Beyond the square-root exponent for depth-three circuits
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs a single language in deterministic polynomial time whose n-bit membership function requires \(2^{\omega(\sqrt n)}\) total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds. (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.
There is a language L\subseteq\bits^*, a deterministic Turing machine deciding L, and fixed positive integers C,a such that the machine runs in time at most C(n+1)^a on every input of length n. If f_n is the membership function of L on \bits^n, then
We construct a language in deterministic polynomial time whose n-bit membership function requires \(2^{\omega(\sqrt n)}\) gates in an unbounded-fan-in OR–AND–OR circuit. The bound holds at every sufficiently large input length and counts all gates, including the bottom layer.
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.
There is a language L\subseteq\bits^*, a deterministic Turing machine deciding L, and fixed positive integers C,a such that the machine runs in time at most C(n+1)^a on every input of length n. If f_n is the membership function of L on \bits^n, then
DepthThree.lean, the release lists it as a main result.
theoremexists_polynomial_time_language_depth_three_lower_bound : ∃ (L : List Bool → Bool) (M : FiniteMultiTapeMachine) (C a : ℕ),0 < C ∧ 0 < a ∧ (∀ w : List Bool, MultiTapeHaltsIn M w (L w) (C * (w.length + 1) ^ a)) ∧ ∀ A : ℝ, 0 < A → ∃ N : ℕ, ∀ n : ℕ, N ≤ n → ∀ D : Circuit3 (Fin n), D.Computes (fun x => L (List.ofFn x)) → (2 : ℝ) ^ (A * Real.sqrt (n : ℝ)) < (D.gateCount : ℝ) := 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.
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.
The source scan also found 1 theorem or lemma declarations whose names contain ‘manuscript’.
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; as the docs say, no fixed exponent n^{1/2+eps} and no unrestricted-depth bound is asserted.
Definitions the group asks readers to compare with the paper:Polynomial time is measured on a hand-rolled FiniteMultiTapeMachine (finite tapes/alphabet/states, Mathlib Turing.Tape, MultiTapeHaltsIn = halts within C*(|w|+1)^a steps), not Mathlib’s TM2ComputableInPolyTime; the circuit class Circuit3 is a custom inductive encoding with free negations/constants (standard)
“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 gives one polynomial-time Boolean language whose exact depth-three OR–AND–OR circuit size eventually exceeds $2^{A\sqrt n}$ for every fixed $A>0$. The language and its polynomial-time algorithm are fixed before $A$ is chosen, while circuits may vary with the input length. Gates have unrestricted finite fan-in and sharing, with free input negations and constants. The result does not assert a fixed exponent $n^{1/2+\varepsilon}$ or a lower bound for unrestricted depth.
AI-drafted comparisons, with no recorded human review
Depth-three circuit lower bound beyond every square-root constantDepthThree
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: one polynomial-time language whose depth-three circuit size eventually exceeds 2^{A√n} for every fixed A > 0.
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.
Depth-three circuit lower bound beyond every square-root constantDepthThree
Two 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 proof and rates its significance, if it holds up, as “Career-defining publication”. 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 112The 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
Address a question raised in an independent assessment
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.