OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every finite n\ge0,\rho\in[-1,1], and f:\Omega_n\to\{-1,1\}, put m=\E f and r_m=\psi^{-1}(H(m)). Then
I(f(X);Y)\le\psi(|\rho|r_m)\le\psi(|\rho|).
For n\ge1, signed coordinate functions attain the bound at m=0. If 0<|m|<1 and 0<|\rho|\le1, the refined upper curve is strictly below the universal curve \psi(|\rho|).
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every n\ge1, every Boolean function f:\{-1,1\}^n\to\{-1,1\}, and every \rho\in[-1,1], writing m=\E f, one has
H(m)-\E H(T_\rho f)\le1-H(\rho).
Both expectations are with respect to the uniform probability measure.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
Two more possible matches
Theorem 1.1(Sharp soft-channel contraction)in Sharp binary-information contraction on the discrete cube, 24 September 2026.
Theorem 1.3(Sharp soft production)in Sharp binary-information contraction on the discrete cube, 24 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At fixed initial information, a noisy coordinate retains the most information. The Boolean specialization resolves the Courtade–Kumar conjecture and gives an output-entropy refinement. We also establish a stronger mean-dependent entropy-production bound. The proof combines an explicit three-point optimizer for the local joining problem, two entropy capacities, and a common-output thinning inequality, followed by dimension induction and integration along the noise semigroup.
We prove the Hellinger conjecture for Boolean functions on the uniform discrete cube, with arbitrary output bias. For a Boolean function of mean m and every \(\rho\in[-1,1]\), the loss \(\sqrt{1-m^2}-\mathbb E\sqrt{1-(T_\rho f)^2}\) is at most \(1-\sqrt{1-\rho^2}\), with equality for signed coordinates. The proof combines asymmetric dimension induction, a calibrated noise-semigroup energy estimate, and finite exact arithmetic certificates. The Hellinger inequality also yields the Courtade–Kumar information bound.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
For every finite n\ge0,\rho\in[-1,1], and f:\Omega_n\to\{-1,1\}, put m=\E f and r_m=\psi^{-1}(H(m)). Then
I(f(X);Y)\le\psi(|\rho|r_m)\le\psi(|\rho|).
For n\ge1, signed coordinate functions attain the bound at m=0. If 0<|m|<1 and 0<|\rho|\le1, the refined upper curve is strictly below the universal curve \psi(|\rho|).
The words they share: attain, bias, boolean, bound, coordinate, functions, output, refined.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:Courtade-Kumar (with attainment) and the soft/biased-Boolean mutual-information contraction are stated; the Hellinger conjecture is not.
Definitions the group asks readers to compare with the paper:Information-theoretic quantities are re-implemented by hand (mutualInformation with an if 0 < joint guard and natural logs divided by ell := Real.log 2, psi, H, psiInv := Function.invFunOn psi (Set.Icc 0 1), production, hybrid), not Mathlib objects; the SoftChannel statement also bundles seven properties (FullMain)
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: 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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:The Hellinger conjecture (the October/September Hellinger paper’s main theorem) is not stated anywhere in the family’s challenges.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization proves sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At each fixed initial information level, a noisy coordinate channel attains the maximum retained information. It also proves the refined Boolean bound that accounts for output bias, with strict improvement for nonconstant biased outputs at nonzero noise correlation, and the selected mean-dependent entropy-production inequality.
For Boolean functions this includes the Courtade–Kumar inequality $I(f(X);Y)\le1-h_2(\varepsilon)$ in bits at crossover probability $0\le\varepsilon\le1/2$, where $h_2$ is binary entropy. A coordinate and its complement attain equality.
AI-drafted comparisons, with no recorded human review
Courtade–Kumar inequality and attainmentCourtadeKumar
Unreviewed AI draft: states a consequence of a paper’s theorem. States the Courtade–Kumar inequality I(f(X);Y) ≤ 1 − h₂(ε) for Boolean functions with attainment by a coordinate, which the paper obtains as the Boolean specialisation of its sharp soft-channel contraction theorem.
Sharp binary-channel contraction and entropy productionSoftChannel204
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (sharp contraction of binary-channel information under symmetric noise) with the bias-refined Boolean corollary and the selected entropy-production inequality.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Courtade–Kumar inequality and attainmentCourtadeKumar
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the scalar certificates of the proof — the induction parameters, the 17 profiles’ five Bernstein sign checks, the three-coordinate base table, the 21-box pair interval certificate, and every finite comparison and Bernstein/table entry of the central, intermediate and large-bias bands; NOT the theorem, whose induction, continuation and norm-inequality arguments are analytic”
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Vu Khac Ky, The Hellinger Conjecture for Antipodal Boolean Functions, 7 October 2026. Source
Vu proves the Hellinger conjecture for antipodal Boolean functions with a self-contained spectral argument and cites OpenAI’s family 119 entry as claiming the inequality for arbitrary output bias and all noise correlations.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 119No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Address a question raised in an independent assessment
Family 119Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Corollary 1.2 (The Boolean bound with output bias)
In Sharp binary-information contraction on the discrete cube, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
coordinate, functions
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Why this passage is shown
Theorem 1.1 (Hellinger conjecture)
In Hellinger contraction with arbitrary Boolean output bias, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
boolean, uniform
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Why this passage is shown
Theorem 1.1 (Sharp soft-channel contraction)
In Sharp binary-information contraction on the discrete cube, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
coordinate
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Why this passage is shown
Theorem 1.3 (Sharp soft production)
In Sharp binary-information contraction on the discrete cube, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“partial”
What their term means
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
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.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 341 s; peak memory 6.4 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the scalar certificates of the proof — the induction parameters, the 17 profiles’ five Bernstein sign checks, the three-coordinate base table, the 21-box pair interval certificate, and every finite comparison and Bernstein/table entry of the central, intermediate and large-bias bands; NOT the theorem, whose induction, continuation and norm-inequality arguments are analytic”
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.