Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set \(E\subset\mathbb R^d\) with Hausdorff dimension greater than \(d/2\) determines a set of Euclidean distances of positive Lebesgue measure. (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
The paper states
In The Falconer distance conjecture in all dimensions, 23 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let d\ge2 be an integer and let E\subset\R^d be compact. If \dim_H E>d/2, then
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FalconerAllDimensionsThe paper states
In The Falconer distance conjecture in all dimensions, 23 September 2026
Let d\ge2 be an integer and let E\subset\R^d be compact. If \dim_H E>d/2, then
The words they share: d/2, dim, compact, integer, conjecture, dimensions, falconer, distance.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
FalconerAllDimensions.lean, the release lists it as a supporting statement.
theorem falconer_distance_conjecture : ∀ (d : ℕ), 2 ≤ d → ∀ E : Set (EuclideanSpace ℝ (Fin d)), IsCompact E → (d : ℝ≥0∞) / 2 < dimH E → 0 < volume {r : ℝ | ∃ x ∈ E, ∃ y ∈ E, dist x y = r} := 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; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
PlanarFalconerThe paper states
In The Falconer distance conjecture in all dimensions, 23 September 2026
Let E\subset\R^d be compact, d\ge2, and \dim_HE>S=d/2. Either Lemma 5.5 already proves that its distance set has positive Lebesgue measure, or the following construction is possible.
After a similarity there are compact separated sets E_1,E_2\subset E and probabilities \mu_i supported on them, each satisfying
for a fixed \chi>0. Their cross directions lie in a fixed arbitrarily small spherical patch. For every sufficiently small fixed \omega>0, there are decreasing Borel pair sets \Gamma_N\subset E_1\times E_2 such that
(\mu_1\otimes\mu_2)(\bigcap_N\Gamma_N)>0;
\Gamma_N is a disjoint union of at most C2^{2d\omega N} products of spatial Borel sets;
for every e>0, there are c(e)>0, C_e<\infty and Borel B_N\subset\Gamma_N with (\mu_1\otimes\mu_2)(B_N)\le C_e2^{-c(e)N}, such that for G_N=\Gamma_N\setminus B_N, all 2^{-N}\le u\le1, and all spherical caps B(w,u),
The last bounds integrate the original opposite marginal, without normalizing its individual fibers. The B_N need not have a product partition or be nested. All constants are independent of N, but may depend on the fixed measures, \omega, and e.
The words they share: proves, plane, compact, dim, distance.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PlanarFalconer.lean, the release lists it as a main result.
def MainTarget : Prop := ∀ E : Set Plane, IsCompact E → 1 < dimH E → 0 < volume (distanceSet E)theorem mainTarget_proved : MainTarget := 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.
MainTarget Model.lean, line 13The source scan also found 2 theorem or lemma declarations whose names contain ‘manuscript’.
MathVet 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.
“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 formalization proves the Falconer distance conjecture in every dimension. For every integer $d\ge2$ and compact set $E\subset\mathbb R^d$ with Hausdorff dimension greater than $d/2$, the set $\{\lVert x-y\rVert:x,y\in E\}$ has positive Lebesgue measure. The linked statements include both this all-dimensional result and the earlier planar case.
Falconer distance theorem in every dimension FalconerAllDimensions
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: in every dimension d ≥ 2, a compact set of Hausdorff dimension above d/2 has a distance set of positive Lebesgue measure.
Planar Falconer distance theorem PlanarFalconer
Unreviewed AI draft: states a weaker form of a paper’s theorem. States the planar case d = 2 of the main theorem, the earlier result that the note keeps beside the all-dimensional statement.
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.
FalconerAllDimensionsTwo 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 |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 658 s; peak memory 7.4 GiB). | 10 October 2026 | Original report |
PlanarFalconerTwo 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 |
Other people’s accounts of running proof-verification software, with links to their reports.
CoolRmal, CoolRmal/falconer-all-dimensions: Comparator replay of the all-dimensional Falconer formalisation, 6 October 2026. Source
Comparator accepted the extracted Falconer challenge on a Linux runner on 6 Oct 2026 (Actions run 37545204340).
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.
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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Offer to write an explanationCompare one Lean statement with its paper: PlanarFalconer, FalconerAllDimensionsSuggest 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
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the Falconer distance results for the plane and all dimensions.
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 Falconer distance conjecture in all dimensions, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, FalconerAllDimensions
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, FalconerAllDimensions
Proof-verification report
cert-machine, PlanarFalconer
Proof-verification report
MathVet, PlanarFalconer
Lean statement
FalconerAllDimensions
Lean statement
PlanarFalconer