Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
The paper states
In Smooth isometric immersions of closed surfaces into Euclidean four-space
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.
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.
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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.
SurfaceImmersionThe paper states
In Smooth isometric immersions of closed surfaces into Euclidean four-space
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.
The words they share: isometric, admits, riemannian, surface, smooth, immersion, space, euclidean, four, closed.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SurfaceImmersion.lean, the release lists it as a supporting statement.
theorem smooth_isometric_immersion (g : SmoothMetric M) : ∃ F : M → Space, IsSmoothIsometricImmersion M g F := by sorryThis 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.
One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program
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: 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.
“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.
cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean 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.
Smooth isometric immersion of every closed surface into four-space SurfaceImmersion
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every closed smooth Riemannian surface admits a smooth isometric immersion into R⁴.
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.
SurfaceImmersionOne report, 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. | 10 October 2026 | Original report |
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No paper elsewhere in the release cites this family’s papers.
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
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Why this passage is shown
Theorem 1.1
In Smooth isometric immersions of closed surfaces into Euclidean four-space
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, SurfaceImmersion
Lean statement
SurfaceImmersion