Three-state reconstruction above the Kesten–Stigum threshold
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.TreeClauses.Tree.regular_reconstructs_above_KSOAI.ThreeState.TreeClauses.Tree.poisson_reconstructs_above_KS
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.
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: three, state and 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.
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 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, count, integer, offspring, regular and three more.
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: integer, offspring, regular, poisson, posterior and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: covers.
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, root, laws and two more.
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, regular and tree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: non, threshold and models.
Proposition 2.3 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: measurable, mass, one, 1/3, marginal, root and one more.
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, rooted, poisson, tree and channel. Found in the paper’s title, which its main theorem states: state. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 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.
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.
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).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 126 s; peak memory 6.2 GiB).
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 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.
Finds part of it stated, or a narrower or weaker statementTheir 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).
Finds part of it stated, or a narrower or weaker statementTheir 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.
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.”
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
104 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: 52 to its own declarations, 32 to Mathlib’s documentation. 19 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.
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
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
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.
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.
Questions to resolve when comparing this statement with the paper
The AI draft finds no Lean statement in its family that states a paper's theorem as the paper does.