Lean statement

Polynomial-time construction of strong thin trees

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.AlgorithmicThinTrees.algorithmic_strong_thin_trees

Lean statement file: AlgorithmicThinTrees.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 174 link it.

See the closest candidates 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 AlgorithmicStrongThinTrees : Prop :=  ∃ C : ℝ, 0 < C ∧ ∃ M : CurrentKS.Machine, ∃ a degree : ℕ, 0 < a ∧    ∀ x : Input, Valid x →      let result := M.run x.encode (a * (x.length + 1) ^ degree)      result.state = none ∧ GoodOutput C x (result.stacks M.outputStack)⋮theorem algorithmic_strong_thin_trees : AlgorithmicStrongThinTrees := 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.

structure OAI.StrongThinTree.MultiGraph line 9

structure MultiGraph (n m : ℕ) where  left : Fin m → Fin n  right : Fin m → Fin n  loopless : ∀ e, left e ≠ right e

definition OAI.StrongThinTree.MultiGraph.cut line 16

def cut {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)

definition OAI.StrongThinTree.MultiGraph.Connected line 21

def Connected {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).Nonempty

definition OAI.StrongThinTree.MultiGraph.SpanningTree line 24

def SpanningTree {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop :=  G.Connected T ∧ ∀ e ∈ T, ¬ G.Connected (T.erase e)

definition OAI.StrongThinTree.MultiGraph.EdgeConnected line 27

def EdgeConnected {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

definition OAI.CurrentKS.frame line 36

def frame : List Bool → List Bool  | [] => [true]  | b :: bs => false :: b :: frame bs

definition OAI.CurrentKS.encodeNat line 40

def encodeNat (n : ℕ) : List Bool := frame (Computability.encodeNat n)

inductive type OAI.CurrentKS.StackAction line 42

inductive StackAction where  | stay  | pop  | push (bit : Bool)  deriving DecidableEq

definition OAI.CurrentKS.StackAction.apply line 48

def StackAction.apply : StackAction → List Bool → List Bool  | .stay, xs => xs  | .pop, xs => xs.tail  | .push b, xs => b :: xs

structure OAI.CurrentKS.Instruction line 53

structure Instruction (k q : ℕ) where  next : Option (Fin q)  actions : Fin k → StackAction

definition OAI.CurrentKS.Instruction.halt line 57

def Instruction.halt (k q : ℕ) : Instruction k q := ⟨none, fun _ => .stay⟩

structure OAI.CurrentKS.Machine line 59

structure Machine where  stackCount : ℕ  states : ℕ  initial : Fin states  inputStack : Fin stackCount  outputStack : Fin stackCount  table : List (Instruction stackCount states)

structure OAI.CurrentKS.Config line 67

structure Config (k q : ℕ) where  state : Option (Fin q)  «stacks» : Fin k → List Bool

definition OAI.CurrentKS.topCode line 71

def topCode : List Bool → ℕ  | [] => 0  | false :: _ => 1  | true :: _ => 2

definition OAI.CurrentKS.headsCode line 76

def headsCode : (k : ℕ) → (Fin k → List Bool) → ℕ  | 0, _ => 0  | k + 1, s => topCode (s 0) + 3 * headsCode k (fun j => s j.succ)

definition OAI.CurrentKS.Machine.step line 80

def Machine.step (M : Machine) (c : Config M.stackCount M.states) :    Config M.stackCount M.states :=  match c.state with  | none => c  | some q =>    let instruction := M.table.getD      (3 ^ M.stackCount * q.val + headsCode M.stackCount c.stacks)      (Instruction.halt M.stackCount M.states)    ⟨instruction.next, fun j => (instruction.actions j).apply (c.stacks j)⟩

definition OAI.CurrentKS.Machine.init line 90

def Machine.init (M : Machine) (input : List Bool) : Config M.stackCount M.states :=  ⟨some M.initial, fun j => if j = M.inputStack then input else []⟩

definition OAI.CurrentKS.Machine.run line 93

def Machine.run (M : Machine) (input : List Bool) (fuel : ℕ) :    Config M.stackCount M.states := (M.step^[fuel]) (M.init input)

structure OAI.AlgorithmicThinTrees.ExplicitInput line 101

structure ExplicitInput where  n : ℕ  m : ℕ  k : ℕ  graph : MultiGraph n m

structure OAI.AlgorithmicThinTrees.BinaryInput line 107

structure BinaryInput where  n : ℕ  p : ℕ  k : ℕ  left : Fin p → Fin n  right : Fin p → Fin n  ordered : ∀ i, left i < right i  distinct : Function.Injective (fun i => (left i, right i))  multiplicity : Fin p → ℕ

abbreviation OAI.AlgorithmicThinTrees.BinaryInput.Edge line 117

abbrev BinaryInput.Edge (x : BinaryInput) := Σ i : Fin x.p, Fin (x.multiplicity i)

definition OAI.AlgorithmicThinTrees.BinaryInput.edgeEquiv line 119

noncomputable def BinaryInput.edgeEquiv (x : BinaryInput) :    x.Edge ≃ Fin (Fintype.card x.Edge) := Fintype.equivFin x.Edge

definition OAI.AlgorithmicThinTrees.BinaryInput.graph line 122

noncomputable def BinaryInput.graph (x : BinaryInput) :    MultiGraph x.n (Fintype.card x.Edge) where  left e := x.left (x.edgeEquiv.symm e).1  right e := x.right (x.edgeEquiv.symm e).1  loopless e := ne_of_lt (x.ordered (x.edgeEquiv.symm e).1)

inductive type OAI.AlgorithmicThinTrees.Input line 128

inductive Input where  | explicit (x : ExplicitInput)  | binary (x : BinaryInput)

9 more definitions are reached the same way; the full file lists them.

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements in A polynomial-time construction of strong thin trees come first because the release’s name for this Lean statement shares words with that paper’s title.

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: list, input, cuts, binary, explicit, thin and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: parallel, deterministic, algorithm, constant and polynomial.
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: output, compute, encoding, explicit, length, trees and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: time, running, one, multigraph, constants, polynomial and one more.
Theorem A.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: encoding, binary and length. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gives, time, running, deterministic, algorithm, one and one more.
Theorem 1.1 of The strong thin tree conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: tree, finite, 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 and constant.
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: edges, tree, trees, graph, spanning, edge and one more.

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 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.

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: 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.
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 gives one deterministic polynomial-time algorithm and an absolute constant $C>0$ such that a finite $k$-edge-connected loopless multigraph yields a spanning tree using at most a $C/k$ fraction of the edges of every cut. The input may encode parallel-edge multiplicities in binary, and the running time is polynomial in that binary input length.”
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

196 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: 62 to its own declarations, 112 to Mathlib’s documentation. 37 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 OAI/-! Deterministic polynomial-time construction of strong thin trees. -/namespace StrongThinTreestructure MultiGraph (n m : ℕ) where  left : Fin m → Fin n  right : Fin m → Fin n  loopless : ∀ e, left e ≠ right enamespace MultiGraphdef cut {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)def Connected {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).Nonemptydef SpanningTree {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop :=  G.Connected T ∧ ∀ e ∈ T, ¬ G.Connected (T.erase e)def EdgeConnected {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 MultiGraphend StrongThinTreenamespace CurrentKSdef frame : List Bool → List Bool  | [] => [true]  | b :: bs => false :: b :: frame bsdef encodeNat (n : ℕ) : List Bool := frame (Computability.encodeNat n)inductive StackAction where  | stay  | pop  | push (bit : Bool)  deriving DecidableEqdef StackAction.apply : StackAction → List Bool → List Bool  | .stay, xs => xs  | .pop, xs => xs.tail  | .push b, xs => b :: xsstructure Instruction (k q : ℕ) where  next : Option (Fin q)  actions : Fin k → StackActiondef Instruction.halt (k q : ℕ) : Instruction k q := ⟨none, fun _ => .stay⟩structure Machine where  stackCount : ℕ  states : ℕ  initial : Fin states  inputStack : Fin stackCount  outputStack : Fin stackCount  table : List (Instruction stackCount states)structure Config (k q : ℕ) where  state : Option (Fin q)  «stacks» : Fin k → List Booldef topCode : List Bool → ℕ  | [] => 0  | false :: _ => 1  | true :: _ => 2def headsCode : (k : ℕ) → (Fin k → List Bool) → ℕ  | 0, _ => 0  | k + 1, s => topCode (s 0) + 3 * headsCode k (fun j => s j.succ)def Machine.step (M : Machine) (c : Config M.stackCount M.states) :    Config M.stackCount M.states :=  match c.state with  | none => c  | some q =>    let instruction := M.table.getD      (3 ^ M.stackCount * q.val + headsCode M.stackCount c.stacks)      (Instruction.halt M.stackCount M.states)    ⟨instruction.next, fun j => (instruction.actions j).apply (c.stacks j)⟩def Machine.init (M : Machine) (input : List Bool) : Config M.stackCount M.states :=  ⟨some M.initial, fun j => if j = M.inputStack then input else []⟩def Machine.run (M : Machine) (input : List Bool) (fuel : ℕ) :    Config M.stackCount M.states := (M.step^[fuel]) (M.init input)end CurrentKSnamespace AlgorithmicThinTreesopen StrongThinTreestructure ExplicitInput where  n : ℕ  m : ℕ  k : ℕ  graph : MultiGraph n mstructure BinaryInput where  n : ℕ  p : ℕ  k : ℕ  left : Fin p → Fin n  right : Fin p → Fin n  ordered : ∀ i, left i < right i  distinct : Function.Injective (fun i => (left i, right i))  multiplicity : Fin p → ℕabbrev BinaryInput.Edge (x : BinaryInput) := Σ i : Fin x.p, Fin (x.multiplicity i)noncomputable def BinaryInput.edgeEquiv (x : BinaryInput) :    x.Edge ≃ Fin (Fintype.card x.Edge) := Fintype.equivFin x.Edgenoncomputable def BinaryInput.graph (x : BinaryInput) :    MultiGraph x.n (Fintype.card x.Edge) where  left e := x.left (x.edgeEquiv.symm e).1  right e := x.right (x.edgeEquiv.symm e).1  loopless e := ne_of_lt (x.ordered (x.edgeEquiv.symm e).1)inductive Input where  | explicit (x : ExplicitInput)  | binary (x : BinaryInput)def ExplicitInput.encode (x : ExplicitInput) : List Bool :=  CurrentKS.encodeNat x.n ++ CurrentKS.encodeNat x.m ++ CurrentKS.encodeNat x.k ++    (List.ofFn fun i : Fin x.m =>      CurrentKS.encodeNat (x.graph.left i).val ++      CurrentKS.encodeNat (x.graph.right i).val).flattendef BinaryInput.encode (x : BinaryInput) : List Bool :=  CurrentKS.encodeNat x.n ++ CurrentKS.encodeNat x.p ++ CurrentKS.encodeNat x.k ++    (List.ofFn fun i : Fin x.p =>      CurrentKS.encodeNat (x.left i).val ++ CurrentKS.encodeNat (x.right i).val ++      CurrentKS.encodeNat (x.multiplicity i)).flattendef Input.encode : Input → List Bool  | .explicit x => false :: x.encode  | .binary x => true :: x.encodedef Input.length (x : Input) : ℕ := x.encode.lengthdef Input.n : Input → ℕ  | .explicit x => x.n  | .binary x => x.ndef Valid : Input → Prop  | .explicit x => 1 ≤ x.n ∧ 1 ≤ x.k ∧ x.graph.EdgeConnected x.k  | .binary x => 1 ≤ x.n ∧ 1 ≤ x.k ∧ x.graph.EdgeConnected x.kdef ThinTree {n m : ℕ} (C : ℝ) (k : ℕ) (G : MultiGraph n m)    (T : Finset (Fin m)) : Prop :=  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).carddef ExplicitInput.encodeEdges (x : ExplicitInput) (edges : List (Fin x.m)) : List Bool :=  edges.flatMap (fun e => CurrentKS.encodeNat e.val)def BinaryInput.encodeEdges (x : BinaryInput) (edges : List x.Edge) : List Bool :=  edges.flatMap (fun e => CurrentKS.encodeNat e.1.val ++ CurrentKS.encodeNat e.2.val)noncomputable def BinaryInput.selected (x : BinaryInput) (edges : List x.Edge) :    Finset (Fin (Fintype.card x.Edge)) := edges.toFinset.map x.edgeEquiv.toEmbeddingdef TreeOutput (C : ℝ) : Input → List Bool → Prop  | .explicit x, bits => ∃ edges : List (Fin x.m), edges.Nodup ∧      bits = x.encodeEdges edges ∧ ThinTree C x.k x.graph edges.toFinset  | .binary x, bits => ∃ edges : List x.Edge, edges.Nodup ∧      bits = x.encodeEdges edges ∧ ThinTree C x.k x.graph (x.selected edges)def GoodOutput (C : ℝ) (x : Input) (bits : List Bool) : Prop :=  TreeOutput C x bits ∧ (x.n = 1 → bits = [])def AlgorithmicStrongThinTrees : Prop :=  ∃ C : ℝ, 0 < C ∧ ∃ M : CurrentKS.Machine, ∃ a degree : ℕ, 0 < a ∧    ∀ x : Input, Valid x →      let result := M.run x.encode (a * (x.length + 1) ^ degree)      result.state = none ∧ GoodOutput C x (result.stacks M.outputStack)end AlgorithmicThinTreesnamespace AlgorithmicThinTreestheorem algorithmic_strong_thin_trees : AlgorithmicStrongThinTrees := by  sorryend AlgorithmicThinTreesend 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.

The strong thin tree conjecture

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.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

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
\[
 |\delta_T(S)|\le\frac Ck\,|\delta_G(S)|
 \qquad\text{for every }\varnothing\ne S\subsetneq V.
\]

A polynomial-time construction of strong thin trees

Theorem (Hierarchy bound for the metric test cone)

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{aligned}\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{aligned}

The cutoff is independent of the graph and the hierarchy depth.

Section 2, Metric-cone shortcuts for thin trees Read it in the paper TeX source, line 52

Show the TeX

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.

Configuration and file facts

File: lean/ComparatorChallenges/AlgorithmicThinTrees.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 174 link it.
Solution module: OAI.Combinatorics.ThinTrees.AlgorithmicMain
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: 37 local vocabulary declarations
Lines of Lean, without comments or blank lines: 131
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