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Uniform black-box noncommutative identity testing across characteristics

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation. (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 independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

What the papers state

Theorem 1.1

In Polynomial Hitting Lists for Noncommutative Rational Formulas, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

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.

Theorem 1.1

In Uniform Matrix Hitting Points in Every Positive Characteristic, 4 October 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

There is a deterministic Turing machine and absolute positive integers C,k with the following property. Given a prime p in binary and positive integers n,s in unary, the machine outputs a positive integer d=O(n^3s^6) and a tuple

(T_1,\ldots,T_n)\in\Mat_d(\F_p)^n.

For every field F of characteristic p and every nonzero f\in F\langle x_1,\ldots,x_n\rangle computed by a formula of size at most s, the same tuple, viewed over F, satisfies

f(T_1,\ldots,T_n)\ne0.

Writing L=\lceil\log_2p\rceil, the total running time and output length are at most C(n+s+L+1)^k. All matrix entries are output densely as ordinary binary residues in \{0,\ldots,p-1\}, with dimensions and separators included. Primality of p is promised.

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

One more possible match

  • Theorem 1.1 in One Rational Matrix Hitting Point for Noncommutative Formulas, 24 September 2026.
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.
Papers and explanations
Three current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Uniform Matrix Hitting Points in Every Positive Characteristic

4 October 2026, current edition.

Read it herePDF

We construct a single matrix substitution that detects every nonzero size-s division-free noncommutative formula in n variables over every field of a given positive characteristic. One deterministic machine, given a promised prime p in binary and n, s in unary, outputs matrices over 𝔽p of dimension \(O(n^3s^6)\) in polynomial bit time. The same tuple works with arbitrary extension-field coefficients, including in characteristic two. The construction also applies to the stated acyclic algebraic path programs.

One Rational Matrix Hitting Point for Noncommutative Formulas

24 September 2026, current edition.

Read it herePDF

We construct, in deterministic polynomial bit time, one tuple of rational matrices that detects every nonzero polynomial computed by a noncommutative division-free formula of a prescribed size. The matrices have dimension \(O(ns^2)\) for n variables and formula size s, and the same tuple works over every field of characteristic zero.

Polynomial Hitting Lists for Noncommutative Rational Formulas

24 September 2026, current edition.

Read it herePDF

We construct, in deterministic polynomial bit time, a polynomial-size list of rational matrix tuples for noncommutative rational formulas over ℚ of bounded tree size. Every nonzero admissible formula has a defined, invertible value at one tuple, with no separate bounds on inverse nesting or rational constant heights. Matrix dimensions, entry bit lengths, and total output length are polynomially bounded.

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.

RationalHitting

The paper states

Theorem 1.1

In Polynomial Hitting Lists for Noncommutative Rational Formulas, 24 September 2026

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.

The words they share: advice, signed, works, oracle, denominators, finite, numerators, inverse, admissible, list, output, turing, machine, positive, binary, evaluation, rational, bits, size, lengths, formula, deterministic, integers, tuples, nonzero, polynomial, variables, invertible.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Polynomial hitting lists for rational formulas

RationalHitting.lean, the release lists it as a main result.

def MainStatement : Prop :=  ∃ (m : ℕ) (G : Machine m) (C k : ℕ), 0 < C ∧ 0 < k ∧    ∀ n s : ℕ, 1 ≤ n → 1 ≤ s → ∃ H : Output n,      H.dimension ≤ C * (n + s + 1) ^ k ∧      (encodeOutput H).length ≤ C * (n + s + 1) ^ k ∧      OutputsWithin G s (C * (n + s + 1) ^ k) H ∧ Hits s Htheorem main : MainStatement := by  sorry

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”.

Read the complete Lean statement

FormulaHitting

The paper states

Theorem 1.1

In One Rational Matrix Hitting Point for Noncommutative Formulas, 24 September 2026

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.

The words they share: entries, zero, integers, field, hitting, rational, formulas, matrix, ns, construction, bit, deterministic, tuple, polynomial.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

One rational matrix tuple hitting all bounded-size noncommutative formulas

FormulaHitting.lean, the release lists it as a supporting statement.

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

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”.

Read the complete Lean statement

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?

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

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 part of the advertised result (their term: “partial”).

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)

“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.

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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).

The gap they name: 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.

“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization gives one explicit tuple of rational matrices that simultaneously detects every nonzero division-free noncommutative formula with at most $s$ gates in $n$ variables, for $n,s\ge1$. Evaluation at that tuple is a nonzero matrix over every characteristic-zero field.

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. The formalized result constructs polynomial-size hitting lists for noncommutative rational formulas with rational constants, addition, multiplication, and inverse gates. Given the number of variables and a formula-size bound, one deterministic polynomial-time machine outputs rational matrix tuples of a common positive dimension. Every admissible nonzero formula within the size bound evaluates to an invertible matrix on some listed tuple. The matrix dimension, complete binary output length, and running time are all polynomially bounded.

AI-drafted comparisons, with no recorded human review

  • Polynomial hitting lists for rational formulas RationalHitting

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the rational-formula paper: one deterministic polynomial-time machine outputs hitting lists of rational matrix tuples for formulas with inverse gates, with polynomial dimension, output length and running time.

  • One rational matrix tuple hitting all bounded-size noncommutative formulas FormulaHitting

    Unreviewed AI draft: states a weaker form of a paper’s theorem. States the hitting property of the paper’s explicit matrix tuple for all nonzero formulas of bounded size over fields of characteristic zero. The main theorem’s polynomial-time construction and its dimension and bit-length bounds are not asserted.

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.

Polynomial hitting lists for rational formulas RationalHitting

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal 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.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 188 s; peak memory 6.6 GiB).11 October 2026Original report

One rational matrix tuple hitting all bounded-size noncommutative formulas FormulaHitting

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal 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.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 40 s; peak memory 2.5 GiB).10 October 2026Original report

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.

Within this family

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.

Offer to write an explanationCompare one Lean statement with its paper: FormulaHitting, RationalHittingSuggest a source attributionSuggest a change

What people have said

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 algorithm and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

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 116 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 116 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form