OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves Moore’s parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li’s reductions give the same bounded-error obstruction for strict majority. (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 two independent assessments disagree on how much of the result described above is expressed by the Lean statements.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.
Fix a nonnegative integer d, a real number c\ge1, and 0<\varepsilon\le1/2. For all sufficiently large n, no circuit of depth at most d on n input qubits and at most n^c total qubits, using arbitrary one-qubit unitaries and unbounded-arity Toffoli gates, computes \parity(x) with probability at least 1/2+\varepsilon for every x\in\{0,1\}^n. Gates in each layer have disjoint supports, ancillas start in \ket0, and one output qubit is measured; the other qubits may be discarded.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Fix positive integers d,k,C and a real 0<\varepsilon\le1/2. There is an integer n_0=n_0(d,k,C,\varepsilon) such that, for every n\ge n_0, every allowed circuit of depth at most d on at most C(n+1)^k total qubits has an input x\in\{0,1\}^{n} on which its answer equals \Parity_n(x) with probability strictly less than 1/2+\varepsilon.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.
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 that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves Moore’s parity conjecture in the measured-output model.
We prove that constant-depth quantum circuits with arbitrary one-qubit gates and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many total qubits. Ancillary qubits are initialized to \(|0\rangle\), one output qubit is measured, and all other final registers may be discarded without restriction. This resolves Moore’s parity conjecture in the measured-output model.
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.
Fix a nonnegative integer d, a real number c\ge1, and 0<\varepsilon\le1/2. For all sufficiently large n, no circuit of depth at most d on n input qubits and at most n^c total qubits, using arbitrary one-qubit unitaries and unbounded-arity Toffoli gates, computes \parity(x) with probability at least 1/2+\varepsilon for every x\in\{0,1\}^n. Gates in each layer have disjoint supports, ancillas start in \ket0, and one output qubit is measured; the other qubits may be discarded.
QACParity.lean, the release lists it as a main result.
defParityStatement : Prop := ∀ (d : ℕ) (c ε : ℝ), 1 ≤ c → 0 < ε → ε ≤ 1 / 2 → ∃ n₀ : ℕ, ∀ n ≥ n₀, ∀ N : ℕ, n ≤ N → (N : ℝ) ≤ (n : ℝ) ^ c → ∀ layers : List (PhysicalLayer N), layers.length ≤ d → ∀ out : Fin N, ¬ (∀ x : Word n,1 / 2 + ε ≤ successProbability (physicalCircuitMatrix layers) out x)⋮theoremQAC.parity_lower_bound : QAC.ParityStatement := 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”.
Fix a nonnegative integer d, a real number c\ge1, and 0<\varepsilon\le1/2. For all sufficiently large n, no circuit of depth at most d on n input qubits and at most n^c total qubits, using arbitrary one-qubit unitaries and unbounded-arity Toffoli gates, computes \parity(x) with probability at least 1/2+\varepsilon for every x\in\{0,1\}^n. Gates in each layer have disjoint supports, ancillas start in \ket0, and one output qubit is measured; the other qubits may be discarded.
RegularParity.lean, the release lists it as a main result.
theoremparity_lower_bound_polynomial_size (d k C : ℕ) (_hd : 0 < d) (_hk : 0 < k) (_hC : 0 < C) : ∃ n₀ : ℕ, ∀ n ≥ n₀, ∀ N : ℕ, n ≤ N → N ≤ C * (n + 1) ^ k → ∀ layers : List (PhysicalLayer N), layers.length ≤ d → ∀ out : Fin N, ∃ x : Word n, successProbability (physicalCircuitMatrix layers) out x < (2 : ℝ) / 3 := 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 disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
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:QAC.parity_lower_bound : ParityStatement = ∀ d, c ≥ 1, 0 < ε ≤ 1/2, ∃ n₀, ∀ n ≥ n₀, ∀ N, n ≤ N ≤ n^c, ∀ layers (≤ d layers of disjoint one-qubit unitaries / unbounded-arity Toffolis), ∀ out, ¬ (∀ x : Word n, 1/2 + ε ≤ successProbability (physicalCircuitMatrix layers) out x) with ancillas initialized to zero (inputWord) and successProbability the Born probability of the measured output qubit equalling the parity, summing over all garbage. This is exactly Moore’s parity conjecture in the measured-output model; RegularParity adds the explicit ‘< 2/3’ polynomial-size specialization. Xu-Li’s majority reduction is only cited in the summary.
“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 rules out bounded-error parity computation by constant-depth quantum circuits with polynomially many zero-initialized ancillary qubits. For every fixed depth, polynomial bound on the total number of qubits, and $0<\varepsilon\le1/2$, every sufficiently large input length has an input on which the measured-output parity success probability is less than $1/2+\varepsilon$. A companion specialization gives success strictly below $2/3$. The formalization also includes the product-projection localization estimate supporting this bound.
The formalized result supplies the polynomial-size parity consequence of the paper. For every fixed circuit depth and polynomial bound on the number of qubits, all sufficiently large input lengths have an input on which any such circuit computes measured-output parity with probability strictly below $2/3$. Ancillary qubits start at zero. The formalization also contains the general positive-advantage parity bound and its product-projection localization estimate; regular-trajectory propagation and pruning are not separate formalized statements here.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every fixed depth, polynomial qubit bound and ε in (0, 1/2], constant-depth quantum circuits compute parity with success below 1/2 + ε on some input of every large length.
Unreviewed AI draft: states a weaker form of a paper’s theorem. States the case of success probability below 2/3 (ε = 1/6) of the parity theorem, which the note calls the polynomial-size consequence; the theorem itself covers every ε in (0, 1/2].
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.
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.
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.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The doc lists further results as formalised that no Comparator theorem name covers.
Isaac Kim, Post listing quantum information results in the release, 6 October 2026. Source
Names parity not in QAC0.
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 274The 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 274Atlas 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 (Parity lower bound)
In Product-projection localization and the QAC0 parity lower bound
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 less than the headline, and OpenAI’s Lean note does not say so.
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 383 s; peak memory 6.3 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.
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 136 s; peak memory 6.3 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.