Lean statement

Global uniqueness of smooth isotropic Lamé moduli

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.Elasticity.global_uniqueness

Lean statement file: ElasticityUniqueness.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 372 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.

def MainClaim : Prop :=  ∀ (Ω : Set X), Domain Ω →  ∀ (lam₁ mu₁ lam₂ mu₂ : X → ℝ),    Admissible Ω lam₁ mu₁ → Admissible Ω lam₂ mu₂ →    DN Ω lam₁ mu₁ = DN Ω lam₂ mu₂ →    (∀ x ∈ Ω, lam₁ x = lam₂ x) ∧ (∀ x ∈ Ω, mu₁ x = mu₂ x)⋮theorem global_uniqueness : MainClaim := 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.Elasticity.X line 19
```
abbrev X := EuclideanSpace ℝ (Fin 3)
```
abbreviation OAI.Elasticity.V line 20
```
abbrev V := X
```

definition OAI.Elasticity.coordVector line 22

def coordVector (i : Fin 3) : X := EuclideanSpace.single i 1

definition OAI.Elasticity.SmoothBoundary line 24

def SmoothBoundary (Ω : Set X) : Prop :=  ∀ x ∈ frontier Ω, ∃ e : OpenPartialHomeomorph X X,    x ∈ e.source ∧    ContDiffOn ℝ (⊤ : ℕ∞) e e.source ∧    ContDiffOn ℝ (⊤ : ℕ∞) e.symm e.target ∧    e x 0 = 0 ∧    ∀ y ∈ e.source, y ∈ Ω ↔ 0 < e y 0

definition OAI.Elasticity.Domain line 32

def Domain (Ω : Set X) : Prop :=  IsOpen Ω ∧ IsConnected Ω ∧ Bornology.IsBounded Ω ∧ SmoothBoundary Ω

definition OAI.Elasticity.SmoothUpTo line 35

def SmoothUpTo (Ω : Set X) (f : X → ℝ) : Prop :=  ∃ U : Set X, IsOpen U ∧ closure Ω ⊆ U ∧ ContDiffOn ℝ (⊤ : ℕ∞) f U

definition OAI.Elasticity.Admissible line 38

def Admissible (Ω : Set X) (lam mu : X → ℝ) : Prop :=  SmoothUpTo Ω lam ∧ SmoothUpTo Ω mu ∧    ∀ x ∈ closure Ω, 0 < mu x ∧ 0 < 3 * lam x + 2 * mu x

abbreviation OAI.Elasticity.Ambient line 42

abbrev Ambient (Ω : Set X) :=  Lp V 2 (volume.restrict Ω) × (Fin 3 → Lp V 2 (volume.restrict Ω))

definition OAI.Elasticity.coordDeriv line 45

def coordDeriv (f : X → V) (i : Fin 3) (x : X) : V :=  fderiv ℝ f x (coordVector i)

definition OAI.Elasticity.SmoothJets line 48

def SmoothJets (Ω : Set X) : Set (Ambient Ω) :=  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),    ContDiff ℝ (⊤ : ℕ∞) f ∧    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}

definition OAI.Elasticity.TestJets line 54

def TestJets (Ω : Set X) : Set (Ambient Ω) :=  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),    ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ tsupport f ⊆ Ω ∧    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}

abbreviation OAI.Elasticity.H1 line 60

abbrev H1 (Ω : Set X) := {j : Ambient Ω // j ∈ closure (SmoothJets Ω)}

definition OAI.Elasticity.HasZeroTrace line 62

def HasZeroTrace (Ω : Set X) (u : H1 Ω) : Prop :=  u.val ∈ closure (TestJets Ω)

definition OAI.Elasticity.SameTrace line 65

def SameTrace (Ω : Set X) (u v : H1 Ω) : Prop :=  u.val - v.val ∈ closure (TestJets Ω)

abbreviation OAI.Elasticity.BoundaryData line 68

abbrev BoundaryData (Ω : Set X) := Quot (SameTrace Ω)

definition OAI.Elasticity.trace line 70

def trace (Ω : Set X) (u : H1 Ω) : BoundaryData Ω := Quot.mk _ u

definition OAI.Elasticity.div line 72

def div (Ω : Set X) (u : H1 Ω) (x : X) : ℝ :=  ∑ i, (u.val.2 i x) i

definition OAI.Elasticity.strain line 75

def strain (Ω : Set X) (u : H1 Ω) (x : X) (i j : Fin 3) : ℝ :=  ((u.val.2 j x) i + (u.val.2 i x) j) / 2

definition OAI.Elasticity.energy line 78

def energy (Ω : Set X) (lam mu : X → ℝ) (u v : H1 Ω) : ℝ :=  ∫ x, (lam x * div Ω u x * div Ω v x +    2 * mu x * ∑ i, ∑ j, strain Ω u x i j * strain Ω v x i j)    ∂(volume.restrict Ω)

definition OAI.Elasticity.WeakSolution line 83

def WeakSolution (Ω : Set X) (lam mu : X → ℝ) (u : H1 Ω) : Prop :=  ∀ v : H1 Ω, HasZeroTrace Ω v → energy Ω lam mu u v = 0

definition OAI.Elasticity.dirichletSolution line 86

def dirichletSolution (Ω : Set X) (lam mu : X → ℝ)    (f : BoundaryData Ω) : H1 Ω := by  classical  exact if h : ∃ u : H1 Ω, trace Ω u = f ∧ WeakSolution Ω lam mu u  then Classical.choose h  else f.out

definition OAI.Elasticity.DN line 93

def DN (Ω : Set X) (lam mu : X → ℝ) : BoundaryData Ω → BoundaryData Ω → ℝ :=  fun f g => energy Ω lam mu (dirichletSolution Ω lam mu f) g.out

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: mu, connected, partial, domain, boundary and bounded. Found in the paper’s title, which its main theorem states: global, uniqueness and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: throughout and lambda.

Theorem 1.1 of Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Theorem

Let \Omega\subset\R^3 be any bounded connected domain with C^\infty boundary. For j=1,2, let \lambda_j,\mu_j\in C^\infty(\overline\Omega;\R) satisfy \mu_j>0 and 3\lambda_j+2\mu_j>0 on \overline\Omega. If

\Lambda_{\lambda_1,\mu_1} =\Lambda_{\lambda_2,\mu_2} \colon H^{1/2}(\partial\Omega;\R^3) \longrightarrow H^{-1/2}(\partial\Omega;\R^3),

then \lambda_1=\lambda_2 and \mu_1=\mu_2 throughout \Omega.

Section 1, Introduction Read it in the paper TeX source, line 31

Show the TeX

Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.

Other possible paper matches, closest wording first

Proposition 5.3 of Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem. Words of the Lean statement, its names split into words, that also appear in the theorem: compact, solution, support and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two, maps, displacement, traction and pairs.
Proposition 4.2 of Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem. Words of the Lean statement, its names split into words, that also appear in the theorem: vector, solutions and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: three.
Proposition 2.2 of Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem. Words of the Lean statement, its names split into words, that also appear in the theorem: solution, supported and smooth.
Proposition 6.3 of Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem. Words of the Lean statement, its names split into words, that also appear in the theorem: vector.
Proposition 3.2 of Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem. Words of the Lean statement, its names split into words, that also appear in the theorem: bounded and supported.

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.

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

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 597 s; peak memory 6.9 GiB).
Reported 11 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 372’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: MainClaim: for every open connected bounded smooth domain Ω ⊂ ℝ³ and two pairs of Lamé moduli smooth in a neighbourhood of the closure with μ > 0 and 3λ+2μ > 0 on the closure, equality of the full displacement-to-traction pairing DN as a function on all pairs of boundary data forces λ₁ = λ₂ and μ₁ = μ₂ on Ω; no analyticity, near-constant or near-boundary assumption. The Classical.choose in dirichletSolution picks the weak Dirichlet solution but only enters DN through energy against it, which is well defined because that solution is a weak solution (unique under the stated positivity); the fallback f.out is used only when no solution exists.
Definitions the group asks readers to compare with the paper: dirichletSolution Ω lam mu f := if h : ∃ u, trace u = f ∧ WeakSolution u then Classical.choose h else f.out and DN f g := energy (dirichletSolution f) g.out (Quot.out representatives of the trace classes); H¹ and traces hand-built as closures of smooth jets in L² and Quot (SameTrace), strain and divergence from jets, energy = ∫ λ div u div v + 2μ ε(u):ε(v)
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 global uniqueness in the three-dimensional static isotropic elasticity inverse problem. On a bounded connected smooth domain, let two pairs of smooth real Lamé moduli satisfy $\mu>0$ and $3\lambda+2\mu>0$ on the closure. If their full displacement-to-traction maps agree, then both Lamé moduli agree throughout the domain. The selected statement is uniqueness; it does not supply a reconstruction algorithm.”
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

110 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: 88 to its own declarations, 62 to Mathlib’s documentation. 24 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. 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 global_uniqueness 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 23 other declarations of the file; with the theorem itself they span 64 lines. 1 of them is named in the statement itself.

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

abbrev Xline 19library
abbrev Vline 20library
def coordVectorline 22library
def SmoothBoundaryline 24library
def Domainline 32library
def SmoothUpToline 35library
def Admissibleline 38library
abbrev Ambientline 42library
def coordDerivline 45library
def SmoothJetsline 48library
def TestJetsline 54library
abbrev H1line 60library
def HasZeroTraceline 62library
def SameTraceline 65library
abbrev BoundaryDataline 68library
def traceline 70library
def divline 72library
def strainline 75library
def energyline 78library
def WeakSolutionline 83library
def dirichletSolutionline 86library
def DNline 93library
def MainClaimline 96in the statementlibrary
theorem global_uniquenessline 103selected theoremlibrary

import Mathlibnamespace OAI/-!# Global uniqueness for smooth isotropic elasticityThe physical Dirichlet-to-Neumann pairing acts on the variational tracequotient. Its quotient norm is not identified here with boundary-chartfractional Sobolev norms.-/noncomputable sectionopen MeasureTheory Setopen scoped BigOperatorsnamespace Elasticityabbrev X := EuclideanSpace ℝ (Fin 3)abbrev V := Xdef coordVector (i : Fin 3) : X := EuclideanSpace.single i 1def SmoothBoundary (Ω : Set X) : Prop :=  ∀ x ∈ frontier Ω, ∃ e : OpenPartialHomeomorph X X,    x ∈ e.source ∧    ContDiffOn ℝ (⊤ : ℕ∞) e e.source ∧    ContDiffOn ℝ (⊤ : ℕ∞) e.symm e.target ∧    e x 0 = 0 ∧    ∀ y ∈ e.source, y ∈ Ω ↔ 0 < e y 0def Domain (Ω : Set X) : Prop :=  IsOpen Ω ∧ IsConnected Ω ∧ Bornology.IsBounded Ω ∧ SmoothBoundary Ωdef SmoothUpTo (Ω : Set X) (f : X → ℝ) : Prop :=  ∃ U : Set X, IsOpen U ∧ closure Ω ⊆ U ∧ ContDiffOn ℝ (⊤ : ℕ∞) f Udef Admissible (Ω : Set X) (lam mu : X → ℝ) : Prop :=  SmoothUpTo Ω lam ∧ SmoothUpTo Ω mu ∧    ∀ x ∈ closure Ω, 0 < mu x ∧ 0 < 3 * lam x + 2 * mu xabbrev Ambient (Ω : Set X) :=  Lp V 2 (volume.restrict Ω) × (Fin 3 → Lp V 2 (volume.restrict Ω))def coordDeriv (f : X → V) (i : Fin 3) (x : X) : V :=  fderiv ℝ f x (coordVector i)def SmoothJets (Ω : Set X) : Set (Ambient Ω) :=  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),    ContDiff ℝ (⊤ : ℕ∞) f ∧    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}def TestJets (Ω : Set X) : Set (Ambient Ω) :=  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),    ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ tsupport f ⊆ Ω ∧    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}abbrev H1 (Ω : Set X) := {j : Ambient Ω // j ∈ closure (SmoothJets Ω)}def HasZeroTrace (Ω : Set X) (u : H1 Ω) : Prop :=  u.val ∈ closure (TestJets Ω)def SameTrace (Ω : Set X) (u v : H1 Ω) : Prop :=  u.val - v.val ∈ closure (TestJets Ω)abbrev BoundaryData (Ω : Set X) := Quot (SameTrace Ω)def trace (Ω : Set X) (u : H1 Ω) : BoundaryData Ω := Quot.mk _ udef div (Ω : Set X) (u : H1 Ω) (x : X) : ℝ :=  ∑ i, (u.val.2 i x) idef strain (Ω : Set X) (u : H1 Ω) (x : X) (i j : Fin 3) : ℝ :=  ((u.val.2 j x) i + (u.val.2 i x) j) / 2def energy (Ω : Set X) (lam mu : X → ℝ) (u v : H1 Ω) : ℝ :=  ∫ x, (lam x * div Ω u x * div Ω v x +    2 * mu x * ∑ i, ∑ j, strain Ω u x i j * strain Ω v x i j)    ∂(volume.restrict Ω)def WeakSolution (Ω : Set X) (lam mu : X → ℝ) (u : H1 Ω) : Prop :=  ∀ v : H1 Ω, HasZeroTrace Ω v → energy Ω lam mu u v = 0def dirichletSolution (Ω : Set X) (lam mu : X → ℝ)    (f : BoundaryData Ω) : H1 Ω := by  classical  exact if h : ∃ u : H1 Ω, trace Ω u = f ∧ WeakSolution Ω lam mu u  then Classical.choose h  else f.outdef DN (Ω : Set X) (lam mu : X → ℝ) : BoundaryData Ω → BoundaryData Ω → ℝ :=  fun f g => energy Ω lam mu (dirichletSolution Ω lam mu f) g.outdef MainClaim : Prop :=  ∀ (Ω : Set X), Domain Ω →  ∀ (lam₁ mu₁ lam₂ mu₂ : X → ℝ),    Admissible Ω lam₁ mu₁ → Admissible Ω lam₂ mu₂ →    DN Ω lam₁ mu₁ = DN Ω lam₂ mu₂ →    (∀ x ∈ Ω, lam₁ x = lam₂ x) ∧ (∀ x ∈ Ω, mu₁ x = mu₂ x)theorem global_uniqueness : MainClaim := by  sorryend Elasticityendend 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.

Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Theorem

\Lambda_{\lambda_1,\mu_1} =\Lambda_{\lambda_2,\mu_2} \colon H^{1/2}(\partial\Omega;\R^3) \longrightarrow H^{-1/2}(\partial\Omega;\R^3),

then \lambda_1=\lambda_2 and \mu_1=\mu_2 throughout \Omega.

Section 1, Introduction Read it in the paper TeX source, line 31

Show the TeX

Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.

Configuration and file facts

File: lean/ComparatorChallenges/ElasticityUniqueness.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 372 link it.
Solution module: OAI.MathematicalPhysics.Elasticity.Uniqueness
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: 23 local vocabulary declarations
Lines of Lean, without comments or blank lines: 64
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