A smooth surface metric with no local isometric immersion in ℝ3
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs a smooth positive-definite metric on \((-1,1)^2\) for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above.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.
There is a C^\infty positive definite Riemannian metric g on (-1,1)^2 such that no open neighborhood U of the origin admits a C^\infty isometric immersion
F:(U,g)\longrightarrow\R^3.
In particular, no such neighborhood admits a smooth isometric embedding.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We construct a smooth positive-definite Riemannian metric on \((-1,1)^2\) that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.
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.
There is a C^\infty positive definite Riemannian metric g on (-1,1)^2 such that no open neighborhood U of the origin admits a C^\infty isometric immersion
F:(U,g)\longrightarrow\R^3.
In particular, no such neighborhood admits a smooth isometric embedding.
The words they share: immersion, positive, smooth, metric, space, local, admits, definite, origin, isometric, neighborhood.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
IsometricImmersion.lean, the release lists it as a main result.
theoremexists_local_metric_without_local_immersion : ∃ g : coordinateSquareOpen → Matrix (Fin 2) (Fin 2) ℝ, LocalSmoothPositive coordinateSquareOpen g ∧ ∀ U : Opens Coord, ∀ hUS : (U : Set Coord) ⊆ square, (0 : Coord) ∈ U → ∀ F : U → Ambient, ¬ (ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) ∞ F ∧ ∀ p : U, ∀ v w : Coord, (inner ℝ : Ambient → Ambient → ℝ) (mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p v) (mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p w) = v ⬝ᵥ (g ⟨p, hUS p.property⟩ *ᵥ w)) := 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.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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 or imply the whole advertised result (their term: “full”).
Their note:There is a smooth (C^∞, positive-definite) metric g on (-1,1)² such that for every open U ⊆ the square containing 0 and every F : U → ℝ³ it is false that F is C^∞ with ⟨dF v, dF w⟩ = vᵀ g w at every point of U, so no neighbourhood of the origin admits a smooth isometric immersion, as in the summary.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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.
The local isometric-immersion problem asks whether every smooth surface metric can be realized locally in Euclidean three-space. The formalized counterexample is a smooth positive-definite metric on $(-1,1)^2$ for which no neighborhood of the origin admits a smooth isometric immersion into $\mathbb R^3$. The obstruction applies to every smooth candidate map on every such neighborhood.
AI-drafted comparisons, with no recorded human review
Smooth local isometric-immersion obstructionIsometricImmersion
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a smooth metric on (−1, 1)² such that no neighbourhood of the origin admits a smooth isometric immersion into R³.
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.
Smooth local isometric-immersion obstructionIsometricImmersion
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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
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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.
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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 334The 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 334Atlas 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 A (the paper prints no number) (Theorem A)
In A Smooth Metric with No Local Isometric Immersion into Three-Space
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 the headline as OpenAI wrote it.
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.