Power-saving bounds for halving pairs and level switches
Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.PlanarHalving.power_bounds
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theorempower_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)) :=sorry
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
noncomputabledefhalvingCount {n : ℕ} (P : Fin n → Point) : ℕ := (Finset.univ.filter fun ij : Fin n × Fin n => ij.1 < ij.2 ∧ sideCount P ij.1 ij.2 = (n - 2) / 2 ∧ sideCount P ij.2 ij.1 = (n - 2) / 2).card
defGeneric {n : ℕ} (P : Fin n → Point) : Prop := GeneralPosition P ∧ (∀ i j, i ≠ j → (P i).1 ≠ (P j).1) ∧ (∀ i j a b, i < j → a < b → slope (P i) (P j) = slope (P a) (P b) → i = a ∧ j = b) ∧ (∀ i j a b c d, i < j → a < b → c < d → (i, j) ≠ (a, b) → (c, d) ≠ (i, j) → (c, d) ≠ (a, b) → ∀ z : Point, z ∈ openSegment ℝ (P i) (P j) → z ∈ openSegment ℝ (P a) (P b) → z ∉ segment ℝ (P c) (P d))
noncomputabledefbelowCount {n : ℕ} (P : Fin n → Point) (i j : Fin n) : ℕ :=let s := slope (P i) (P j) (Finset.univ.filter fun l => (P l).2 - s * (P l).1 < (P i).2 - s * (P i).1).card
noncomputabledefswitchCount {n : ℕ} (P : Fin n → Point) (k : ℕ) : ℕ := (Finset.univ.filter fun ij : Fin n × Fin n => ij.1 < ij.2 ∧ 0 < k ∧ k < n ∧ belowCount P ij.1 ij.2 = k - 1).card
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: even, real, 4/3 and point. Found in the paper’s title, which its main theorem states: halving and power. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: collinear, three, constants, varepsilon and absolute.
Theorem 1.1 of A power saving for planar halving lines
Theorem
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,
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,
\[
h(P)\le C n^{4/3-\varepsilon}.
\]
Other possible paper matches, closest wording first
Corollary 1.3 of A power saving for planar halving lines. Words of the Lean statement, its names split into words, that also appear in the theorem: bound, real, integer and point. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: levels, pair, collinear, three, number, constants and two more.
Proposition 8.1 of A power saving for planar halving lines. Words of the Lean statement, its names split into words, that also appear in the theorem: switches, generic, 4/3, integer and points. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: cn, number, constants, varepsilon and absolute.
Theorem 1.2 of A power saving for planar halving lines. Words of the Lean statement, its names split into words, that also appear in the theorem: bound, generic, 4/3, integer and point. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constants, varepsilon and absolute.
Theorem 2.1 of A power saving for planar halving lines. Words of the Lean statement, its names split into words, that also appear in the theorem: generic, 4/3, integer and point.
Proposition 6.1 of A power saving for planar halving lines. Words of the Lean statement, its names split into words, that also appear in the theorem: position and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: additional.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 600 s; peak memory 6.5 GiB).
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 183’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
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').
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: 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.
OpenAI’s own note on what this Lean statement covers
“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.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
56 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 24 to its own declarations, 23 to Mathlib’s documentation. 10 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
import Mathlibnamespace OAInamespace PlanarHalvingabbrevPoint := ℝ × ℝdeforient (p q r : Point) : ℝ := (q.1 - p.1) * (r.2 - p.2) - (q.2 - p.2) * (r.1 - p.1)defGeneralPosition {n : ℕ} (P : Fin n → Point) : Prop :=Function.Injective P ∧ ∀ i j l, i ≠ j → j ≠ l → i ≠ l →orient (P i) (P j) (P l) ≠ 0noncomputabledefsideCount {n : ℕ} (P : Fin n → Point) (i j : Fin n) : ℕ := (Finset.univ.filter fun l => 0 < orient (P i) (P j) (P l)).cardnoncomputabledefhalvingCount {n : ℕ} (P : Fin n → Point) : ℕ := (Finset.univ.filter fun ij : Fin n × Fin n => ij.1 < ij.2 ∧ sideCount P ij.1 ij.2 = (n - 2) / 2 ∧sideCount P ij.2 ij.1 = (n - 2) / 2).cardnoncomputabledefslope (p q : Point) : ℝ := (q.2 - p.2) / (q.1 - p.1)defGeneric {n : ℕ} (P : Fin n → Point) : Prop :=GeneralPosition P ∧ (∀ i j, i ≠ j → (P i).1 ≠ (P j).1) ∧ (∀ i j a b, i < j → a < b → slope (P i) (P j) = slope (P a) (P b) → i = a ∧ j = b) ∧ (∀ i j a b c d, i < j → a < b → c < d → (i, j) ≠ (a, b) → (c, d) ≠ (i, j) → (c, d) ≠ (a, b) → ∀ z : Point, z ∈ openSegmentℝ (P i) (P j) → z ∈ openSegmentℝ (P a) (P b) → z ∉ segmentℝ (P c) (P d))noncomputabledefbelowCount {n : ℕ} (P : Fin n → Point) (i j : Fin n) : ℕ :=let s := slope (P i) (P j) (Finset.univ.filter fun l => (P l).2 - s * (P l).1 < (P i).2 - s * (P i).1).cardnoncomputabledefswitchCount {n : ℕ} (P : Fin n → Point) (k : ℕ) : ℕ := (Finset.univ.filter fun ij : Fin n × Fin n => ij.1 < ij.2 ∧ 0 < k ∧ k < n ∧ belowCount P ij.1 ij.2 = k - 1).cardtheorempower_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)) :=sorryend PlanarHalvingend OAI
Every extracted statement of the accompanying papers
The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.
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,
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,
\[
h(P)\le C n^{4/3-\varepsilon}.
\]
Theorem(Uniform bound over all ranks)
There are absolute constants \varepsilon>0 and C<\infty such that for every integer N\ge1, every generic N-point planar configuration P, and every integer 0\le k\le N,
There are absolute constants $\varepsilon>0$ and $C<\infty$ such that
for every integer $N\ge1$, every generic $N$-point planar configuration
$P$, and every integer $0\le k\le N$,
\[
S_k(P)\le C N^{4/3-\varepsilon}.
\]
Corollary(Planar k-sets and straight-line levels)
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
C_0 n(k+1)^{1/3-\varepsilon_0}.
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.
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
\begin{equation}
C_0 n(k+1)^{1/3-\varepsilon_0}.
\end{equation}
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.
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 183 link it.
Solution module
OAI.Combinatorics.HalvingLines.Main
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
9 local vocabulary declarations
Lines of Lean, without comments or blank lines
37
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Power-saving bounds for halving pairs and level switches
Family 183: Power savings for planar halving lines and k-sets
Candidate pairing
Theorem
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,
h(P)\le C n^{4/3-\varepsilon}.
A power saving for planar halving lines
TeX label thm:halving-power, build/sections/introduction.tex, line 9
The Lean statement says
theorempower_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)) :=sorry
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.