Ring-theoretic properties of the completed surface-cone candidate
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.SmallCM.candidate_ring_properties
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.
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: complete, residue, field, noetherian, domain and local. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimensional and three.
Theorem 1.1 of A Complete Local Domain without a Small Cohen–Macaulay Module
Theorem
There exists a three-dimensional complete Noetherian normal local domain R containing \C, with residue field \C, such that no nonzero finitely generated R-module has depth three.
There exists a three-dimensional complete Noetherian normal local domain
$R$ containing $\C$, with residue field $\C$, such that no nonzero finitely
generated $R$-module has depth three.
Other possible paper matches, closest wording first
Corollary 7.1 of A Complete Local Domain without a Small Cohen–Macaulay Module. Words of the Lean statement, its names split into words, that also appear in the theorem: maximal, domain, local and surface. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimensional and three.
Proposition 6.1 of A Complete Local Domain without a Small Cohen–Macaulay Module. Words of the Lean statement, its names split into words, that also appear in the theorem: complex and surface.
Theorem 1.2 of A Complete Local Domain without a Small Cohen–Macaulay Module. Words of the Lean statement, its names split into words, that also appear in the theorem: surface.
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 282 s; peak memory 6.4 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 195’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 a supporting or a different result statedTheir term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
Their note: Lean states only that some C-algebra R is a three-dimensional, m-adically complete, Noetherian, integrally closed local domain with residue field C; it never mentions modules, depth or Cohen-Macaulayness, so the defining property of the headline (no nonzero finitely generated module of depth 3) and the essentially-finite-type corollary are not stated. As typed the existential is also satisfied by C[[x,y,z]] and does not name the surface cone, so it records only the ring-theoretic half of the construction.
Definitions the group asks readers to compare with the paper: candidate_ring_properties (the statement itself): a bare existence claim over an unconstrained R; it is not tied to the paper's completed section ring R(X,H) (the proof uses ExplicitCone.completedRing, the statement does not mention it) and is satisfied by the regular ring C[[x,y,z]]; the key property (no nonzero finitely generated module of depth three) is absent. ; IsIntegrallyClosed R: R is integrally closed in its fraction field (normal domain); standard. ; IsAdicComplete (IsLocalRing.maximalIdeal R) R: R is Hausdorff and complete for the maximal-ideal-adic topology; standard. ; ringKrullDim R = 3: Krull dimension 3 (value in WithBot N-infinity); standard. ; Function.Bijective (algebraMap C (IsLocalRing.ResidueField R)): the structure map C -> R/m is bijective, i.e. residue field C; standard. ; Algebra C R: the paper's 'containing C' is encoded as an algebra structure over C; standard.
Finds a supporting or a different result statedTheir term: “SUPPORT-ONLY”.
The challenge statements state only supporting results. Their record
Their note: The counterexample itself (no nonzero finitely generated MCM module) and the essentially-of-finite-type example are not stated; only properties of the candidate ring are, and those properties are existential.
OpenAI’s own note on what this Lean statement covers
“The linked formalization verifies only the ring-theoretic candidate: there exists a three-dimensional complete Noetherian integrally closed local domain over $\mathbb C$ whose residue field is $\mathbb C$.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
18 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, 12 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.
There exists a three-dimensional complete Noetherian normal local domain R containing \C, with residue field \C, such that no nonzero finitely generated R-module has depth three.
There exists a three-dimensional complete Noetherian normal local domain
$R$ containing $\C$, with residue field $\C$, such that no nonzero finitely
generated $R$-module has depth three.
Theorem
Let X be a smooth integral projective surface over \C, and let H be an ample globally generated divisor. If R(X,H) admits a small Cohen–Macaulay module, then
Let $X$ be a smooth integral projective surface over $\C$, and let $H$ be
an ample globally generated divisor. If $R(X,H)$ admits a small
Cohen--Macaulay module, then
\begin{equation}
K_X^2-2c_2(X)\leq 6H^2.
\end{equation}
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
Imports only Mathlib; no local vocabulary added
Lines of Lean, without comments or blank lines
7
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Ring-theoretic properties of the completed surface-cone candidate
Family 195: A counterexample to the small Cohen–Macaulay module conjecture
Candidate pairing
Theorem
There exists a three-dimensional complete Noetherian normal local domain R containing \C, with residue field \C, such that no nonzero finitely generated R-module has depth three.
A Complete Local Domain without a Small Cohen–Macaulay Module
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.
Questions to resolve when comparing this statement with the paper
The AI draft finds no Lean statement in its family that states a paper's theorem as the paper does.