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.StrongThinTree.strongThinTree
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
defMainStatement : Prop := ∃ C : ℝ, 0 < C ∧ ∀ (k : ℕ), 1 ≤ k → ∀ (n m : ℕ), 2 ≤ n → ∀ G : MultiGraph n m, G.EdgeConnected k → ∃ T : Finset (Fin m), G.SpanningTree T ∧ ∀ S : Finset (Fin n), S.Nonempty → S ≠ Finset.univ → ((G.cut T S).card : ℝ) ≤ (C / (k : ℝ)) * (G.cut Finset.univ S).card⋮theoremstrongThinTree : 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.
defcut {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) (S : Finset (Fin n)) : Finset (Fin m) := T.filter fun e => (G.left e ∈ S ∧ G.right e ∉ S) ∨ (G.right e ∈ S ∧ G.left e ∉ S)
defEdgeConnected {n m : ℕ} (G : MultiGraph n m) (k : ℕ) : Prop := ∀ S : Finset (Fin n), S.Nonempty → S ≠ Finset.univ → k ≤ (G.cut Finset.univ S).card
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: tree, connected, spanning, edge and integer. Found in the paper’s title, which its main theorem states: strong and thin. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: multigraph, constant, two, least, vertices and finite.
Statements in The strong thin tree conjecture come first because the release’s name for this Lean statement shares words with that paper’s title.
There is a universal constant C>0 such that, for every integer k\ge1, every finite k-edge-connected undirected multigraph without loops and with at least two vertices has a spanning tree T satisfying
|\delta_T(S)|\le\frac Ck\,|\delta_G(S)|
\qquad\text{for every }\varnothing\ne S\subsetneq V.
There is a universal constant \(C>0\) such that, for every integer
\(k\ge1\), every finite \(k\)-edge-connected undirected multigraph
without loops and with at least two vertices has a spanning tree \(T\)
satisfying
\[
|\delta_T(S)|\le\frac Ck\,|\delta_G(S)|
\qquad\text{for every }\varnothing\ne S\subsetneq V.
\]
Other possible paper matches, closest wording first
Corollary 7.1 of A polynomial-time construction of strong thin trees. Words of the Lean statement, its names split into words, that also appear in the theorem: cuts, thin, tree, graph, edge, spanning and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: parallel, algorithm, constant, vertices and finite.
Theorem 4.1 of The strong thin tree conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: tree, graph, spanning, edge and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: edges.
Proposition 6.2 of The strong thin tree conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: nonempty, spanning, edge and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two, least, vertices and absolute.
Theorem 3.2 of The strong thin tree conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: connected, graph and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound and absolute.
Proposition 3.1 of A polynomial-time construction of strong thin trees. Words of the Lean statement, its names split into words, that also appear in the theorem: spanning and edge. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one, multigraph, constants, two, least, vertices and one more.
Theorem 2.3 of The strong thin tree conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: graph, spanning, edge and integer.
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 174’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.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: StrongThinTree.MainStatement (∃ C > 0, ∀ k ≥ 1, ∀ n m, 2 ≤ n → ∀ G : MultiGraph n m, G.EdgeConnected k → ∃ T, G.SpanningTree T ∧ ∀ S, |cut T S| ≤ (C/k) * |cut univ S|) is the existence theorem; AlgorithmicStrongThinTrees adds ∃ M : CurrentKS.Machine, ∃ a degree, ... result.state = none ∧ GoodOutput C x ... for explicit and binary-multiplicity inputs. Both parts of the summary are stated; binary multiplicities are really binary (encodeNat n = frame (Computability.encodeNat n) and Mathlib's encodeNat is binary).
Definitions the group asks readers to compare with the paper: CurrentKS.Machine (hand-rolled multi-stack machine with a lookup table, Machine.run with a fuel argument) and the notion of polynomial time M.run x.encode (a * (x.length + 1) ^ degree) halting with state = none: a custom model, not Mathlib's TM2ComputableInPolyTime; multi-stack machines are polynomially equivalent to Turing machines so this looks faithful, but 'polynomial time' is a bespoke definition.
OpenAI’s own note on what this Lean statement covers
“The formalized result gives one absolute constant $C>0$ such that every finite loopless $k$-edge-connected multigraph with at least two vertices has a spanning tree crossing each nontrivial cut at most $(C/k)$ times the original cut size. Parallel edges remain distinct. No numerical value of $C$ or construction algorithm is asserted.”
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
43 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, 27 to Mathlib’s documentation. 6 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.
import Mathlibnamespace OAInamespace StrongThinTreestructureMultiGraph (n m : ℕ) whereleft : Fin m → Fin nright : Fin m → Fin nloopless : ∀ e, left e ≠ right enamespace MultiGraphdefcut {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) (S : Finset (Fin n)) : Finset (Fin m) := T.filter fun e => (G.left e ∈ S ∧ G.right e ∉ S) ∨ (G.right e ∈ S ∧ G.left e ∉ S)defConnected {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop := ∀ S : Finset (Fin n), S.Nonempty → S ≠ Finset.univ → (G.cut T S).NonemptydefSpanningTree {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop := G.Connected T ∧ ∀ e ∈ T, ¬ G.Connected (T.erase e)defEdgeConnected {n m : ℕ} (G : MultiGraph n m) (k : ℕ) : Prop := ∀ S : Finset (Fin n), S.Nonempty → S ≠ Finset.univ → k ≤ (G.cut Finset.univ S).cardend MultiGraphdefMainStatement : Prop := ∃ C : ℝ, 0 < C ∧ ∀ (k : ℕ), 1 ≤ k → ∀ (n m : ℕ), 2 ≤ n → ∀ G : MultiGraph n m, G.EdgeConnected k → ∃ T : Finset (Fin m), G.SpanningTree T ∧ ∀ S : Finset (Fin n), S.Nonempty → S ≠ Finset.univ → ((G.cut T S).card : ℝ) ≤ (C / (k : ℝ)) * (G.cut Finset.univ S).cardtheoremstrongThinTree : MainStatement := bysorryend StrongThinTreeend 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.
There is a universal constant C>0 such that, for every integer k\ge1, every finite k-edge-connected undirected multigraph without loops and with at least two vertices has a spanning tree T satisfying
|\delta_T(S)|\le\frac Ck\,|\delta_G(S)|
\qquad\text{for every }\varnothing\ne S\subsetneq V.
There is a universal constant \(C>0\) such that, for every integer
\(k\ge1\), every finite \(k\)-edge-connected undirected multigraph
without loops and with at least two vertices has a spanning tree \(T\)
satisfying
\[
|\delta_T(S)|\le\frac Ck\,|\delta_G(S)|
\qquad\text{for every }\varnothing\ne S\subsetneq V.
\]
There is an absolute integer h_0 such that, for every integer h\ge h_0, every locally h-connected hierarchy of G, and every collection \mathcal M of nonroot nodes satisfying |O(A)|\ge |P(A)|/h, there is a matrix D\succ0 on \mathsf E with
There is an absolute integer $h_0$ such that, for every integer
$h\ge h_0$, every locally $h$-connected hierarchy of $G$, and every
collection $\mathcal M$ of nonroot nodes satisfying
$|O(A)|\ge |P(A)|/h$, there is a matrix $D\succ0$ on $\mathsf E$ with
\begin{align}
\ip{D}{Z}&\le\ip{L_G}{Z} &&(Z\in\mathcal N),\\
\frac1{|O(A)|}\sum_{e\in O(A)}b_e^TD^{-1}b_e
&\le h^{-31/40} &&(A\in\mathcal M).
\end{align}
The cutoff is independent of the graph and the hierarchy depth.
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
6 local vocabulary declarations
Lines of Lean, without comments or blank lines
23
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
Strong thin-tree bound
Family 174: Deterministic construction of strong thin spanning trees
Candidate pairing
Theorem
There is a universal constant C>0 such that, for every integer k\ge1, every finite k-edge-connected undirected multigraph without loops and with at least two vertices has a spanning tree T satisfying
|\delta_T(S)|\le\frac Ck\,|\delta_G(S)|
\qquad\text{for every }\varnothing\ne S\subsetneq V.
The strong thin tree conjecture
TeX label thm:main, build/source/sections/introduction.tex, line 18
The Lean statement says
defMainStatement : Prop := ∃ C : ℝ, 0 < C ∧ ∀ (k : ℕ), 1 ≤ k → ∀ (n m : ℕ), 2 ≤ n → ∀ G : MultiGraph n m, G.EdgeConnected k → ∃ T : Finset (Fin m), G.SpanningTree T ∧ ∀ S : Finset (Fin n), S.Nonempty → S ≠ Finset.univ → ((G.cut T S).card : ℝ) ≤ (C / (k : ℝ)) * (G.cut Finset.univ S).card⋮theoremstrongThinTree : 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.