Lean statement

Sharp terminal edge count in random triangle removal

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.SharpTerminalLeave.sharp_terminal_leave

Lean statement file: TriangleRemoval.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 188 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.

theorem sharp_terminal_leave :    Tendsto (fun n => expectation n (fun G =>      (normalizedLeave n G - sharpConstant) ^ 2)) atTop (𝓝 0) ∧    (∀ ε : ℝ, 0 < ε →      Tendsto (fun n => probability n (fun G =>        ε < |normalizedLeave n G - sharpConstant|)) atTop (𝓝 0)) ∧    Tendsto (fun n => expectation n (fun G => (G.card : ℝ)) / normalization n)      atTop (𝓝 sharpConstant) := 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.SharpTerminalLeave.Graph line 13

abbrev Graph (n : ℕ) := Finset (Finset (Fin n))

definition OAI.SharpTerminalLeave.completeGraph line 15

def completeGraph (n : ℕ) : Graph n := Finset.univ.powersetCard 2

definition OAI.SharpTerminalLeave.triangles line 17

def triangles {n : ℕ} (G : Graph n) : Finset (Finset (Fin n)) :=  (Finset.univ.powersetCard 3).filter (fun t => t.powersetCard 2 ⊆ G)

definition OAI.SharpTerminalLeave.step line 20

noncomputable def step {n : ℕ} (G : Graph n) : PMF (Graph n) :=  if h : (triangles G).Nonempty then    (PMF.uniformOfFinset (triangles G) h).map (fun t => G \ t.powersetCard 2)  else PMF.pure G

definition OAI.SharpTerminalLeave.evolve line 25

noncomputable def evolve {n : ℕ} (G : Graph n) : ℕ → PMF (Graph n)  | 0 => PMF.pure G  | k + 1 => (evolve G k).bind step

definition OAI.SharpTerminalLeave.terminalLaw line 29

noncomputable def terminalLaw (n : ℕ) : PMF (Graph n) :=  evolve (completeGraph n) (n.choose 2)

definition OAI.SharpTerminalLeave.expectation line 32

noncomputable def expectation (n : ℕ) (f : Graph n → ℝ) : ℝ :=  ∑ G, (terminalLaw n G).toReal * f G

definition OAI.SharpTerminalLeave.probability line 35

noncomputable def probability (n : ℕ) (P : Graph n → Prop) : ℝ := by  classical  exact expectation n (fun G => if P G then 1 else 0)

definition OAI.SharpTerminalLeave.normalization line 39

noncomputable def normalization (n : ℕ) : ℝ := (n : ℝ) ^ (3 / 2 : ℝ)

definition OAI.SharpTerminalLeave.normalizedLeave line 41

noncomputable def normalizedLeave (n : ℕ) (G : Graph n) : ℝ :=  (G.card : ℝ) / normalization n

definition OAI.SharpTerminalLeave.sharpConstant line 44

noncomputable def sharpConstant : ℝ := 1 / (2 * Real.sqrt 2)

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: 3/2, sqrt, triangle and uniform. Found in the paper’s title, which its main theorem states: sharp, terminal and leave. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: removal.

Theorem 1.1 of The sharp terminal leave in random triangle removal

Theorem

For the uniform triangle-removal process starting from K_n,

\E\left[\left(\frac{F_n}{n^{3/2}}- \frac1{2\sqrt2}\right)^2\right]\longrightarrow0.

In particular,

\frac{F_n}{n^{3/2}}\xrightarrow{\Prb}\frac1{2\sqrt2}, \qquad \frac{\E F_n}{n^{3/2}}\longrightarrow\frac1{2\sqrt2}.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

For the uniform triangle-removal process starting from $K_n$,
\[
 \E\left[\left(\frac{F_n}{n^{3/2}}-
                   \frac1{2\sqrt2}\right)^2\right]\longrightarrow0.
\]
In particular,
\[
 \frac{F_n}{n^{3/2}}\xrightarrow{\Prb}\frac1{2\sqrt2},
 \qquad
 \frac{\E F_n}{n^{3/2}}\longrightarrow\frac1{2\sqrt2}.
\]

Other possible paper matches, closest wording first

Proposition 2.2 of The sharp terminal leave in random triangle removal. Words of the Lean statement, its names split into words, that also appear in the theorem: step, constants, uniform and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: number and edges.
Proposition 6.3 of The sharp terminal leave in random triangle removal. Words of the Lean statement, its names split into words, that also appear in the theorem: triangle, probabilities, uniform and graphs. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: edges.
Proposition 4.1 of The sharp terminal leave in random triangle removal. Words of the Lean statement, its names split into words, that also appear in the theorem: constants and graph.
Proposition 3.4 of The sharp terminal leave in random triangle removal. Words of the Lean statement, its names split into words, that also appear in the theorem: triangle and probability.
Proposition 4.2 of The sharp terminal leave in random triangle removal. Words of the Lean statement, its names split into words, that also appear in the theorem: probabilities and graph.

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, Classical.choice, Quot.sound

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 188’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: sharp_terminal_leave states, for the Mathlib-PMF Markov chain evolve (completeGraph n) (n.choose 2) that deletes a uniformly chosen remaining triangle (PMF.uniformOfFinset (triangles G)), (i) E[(|G|/n^(3/2) - 1/(2√2))^2] → 0, (ii) convergence in probability, (iii) E|G|/n^(3/2) → 1/(2√2). This is the summary's mean-square convergence; the process is absorbing at triangle-free graphs so running C(n,2) steps gives the terminal law.
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 that the number of edges at termination, divided by $n^{3/2}$, converges in $L^2$ to $1/(2\sqrt2)$. It also states convergence in probability and convergence of the normalized expectation to the same constant. This is the triangle case of the sharp terminal-leave conjecture of Joos and Kühn.”
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

67 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: 28 to its own declarations, 21 to Mathlib’s documentation. 12 declared names also link to the same declaration in OpenAI’s library. 3 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 sharp_terminal_leave 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 11 other declarations of the file; with the theorem itself they span 31 lines. 5 of them are named in the statement itself.

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

abbrev Graphline 13library
def completeGraphline 15library
def trianglesline 17library
def stepline 20library
def evolveline 25library
def terminalLawline 29library
def expectationline 32in the statementlibrary
def probabilityline 35in the statementlibrary
def normalizationline 39in the statementlibrary
def normalizedLeaveline 41in the statementlibrary
def sharpConstantline 44in the statementlibrary
theorem sharp_terminal_leaveline 55selected theoremlibrary

import Mathlibnamespace OAIsectionsectionopen Filteropen scoped BigOperators Topologynamespace SharpTerminalLeaveabbrev Graph (n : ℕ) := Finset (Finset (Fin n))def completeGraph (n : ℕ) : Graph n := Finset.univ.powersetCard 2def triangles {n : ℕ} (G : Graph n) : Finset (Finset (Fin n)) :=  (Finset.univ.powersetCard 3).filter (fun t => t.powersetCard 2 ⊆ G)noncomputable def step {n : ℕ} (G : Graph n) : PMF (Graph n) :=  if h : (triangles G).Nonempty then    (PMF.uniformOfFinset (triangles G) h).map (fun t => G \ t.powersetCard 2)  else PMF.pure Gnoncomputable def evolve {n : ℕ} (G : Graph n) : ℕ → PMF (Graph n)  | 0 => PMF.pure G  | k + 1 => (evolve G k).bind stepnoncomputable def terminalLaw (n : ℕ) : PMF (Graph n) :=  evolve (completeGraph n) (n.choose 2)noncomputable def expectation (n : ℕ) (f : Graph n → ℝ) : ℝ :=  ∑ G, (terminalLaw n G).toReal * f Gnoncomputable def probability (n : ℕ) (P : Graph n → Prop) : ℝ := by  classical  exact expectation n (fun G => if P G then 1 else 0)noncomputable def normalization (n : ℕ) : ℝ := (n : ℝ) ^ (3 / 2 : ℝ)noncomputable def normalizedLeave (n : ℕ) (G : Graph n) : ℝ :=  (G.card : ℝ) / normalization nnoncomputable def sharpConstant : ℝ := 1 / (2 * Real.sqrt 2)end SharpTerminalLeaveendendopen Filteropen scoped BigOperators Topologynamespace SharpTerminalLeavetheorem sharp_terminal_leave :    Tendsto (fun n => expectation n (fun G =>      (normalizedLeave n G - sharpConstant) ^ 2)) atTop (𝓝 0) ∧    (∀ ε : ℝ, 0 < ε →      Tendsto (fun n => probability n (fun G =>        ε < |normalizedLeave n G - sharpConstant|)) atTop (𝓝 0)) ∧    Tendsto (fun n => expectation n (fun G => (G.card : ℝ)) / normalization n)      atTop (𝓝 sharpConstant) := by  sorryend SharpTerminalLeaveend 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 sharp terminal leave in random triangle removal