Lean statement

One rational matrix tuple hitting all bounded-size noncommutative formulas

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.NCHitting.universal_hitting

Lean statement file: FormulaHitting.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 116 link it.

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.

theorem universal_hitting (n s : ℕ) (hn : 1 ≤ n) (hs : 1 ≤ s)    (F : Type u_2) [Field F] [CharZero F] (f : Formula F n)    (hsize : f.size ≤ s) (hf : ∃ w, f.coeff w ≠ 0) :    f.eval (Matrix.scalar (Fin (dimension n s)))      (fun i r q => (generator n s i r q : F)) ≠ 0 := 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.

abbreviation OAI.NCHitting.Word line 8
```
abbrev Word (n : ℕ) := List (Fin n)
```

abbreviation OAI.NCHitting.WordSeries line 9

abbrev WordSeries (F : Type*) (n : ℕ) := Word n → F

definition OAI.NCHitting.convolution line 11

def convolution {F : Type*} {n : ℕ} [Semiring F]    (f g : WordSeries F n) : Word n → F  | [] => f [] * g []  | i :: w => f [] * g (i :: w) + convolution (fun v => f (i :: v)) g w

inductive type OAI.NCHitting.Formula line 16

inductive Formula (F : Type*) (n : ℕ) where  | scalar : F → Formula F n  | var : Fin n → Formula F n  | add : Formula F n → Formula F n → Formula F n  | mul : Formula F n → Formula F n → Formula F n

definition OAI.NCHitting.Formula.size line 24

def size {F : Type*} {n : ℕ} : Formula F n → ℕ  | scalar _ => 1  | var _ => 1  | add f g => f.size + g.size + 1  | mul f g => f.size + g.size + 1

definition OAI.NCHitting.Formula.coeff line 30

noncomputable def coeff {F : Type*} {n : ℕ} [Semiring F] :    Formula F n → WordSeries F n  | scalar a => fun w => if w = [] then a else 0  | var i => fun w => if w = [i] then 1 else 0  | add f g => fun w => coeff f w + coeff g w  | mul f g => convolution (coeff f) (coeff g)

definition OAI.NCHitting.Formula.eval line 37

def eval {F A : Type*} {n : ℕ} [Semiring A]    (scalarMap : F → A) (T : Fin n → A) : Formula F n → A  | scalar a => scalarMap a  | var i => T i  | add f g => eval scalarMap T f + eval scalarMap T g  | mul f g => eval scalarMap T f * eval scalarMap T g

definition OAI.NCHitting.dimension line 46

def dimension (n s : ℕ) : ℕ := if n = 1 ∧ s = 1 then 1 else 2 * n * s ^ 2

definition OAI.NCHitting.matrixEntry line 48

def matrixEntry (i r q : ℕ) : ℚ :=  if q < r then (-1 : ℚ) ^ (r - q - 1) / ((r : ℚ) * (i : ℚ) ^ (r - q))  else 0

definition OAI.NCHitting.generator line 52

def generator (n s : ℕ) : Fin n → Matrix (Fin (dimension n s)) (Fin (dimension n s)) ℚ :=  fun i r q => if n = 1 ∧ s = 1 then 1 else matrixEntry (i.val + 1) r.val q.val

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: entries, zero, integers and field. Found in the paper’s title, which its main theorem states: hitting, rational, formulas and matrix. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ns, construction, bit, deterministic, tuple and polynomial.

Statements in One Rational Matrix Hitting Point for Noncommutative Formulas come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of One Rational Matrix Hitting Point for Noncommutative Formulas

Theorem

There is a deterministic algorithm which, given integers n,s\ge1, constructs a tuple

\mathbf T_{n,s}\in\Mat_d(\Q)^n, \qquad d\le 2ns^2,

in time polynomial in n,s, measured in bit operations. The numerators and denominators of its entries have bit length O(\log(ns+1)+ns^2\log(n+1)). For every field F of characteristic zero and every 0\ne f\in\mathsf{Form}_{n,s}(F),

f(\mathbf T_{n,s})\ne0,

where the tuple is viewed over F through the canonical embedding \Q\hookrightarrow F. The construction uses d=2ns^2 when ns\ge2 and d=1 when n=s=1.

Section 1, Introduction Read it in the paper TeX source, line 24

Show the TeX

There is a deterministic algorithm which, given integers $n,s\ge1$,
constructs a tuple
\[
 \mathbf T_{n,s}\in\Mat_d(\Q)^n,
 \qquad d\le 2ns^2,
\]
in time polynomial in $n,s$, measured in bit operations. The numerators and
denominators of its entries have bit length
$O(\log(ns+1)+ns^2\log(n+1))$.
For every field $F$ of characteristic zero and every
$0\ne f\in\mathsf{Form}_{n,s}(F)$,
\[
 f(\mathbf T_{n,s})\ne0,
\]
where the tuple is viewed over $F$ through the canonical embedding
$\Q\hookrightarrow F$. The construction uses $d=2ns^2$ when $ns\ge2$
and $d=1$ when $n=s=1$.

Other possible paper matches, closest wording first

Theorem 1.1 of Uniform Matrix Hitting Points in Every Positive Characteristic. Words of the Lean statement, its names split into words, that also appear in the theorem: dimensions, entries, size, formula, integers, matrix and one more. Found in the paper’s title, which its main theorem states: hitting. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: property, deterministic and tuple.
Proposition 3.2 of One Rational Matrix Hitting Point for Noncommutative Formulas. Words of the Lean statement, its names split into words, that also appear in the theorem: size, zero, formula and matrix. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: polynomial.
Theorem 1.1 of Polynomial Hitting Lists for Noncommutative Rational Formulas. Words of the Lean statement, its names split into words, that also appear in the theorem: generator, list, rational, size, formula and integers. Found in the paper’s title, which its main theorem states: hitting. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: property, bits, deterministic and tuples.
Proposition 5.1 of One Rational Matrix Hitting Point for Noncommutative Formulas. Words of the Lean statement, its names split into words, that also appear in the theorem: hitting, rational, zero, integers and field. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bit, tuple and polynomial.
Proposition 2.2 of Uniform Matrix Hitting Points in Every Positive Characteristic. Words of the Lean statement, its names split into words, that also appear in the theorem: series, word, integer and field.
Corollary 6.1 of Polynomial Hitting Lists for Noncommutative Rational Formulas. Words of the Lean statement, its names split into words, that also appear in the theorem: generator, dimension, list, rational, entries, zero and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounded, bit, deterministic and polynomial.

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 40 s; peak memory 2.5 GiB).
Reported 10 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 116’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Characteristic-zero existence of one hitting tuple (no complexity claim) and the poly-time rational hitting lists are stated; positive-characteristic constructions and the polynomial-bit-time bound for the single tuple are not.
Definitions the group asks readers to compare with the paper: Hits quantifies over Admissible/Nonzero formulas via a hand-rolled relational Eval (inverse requires an actual two-sided inverse of the evaluated operand); machine is Mathlib Turing.TM0.Machine (Fin 4) (Fin (m+1)) with unary input n,1,s (time polynomial in n + s, not in a binary length)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: Every positive characteristic: no challenge states a hitting tuple over 𝔽_p or over fields of characteristic p (the October-4 manuscript 'Uniform Matrix Hitting Points in Every Positive Characteristic', which the scope note does not link). Also not asserted, but declared: the polynomial bit-time construction of the characteristic-zero tuple.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The selected theorem states this universal hitting property. The paper's deterministic polynomial bit-construction bound and matrix-dimension bound $O(ns^2)$ are not separately asserted in it.”
Quoted from OpenAI’s Lean note for this family

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

66 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: 43 to its own declarations, 15 to Mathlib’s documentation. 11 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 universal_hitting 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 40 lines. 6 of them are named in the statement itself.

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

abbrev Wordline 8library
abbrev WordSeriesline 9library
def convolutionline 11library
inductive Formulaline 16in the statementlibrary
def Formula.sizeline 24in the statementlibrary
def Formula.coeffline 30in the statementlibrary
def Formula.evalline 37in the statementlibrary
def dimensionline 46in the statementlibrary
def matrixEntryline 48library
def generatorline 52in the statementlibrary
theorem universal_hittingline 57selected theoremlibrary

import Mathlib.Data.Matrix.Basicimport Mathlib.Data.Rat.Defsnamespace OAInamespace NCHittingabbrev Word (n : ℕ) := List (Fin n)abbrev WordSeries (F : Type*) (n : ℕ) := Word n → Fdef convolution {F : Type*} {n : ℕ} [Semiring F]    (f g : WordSeries F n) : Word n → F  | [] => f [] * g []  | i :: w => f [] * g (i :: w) + convolution (fun v => f (i :: v)) g winductive Formula (F : Type*) (n : ℕ) where  | scalar : F → Formula F n  | var : Fin n → Formula F n  | add : Formula F n → Formula F n → Formula F n  | mul : Formula F n → Formula F n → Formula F nnamespace Formuladef size {F : Type*} {n : ℕ} : Formula F n → ℕ  | scalar _ => 1  | var _ => 1  | add f g => f.size + g.size + 1  | mul f g => f.size + g.size + 1noncomputable def coeff {F : Type*} {n : ℕ} [Semiring F] :    Formula F n → WordSeries F n  | scalar a => fun w => if w = [] then a else 0  | var i => fun w => if w = [i] then 1 else 0  | add f g => fun w => coeff f w + coeff g w  | mul f g => convolution (coeff f) (coeff g)def eval {F A : Type*} {n : ℕ} [Semiring A]    (scalarMap : F → A) (T : Fin n → A) : Formula F n → A  | scalar a => scalarMap a  | var i => T i  | add f g => eval scalarMap T f + eval scalarMap T g  | mul f g => eval scalarMap T f * eval scalarMap T gend Formuladef dimension (n s : ℕ) : ℕ := if n = 1 ∧ s = 1 then 1 else 2 * n * s ^ 2def matrixEntry (i r q : ℕ) : ℚ :=  if q < r then (-1 : ℚ) ^ (r - q - 1) / ((r : ℚ) * (i : ℚ) ^ (r - q))  else 0def generator (n s : ℕ) : Fin n → Matrix (Fin (dimension n s)) (Fin (dimension n s)) ℚ :=  fun i r q => if n = 1 ∧ s = 1 then 1 else matrixEntry (i.val + 1) r.val q.valuniverse u_2theorem universal_hitting (n s : ℕ) (hn : 1 ≤ n) (hs : 1 ≤ s)    (F : Type u_2) [Field F] [CharZero F] (f : Formula F n)    (hsize : f.size ≤ s) (hf : ∃ w, f.coeff w ≠ 0) :    f.eval (Matrix.scalar (Fin (dimension n s)))      (fun i r q => (generator n s i r q : F)) ≠ 0 := by  sorryend NCHittingend 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.

One Rational Matrix Hitting Point for Noncommutative Formulas

Theorem

There is a deterministic algorithm which, given integers n,s\ge1, constructs a tuple

\mathbf T_{n,s}\in\Mat_d(\Q)^n, \qquad d\le 2ns^2,

f(\mathbf T_{n,s})\ne0,

where the tuple is viewed over F through the canonical embedding \Q\hookrightarrow F. The construction uses d=2ns^2 when ns\ge2 and d=1 when n=s=1.

Section 1, Introduction Read it in the paper TeX source, line 24

Show the TeX

There is a deterministic algorithm which, given integers $n,s\ge1$,
constructs a tuple
\[
 \mathbf T_{n,s}\in\Mat_d(\Q)^n,
 \qquad d\le 2ns^2,
\]
in time polynomial in $n,s$, measured in bit operations. The numerators and
denominators of its entries have bit length
$O(\log(ns+1)+ns^2\log(n+1))$.
For every field $F$ of characteristic zero and every
$0\ne f\in\mathsf{Form}_{n,s}(F)$,
\[
 f(\mathbf T_{n,s})\ne0,
\]
where the tuple is viewed over $F$ through the canonical embedding
$\Q\hookrightarrow F$. The construction uses $d=2ns^2$ when $ns\ge2$
and $d=1$ when $n=s=1$.

Polynomial Hitting Lists for Noncommutative Rational Formulas

Theorem

There are a deterministic Turing machine G and fixed positive integers C,k with the following property. Given 1^n,1^s, where n,s\geq1, the machine outputs a finite list \mathcal H_{n,s} of n-tuples of rational square matrices, in at most C(n+s+1)^k bit operations. The entire output, with signed binary numerators and positive binary denominators, has at most C(n+s+1)^k bits. For every nonzero admissible rational formula \Phi in n variables of size at most s, some tuple in \mathcal H_{n,s} is a defined evaluation of \Phi with invertible value.

The same list works for all such formulas, regardless of the bit lengths of their rational constants or their inverse depth. The generator receives no formula, oracle, randomness, or advice.

Section 1, Introduction Read it in the paper TeX source, line 20

Show the TeX

There are a deterministic Turing machine $G$ and fixed positive integers $C,k$ with the following property. Given $1^n,1^s$, where $n,s\geq1$, the machine outputs a finite list $\mathcal H_{n,s}$ of $n$-tuples of rational square matrices, in at most $C(n+s+1)^k$ bit operations. The entire output, with signed binary numerators and positive binary denominators, has at most $C(n+s+1)^k$ bits. For every nonzero admissible rational formula $\Phi$ in $n$ variables of size at most $s$, some tuple in $\mathcal H_{n,s}$ is a defined evaluation of $\Phi$ with invertible value.

The same list works for all such formulas, regardless of the bit lengths of their rational constants or their inverse depth. The generator receives no formula, oracle, randomness, or advice.

Configuration and file facts

File: lean/ComparatorChallenges/FormulaHitting.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 116 link it.
Solution module: OAI.Algebra.FormulaHitting.Hitting
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.Data.Matrix.Basic, Mathlib.Data.Rat.Defs
Imports from outside Mathlib: None
Local vocabulary: 10 local vocabulary declarations
Lines of Lean, without comments or blank lines: 40
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.

Plectis Atlas