\ldots,z_n]$ such that
$\CC[z_1,\ldots,z_n]/(f)\cong\CC^{[n-1]}$ and $f$ is not a coordinate.
In $R=\C[x_1,x_2,x_3,x_4]$, put
\begin{equation}
Q=x_2^2-x_4^2+x_1x_3,
\qquad
f=x_1-2Q\bigl(Q(x_2+x_4)+x_1x_4\bigr).
\end{equation}
There is a $\C$-algebra automorphism of $R[w]$ sending $f$ to $x_1$, but there is no such automorphism of $R$.
Corollary
For every \lambda\in\C, the hypersurface f^{-1}(\lambda)\subset\A^4 is isomorphic to \A^3 and its embedding in \A^4 is not rectifiable.
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
More about this statement
Commentary The headline claims a stable-coordinate counterexample, but the formalised polynomial has a critical point, so it is not a stable coordinate. Robert Joseph George, Do the Lean statements in openai/math match their papers?, 2026-10-07Source
Commentary At the 6 October commit the family headline (a stable-coordinate counterexample) was not what AbhyankarSathaye states, and the scope note did not link the 5 October manuscript. Both were resolved at fd4aeeb2, which links that manuscript and adds StableCoordinateFour. VolMax-Studio, VolMax-Studio/p10-openai-math-verification-pilot: pilot verification of one family 185 theorem, 2026-10-07Source