Nonsingular systems of equations over arbitrary groups
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie’s conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire’s conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group. (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
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.
The other possible matches are Corollary 1.2(The Kervaire conjecture) in The Kervaire theorem for groups () and Theorem 1.1 in The Kervaire theorem for groups ().
Let G be any group, let 1\le m\le n be integers, and let w_1,\ldots,w_m\in G*F_n. If their exponent-sum matrix has row rank m over \mathbb Q, then the canonical homomorphism
Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie’s conjecture on nonsingular systems.
We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
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.
Kervaire.lean, the release lists it as a supporting statement.
theoremcoefficient_injective (A : Type u) [Group A] (w : WordGroup A) (hw : Unimodular w) : Function.Injective (coefficientMap w) := 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?
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:Only the Kervaire special case is stated: coefficient_injective (A) (w : WordGroup A) (hw : Unimodular w) : Function.Injective (coefficientMap w) with WordGroup A := Monoid.Coprod A (Multiplicative ℤ) and Unimodular w := exponentSum w = 1 ∨ exponentSum w = -1 (one equation in one unknown with exponent sum ±1; docs: ‘the selected unimodular-relator result underlying the conjecture’). The summary’s headline, Howie’s conjecture for arbitrary finite systems whose exponent-sum matrix has full row rank over ℚ (simultaneous solution in an overgroup), is not stated; the Kervaire companion claim is covered (coefficient injectivity implies nontriviality).
“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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:Howie’s theorem on finite nonsingular systems (the headline’s lead clause, from the October 5 manuscript) is not stated, not even the general one-equation case (exponent sum != 0). The coefficient-embedding clause is stated only for one unimodular relator.
“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.
Kervaire’s conjecture states that the free product of a nontrivial group with an infinite cyclic group cannot be normally generated by one element. The linked formalization proves the stronger coefficient-injectivity statement used in the paper: for any group $A$ and any relator in $A*\mathbb Z$ whose exponent sum in the cyclic generator is $1$ or $-1$, the natural map from $A$ into the quotient by that relator is injective. This is the selected unimodular-relator result underlying the conjecture.
AI-drafted comparisons, with no recorded human review
Coefficient-group injectivity for unimodular one-relator quotientsKervaire
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every group A and every relator of exponent sum ±1 in the free generator, A injects into the one-relator quotient. Kervaire’s conjecture is the paper’s corollary of it.
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.
Coefficient-group injectivity for unimodular one-relator quotientsKervaire
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original 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.
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 “Career-defining publication”. The page says that nothing on it is verified.
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
Identify the theorem the catalogue describes
Family 256The 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 256Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Corollary 1.2 (The Kervaire conjecture)
In The Kervaire theorem for groups, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
kervaire
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
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. 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; 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; 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 719 s; peak memory 6.6 GiB).
Date
11 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.
Coefficient-group injectivity for unimodular one-relator quotients
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.