OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves Khot’s 2-to-1 Games Conjecture with perfect completeness: for every fixed rational \(\delta\in(0,1)\), it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above.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 1.3(Satisfiable Not-Two and three-query PCPs) in the same paper () and Corollary 1.2(Maximum k-colorable subgraph) in the same paper ().
For every fixed rational \delta\in(0,1) there are an integer q=q(\delta)\ge2 and a deterministic polynomial-time reduction from 3-SAT to 2-to-1 games with alphabets [2q] and [q] such that:
(i)
a satisfiable formula produces a game of value 1;
(ii)
an unsatisfiable formula produces a game of value at most \delta.
The output contains a nonempty explicit list of unweighted edge occurrences and an explicit table for every projection map. Every table has exactly two preimages for each right answer. The polynomial running-time bound is in the ordinary binary input length; its degree and constants may depend on the fixed \delta.
Reported proof verification
One Lean statement in the release; one independent group reports that Comparator accepted a run for this file at the version shown here.
We prove the 2-to-1 Games Conjecture with perfect completeness. For every fixed rational \(\delta\in(0,1)\), it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most δ, with a fixed alphabet and an explicitly listed unweighted multiset of constraints.
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.
For every fixed rational \delta\in(0,1) there are an integer q=q(\delta)\ge2 and a deterministic polynomial-time reduction from 3-SAT to 2-to-1 games with alphabets [2q] and [q] such that:
(i)
a satisfiable formula produces a game of value 1;
(ii)
an unsatisfiable formula produces a game of value at most \delta.
The output contains a nonempty explicit list of unweighted edge occurrences and an explicit table for every projection map. Every table has exactly two preimages for each right answer. The polynomial running-time bound is in the ordinary binary input length; its degree and constants may depend on the fixed \delta.
The words they share: rational, reduction, projection, exactly, formula, edge, map, list, table, time, length, alphabets, binary, input, value, two, game, nonempty, satisfiable, completeness, perfect, preimages, produces, sat, delta, games, deterministic, unweighted, unsatisfiable, polynomial, output, depend.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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.
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?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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 or imply the whole advertised result (their term: “full”).
Their note:None; NP-hardness is encoded as a polynomial-time Karp gap reduction from binary 3SAT (Mathlib TM2 model, finite stack alphabets).
Definitions the group asks readers to compare with the paper:Same Mathlib TM2ComputableInPolyTime + FiniteAlphabet encoding as family 102; instances are multigraphs (edge list with repeats), outputs written with unary encodeWord
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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.
The formalized result establishes perfect-completeness hardness for 2-to-1 games. For every fixed rational $0<\delta<1$, a deterministic polynomial-time reduction from binary 3SAT produces a nonempty unweighted game of value exactly $1$ on satisfiable inputs and at most $\delta$ on unsatisfiable inputs. Each constraint projection has exactly two preimages for each output label. The alphabet depends only on $\delta$, and the runtime is measured in the original input bit length.
AI-drafted comparisons, with no recorded human review
Perfect-completeness reduction for 2-to-1 gamesPerfectCompleteness
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every rational δ in (0, 1), a polynomial-time reduction from 3SAT to 2-to-1 games with perfect completeness and soundness δ. The paper’s corollaries are not part of 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.
Perfect-completeness reduction for 2-to-1 gamesPerfectCompleteness
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“REFUSED”
Re-run through Comparator did not reach a verdict (Lean’s default kernel accepts the solution; the nanoda replay stopped: fatal runtime error: stack overflow, aborting).
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
PerfectCompleteness: Tracked source lean/OAI/Computability/PerfectCompleteness/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
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
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 “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 105The 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
Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 the headline as OpenAI wrote it.
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 did not reach a verdict (Lean’s default kernel accepts the solution; the nanoda replay stopped: fatal runtime error: stack overflow, aborting).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
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.