Lean statement

Subsequential savings for exact Fourier circuits

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.ExactFourier.main_theorem

Lean statement file: ExactFourier.leanShow on the mapFamily 130. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

def MainStatement : Prop :=  ∀ c : ℝ, 0 < c → ∀ N₀ : ℕ, 2 ≤ N₀ → ∃ n : ℕ, N₀ ≤ n ∧    ∃ C : Circuit n, C.Computes (fourierMatrix n) ∧      (C.size : ℝ) < c * (n : ℝ) * Real.logb 2 (n : ℝ)⋮theorem main_theorem : MainStatement := by  sorry

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.

inductive type OAI.ExactFourier.Gate line 13

inductive Gate (w : ℕ) where  | add (i j : Fin w)  | sub (i j : Fin w)  | scale (c : ℂ) (i : Fin w)

definition OAI.ExactFourier.Gate.eval line 18

def Gate.eval {w : ℕ} (v : Fin w → ℂ) : Gate w → ℂ  | .add i j => v i + v j  | .sub i j => v i - v j  | .scale c i => c * v i

inductive type OAI.ExactFourier.Program line 24

inductive Program (n : ℕ) : ℕ → Type where  | nil : Program n 0  | step {k : ℕ} (p : Program n k) (g : Gate (n + 1 + k)) : Program n (k + 1)

structure OAI.ExactFourier.Circuit line 36

structure Circuit (n : ℕ) where  size : ℕ  program : Program n size  outputs : Fin n → Fin (n + 1 + size)

definition OAI.ExactFourier.zeta line 44

noncomputable def zeta (n : ℕ) : ℂ := Complex.exp (2 * Real.pi * Complex.I / (n : ℂ))

definition OAI.ExactFourier.fourierMatrix line 45

noncomputable def fourierMatrix (n : ℕ) : Matrix (Fin n) (Fin n) ℂ :=  fun j k => zeta n ^ (j.val * k.val)

definition OAI.ExactFourier.Circuit.Computes line 48

def Circuit.Computes {n : ℕ} (C : Circuit n) (A : Matrix (Fin n) (Fin n) ℂ) : Prop :=  ∀ x, C.eval x = A.mulVec x

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: charging, gate, circuits, complex, scalar and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subtractions, additions, multiplication and log.

Statements in Finite tensor savings and exact Fourier circuits come first because the release’s name for this Lean statement shares words with that paper’s title.

Proposition 3.4 of Finite tensor savings and exact Fourier circuits

Proposition (Transfer of a finite win)

If a finite win exists, there are integers n_m\to\infty and exact linear circuits for F_{n_m} whose scalar gate counts satisfy

L(n_m)=o(n_m\log_2n_m).

In particular the conclusion is in the model charging additions, subtractions, and multiplication by every prechosen complex scalar.

Section 3, From a finite win to exact Fourier circuits Read it in the paper TeX source, line 310

Show the TeX

If a finite win exists, there are integers $n_m\to\infty$ and exact
linear circuits for $F_{n_m}$ whose scalar gate counts satisfy
\[
 L(n_m)=o(n_m\log_2n_m).
\]
In particular the conclusion is in the model charging additions,
subtractions, and multiplication by every prechosen complex scalar.

Other possible paper matches, closest wording first

Theorem 1.1 of Finite tensor savings and exact Fourier circuits. Words of the Lean statement, its names split into words, that also appear in the theorem: circuit and integer. Found in the paper’s title, which its main theorem states: fourier. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: fewer, gates and log.
Corollary 5.6 of Finite tensor savings and exact Fourier circuits. Words of the Lean statement, its names split into words, that also appear in the theorem: dag, one, matrix and scalar. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gates and bound.
Theorem A.4 of An explicit power saving for the exact discrete Fourier transform. Words of the Lean statement, its names split into words, that also appear in the theorem: pi, exp, output, program, zeta, fourier and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sequence, bit, gates, bounded and log.
Corollary 4.3 of Finite tensor savings and exact Fourier circuits. Words of the Lean statement, its names split into words, that also appear in the theorem: values, one and matrix. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound.
Proposition 2.4 of An explicit power saving for the exact discrete Fourier transform. Words of the Lean statement, its names split into words, that also appear in the theorem: steps, permutations, arbitrary, one and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bits.
Theorem 1.1 of An explicit power saving for the exact discrete Fourier transform. Words of the Lean statement, its names split into words, that also appear in the theorem: zeta, computes, scalar and integer. Found in the paper’s title, which its main theorem states: fourier. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: log.

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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 195 s; peak memory 6.6 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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 130’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 agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: UniformFourier.transform_main (DFTGoal) states that one fixed pair of closed RAM programs (order, solve) outputs dft n x exactly for every n > 0 and every x, with total work including root-order selection (d.work) and scalar preparation at most W n, where W =O n(log n)^(1−10^-13) (also the paper's θ-bound and o(n log n)); unfolding DFTGoal, DFTProgram and TimeBounds is the only step, and convolution_main likewise states Corollary 1.3. ExactFourier is the separate subsequential circuit result (for each c and N₀ some n ≥ N₀ has an exact circuit with fewer than c n log₂ n gates), which is the other paper's theorem and not the headline claim.
Definitions the group asks readers to compare with the paper: UniformFourier RAM model (Tape, Bill, Atom, Code, Code.run, Prog, run): bespoke typed point-free RAM with unit-cost exact complex field operations and unit-cost word operations; the meaning of 'O(...)' rests entirely on its cost semantics. I found no undercharging: every Code node and Atom pays at least 1, tab/sow pay 1+len, loops and bounded descents pay per iteration/level, and arrays are never copied for free ; UniformFourier.DFTProgram / ConvProgram: the program itself selects the root order d = run order n (0 < d < 1024 n³, its work counted) and then receives root d := ExactFourier.zeta d for free; the target is dft n with ζ_n, and every scalar the program can form lies in ℚ(ζ_d), so d must give ζ_n ∈ ℚ(ζ_d) (n | d, or d odd and n | 2d); matches the paper's 'specified root ζ_{D_*}, D_* < 1024 n³' ; UniformFourier Atom set: the only complex constants are 0 (cz) and 1 (cone); every other scalar must be prepared from the supplied root by counted add/sub/scale/inv on scalar registers; data registers (Paint.left/right/both) cannot become scalars or integers, and ifz tests only integers, so control flow is data-independent; 'cross' (data × data) exists only when allow = true (convolution), and only left × right ↦ both ; UniformFourier Bill.valid (a Prop): the side condition that every inv has a nonzero argument on the executed path; DFTProgram and ConvProgram require d.valid and result.valid, so division by zero is excluded rather than tested ; UniformFourier cap := (n+2)^cBound with ∃ cBound: forces work, root order and every produced integer (peak) to be polynomial in n, i.e. O(log n)-bit words; this is the 'logarithmic-word indexing' of the summary ; UniformFourier TimeBounds: W =O paperTime ∧ W =O decimalTime ∧ W =o nlogn with natural logs; asymptotic only (finitely many n are constrained only by correctness and cap); paperTime uses the paper's constants m = 10^6, W_* = 2^71, Δ = 6871402692000000 and θ = log(m − Δ/W_*)/log m; decimalTime n = n (log n)^(1 − 10^-13) ; UniformFourier dft / conv: dft n x j = Σ_k ζ_n^(jk) x_k with ζ_n = exp(+2πi/n) (unnormalized, same sign convention as the paper's F_n); conv is the full linear convolution on Fin (2n−1) ; ExactFourier Gate / Program / Circuit: linear scalar DAG with n inputs, an extra constant 0, free reuse and fan-out, outputs naming any available value (permutations free), and scale (c : ℂ) with arbitrary complex c at cost one; coefficient generation is uncharged (nonuniform); standard for the paper's model ; ExactFourier MainStatement: subsequential (∃ n ≥ N₀ for each c and N₀), not an all-length bound; this is the intended statement of the finite-tensor paper's thm:main
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result gives exact discrete Fourier transforms with arbitrarily small normalized circuit cost along an unbounded sequence of lengths. For every $c>0$ and every cutoff $N_0\ge2$, some $n\ge N_0$ has a circuit computing the unnormalized DFT with fewer than $cn\log_2 n$ gates. Addition, subtraction, and multiplication by a predetermined complex scalar each cost one gate; diagonal scalings are charged. The result is subsequential, with no all-length, bounded-coefficient, conditioning, or bit-complexity claim.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

62 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: 13 to its own declarations, 22 to Mathlib’s documentation. 10 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_theorem exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 10 other declarations of the file; with the theorem itself they span 34 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.ExactFourier are listed without it.

inductive Gateline 13library
def Gate.evalline 18library
inductive Programline 24library
def Program.evalline 28library
structure Circuitline 36library
def Circuit.evalline 41library
def zetaline 44library
def fourierMatrixline 45library
def Circuit.Computesline 48library
def MainStatementline 52in the statementlibrary
theorem main_theoremline 57selected theorem

import Mathlibnamespace OAI/-!Scalar circuits for exact discrete Fourier transforms.No bounds on predetermined complex coefficients, reuse, storage, or conditioning.-/namespace ExactFourier/-- One charged scalar gate, referencing only available values. -/inductive Gate (w : ℕ) where  | add (i j : Fin w)  | sub (i j : Fin w)  | scale (c : ℂ) (i : Fin w)def Gate.eval {w : ℕ} (v : Fin w → ℂ) : Gate w → ℂ  | .add i j => v i + v j  | .sub i j => v i - v j  | .scale c i => c * v i/-- A topologically ordered scalar DAG; available values are never consumed. -/inductive Program (n : ℕ) : ℕ → Type where  | nil : Program n 0  | step {k : ℕ} (p : Program n k) (g : Gate (n + 1 + k)) : Program n (k + 1)def Program.eval {n : ℕ} : {k : ℕ} → Program n k → (Fin n → ℂ) →    (Fin (n + 1 + k) → ℂ)  | 0, .nil, x => Fin.snoc x 0  | _ + 1, .step p g, x =>      let v := p.eval x      Fin.snoc v (g.eval v)/-- Outputs name available values, so permutations and arbitrary fanout are uncharged. -/structure Circuit (n : ℕ) where  size : ℕ  program : Program n size  outputs : Fin n → Fin (n + 1 + size)def Circuit.eval {n : ℕ} (C : Circuit n) (x : Fin n → ℂ) : Fin n → ℂ :=  fun i => C.program.eval x (C.outputs i)noncomputable def zeta (n : ℕ) : ℂ := Complex.exp (2 * Real.pi * Complex.I / (n : ℂ))noncomputable def fourierMatrix (n : ℕ) : Matrix (Fin n) (Fin n) ℂ :=  fun j k => zeta n ^ (j.val * k.val)def Circuit.Computes {n : ℕ} (C : Circuit n) (A : Matrix (Fin n) (Fin n) ℂ) : Prop :=  ∀ x, C.eval x = A.mulVec x/-- The precise quantified formulation of source theorem `thm:main`. -/def MainStatement : Prop :=  ∀ c : ℝ, 0 < c → ∀ N₀ : ℕ, 2 ≤ N₀ → ∃ n : ℕ, N₀ ≤ n ∧    ∃ C : Circuit n, C.Computes (fourierMatrix n) ∧      (C.size : ℝ) < c * (n : ℝ) * Real.logb 2 (n : ℝ)theorem main_theorem : MainStatement := by  sorryend ExactFourierend 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.

Finite tensor savings and exact Fourier circuits

Theorem

In this model,

\liminf_{n\to\infty}\frac{L(n)}{n\log_2 n}=0.

Equivalently, for every c>0 and every integer N_0\ge2, there is an integer n\ge N_0 and an exact circuit for F_n with fewer than c n\log_2 n gates.

Section 1, Introduction Read it in the paper TeX source, line 46

Show the TeX

In this model,
\[
 \liminf_{n\to\infty}\frac{L(n)}{n\log_2 n}=0.
\]
Equivalently, for every $c>0$ and every integer $N_0\ge2$, there is an
integer $n\ge N_0$ and an exact circuit for $F_n$ with fewer than
$c n\log_2 n$ gates.

An explicit power saving for the exact discrete Fourier transform

Theorem

There is a single deterministic algorithm which, for every positive integer n and every x\in\C^n, computes F_nx exactly. In the model above, including scalar preparation, schedule construction, and index work, its running time is

O\!\left(n(\log n)^\theta(\log\log n)^{4-\theta}\right)

as n\to\infty. It suffices to supply the specified root \zeta_{D_*} for an explicitly computable integer D_*<1024n^3.

Section 1, Introduction Read it in the paper TeX source, line 85

Show the TeX

There is a single deterministic algorithm which, for every positive integer $n$ and every $x\in\C^n$, computes $F_nx$ exactly. In the model above, including scalar preparation, schedule construction, and index work, its running time is
\begin{equation}
 O\!\left(n(\log n)^\theta(\log\log n)^{4-\theta}\right)
\end{equation}
as $n\to\infty$. It suffices to supply the specified root $\zeta_{D_*}$ for an explicitly computable integer $D_*<1024n^3$.

Corollary

In the same model, the transform is computable at every length in

O\!\left(n(\log n)^{1-10^{-13}}\right)=o(n\log n)

time.

Section 1, Introduction Read it in the paper TeX source, line 93

Show the TeX

In the same model, the transform is computable at every length in
\[
 O\!\left(n(\log n)^{1-10^{-13}}\right)=o(n\log n)
\]
time.

Corollary (Exact convolution and polynomial multiplication)

There is a single deterministic algorithm which, for every positive integer n and every a,b\in\C^n, computes the linear convolution

\begin{gathered}c_k=\sum_{\substack{0\le r,s<n\\r+s=k}}a_rb_s, \qquad 0\le k\le 2n-2.\end{gathered}

Equivalently, it multiplies two complex polynomials of degree less than n. In the exact complex field-operation, specified-root, and address cost model above, with general field multiplication permitted for the pointwise products, the total running time, including scalar preparation, schedule construction, padding, and index work, is

O\!\left(n(\log n)^\theta(\log\log n)^{4-\theta}\right) \qquad(n\to\infty),

and hence is eventually O(n(\log n)^{1-10^{-13}}). This is a bilinear arithmetic algorithm, not an input-linear circuit: its pointwise products multiply a value linear in a by a value linear in b.

Section 1, Introduction Read it in the paper TeX source, line 103

Show the TeX

There is a single deterministic algorithm which, for every positive integer
$n$ and every $a,b\in\C^n$, computes the linear convolution
\[
 c_k=\sum_{\substack{0\le r,s<n\\r+s=k}}a_rb_s,
 \qquad 0\le k\le 2n-2.
\]
Equivalently, it multiplies two complex polynomials of degree less than $n$.
In the exact complex field-operation, specified-root, and address cost model
above, with general field multiplication permitted for the pointwise products,
the total running time, including scalar preparation, schedule construction,
padding, and index work, is
\[
 O\!\left(n(\log n)^\theta(\log\log n)^{4-\theta}\right)
 \qquad(n\to\infty),
\]
and hence is eventually $O(n(\log n)^{1-10^{-13}})$.
This is a bilinear arithmetic algorithm, not an input-linear circuit: its
pointwise products multiply a value linear in $a$ by a value linear in $b$.

Configuration and file facts

File: lean/ComparatorChallenges/ExactFourier.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Computability/FourierCircuit/Main.lean
Allowed axioms: 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: 10 local vocabulary declarations
Lines of Lean, without comments or blank lines: 34
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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