Improves the planar halving-line bound to \(O(n^{4/3-\varepsilon})\) for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has \(O(n(k+1)^{1/3-\varepsilon_0})\) strictly separable k-subsets for \(1\le k\le n/2\), with an absolute \(\varepsilon_0\gt 0\). The constants and positive exponents are nonquantitative. (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
In A power saving for planar halving lines, 25 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 exist absolute constants \varepsilon>0, C<\infty, and n_0 such that, for every even n\ge n_0 and every n-point set P\subset\mathbb R^2 with no three collinear,
In A power saving for planar halving lines, 25 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
There are absolute constants \varepsilon_0>0 and C_0<\infty such that, for every integer n\ge2 and every integer 1\le k\le n/2, the number of k-sets of any n-point set Q\subset\mathbb R^2 with no three collinear is at most
The same bound holds for \operatorname{comp}(\Lambda_k(\mathcal L)) for any arrangement \mathcal L of n nonvertical straight lines with no parallel pair and no three concurrent.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.
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.
HalvingLinesThe paper states
In A power saving for planar halving lines, 25 September 2026
There exist absolute constants \varepsilon>0, C<\infty, and n_0 such that, for every even n\ge n_0 and every n-point set P\subset\mathbb R^2 with no three collinear,
The words they share: even, real, 4/3, point, halving, power, collinear, three, constants, varepsilon, absolute.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
HalvingLines.lean, the release lists it as a supporting statement.
theorem power_bounds : (∃ ε : ℝ, 0 < ε ∧ ∃ C : ℝ, ∃ n₀ : ℕ, ∀ n : ℕ, Even n → n₀ ≤ n → ∀ P : Fin n → Point, GeneralPosition P → (halvingCount P : ℝ) ≤ C * (n : ℝ) ^ ((4 : ℝ) / 3 - ε)) ∧ (∃ ε : ℝ, 0 < ε ∧ ∃ C : ℝ, ∀ n : ℕ, 1 ≤ n → ∀ P : Fin n → Point, Generic P → ∀ k : ℕ, k ≤ n → (switchCount P k : ℝ) ≤ C * (n : ℝ) ^ ((4 : ℝ) / 3 - ε)) := 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.
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
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.
GeneralPosition Model.lean, line 12Generic Model.lean, line 26Neither 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: First half is stated: ∃ ε>0, C, n₀, ∀ even n ≥ n₀, ∀ P, GeneralPosition P → halvingCount P ≤ C * n^(4/3-ε). The k-set claim O(n(k+1)^{1/3-ε0}) for 1 <= k <= n/2 is NOT stated: the second conjunct only bounds switchCount P k (level-switch pairs) by C * n^(4/3-ε) uniformly in k, for the stronger hypothesis Generic P (distinct x-coordinates/slopes plus a segment-crossing condition), i.e. a weaker, k-independent bound on a different count (docs: 'a bound of the same form for all level-switch counts under the additional generic-position assumptions').
“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 — UNDECLARED”).
The gap they name: The k-set clause — O(n(k+1)^{1/3−ε₀}) strictly separable k-subsets for 1 ≤ k ≤ n/2 under no-three-collinear — is not stated. The only k-dependent statement is a k-uniform C·n^{4/3−ε} bound on level-switch counts, and it needs additional genericity hypotheses.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
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
A halving pair in an even planar point set is a pair whose line leaves equally many remaining points on each side. The formalization proves that some absolute $\varepsilon>0$ and $C$ bound the number of halving pairs by $Cn^{4/3-\varepsilon}$ for every sufficiently large even $n$ and every $n$-point set with no three collinear. It also proves a bound of the same form for all level-switch counts under the additional generic-position assumptions in the statement. The constants are existential.
Power-saving bounds for halving pairs and level switches HalvingLines
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s halving-pair bound Cn^{4/3−ε} and the bound of the same form for level-switch counts under generic position. The k-set corollary is not separately stated.
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.
HalvingLinesThree 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 600 s; peak memory 6.5 GiB). | 11 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.
anxkhn, Series of 15 issues and 15 draft PRs adding release results to Formal Conjectures (issues 6972 to 6997, PRs 6977 to 7002), 8 October 2026. Source
Proposes adding the release’s power-saving bound for planar halving lines to Formal Conjectures (issue 6974, PR 6979).
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
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.
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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.
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 A power saving for planar halving lines, 25 September 2026
Why this passage is shown
Corollary 1.3 (Planar k-sets and straight-line levels)
In A power saving for planar halving lines, 25 September 2026
Why this passage is shown
Theorem 1.2 (Uniform bound over all ranks)
In A power saving for planar halving lines, 25 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, HalvingLines
Proof-verification report
MathVet, HalvingLines
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, HalvingLines
Lean statement
HalvingLines