Determines the optimal worst-case covering density as \(\Theta(n\log n)\), for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most \(Cn\log n\); centrally symmetric examples in every sufficiently large dimension require at least \(cn\log n\) even without the lattice restriction, for absolute \(c,C\gt 0\). (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
In Translative covering densities of order n log n, 23 September 2026
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.
There are absolute constants c>0 and n_0 such that, for every integer n\ge n_0, some centrally symmetric convex body K_n\subset\mathbb R^n satisfies
In A single-lattice covering bound of order n log n, 23 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
There is an absolute constant C>0 such that, for every integer n\ge2 and every convex body K\subset\R^n, one full-rank lattice \Lambda satisfies
No symmetry or boundary regularity is assumed.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.
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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.
SingleLatticeCoveringThe paper states
In A single-lattice covering bound of order n log n, 23 September 2026
There is an absolute constant C>0 such that, for every integer n\ge2 and every convex body K\subset\R^n, one full-rank lattice \Lambda satisfies
No symmetry or boundary regularity is assumed.
The words they share: lattice, convex, integer, body, log, single, covering, regularity, cn, symmetry, boundary, assumed, rank, full, one, absolute, constant.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SingleLatticeCovering.lean, the release lists it as a main result.
theorem single_lattice_covering : ∃ C : ℝ, 0 < C ∧ ∀ (n : ℕ), 2 ≤ n → ∀ (K : Set (Space n)), IsConvexBody K → ∃ (L : Submodule ℤ (Space n)) (_ : DiscreteTopology L), IsZLattice ℝ L ∧ LatticeCovers K L ∧ (volume K).toReal / ZLattice.covolume L ≤ C * (n : ℝ) * Real.log (n : ℝ) := 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
CoveringDensityThe paper states
In Translative covering densities of order n log n, 23 September 2026
There are absolute constants c>0 and n_0 such that, for every integer n\ge n_0, some centrally symmetric convex body K_n\subset\mathbb R^n satisfies
The words they share: symmetric, centrally, convex, integer, real, theta, body, absolute, constants, log, translative, densities, covering, order.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CoveringDensity.lean, the release lists it as a supporting statement.
def OptimalOrder : Prop := ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∃ n₀ : ℕ, ∀ n : ℕ, n₀ ≤ n → orderBounds c C n (translativeSup n) ∧ orderBounds c C n (symmetricTranslativeSup n) ∧ orderBounds c C n (latticeSup n) ∧ orderBounds c C n (symmetricLatticeSup n)theorem CoveringOrder.optimal_order : CoveringOrder.OptimalOrder := by sorryThis 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
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.
LocallyFiniteCenters Model.lean, line 52Covers Model.lean, line 55ConvexBody Model.lean, line 58CentrallySymmetric Model.lean, line 61orderBounds Model.lean, line 95OptimalOrder Model.lean, line 102MainStatement Model.lean, line 115IsConvexBody Model.lean, line 14LatticeCovers Model.lean, line 17MathVet 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; the lower bound for centrally symmetric bodies is stated for the translative supremum too (hence for the lattice one).
Definitions the group asks readers to compare with the paper: Hand-rolled density notions: euclideanVolume (pushforward of product Lebesgue measure), euclideanCovolume, thetaL, thetaT with upperCenterIntensity := limsup #(X ∩ cube R) / (2R)^n; standard in form
“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.
cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean covers
The formalized result gives an absolute constant $C>0$ such that every convex body $K\subset\mathbb R^n$, $n\ge2$, admits a covering by translates along one full-rank lattice $L$ with density $|K|/\mathrm{covol}(L)\le Cn\log n$. The covering is exact, and no symmetry, boundary regularity, or volume normalization is assumed.
The formalization determines the order of the largest covering density in dimension $n$. For all sufficiently large $n$, the suprema of translative covering density and lattice covering density are each bounded above and below by absolute positive multiples of $n\log n$, both over all convex bodies and over centrally symmetric convex bodies.
Single-lattice covering bound SingleLatticeCovering
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every convex body in dimension n ≥ 2 is covered by translates along one lattice with density at most Cn log n.
Optimal order for translative and lattice covering-density suprema CoveringDensity
Unreviewed AI draft: states a paper’s theorem as the paper does. States matching upper and lower bounds of order n log n for the translative and lattice covering-density suprema, over all and over centrally symmetric bodies. Its lower bound contains the main theorem of the translative paper.
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.
SingleLatticeCoveringThree 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 227 s; peak memory 6.6 GiB). | 11 October 2026 | Original report |
CoveringDensityTwo 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
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.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Hexagon, Hexagon: a repository for research in the mathematical sciences and theoretical computer science, 8 October 2026. Source
hexagon:2610.00165v1 (Scott Kominers) constructs multi-coverings by lattice translates building on OpenAI’s horizontal-vertical covering construction; the family is probably 092 (convex-body covering density) but the abstract does not give an id.
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
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Offer to write an explanationCompare one Lean statement with its paper: SingleLatticeCovering, CoveringDensitySuggest a source attributionSuggest a change
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.
Author not stated. Source
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In Translative covering densities of order n log n, 23 September 2026
Why this passage is shown
Theorem 1.1
In A single-lattice covering bound of order n log n, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, SingleLatticeCovering
Proof-verification report
MathVet, SingleLatticeCovering
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, SingleLatticeCovering
Proof-verification report
cert-machine, CoveringDensity
Proof-verification report
MathVet, CoveringDensity
Lean statement
CoveringDensity
Lean statement
SingleLatticeCovering