Average sensitivity of polynomial threshold functions
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most \(8d\sqrt n\), uniformly for \(1\le d\le n\). Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with \(\mathop{\mathrm{sign}}\nolimits (0)=1\). (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
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Let n\ge1 and 1\le d\le n be integers. Let p be a real multilinear polynomial of degree at most d, and define f:\{-1,1\}^n\to\{-1,1\} by f(x)=\sgn(p(x)), with \sgn(0)=1. Then, for average sensitivity under the uniform law,
I(f)\le 8d\sqrt n.
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.
For every n ≥ 1 and \(1\le d\le n\), we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most \(8d\sqrt n\). This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention \(\mathop{\mathrm{sgn}}\nolimits (0)=1\).
One Lean statement in this family
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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.
Let n\ge1 and 1\le d\le n be integers. Let p be a real multilinear polynomial of degree at most d, and define f:\{-1,1\}^n\to\{-1,1\} by f(x)=\sgn(p(x)), with \sgn(0)=1. Then, for average sensitivity under the uniform law,
I(f)\le 8d\sqrt n.
The words they share: multilinear, average, sensitivity, real, sqrt, degree, polynomial, integers, threshold, functions.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
GotsmanLinial.lean, the release lists it as a main result.
defGotsmanLinialStatement : Prop := ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d → d ≤ n → ∀ p : MvPolynomial (Fin n) ℝ, IsMultilinear p → p.totalDegree ≤ d → averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ)⋮theoremgotsmanLinialStatement : GotsmanLinialStatement := bysorry
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”.
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?
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:None; the exact-extremizer form of the Gotsman-Linial conjecture is not claimed.
“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 formalized result proves the average-sensitivity bound $8d\sqrt n$ for every Boolean threshold function defined by a real multilinear polynomial of degree at most $d$ on the $n$-dimensional cube. The sign convention assigns value $1$ at zero. This establishes the stated asymptotic bound, rather than the separate conjecture identifying exact extremizers.
AI-drafted comparisons, with no recorded human review
Average sensitivity of polynomial threshold functionsGotsmanLinial
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with its constant: average sensitivity at most 8d√n for every degree-d polynomial threshold function on the n-cube, with sign 1 at zero.
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.
Average sensitivity of polynomial threshold functionsGotsmanLinial
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
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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 “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 127The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 127Atlas 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
In Average sensitivity of polynomial threshold functions, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
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 143 s; peak memory 3.3 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.
Average sensitivity of polynomial threshold functions
Role
the release lists it as a main result
Paired with
Theorem 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.