Boolean functions violate the square-root degree bound by arbitrary factors
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Disproves the proposed square-root bound relating a Boolean function’s linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with \(\sum_i\widehat f(\{i\})\gt C\sqrt{\deg(f)}\). Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function \(f:\{-1,1\}^n\to\{-1,1\}\) on a finite sign cube such that \(\displaystyle \sum_{i=1}^n \widehat f(\{i\})\gt C\sqrt{\deg(f)}.\) Here \(\widehat f(\{i\})\) is the linear Fourier coefficient associated with the ith input, and \(\deg(f)\) is the degree of the real multilinear polynomial representing f.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
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.
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”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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 or imply the whole advertised result (their term: “full”).
Their note:main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios with SignedViolations := ∀ C > 0, ∃ n > 0, ∃ f : Cube (Fin n) → ℝ, IsBoolean f ∧ Nonconstant f ∧ C * √(fourierDegree f) < singletonSum f; Fourier coefficients are the uniform-average E[f * χ_s] over the ±1 cube and fourierDegree is the max |s| with nonzero coefficient (real multilinear degree, as the file comment says). Standard definitions, matches the summary exactly.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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.
The formalization disproves the Gopalan–Servedio square-root degree conjecture by an unbounded factor. For every $C>0$, it gives a nonconstant Boolean function on a finite sign cube for which the sum of its linear Fourier coefficients exceeds $C\sqrt{\deg f}$, where $\deg f$ is its real multilinear degree.
It also proves that the ratio of the sum of the absolute values of those coefficients to $\sqrt{\deg f}$ is unbounded among positive-degree Boolean functions.
AI-drafted comparisons, with no recorded human review
Unbounded violations of the square-root Fourier-degree boundSquareRootDegree
Unreviewed AI draft: states a paper’s theorem as the paper does. States both clauses of the main theorem: for every C > 0 a Boolean function whose linear Fourier coefficients sum to more than C√(deg f), and unboundedness of the corresponding ratio.
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.
Unbounded violations of the square-root Fourier-degree boundSquareRootDegree
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.
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 counterexample and rates its significance, if it holds up, as “Field-specific journal”. The page says that nothing on it is verified.
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
Identify the theorem the catalogue describes
Family 192No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Compare the proposed paper and Lean matches
Family 192Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In Unbounded Violations of the Square-Root Degree Bound, 26 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
boolean, function
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
“full”
What their term means
The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 the headline as OpenAI wrote it.
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 423 s; peak memory 6.4 GiB).
Date
10 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.
Unbounded violations of the square-root Fourier-degree bound
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.