Lean statement

Supercritical reconstruction on regular and Poisson 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.

Theorems selected for verification: OAI.ThreeState.regular_supercritical OAI.ThreeState.poisson_supercritical

Lean statement file: ThreeStateSupercritical.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 229 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 regular_supercritical (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)    (hcrit : 1 < (b : ℝ) * lam ^ 2) : Reconstructs (regularAdvantage b lam h) := by  sorry⋮theorem poisson_supercritical (d : ℝ) (hd : 1 < d) (lam : ℝ) (h : Admissible lam)    (hcrit : 1 < d * lam ^ 2) : Reconstructs (poissonAdvantage d hd lam h) := 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.ThreeState.Spin line 7
```
abbrev Spin := Fin 3
```

definition OAI.ThreeState.Admissible line 10

def Admissible (lam : ℝ) : Prop := -(1 / 2 : ℝ) ≤ lam ∧ lam ≤ 1

definition OAI.ThreeState.channel line 13

noncomputable def channel (lam : ℝ) (i j : Spin) : ℝ :=  if i = j then (1 + 2 * lam) / 3 else (1 - lam) / 3

definition OAI.ThreeState.Observation line 18

def Observation : ℕ → Type  | 0 => Spin  | n + 1 => Multiset (Observation n)

definition OAI.ThreeState.channelPMF line 34

noncomputable def channelPMF (lam : ℝ) (h : Admissible lam) (i : Spin) : PMF Spin :=  PMF.ofFintype (fun j => ENNReal.ofReal (channel lam i j)) (by    rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ => channel_nonneg h i j)]    simp [channel_sum])

definition OAI.ThreeState.iidList line 40

noncomputable def iidList {α : Type*} (p : PMF α) : ℕ → PMF (List α)  | 0 => PMF.pure []  | n + 1 => p.bind fun a => (iidList p n).map (List.cons a)

definition OAI.ThreeState.observedLaw line 46

noncomputable def observedLaw (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :    (n : ℕ) → Spin → PMF (Observation n)  | 0, i => PMF.pure i  | n + 1, i => offspring.bind fun k =>      (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h n)) k).map        (fun xs : List (Observation n) => (xs : Multiset (Observation n)))

definition OAI.ThreeState.uniformSpin line 53

noncomputable def uniformSpin : PMF Spin := PMF.ofFintype (fun _ => 1 / 3) (by  simp  exact ENNReal.mul_inv_cancel (by norm_num) (by finiteness))

definition OAI.ThreeState.marginal line 58

noncomputable def marginal {α : Type*} (law : Spin → PMF α) : PMF α :=  uniformSpin.bind law

definition OAI.ThreeState.posterior line 62

noncomputable def posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) : ℝ :=  if marginal law y = 0 then 1 / 3  else (law i y).toReal / (3 * (marginal law y).toReal)

definition OAI.ThreeState.advantage line 68

noncomputable def advantage {α : Type*} (law : Spin → PMF α) : ℝ :=  ∑' y, (marginal law y).toReal *    ((∑ i : Spin, |posterior law y i - 1 / 3|) / 2)

definition OAI.ThreeState.regularLaw line 72

noncomputable def regularLaw (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :=  observedLaw (PMF.pure b) lam h n

definition OAI.ThreeState.poissonOffspring line 76

noncomputable def poissonOffspring (d : ℝ) (hd : 0 ≤ d) : PMF ℕ := by  refine ⟨fun k => ENNReal.ofReal (Real.exp (-d) * d ^ k / (k.factorial : ℝ)), ?_⟩  apply ENNReal.hasSum_coe.2  have hsum : HasSum (fun k : ℕ => Real.exp (-d) * d ^ k / (k.factorial : ℝ)) 1 :=    ProbabilityTheory.hasSum_one_poissonMeasure ⟨d, hd⟩  simpa using hsum.toNNReal (fun k => by positivity)

definition OAI.ThreeState.poissonLaw line 83

noncomputable def poissonLaw (d : ℝ) (hd : 0 ≤ d) (lam : ℝ)    (h : Admissible lam) (n : ℕ) :=  observedLaw (poissonOffspring d hd) lam h n

definition OAI.ThreeState.regularAdvantage line 87

noncomputable def regularAdvantage (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) : ℝ :=  advantage (regularLaw b lam h n)

definition OAI.ThreeState.poissonAdvantage line 90

noncomputable def poissonAdvantage (d : ℝ) (hd : 1 < d) (lam : ℝ)    (h : Admissible lam) (n : ℕ) : ℝ :=  advantage (poissonLaw d (le_trans (by norm_num) hd.le) lam h n)

definition OAI.ThreeState.Reconstructs line 95

def Reconstructs (a : ℕ → ℝ) : Prop :=  ∃ L : ℝ, 0 < L ∧ Filter.Tendsto a Filter.atTop (nhds L)

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: real, admissible, integer, regular and poisson. Found in the paper’s title, which its main theorem states: channel. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: non, threshold, reconstruction and model.

Statements in The exact reconstruction threshold for the three-state symmetric channel come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of The exact reconstruction threshold for the three-state symmetric channel

Theorem (Exact threshold)

For every integer b\ge2 in the regular model and every real d>1 in the Poisson model, reconstruction occurs if and only if

d\lambda^2>1,

with d=b in the regular case. At every admissible equality d\lambda^2=1, there is non-reconstruction.

Section 1, Introduction Read it in the paper TeX source, line 44

Show the TeX

For every integer $b\ge2$ in the regular model and every real $d>1$
in the Poisson model, reconstruction occurs if and only if
\[
 d\lambda^2>1,
\]
with $d=b$ in the regular case. At every admissible equality
$d\lambda^2=1$, there is non-reconstruction.

Other possible paper matches, closest wording first

Proposition 5.1 of The exact reconstruction threshold for the three-state symmetric channel. Words of the Lean statement, its names split into words, that also appear in the theorem: uniform, integer, offspring, regular, poisson, posterior and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: covers.
Proposition 2.8 of The exact reconstruction threshold for the three-state symmetric channel. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, advantage, integer, offspring, regular, poisson and one more.
Proposition 2.4 of The exact reconstruction threshold for the three-state symmetric channel. Words of the Lean statement, its names split into words, that also appear in the theorem: spins, probabilities, advantages, 1/2, laws and posterior.
Proposition 6.3 of The exact reconstruction threshold for the three-state symmetric channel. Words of the Lean statement, its names split into words, that also appear in the theorem: admissible and regular. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: non, threshold and models.
Theorem 1.1 of The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model. Words of the Lean statement, its names split into words, that also appear in the theorem: real, integer, poisson and channel. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: models.
Proposition 3.3 of The exact reconstruction threshold for the three-state symmetric channel. Words of the Lean statement, its names split into words, that also appear in the theorem: uniform, 1/2 and posterior. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: reconstruction and models.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 67 s; peak memory 6.2 GiB).
Reported 10 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 229’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.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: Only the supercritical (Kesten-Stigum) direction for the three-state symmetric channel is stated: regular_supercritical (hcrit : 1 < b * lam^2) : Reconstructs .. and poisson_supercritical (hd : 1 < d) (hcrit : 1 < d * lam^2) : Reconstructs .. (advantage ∑' y, marginal y * (∑ i |posterior y i - 1/3|)/2 converges to L > 0), docs: 'Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them'. The summary's headline is the EXACT threshold dλ²>1 with nonreconstruction at equality, plus the four-state ferromagnetic case and the SBM weak-recovery threshold: none of those is stated (the formalized direction is the classical one, the sharpness is missing).
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their 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: (a) Non-reconstruction at and below dλ² = 1 (the 'exact' part of the threshold, and the new direction — reconstruction above Kesten–Stigum is classical) is not stated for either model; (b) every four-state ferromagnetic result (threshold on regular and Poisson trees, nonreconstruction at equality, and the L³-capacity criterion) is not stated; (c) the three-community SBM weak-recovery threshold is not stated. (a) for three states and (c) are declared; (a) for four states and (b) are not.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The selected statements cover reconstruction above the Kesten–Stigum threshold. Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

108 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: 55 to its own declarations, 36 to Mathlib’s documentation. 21 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name 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 regular_supercritical 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 19 other declarations of the file; with the theorem itself they span 65 lines. 4 of them are named in the statement itself.

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

abbrev Spinline 7library
def Admissibleline 10in the statementlibrary
def channelline 13library
def Observationline 18library
theorem channel_nonnegline 22library
theorem channel_sumline 31library
def channelPMFline 34library
def iidListline 40library
def observedLawline 46library
def uniformSpinline 53library
def marginalline 58library
def posteriorline 62library
def advantageline 68library
def regularLawline 72library
def poissonOffspringline 76library
def poissonLawline 83library
def regularAdvantageline 87in the statementlibrary
def poissonAdvantageline 90in the statementlibrary
def Reconstructsline 95in the statementlibrary
theorem regular_supercriticalline 98selected theoremlibrary
theorem poisson_supercriticalline 102selected theoremlibrary

import Mathlibnamespace OAInamespace ThreeStateabbrev Spin := Fin 3 def Admissible (lam : ℝ) : Prop := -(1 / 2 : ℝ) ≤ lam ∧ lam ≤ 1 noncomputable def channel (lam : ℝ) (i j : Spin) : ℝ :=  if i = j then (1 + 2 * lam) / 3 else (1 - lam) / 3 def Observation : ℕ → Type  | 0 => Spin  | n + 1 => Multiset (Observation n)theorem channel_nonneg {lam : ℝ} (h : Admissible lam) (i j : Spin) :    0 ≤ channel lam i j := by  unfold channel  split_ifs  · dsimp [Admissible] at h    linarith [h.1]  · dsimp [Admissible] at h    linarith [h.2]theorem channel_sum (lam : ℝ) (i : Spin) : ∑ j, channel lam i j = 1 := by  fin_cases i <;> simp [channel, Fin.sum_univ_succ] <;> ringnoncomputable def channelPMF (lam : ℝ) (h : Admissible lam) (i : Spin) : PMF Spin :=  PMF.ofFintype (fun j => ENNReal.ofReal (channel lam i j)) (by    rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ => channel_nonneg h i j)]    simp [channel_sum]) noncomputable def iidList {α : Type*} (p : PMF α) : ℕ → PMF (List α)  | 0 => PMF.pure []  | n + 1 => p.bind fun a => (iidList p n).map (List.cons a) noncomputable def observedLaw (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :    (n : ℕ) → Spin → PMF (Observation n)  | 0, i => PMF.pure i  | n + 1, i => offspring.bind fun k =>      (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h n)) k).map        (fun xs : List (Observation n) => (xs : Multiset (Observation n)))noncomputable def uniformSpin : PMF Spin := PMF.ofFintype (fun _ => 1 / 3) (by  simp  exact ENNReal.mul_inv_cancel (by norm_num) (by finiteness)) noncomputable def marginal {α : Type*} (law : Spin → PMF α) : PMF α :=  uniformSpin.bind law noncomputable def posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) : ℝ :=  if marginal law y = 0 then 1 / 3  else (law i y).toReal / (3 * (marginal law y).toReal) noncomputable def advantage {α : Type*} (law : Spin → PMF α) : ℝ :=  ∑' y, (marginal law y).toReal *    ((∑ i : Spin, |posterior law y i - 1 / 3|) / 2)noncomputable def regularLaw (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :=  observedLaw (PMF.pure b) lam h n noncomputable def poissonOffspring (d : ℝ) (hd : 0 ≤ d) : PMF ℕ := by  refine ⟨fun k => ENNReal.ofReal (Real.exp (-d) * d ^ k / (k.factorial : ℝ)), ?_⟩  apply ENNReal.hasSum_coe.2  have hsum : HasSum (fun k : ℕ => Real.exp (-d) * d ^ k / (k.factorial : ℝ)) 1 :=    ProbabilityTheory.hasSum_one_poissonMeasure ⟨d, hd⟩  simpa using hsum.toNNReal (fun k => by positivity)noncomputable def poissonLaw (d : ℝ) (hd : 0 ≤ d) (lam : ℝ)    (h : Admissible lam) (n : ℕ) :=  observedLaw (poissonOffspring d hd) lam h nnoncomputable def regularAdvantage (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) : ℝ :=  advantage (regularLaw b lam h n)noncomputable def poissonAdvantage (d : ℝ) (hd : 1 < d) (lam : ℝ)    (h : Admissible lam) (n : ℕ) : ℝ :=  advantage (poissonLaw d (le_trans (by norm_num) hd.le) lam h n) def Reconstructs (a : ℕ → ℝ) : Prop :=  ∃ L : ℝ, 0 < L ∧ Filter.Tendsto a Filter.atTop (nhds L)theorem regular_supercritical (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)    (hcrit : 1 < (b : ℝ) * lam ^ 2) : Reconstructs (regularAdvantage b lam h) := by  sorrytheorem poisson_supercritical (d : ℝ) (hd : 1 < d) (lam : ℝ) (h : Admissible lam)    (hcrit : 1 < d * lam ^ 2) : Reconstructs (poissonAdvantage d hd lam h) := by  sorryend ThreeStateend 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 exact reconstruction threshold for the three-state symmetric channel

Theorem (Exact threshold)

For every integer b\ge2 in the regular model and every real d>1 in the Poisson model, reconstruction occurs if and only if

d\lambda^2>1,

with d=b in the regular case. At every admissible equality d\lambda^2=1, there is non-reconstruction.

Section 1, Introduction Read it in the paper TeX source, line 44

Show the TeX

For every integer $b\ge2$ in the regular model and every real $d>1$
in the Poisson model, reconstruction occurs if and only if
\[
 d\lambda^2>1,
\]
with $d=b$ in the regular case. At every admissible equality
$d\lambda^2=1$, there is non-reconstruction.

Corollary (Exact weak-recovery threshold for three communities)

In the model above, the following statements hold.

(i)
If (a-b)^2\le 3(a+2b), then every sequence of estimators \widehat\sigma_n using only G_n and independent randomization satisfies
\lim_{n\to\infty}\E\operatorname{ov}(\sigma,\widehat\sigma_n)=\frac13.
(ii)
If (a-b)^2>3(a+2b), there are \varepsilon>0 and an algorithm, allowed to depend on the fixed parameters a,b, which runs in O(n\log n) time and satisfies
\PP\!\left(\operatorname{ov}(\sigma,\widehat\sigma_n) \ge\frac13+\varepsilon\right)\longrightarrow1.

Thus weak recovery is possible exactly under the strict inequality, and is information-theoretically impossible at equality.

Section 1, Introduction Read it in the paper TeX source, line 203

Show the TeX

In the model above, the following statements hold.
\begin{enumerate}[label=(\roman*)]
\item If $(a-b)^2\le 3(a+2b)$, then every sequence of estimators
$\widehat\sigma_n$ using only $G_n$ and independent randomization satisfies
\[
 \lim_{n\to\infty}\E\operatorname{ov}(\sigma,\widehat\sigma_n)=\frac13.
\]
\item If $(a-b)^2>3(a+2b)$, there are $\varepsilon>0$ and an algorithm,
allowed to depend on the fixed parameters $a,b$, which runs in
$O(n\log n)$ time and satisfies
\[
 \PP\!\left(\operatorname{ov}(\sigma,\widehat\sigma_n)
              \ge\frac13+\varepsilon\right)\longrightarrow1.
\]
\end{enumerate}
Thus weak recovery is possible exactly under the strict inequality,
and is information-theoretically impossible at equality.

Configuration and file facts

File: lean/ComparatorChallenges/ThreeStateSupercritical.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 229 link it.
Solution module: OAI.Probability.ThreeState.Main
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: 17 local vocabulary declarations
Lines of Lean, without comments or blank lines: 65
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