For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least \(n^{-2+c}\) for one absolute c > 0. This disproves the conjectured almost-n−2 upper bound in Heilbronn’s triangle problem, which asks how large the smallest determined triangle can be. (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
The paper states
In A power improvement in the Heilbronn triangle lower bound
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There are absolute constants \eta,c_1>0 and an integer n_0\ge3 such that
The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
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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.
HeilbronnTriangleThe paper states
In A power improvement in the Heilbronn triangle lower bound
There are absolute constants \eta,c_1>0 and an integer n_0\ge3 such that
The words they share: integer, heilbronn, triangle, power, lower, bound, eta.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
HeilbronnTriangle.lean, the release lists it as a supporting statement.
theorem heilbronn_power_lower_bound : 0 < heilbronnExponent ∧ ∃ n : ℕ → ℕ, ∃ P : ℕ → Finset Point, Filter.Tendsto n Filter.atTop Filter.atTop ∧ ∀ j : ℕ, 3 ≤ n j ∧ (P j).card = n j ∧ pointsInUnitSquare (P j) ∧ triangleAreasAtLeast (P j) (Real.rpow (n j : ℝ) (-2 + heilbronnExponent)) := 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.
The file states one more theorem: almost_.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
Neither 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 a weaker statement than the advertised result (their term: “weaker-statement”).
Their note: heilbronn_power_lower_bound only yields an unbounded sequence of sizes (∃ n P, Tendsto n atTop atTop ∧ ∀ j,... triangleAreasAtLeast (P j) ((n j)^(-2 + heilbronnExponent))), not ‘for every sufficiently large n’ (docs: 'The linked construction is for an unbounded sequence of sizes; the paper’s statement for every sufficiently large n is broader'). The disproof of the almost-n^-2 upper bound is stated (almost_n_minus_two_refuted : ¬ eventualAlmostUpperBound (heilbronnExponent/2)) and follows from the subsequence, so only the all-large-n uniformity is missing; the exponent is an astronomically small explicit constant 1/(100000 * K).
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker 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 — DECLARED”).
The gap they name: Only an unbounded sequence of sizes n_j is stated, not every sufficiently large n.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says 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
Heilbronn’s triangle problem asks how large the smallest triangle determined by $n$ points in the unit square can be. The formalization constructs an unbounded sequence of sizes $n$ and point sets for which every triangle has area at least $n^{-2+\eta}$, for one fixed $\eta>0$. It consequently refutes the proposed upper bound of order $n^{-2+\varepsilon}$ for every $\varepsilon>0$. The linked construction is for an unbounded sequence of sizes; the paper’s statement for every sufficiently large $n$ is broader.
Power improvement for the smallest Heilbronn triangle HeilbronnTriangle
Unreviewed AI draft: states a weaker form of a paper’s theorem. States the area lower bound n^{-2+η} along an unbounded sequence of sizes, whereas the main theorem gives it for every sufficiently large n; the refutation of the n^{-2+ε} upper bound follows.
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.
HeilbronnTriangleTwo 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.
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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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.
Author not stated. Source
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Why this passage is shown
Theorem 1.1
In A power improvement in the Heilbronn triangle lower bound
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, HeilbronnTriangle
Proof-verification report
MathVet, HeilbronnTriangle
Lean statement
HeilbronnTriangle