OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite 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 a supporting step or a different result.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 Theorem 1.2 in A negative solution to the finite lattice representation problem () and Theorem 1.1 in A negative solution to the finite lattice representation problem ().
The following statements hold.
(i)
A finite nonempty lattice is representable if and only if it satisfies the finite colored-graph criterion in Theorem 2.1. For each candidate graph size the criterion is decidable by a finite test.
(ii)
There is no deterministic Turing machine that, given the order table of a finite nonempty lattice, always halts and decides \mathcal R(L).
(iii)
There is a nonrepresentable finite lattice L_0 of minimum cardinality, specified by the finite formulas in Section 17. Every finite lattice of smaller cardinality is representable, and \lvert L_0\rvert>7.
(iv)
Given the order table of a finite nonempty lattice L, it is undecidable whether L\cong[H,G] for some finite group G and subgroup H\le G, where [H,G] is the full subgroup interval.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We give an explicit colored-graph characterization of the finite nonempty lattices that occur as full congruence lattices of finite algebras, and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer. We also prove that recognition of full subgroup intervals in finite groups is undecidable.
We give a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.
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. The outside readers find that the release’s Lean states a supporting or different result, so the theorem it is paired with is shown under its own heading.
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.
A finite nonempty lattice is representable if and only if it satisfies the finite colored-graph criterion in Theorem 2.1. For each candidate graph size the criterion is decidable by a finite test.
(ii)
There is no deterministic Turing machine that, given the order table of a finite nonempty lattice, always halts and decides \mathcal R(L).
(iii)
There is a nonrepresentable finite lattice L_0 of minimum cardinality, specified by the finite formulas in Section 17. Every finite lattice of smaller cardinality is representable, and \lvert L_0\rvert>7.
(iv)
Given the order table of a finite nonempty lattice L, it is undecidable whether L\cong[H,G] for some finite group G and subgroup H\le G, where [H,G] is the full subgroup interval.
The words they share: colored, criterion, finite, full, graph, interval, lattice, nonempty, representable, specified, subgroup.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
FiniteCongruenceGraph.lean, the release lists it as a supporting statement.
theoremgraph_criterion (Lat : Type) [Fintype Lat] [Lattice Lat] [OrderBot Lat] : Representable Lat ↔ HasGraphWitness Lat := 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”.
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 a supporting result (their term: “supporting-only”).
Their note:graph_criterion (Lat) [Fintype Lat] [Lattice Lat] [OrderBot Lat] : Representable Lat ↔ HasGraphWitness Lat is only the paper’s colored-graph characterization (docs: 'This selected theorem is the equivalence with the graph criterion; the paper’s undecidability and subgroup-interval conclusions are outside it'). Nothing states that some finite lattice is NOT representable (the headline negative answer to the finite lattice representation problem) or that representability/subgroup-interval membership is undecidable.
“supporting-only”, in their words: The challenge statements state a lemma or auxiliary result, not the headline.
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 different result (their term: “DIFFERENT”).
The gap they name:every clause of the headline: existence of a non-representable finite lattice, undecidability of representability, undecidability of subgroup-interval recognition. The Lean states instead a characterization (representable ⇔ colored-graph witness) that neither implies nor is implied by any headline clause.
“DIFFERENT”, in their words: The challenge statements state a different theorem from the headline.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The finite lattice representation problem asks which finite lattices occur as the full congruence lattice of a finite algebra. The formalization proves the paper’s colored-graph characterization: a finite nonempty lattice is representable exactly when it admits the specified finite nonempty graph witness, whose edge colors encode the required congruence relations. This selected theorem is the equivalence with the graph criterion; the paper’s undecidability and subgroup-interval conclusions are outside it.
AI-drafted comparisons, with no recorded human review
Colored-graph criterion for finite congruence latticesFiniteCongruenceGraph
Unreviewed AI draft: states part of a paper’s theorem. States part (i) of the main theorem, the coloured-graph characterisation of representable finite lattices. The undecidability and subgroup-interval parts are outside the statement.
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.
Colored-graph criterion for finite congruence latticesFiniteCongruenceGraph
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 counterexample and rates its significance, if it holds up, as “Major math journal”. 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 206The 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 206Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.2
In A negative solution to the finite lattice representation problem, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
finite, group, lattice, subgroup
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The challenge statements state a lemma or auxiliary result, not the headline.
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 a different theorem from the headline.
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 31 s).
Date
10 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.
Colored-graph criterion for finite congruence lattices
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.