Lean statement

Smooth isometric immersion of every closed surface into four-space

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.ClosedSurfaceR4.FiniteOrderSmoothing.smooth_isometric_immersion

Lean statement file: SurfaceImmersion.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 333 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem smooth_isometric_immersion (g : SmoothMetric M) :    ∃ F : M → Space, IsSmoothIsometricImmersion M g F := by  sorry

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.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.ClosedSurfaceR4.Plane line 11

abbrev Plane := EuclideanSpace ℝ (Fin 2)

abbreviation OAI.ClosedSurfaceR4.Space line 12

abbrev Space := EuclideanSpace ℝ (Fin 4)

abbreviation OAI.ClosedSurfaceR4.planeModel line 13
```
abbrev planeModel := 𝓘(ℝ, Plane)
```
abbreviation OAI.ClosedSurfaceR4.spaceModel line 14
```
abbrev spaceModel := 𝓘(ℝ, Space)
```

abbreviation OAI.ClosedSurfaceR4.SmoothMetric line 22

abbrev SmoothMetric :=  Bundle.ContMDiffRiemannianMetric planeModel ∞ Plane    (fun p : M => TangentSpace planeModel p)

definition OAI.ClosedSurfaceR4.IsSmoothIsometricImmersion line 26

def IsSmoothIsometricImmersion (g : SmoothMetric M) (F : M → Space) : Prop :=  ContMDiff planeModel spaceModel ∞ F ∧  ∀ (p : M) (v w : TangentSpace planeModel p),    inner ℝ (mfderiv planeModel spaceModel F p v)      (mfderiv planeModel spaceModel F p w) = g.inner p v w

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: isometric, admits, riemannian, surface, smooth and immersion. Found in the paper’s title, which its main theorem states: space, euclidean and four. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: closed.

Theorem 1.1 of Smooth isometric immersions of closed surfaces into Euclidean four-space

Theorem

Every closed smooth Riemannian surface (M,g) admits a C^\infty isometric immersion into (\R^4,\delta_4), where \delta_4=\sum_{i=1}^4 dx_i^2.

Section 1, Introduction Read it in the paper TeX source, line 14

Show the TeX

Every closed smooth Riemannian surface $(M,g)$ admits a $C^\infty$ isometric
immersion into $(\R^4,\delta_4)$, where $\delta_4=\sum_{i=1}^4 dx_i^2$.

Other possible paper matches, closest wording first

Proposition 5.1 of Smooth isometric immersions of closed surfaces into Euclidean four-space. Words of the Lean statement, its names split into words, that also appear in the theorem: metric, riemannian, surface, immersions and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: closed.
Proposition 4.1 of Smooth isometric immersions of closed surfaces into Euclidean four-space. Words of the Lean statement, its names split into words, that also appear in the theorem: compact, smooth and immersion. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: differential.
Proposition 3.1 of Smooth isometric immersions of closed surfaces into Euclidean four-space. Words of the Lean statement, its names split into words, that also appear in the theorem: tangent, surface, smooth and immersion. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: preserved and closed.
Proposition 7.1 of Smooth isometric immersions of closed surfaces into Euclidean four-space. Words of the Lean statement, its names split into words, that also appear in the theorem: tangent, surface, smooth and immersion. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: closed.

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

One independent group reports that Comparator accepted a run for this Lean statement file: Carlos Toledo’s cert-machine. 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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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 333’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.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: smooth_isometric_immersion (g : SmoothMetric M) : ∃ F : M → ℝ⁴, ContMDiff ∞ F ∧ ∀ p v w, ⟪dF v, dF w⟫ = g.inner p v w for every compact Hausdorff second-countable C^∞ 2-manifold without boundary and every C^∞ Riemannian metric (Mathlib's ContMDiffRiemannianMetric), with no orientability or curvature hypothesis; 'immersion' follows from positive-definiteness (injective differential), and the case is global and closed only, as in the summary.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that every closed smooth Riemannian surface admits a smooth isometric immersion into $\mathbb R^4$. The differential preserves the Riemannian inner product on every tangent space. No orientability assumption is imposed, and the statement asks for an immersion rather than an embedding.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

49 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: 21 to its own declarations, 23 to Mathlib’s documentation. 7 declared names also link 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves smooth_isometric_immersion exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 6 other declarations of the file; with the theorem itself they span 15 lines. 3 of them are named in the statement itself.

Names in the namespace OAI.ClosedSurfaceR4 are listed without it.

abbrev Planeline 11library
abbrev Spaceline 12in the statementlibrary
abbrev planeModelline 13library
abbrev spaceModelline 14library
abbrev SmoothMetricline 22in the statementlibrary
def IsSmoothIsometricImmersionline 26in the statementlibrary
theorem FiniteOrderSmoothing.smooth_isometric_immersionline 40selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen Bundle Manifoldopen scoped ContDiffnamespace ClosedSurfaceR4abbrev Plane := EuclideanSpace ℝ (Fin 2)abbrev Space := EuclideanSpace ℝ (Fin 4)abbrev planeModel := 𝓘(ℝ, Plane)abbrev spaceModel := 𝓘(ℝ, Space)sectionvariable (M : Type*) [TopologicalSpace M] [T2Space M]  [SecondCountableTopology M] [CompactSpace M]  [ChartedSpace Plane M] [IsManifold planeModel ∞ M]abbrev SmoothMetric :=  Bundle.ContMDiffRiemannianMetric planeModel ∞ Plane    (fun p : M => TangentSpace planeModel p)def IsSmoothIsometricImmersion (g : SmoothMetric M) (F : M → Space) : Prop :=  ContMDiff planeModel spaceModel ∞ F ∧  ∀ (p : M) (v w : TangentSpace planeModel p),    inner ℝ (mfderiv planeModel spaceModel F p v)      (mfderiv planeModel spaceModel F p w) = g.inner p v wendnamespace FiniteOrderSmoothingvariable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M]  [IsManifold planeModel ∞ M] [CompactSpace M] [T2Space M] [SecondCountableTopology M]/-- Every compact smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space. -/theorem smooth_isometric_immersion (g : SmoothMetric M) :    ∃ F : M → Space, IsSmoothIsometricImmersion M g F := by  sorryend FiniteOrderSmoothingend ClosedSurfaceR4endend 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.

Smooth isometric immersions of closed surfaces into Euclidean four-space

Theorem

Every closed smooth Riemannian surface (M,g) admits a C^\infty isometric immersion into (\R^4,\delta_4), where \delta_4=\sum_{i=1}^4 dx_i^2.

Section 1, Introduction Read it in the paper TeX source, line 14

Show the TeX

Every closed smooth Riemannian surface $(M,g)$ admits a $C^\infty$ isometric
immersion into $(\R^4,\delta_4)$, where $\delta_4=\sum_{i=1}^4 dx_i^2$.

Configuration and file facts

File: lean/ComparatorChallenges/SurfaceImmersion.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 333 link it.
Solution module: OAI.Geometry.SurfaceImmersion.Main
Allowed axioms: 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: 6 local vocabulary declarations
Lines of Lean, without comments or blank lines: 18
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas