OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items. (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 Theorem 1.3(Constant-additive hardness) in the same paper () and Corollary 1.4 in the same paper ().
For every integer c\ge0, there is a finite rational bin-packing instance I and an integer B such that every item size exceeds 1/6 and
\LP(I)=B,
\qquad \OPT(I)>B+c.
The equality for the fractional optimum also holds for the configuration LP whose patterns are subsets of the individual item copies.
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 disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer c, distinguishing instances that fit in B bins from those requiring more than \(B+c\) bins is NP-hard. Both results hold with rational item sizes greater than 1/6, so each bin contains at most five items.
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.
BinPackingGap.lean, the release lists it as a main result.
theoremmain_results : (∀ c : ℕ, ∃ I : Instance, ∃ B : ℕ, I.n = 5 * B ∧ (∀ i, (1 / 6 : ℚ) < I.size i ∧ I.size i < 1) ∧ B + c < opt I ∧ individualLP I = (B : ℝ) ∧ typeLP I = (B : ℝ)) ∧ (∀ c : Nat, PackingGapNPHard c) ∧ ((∃ c : Nat, Nonempty (AbsoluteAdditiveAlgorithm c)) ↔ PEqualsNP) ∧ (PEqualsNP → ∃ algorithm : AbsoluteAdditiveAlgorithm 0, (∀ (items : RawInstance) (valid : items.Valid), ∃ packing : RawPacking, algorithm.run (BinaryEncoding.rawInstanceBits items) = some packing ∧ packing.Feasible items ∧ packing.bins = opt (items.toInstance valid)) ∧ algorithm.run (BinaryEncoding.rawInstanceBits []) = some ⟨0, []⟩ ∧ (∀ input : List Bool, BinaryEncoding.decodeRawInstance input = none → algorithm.run input = none) ∧ (∀ items : RawInstance, ¬ items.Valid → algorithm.run (BinaryEncoding.rawInstanceBits items) = none)) := by⋮
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; P/NP are defined by Mathlib TM2 poly-time deciders and the certificate definition (InNP).
Definitions the group asks readers to compare with the paper:PEqualsNP := ClassP = ClassNP with ClassP via TM2ComputableInPolyTime deciders (+ finite alphabet) and ClassNP via CookLevin.InNP certificate verifiers; LP values are sInf over fractional covers (nonempty, bounded below)
“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 results rule out a universal additive bound for the configuration linear program in bin packing. For every integer $c\ge0$, there are an integer $B$ and a rational instance with $5B$ items whose individual-copy and size-type configuration-LP values both equal $B$, but whose integral optimum is greater than $B+c$. Distinguishing a packing in $B$ bins from the absence of one in $B+c$ bins is NP-hard for each fixed $c$. A deterministic polynomial-time algorithm with a fixed absolute additive allowance exists exactly when $P=NP$; under that equality, an optimal algorithm is constructed.
AI-drafted comparisons, with no recorded human review
Unbounded configuration gaps and additive hardnessBinPackingGap
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s three headline statements: unbounded configuration-LP gaps, NP-hardness of distinguishing B from B + c bins, and the equivalence of a constant-additive polynomial-time algorithm with P = NP.
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.
Unbounded configuration gaps and additive hardnessBinPackingGap
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).
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Theorem 1.2 of Additive hardness and unbounded configuration gaps in bin packing, 24 September 2026
Reported by
cert-machine
Method
Exact computation
Their verdict
“NEEDS DATA” Cannot be decided from the release: the specific object the argument relies on is not published.
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 “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 118The 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.
Theorem 1.2 of Additive hardness and unbounded configuration gaps in bin packing, 24 September 2026
Exact calculations for finite claims, cert-machine
Their verdict
“NEEDS DATA”
Claim tested or proved
Cannot be decided from the release: the specific object the argument relies on is not published.
Detail
preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/build/source/consequences.tex l.9-24 (proof of Theorem thm:unbounded-gap), /build/source/instance.tex l.26-60 (parameters eq:parameters and geometry). Instance itself NOT PUBLISHED (only its prescription). Lean: lean/ComparatorChallenges/BinPackingGap.lean; proof lean/OAI/Computability/BinPacking/ (284 files).