Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but \(o(n)\) points of any n-point planar set determine at least \(n^{1-\varepsilon}\) distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by \(O(n^{4/3-\delta})\) for an absolute δ > 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
It bears on Erdős problem #90 and Erdős problem #604, as listed by Thomas Bloom, erdosproblems.com.
In The weak pinned planar distance theorem, 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.
For every fixed s>0, one has F_n(s)\longrightarrow0 as n\longrightarrow\infty.
No Lean statement in the release is paired with this theorem.
In The weak pinned planar distance theorem, 23 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
For every fixed \varepsilon>0,
In particular, every sufficiently large n-point planar set has a pin determining at least n^{1-\varepsilon} distinct nonzero distances.
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.
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Choose one to see it beside the paper theorem it is paired with.
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.
PinnedDistancesThe paper states
In The weak pinned planar distance theorem, 23 September 2026
For every fixed \varepsilon>0,
In particular, every sufficiently large n-point planar set has a pin determining at least n^{1-\varepsilon} distinct nonzero distances.
The words they share: determining, distances, distinct, fixed, large, least, planar, point.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PinnedDistances.lean, the release lists it as a main result.
theorem main (s : ℝ) (hs : 0<s) : Tendsto (fun n : ℕ => F n s) atTop (𝓝 0) := 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.
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
PlanarUnitDistancesThe paper states
In A power saving for planar unit distances, 23 September 2026
There are absolute constants 0<C<\infty and 1\le\beta<4/3 such that u(n)\le Cn^\beta for every nonnegative integer n.
The words they share: 4/3, integer, unit, cn, beta, absolute, constants.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PlanarUnitDistances.lean, the release lists it as a supporting statement.
theorem main : ∃ C β : ℝ, 0 < C ∧ 1 ≤ β ∧ β < (4 : ℝ) / 3 ∧ ∀ n : ℕ, (u n : ℝ) ≤ C * (n : ℝ) ^ β := 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.
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.
MainStatement Model.lean, line 43PinsStatement Model.lean, line 46MathVet 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: WeakPinned.main : Tendsto (fun n => F n s) atTop (𝓝 0) with F n s = sSup {pairFraction P s | P.card = n} states the rich-pair-fraction form; the summary’s 'all but o(n) points determine >= n^{1-eps} distinct distances’ follows by a routine pigeonhole step (docs: ‘Consequently’) that is not itself stated in Lean. The unit-distance theorem IS also stated (∃ C β, 0 < C ∧ 1 ≤ β ∧ β < 4/3 ∧ ∀ n, (u n:ℝ) ≤ C * n^β, u = sSup over n-point sets, bounded so a true max); docs paragraph 1 says it ‘is not included’ but the Lean contradicts that (Lean wins).
“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 shows that large repeated distance fibers are rare in arbitrary finite planar point sets. For each fixed $s>0$, the largest possible fraction of ordered distinct pairs $(x,y)$ whose distance from the pin $x$ occurs at least $n^s$ times tends to zero as the set size $n$ grows. Consequently, for every $\varepsilon>0$, the fraction of pins determining fewer than $n^{1-\varepsilon}$ distances tends uniformly to zero. The separate unit-distance power-saving theorem is not included.
The planar unit-distance problem asks how many pairs at distance one can occur among $n$ points. The formalization proves that there are absolute constants $C>0$ and $1\le\beta<4/3$ such that every finite planar point set of size $n$ determines at most $Cn^\beta$ unordered unit-distance pairs. The same constants work for every $n$, giving a fixed power improvement over the classical exponent $4/3$.
Weak pinned planar distance theorem PinnedDistances
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (pins with many repeated distances are rare) and its corollary on pins determining few distances.
Power saving for planar unit distances PlanarUnitDistances
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: absolute constants C and β < 4/3 with at most Cn^β unit distances among n points in the plane.
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.
PinnedDistancesThree 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 376 s; peak memory 6.5 GiB). | 11 October 2026 | Original report |
PlanarUnitDistancesThree 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 447 s; peak memory 6.6 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.
Hexagon, Hexagon: a repository for research in the mathematical sciences and theoretical computer science, 1 October 2026. Source
hexagon:2610.00016v2 (Eric Naslund) reaches a unit-distance exponent of 1.043 with the number-field construction of OpenAI and Sawin; it predates the October release and relates to the earlier unit-distance disproof.
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.
Offer to write an explanationCompare one Lean statement with its paper: PinnedDistances, PlanarUnitDistancesSuggest 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 “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source
Compiler not recorded (Atlas survey), Atlas survey of Erdős problem pages, recorded on 11 October 2026, 11 October 2026. Source
The imported Atlas survey reports: Problem 90 page says the unit-distance conjecture was disproved by an internal OpenAI model (May 2026 PDF on cdn.openai.com); it does not mention the 6 Oct power-saving upper bound.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 604 on pinned distances, a result with a weak exponent, with the formalisation linked and no earlier work named.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 90, a power saving over the Spencer, Szemerédi and Trotter unit distance bound, notes the PDF gives no value for the saving, and links the formalisation.
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 The weak pinned planar distance theorem, 23 September 2026
Why this passage is shown
Corollary 1.2
In The weak pinned planar distance theorem, 23 September 2026
Why this passage is shown
Theorem 1.1
In A power saving for planar unit distances, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, PinnedDistances
Proof-verification report
MathVet, PinnedDistances
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, PinnedDistances
Proof-verification report
cert-machine, PlanarUnitDistances
Proof-verification report
MathVet, PlanarUnitDistances
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, PlanarUnitDistances
Lean statement
PinnedDistances
Lean statement
PlanarUnitDistances