Lean statement

Planar graph metrics in $L_1$

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.PlanarL1.planar_graph_metrics_embed_L1

Lean statement file: PlanarL1.leanShow on the mapFamily 089. OpenAI lists this as a main result in its Lean catalogue.

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 MainStatement : Prop :=  ∃ C : ℝ, 1 ≤ C ∧    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (length : Fin n → Fin n → ℝ),      G.Connected → IsPlanar G →      (∀ u v, length u v = length v u) →      (∀ u v, G.Adj u v → 0 < length u v) →      ∃ (Ω : Type) (measurable : MeasurableSpace Ω)          (μ : @MeasureTheory.Measure Ω measurable)          (f : Fin n → @MeasureTheory.Lp Ω ℝ measurable _ 1 μ),        ∀ x y, graphDistance G length x y ≤ ‖f x - f y‖ ∧          ‖f x - f y‖ ≤ C * graphDistance G length x y⋮theorem planar_graph_metrics_embed_L1 : MainStatement := 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.

definition OAI.PlanarL1.IsPlanar line 9

def IsPlanar {V : Type} (G : SimpleGraph V) : Prop :=

Show the remaining 10 lines

  ∃ (point : V → ℝ × ℝ) (arc : ∀ u v : V, G.Adj u v → C(unitInterval, ℝ × ℝ)),    Function.Injective point ∧    (∀ u v h, (arc u v h) 0 = point u ∧ (arc u v h) 1 = point v) ∧    (∀ u v h, Function.Injective (arc u v h)) ∧    (∀ u v h (t : unitInterval) w, 0 < (t : ℝ) → (t : ℝ) < 1 →      (arc u v h) t ≠ point w) ∧    (∀ u v h x y k (s t : unitInterval),      0 < (s : ℝ) → (s : ℝ) < 1 → 0 < (t : ℝ) → (t : ℝ) < 1 →      (arc u v h) s = (arc x y k) t →      (u = x ∧ v = y) ∨ (u = y ∧ v = x))

definition OAI.PlanarL1.walkLength line 22

def walkLength {V : Type} {G : SimpleGraph V} (length : V → V → ℝ)    {u v : V} : G.Walk u v → ℝ  | .nil => 0  | @SimpleGraph.Walk.cons _ _ u w _ _ p => length u w + walkLength length p

definition OAI.PlanarL1.graphDistance line 28

noncomputable def graphDistance {V : Type} (G : SimpleGraph V)    (length : V → V → ℝ) (u v : V) : ℝ :=  sInf (Set.range (fun p : G.Walk u v => walkLength length p))

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: universal, planar, real, positive, connected, measure, edge, space and three more. Found in the paper’s title, which its main theorem states: embed, distortion and metrics. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.

Statements in Planar Graph Metrics Embed into L1 with Constant Distortion come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Planar Graph Metrics Embed into L1 with Constant Distortion

Theorem

There is a finite universal constant C\ge1 with the following property. For every finite connected planar graph G=(V,E) with strictly positive real edge lengths, there exist a measure space (\Omega,\mathcal F,\mu) and a map f:V\to L_1(\Omega,\mathcal F,\mu;\R) such that

d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y) \qquad(x,y\in V).

Section 1, Introduction Read it in the paper TeX source, line 13

Show the TeX

There is a finite universal constant $C\ge1$ with the following property.
For every finite connected planar graph $G=(V,E)$ with strictly positive real
edge lengths, there exist a measure space $(\Omega,\mathcal F,\mu)$ and a map
$f:V\to L_1(\Omega,\mathcal F,\mu;\R)$ such that
\[
 d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y)
 \qquad(x,y\in V).
\]

Other possible paper matches, closest wording first

Proposition 2.3 of Planar Graph Metrics Embed into L1 with Constant Distortion. Words of the Lean statement, its names split into words, that also appear in the theorem: planar, realization, distances, positive, connected, edge and four more.
Corollary 10.1 of Planar Graph Metrics Embed into L1 with Constant Distortion. Words of the Lean statement, its names split into words, that also appear in the theorem: simple, distortion, real, positive, integers, connected and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Theorem 1.1 of L1 Embeddings of Graphs of Bounded Treewidth. Words of the Lean statement, its names split into words, that also appear in the theorem: vertices, real, integer, function, connected, edge and four more. Found in the paper’s title, which its main theorem states: bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ratios, number, independent and constant.
Proposition 9.1 of Planar Graph Metrics Embed into L1 with Constant Distortion. Words of the Lean statement, its names split into words, that also appear in the theorem: vertex, distance, vertices, point, measure, one and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constants.
Corollary 4.6 of Planar Graph Metrics Embed into L1 with Constant Distortion. Words of the Lean statement, its names split into words, that also appear in the theorem: arbitrary, intervals, functions, point, metrics, one and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: paths, independent and constants.
Theorem 4.1 of Planar Graph Metrics Embed into L1 with Constant Distortion. Words of the Lean statement, its names split into words, that also appear in the theorem: realization, function, points, one, spaced and length. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: including, paths and independent.

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.

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

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 089’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 disagree 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: Both L1-embedding theorems are stated; the flow-cut-gap corollary is not. Planarity is a hand-rolled topological drawing predicate.
Definitions the group asks readers to compare with the paper: IsPlanar is a custom 'drawing by simple plane arcs' predicate (injective points, arc u v h : C(unitInterval, ℝ × ℝ) per oriented edge, interiors of arcs of distinct unoriented edges disjoint); HasTreeDecomposition and graphDistance := sInf (range walkLength) are also hand-rolled; both read as standard
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: Only the flow–cut-gap sentence, a classical corollary of the stated embeddings (LLR 1995 / Aumann–Rabani 1998); no flow or cut object appears in either challenge.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result gives one constant $C$ for every finite connected planar graph with arbitrary positive real edge lengths: its weighted shortest-path metric embeds into real $L_1$ with distortion at most $C$. The constant is independent of the number of vertices and the ratios between edge lengths; singleton graphs are included.”
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

51 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: 6 to its own declarations, 19 to Mathlib’s documentation. 4 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 planar_graph_metrics_embed_L1 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 4 other declarations of the file; with the theorem itself they span 31 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.PlanarL1 are listed without it.

def IsPlanarline 9library
def walkLengthline 22library
def graphDistanceline 28library
def MainStatementline 34in the statementlibrary
theorem planar_graph_metrics_embed_L1line 46selected theorem

import Mathlibnamespace OAInamespace PlanarL1/-- A topological drawing by simple plane arcs, with injective vertex placement,no vertex in an edge interior, and disjoint interiors for distinct unoriented edges. -/def IsPlanar {V : Type} (G : SimpleGraph V) : Prop :=  ∃ (point : V → ℝ × ℝ) (arc : ∀ u v : V, G.Adj u v → C(unitInterval, ℝ × ℝ)),    Function.Injective point ∧    (∀ u v h, (arc u v h) 0 = point u ∧ (arc u v h) 1 = point v) ∧    (∀ u v h, Function.Injective (arc u v h)) ∧    (∀ u v h (t : unitInterval) w, 0 < (t : ℝ) → (t : ℝ) < 1 →      (arc u v h) t ≠ point w) ∧    (∀ u v h x y k (s t : unitInterval),      0 < (s : ℝ) → (s : ℝ) < 1 → 0 < (t : ℝ) → (t : ℝ) < 1 →      (arc u v h) s = (arc x y k) t →      (u = x ∧ v = y) ∨ (u = y ∧ v = x))/-- The sum of edge lengths along a finite graph walk. -/def walkLength {V : Type} {G : SimpleGraph V} (length : V → V → ℝ)    {u v : V} : G.Walk u v → ℝ  | .nil => 0  | @SimpleGraph.Walk.cons _ _ u w _ _ p => length u w + walkLength length p/-- The infimum of lengths of actual walks between the two vertices. -/noncomputable def graphDistance {V : Type} (G : SimpleGraph V)    (length : V → V → ℝ) (u v : V) : ℝ :=  sInf (Set.range (fun p : G.Walk u v => walkLength length p))/-- One universal distortion bound for finite connected planar graphs witharbitrary symmetric positive edge lengths, realized in a real `L¹` space. -/def MainStatement : Prop :=  ∃ C : ℝ, 1 ≤ C ∧    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (length : Fin n → Fin n → ℝ),      G.Connected → IsPlanar G →      (∀ u v, length u v = length v u) →      (∀ u v, G.Adj u v → 0 < length u v) →      ∃ (Ω : Type) (measurable : MeasurableSpace Ω)          (μ : @MeasureTheory.Measure Ω measurable)          (f : Fin n → @MeasureTheory.Lp Ω ℝ measurable _ 1 μ),        ∀ x y, graphDistance G length x y ≤ ‖f x - f y‖ ∧          ‖f x - f y‖ ≤ C * graphDistance G length x ytheorem planar_graph_metrics_embed_L1 : MainStatement := by  sorryend PlanarL1end 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.

Planar Graph Metrics Embed into L1 with Constant Distortion

Theorem

d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y) \qquad(x,y\in V).

Section 1, Introduction Read it in the paper TeX source, line 13

Show the TeX

There is a finite universal constant $C\ge1$ with the following property.
For every finite connected planar graph $G=(V,E)$ with strictly positive real
edge lengths, there exist a measure space $(\Omega,\mathcal F,\mu)$ and a map
$f:V\to L_1(\Omega,\mathcal F,\mu;\R)$ such that
\[
 d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y)
 \qquad(x,y\in V).
\]

L1 Embeddings of Graphs of Bounded Treewidth

Theorem

For every integer k\ge2 there is a finite constant C_k\ge1 with the following property. If a finite connected graph G=(V,E) admits a tree decomposition with bags of size at most k, then, for every edge-length function \ell:E\to(0,\infty), there is a map F from V into a real L_1 space such that

d_{G,\ell}(u,v) \le \|F(u)-F(v)\|_1 \le C_k\,d_{G,\ell}(u,v) \qquad(u,v\in V).

The constant is independent of the number of vertices, the depth of the decomposition, and all ratios of edge lengths.

Section 1, Introduction Read it in the paper TeX source, line 27

Show the TeX

For every integer $k\ge2$ there is a finite constant $C_k\ge1$ with the
following property. If a finite connected graph $G=(V,E)$ admits a tree
decomposition with bags of size at most $k$, then, for every edge-length
function $\ell:E\to(0,\infty)$, there is a map $F$ from $V$ into a real
$L_1$ space such that
\begin{equation}
 d_{G,\ell}(u,v)
 \le \|F(u)-F(v)\|_1
 \le C_k\,d_{G,\ell}(u,v)
 \qquad(u,v\in V).
\end{equation}
The constant is independent of the number of vertices, the depth of the
decomposition, and all ratios of edge lengths.

Configuration and file facts

File: lean/ComparatorChallenges/PlanarL1.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/PlanarL1/Embedding.lean
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: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 31
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.

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