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.ScalarConductivity.main_nonuniqueness
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
defMainStatement : Prop := ∃ (γ₀ γ₁ : R3 → ℝ) (c C : ℝ) (Λ : DNOperator),0 < c ∧ c < C ∧ Measurable γ₀ ∧ Measurable γ₁ ∧ MemLp γ₀ ∞ ballMeasure ∧ MemLp γ₁ ∞ ballMeasure ∧ (∀ᵐ x ∂ballMeasure, c ≤ γ₀ x ∧ γ₀ x ≤ C) ∧ (∀ᵐ x ∂ballMeasure, c ≤ γ₁ x ∧ γ₁ x ≤ C) ∧ EqualOneNearBoundary γ₀ ∧ EqualOneNearBoundary γ₁ ∧0 < ballMeasure {x | γ₀ x ≠ γ₁ x} ∧ IsDirichletToNeumann γ₀ Λ ∧ IsDirichletToNeumann γ₁ Λ⋮theoremmain_nonuniqueness : MainStatement := bysorry
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.
defEqualOneNearBoundary (γ : R3 → ℝ) : Prop := ∃ r : ℝ, 0 < r ∧ r < 3 ∧ ∀ᵐ x ∂ballMeasure, r < ‖x‖ → γ x = 1
Possible paper match, with no recorded review
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: bounded, conductivities, differ, dirichlet, full, main, neumann, nonuniqueness and three more.
Statements in Nonuniqueness for bounded measurable scalar conductivities in three dimensions come first because the release lists only that paper for this Lean statement.
Corollary 1.2 of Nonuniqueness for bounded measurable scalar conductivities in three dimensions
Corollary(Localized nonuniqueness)
Let c,C be the constants in Theorem thm:main. For every bounded connected Lipschitz domain \Omega\subset\R^3 and every finite family of pairwise disjoint balls B_1,\ldots,B_m\Subset\Omega, m\geq1, there are 2^m pairwise distinct real scalar conductivities \gamma_\varepsilon\in L^\infty(\Omega), indexed by \varepsilon\in\{0,1\}^m, such that
and all their full weak Dirichlet-to-Neumann operators from H^{1/2}(\partial\Omega) to H^{-1/2}(\partial\Omega) agree. Distinct conductivities differ on a set of positive measure.
Let $c,C$ be the constants in Theorem~\ref{thm:main}.
For every bounded connected Lipschitz domain $\Omega\subset\R^3$
and every finite family of pairwise disjoint balls
$B_1,\ldots,B_m\Subset\Omega$, $m\geq1$, there are $2^m$
pairwise distinct real scalar conductivities
$\gamma_\varepsilon\in L^\infty(\Omega)$, indexed by
$\varepsilon\in\{0,1\}^m$, such that
\[
c\leq\gamma_\varepsilon\leq C\quad\text{almost everywhere},
\qquad
\gamma_\varepsilon=1\quad\text{on }
\Omega\setminus\bigcup_{i=1}^m B_i,
\]
and all their full weak Dirichlet-to-Neumann operators from
$H^{1/2}(\partial\Omega)$ to $H^{-1/2}(\partial\Omega)$ agree.
Distinct conductivities differ on a set of positive measure.
Other possible paper matches, closest wording first
Proposition 5.6 of Nonuniqueness for bounded measurable scalar conductivities in three dimensions. Words of the Lean statement, its names split into words, that also appear in the theorem: neumann, equal, weak, operators and dirichlet. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: full.
Theorem 1.1 of Nonuniqueness for bounded measurable scalar conductivities in three dimensions. Words of the Lean statement, its names split into words, that also appear in the theorem: equal, real, operators and boundary. Found in the paper’s title, which its main theorem states: nonuniqueness and measurable. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: conductivities, scalar and bounded.
Theorem 3.1 of Nonuniqueness for bounded measurable scalar conductivities in three dimensions. Words of the Lean statement, its names split into words, that also appear in the theorem: gradients, measurable, real, one, zero and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: full, scalar, two and bounded.
Proposition 4.1 of Nonuniqueness for bounded measurable scalar conductivities in three dimensions. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, weak, measure, boundary, harmonic, zero and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: conductivity, scalar and bounded.
Theorem 3.1 of Smooth anisotropic uniqueness in the Calderón problem from one boundary patch. Words of the Lean statement, its names split into words, that also appear in the theorem: span, real, near, one, compact, dirichlet and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two.
Proposition 3.7 of Nonuniqueness for bounded measurable scalar conductivities in three dimensions. Words of the Lean statement, its names split into words, that also appear in the theorem: volume, weak and near. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: scalar.
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
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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.
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 365’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.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: main_nonuniqueness covers only the contrast claim: on the ball B(0,3) ⊂ ℝ³ there are two bounded measurable conductivities in [c, C], both equal to 1 near the boundary, differing on a positive-measure set, with one and the same weak Dirichlet-to-Neumann operator Λ (IsDirichletToNeumann: for every trace a unique harmonic H¹ extension with Λ f (trace v) = ∫γ∇u·∇v). The summary's first-named result, recovery of a smooth metric and unitary connection on a rank-two Hermitian bundle over manifolds of dimension ≥ 3 from zero-frequency data on one boundary patch (up to diffeomorphism and gauge), has no statement; the docs scope also only describes the conductivity result.
Definitions the group asks readers to compare with the paper: H¹(B) is hand-built as the closure of smooth jets (f, ∇f) in L²(B; ℝ⁴), H¹₀ as the closure of compactly supported jets, TraceSpace := H1 ⧸ H10, and the DN operator is a continuous bilinear map TraceSpace → TraceSpace*; EqualOneNearBoundary is only a.e. in r < ‖x‖ < 3
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The headline's main theorem — joint recovery of a smooth metric and unitary connection from one boundary patch (and the metric-only same-patch Calderón uniqueness) — is not formalized; only the bounded-measurable scalar nonuniqueness example is.
OpenAI’s own note on what this Lean statement covers
“The formalized result gives nonuniqueness for bounded measurable scalar conductivities on the ball $B(0,3)\subset\mathbb R^3$: two uniformly positive conductivities differ on a set of positive volume, equal $1$ near the boundary, and have the same full weak Dirichlet-to-Neumann operator. Weak solutions exist uniquely for every boundary trace.”
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
83 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: 77 to its own declarations, 36 to Mathlib’s documentation. 24 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.
There exist real scalar functions $\gamma_0,\gamma_1\in
L^\infty(B(0,3);\R)$ and constants $0<c<C<\infty$ such that
\begin{enumerate}[label=\textup{(\roman*)},itemsep=2pt]
\item $c\leq\gamma_j\leq C$ almost everywhere, for $j=0,1$;
\item both conductivities equal $1$ in a neighborhood of the boundary;
\item $\abs{\{x:\gamma_0(x)\ne\gamma_1(x)\}}>0$;
\item $\Lambda_{\gamma_0}=\Lambda_{\gamma_1}$ as bounded operators
from $H^{1/2}(\partial B(0,3))$ to $H^{-1/2}(\partial B(0,3))$.
\end{enumerate}
Corollary(Localized nonuniqueness)
Let c,C be the constants in Theorem thm:main. For every bounded connected Lipschitz domain \Omega\subset\R^3 and every finite family of pairwise disjoint balls B_1,\ldots,B_m\Subset\Omega, m\geq1, there are 2^m pairwise distinct real scalar conductivities \gamma_\varepsilon\in L^\infty(\Omega), indexed by \varepsilon\in\{0,1\}^m, such that
and all their full weak Dirichlet-to-Neumann operators from H^{1/2}(\partial\Omega) to H^{-1/2}(\partial\Omega) agree. Distinct conductivities differ on a set of positive measure.
Let $c,C$ be the constants in Theorem~\ref{thm:main}.
For every bounded connected Lipschitz domain $\Omega\subset\R^3$
and every finite family of pairwise disjoint balls
$B_1,\ldots,B_m\Subset\Omega$, $m\geq1$, there are $2^m$
pairwise distinct real scalar conductivities
$\gamma_\varepsilon\in L^\infty(\Omega)$, indexed by
$\varepsilon\in\{0,1\}^m$, such that
\[
c\leq\gamma_\varepsilon\leq C\quad\text{almost everywhere},
\qquad
\gamma_\varepsilon=1\quad\text{on }
\Omega\setminus\bigcup_{i=1}^m B_i,
\]
and all their full weak Dirichlet-to-Neumann operators from
$H^{1/2}(\partial\Omega)$ to $H^{-1/2}(\partial\Omega)$ agree.
Distinct conductivities differ on a set of positive measure.
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
52
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
Scalar conductivity nonuniqueness
Family 365: Joint metric and connection recovery from one boundary patch
Candidate pairing
Corollary (Localized nonuniqueness)
Let c,C be the constants in Theorem thm:main. For every bounded connected Lipschitz domain \Omega\subset\R^3 and every finite family of pairwise disjoint balls B_1,\ldots,B_m\Subset\Omega, m\geq1, there are 2^m pairwise distinct real scalar conductivities \gamma_\varepsilon\in L^\infty(\Omega), indexed by \varepsilon\in\{0,1\}^m, such that
and all their full weak Dirichlet-to-Neumann operators from H^{1/2}(\partial\Omega) to H^{-1/2}(\partial\Omega) agree. Distinct conductivities differ on a set of positive measure.
Nonuniqueness for bounded measurable scalar conductivities in three dimensions
TeX label cor:localized, build/paper.tex, line 98
The Lean statement says
defMainStatement : Prop := ∃ (γ₀ γ₁ : R3 → ℝ) (c C : ℝ) (Λ : DNOperator),0 < c ∧ c < C ∧ Measurable γ₀ ∧ Measurable γ₁ ∧ MemLp γ₀ ∞ ballMeasure ∧ MemLp γ₁ ∞ ballMeasure ∧ (∀ᵐ x ∂ballMeasure, c ≤ γ₀ x ∧ γ₀ x ≤ C) ∧ (∀ᵐ x ∂ballMeasure, c ≤ γ₁ x ∧ γ₁ x ≤ C) ∧ EqualOneNearBoundary γ₀ ∧ EqualOneNearBoundary γ₁ ∧0 < ballMeasure {x | γ₀ x ≠ γ₁ x} ∧ IsDirichletToNeumann γ₀ Λ ∧ IsDirichletToNeumann γ₁ Λ⋮theoremmain_nonuniqueness : MainStatement := bysorry
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
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.