Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.QAC.parity_lower_bound_polynomial_size
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
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
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
defmatrix {N : ℕ} : Gate N → Operator N | .local q u => tensorMatrix fun i => if i = q then (u : QubitOperator) else1 | .toffoli c t _ => fun x y => if x = toffoliWord c t y then1else0
defsuccessProbability {n N : ℕ} (W : Operator N) (out : Fin N) (x : Word n) : ℝ := ∑ y : Word N, if y out = parity x then Complex.normSq (W y (inputWord x)) else0
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: toffoli, unbounded, unitaries, arity, supports, disjoint, gates, lower and 14 more. Found in the paper’s title, which its main theorem states: product. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measured, start, number, large, sufficiently and output.
Statements in Product-projection localization and the QAC0 parity lower bound come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1 of Product-projection localization and the QAC0 parity lower bound
Theorem(Parity lower bound)
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.
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.
Other possible paper matches, closest wording first
Corollary 4 of Product-projection localization and the QAC0 parity lower bound. Words of the Lean statement, its names split into words, that also appear in the theorem: success, polynomially, many, zero, computes, least and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measured, large, sufficiently and output.
Corollary 1 of Product-projection localization and the QAC0 parity lower bound. Words of the Lean statement, its names split into words, that also appear in the theorem: remaining, final, zero, integer, depth, total and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: large and sufficiently.
Corollary 3 of Product-projection localization and the QAC0 parity lower bound. Words of the Lean statement, its names split into words, that also appear in the theorem: polynomially, functions, many, computes, least, probability and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: large and sufficiently.
Theorem 2 of Product-projection localization and the QAC0 parity lower bound. Words of the Lean statement, its names split into words, that also appear in the theorem: lower, real, bound, integer, norm, product and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: numbers, large and sufficiently.
Theorem 1.1 of Regular trajectories, pruning and quantum parity. Words of the Lean statement, its names split into words, that also appear in the theorem: real, probability, parity, integers, depth, total and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: strictly.
Corollary 2 of Product-projection localization and the QAC0 parity lower bound. Words of the Lean statement, its names split into words, that also appear in the theorem: function, integers, depth, total, qubits, inputs and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: quantum, large, sufficiently and output.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 136 s; peak memory 6.3 GiB).
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 274’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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.
OpenAI’s own note on what this Lean statement covers
“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.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a weaker form of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
75 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 36 to its own declarations, 24 to Mathlib’s documentation. 16 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
import Mathlibnamespace OAInoncomputablesectionopenscoped BigOperators Matrix.Norms.L2Operatornamespace QACabbrevBit := Fin2abbrevWord (N : ℕ) := Fin N → BitabbrevOperator (N : ℕ) := Matrix (Word N) (Word N) ℂabbrevQubitOperator := MatrixBitBitℂ/-- Tensor product in the computational basis. -/deftensorMatrix {N : ℕ} (a : Fin N → QubitOperator) : Operator N :=fun x y => ∏ i, a i (x i) (y i)/-- A one-qubit unitary or an unbounded-arity Toffoli gate. -/inductiveGate (N : ℕ) where | local (q : Fin N) (u : Matrix.unitaryGroupBitℂ) | toffoli (controls : Finset (Fin N)) (target : Fin N) (hne : target ∉ controls)namespace Gatedefsupport {N : ℕ} : Gate N → Finset (Fin N) | .local q _ => {q} | .toffoli c t _ => insert t c/-- A Toffoli flips its target iff all controls are one. -/deftoffoliWord {N : ℕ} (c : Finset (Fin N)) (t : Fin N) (x : Word N) : Word N :=if ∀ i ∈ c, x i = 1thenFunction.update x t (1 - x t) else xdefmatrix {N : ℕ} : Gate N → Operator N | .local q u => tensorMatrixfun i => if i = q then (u : QubitOperator) else1 | .toffoli c t _ => fun x y => if x = toffoliWord c t y then1else0end GatestructurePhysicalLayer (N : ℕ) wheregates : List (Gate N)disjoint : gates.Pairwise (fun A B => Disjoint A.support B.support)defphysicalLayerMatrix {N : ℕ} (L : PhysicalLayer N) : Operator N := (L.gates.map Gate.matrix).prod/-- Chronological order: the rightmost matrix acts first. -/defphysicalCircuitMatrix {N : ℕ} (layers : List (PhysicalLayer N)) : Operator N := ((layers.map physicalLayerMatrix).reverse).prod/-- The computational-basis input with the remaining qubits initialized to zero. -/definputWord {n N : ℕ} (x : Word n) : Word N :=fun i => if h : i.val < n then x ⟨i.val, h⟩ else0defparity {n : ℕ} (x : Word n) : Bit := ∑ i, x i/-- Born probability, summing all final garbage with no cleanliness assumption. -/defsuccessProbability {n N : ℕ} (W : Operator N) (out : Fin N) (x : Word n) : ℝ := ∑ y : Word N, if y out = parity x thenComplex.normSq (W y (inputWord x)) else0/-- Fixed-depth circuits with polynomially many total qubits cannot compute paritywith worst-case success at least two thirds. -/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 := bysorryend QACendend OAI
Every extracted statement of the accompanying papers
The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.
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.
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.
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.
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 $1/2+\varepsilon$.
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
15 local vocabulary declarations
Lines of Lean, without comments or blank lines
36
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Polynomial-size parity specialization
Family 274: Parity is not in QAC0
Candidate pairing
Theorem (Parity lower bound)
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.
Product-projection localization and the QAC0 parity lower bound
TeX label thm:parity, build/sections/introduction.tex, line 25
The Lean statement says
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
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Two independent readers, MathVet and cert-machine, disagree on whether this family's challenges state its headline; a specialist reading would settle it.